agraiments.tex
\paginablanca
\chapter*{Agra\"\i ments, Danksagung
\markboth{Agra\"\i ments, Danksagung}{Agra\"\i ments, Danksagung}}
\thispagestyle{empty}
\begin{spacing}{0.98}

\inicialfada Heidelberg is a little city in the south of Germany. It 
has a nice old bridge, a romantic castle, a very concurred old town
and -I was told- a couple of native inhabitants. I spent in this city
about five years of my life, and I daresay they were the most intesive
ones. A lot of things happened. ``For instance'', I got to know
Sabine after one and a half weeks of my arrival. Very soon we started
{\em the} relationship that would conduct us in a inavoidably retoric
way to marriage.

\noindent
If somebody asked me to tell the first thing that comes to my mind
when I think about these years, I would answer the following...

\begin{quote}
  \footnotesize{ sabine bridge potatoes kitchen Schwalbe Jorge Mayer
  {\"A}rger fill de sa mare Francine Vienna bridge cold cold cold cold
  mountains sauerkraut Frank Wixenpa'aslocas wine Oberseminar hin und
  her fahren Wien cold cold cold Hauptstra{\ss}e Emil potato Mike
  Deiters 4{$\mid$} {[}{$\mid$}n4 g3nd m{\'3}\$ r4r374 Teeminar
  teaminar teminable @Weberstra{\ss}e 13 Sant Petersburg Masha Sergei
  Russian Russian Mahos Tilemahos Vienna Sabine Gerhabt-Hauptmann
  Stra{\ss}e Mutter Umzug moving moving CRACK oh my goodness that was
  my back Basel Alfonso shop buy smoke Dante Inferno work work woman
  in pink tracksuit yellow umbrella Nickel Nickel what for und viele
  iuius for a long while escape go back to Spain and walk walk walk in
  the night Danzigerstra{\ss}e John Lucie tinto Sabine is back
  al$\cdot$leluia per qu{\`e} peta borinot turbi (A) anxious -xit
  anxiety dean- and uprooting syndrome quatsch better cold dark night
  Vienna Russian Sprachlabor Marc Russian Silke Eva Marion work work
  Marc wark Mork Russian workian Freitag Bl{\"o}dsinn in der K{\"u}che
  paper light see end pictures marriage vacherin gluant Marc's
  supermongo subroutines again Dante Inferno but smoothed end}
\end{quote}

This section cannot be short at all because I am in hock to a lot of
people. I cannot help feeling very grateful to them and they deserve
it. On the other hand, being a social animal, as only a genetically
galician but in Val{\`e}ncia born person can be, it would be a crass
error and an unpardonable insult not to talk about the friends I made
during these years.

\vskip0.2cm
\noindent
First of all, I would like to thank here my advisor, {\bf Rainer
Spurzem}, for giving me the opportunity of coming to Germany to do a
Ph.D. He encouraged me always when things were not working as they
should (a bad habit of the process of becoming a doctor, according to
a lot of people). He and {\bf Andreas Just} treated me as if I had
been in the group for years from the very beginning.

\vskip0.2cm
\noindent
%\parpic[r][r]{\mbox{\resizebox{0.35\hsize}{!}
%	{\includegraphics{diag_agraiments/last_min1.eps}}}}
\noindent
The French-speaking commonwealth, {\bf Marc Freitag} and {\bf
Francine Leeuwin}, carried out a very important role during these
years... I daresay Marc is responsible for me becoming a scientist, if
you happen to have any complaint, please refer to him. They gave me
also something much more valuable than their help, namely their
friendship. I am very grateful to both of them for a lot of
things... Francine, for instance, introduced me into the wonderful
world of french spirits (I am {\em not} talking about the agent or
subject of vital and spiritual functions, whether spiritual or
material). It helped a lot to survive the absolutely sun-less winter
evenings of Heidelberg after a long full working day. I got to know
Marc when he visited our group in Heidelberg in March 2001. I felt as
if we had been friends since always from the very first beer at the
Markt St{\"u}bel (a place which we haunted later). When he moved to
Heidelberg we became inseparable and even neighbours. Among
innumerable things, Marc made me rediscover {\'E}dika in his full
splendour. It would not be an exaggeration at all to say that both of
them, Marc and Francine, have become a part of my family (in its full
southern meaning).

\vskip0.2cm
\noindent
Elaborating on nice people, I have to mention here two very nice
aussies, Bruce and Sheila... I mean {\bf John} and {\bf Lucie
Benz} (n{\'e}e Chapman), who made me feel ``at home'' from the very
beginning. Everything began with a red wine, carried on with tinto and
jam{\'o}n (pata negra, of course) and finished with two marriages,
theirs (on next Saturday, by the way) and mine.

\vskip0.2cm
\noindent
As regards the {\em Kersche} community, it is a must to say how nice
it was for me to share a place in the Kirchheim community with our
fellow-beings {\bf Eva}, {\bf Marion}, {\bf Silke} and -inevitably-
Marc.
%\vspace{-0.4cm}
%\parpic[l][r]{\mbox{\resizebox{0.30\hsize}{!}
%	{\includegraphics{diag_agraiments/last_min2.eps}}}}
The iberians would not be happy if I forgot them here and so I have to
address a few words to my two favourite tableland people, {\bf Jorge
Pe{\~n}arrubia}, {\em el Doctor Xungal{\'i}n}, compa{\~n}ero
infatigable de diversas peripecias acontecidas en el consuetudinario
devenir, por no llamarlo divagar, del instituto (l{\'e}ase cocina,
broncas diversas con el aparato administrativo ariense, puerta
corrediza y un sinf{\'i}n m{\'a}s de cosas) and el lobo mesetario {\bf
David L{\'o}pez} and parienta, {\bf Elise Schieck}, for his spirituous
southern chocolate, republican ideas and swedish food.  And here I
cannot -I must not- forget {\bf Pablo Gonz{\'a}lez}, {\em Polito}, den
entspanntesten Canario der Welt, for his natural sedateness, papas
arrugadas con mojo rojo (y verde) and ron canario.

\vskip0.2cm
\noindent
Respect to the ARI people, {\bf Gernot Burkhardt}'s predisposition
to make any kind of favour I asked for nicely surprised me
everytime. He supported us, Jorge and me, in our little ``domestic''
conflicts at the Institute and was always very friendly to us. 
Somebody like he can make you see things in a different way 
when you start as a newbie and have to confront difficulties and
people who are not as nice as he is.

{\bf Christoph Eichhorn}, {\bf Kristin Warnick}, {\bf Chingis Omarov},
{\bf Toshio Tsuchiya}, {\bf Eliani Ardi}, {\bf Ikbal Aryfanto}, {\bf
Jos{\'e} Fiestas}, {\bf Emil Khalisi}, {\bf Michael Fellhauer}, {\bf
Stefan Deiters}, {\bf Andreas Ernst}, {\bf Patrick Glaschke}, {\bf
Pavel Kroupa} and {\bf Christian Boily} were very cordial and amiable
to me during my stay at ARI.

\vskip0.2cm
\noindent
{\bf Ernst Pendl}, {\em Eanstl}, bin i d{\aa}ngba fia dem V{\aa}da sei
Dochda Sabine und seine Begeisterung f{\"u}r Astronomie. Im Kosmos
umanaundstrawanzn... per aspera ad astra.  -- Heast! geh gibtsn des
{\aa}, Spania de wos Weanerisch kennan! But yes, there are, and I am
one. D{\aa} h{\aa}b i eams einegs{\aa}t, net!  

\vskip0.2cm
\noindent
Ich wei{\ss} {\bf Christine} ob der nach einer Woche meines Daseins in 
Deutschland geschlossenen Bekannt-\\schaft zu ihrer Nachfahrin sehr zu
sch{\"a}tzen.

\vskip0.2cm
\noindent
{\bf Margaret Mehls}, {\em Margarita} war immer {\"a}u{\ss}erst
liebensw{\"u}rdig zu mir in der Weberstra{\ss}e 13 (Aussenstelle vom
ARI). Es macht einen riesengro{\ss}en Unterschied, ob deine
``Nachbarn'' (eigentlich Mitbewohner, nachdem ich fast den ganzen Tag
hier bin) nett sind oder nicht. In ihrem Falle (und in dem von Herrn
Mehls, selbstredend) sollte man eigentlich eher {\"u}ber Herzlichkeit
als Nettigkeit reden.

\vskip0.2cm
\noindent
%\parpic[r][r]{\mbox{\resizebox{0.65\hsize}{!}
%	{\includegraphics{diag_agraiments/last_min3.eps}}}}
{\bf Matt Bonner} em va ajudar amb l'Angl{\`e}s de la petita introducci{\'o}
que vaig fer, car se'm feia molt envitricollat de traduir l'Homer i
les meues paraules a la seua vernacla saxona. Ell, habituat com
est{\`a} a traduir coses al Catal{\`a} per a {\tt Debian}, ho f{\'e}u
en un tres i no-res i, el millor, perqu{\`e} s{\'\i}, sense
con{\`e}ixer-me de quasi res, nom{\'e}s del f{\`o}rum de
traducci{\'o}. Amb persones com {\em Mateu Bonet} (com a ell li agrada
que li diguen), hom pot creure que es pot recuperar les esperances. No
vaig gosar de demanar ning{\'u} de Micro{\$}oft. S{\'o}c optimista,
per{\`o} no tant.

\vskip0.2cm
\noindent
Going back a bit in history, just over a couple of years, when I was
studying at the Universty of Val{\`e}ncia, {\bf Vicent Mart{\'i}nez Sancho}
played a decisive role in a lot a different aspects of my
education. He not only managed to unfailingly wake up in all his
students an enthusiastic interest for the subjects he teaches, but
also won our trust and affection. Besides Physics, he taught us how to
tackle with life, which is a much more difficult subject. Putting it
into his words, ``{\'E}sser dotat per a les matem{\`a}tiques no t{\'e}
cap m{\`e}rit. Car m{\`e}rit ve de meritum, del participi de merere
'mer{\`e}ixer'. {\'E}s aix{\`o}, una qualitat natural... Com el beneit
del cab{\`a}s del meu poble, que fa trucs molt enginyosos amb un
palet. El que faces amb la teua vida, com l'enfoques i quin tipus
d'{\'e}sser hum{\`a} esdevingues, aix{\`o} s{\'i} que {\em
mereix}''. His concern and attentiveness for culture (understood in
all its broad meaning) influenced, {\em influences}, me until now.

\vskip0.2cm
\noindent
Cuando, en los {\'u}ltimos d{\'\i}as de la redacci{\'o}n de la tesis,
me puse a repasar estas p{\'a}ginas, me di cuenta de que el nombre de
mi madre no aparec{\'\i}a en ninguna de ellas. ?`Me hab{\'\i}a
olvidado de ella? !`?` De mi madre?!  No era posible. En ese momento,
en que estaba pensando (contecimiento extraordinario), me di cuenta de
que hab{\'\i}a sido porque para m{\'\i} es tan evidente todo lo que le
debo -que es, eso, todo- y lo mucho que la quiero -que es un
much{\'\i}simo mucheante mucheado- que, de hecho, el nombre, no es
importante. Lo que s{\'\i} es importante es todo lo que ella me ha
aportado y aporta en la vida. Gracias, {\bf {\em Luisa}}.

\vskip0.2cm
\noindent
{\"U}ber Dich brauche ich wenig zu sagen, {\bf Sabine}, denn es ist nicht nur
f{\"u}r mich, sondern f{\"u}r alle klar, wie sehr ich Dich liebe. Wir haben
ein ganzes Leben vor uns, das gerade anf{\"a}ngt.

\vskip0.2cm
\noindent
\'{3}\$ gl\`{4} g'4\${\'\i} g4l 3\$m3nd\`{4} 3l n0m d3l(\$) \$4ny0(\$)
{\bf Els Etsus}: 3rr${\mid}$gu${\mid}$, 3rm${\mid}$[${\mid}$ ${\mid}$ 3rb3r3
d'b4n${\mid}$4 b3r d0d 3r g0n r0dll1 g4'n\$ h4m b1rd4d 4gu3\$d\$
4ny\$. 

\vskip.1cm
\hspace{5.5cm}{
\large G0n d${\mid}$-D-4rl!!!\,{-}{-}{-}{-}1
} \hspace{0.4cm}{\footnotesize (sorry for that, but it was {\em also} a
must).}

\end{spacing}

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{\sf I think this is really the end. I do not know what else I could tell
you, after \thepage ~pages of wanton hullabaloo... Now it is time to
have a carajillo... Let me pythonise it...}
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\mbox{\footnotesize {\sf This work was written with 
\LaTeX ~(what else?) and processed with a {\tt Debian Sid}-- based system,} 
{\sc vindobona}}

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contents.tex
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\renewcommand{\contentsname}{Structure of the thesis} % nom original = Contents 
\begin{spacing}{1.05} % Per a fer m�s f�cil la lectura
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dedicatoria.tex
\thispagestyle{empty}


\noindent
\begin{flushright}
\small
\begin{quote}
Antonio Amaro Pita, my father, never could see finished this work,
because he passed away in November of last year. It is very difficult
to write these lines, because I miss him a lot. He lived intensively,
much more than all people I know. He squeezed the nice things out of
life until the very last drop and then he decided to go away and carry
on with the fun somewhere else because here it was damned boring for
him. \\

\vskip0.2cm
\noindent When my father died, Marc Freitag, who was in the north of 
America, wrote:
\vskip0.3cm

\noindent
{\em En Pennsylvanie c'est un automne magnifique avec des 
arbres qui deviennent tr{\`e}s rouges avant de se depouiller
totalement.  J'imagine qu'il est normal que les grands arbres, devenus
vieux, soient terass{\'e}s par le vent... Cela n'enl{\`e}ve rien a la
beaut{\'e} de la for{\^e}t}.\\
\vskip0.3cm
\noindent
There is death because there is life, and there is life because there
is death. My father died, and so will I someday; this is not a reason
to be upset, it is just the flowing of life. It steals us the surface
of the sand and allows us to walk in the new surface and leave our
prints. Sabine is now pregnant and I hope our child will be as full of
life as his grandfather was and able to enjoy every second of his new
life, just as my father did.\\
\end{quote}
\end{flushright}

\vskip1cm
\begin{flushright}
{Iste traballo est{\'a} adicado a il}
\end{flushright}

\normalsize

diffusion_model.tex
\chapter{A diffusion model for accretion of stars}
\label{ch.diff_model}

%\vspace{-11cm}
%\begin{quote}
%\begin{flushright}
%{{\sf \footnotesize l'univers fa xa-xa i la guida es una
%tombola-tom-tombola ....... de l1 i de col0!!!!!!!!1
% \\
%{\vspace{0.15cm}\em Dr. Miquel Ardid Ramirez }}}
%\end{flushright}
%\end{quote}
%\vspace{11cm}

\inicial{V}{iewing} the problem of the joint evolution of a
spherical star cluster with a central BH making feasible
anisotropy, we shall introduce in these pages
\footnote{We employed some of the results 
of this chapter for \citet{ASEtAl04} and 
\citet{SpurzemEtAl03}} the
simulation method to follow to analyse such a scenario.
This procedure, as we exposed in former chapters, is based
on moments of the Boltzmann equation with relaxation. The cluster is
modelled like a self-gravitating, conducting gas sphere, according to
the methods presented in
\citet{LS91} and
\citet{GS94}. These models improved earlier gas models of
\citet{Bettwieser83} and \citet{Heggie84}. Much as the structure of
the numerical method is for the sake of computational efficiency on
account of physical accuracy, it allows for all the most important
physical ingredients that may carry out a role in the evolution of a
spherical cluster. These include, among others, self-gravity, two-body
relaxation, stellar evolution, collisions, binary stars etc and,
undoubtedly, the interaction with a central BH and the role of a mass
spectrum. The specific advantage of the so-called ``gaseous model'' to
other simulations methods (see chapter \ref{ch.th_model} for a
description) is that the simulations are comparatively much faster,
since they are grounded on numerical integration of a relatively small
set of partial differential equations with just one spatial variable,
the radius $r$. In addition, all quantities of interest are accessible
as smooth functions of $r$ and time and this allows one to investigate
in detail clean-cut aspects of the dynamics without being hindered by
the important numerical noise particle-based methods ($N$--body and
Monte Carlo) suffer from.

In this chapter we concentrate on the simplest version of the gaseous
model which includes the interaction between a central BH and its host
cluster. In particular, we assume that all stars are sun-like, neglect
stellar evolution, direct collisions between stars and the role of
binary stars. Also, the only interaction between BH and the stellar
system is tidal disruptions (besides the BH's contribution to the
gravitational field), and we undertake that the BH stays fixed at the
centre of the cluster. While admittedly very simplified, we reckon this
idealised situation warrants consideration. First, it helps us to
establish that the gaseous model is also able to treat more complex
situations in phase space than self-gravitating star clusters, such as
those caused by {loss-cone} accretion and a central BH; second,
experience with the gaseous model and other methods showed us how
intricate the interplay between various physical effects can become
during the evolution of clusters and so we feel compelled to first
consider the simplest models to develop a robust understanding of the
mechanisms at play.

The structure of this chapter is as follows: In section
\ref{sec.diff_model} we explain the physics of the problem. In 2.2 we
introduce briefly the various analytical and numerical methods used so
far to investigate spherical clusters with central BH and summarise
their key results.  In section \ref{sec.diff_model_th} we give a
description of the theoretical diffusion model.  Section
\ref{sec.diff_model_results} is devoted to results obtained in a first
set of simulations. Finally, in \ref{sec.diff_model_disc}, we draw
conclusions about this first use of this new code version and present
what our future work with it will likely consist of.

\section{{Loss-cone} accretion on to massive BHs}
\label{sec.diff_model}

\subsection{Previous theoretical and numerical studies}

If we arrange numerical methods for stellar dynamics in order of
validity and both increasing the spatial resolution and decreasing the
required computational time, we can distinguish four general classes.
The most direct approach is the so-called $N$--body method
\citep{Aarseth99a,Aarseth99b,Spurzem99}. Monte Carlo codes are also
particle-based, but rely on the assumptions that the system is
spherically symmetric and in dynamical equilibrium and treat the
relaxation in the Fokker-Planck approximation (see 3.2)
(\citealt{FB04,FregeauEtAl03,FB01a,GS03,Giersz98,JoshiEtAl01}).
Ensuingly, we have the two-dimensional numerical direct solutions of
the Fokker-Planck equation
\citep{Takahashi97,Takahashi96,Takahashi95}, and the gaseous models.
The idea of this model goes back to \citet{HachisuEtAl78} and
\citet{LBE80}, who first proposed to treat the two-body relaxation as
a transport process like in a conducting plasma. They had been
developed further by \citet{Bettwieser83,BS84,Heggie84,HR89}. Their
present form, published in \citet{LS91,GS94,ST95} improves the
detailed form of the conductivities in order to yield high accuracy
(for comparison with $N$--body) and correct multi-mass models. This
point has been made already in \citet{Spurzem92}.

\citet{Peebles72,SL76} and especially \citet{FR76} and \citet{BW76} addressed
the problem of a stationary stellar density profile around a massive
star accreting BH. They found that, under certain conditions, the
density profile $\rho\propto r^{-{7/4}}$ is established in the region
where the BH's gravitational potential well dominates the self-gravity
of the stars \footnote{We must mention here the legwork done twelve
  years before this analysis by \citet{Gurevich64}, since he got an
  analogous solution for the distribution of electrons in the vicinity
  of a positively charged Coulomb centre.}.

The problem of a star cluster with a massive central star-accreting BH
has been widely coped with Fokker-Planck numerical models.  This
approach was useful in order to test the solidness of the method to
reproduce the $\rho \propto r^{-{7/4}}$ stationary density profile,
since {loss-cone} accretion disturbs such a density cusp \citep{OR78}.
The authors show that the stationary density profile follows from
their stellar-dynamical equation of heat transfer by scaling arguments
which are analogous to those given in \citet{SL76}.

\subsection{The diffusion model}

We can express the {tidal radius} in terms of the internal stellar 
structure re-writing Eq.\,(\ref{eq.r_tid_bind}), 

\be
r_{\rm t} \propto\Bigg( 
\frac{\mbh}{x_{\rm b} \pi \bar{\rho}} \Bigg)^{1/3},
\label{eq.r_tid_bind2}
\ee

\noindent
where $\mbh$ denotes the mass of the central BH, $\bar{\rho}$ the mean
stellar internal density, $n$ is the polytropic index (stars are
supposed to be polytropes) and $x_{\rm b}$ is a parameter
proportional to the gravitational binding energy of the star that
describes effects of the internal stellar structure. We assume that a
star is disrupted by tidal forces when it crosses the tidal
radius. The free parameter $\epsilon_{\rm eff}$ (accretion efficiency)
determines the mass fraction of the gaseous debris being accreted on
to the central BH ($\epsilon_{\rm eff}=1$ corresponds to 100\%
efficiency).

There are two concurrent processes driving stars towards the tidal
radius; namely the {\em energy diffusion} and the {\em {loss-cone}
accretion}. In the first case, stars on nearly circular orbits lose
energy by distant gravitational encounters with other stars and in the
process their orbits get closer and closer to the central BH.  The
associated energy diffusion time-scale can be identified with the
local stellar-dynamical relaxation time, Eq.\,(\ref{eq.relax_t}), but
generalised for anisotropy as in \citet{Bettwieser83},

\be 
t_{\rm relax}= \frac {9}{16 \sqrt {\pi}}
\frac {\sr (\s_t^2/2)} {G^2 m_\star \rho_\star(r) \ln \Lambda}.  
\label{eq.relax_t2}
\ee

\noindent
Here $\sr$, $\s_t$ are the radial and tangential velocity dispersions
(in case of isotropy $2\sr^2 = \s_t^2 $), $\rho_\star (r)$ is the mean
stellar mass density, $N$ the total particle number, $G$ the
gravitational constant, $m_\star$ the individual stellar mass and

\be
\ln \Lambda \equiv \ln\,(p_{\rm max}/p_0)=\ln\,(\gamma N) 
\label{eq.lambda}
\ee

\noindent
is the Coulomb logarithm. We set $\gamma=0.11$ \citep{GH94}. In this
expression $p_{\max}$ is an upper limit of $p$, the {\em impact
  parameter}; $p_0$ is the value of $p$ that corresponds to an
encounter of angle $\psi=\pi/4$, where $\psi= (\pi - \xi)/2$ if $\xi$
is defined to be the {\em deflection angle} of the encounter
\citep{Spitzer87}. In the vicinity of the BH ($r<r_{\rm h}$, see 
Sec.~\ref{subsec:units}), one
should be aware that $\Lambda \approx \mbh/m_{\star}$
\citep{BW76,LS77} but, for simplification, here we shall use
Eq.\,\ref{eq.lambda} (strictely speaking only valid at distances
$r>r_{\rm h}$) everywhere.

For a more detailed discussion of the energy
diffusion process and its description in the context of the moment
model see \citet{BS86}.

As regards the second process, the {\em {loss-cone} accretion}, stars
moving on radially elongated orbits are destroyed by tidal forces when
they enter the {tidal radius} $r_{\rm t}$. A star will belong to the
{loss-cone} when its {\em peribarathron} (distance of closest approach
to the BH, see Fig.\ref{fig.peribarath}) is less than or equal to the
{tidal radius} $r_{\rm t}$, provided that its orbit is not disturbed
by encounters.  Thus, the {loss-cone} can be defined as that part of
stellar velocity space at radius $r$, which is given by

\be 
|v_t| < v_{\rm lc}(r)=\frac {r_{\rm t}}{\sqrt {r^2-r_{\rm t}^2}} 
\cdot \sqrt {2[ \phi (r_{\rm t}) - \phi (r)] + v_{\rm r}(r)^2} 
\label{eq.v_lc}
\ee

\noindent 
(see Eq. \ref{eq.lc_vel_d}).
\noindent 
In the last formula $v_r$, $v_t$ are the radial and tangential
velocity of a star and

\be
\phi (r_{\rm t}) - \phi (r)=\frac{G\mbh}{r_{\rm t}} + \phi_{\star}(r_{\rm t})-
\frac {G\mbh}{r}- \phi_{\star}(r)
\label{eq.diff_potentials}
\ee

\noindent
At distances $r \gg r_{\rm t}$ we can approximate this expression taking
into account that ${G\mbh}/{r_{\rm t}} \gg \phi(r)~{\rm and}~\phi(r_{\rm t})$
(see Eq. \ref{eq.lc_vel_k}).

\noindent
For a deeper analysis on {loss-cone} phenomena see chapter \ref{ch.semi_analytical}.

Similar to \citet{daCosta81}, we define two time-scales: $t_{\rm out}$
to account for the depletion of the {loss-cone} and $t_{\rm in}$ for
its replenishment. For a BH, $t_{\rm out}$ is equivalent to one
crossing time, since it is assumed that a star is destroyed by just
one crossing of the {tidal radius}. In chapter \ref{ch.semi_analytical}
we mentioned that in the special case that the central object is a
super-massive star, $t_{\rm out}=n\,t_{\rm cross}$.  Here $n>1$ is the
number of passages until a star is trapped in the central object.

The {loss-cone} is replenished by distant gravitational encounters that
change the angular momentum vectors of the stars. To estimate $t_{\rm
  in}$, we make the assumption that random gravitational encounters
thermalise the whole velocity space at some given radius $r$ after a
time-scale of the order of $t_{\rm relax}$. As a first approximation,
the fraction $\Omega$ of the three-dimensional velocity space is
re-populated within a time-scale

\be
t_{\rm in}=\Omega \,t_{\rm relax}
\label{eq.t_in_omega}
\ee

\noindent
where 

\be
\Omega := \int_{\rm lc}f d^3v\,/\int_{\infty} f d^3v; 
\label{eq.omega}
\ee

\noindent
the subscript $\infty $ denotes an integration over the velocity space
as a whole, and the subscript $lc$ means that the integration over the
{loss-cone} part of the velocity space is given by Eq. (\ref{eq.v_lc}).
As a matter of fact, $\Omega$ can be envisaged as the fraction of the
surface of a velocity ellipsoid which is cut out by the {loss-cone}.
Close to the {tidal radius} $r_{\rm t}$, and for appreciable amounts of
stellar-dynamical velocity dispersion anisotropies, our method
describes the {loss-cone} size $\Omega $ more exactly than those given
by \citep{FR76,daCosta81}.  On the other hand, their models can be
''recovered'' in the limit of $r\gg r_{\rm t}$ and isotropy,
$2\sr^2\approx\s_t^2$ (denoted as the ``small {loss-cone}
approximation''), where

\be
\Omega \approx v_{\rm lc}^2/\s_t^2 \approx r_{\rm t}/r.
\label{eq.omega_app}
\ee

\noindent
This is equivalent to their definition of $\theta_{\rm lc}$ if $\Omega
= \theta_{\rm lc}^2 /4$ is adopted.
 
Where the {loss-cone} effects can be neglected, a
Schwarz\-schild-\-Boltz\-mann type distribution function can be
assumed,

\be
f\propto\exp\Bigl(-{(v_r-\langle v_r\rangle )^2\over
2\sr^2} -{v_t^2\over\s_t^2}\Bigr).
\label{eq.sb_DF}
\ee

\noindent
Third order moments of the velocity distribution that represent the
stellar-dynamical energy flux do not alter such a distribution
function significantly \citep{BS85}.
 
An important quantity is the {\em critical radius} $r_{\rm crit}$. Let
$\theta_{\rm D}^2$ be the average quadratic deflection angle produced
by relaxation during $t_{\rm out}$ ($=t_{\rm cross}$ here),
\be
\theta_{\rm D}^2 = \frac{t_{\rm out}}{t_{\rm relax}}.   
\ee
Then, by definition,
\be
\theta_{\rm lc}(r_{\rm crit}) = \theta_{\rm D}(r_{\rm crit}), 
\ee
which is equivalent to $t_{\rm out}(r_{\rm crit}) = 4\,t_{\rm
in}(r_{\rm crit})$.  For most clusters where such a radius can be
defined, $\theta_{\rm lc} > \theta_{\rm D}$ inside $r_{\rm crit}$
while the opposite holds outside. This means that, at large radii,
relaxation is efficient enough to make stars diffuse into and out of
loss cone orbits over a time scale $t_{\rm out}$ so that the
distribution function is not appreciably depleted in the loss
cone. Conversely, deep inside $r_{\rm crit}$, the loss cone orbits are
essentially empty and the flux of stars into this domain of phase-space
(and into the BH) can be treated as a diffusive process because the
size of one individual step of the velocity random walk process,
$\theta_{\rm D}$, is (much) smaller that the characteristic size of the
problem, $\theta_{\rm lc}$.

Note that a critical radius does not necessarily exist (see
chapter \ref{ch.semi_analytical}). For instance, if one assumes that
gravity of the BH dominates the stellar self-gravitation and that the
density profile follows a power-law, $\rho\propto r^{-\alpha}$, one
has $\theta_{\rm lc}^2 \propto r^{-1}$, $\theta_{\rm D}^2 \propto
r^{3-\alpha}$ and a critical radius would not exist for $\alpha >4$.

Now we want to generalise the stationary model (see
chapter\ref{ch.semi_analytical}), which assumes an empty {loss-cone}
within $r_{\rm crit}$ and a full {loss-cone} elsewhere, by means of a
simple ``diffusion'' model, which is derived from the above
considerations; this means that the filling degree of the {loss-cone}
$K$ can be continuously estimated within its limiting values,

\be 
K \in [0,1].
\label{eq.K_limits}
\ee

\noindent
Let $f$ be the unperturbed velocity distribution (without {loss-cone}
accretion); if the {loss-cone} is empty and we neglect the angular
momentum diffusion, $f=0$ inside the {loss-cone} (and unchanged
elsewhere in the velocity space).  In point of fact, $f$ will have a
continuous transition from nearly unperturbed values at large angular
momenta to a partially depleted value within the {loss-cone}. This value
is determined by the ratio of $t_{\rm in}$ and $t_{\rm out}$.  Such a
smooth transition of the distribution function given as a function of
angular momentum $f(J)$ has been derived from self-consistent models
of angular momentum diffusion (e.g.  \citealt{CK78} or
\citealt{MS80}). We approximate $f(J)$ by a distribution function
that has a sudden jump just at the value $J_{\rm min}= m_\star v_{\rm
  lc}$ from an unperturbed value $f_0$ given by the moment equations
(assuming a Schwarzschild-Boltzmann distribution) to the constant
lowered value

\be
f=Kf_0,~{\rm with}~0\leq K \leq 1
\label{eq.f_K_f0}
\ee

\noindent
within the {loss-cone} (i.e. $J<J_{\rm min}$ or $\vert v_t\vert <
v_{\rm lc}$). This implies that, as means to compute the mean mass
density of {loss-cone} stars, we have to calculate the integral

\be
\rho_{\rm lc}=\int_{\rm lc} K \,f_0\, d^3v.
\label{eq.rho_lc}
\ee

\noindent
And then, accordant with the definition of $\Omega $,

\be 
\rho_{\rm lc, full}=\rho\,\Omega,
\label{eq.rho_omega}
\ee

\noindent
in the case that we have a full {loss-cone}.

\noindent
In regard to the radial and tangential stellar velocity dispersions in
the {loss-cone} $\sigma_{{\rm lc},\,r}$ and $\sigma_{{\rm lc},\,t}$, we can
compute them using second moments integrated over the {loss-cone} part
of velocity space. As for the definition of the quantities $E_r$ and
$E_t$ used in Sect. 3,
\begin{eqnarray}
\sigma_{{\rm lc},\,r}^2 &=& E_r\,\sr^2, \label{eq.E_r,t} \nonumber \\
\sigma_{{\rm lc},\,t}^2&=&E_t\,\s_t^2, 
\end{eqnarray}

\noindent
in the small {loss-cone} approximation we have that
$E_r\approx 1$ and $E_t \ll 1$.


The arguments about the time-scales that have led us to the derivation
of $t_{\rm in}$ and $t_{\rm out}$ guide us also to the following {\em
  diffusion equation} for the time evolution of the spatial density
$\rho_{\rm lc}=K\rho\,\Omega $ of {loss-cone} stars:

\be 
{d\rho_{\rm lc}\over dt}= -{\rho_{\rm lc}\,P_{\rm lc}\over t_{\rm
    out}} + \frac{\rho \,\Omega -\rho_{\rm lc}}{t_{\rm in}}.
\label{eq.diff_eq_rho_lc}
\ee

\noindent
In this equation, the second term on the right hand is the refilling term
due to relaxation.

{As we assume relaxation is due to a large number of small-angle
  deflections and can thus be seen as a diffusive process in velocity
  space,} the probability $P(\theta)$ that a star is scattered in an
angle $\theta$ in a time $t_{\rm out}$ is

\be
P({\theta})=\frac{2}{\sqrt{\pi}\,\theta_{D}}\exp\,(-{\theta^2}/
{\theta_{D}^2}),
\label{eq.P_theta}
\ee

\noindent
The distribution is normalised to one,

\be
\int^{\infty}_{0}P(\theta)\,d\theta=1
\label{eq.normP}
\ee

\noindent 
and has the property that its mean square value is
$\theta_{D}^2$. A star remains in the {loss-cone} during a time $t_{\rm
  out}$ if its RMS diffusion angle is smaller than the {loss-cone}
angle $\theta_{lc}$. The probability for this to happen is

\be
P_{lc}=\int^{\theta_{lc}}_{0}P(\theta)\,d\theta={\rm erf}(\sqrt{4\,{
t_{\rm in}}/{t_{\rm out}}}).
\label{eq.Plc}
\ee

In the case that a star is unperturbed by the rest of the stellar
system, it will sink on to the central BH in a time $t_{\rm out}$.
Actually, this is a somehow simplified description of the physical
process, for part of the {loss-cone} stars will be scattered out of it
before they slump. The required time for this event is $t_{\rm in}$,
since in this time-scale the angular momentum vector will change
(due to distant encounters) on an amount that is comparable with the
size of the {loss-cone} in the angular momentum space. For this reason
we have introduced the quantity $P_{\rm lc}$ in Eq.
(\ref{eq.diff_eq_rho_lc}).

Bluntly speaking, the {\em effective} time-scale that describes the
{loss-cone} depletion allowing for perturbation due to angular
momentum diffusion is

\be
t_{\rm out,\,eff} = t_{\rm out} / P_{\rm lc}.
\label{eq.t_out,eff}
\ee

\noindent
As a matter of fact, this definition ensures us that far outside of
the critical radius the {loss-cone} depletes in a time that grows
infinitely, as it is physically expected. In the regime where $r\ll
r_{\rm crit}$, $P_{\rm lc}$ tends asymptotically to 1 and to 0 where
$r\gg r_{\rm crit}$, passing through a transition zone at $r=r_{\rm
  crit}$.

We can consider Eq. (\ref{eq.diff_eq_rho_lc}) as an ordinary
differential equation for $K=\rho_{\rm lc}/(\rho\,\Omega)$ if we assume
that the stellar density and the {loss-cone} size are time-independent.
Transport phenomena can be neglected, for they are related to the
relaxation time, and $t_{\rm relax}\gg t_{\rm in},t_{\rm out}$.
Bearing this in mind we can get an analytical solution $K(t)$ for the
differential equation with the initial condition that
$K(t)|_{t_0}=K_0$,

\begin{eqnarray}
\lefteqn{K(t)=K_0\exp\Bigl(-{P_{\rm lc}\,\xi\,(t-t_0)\over t_{\rm out}}
\Bigr)+}\label{eq.K(t)} \nonumber \\
& & {} \allowbreak {t_{\rm out}\over P_{\rm lc}\,t_{\rm in}\,\xi}\cdot
\bigg( 1-\exp\Big(-{P_{\rm lc}\,\xi\,(t-t_0)\over t_{\rm
    out}}\Big)\bigg).
\end{eqnarray} 

\noindent
In the last equation we have defined $\xi$ for legibility reasons as
follows,

\be
\xi:=1+(t_{\rm out}/P_{\rm lc}\,t_{\rm in}).
\ee

\noindent
For $r=r_{\rm crit}$, with $t_{\rm in}= t_{\rm out}$, the stationary
filling degree of the {loss-cone} turns out to be

\be
K_{\infty}:=\lim_{t\to\infty}K(t)={1\over 2}.
\label{eq.K_infty}
\ee

\noindent
Note that \citet{MM03} recently gave a detailed summary of {loss-cone}
effects. They derived expressions for non-equilibrium 
configurations. They employ a rather different treatment for the
diffusion since they tackle the problem of binary BHs scattering.

\section{Inclusion of the central BH in the system}
\label{sec.diff_model_th}

In this subsection we discuss the way we cope with the {loss-cone} in
our approach. For this aim we accept the following:
  
\vskip0.2cm
\noindent
{1.}\,The system has central ($r=0$) fixed BH 

\vskip0.2cm
\noindent
{2.}\,Stars are totally destroyed when they enter $r_{\rm t}$

\vskip0.2cm
\noindent
{3.}\,Gas is completely and immediately accreted on to the BH

\vskip0.2cm

\noindent
%\modif{
%As regards the first point, one should mention that the role of
%brownian motion of the central BH can be important; as a matter of
%fact, for a cluster with core radius $R_{\rm core}$, equipartition
%predicts a wandering radius $R_{\rm wan}$ of order

%\be
%R_{\rm wan}\approx R_{\rm core}\,\sqrt{M_{\star}/\mbh}, 
%\label{eq.wandering_radius}
%\ee

%\noindent
%which is larger than the tidal disruption radius for BHs less massive
%than $10^9 M_{\odot}$ if the core radius is 1 pc
%\citep{BW76,ChatterjeeEtAl02,LT80}.  In sytems with cusps, the value
%of $R_{\rm wan}$ is not evident (see \citealt{DHM03}). The effect of
%this motion on disruption and capture rates is not known yet, but it
%seems likely that these rate can be either increased or decreased
%\citep{MT99}.  }
%\noindent

{
As regards the first point, one should mention that the role of
brownian motion of the central BH can be important; as a matter of
fact, for a cluster with core radius $R_{\rm core}$, equipartition
predicts a wandering radius $R_{\rm wan}$ of order

\be
R_{\rm wan}\approx R_{\rm core}\,\sqrt{M_{\star}/\mbh},
\label{eq.wandering_radius}
\ee

\noindent
which is larger than the tidal disruption radius for BHs less massive
than $10^9 M_{\odot}$ if the core radius is 1 pc
\citep{BW76,LT80,ChatterjeeEtAl02}. The wandering of a MBH at the
centre of a cuspy cluster has been simulated by \citet{DHM03} with a
$N$-body code allowing $N=10^6$. They find that RMS velocity of the
quickly reaches equipartition with the stars but do not comment on the
wandering radius. In Addendum A, at the end of this chapter 
(section~\ref{add.wander}), we present a simple
estimate suggesting that, in a cusp $\rho \propto r^{-\alpha}$, the
wandering radius may be much reduced, $R_{\rm wan} \propto a
(m_{\star}/\mbh)^{1/(2-\alpha)}$, where a is typical length scale for
the central parts of the cluster. For $\alpha \ge 1.5$, one would then
expect $R_{\rm wan}$ to be smaller than $R_{\rm t}$ for black holes as
light as $2000\,\msol$, but this arguments neglects the flattening of
the density profile due to {loss-cone} accretion. Further $N$-body
simulations are clearly required to settle the question and, in
particular, if $R_{\rm wan}$ is larger than $R_{\rm t}$, to establish
the effect of the motion of the MBH on disruption rates, which can be
either increased or decreased \citep{MT99}.
}

We arrogate that at any sphere of radius $r$ the transport of
{loss-cone} stars in the time-scale $t_{\rm out,eff}$ towards the
centre happens instantaneously compared with the time step used for 
the time evolution.

Hence, the local density ``loss'' at $r$ is 

\be 
\bigg( \frac{\delta \rho}{\rm \delta t} \bigg)_{\rm lc}=
 -{\rho_{\rm lc}P_{\rm lc}\over t_{\rm out}} ;
\label{eq.drho_dt_lc}
\ee

\noindent
with $\rho_{\rm lc}=K\Omega\rho$.

\noindent
This corresponds to a local energy {\em loss} \footnote{By {\em loss}
  we mean here transport of mass and kinetic energy toward the central
  BH, for it is lost for the stellar system.} of

\begin{eqnarray}
\bigg( \frac{\delta \rho\,\sr^2}{\delta t} \bigg)_{\rm lc}&=&-\bigg( 
\frac{\delta \rho}{\delta t}\bigg)_{\rm lc} \cdot\Big( E_r\,\sr^2+u^2 \Big) 
\nonumber \\
\bigg( \frac {\delta \rho\,\s_t^2}{\delta t}\bigg)_{\rm lc}&=&-\bigg( 
\frac{\delta \rho}
{\delta t}\bigg)_{\rm lc}E_t\,\s_t^2.
\label{eq.loc_energy_loss}
\end{eqnarray}

\noindent
$E_r$ and $E_t$ are worked out integrating over the velocity
distribution part that corresponds to the {loss-cone} with the
approximation $u\ll\sr$ and $v_{\rm lc}\ll\s_t$,

\begin{eqnarray}
E_r&\approx& 1 \nonumber \\
E_t&\approx& v_{\rm lc}^2/\s_t^2 \ll 1
\label{eq.Er_Et}
\end{eqnarray}

\noindent
Thereupon, the mass accretion rate of the central BH can be calculated as 

\be {\dot M}=-\epsilon_{\rm eff} \int_{r_{\rm t}}^{R_{\rm tot}} \bigg(
\frac{\delta \rho}{\delta t} \bigg)_{\rm lc} 4\pi r^2 dr.
\label{eq.m_dot}
\ee

\noindent
Here $R_{\rm tot}$ stands for the total radius of the stellar system.
The accretion efficiency has been set throughout our calculations to
$\epsilon_{\rm eff}=1$; for a discussion on different $\epsilon_{\rm
  eff}$-values see \citet{MS80}.
 
The complete set of Eqs. (\ref{eq.set_of_eqs}) including the local
accretion terms of the type $(\delta/\delta t)_{\rm lc}$ for energy
diffusion and {loss-cone} accretion are solved implicitly. For every
time step the mass of the BH and the filling degree $K$ of the
{loss-cone} are brought up to the new state of the system.  The time
step is chosen in order to keep the maximum changes of the variables
below 5\% .

\noindent
For the model calculation we have utilised for the boundary conditions
that at the outer limit, $R_{\rm tot}=10^4$\,pc, we impose $u=F=0$ and
$M_{\rm r}=M_{\rm tot}$. No stellar evaporation is allowed. At the
centre, the usual boundary conditions for the gaseous model are
$u=M_{\rm r}=F=0$ but the central point $r_1=0$ is not explicitly
used when there is a BH, for obvious reasons. Instead, one imposes
that all quantities vary as power-laws, ${\rm d}\,\ln\,x/{\rm
  d}\,\ln\,r = C^{\rm st}$ inside the first non-zero radius of the
discretisation mesh, $r_2=1.7\times 10^{-6}\,{\rm pc}$ (see chapter
 \ref{ch.th_model}).

\section{Units and useful quantities}
\label{subsec:units}

The units used in the computations correspond to the so-called
$N$-body unit system, in which $G=1$, the total initial mass of the
stellar cluster is 1 and its initial total energy is $-1/2$
\citep{Henon71a,HM86}. For the simulations presented here, the 
initial cluster structure corresponds to the Plummer model whose
density profile is $\rho(r) =
\rho_0{\left(1+\left({r}/{R_{\rm Pl}}\right)^2\right)^{-5/2}}$, where
$R_{\rm Pl}$ is the Plummer scaling length.  For such a model the
$N$-body length unit is $\mathcal{U}_\mathrm{l}
={16}/(3\pi)\,R_{\rm Pl}$.

In the situations considered here, the evolution of the cluster is
driven by 2-body relaxation. Therefore, a natural time scale is the
(initial) {\em half-mass relaxation time}. We use the definition of
\citet{Spitzer87}, 
\be
\label{eq_rel_time}
T_{\rm rh}(0) = \frac{0.138 N}{\ln \Lambda}
\left(\frac{R_{1/2}^3}{G{\cal M}_{\rm cl}}\right)^{1/2}.
\ee
For a Plummer model, the half-mass radius is $R_{1/2}
=0.769\,\mathcal{U}_\mathrm{l} = 1.305\, R_{\rm Pl}$. 
${\cal M}_{\rm cl}$ is the total stellar mass.

For a cluster containing a central BH, an important quantity is the
{\em influence radius}, enclosing the central region inside of which
the gravitational influence of the BH dominates over the self-gravity
of the stellar cluster. The usual definition is $r_{\rm
h}=G\mbh/\sigma_0^2$, where $\sigma_0$ is the velocity dispersion in
the cluster at a large distance from the BH. As the latter quantity is
only well defined for a cluster with an extended core, we use here the
alternate and approximate definition $M_{\rm r}(r_{\rm h})=\mbh$,
i.e. $r_{\rm h}$ is the radius of that encloses a total stellar mass
equal to the mass of the BH.

\section{Results}
\label{sec.diff_model_results}

We study the evolution of a stellar cluster with a so-called ``seed
BH'' at its centre. We consider two possible configurations for the
stellar system; one of a total mass of $M_{\rm tot}=10^5 \msol$ and
another of $10^6 \msol$. For the initial BH mass, we have chosen $\mbh
(0)=50\msol$ and $500\msol$ and we model it as a Plummer of $R_{\rm
  Pl}=1$pc. Even though it would be more realistic to set the
efficiency parameter $\epsilon_{\rm eff}=1/2$, we choose here
$\epsilon_{\rm eff}=1$ for historical reasons. Nonetheless, here we
study additionally the influence of a variation of the stellar
structure parameter $x_b$, since it influences the {tidal radius} and
hence the accretion rates (see Eq. \ref{eq.r_tid_bind}). For this
intent we compare case a ($x_b=1$) with another one in which we choose
the value $x_b=2$ (case b). As regards the physical meaning, the stars
of case b have twice as much internal binding energy than case a.

The cluster evolves during its pre-collapse phase up to a maximum
central density from which the energy input due to star accretion near the
{tidal radius} becomes sufficient in order to halt and reverse the core
collapse. Immediately afterwards, the post-collapse evolution starts.
At the beginning of the re-expansion phase, the BH significantly grows
to several $10^3$ solar masses. Thereon, a slow further expansion and
growth of the BH follow.
 
In Fig.\,\ref{fig.mbh_dmbh_1e5}, we follow the evolution of the mass
of a central BH in a globular cluster of $10^5$ stars of $1\,{\rm
  M}_\odot$. Panel (a) shows the mass of the BH
as function of time. On panel (b), we present
the accretion rate on to the BH, i.e. its growth rate. For
$\mathcal{M}_{\rm bh}(0)=50\,{\rm M}_\odot$, the early cluster's
evolution is unaffected by the presence of the BH which starts growing
suddenly at the moment of deep core collapse, around $T\simeq
14.5\,T_{\rm rh}(0)$. In Fig.\,\ref{fig.mbh_dmbh_1e6} we follow the same
evolution for the case of a stellar cluster of $10^6$ stars.

\bfig 
\begin{center}
\resizebox{0.6\hsize}{!}{\includegraphics[bb=33 183 280 667,clip]
	{diag_diffusion_model/mbh_dmbh_1e5.eps}}
\caption{Evolution of the mass of a central BH in a globular cluster
  of $10^5$ stars of $1\,{\rm M}_\odot$. We considered three cases. In
  case a (solid line), the initial BH mass is $\mbh(0)=50\,{\rm
  M}_\odot$ and $x_b=1$, case b (dashes) has the same initial BH mass
  but $x_b=2$ while case c (dash-dot) corresponds to
  $\mbh(0)=500\,{\rm M}_\odot$ and $x_b=1$. An accretion efficiency of
  $\epsilon_{\rm eff}=1$ is assumed.  Panel (a) shows the mass of the
  BH as function of time and panel (b) the accretion rate on to the
  BH. At late times, the mass of the central BH increases like $\mdot
  \propto T^{-1.2}$ as predicted by simple scaling arguments (see
  text)
\label{fig.mbh_dmbh_1e5} }
\end{center}
\efig

From Figs.\,(\ref{fig.mbh_dmbh_1e5}) and (\ref{fig.mbh_dmbh_1e6}) we can see
that the differences between the cases a, b and c are nearly
negligible after core collapse. In general, the structure of the cluster
at late times is nearly independent of $\mbh(0)$ and $x_{\rm b}$.
From these plots we can infer that this occurs since core collapse
leads to higher densities if the initial BH mass is smaller and
thus the integrated accreted stellar mass increases.

\bfig 
\begin{center}
\resizebox{0.7\hsize}{!}{\includegraphics[bb=33 183 280 667,clip]
	{diag_diffusion_model/mbh_dmbh_1e6.eps}}
\caption{Same as Fig.\,\ref{fig.mbh_dmbh_1e5} but for a cluster of $10^6$ stars 
with the same size ($R_{\rm P}=1\,{\rm pc}$).
  \label{fig.mbh_dmbh_1e6} }
\end{center}
\efig

We exhibit the evolution of the structure of the cluster for case a
with $10^5$ stars in Fig.\,\ref{fig.LagrRad_a1e5}. With dotted lines
we plot various Lagrangian radii, for mass fractions ranging between
$10^{-3}$\,\% (which formally corresponds to only one star) to
$90$\,\%. Only the mass still in the stellar component at a given time
is taken into account. Moreover, the evolution of the influence radius
(solid line, defined as the radius enclosing a stellar mass equal to
the BH mass) and critical radius (dashed line) are shown, so that one
can infer the percentage of the stellar mass embodied within them at a
certain moment. For late time, one obtains self-similar evolution with
size increasing like $R\propto T^{2/3}$, as expected for a system in
which the central object has a small mass and the energy production is
confined to a small central volume.
\citep{Henon65,Shapiro77,McMLC81,Goodman84}. We consider too the case
of a $10^6$ stars in Fig.\,\ref{fig.LagrRad_a1e6}, for which the
$R\propto T^{2/3}$ expansion is a poor approximation because, at late
times, the BH comprises of order 40\,\% of the system mass.


\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics[bb=34 183 572 685,clip]{diag_diffusion_model/LagrRad_a1e5.eps}}
\caption{Evolution of the radii of spheres enclosing
  the indicated fraction of the total mass, Lagrangian radii, for case
  a. The mass fractions range from $10^{-3}$\,\% to $90$\,\%.  The
  influence and critical radius are displayed (solid and broken line).
  See text for further explanation.
\label{fig.LagrRad_a1e5} }
\EC
\efig


\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics[bb=34 183 572 685,clip]{diag_diffusion_model/LagrRad_a1e6.eps}}
\caption{Same as Fig.\,\ref{fig.LagrRad_a1e5} but for a cluster of $10^6$ stars. 
 \label{fig.LagrRad_a1e6} }
\EC
\efig

We observe in Fig.\,\ref{fig.dens_a1e5_10Gyr} that for the evolved
post-collapse model the spatial profile of the stellar density has a
power law slope of $\rho\propto r^{-{7/4}}$ in the region $r_{\rm
  crit}<r<r_{\rm h}$, where $r_{\rm h}$ is the influence radius. The
density profile flattens for $r<r_{\rm crit}$ due to the effective
{loss-cone} accretion. For the same post-collapse moment we display the
surface density for case a in Fig.\,\ref{fig.projdens_a1e5_10Gyr}.


Arguments based on constant accretion rate lead to the result that one
should expect a stellar density profile proportional to $r^{-1/2}$
inside $r_{\rm crit}$. If one takes Eq.  (\ref{eq.diff_eq_rho_lc}) and
considers that ${\dot M}=C^{\rm st}$, we get that

\be
\frac{d\rho_{\rm lc}}{dt} \propto r^{-\alpha -5/2}.
\label{eq.1st_arg_alpha_0.5}
\ee

\noindent
Where we have assumed that $P_{\rm lc} \approx 1$, $\rho \propto
r^{-\alpha}$ and suppose that the gravity of the BH totally dominates
this region, so that we have a Keplerian profile, $\sigma \propto
r^{-1/2}$. We have also taken into account the fact that $t_{\rm
  dyn}=r/\sigma$. Thus,

\be
{\dot M}=r^{1/2-\alpha}.
\ee

\noindent
Whence, $\alpha=-1/2$.

Another possible way to derive this result is based on the Eq. (4-137)
of \citet{BT87},

\be
\rho(r)=4\pi \int^{\psi}_{0}f(\epsilon)\sqrt{2(\psi -\epsilon)}d\epsilon
\label{eq.BT87_4_137}
\ee

\noindent
In this equation $\psi:=-\phi+\phi_0$ and $\epsilon:=-E+\phi_0$, where
$\phi_0$ is a constant that makes $f>0$ for $\epsilon>0$ and $f=0$ for
$\epsilon \leq 0$. If we assume that f\,($\epsilon$)=0 for $\epsilon >
\epsilon_0$ (where $\epsilon_0$ is around $r_{\rm crit}$) and consider
that we are in the region where $r\ll r_{\rm crit}$, $\psi=G\mbh /r \gg \epsilon$ and then, from 

\be
\rho(r) \approx 4\pi \sqrt{2 \psi} \int^{\psi_{\epsilon_0}}_0 f(\epsilon)d\epsilon
\label{}
\ee

\noindent
one can conclude that $\rho(r) \propto r^{-1/2}$.

Nevertheless, none of these derivations can be applied to the
interpretation of our results, since it would contradict the basis of
our model's idea. We cannot expect this ``classical'' result in our
results, since it would imply that the filling degree $K$ is constant
in the region where $P_{\rm lc} \approx 1$ ($r \ll r_{\rm crit}$ see
Fig.  \ref{fig.LC_c1e5_10Gyr}), which is not the case at all.

As a matter of fact, one does not know a priori which kind of slope
one will have in this region. It is rather an initial condition that
we introduce and it is totally disconnected with our diffusion model.
We could bargain for such a slope in the case that we just have stars
that come from unbound orbits and fall on to the central BH
disappearing immediately from the stellar system. In the model
presented here, we have also stars with bounded orbits around the BH
that will perturbe the slope of $\alpha=-1/2$. \citet{LS77} proved
that within $r_{\rm crit}$, according to their Eq. (71) (where they assume
$\Omega \ll 1$, small loss-cone), $\alpha$ will continously vary from 1.75
to $-\infty$.


\bfig
\C 
\resizebox{0.65\hsize}{!}{\includegraphics[bb=38 160 578 712,clip]
	{diag_diffusion_model/dens_a1e5_10Gyr.eps}}
\caption{Density profile for a $10^5{\rm M}_\odot$ globular cluster 
  with $\mathcal{M}_{\rm bh}(0)=50\,{\rm M}_\odot$ (case a) at 10
  Gyrs. The round dot indicates the influence radius
  ($M_r=\mathcal{M}_{\rm bh}$), the star the critical radius and the
  square the radius below which the description of the cluster as a
  continuum loses significance because the enclosed mass is smaller
  than $1\,{\rm M}_\odot$. As expected, for the zone between the
  critical and influence radii, the density profile closely
  reassembles a power-law of exponent $-7/4$. At that stage, the
  structure of case c ($\mathcal{M}_{\rm bh}(0)=500\,{\rm M}_\odot$)
  is extremely similar. We can see that from the ``1-star'' radius
  onwards the slope of the curve shows a tendence to increase. This is
  due to the fact that the {loss-cone} size is artificially limited
  for stability purposes. On the other hand, the slope comprised
  between the critical radius and the 1-star radius is consistent with
  the arguments given in the work of \citet{LS77} (see text for
  further explanation).
\label{fig.dens_a1e5_10Gyr} }
\EC
\efig

\bfig 
\C
\resizebox{0.6\hsize}{!}{\includegraphics[bb=38 160 578 712,clip]
	{diag_diffusion_model/disp_a1e5_10Gyr.eps}}
\caption{Profile of the three-dimensional velocity dispersion for 
the same case and time as Fig.\,\ref{fig.dens_a1e5_10Gyr}. See text
for comments.
\label{fig.disp_a1e5_10Gyr} }
\EC
\efig

\bfig
\C 
\resizebox{0.8\hsize}{!}{\includegraphics[bb=38 160 578 712,clip]
	{diag_diffusion_model/ani_a1e5_10Gyr.eps}}
\caption{Profile of the anisotropy parameter for 
  the same case and time as Fig.\,\ref{fig.dens_a1e5_10Gyr}. The
  decrease of at the border is an artefact of the inappropriate
  boundary condition. An outer boundary with radial anisotropy should
  be open, but here we enforce the adiabatic wall. If one ``opens''
  the wall, but it would be at the expense of the stability of the
  program. All this does affect only a very small fraction of the
  total mass.
\label{fig.ani_a1e5_10Gyr} }
\EC
\efig

\bfig
\C 
\resizebox{0.8\hsize}{!}{\includegraphics[bb=38 160 578 712,clip]
	{diag_diffusion_model/proj_dens_a1e5_10Gyr.eps}}
\caption{Projected density for the same case and time as Fig.\,\ref{fig.dens_a1e5_10Gyr}. 
  In the interval between $r_{\rm h}$ and $r_{\rm crit}$ we get a slope of
  $-3/4$, as expected.
\label{fig.projdens_a1e5_10Gyr} }
\EC
\efig

\bfig
\C 
\resizebox{0.8\hsize}{!}{\includegraphics[bb=38 160 578 712,clip]
	{diag_diffusion_model/proj_disp_a1e5_10Gyr.eps}}
\caption{Projected velocity dispersions for 
the same case and time as Fig.\,\ref{fig.dens_a1e5_10Gyr}. The line of
sight component is represented with a solid line and the proper motion
component with a dashed one for the radial and dashed-dot for the
tangential contribution.
\label{fig.projdisp_a1e5_10Gyr} }
\EC
\efig

Regarding the three dimensional velocity dispersion, in
Fig.\,\ref{fig.disp_a1e5_10Gyr} we can see coming a slope of -1/2 at
the inner region; but extending below the one-star radius does not
make much sense. A slope of -1/2 is what one would expect from
Kepler's third law and a simple application of Jeans equation, with
the assumptions that (1) dynamical equilibrium holds, (2) the gravity
is dominated by the central BH, (3) the density follows a pure
power-law and (4) the anisotropy $\sigma_t/\sigma_r$, is constant,
indeed predicts $\sigma \propto r^{-1/2}$. At the end of this chapter,
in Addendum B (section~\ref{add.sigma})
, we show that the Jeans equation for
stationary equilibrium actually describes the central regions of the
cluster quite well. The reason why the velocity dispersion does not
follow closely the ``Keplerian'' profile has to do with the fact that
none of assumptions (2)-(4) exactly holds all the way from the
influence radius inward.

Figures\,\ref{fig.projdisp_a1e5_10Gyr} and
\ref{fig.projdens_a1e5_10Gyr} give the plots of the projected density
and velocity dispersions for the late post-collapse model. See 
Addendum C (section \ref{add.proj_vel}).

The effects of anisotropy are studied in Fig.\,\ref{fig.ani_a1e5_10Gyr},
where we can see that the external parts of the cluster are dominated
by radial orbits. Inside the critical radius (indicated by a star
symbol), one notices a slight tangential anisotropy, an effect of the
depletion of {loss-cone} orbits. At large radii, the velocity
distribution tends to isotropy as an effect of the outer bounding
condition imposed at $10^4$\,pc.

\noindent
The effects of anisotropy in the stellar system can also be seen in
Fig.\,\ref{fig.projdisp_a1e5_10Gyr}, where we plot the components
along the line of sight $\sigma_{\rm LOS}$ (solid line) and on the sky
(``proper motions''). The latter is decomposed into the radial
direction (i.e. towards/away from the position of the cluster's
centre) component, $\sigma_{\rm PM,r}$ (dashes) and the tangential
component, $\sigma_{\rm PM,t}$ (dash-dot). Note that the radial
anisotropy in the outskirts of the cluster reveals itself as a radial
``proper motion'' dispersion slightly larger than the other
components. For an isotropic velocity dispersion, all three components
would be equal. Despite {loss-cone} effects, there is no measurable
anisotropy at the centre.

The {loss-cone} induced anisotropy could be
detected only if one could select the stars that are known to be
spatially close to the centre (and not only in projection) as would be
feasible these stars happened to be of a particular population. An
interesting possibility that we'll soon investigate with multi-mass
models is the concentration at the centre of more massive stars,
i.e. mass segregation.

Note that some anisotropy has been detected among the stars orbiting
the central massive black hole of the Milky Way, Sgr\,A$^\ast$, at
distances closer than 1, i.e. 0.04\,pc which is well inside the
critical radius ($>1$\,pc) \citep{SchoedelEtAl03}. However, the
detected anisotropy is in the radial direction rather than
tangential. It is probably not connected to {loss-cone} effects but to
particular history of these seemingly very young stars \citep{GhezEtAl03}
which remains a puzzle.

\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics[bb=26 176 580 687,clip]
	{diag_diffusion_model/evol_dens_Rinfl_a1e5.eps}}
\caption{Evolution of the stellar density in the central region for our model with $10^5$ 
stars and $\mathcal{M}_{\rm bh}(0)=50\,{\rm M}_\odot$ (case a).  The
solid line depicts the density at the influence radius $R_{\rm
h}$. The dashed line shows the average density {\em within} $R_{\rm
h}$.
\label{fig.evol_dens_Rinfl_a1e5} }
\EC
\efig


\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics[bb=26 176 580 687,clip]
	{diag_diffusion_model/evol_disp_Rinfl_a1e5.eps}}
\caption{Same as Fig.\,\ref{fig.evol_dens_Rinfl_a1e5} but 
for the three dimensional velocity dispersion.
\label{fig.evol_disp_Rinfl_a1e5} }
\EC
\efig

In Fig.\,\ref{fig.LC_c1e5_10Gyr} the diffusion model is layed out for
the {loss-cone} as evaluated in previous sections. We evaluate a
model close to the post-collapse moment analysed in the other plots.
We depict here the {loss-cone} filling factor $K$ (upper panel),
the {loss-cone} and diffusion angles $\theta_D$ and $\theta_{\rm lc}$
(middle panel) and the local contributions to the total
{loss-cone} star accretion rate (lower panel). Our diffusion
model reproduce well the picture of \citet{FR76}: the critical radius
is defined by $\theta_D =
\theta_{\rm lc}$ and coincides with the radius where the local
contribution to the {loss-cone} accretion rate has its peak value.
These two angles are connected to the time-scales $t_{\rm in}$ and
$t_{\rm out}$, $\theta_D^2\propto t_{\rm out}/t_{\rm relax}$ and
$\theta_{\rm lc}^2\propto t_{\rm in}/t_{\rm relax}$. The figures show
that the maximum contribution to the mass accretion rate stems at the
radius where $\theta_D =\theta_{\rm lc}$. Consistently, \citet{FR76}
estimated the total mass accretion rate as $\mdot\propto
\rho(r_{\rm crit})\,r_{\rm crit}^3/ t_{\rm relax}(r_{\rm crit})$.

The dependence of {loss-cone} accretion rate on time during the late
re-expansion phase can be estimated through simple scaling laws. One
starts with the relation of \citet{FR76} mentioned above,
\be
\mdot\simeq
\frac{\rho(r_{\rm crit})\,r_{\rm crit}^3}{t_{\rm relax}(r_{\rm crit})},
\label{eq:MdotFR76}
\ee 
and the definition of the critical radius,
\be
\begin{aligned}
\theta_{\rm lc}^2(r_{\rm crit}) &= \theta_{\rm D}^2(r_{\rm crit}) \\
\Rightarrow \frac{r_{\rm t}}{r_{\rm crit}} &\simeq 
\frac{t_{\rm cross}(r_{\rm crit})}{t_{\rm relax}(r_{\rm crit})}.
\end{aligned}
\label{eq:RtoverRcrit}
\ee
One substitutes the following relations into Eq.~\ref{eq:RtoverRcrit},

\be
\begin{aligned}
r_{\rm t} &\propto \mbh^{1/3},\\
t_{\rm cross}(r) & \propto \frac{r^{3/2}}{\mbh^{1/2}}, \\
t_{\rm relax}(r) & \propto \frac{\sigma(r)^{3}}{n(r)} \propto \frac{\mbh^{3/2}}{r^{3/2}\,n(r)},
\end{aligned}
\label{eq:RelRcrit}
\ee

where we have made use of the fact that the potential is dominated by
the BH in the region of interest ($r_{\rm crit} < r_{\rm
h}$). Finally one needs the dependence of the density of stars on time
and radius, $n(r,T)$. We have seen that, to a good approximation, the
re-expansion of the cluster is homologous with Lagrange radii
expanding like $R \propto T^{2/3}$. In the region between $r_{\rm
crit}$ and $r_{\rm h}$, the density profile resembles a power-law cusp;
hence, a general self-similar evolution can be described by
\be
n(r,T) = n_0(T)\left(\frac{r}{r_0(T)}\right)^{-\alpha},
\ee
where $r_0$ is some Lagrange radius. Hence, from conservation of mass inside $r_0$,
\be
n(r,T) \propto T^{\frac{2\alpha-6}{3}}r^{-\alpha}.
\label{eq:DensOfTR}
\ee
Combining relations \ref{eq:RtoverRcrit},
\ref{eq:RelRcrit} and \ref{eq:DensOfTR}, one finds

\be
r_{\rm crit} \propto \mbh^\frac{7}{3(4-\alpha)} T^\frac{2(3-\alpha)}{3(4-\alpha)}
\ee

and, inserting this into Eq.~\ref{eq:MdotFR76},

\be
\mdot \propto \mbh^\frac{27-19\alpha}{6(4-\alpha)} T^\frac{7(\alpha-3)}{3(4-\alpha)},
\ee

which, by integration, yields

\begin{align}
\mbh &\propto T^\frac{2(4\alpha-9)}{13\alpha-3},\\
\mdot &\propto T^\frac{-5(\alpha-3)}{13\alpha-3}.
\end{align}

For $\alpha=7/4$, which is appropriate here (again because $r_{\rm crit} <
r_{\rm h}$), the exponent in the last relation turns out to be $-95/79
\simeq -1.20$, in remarkable agreement with figures \ref{fig.mbh_dmbh_1e5}b 
and \ref{fig.mbh_dmbh_1e6}b.

\bfig 
\C
\resizebox{0.6\hsize}{!}{\includegraphics[bb=104 144 529 712,clip]
	{diag_diffusion_model/LC_a1e5_10Gyr.eps}}
\caption{In this triple diagram we evince the dependence on radius of the mass 
  accretion rate, the {loss-cone} and diffusion angle which are
  related to the filling and depletion time-scales of the
  {loss-cone} (see text) and the filling factor $K$. The
  critical radius is defined by the condition $\theta_D = \theta_{\rm
  lc}$ (broken line).  \label{fig.LC_c1e5_10Gyr} }
\EC
\efig

\section{Discussion}
\label{sec.diff_model_disc}

We have presented in this chapter a method to follow the evolution of a
spherical stellar cluster with a central accreting BH in a fully
self-consistent manner concerning the spatial resolution. As regards
the velocity space, we use a simplified model based on ideas of
\citet{FR76} in order to describe the behaviour of the distribution
function inside and outside the {loss-cone} by a simple diffusion
equation. This numerical method is an extension of the ``gaseous
model'' which has been successfully applied to a variety of aspects of
the evolution of globular clusters without central BH \citep[][amongst
others]{ST95,SA96,GS00,GS03,DS01}. With this new version, the
simulation of galactic nuclei is also feasible.

In addition to an explanation of the physical and numerical principles
underlying our approach, we have concentrated on a few simple test
computations, aimed at checking the proper behaviour of the code. We
considered a system where all stars are and remain single, have the
same mass, stellar evolution and collisions are neglected and a seed
central BH is allowed to grow by accreting stellar matter through
tidal disruptions. The present version of the code already allows for
a (discretised) stellar mass spectrum and stellar evolution and we are
in the process of including stellar collisions because they are
thought to dominate over tidal disruption in most galactic nuclei, as
far as accretion on to the BH is concerned
\citep{DDC87a,DDC87b,MCD91,FB02b}. In a subsequent chapter, we shall
increase complexity and realism one step further and consider systems
with a mass spectrum. Using both this gaseous code and the Monte Carlo
algorithm \citep{FB01a,FB02b}, we will investigate the role of mass
segregation around a massive black hole (Amaro-Seoane, Freitag \&
Spurzem, in preparation), a mechanism which may have important
observational consequences as it probably affects the structure of the
central cluster of the Milky Way
\citep{Morris93,MEG00,Freitag03,Freitag03b,PL03} and impacts rates of
tidal disruptions and capture of compact stars by emission of
gravitational waves in dense galactic nuclei \citep[][and references
therein]{MT99,SU99,Sigurdsson03}.

Unfortunately, the literature has relatively little to offer to check
our models. The most robust predictions are probably the analytical
and semi-analytical analysis for the regime where the gravity of the
BH dominates and the Fokker-Planck treatment of relaxation holds
\citep{BW76,SL76,LS77,CK78}. The most important feature of these
solutions is that, provided the system is well relaxed and one stands
beyond the critical radius (inside of which {loss-cone} effects
complicate the picture), a cuspy density distribution is established,
$\rho\propto r^{-\alpha}$ with $\alpha=7/4$. Our code nicely agrees
with this prediction. 

Concerning the evolution of the system, we first note that, initially,
the cluster follows the usual and well understood route to
core-collapse. That the gaseous model can successfully simulate this
phase has been clearly established in previous works
\citep{GS94,SA96}. When the core has become dense enough, the BH starts 
growing quite suddenly. As it accretes stars that are deeply bound,
i.e. with very negative energies, the BH creates a outward flux of
energy and allows the cluster to re-expand. As long as the source of
energy is centrally concentrated and that the mass of the BH remains
relatively slow, one expect the re-expansion to become self-similar, a
regime during which the size of the cluster increases like $R\propto
T^{2/3}$ \citep[][among
others]{Henon65,Shapiro77,McMLC81,Goodman84}. This is again well
reproduced by the gaseous model.  Solving the Fokker-Planck equation
with a Monte Carlo method, \citet{MS80} and \citet{DS82} have realised
a series of simulations of single-mass globular clusters with a
central BH. Because their resolution was quite low and because they
used ``initial'' conditions difficult to implement (in most of their
runs the central BH is not present initially but introduced at some
instant during deep collapse), we do not attempt a quantitative
comparison with their results. An added difficulty is that we do not
include tidal truncation of the cluster. However, an important finding
of \citet{MS80} is reproduced by our computations, namely that the
initial mass of the seed black hole has little effect on the
post-collapse evolution, provided it represents only a small fraction
of cluster mass. In particular, the BH mass at late times converges to
the same value which only depends on the size and mass of the
cluster. We note that such convergence was also obtained with the
Monte Carlo algorithm and that a comparison between results obtained
with that code and an early version of the program described here was
presented in \citet{FB02b}. More comparisons between the two methods
are planned (Amaro-Seoane, Freitag \& Spurzem, in preparation).

Here one has to mention that the energy input of the BH
star accretion causes a ``temperature'' increase in the central region
which is followed by a thermal expansion. Therefore, the system is a
normal thermal system with positive specific heat in contrast to the
cores of self-gravitating systems, where the energy input due to
binary hardening causes a core expansion and decrease of temperature
\citep{BS84}. Afterwards, an inversion in the radial temperature
profile follows and the expansion is a reverse gravothermal
instability \citep{Spurzem91}. Since our system is dominated by
external gravitation in the centre we cannot expect such a
behaviour. Furthermore, as the central BH grows irreversibly, it
continues accreting stars in spite of the re-expansion and continuous
decrease of the density of stars. Hence, a second core collapse is
impossible and oscillations of the central cluster density do not
occur.

Among the aspects of our results that require further investigation,
we mention the shape of the density profile inside the critical
radius.  Although the well known $\rho \propto r^{-7/4}$ Bahcall-Wolf
solution only strictly applies for $r_{\rm crit}<r<r_{\rm h}$, this
cusp extends inward nearly down to the tidal disruption radius in the
stationary models of \cite{MS79}, which also include {loss-cone}
physics. This result is in disagreement with the analysis of
\citet{DO77a} (see also \citealt{OR78}) which predicted $\rho \propto
r^{-1/2}$ in this region.  As shown on Fig.~\ref{fig.dens_a1e5_10Gyr},
we obtain an even stronger flattening of the density law inside
$r_{\rm crit}$. At the present time it is unknown to us which
solution, if any, is the correct one. A possibility to be considered
is that this a consequence of the truncation of the moment equations
to the second order. In other words, in regions where the {loss-cone} is
significantly depleted, representing the velocity distribution by a
simple dispersion ellipsoid and using the velocity dispersion to
determine an ``effective'' {loss-cone} aperture (Eq.~\ref{eq.lc_vel_k})
is clearly quite a strong approximation. This may impact the density
distribution as the system adjusts its central structure to produce
the heating rate required by the overall expansion.

Fortunately,
numerical inaccuracies at very small radii are unlikely to affect
the overall structure and evolution of the cluster because the
{loss-cone} accretion physics are essentially determined by the
conditions at the critical radius, not in the immediate vicinity of
the BH.


\section{Addendum A: MBH wandering in a cuspy cluster.}
\label{add.wander}

\def\mcl{{\cal M}_{\rm cl}} 
\def\m{{\cal M}} 

Here we present a simple estimate of the wandering radius $R_{\rm
  wan}$ of a MBH embeded in a stellar cluster whose density posseses a
power-law cusp in the inner regions. We assume that, were it not for
the effect of the MBH itself, the stellar cluster would be described
by an eta-model \citep{Dehnen93,Tremaine94} with enclosed stellar mass
% 
\be 
\m_\ast(r) = \mcl \left(\frac{r/a}{1+r/a}\right)^{\eta},
\label{eq.wand_1} 
\ee
%
where $\mcl$ is the total mass in stars and $a$ the break radius. For
$r\ll a$, the density is $\rho \propto r^{-\alpha}$ with
$\alpha=3-\eta$. Inside $R_{\rm wan}$, the MBH strongly perturbates
stellar orbits and we suppose the density is rendered more or less
constant. Hence, the potential felt by the MBH is approximately
harmonic,
%
\be 
\Phi(r) = \Phi_0 + \frac{1}{2}\omega^2r^2\ \ \mbox{, with}\ 
\omega^2=\frac{G\m_\ast(R_{\rm wan})}{R_{\rm wan}^3}.
\label{eq.wand_2}  
\ee
%
For a harmonic oscillator, the RMS amplitude of the oscillations in
velocity and space are linked to each other,
%
\be
 V_{\rm RMS}^2 = \omega^2 R_{\rm RMS}^2 \approx \omega R_{\rm wan}^2 = 
\frac{G\m_\ast(R_{\rm wan})}{R_{\rm wan}}.
\label{eq:Vrms}
\ee
%
\citet{DHM03} have verified with $N$-body simulations that
equipartition of kinetic energy between the MBH and the stars is
established, at least in the case of $\eta=1.5$. Namely,
%
\be
\mbh V_{\rm RMS}^2 \simeq m_\ast \sigma^2,
\label{eq:Vrms2}
\ee
%
where $\sigma$ is the stellar velocity dispersion at $r=a$. For
$\eta=1.5$ \citep{Tremaine94},
%
\be 
\sigma^2 \simeq 0.1 \frac{G\mcl}{a}.
\label{eq:sigma}
\ee
%
Finally, assuming $R_{\rm wan} \ll a$ and, hence, $\m_\ast(R_{\rm
  wan}) \simeq \mcl \left(R_{\rm wan}/a\right)^\eta$ and combining
equations~\ref{eq:Vrms}, \ref{eq:Vrms2} and \ref{eq:sigma}, we obtain
%
\be
R_{\rm wan} \approx 0.01 a\left(\frac{m_\ast}{\mbh}\right)^2\ \ \mbox{for}\ \eta=1.5
\label{wand_3}
\ee
%
and
%
\be
R_{\rm wan} \propto a\left(\frac{m_\ast}{\mbh}\right)^{1/(\eta-1)}
\label{eq.wand_4}
\ee
%
for general eta-models.


\section{Addendum B: Velocity dispersion in the central regions}
\label{add.sigma}


\bfig 
\C
\centerline{\resizebox{0.55\hsize}{!}{\includegraphics
	{diag_diffusion_model/JeansEqn_1e5a.eps}}}
\caption {
  Check of stationary Jeans equation for our ``standard'' model
  ($10^5$ stars, case a) at $T=10\,Gyrs$ (same model and time as
  Fig.\,\ref{fig.disp_a1e5_10Gyr}).  Panel (a) depicts the logarithmic
  derivatives of the stellar density $n$ and radial component of the
  velocity dispersion well as the anisotropy parameter,
  $2\beta=2-\sigma_t^2/\sigma_r^2$. Horizontal lines corresponding to
  values of $0$, $-1$, $-1.75$ and $-2.23$ are present to guide the
  eye. In particular, the ``Keplerian'' velocity profile is ${\rm
    d}\ln\,\sigma_r^2/{\rm d}\ln\,r\equiv -1$. The round dot indicates
  the influence radius, the star the critical radius and the square
  the ``1-star'' radius. On panel (b), we plot the three terms of the
  right side of the stationary Jeans equation and check that their sum
  is (nearly) equal to the left side term, i.e.\ $GM_{\rm r}$. 
  \label{fig.jeans}}
\EC
\efig 

\bfig
\C 
\centerline{\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_diffusion_model/JeansEq_1e5a_10Gyr_ZoomInNew.eps}}}
\caption {
  Blow-up of panel (b) of Fig.\,\ref{fig.jeans} for the region between
  the 1-star and the influence radius. 
  \label{fig.jeansb}}
\EC
\efig 

In the region dominated by the central BH, one may expect a
``Keplerian'' profile for the velocity dispersion, $\sigma \propto
r^{-1/2}$. However, in Sec.\,\ref{sec.diff_model_results}, we have seen that in our
standard model ($10^5$ stars, case a this relation does not really
apply where expected, i.e.\ between the ``1-star'' and the influence
radius. Here we show that the spherical Jeans equation for a system in
dynamical equilibrium is nevertheless obeyed.

In spherical symmetry the assumption of dynamical equilibrium (or
stationarity) amounts to $u = \left< v_{\rm r} \right> \equiv 0$. We
note that the gaseous model can cope with $u\neq 0$. On the other
hand, for a system whose evolution is driven by relaxation, one
expects $\sigma_{\rm r,t} \gg u$. The Jeans equation then reads
(\citealt{BT87}, Eq.\,4-55)

\be
GM_{\rm r} = -r\sigma_{\rm r}^2 \left(\frac{{\rm d}\ln n}{{\rm d}\ln r}
+\frac{{\rm d}\ln \sigma_{\rm r}^2}{{\rm d}\ln r}+2\beta\right).
\label{eq.jeans}
\ee $M_{\rm r}$ is the mass enclosed by the radius $r$, $n$ the number
density of stars and $2\beta=2-\sigma_t^2/\sigma_r^2$ is the
anisotropy parameter; other quantities have been defined previously.
One sees easily that if $M_{\rm r}\equiv \mbh$ and the first and third
term in the brackets are both constant, $\sigma \propto r^{-1/2}$ at
small $r$. But, as figures~\ref{fig.jeans} and \ref{fig.jeansb}
demonstrate, none of these assumptions exactly applies in the range of
radius under consideration. Consequently, we do not get a clean
Keplerian velocity profile although (or {\em because})
Eq.\,\ref{eq.jeans} is satisfied. Finally, we mention that our models
with $10^6$ and $10^7$ stars (and same initial size) exhibit a
Keplerian velocity cusp outside the 1-star radius, during their
post-collapse evolution. This is partly due to the relatively more
massive black hole (larger influence radius) and partly to the much
smaller 1-star radius.

\section{Addendum C: Projected velocity dispersions}
\label{add.proj_vel}

For the Figures\,\ref{fig.projdisp_a1e5_10Gyr} and
\ref{fig.projdens_a1e5_10Gyr} we have integrated the density along the
z-axis for the projection,
\begin{eqnarray}
\lefteqn{
\Sigma (r)= \int_{z=0}^{z=R_{\rm max}} \rho(\sqrt{r^2+z^2})\,dz ={}} 
                                                 \nonumber \label{eq.sigma_proj} \\
& & {} 2 \int_{\rm r}^{z={R_{\rm max}}} \rho(R)\frac{R}{\sqrt{R^2-r^2}}\,dR
\end{eqnarray} 

\noindent
If one observes the cluster along the z-axis, the contributions to the
projected velocity dispersions are as indicated in
Fig.\,\ref{fig.proj_vel_disp}. We have that $R=\sqrt{r^2+z^2}$ and
$\sigma_{\theta}= \sigma_{\phi}=\sigma_{\rm t}$, where the subscript t
stands for tangential. We can reckon that

\begin{eqnarray}
\sigma^2_{\rm z}=\sigma^2_{\rm R}\,\cos^2\theta + \myH{\sigma}^2_{\rm t}\sin^2 \theta \nonumber \\
\sigma^2_{\rm r}=\sigma^2_{\rm R}\,\sin^2\theta + \myH{\sigma}^2_{\rm t}\cos^2 \theta,
\end{eqnarray}

\noindent
where we have defined $\myH{\sigma}^2_{\rm t} \equiv \sigma^2_{\rm
  t}/2$, to be consistent with the notation used until now.

\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics[clip]{diag_diffusion_model/proj_vel_disp.eps}}
\caption{Different contributions to the projected velocity dispersions.
  The semi-major axis of the ellipsoid perpendicular to the page
  corresponds to $\sigma_{\rm t}=\sigma_{\phi}$.
\label{fig.proj_vel_disp} 
}
\EC
\efig

\noindent
In Fig.\,\ref{fig.proj_vel_disp} we can see that $\sigma_{\rm z}$
contributes to $\sigma_{\rm LOS}$, $\sigma_{\rm r}$ to $\sigma_{\rm
  pm,r}$ and $\myH{\sigma}_{\rm t}^2$ to $\sigma_{\rm pm,t}$.

Thus, we obtain the projected velocity dispersions,

\begin{eqnarray}
\lefteqn{
\sigma^2_{\rm LOS}(r)=\frac{2}{\Sigma(R)}\int^{R_{\rm max}}_{z=0}(\sigma^2_{\rm R}(R)\cos^2\theta+
\myH{\sigma}^2_{\rm t}(R)\sin^2\theta)\rho(R)\,dz={}}
\nonumber \\
&& {} \frac{2}{\Sigma(R)}\int^{R_{\rm max}}_{z=0}(\frac{z^2}{R^2}(\sigma^2_{\rm R}(R)-\myH{\sigma}^2_{\rm t}(R))+
\myH{\sigma}^2_{\rm t}(R))\rho(R)\,dz \nonumber
\end{eqnarray}
\begin{eqnarray}
\lefteqn{
\sigma^2_{\rm pm,r}(r)=\frac{2}{\Sigma(R)}\int^{R_{\rm max}}_{z=0}(\sigma^2_{\rm R}(R)\sin^2\theta+
\myH{\sigma}^2_{\rm t}(R)\cos^2\theta)\rho(R)\,dz={}} \nonumber \\ 
&& {} \frac{2}{\Sigma(R)}\int^{R_{\rm max}}_{z=0}(\frac{z^2}{R^2}(\myH{\sigma}^2_{\rm t}(R)-{\sigma}^2_{\rm R}(R))+
{\sigma}^2_{\rm R}(R))\rho(R)\,dz \nonumber
\end{eqnarray}
\begin{eqnarray}
\lefteqn{
\sigma^2_{\rm pm,t}(r)=\frac{2}{\Sigma(R)}\int^{R_{\rm max}}_{z=0}\myH{\sigma}^2_{\rm t}(R)\rho(R)\,dz,}
\label{eq.proj_vel_disp}
\end{eqnarray}

\noindent
since $\sin^2\theta=r^2/R^2=1-\cos^2\theta$ and $\cos^2\theta=z^2/R^2$.

%%%%%%% Bibliografia del cap�tol
\newpage
\thispagestyle{empty}
\begin{spacing}{0.95}
\cleardoublepage
\bibliographystyle{apalike}
\section{Literature of chapter \thechapter}
\bibliography{aamnem99,biblio_tesi}
\end{spacing}
\newpage


mass_seg.tex
\chapter{Multi-components clusters with/out a central BH: 
Mass segregation}
\label{ch.mass_seg}

%\vspace{-11cm}
%\begin{quote}
%\begin{flushright}
%{{\tt Wovon man nicht sprechen kann,\\ 
%dar{\"u}ber mu{\ss} man
%schweigen} \\
%{\vspace{0.15cm}\em L. Wittgenstein, Tractatus (7)}}
%\end{flushright}
%\end{quote}
%\vspace{11cm}

\section{An academic exercise: Mass segregation in two mass-component 
clusters}

\inicial{S}{ince} we have arduously examined the idealised situation for
a stellar cluster in which all stars possess the same mass in order to
subject to scrutiny our model, we have morally the right to extend the
analysis a further step. Here we tackle with more realistic
configurations in which the stellar system is splitted into various
components \footnote{This chapter is part of present work and will be
published like Freitag, Amaro-Seoane and Spurzem (2004). Some part of it
has been employed for Khalisi, Amaro-Seoane and Spurzem (2004)
-submitted to MNRAS- and an initial concept was employed for
\citet{AS03}}. The second integer number immediately after one is two
and so will we first extend, cautious and wary as we are, our models
to two-components star clusters. The initial motivation for the study
of such physical systems was the better resemblance to real
clusters. The processes that one-component clusters bring about is
nowadays well understood and has been plenteously studied by
different authors to check for the goodness of their approaches. New
features of these systems' behaviour arise when we consider a stellar
system in which masses are divided into two groups. Depending on how
the system taken into consideration is configurated will we exclude
{\em dynamical equilibrium} (here it is meant that the system is not
stable on dynamical time-scales) or equipartition of different
components kinetic energies is not allowed ({\em thermal
equilibrium}). 

\citet{Spitzer69} gave the analysis an initial shove
with his study. For some clusters it seemed impossible to find a
configuration in which they enjoy dynamical and thermal equilibrium
altogether. The heavy component sink into the centre because they cede
kinetic energy to the light one when reaching equipartition. The
process will carry on until equipartition is fully gained on and the
heavy component self-gravity moves out in the core. In the most of the cases,
equipartition happens to be impossible, because the subsystem of massive stars 
will undergo core collapse before equipartiton is reached.
Anon, a {\em
gravothermal collapse} will happen upon this component and, as a
result, a small dense core of heavies is formed
\citep{Spitzer69,LF78}. 
This gravothermal contraction is a product of negative heat capacity,
a typical property of gravitationally bound systems
\citep{ElsonEtAl87}. 

Disparate authors have addressed the problem
of thermal and dynamical equilibrium in such systems, from recent
direct $N$-body simulations \citep{PZMM00} and Monte Carlo simulations 
\citep{WJR00} to direct integration of
the Fokker-Planck equation \citep{IW84,KLG98} including Monte Carlo
approaches to the numerical integration of this equation
\citep{SH71b}. For a general and complete overview of the historical evolution
of two-stars stellar components, see \citet{WJR00} and references
therein.

If we do not have any energy source in the cluster and stars do not
collide (physically), the contraction carries on self-similarly
indefinitely; in such a case, one says that the system undergoes {\em
core-collapse}. This phenomenon has been observed in a big number of
works using different methods
\citep[etc]{Henon73,Henon75,SS75a,Cohn80,MS80,Stodol82,
Takahashi93,GH94b,Takahashi95,SA96,Makino96,Quinlan96,DCLY99,JRPZ00}
Core collapse is not just a characteristic of multi-mass systems, but has
been also observed in single mass analysis.

\citet{Spitzer69} gave the analytical criterion to determine whether
a two-component system has achieved energy equipartition. According to
his analysis, energy equipartition between the light and heavy
component exists if

\be
{S}:= \Big(\frac{{\cal M}_{\rm h}}{{\cal M}_{\rm l}}\Big)
\Big(\frac{m_{\rm h}}{m_{\rm l}}\Big)^{3/2} < 0.16
\label{eq.spitzer_stab}
\ee

\noindent
Where ${\cal M}_{\rm l}$ and ${\cal M}_{\rm h}$ are the total numbers
of light and heavy components, respectively. More recent numerical
calculations \citep{WJR00} have settled this criterion to $\Lambda$,

\be
{\Lambda}:= \Big(\frac{{\cal M}_{\rm h}}{{\cal M}_{\rm l}}\Big)
\Big(\frac{m_{\rm h}}{m_{\rm l}}\Big)^{2.4} < 0.16
\label{eq.spitzer_stab_new}
\ee

When we modify the ratio ${\cal M}_{\rm max}/{\cal M}$, the time required to
reach core-collapse is different. In a cluster with, for instance, a broad
Salpeter IMF between $[0.2\,\msol,\,120\,\msol]$ core-collapse takes
place after $\lesssim 0.1\,t_{\rm rh}(0)$, whereas for a single-mass
Plummer model it occurs after $\gtrsim 10\,t_{\rm rh}(0)$ (this
example was taken from the Monte Carlo-based calculations of
\citealt{GFR04})

There is an ample evidence for mass-segregation in observed clusters.
\citet{McMS94} \citet{HH98} provided
a new deep infrared observations of the Trapezium cluster in Orion
that clearly show the mass segregation in the system, with the highest
mass stars segregated into the centre of the cluster. To test whether
there is evidence for more general mass segregation, they show in
Fig.\,(\ref{fig.hillen_mass_seg}) cumulative distributions with radius
of stars contained within different mass intervals. They include in
the plot four different panels in order to manifest the sensitivity
to the limiting radius. They find that, inside 1.0 pc, general mass
segregation appears to be established in the cluster, with stars of
masses less than 0.3, 0.3-1.0, 1.0-5.0, and greater than 5 $\msol$
progressively more centrally concentrated with increasing mass. 

At this point, the question looms up whether for very young clusters
mass segregation is due to relaxation, like in our models, or rather
it reflects the fact that massive stars are formed preferentially
towards the centre of the cluster, as some models predict.

\bfig
\C
\resizebox{0.6\hsize}{!}{\includegraphics[bb=173 240 420 557,clip]
	{diag_mass_seg/hillenbrand_98_fig6.eps}}
\caption{In this plot by \citet{HH98} we have a clear-cut evidence for
mass-segregation of stars more massive than 5 $\msol$ (long-dashed
lines) toward the cluster centre and some evidence for general mass
segregation persisting down to 1-2 $\msol$ in the Orion Nebula
cluster. The cumulative radial distributions of source counts over
different mass intervals are shown. To clarify the sensitivity of the
cumulative plots to the outer radius they have shown here four panels
with four different limiting radii 
\label{fig.hillen_mass_seg} }
\EC
\efig

\citet{RM98} study the radial structure of
Praesepe and of the very young open cluster NGC 6231. There they find
evidence for mass segregation among the cluster members and between
binaries and single stars. They put it down to the greater average
mass of the multiple systems. In Fig.\,(\ref{fig.raboud_mass_seg}) of
\citet{RM98} again we have clear evidence for mass segregation in NGC
6231. In the two first panels the mass intervals are set in a
different way to those in the bottom.

The two left-hand panels include the 9 bright stars of the cluster
Corona, while the right hand two diagrams do not. The manifestation of
mass segregation for massive stars (triangles) is clearly nailed
down, whereas for stars with masses $\in\, [5,\,20]\,\msol$ are
spatially well mixed (open squares and crosses); i.e., mass
segregation is not yet established over a rather large mass
interval. This population is more concentrated than the lower-mass
population (here shown with filled squares). They make out from this
Fig.\,(\ref{fig.raboud_mass_seg}) that only a dozen, bright, massive,
mainly binary stars are well concentrated toward the cluster centre.

\bfig
\C
\resizebox{0.8\hsize}{!}{\includegraphics[bb=138 228 480 567,clip]
	{diag_mass_seg/raboud_98_fig12.eps}}
\caption{Mass segregation in NGC 623 for two mass interval sets. The two
left panels include all sample stars, whereas the right ones do not
include the 9 bright stars of the cluster corona. For the two top
figures $M<5\,\msol$ (filled squares), $M \in \,[5,\,10[\,\msol$ (open
squares), $M \in \,[10,\,20[\,\msol$ (crosses) and $M \geq 20\,\msol$
(triangles).  For the two bottom figures, $M<2.5\,\msol$ (filled
squares), $M \in \,[2.5,\,6.3[\,\msol$ (open squares), $M \in
\,[6.3,\,15.8[\,\msol$ (crosses) and $M \geq 15.8\, \msol$ (triangles)
\label{fig.raboud_mass_seg} }
\EC
\efig

It seems therefore interesting to set out for multi-mass models with
the two-component ones as a starting point to take care of, since it
is well-studied and we have robust observational proofs of this
phenomenon. On the other hand, observations do not tell us whether
mass segregation is due to relaxation.

For this aim, and as the title of this section indicates, we have
performed here a somehow homiletic but in any case interesting
exercise in order to make an acid test of the gaseous model. In less
than three days we performed a whole set of $10^4$ simulations
for two-component models. We define two parameters now that describe 
the physics of the system,

\be
q := {{\cal M}_{\rm h}}/{\cal M},~
\mu := {m_{\rm h}}/{m_{\rm l}}
\ee

\noindent
In this definition, ${\cal M}$ is the total mass of the system, 
${\cal M}_{\rm h}$ the total mass in heavies and ${m_{\rm h,\,l}}$
the mass of one heavy (light) star.

If $\zeta \equiv 1-q$, we have let
$\zeta$ vary from $10^{-4}$ to $9.99 \cdot 10^{-1}$. For each
$\zeta$ value, we let $\mu$ vary between $1.03$ and $10^3$.

The values for $q$ are regularly distributed in $\log{(\zeta)}$. For
$\zeta \approx 1$ we have added a series of values in
$\log{(\zeta-1)}$. The mean particle mass is $1
\msol$ and the total mass $10^6 \,\msol$, but this has not importance
for our study, because the physics of the system is ruled by
relaxation and therefore the only relevant thing is the relaxation
time. If we are using Fokker-Planck or half-relaxation units, we can
always extend the physics to any other system containing more
particles (if only relaxation is on play). As a matter of fact, one
relates the $N$-body units to Fokker-Planck units as follows:

\be
{\cal U}_{\rm FP} = {\cal U}_{\rm NB} \cdot {\cal
N}_{\star} / \ln(\gamma \cdot {\cal N}_{\star})
\ee

\noindent
Therefore, the relation between FP- and $N$-body- time units is

\be
{\cal U}_{\rm FP}|_{\rm time} = {\cal U}_{\rm NB}|_{\rm time}\cdot {\cal
N}_{\star} / \ln(\gamma \cdot {\cal N}_{\star})
\label{eq.FP_Nbody_units}
\ee

\noindent
In order to be able to compare our results with $N$-body, this
relation is very useful.  The mean mass is therefore just a
normalisation. What really determines the dynamics of the system are
the mass ratios, $q$ and $\mu$.

In Fig.\,(\ref{fig.mass_seg_10_4}) we show the whole $(q,\,\mu)$-parameter
space in a plot where the time at which the core-collapse begins is
also included. The green zone corresponds to the quasi single-mass case.
In the red zone we have the largest difference between masses and blue
is an intermediate case.

\bfig
\C
\resizebox{0.9\hsize}{!}{\includegraphics[clip]
	{diag_mass_seg/2Comp_ColorMap.eps}}
\caption{Parameter space for the set of $10^4$ simulations. 
Here $t_{\rm end}$ stands for the core collapse time and is expressed
in FP units (see text); time at which the simulation ended. $q$ and
$\mu$ are plotted logarithmically.
\label{fig.mass_seg_10_4}}
\EC
\efig

In Fig.\,(\ref{fig.Tcc_q_diff_mu}) we show collapse times for clusters
models with two mass components normalised to the single-mass
core-collapse time $T_{\rm cc}({\rm s.s.})$ for different values of
$\mu$. The initial clusters are Plummer spheres without segregation.
The collapse times are displayed as a function of the mass fraction of
the heavy component in the cluster. When compared to single-mass
component systems, we see that the core-collapse time is accelerated
notably for a wide range of the heavy component ${\cal M}_{\rm h}$
($M_2$). Even a small number of heavies accelerate the core-collapse
time.

\bfig
\C
\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_mass_seg/Tcc_q_diff_mu.eps}}
\caption{Core-collapse time for different values of $q$ and $\mu$
\label{fig.Tcc_q_diff_mu}}
\EC
\efig

It is really interesting to compare the capacity of our approach by
comparing the results of this set of simulations to the $N$-body
calculations of star clusters with two-mass components performed by
\citet{Khalisi02} during the completition of his doctoral thesis. 
For this aim, we plot the evolution of the average mass in Lagrangian
shells of the cluster from the averaged mass in Lagrangian {\em
spheres} containing the following mass percentages
$[0-1],\,[2-5],\,[10-20],\,[40-50],\,[75-95]~\%$, among others, to be
able to compare with the results of Khalisi. These are the comprised
volume between two Lagrangian radii, which contain a fixed mass
fraction of the bound stars in the system.

We have calculated the average mass as follows: If $M_{\rm r}^{({\rm
i})}$ is the total mass for the component ${\rm i}$ comprised at the
radius r and $\bar{m}_{\star}^{({\rm i})}$ is the average mass for
this component within that radius, we can find out what is the value
of $\bar{m}_{\star}^{({\rm i};\,{\rm i}+1)}$ (the average mass between
$\bar{m}_{\star}^{({\rm i})}$ and $\bar{m}_{\star}^{({\rm i}+1)}$)
knowing $M_{\rm r}^{({\rm i})}$, $M_{\rm r}^{({\rm i}+1)}$,
$\bar{m}_{\star}^{({\rm i})}$ and $\bar{m}_{\star}^{({\rm
i}+1)}$. This is schematically shown in
Fig.\,(\ref{fig.lagr_av_mass}). Indeed,

\bfig
\C
\resizebox{0.4\hsize}{!}{\includegraphics
	{diag_mass_seg/lagr_av_mass.eps}}
\caption{Average mass in Lagrangian shells from 
averaged mass in Lagrangian spheres
\label{fig.lagr_av_mass}}
\EC
\efig



\be
M_{\rm r}^{({\rm i}+1)}=N_{\rm r}^{({\rm i})}\cdot
\bar{m}_{\star}^{({\rm i})} + N_{\rm r}^{({\rm i};\,{\rm i}+1)} \cdot
\bar{m}_{\star}^{({\rm i};\,{\rm i}+1)}= N_{\rm r}^{({\rm i}+1)} \cdot
\bar{m}_{\star}^{({\rm i}+1)}.
\label{eq.aver_mass_Lag1}
\ee

\noindent
Since

\be
N_{\rm r}^{({\rm i}+1)}=N_{\rm r}^{({\rm i})}+N_{\rm r}^{({\rm
i};\,{\rm i}+1)},
\ee

\noindent
where

\be
N_{\rm r}^{({\rm i})}=\frac{M_{\rm r}^{({\rm
i})}}{\bar{m}_{\star}^{({\rm i})}}
\ee

\noindent
we have that, from Eq.\,(\ref{eq.aver_mass_Lag1}),

\be
\bar{m}_{\star}^{({\rm i};\,{\rm i}+1)}=
\frac{M_{\rm r}^{({\rm i}+1)}-M_{\rm r}^{({\rm i})}}
{{M_{\rm r}^{({\rm i}+1)}}/{\bar{m}_{\star}^{({\rm i}+1)}} -
{M_{\rm r}^{({\rm i})}}/{\bar{m}_{\star}^{({\rm i})}}}
\ee

We show in Figs.\,(\ref{fig.nbody_spedi_2comp}) and
(\ref{fig.spedi_spedi_2comp}) we show the curves corresponding to the
values shown in table
\ref{tab.mu}.

\bfig
\C
\resizebox{1\hsize}{!}{\includegraphics
	{diag_mass_seg/fallshell.ps}}
\caption{Average Lagrangian radii shells for the 
$N$-body models of \citet{Khalisi02} (see text for further explanation)
\label{fig.nbody_spedi_2comp}}
\EC
\efig

\bfig
\C
\resizebox{0.85\hsize}{!}{\includegraphics
	{diag_mass_seg/avrgmass_q0_1_6mu.eps}}
\caption{Average Lagrangian radii shells for our models, equivalents 
to those of Fig.\,(\ref{fig.nbody_spedi_2comp}) 
\label{fig.spedi_spedi_2comp}}
\EC
\efig


\begin{table}
\C
\begin{tabular}{c |c c} 
$\mu$ in \citet{Khalisi02} & 
\multicolumn{2}{c}{$\mu$ in this work} \\ 
\hline \hline 
1.25 & 1.27 \\ 
1.5 & 1.56 \\ 
2 & 2.06 \\
3 & 2.92 \\
5 & 5.09 \\
10 & 10.2 \\
\hline
\end{tabular} 
\caption{Different $\mu$ values used in the $N$-body calculations 
and in our models for Fig.\,(\ref{fig.nbody_spedi_2comp})}
\label{tab.mu}
\EC
\end{table}

We have followed in the curves the evolution of the system until a deep collapse
of the system. They show the evolution until the most massive component dominates
the centre.

In order to compare our plots with those of \citet{Khalisi02}, one
should look in his diagrams in the region {\em during} core
contraction. At this point, we can observe in
Fig.\,(\ref{fig.nbody_spedi_2comp}) a self-similarity after
core-collapse \citep{GH96}. Binaries are responsible for interrupting
core-collapse and driving core re-expansion in the $N$-body
simulations. The flattening in
the $N$-body plots at the moment of core-collapse is due to the binary
energy generation. This means that we can only compare the steep rise,
but not the saturation.

For instance, in the second plot of the $N$-body set (second column on
the top), we have to look at the point in which the average mass of
the $N$-body system is about 1.20 in the $0 - 1 \%$ shell. This
establishes the limit until which we can really compare the behaviour
as given by both methods. Our simulations yield a very similar
evolution until that point. The gaseous model behaves (it clearly
shows the tendency) like $N$-body.

From Eq.\,(\ref{eq.FP_Nbody_units}), we get that the conversion factor
is the same; namely, for $\gamma =0.11$, $\ln (\gamma \cdot {\cal
N}_{\star})/ {\cal N}_{\star}= 0.0022$. On the other hand, the value
of $\gamma$ is not so well defined and depends on the cluster's mass
spectrum
\citep{Henon75}. This means that potentially it is not the same for
the different models. For a broader mass spectrum, $\gamma$ is about
0.01 and, unfortunately, in the case of having a small particle
number, it will definitively make an important difference despite the
``smoothing'' effect of the logarithm, viz $\ln (\gamma \cdot {\cal
N}_{\star})/ {\cal N}_{\star} = 0.0013$. Thus, in order to be able to
compare the different models, one should consider $\gamma$ as a free
parameter ranging between 0.01 to 0.2 and look for the best fit for
the most of the cases. On the other hand, we must bear in mind that
the $N$-body simulations of \citet{Khalisi02} do not go in deep core
collapse and, so, the moment at which the core radius reaches a
minimum is not the same as for our model.  To sum up, we cannot really
exactly say until which point we can compare, because the core
collapse time will be different.

%\begin{table}
%\C
%\begin{tabular}{c| c c c c c c c} 
%\hline
%${\cal N}_{\star}$ & A & B & C & D & E & F & G\\
%\hline \hline 
%$10^3$ & 82 & 69 & 53 & 36 & 22 & 11 & 4 \\
%$2.5\cdot 10^3$ & 204 & 172 & 132 & 89 & 54 & 27 & 11 \\ 
%$5 \cdot 10^3$ & 408 & 345 & 263 & 179 & 109 & 55 & 22 \\
%$10^4$ & 816 & 690 & 526 & 357 & 217 & 110 & 44 \\
%$2\cdot 10^4$ & 1633 & 1379 & 1053 & 714 & 435 & 220 & 88\\
%\hline
%\end{tabular} 
%\caption{Absolute numbers of heavies in Fig.\,(\ref{fig.emil_9p2})}
%\label{tab.heavies}
%\EC
%\end{table}

%\bfig
%\C
%\resizebox{0.7\hsize}{!}{\includegraphics
%	{diag_mass_seg/emil_9p2.eps}}
%\caption{Core radius of the different $N$-body set of simulations of 
%\citet{Khalisi02} (see text)
%\label{fig.emil_9p2}}
%\EC
%\efig

\section{Clusters with a broader ($>2$) mass spectrum {\em without} a BH}

So as to be able to interpret observations of young stellar clusters
extending to a larger number of mass components, it is of paramount
relevance to understand the physics behind clusters without a central
BH first. It has been shown that for a cluster with a realistic IMF,
equipartition cannot be reached, for the most massive stars build a
subsystem in the cluster's centre as the process of segregation goes
on thanks to the kinetic energy transfer to the light mass components
until the cluster undergoes core collapse \citep{Spitzer69,IW84,IS85}.

Whereas the case in which the BH ensconces itself at the centre of the
host cluster is more attractive from the dynamical point of view, one
should study, in a first step, more simple models.

In this section we want, thus, to go a step further and evaluate
stellar clusters with a broad mass function (MF hereafter). We study
those clusters for which the relaxation time is short enough, because
this will lead the most massive stars to the centre of the system due
to mass segregation before they have time to leave the main sequence
(MS). In this scenario, we can consider that stellar evolution plays
no role; stars did not have time to start evolving. The configuration
is similar to that of \citet{GFR04}, but they employ a rather
different approach based on a Monte Carlo code, using the ideas of
\citet{Henon73} that allows one to study various aspects of
the stellar dynamics of a dense stellar cluster with our without a
central MBH. Our scheme, although being less realistic than MC codes
(and $N$-body ones) and unable, in its present version, to account for
collision has the advantage, as already mentioned in fore-going
sections, of being much faster to run, and providing data that has no
numerical noise. It captures the essential features of the physical
systems considered in our analysis.

\subsection{Mass segregation in realistic clusters}

One of the first questions we should dope out is up to how many
components one should take into consideration when performing our
calculations. Since the computational time becomes larger and larger
when adding more and more components to the system, we should first
find out what is a realistic number of components in our case. For
this end we performed different computations with different number of
stellar components. 

For the simulations shown here, the initial cluster models are Plummer
models with a Salpeter MF 

\be
\frac{dN_{\star}}{dM_{\star}} \propto M_{\star}^{-\alpha}
\label{eq.salpeter}
\ee
\noindent
between $0.2$ and $120\,\msol$. In this equation $\alpha$ takes
$\alpha=2.35$ (Salpeter). There is no initial mass segregation.

The discretisation of the mass components has been done as follows:

\be
\log (M_{\rm comp}|_i)=\log (M_{\rm min})+
\log \Big(\frac{M_{\rm max}}{M_{\rm min}}\Big) \cdot 
\Big(  \frac{i}{N_{\rm comp +1}} \Big)^{\delta}
\label{eq.discIMF}
\ee

In this equation $\delta$ is the discretisation exponent. If $\delta
>1$ we have more bins at low mass; for $\delta <1$, we have more bins
at high mass. I.e., $\delta$ allows one to put more discretised mass
components at low masses ($\delta>1$) or at high masses ($\delta <1$),
$\delta$=1 gives the logarithmical equal spacing. $M_{\rm max,\,min}$
are, respectively, the maximum and minimum individual stellar masses
for the components. For all simulations the number of mass bins has
been typically set to 15. We have chosen a Plummer model by default
and the models have $10^6$ stars. The model radius by default is
$R_{\rm Pl}=1$ pc. The default initial mass function is a Salpeter.

In Fig.\,(\ref{fig.how_many_comp}) we show the Lagrangian radii for ten
different models and look for main dynamical characteristics of the
system's behaviour, the core collapse time, the Lagrangian radii
containing 90, 70, 50, 20, 10, 3, 1, 0.3, 0.1, $3\cdot 10^{-2}$,
$10^{-2}$, $3\cdot 10^{-3}$ and $10^{-3}$ \% of the stellar mass.

In this plot, $N_{\rm comp}$ stands for the mass spectrum different
components number.  For $N_{\rm comp}=6$ we have performed three
simulations varying the $\delta$ parameter between 1.0 (equal
logarithmic spacing of components), 0.75 (more massive components) and
0.5 (even more). For $N_{\rm comp}=12$ we have performed one only
simulation (with $\delta=1$, by default); for the $N_{\rm comp}=20$
case we have repeated the same procedure as with six components, the
last but one that we have chosen is $N_{\rm comp}=20$ and, in this
case, we studied two grid resolutions, $N_{\rm sh}=200$ (the default
value) and 400 grid points, in order to check whether this could
influence the results (see chapter \ref{ch.th_model}). To finish
with, a big simulation with $N_{\rm comp}=50$ was performed and
included in the analysis. Whereas we can see an important difference
between models of 6 and 12 components, we see that the global
behaviour from 12 components onwards is very similar. Therefore,
unless indicated, we choose 15 components in our study in this
section, since a higher number would not contribute anything
essential.

\bfig
\C
\resizebox{0.48\hsize}{!}{\includegraphics
	{diag_mass_seg/spedi_compare_Ncomp_MavLag.eps}}
\resizebox{0.48\hsize}{!}{\includegraphics
	{diag_mass_seg/spedi_compare_Ncomp_RayLag.eps}}
\caption{Lagrangian radii and average stellar mass for 10 models 
with different mass spectrum (see text)
\label{fig.how_many_comp}}
\EC
\efig

To see this in more detail, in Fig.\,(\ref{fig.how_many_2figs}) we
show the Lagrangian radii for each stellar mass $m_i$ and the
corresponding mass fraction $f_m$ for the 25 and 15 components
simulations. Again, we cannot see any substantial difference
between the 25 and 15 cases.

\bfig
\C
\resizebox{.48\hsize}{!}{\includegraphics
	{diag_mass_seg/spedi_0p2to120_15comp_RayLagComp.eps}}
\resizebox{.48\hsize}{!}{\includegraphics
	{diag_mass_seg/spedi_0p2to120_25comp_RayLagComp.eps}}
\caption{Lagrangian radii for each stellar mass $m_i$ in the cases of 25 
and 15 mass components
\label{fig.how_many_2figs}}
\EC
\efig

Taking the last arguments into account, we have done an analysis of
mass segregation in multi-mass models with more than two stellar
components without BH. In Figs.\,(\ref{fig.mass_seg_multi_noBH_1}) and
(\ref{fig.mass_seg_multi_noBH_2}), we show in the evolution of a
stellar cluster of 15 components (in colours); $m$ is the mass (in
$\msol$) of the stars in each component and $f_m$ the corresponding
fraction of the total mass. In the upper box we have the density
profile, where the solid black line represents the total
density; below, we have the average total mass for the system. We show
different moments of the system. At $T=0$ we have the initial model,
which duly shows no mass segregation. As time elapses, at $T=5.30\cdot
10^{-2}\,T_{\rm rh}(0)$, we observe how mass segregation has
fragmented the initial configuration; the heavy components have sunken
into the central regions of the stellar cluster and, thus, risen the
mean average mass. The outer parts of the system start losing their
heavy stars quickly and, consequently, their density profile
retrogresses. This becomes more acute for later times at $T=6.75\cdot
10^{-2}\,T_{\rm rh}(0)$, as the right plots of
Fig.\,(\ref{fig.mass_seg_multi_noBH_2}) show. In these plots and, more
markedly in the right panel of density profile, we can observe a
depletion at intermediate radius.

\bfig
\C
\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_mass_seg/DensProf_15CompAlpha2p35Mmax450_0.eps}}
\caption{Initial density profile for a stellar cluster with 15 
components (upper panel) in $N$-body units and average total 
stellar mass in $\msol$ (lower panel)
\label{fig.mass_seg_multi_noBH_1}}
\EC
\efig

\bfig
\C
\resizebox{0.65\hsize}{!}{\includegraphics
	{diag_mass_seg/DensProf_15CompAlpha2p35Mmax450_30.eps}}
\resizebox{0.65\hsize}{!}{\includegraphics
	{diag_mass_seg/DensProf_15CompAlpha2p35Mmax450_100.eps}}
\caption{Same situation as in Fig.\,(\ref{fig.mass_seg_multi_noBH_1}) but 
at later times. See text for explanations.
\label{fig.mass_seg_multi_noBH_2}}
\EC
\efig

\subsection{Core-collapse evolution}

\citet{GFR04} shown that for a broad MF --which can be a Salpeter 
or a Kroupa--, mass segregation produces a core-collapse of the system
that happens very fast. For clusters of moderate initial
concentration, they find that this is in about 10\,\% of the $T_{\rm
rh}(0)$, the initial half-mass relaxation time (i.e. the
half-relaxation time that the cluster had when time started, at
$t=0$). A good and clear illustration of this is
Fig.\,(\ref{fig.msegr}) and Fig.\,(\ref{fig.core_coll_spedi_king}). In
the former one, on the left panel we have the initial configuration of
the system. On the right one, we have the cluster at the moment of
core-collapse. In the figure, all stars within a slice containing the
centre has been depicted. On the other hand, this does not represent a
real physical system, because all stars' radii have been magnified
(see bottom of each panel). The dashed circles represent spheres
containing 1, 3 and 10\,\% of the total cluster mass (from the
centre). We can clearly see how the massive, large stars are
segregated towards the centre. In
Fig.\,(\ref{fig.core_coll_spedi_king}), we show the core-collapse
evolution of a multi-mass stellar cluster simulated with the gaseous
model. As usual, $m$ is the mass (in $\msol$) of the stars in each
component and $f_m$ the corresponding fraction of the total mass. On
the left panel we display the time evolution of the central density
for a model in which we have employed 15 individual mass
components. The total density is given by the dotted line. On the
right panel we have the evolution of the central velocity
dispersions. The dotted black line shows the mass-averaged value

\be
\bar{\sigma}^2=\sqrt{\frac{\sum_{i=1}^{15}m_i\,\sigma_i^2}{\bar{m}}}. 
\ee

\noindent
$N$-body units are used for the $y$-axes.


\bfig
\C
\resizebox{0.9\hsize}{!}{\includegraphics
	{diag_mass_seg/CoreCollapse.jpg.eps}}
\caption{Illustration of core-collapse in multi-mass systems treated with
a Monte Carlo approach (courtesy of M. Freitag)
\label{fig.msegr}}
\EC
\efig

\bfig
\C
\resizebox{\hsize}{!}{\includegraphics
	{diag_mass_seg/spedi_king_Wo3_15comp_0p2to120_CtrRhoSigma_hor.eps}}
\caption{Evolution of the central density and $3D$-velocity dispersion 
in a model with 15 components (see text for further explanations)
\label{fig.core_coll_spedi_king}}
\EC
\efig

One notes that, during core collapse, the central regions of the
cluster become completely dominated by the most massive stars. But,
contrary to the case of single-mass clusters, the central velocity
dispersion {\em decreases} \footnote{Priv. comm. M. Freitag} (see
Fig.\,\ref{fig.core_coll_spedi_king}).

\section{Clusters with a broader mass spectrum {\em with} a BH}

The logical next step to the systems studied in previous sections is
that of a multi-mass component cluster harbouring a central seed BH
that grows due to stellar accretion. 

In this final section we extend our analysis to systems for which we
use an evolved mass function of an age of about 10 Gyr. We consider a
mass spectrum with stellar remnants. We employ a Kroupa IMF
\citep{KroupaEtAl93,Kroupa01} with Mzams\footnote{The zero age main
sequence (ZAMS) corresponds to the position of stars in the
Hertzsprung-Russell diagram where stars begin hydrogen fusion.} from
$0.1$ to $120\,\msol$ with the turn-off mass of $1\,\msol$.
We have chosen the following values for the exponent according to the
mass interval,

\be
 \alpha = \left\{ \begin{array}{lll}
 1.3, & 0.008 \leq m_{\star}/\msol < 0.5  \\
 2.2, & 0.5 \leq m_{\star}/\msol < 1 \\
 2.7, & 1 \leq m_{\star}/\msol \leq 120.
                              \end{array}
                              \right. 
\ee

\noindent
And with the following component's distribution,

\begin{itemize}
\addtolength{\itemsep}{-0.5\baselineskip}

\item Main sequence stars of $0.1-1 \,\msol$ ($\sim$ 7 components)

\item White dwarves of $\sim \, 0.6\,\msol$ (1 component)

\item Neutron stars of $\sim \,1.4\,\msol$ (1 component)

\item Stellar black holes of $\sim\,10\msol$ (1 component)

\end{itemize}

The defined IMF evolves and gives us at later times an evolved populations
with compact remnants. This means that main sequence stars can
transform into white dwarves, neutron stars or stellar black holes
according to their masses. If $m_{\rm MS}$ is the mass of a MS star,
we have defined the following mass ranges for the evolution into
compact remnants:

\begin{description}
\item[White dwarves] in the range of $1 \leq m_{\rm MS}/\msol < 8$ 
\item[Neutron stars] for masses $8 \leq m_{\rm MS}/\msol < 30$
\item[Stellar black holes] for bigger masses,  $\geq 30 \msol$
\end{description}

As we already mentioned, we put at the centre a seed BH whose
initial mass is of $50\,\msol$. The initial model for the cluster is a
Plummer sphere with a Plummer radius $R_{\rm Pl}=1$ pc. The total
number of stars in the system is ${\cal N}_{\rm
cl}=10^6$. 

The presence of a small fraction of stellar remnants may greatly
affect the evolution of the cluster and growth of the BH because they
segregate to the centre from which they expel MS stars but, being
compact cannot be tidally disrupted. This kind of evolution is shown
in Figs.\,(\ref{fig.norm10comp_raylag}) and
(\ref{fig.before_afterCC_multiBH}).

Fig.\,(\ref{fig.norm10comp_raylag}) shows us the time
evolution of different Lagrange radii with $0.1,\,10,\,50,\,80\,\%$ of
the mass of each component. Here the core collapse happens at about 
$T=0.18\,T_{\rm rh}(0)$.
The henceforth re-opening out is due BH accretion.

In Fig.\,(\ref{fig.before_afterCC_multiBH}), we plot the density
profiles of the system before and after the post-collapse phase. We
can see that the slope of $\rho\propto R^{-7/4}$ on account of the
cusp of stellar BHs that has formed around the central BH. We can see
how the different components redistribute in the process. We can see
how the BH dominates the dynamics at the centre.
\bfig
\C
\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_mass_seg/norm10comp_raylag.eps}}
\caption{Lagrange radii evolution for a 10 components 
calculation with a seed BH and stellar remnants
\label{fig.norm10comp_raylag}}
\EC
\efig


\bfig
\C
\resizebox{0.65\hsize}{!}{\includegraphics
	{diag_mass_seg/norm_density_T0p15.eps}}
\resizebox{0.65\hsize}{!}{\includegraphics
	{diag_mass_seg/norm_density_T0p253.eps}}
\caption{Density profiles in a multi-mass system with seed 
BH before and after core-collapse (see text)
\label{fig.before_afterCC_multiBH}}
\EC
\efig

We can study how the system evolves from the point of view of the
distribution of kinetic energies between the different components of
the clusters during the process of mass segregation. 

In Fig.\,(\ref{fig.temp_10comp_BH}) we show the evolution of the
``temperature'' of the system, defined as the mean kinetic energy per
star divided by the global mean mass (in order to have a
``temperature'' expressed in square velocity units).

In this plot we show the core collapse situation corresponding to
Figs.\,(\ref{fig.norm10comp_raylag}) and
(\ref{fig.before_afterCC_multiBH}): We consider a 10 component cluster
with the characteristics explained before. The mean temperature is
defined like $\sum n_{\rm i} \cdot T_{\rm i}/ \sum n_{i}$, where
$n_{\rm i}$ is the numerical local density for component i. This
corresponds to the mean kinetic energy per star. We can see in the
plot that it is about the same as the heaviest component in the inner
regions, even though one could think that segregation should not have
set in in the beginning. This is due to the fact that the moment does
not correspond to exactly the initial moment, $T=0$. On the right, we
can already see how the mean central temperature moves back (solid
black line) and the most massive component (dashed red line)
increases. For later times, the kinetic energies of the different
components rise at the inner part of the cluster and the most massive
one approaches the sum of all of them. This is even more evident in
the last plot, where all components' temperatures have sunken except
for the corresponding to the most massive one. At the exterior zones,
the mean temperature is much lower than the most heavy component
because the system did not have evolution towards equipartition and,
so, the velocity dispersion $\sigma_{\rm v}$ of the heavy components
is the same that the other's component velocity dispersion but their
kinetic energy is much larger (in terms of mass).

At the current moment we are tackling with these and another
configurations, but we have to first resolve the problem of BH growth
and grid resolution, because as the BH grows, the grid gets too close
to the centre. This problem is being now studied to carry on with new
calculations from the cluster re-expansion onwards caused by the 
BH growth.




\bfig
\C
\resizebox{\hsize}{!}{\includegraphics[bb=1 101 583 816,clip]
	{diag_mass_seg/temp_10comp_BH.ps}}
\caption{Different moments in the evolution of the cluster's
temperature for a 10 stellar components system with a seed BH
\label{fig.temp_10comp_BH}}
\EC
\efig


  
%%%%%%% Bibliografia del cap�tol
\newpage
\thispagestyle{empty}
\begin{spacing}{0.95}
\cleardoublepage
\bibliographystyle{apalike}
\section{Literature of chapter \thechapter}
\bibliography{aamnem99,biblio_tesi}
\end{spacing}
\newpage

motiv.tex
\chapter{Motivation}

\section{What is this all about? First words}

\inicial{M}{assive} objects avoiding light to escape from it 
is a concept that goes back to the 18th century, when John Michell
(1724 -- 1793), an English natural philosopher and geologist overtook
Laplace with the idea that a very massive object could be able to stop
light escaping from it thanks to its overwhelming gravity. Such an
object would be {\em black}, that is, invisible, precisely because of
the lack of light \citep{Michell1784,Schaffer79}. He wrote:

\begin{quote}
\em{
``If the semi-diameter of a sphere of the same density as the sun is
in the proportion of five hundred to one, and by supposing light to be
attracted by the same force in proportion to its mass with other
bodies, all light emitted from such a body would be made to return
towards it, by its own proper gravity.''}
\end{quote}

Even thought we shall sift through this concept oftentimes in the work
to be presented along the next chapters, it appears to be necessary to
give an initial, short overview of the problem's subject we shall
tackle in this thesis.

A ``black hole''\footnote{This term was first employed by John
Archibald Wheeler (b. 1911), an American theoretical physicist}
embraces the observation of phenomena which are associated with matter
accretion on to it, for we are not able to directly observe
it. Emission of electromagnetic radiation, accretion discs and
emerging jets are some, among others, evidences we have for the
existence of such massive dark objects, lurking at the centre of
galaxies. 

On the other hand, spectroscopic and photometric studies of the
stellar and gas dynamics in the inner regions of local spheroidal
galaxies and prominent bulges give us the certainty that nearly all
galaxies should harbour a central massive dark object, with a tight
relationship between its mass and the mass or the velocity dispersion
of the host galaxy spheroidal component (as we will see
ahead). Nonetheless, even though we should mention that we do not
dispose of any direct evidence that such massive dark objects are
black holes, alternative explanations are sorely constrained (see, for
instance, \citealt{Kormendy03}).

Super-massive black holes are ensconced at the centre of {\em active
galaxies}. What we understand with {\em active} is a galaxy in which
we can find an important amount of emitted energy which cannot be
attributed to its ``normal'' components. AGNs, as they are usually
denominated, have the powerhouse for their observed phenomena in a
compact region in the centre.\\

We will embark in the next chapters on an analytical and numerical
study of the dynamics of stellar systems harbouring a central massive
object in order to extract the dominant physical processes and their
parameter dependences like dynamical friction and mass accretion.

This chapter is devoted to a general description of the scenario with
which we shall deal with. Firstly, we give a short bird's-eye view of
active galactic nuclei in order to clearly exhibit the boundaries of
our problem and then, secondly, we will give an extending to the
present time summary on the possible nature of the central dark object
and its possible origin and formation.

\section{What are AGNs and what makes them interesting?}

The expression ``active galactic nucleus'' of a galaxy (AGN
henceforth) is referred to the energetic phenomena occurring at the
central regions of galaxies which cannot be explained in terms of
stars, dust or interstellar gas. The released energy is emitted across
most of the electromagnetic spectrum, UV, X-rays, as infrared, radio
waves and gamma rays. Such objects have big luminosities ($10^4$ times
that of a typical galaxy) coming from tiny volumes ($\ll 1 {\rm
pc}^3$); in the case of a typical Seyfert galaxy the luminosity is
about $\sim 10^{11} ~L_{\odot}$ ($L_{\odot} =3.83 \cdot 10^{33}$ erg/s
is the luminosity of the sun), whilst for a typical quasar it is
brighter by a factor $100$ or even more; actually they can emit as
much as some thousand galaxies like our Milky-Way. They are thus the
most powerful objects in the universe. There is a connexion between
young galaxies and the creation of active nuclei, because this
luminosity can strongly vary with the red-shift.\\

Anticipating something on which we will elaborate a bit ahead,
nowadays one explains (although this is still not accepted
ubiquitously \footnote{For some alternative and interesting schemes,
see \citet{GO64} for spinars, \citet{AKO75} for clusters of stellar
mass BHs or neutron stars and \citet{Terlevich89} for {\em warmers}:
massive stars with strong mass-loss spend a significant amount of
their He-burning phase to the left of the ZAMS on the HR diagram. The
ionisation spectrum of a young cluster of massive stars will be
strongly influenced by extremely hot and luminous stars}) the
generation of energy resorting to matter accreting on to a
super-massive black hole in the range of mass $\mbh \sim
10^{\,6-10}\,\msol$ (where $\mbh$ is the black hole mass). In this
process, angular momentum flattens the structure of the in-falling
material to a so-called {\em accretion disc}. 

It is frequent to observe jets, which may be arising from the
accretion disc (see Fig.\,\ref{fig.uni_agn}), although we do not
dispose of direct direct observations that corroborate this. Accretion
is a very efficient channel of turning matter into energy. Whereas
nuclear fusion reaches only a few percent, accretion can breeze in
almost 50\% of the mass-energy of a star into energy.\\

Being a bit more punctilious, we should say that hallmarks for AGNs
are the frequency of their electro-magnetic emission frequencies,
observed from $\lesssim 100$ MHz (as low frequency radio sources) to
$\gtrsim 100$ MeV (which corresponds to $\sim 2 \cdot 10^{22}$ Hz
gamma ray sources). Giant jets give the upper size of manifest
activity $\lesssim 6~{\rm Mpc} \sim 2
\cdot10^{25}$ cm \footnote{If we do not take 
into account the ionising radiation on intergalactic medium}, and the
lower limit is given by the covered distance by light in the shortest
X-ray variability times, which is $\sim 2\cdot 10^{12}$ cm.\\

As regards the size, and resorting again to Fig.\,(\ref{fig.uni_agn}),
we can envisage this as a radial distance from the very centre of the
AGN where, ostensibly, a SMBH is harboured, and the different observed
features of the nucleus.  From the centre outwards, we have first a UV
ionising source amidst the optical continuum region. This, in turn, is
enclosed by the emission line clouds (NLR, BLR \footnote{See
subsection \ref{subs.tax}}) and the compact radio sources and these
betwixt another emitting region etc.\\

The radiated power at a certain frequency per {\em dex} \footnote{A
``unit'' in the logarithmic axis} frequency ranges from $\sim 10^{39}$
erg/s (radio power of the MW) to $\sim 10^{48}$ erg/s, the emitted UV
power of the most powerful, high-redshifted quasars. Such broad
frequencies and radius ranges for emission makes us to duly note that
they are far out of thermal equilibrium. This manifests in two ways:
first, smaller regions are hotter; second, components of utterly
different temperature can exist together, even though components
differ in one or two orders of magnitude in size.

\bfig
\C
\resizebox{1\hsize}{!}{\includegraphics
	{diag_motiv/unified_AGN.eps}}
\caption{Unified model of an AGN. A relativistic jet is only to be found 
in radio-loud sources. In the vicinity of the central BH we may find a
flared-up disc due to X-rays. 
\label{fig.uni_agn}}
\EC
\efig

\subsection{Outstanding features of AGNs}

We have to somehow specify what we mean with the term AGN. For this
scope we should name the observable phenomena which are used to detect
them; on the other hand, nevertheless, AGNs can be noticed in many
ways, but this does not mean that all of them partake the same
features. In fact, an AGN ``picks up'' some of these defining
qualities to be such an object.

\paragraph{Very small angular size}
If we have optical images of the host galaxy, the nucleus happens to
be a very bright point with a flux that can touch or even surpass the
rest one of the host galaxy. In Fig.\,(\ref{fig.ngc}) we can see an
example of this: NGC 1566, a nearby AGN \citep{JarretEtAl03}.

\paragraph{Radio astronomical observations} were the first proof for the
existence of non-stellar activity in external galaxies. Early
observations revealed that many bright radio sources have the shape of
two lobes with a galaxy situated at half the distance (see
Fig.\,\ref{fig.cygnus}). Even though radio astronomical techniques are
powerful, they might be misguiding, since the radio band never
accounts for more than $\sim 1\%$ of the bolometric luminosity; to
boot, less biased surveys bore witness that the most AGNs emit a much
smaller portion of their total power in the radio.

\bfig
\C
\resizebox{0.5\hsize}{!}{\includegraphics
	{diag_motiv/2MASS_NGC_1566_JHK_petita.eps}}
\caption{NGC 1566 \citep{JarretEtAl03}
\label{fig.ngc}}
\EC
\efig

\paragraph{High luminosity} 
This can run in the case of an AGN from about $1\%$ of a typical
galaxy up to $\sim 10^4$ times as great. Of course these limits are
not completely fixed, for there can be ``hidden'' a huge population
with lower, non observable luminosities; another possible reason for
the incapacity of observation is relativistic beaming, or obscuration,
due to thick dust extinction, which can misguide our
measurements. Compared with the galaxy spectra, an AGN continuum
spectra looks stunningly different. The energy flux per logarithmic
bandwidth produces a spectrum far broader compared with that of a
galaxy. In comparison with a galaxy, AGNs give out in the radio band a
fraction of bolometric luminosity that is about an order of magnitude
greater, and the corresponding in the x-ray band is between three and
four larger.

\paragraph{Variability}
is often said to be a distinctive characteristic of AGNs; this is
relative for, even though the most of them can be seen to vary in the
optical band, the typical amplitude over time-scales is frequently
$10\%$. There is no a fixed time-scale on which AGNs vary, and
therefore it is hard to measure their amplitude of variability.  We
can find a subgroup of these objects that can be observed to vary even
from night to night, and cumulative changes of factors of hundred have
happened over year time-scales.  In Fig.\,(\ref{fig.pks}) we can see
an example of this rapid variation \citep{PianEtAl97}.

\bfig
\C
\resizebox{0.7\hsize}{!}{\includegraphics
	{diag_motiv/cygnusA.eps}}
\caption{Cygnus A at 6 cm 
wavelength \citep{PerleyEtAl84}.
The total extent of this source is $\sim 120$ kpc
\label{fig.cygnus}}
\EC
\efig

\paragraph{Polarisation}
Like in the case of galaxies, even though stars are in itself
unpolarised, the light we observe is by and large polarised by about
$0.5\%$ because of interstellar dust transmission polarisation. As for
the AGNs, they are also polarised, linearly and with a fractional
polarisation of between $0.5\--2\%$. We find a minority that is much
more polarised, frequently $\sim 10\%$. We have a strong variation on
both the magnitude and the direction of the polarisation for those
objects that are strongly polarised and have strongly variable total
flux. Nonetheless, variability does have bounds, for circular
polarisation has never been detected.

\paragraph{Emission lines} 
This draws our attention, for it is easy and productive to study them.
On the contrary to most of galaxies, these are very prominent for this
kind of objects.  The observed emission lines are stereotypic from one
AGN to the other; almost at all times we observe Ly$\alpha$, the
Balmer lines, the $C_{\rm IV}$ 1549 doublet etc. Oftentimes is seen
the FeK${\alpha}$ x-ray line near 6.4 KeV.

\bfig
\C
\resizebox{0.9\hsize}{!}{\includegraphics[bb=108 230 504 528,clip]
	{diag_motiv/pks2155.eps}}
\caption{Two ultraviolet light curves for PKS 2155-304. 
The open circles show the 
continuum flux at 2800 {\AA}; the filled ones at 1400 {\AA}. We
can see that within one day changes of several tens of 
percent occurred
\label{fig.pks}}
\EC
\efig


\subsection{AGN taxonomy}
\label{subs.tax}

To key out the sub-variety of AGNs we are talking about we have
several terms. Whether or not the distinctions is beneficial is
something we could call into question, but we have to know it if we
want to follow ``the conversation''. A deep description of all AGNs
nomenclatures with their typical characteristics is beyond the scope
of this work. Here we just want to give a general idea of the vast
variety of AGNs and main features.  We have different ways of
categorisation, depending on by which criteria we want to
classify. Some of the terms are self-explanatory; like, for instance
{\em radio loud} or {\em radio quiet}. OVV is the acronym for {\em
Optically Violently Variable}, for in the optic band we have very
rapid and large amplitude variability, as the name itself
reveals. Others are coined terms; {\em Quasar} is simply the
pronounced form of ``QSR'' ({\em Quasi-Stellar Radio source}), but
after a time the meaning changed and evolved to ``generic AGN'';
nowadays it has nothing to do with the radio luminosity. In fact, for
instance, very low luminosity AGNs are called {\em
micro-quasars}. Some terms bring up the name of the first people that
identified the class: Carl Seyfert remarked the first {\em Seyfert
galaxies}, and they were split up later into two types, depending on
the existence or not of broad wings in the emission lines. In an equal
manner, Fanaroff and Riley pointed out a distinction in luminosity and
morphology among the radio galaxies, and so they were named {\em FR
galaxies} after them. We find also far-out histories in the
nomenclature of AGNs: The type {\em BL Lac Objects} were initially
identified as variable stars in the Lacerta constellation and, thus,
they were named ``BL Lac''. The term {\em Blazar} springs up from the
fact that the power output of the classes OVV and BL Lac ``blazes''
dramatically and was thought to unify them, because they are very
similar.

In table \ref{table.tax} (taken from \citealt{Krolik99}) an object is
said to be point-like if an optical point source can be seen. By
broad-band we mean that there is a comparable luminosity in the
infrared, optical and x-ray bands. The existence in the optical and
ultraviolet spectra of lines several thousand (hundred) km/s in width
is meant with broad-lines (narrow-lines). By radio we understand that
for the object the fraction of luminosity emitted in the radio is
relatively large, perhaps $\sim 10^{-3}$ of the bolometric. The
members of a class should vary by an order of magnitude or more in the
optical band over a human time life for the variable entrance and for
polarisation the optical light should be at least a few percent
linearly polarised. In the table we have added the subdivision {\em
LINERs}, which stands for ``Low-Ionisation Nuclear Emission Region''
\citep{Heckman80}, but this category is on the verge of activity; it
is not clear whether or not these galaxies are really active.

This classification allows us to arrange the types in a
three-dimensional parameter space, since we find that we have the
groups radio-loud against radio-quiet, strong variable against all the
others and narrow emission lines against broad emission lines. We can
display a three-parameter space in which we locate each of the types
of AGNs (after \citealt{Krolik99}), as we can see in Fig.\,(\ref{fig.class}).

\bfig
\C
\resizebox{0.7\hsize}{!}{\includegraphics
	{diag_motiv/classif.eps}}
\caption{Classification of AGNs in a three-dimensional parameter space
\label{fig.class}}
\EC
\efig

\begin{sidewaystable}[p]
\renewcommand{\arraystretch}{1.2} % enlarge line spacing
\begin{tabular}{l l l l l l l l } 
Type & Point-like & Broad-band & Broad lines & Narrow lines & Radio & Variable 
& Polarised \\ \hline
Radio-loud quasars & yes & yes & yes & yes & yes & some & some \\
Radio-quiet quasars & yes & yes & yes & yes & weak & weak & weak \\
Broad line radio galaxies & yes & yes & yes & yes & yes & weak & weak \\
(FR2 only) \\
Narrow line radio galaxies & no & no & no & yes & yes & no & no \\
(FR1 and FR2) \\
OVV quasars & yes & yes & yes & yes & yes & yes & yes \\
BL Lac objects & yes & yes & no & no & yes & yes & yes \\ 
Seyferts type 1 & yes & yes & yes & yes & weak & some & weak \\
Seyferts type 2 &  no & yes & no & yes & weak & no & some \\
LINERs & no & no & no & yes & no & no & no\\ \hline
\end{tabular}
\caption{The AGN taxonomy}
\label{table.tax}
\end{sidewaystable}

\subsection{The unified model}

After a study of almost 20 years, as we commented at the beginning of
this chapter, a unification scheme for AGNs has appeared in the
community. According to it, a well-mannered AGN should have the
following identifying characteristics (see Fig\,\ref{fig.uni_agn}),

\begin{description}
\addtolength{\itemsep}{-0.5\baselineskip}
\item[Super-massive black hole]-- in the range of $10^{\,6-10} \,\msol$

\item[Accretion disc and corona]-- heated by magnetic and/ or viscous
processes with optical radiation though soft X-ray energy

\item[Broad line region]-- with high velocity gas

\item[Narrow line region]-- corresponding to lower velocity gas 

\item[Obscuring medium]-- which can adopt a torus form or another geometry.
This has the characteristic, as its names pinpoints, of hiding the broad-line
region from some directions owing to its material (dust). It is located at
about 10 -- 100 pc between the smaller inner region from which the
broad emission lines come from and the more external zone where the narrow
emission lines emerge 
 
\item[Relativistic jet]-- emerging at distances of about $\sim 100 ~ 
{\cal R}_{\rm Schw}$ (Schwarzschild radius) of the BH. The extension of
these objects can be out to tens of hundreds of parsecs or even Mpc
\end{description}

Excluding intrinsic variances in BH's differentia, like $\mbh$,
ionisation parameter, size, density, luminosity etc, the unification
model properties (except for, maybe, relativistic jets) stay staunch
to all AGNs. In conformity with it, many of the main observational
characteristics that we have described before are due to orientation
and are not intrinsic differences. The galaxy appears as a Seyfert 2
or NLRG at the regions where the obscuring medium stops the direct
view of the central parts. Facing the AGN, as we look into the inner
regions and it appears as Seyfert 1s or, in radio-loud, BLRGs to
radio-loud quasars and at inferior luminosities, BL Lacs.

\subsection{Appraising $\mbh$}

A direct consequence of the paradigm of SMBHs at the centre of ancient
galaxies to explain the energy emitted by quasars is that relic SMBHs
should inhabit at least a fraction of present-day galaxies
\citep{Rees90}. This conclusion was first made quantitative by
\citet{Soltan82} and has recently be revisited in more detail and in
the light of recent observations by \citet{YT02}.

In a system of luminosity $L$ with a lifetime $T_{\rm lum}$, if
$\epsilon$ is the conversion of matter to energy (accretion)
efficiency, one should expect that in the system, an amount of
material ${L T_{\rm lum}}/{\epsilon c^2}$ has been gathered at some
place, where c is the light velocity. We are referring to the central
super-massive black hole, harboured by the host galaxy. We will give a
description of such an object in the next section and pore over it in
the next chapters.

If we are capable of determining $T_{\rm lum}$ and $\epsilon$, we will
have a rough idea of what ${\cal M}$ is. Usually, for $\epsilon$ one can
prove that $\epsilon \sim 1/10$ is a suitable value
\citep{McCray79}. Assuming that the lobes are powered by the central
engine, we have that, for Cyg A, with a lobe separation of 80 kpc,
$T_{\rm lum} \sim 4 \cdot 10^7$ yr and, so, $\mbh \sim 10^8 \msol$. If
we do not have at our disposal double radio sources, we can reckon in
a different way: if the lifetime of the galaxy is $\sim 10^{10}$ yr
and 1\% of galaxies are active and they all go through activity
phases, $T_{\rm lum} \sim 10^8$ yr; this means that, for the more
luminous sources ($L \geq 10^{47}$ erg/s), $\mbh \sim 10^{10}
\msol$. On the other hand, if we had resorted to the luminosity per
unit volume from quasars from observations, we would have found out
that, since this gives the energy output per unit volume over the age
of the universe, the mass density of ``recycled'' material is
consistent with an average dead mass of $10^8 \msol$. A number of
independent deductions leads to a similar result \citep{FKR02}.


\section{Massive black holes and their possible progenitors}

\subsection{(Super-) massive black holes}

The quest for the source of the luminosities of $L \approx 10^{12}
{\rm L}_{\odot }$ produced on such small scales, jets and other
properties of quasars and other types of active galactic nuclei led in
the 60's and 70's to a thorough research that hint to the inkling of
``super-massive central objects'' harboured at their centres. These
were suggested to be the main source of such characteristics
\citep{LyndenBell67,LR71,Hills75}. \citet{LB69}
showed that the release of gravitational binding energy by stellar
accretion on to a SMBH could be the primary powerhouse of an AGN
\citep{LB69}. 

In the last decade, observational evidences have been accumulating
that strongly suggest that massive BHs are indeed present at the
centre of most galaxies, with a significant spheroidal component.
Mostly thanks to the {\em HST}, the kinematics in present-day universe
of gas or stars has been measured in the central parts of tens of
nearby galaxies. In almost all cases \footnote{With, notably, the
  possible exception of M33 \citep{GebhardtEtAl01,MFJ01}}, proper
modelling of the measured motions requires the presence of a central
compact dark object with a mass of a few $10^6$ to $10^9\msol$
\citep[][and references
therein]{FPPMWJ01,GRH02,PinkneyEtAl03,Kormendy03}. Note, however, that
the conclusion that such an object is indeed a BH rather than a
cluster of smaller dark objects (like neutron stars, brown dwarves
etc) has only been reached for two galaxies. The first one is the
Milky Way itself at the centre of which the case for a $3$--$4\times
10^6 \msol$ MBH has been clinched, mostly through ground-based IR
observations of the fast orbital motions of a few stars
\citep{GhezEtAl03,SchoedelEtAl03}. The second case is NGC4258, which
passes a central Keplerian gaseous disc with $\rm{H_2O}$ MASER strong
sources allowing high resolution VLBI observations down to 0.16\,pc of
the centre \citep{Miyoshi95,Herrnstein99,MoranEtAl99}.

In any case, it is nowadays largely accepted that the central dark
object required to explain kinematics data in local active and
non-active galaxies is an MBH. The large number of galaxies surveyed
has allowed to study the demographics of the MBHs and, in particular,
look for correlations with properties of the host galaxy. The most
remarkable ones are the fact that the MBH has a mass which is roughly
about $0.1\%$ of the stellar mass of the spheroidal component of the
galaxy and that the mass of the BH, $\mbh$, correlates even more
tightly with the velocity of this component.  These facts certainly
strike a close link between the formation of the galaxy and the
massive object harboured at its centre. 

Of particular importance are the following channels for interactions
between stars and the DCO (assumed to be a MBH).

The centre-most part of a galaxy, its {\em nucleus} consist of a
cluster of a few $10^7$ to a few $10^8$ stars surrounding the DCO,
with a size of a few pc. The nucleus is naturally expected to play a
major role in the interaction between the DCO and the host galaxy. In
the nucleus, stellar densities in excess of $10^6\,{\rm pc}^{-3}$ and
relative velocities of order a few 100 to a few $1000\,\kms$ are
reached. In these exceptional conditions and unlike anywhere else in
the bulk of the galaxy, collisional effects come into play. These
include 2-body relaxation, i.e. mutual gravitational deflections, and
genuine contact collisions between stars.

Stars can produce gas to be accreted on to the MBH, through normal
stellar evolution, collisions or disruptions of stars by the strong
central tidal field. When the massive central object acts on the
gravity of a body, a star, the {\em tidal forces} set in and the
difference of gravitational forces can prodigally vary between the
diametrically separated points of the star. The result is that the
star will be altered in its shape, from its initial approximately
spherical architecture to an ellipsoidal one, splitted into two lobes,
since the volume remains the same. In the end, as tidal forces
increase, the star will be {\em tidally disrupted}. See section
\ref{add.tid_radius}. In Fig.\,(\ref{fig.bhstar_chandra}) we give an
intuitive image of this scenario, where distortions due to
gravitational-lens have not been taken into consideration. In
Fig.\,(\ref{fig.rxj1242}), on the left we show a Chandra x-ray image
of J1242-11 with a scale of 40 arcsec on a side. This Fig. pinpoints
one of the most extreme variability events ever detected in a
galaxy. One plausible explanation for the extreme brightness of the
ROSAT source could be accretion of stars on to a super-massive black
hole. On the right we have its optical companion piece, obtained with
the 1.5 m Danish telescope at ESO/La Silla. The right circle indicates
the position of the Chandra source in the centre of the brighter
galaxy.

\bfig
\C
\resizebox{0.7\hsize}{!}{\includegraphics
	{diag_motiv/bhstar_chandra.eps}}
\caption{Schematic representation of the tidal disruption process.
As the star approaches the central BH, tidal forces act on it and tear
it apart. Illustration credit M. Weiss
\label{fig.bhstar_chandra}}
\EC
\efig

These processes may contribute significantly to the mass of the MBH
\citep{MCD91,FB02b}. Tidal disruptions trigger phases of bright
accretion that may reveal the presence of a MBH in an otherwise
quiescent, possibly very distant, galaxy
\citep{Hills75,GezariEtal03}.


On the other hand, stars can be swallowed whole if they are kicked
directly through the horizon or are captured by emission of
gravitational waves (GWs). The later process is one of the main
targets of the future space-borne GW antenna, LISA (Laser
Interferometer Space Antenna).


For a spherical nucleus in dynamical equilibrium, only collisional
effects can bring stars on to the ``loss-cone'', i.e. the very
elongated orbits which allow close interaction between a star and the
DCO \citep{AS01}.

\bfig
\C
\resizebox{0.4\hsize}{!}{\includegraphics
	{diag_motiv/rxj1242_opt_cercle.eps}}
\resizebox{0.4\hsize}{!}{\includegraphics
	{diag_motiv/rxj1242_xray.eps}}
\caption{Optical and x-ray images of RX J1242-11. 
Credits: ESO/MPE/S.Komossa (left) and 
NASA/CXC/MPE/S.Komossa et al.(right)     
\label{fig.rxj1242}}
\EC
\efig

So far, galactic nuclei have been modelled as isolated spherical
clusters (i.e. \cite{MCD91,FB02b}).  {However, non-spherical
structures such as triaxial bulges, bars or stellar discs are common
on scales of 100-1000\,pc, and also the nucleus itself may be
non-spherical. E.g. it could be rotating, as a result of a merger with
another nucleus \citep{MM01} or due to dissipative interactions between
the stars and a dense accretion disc
\citep{Rauch95}}. The influence of non-sphericity at small and
intermediate scales on the structure and evolution of the nucleus has
been little explored {but it could boost the estimates of capture and
disruption rates by order{s} of magnitudes} \cite{MP03}.


\subsection{Intermediate mass black holes}

If one extends the relations we mentioned above to smaller stellar
systems, one could expect that globular clusters host so-called {\em
intermediate-mass black holes}, i.e. BHs whose mass is in the range of
$10^2$--$10^4
\msol$ (see, for illustration, Fig. {\ref{fig.imbh_extend}}).

\bfig
\C
\resizebox{0.7\hsize}{!}{\includegraphics
	{diag_motiv/mass_rel.eps}}
\caption{Extension of galactic correlations to smaller systems 
\label{fig.imbh_extend}}
\EC
\efig


After having been suggested in
the 70's to explain the x-ray sources observed in globular clusters,
later discovered to be stellar-mass binaries, this possibility has
recently be revived by two lines of observations.  First IMBHs may
explain the ultra-luminous x-ray sources (ULXs) that are present in
regions of strong stellar formation in interacting galaxies and hence
suggesting a link with young ``super stellar clusters'' (SSC),
although ULXs are typically not found at the centre of luminous SSCs.
On IMBHs and their possible link to ULXs, see the review by
\cite{MC03}. Second, recent {\em HST} observations of the stellar
kinematics at the centre of M15 around the Milky Way and G1 around M31
have been interpreted as indications of the presence of an IMBH in
both clusters \citep{vdMEtAl02,GerssenEtAl02,GerssenEtAl03,GRH02}.
However, in the case of M15, the mass of the point masses required by
the observations is compatible with zero and $N$--body models have
been made of both clusters that lack a central IMBH but are compatible
with the observations 
\citep{BaumgardtEtAl03,BaumgardtEtAl03b}
. We
note that scenarios have been proposed that would quite naturally
explain the formation of an IMBH at the centre of a stellar cluster,
through run-away stellar collisions, provided that the relaxation time
is short enough and that very massive stars ($10^2\msol < M_{\star} <
10^4 \msol$) evolve into IMBH
\citep{EbisuzakiEtAl01,PZMcM02,RFG03}.

The theoretical study of the structure and evolution of a stellar
cluster (galactic nucleus or globular cluster) harbouring a central
MBH started 30 years ago. However, due to the complex nature of the
problem which includes many physical processes and span a huge range
of time and length scales, our understanding of such systems is still
incomplete and, probably, subjected to revision. As in many fields of
astrophysics, analytical computations can only been applied to highly
idealised situations and only a very limited variety of numerical
methods have been developed so far that can tackle this problem.

\section{Super-massive stars: Possible SMBHs progenitors?}

Since it became clear relatively early that most super-massive black
holes cannot be formed fast enough from stellar mass seed black holes
in nuclei (\citealt{DS82,DS83}, but see also \citealt{Lee95} for a
somewhat differing view), they must have been formed during the galaxy
formation process directly, which is linked to cosmological boundary
conditions. \citet{Rees84} argued that galactic nuclei in their
formation process inevitably produce a dense core consisting of a
star-gas system or a cluster of compact stellar evolution remnants,
both ultimately collapsing to a super-massive black hole.

The concept of central super-massive stars (\smss~henceforth) (${\cal
M} \geq 5 \times 10^4 \msol$)
\footnote{Very massive stars are usually 
defined in the related literature to be stars so massive that at some
point they collapse on an electron-positron pair instability;
according to \citet{BA84,WW82,ZN71}, this is the situation of a star whose
mass is larger than $\sim 100 \msol$ but less than $5
\times 10^4 \msol$. Stars even more massive 
will collapse on the general relativistic gravitational instability
before they start burning H \citep{Chandra64,HF63}; here we denominate
such objects after them as {\em super-massive stars}, \smss.} embedded
in dense stellar systems was suggested as a possible explanation for
high- energy emissions phenomena occurring in AGN and quasars
\citep{Hara78,Vilk78}, such as X-ray emissions \citep{BO75}.
\smss~ and super-massive black holes (SMBHs) are two possibilities 
to explain the nature of these SMOs, and the super-massive stars may
be an intermediate step towards the formation of these \citep{Rees84}:
\citet{SP84} argued that a $10^6 \msol$ central gas
cloud may be a possible source of energy emission of some Seyfert
galaxies. For a more up-to-date documentation on the formation of such
an \sms~see \citet{QS90} (and references therein). On the
other hand, such a system is interesting not just because it
generalises the BH accretion problem to a massive central gas cloud,
but also because such clouds can be the massive BH progenitors or a
possible stock-point to the gas being piled up at the centres of GN.
Nowadays we know that post-Newtonian instabilities make the lifetime
of the star lay in the range $10^5-10^6$ yr \citep{FW86}. On
the other hand, the active phenomena related to QSOs and giant radio
sources have a lifetime of about $\sim 10^8$ yr. This limit caused the
idea that accretion on super-massive BH could be the engine of these
phenomena \citep{Hills75,LB69,Salpeter64,LBR71}.

Finally, the stability of compact dense star clusters was examined
\citep{ZP66,QS90}. All these papers as a
common feature conclude that, provided the central object, star
cluster, \sms~or a mixture of both, becomes smaller than a certain
critical radius, it is able to undergo catastrophic collapse in a
dynamical time scale due to an instability caused by Post-Newtonian
relativistic corrections of hydrostatic equilibrium. The question,
however, whether and how that final unstable state can be reached, is
much less clear. Angular momentum of the protogalaxy or its dark
matter halo, self-enrichment during the dissipative collapse providing
opacity through lines which prevents collapse as compared to radiation
driven expansion, and star-gas interactions heating the central
massive gas object could all at least for some time prevent the
ultimate collapse. Given the complex physical nature of the
interstellar matter, star formation, and stellar interactions alone
this is a complicated question and the conditions under which a
super-massive object can form in a spherical, isolated star-forming
and collapsing gas cloud, rotating or not, has to our knowledge never
been exhaustively studied and answered. Some pioneering approaches
were however done by
\citet{SS66,SS67,Colgate67,LangbeinEtAl90,QS87,Sanders70b}. The
question has gained even more complexity, since we now know that the
baryonic matter of galaxies collapses in their dominating dark halo,
and that most galaxies and their dark haloes experience merging with
other dark haloes and large and small galaxies during the hierarchical
gravitational structure formation
\citep{DiaferioEtAl99,KauffmanEtAl99a,KauffmanEtAl99b}.

The general idea of how these \smss~have formed is that of massive
clusters collapse and coalescence nearby the galactic
centre. According to \citet{BR78}, a dense cluster of $1 \msol$
main sequence stars forms a cloud of about $10^5-10^6 M_
{\odot}$. When the post-Newtonian instability occurs we can have
either a nuclear explosion, a collapse to BH, fragmentation or a
collapse to a BH ``seed''. \citep{Rees84} studies the possible runaway
evolution for active GN and includes a scheme in which the \sms~
formation and fate are given (see
Fig.\,\ref{fig.sms_form_fate}). \citet{FW86} find for their
\smss~sample that the collapse, bounce, expansion and/or explosion are
homologous. They attribute this to the fact that the adiabatic index
is so close to $4/3$ (and the polytropic index $n=3$). \citet{GW80}
claim that such a configuration is scale invariant and that it is
essentially right at the Jeans mass so that collapse or expansion will
remain self-similar.

\bfig
\C
\resizebox{1\hsize}{!}{\includegraphics
	{diag_motiv/sms_form_fate.eps}}
\caption{Schematic diagram for the \sms~ formation and fate 
(source \citealt{Rees84})
\label{fig.sms_form_fate}}
\EC
\efig

This has led to another type of study of black hole statistics.  As we
explained in the last section, the black hole masses are well
correlated with the bulge masses of their mother galaxies. As a matter
of fact, these correlations support the idea that black hole formation
is linked to galaxy formation.

The detailed physics and parameters describing how these processes
work in a self-consistent model of black hole formation, however, are
much less understood. This means that we, poor mortals, lack any idea
of what are the signatures of the black hole formation process in the
morphology and kinematics of the innermost core and cusp regions, and
to what extent they survive the merging history. Brave attempts to
advance modelling in that domain \citep{Rauch99} demonstrate in our view
more the problems which still prevail originating from the large
dynamical range of the problem and the complexity of the treatment of
relaxation in a stellar system, rather than that they provide much
reliable new insight.

Galaxy merging poses another serious problem due to the possibility
that it can lead to two or more black holes in one nucleus, and the
structure and kinematics is critically dependent on the evolution and
possible gravitational radiation merger of the resulting black hole
binary.  The most direct approach to study the dynamical evolution of
a system containing a large number of stars is the so-called
``direct'' $N$-body methods in which the trajectories of $N$ particles
are explicitly integrated by computing all $N(N-1)/2$ forces at each
time step. With present-day hardware and software, one is limited to
$N$ {considerably} smaller than $10^6$ and rescaling the results to
larger number of stars ($10^7-10^8$ for a galactic nucleus) presents
intractable difficulties as soon as many different physical processes
are at play (relaxation, collision, interaction with the central MBH).

This is the only available possibility here using special GRAPE
supercomputers \citep{Makino97,ME96}. The use of a GRAPE computer
for such simulations is particularly well suited because, as a
built-in feature, the list of the 50 closest neighbours is returned
for each particle, which permits an easy local density estimation
\cite{Burkert00}.

The principle behind GRAPE systems is to hardwire on a special purpose
chip the most time consuming part of an $N$-body simulation : the
calculation of the accelerations between the particles. The remainder
is calculated on a normal computer which serves as host to the
accelerator board(s) containing the special purpose chips. The history
of the GRAPE project (short for GRAvity PipE) is detailed in a book by
\citet{MT98}, and more information is available on the
GRAPE project website
\footnote{\url{http://grape.c.u-tokyo.ac.jp/grape}} at the University
of Tokyo. 

Another possibility are general purpose supercomputers, or a suitable
hybrid method between direct and approximate $N$-body codes
\citep{HSS01}. From these models it is yet unclear how fast in the
real system dynamical friction, stochastic three-body interactions and
external perturbations work together to produce eventually a single
black hole again.

On the other hand, due to the ever increasing observational
capabilities with ground and space based telescopes we get more and
more detailed dynamical and photometric data of the structure of
stellar systems around black holes. Therefore, we find it worthwhile
to reconsider with present day numerical possibilities and increased
knowledge about galaxy formation and evolution a detailed study of the
evolution of dense star clusters, with gas, forming an \sms~and its
further evolution.

Furthermore, independently on the importance of the role of an \sms~ 
in the process of formation of SMBHs, these objects have recently drawn
the general attention, for they could be good candidates for detection by 
proposed space-based gravitational wave detectors like LISA (the Laser
Interferometer Space Antenna). LISA would be very sensitive to long wavelengths
and low frequency radiation. Thus, \smss~ are among the most probable
sources (see, for instance \citealt{Thorne95}).

\section{Time-scales}

We introduce in this subsection some useful time-scales to which we
will refer often throughout this thesis; namely {\sl the crossing
  time}, {\sl the relaxation time} and {\sl the dynamical time}. These
three time-scales allow us to delimit our physical system.

\subsection{The relaxation time}

\citet{Chandra42} defined a time-scale which stems from the 2-body
small-angle encounters and gives us a typical time for the evolution
of a stellar system.

This relaxation time could be regarded as an analogy of the shock time
of the gas dynamics theory, by telling us when a particle (a star) has
forgotten its initial conditions or, expressed in another way, when
the local thermodynamical equilibrium has been reached. Then we can
roughly say that the most general idea is that this is the time over
which the star ``forgets'' its initial orbit due to the series of
gravitational tugs caused by the passing-by stars. After a relaxation
time the system has lost all information about the initial orbits of
all the stars.  This means that the encounters alter the star orbit
from that one it would have followed if the distribution of matter
were smooth. Therewith can we regard {\sl the relaxation time} as the
time interval required for the velocity distribution to reach the
Maxwell-Boltzmann form.

In galactic nuclei the relaxation time is \citep{Chandra42,Larson70a},
\begin{equation}
t_{\rm relax}= \frac {9}{16 \sqrt {\pi}} \frac {\sigma ^3 }{G^2m \rho 
\ln(\gamma N)}.
\label{eq.relax_t}
\end{equation}
\noindent
In this expression $G$ is the gravitational constant, $\rho$ the mean
stellar mass density, $\sigma$ the local 1D velocity dispersion, $N=
\frac{4}{3}\pi n_ {c}r_{c}^3$ the total particle number and $\gamma$ a
parameter of order unity whose exact value cannot be defined easily and
depends on the initial model and the anisotropy. We will elaborate
on this in the next chapters.

\subsection{The crossing time}

As the name suggests, this is the required time for a star to pass
through the system, to {\sl cross} it. Obviously, its value is given
by the ratio between space and velocity,
\begin{equation}
t_{\rm cross}= \frac {R}{v},
\end{equation}
\noindent
where $R$ is the radius of the physical system and $v$ the velocity of the 
star crossing it.

For instance, in a star cluster it would be:
\begin{equation}
t_{\rm cross}= \frac{r_{\rm h}}{\sigma_{\rm h}};
\end{equation}
\noindent
where $r_{\rm h}$ is the radius containing 50 \% of the total mass and
$\sigma_{\rm h}$ is a typical velocity taken at $r_{\rm h}$. One
denominates it {\em velocity dispersion} and is introduced by the
statistical concept of root mean square (RMS) dispersion; the {\em
variance} $\sigma^2$ gives us a measure for the dispersion, or
scatter, of the measurements within the statistical population, which
in our case is the star sample,
\begin{equation}
\sigma^2= \frac{1}{N} \sum_{i=1}^{N} (x_{i}- \mu_{a})^2. \nonumber
\end{equation}
\noindent
Where $x_{i}$ are the individual stellar velocities and $\mu_{a}$ is the arithmetic mean,
\begin{equation}
\mu_{a} \equiv  \frac{1}{N} \sum_{i=1}^{N}x_{i}. \nonumber
\end{equation}
If Virial equilibrium prevails, we have
$\sigma_{\rm h}
\approx \sqrt{GM_{\rm h}/r_{\rm h}}$, then we get the dynamical time-scale
\begin{equation}
t_{\rm dyn} \approx \sqrt {\frac{r_{\rm h}^3}{GM_{\rm h}}}\approx
\frac {1}{\sqrt{G\rho_{\star}}},
\end{equation}
\noindent
where $\rho_{\star}$ is the mean stellar density.

On the contrary to the gas dynamics, the thermodynamical equilibrium time-scale 
$t_{\rm relax}$ in a stellar system is large compared with the crossing time 
$t_{\rm cross}$. In a homogeneous, infinite stellar system, we expect in the 
limit $t\to \infty$ some kind of stationary state to be established. 
The decisive feature for such a Virial equilibrium is how quick a perturbation 
of the system will be smoothed down. 

The dynamical time in Virial equilibrium is \citep[cf e.g.][]{Spitzer87}:
\begin{equation}
t_{\rm dyn}\propto\frac{\log (\gamma N)}{N}\cdot t_{\rm relax} 
\ll t_{\rm relax}.
\end{equation}

If we have perturbations in the system because of the heat conduction,
star accretion on the BH, etc. a new Virial equilibrium will be
established within a $t_{\rm dyn}$, which is short. This means that we
get again a Virial-type equilibrium in a short time. This situation
can be considered not far from a Virial-type equilibrium. We say that
the system {\sl changes in a quasi-stationary way}.

\subsection{Collision time}

\noindent
${t_{\rm coll}}$ is defined as the mean time which has passed 
when the number of stars within a volume $V=\Sigma \cdot v_{\rm rel} 
\cdot \triangle t$ is one (see Fig.\,\ref{fig.coll_time}), 
where $v_{\rm rel}$ is the relative velocity at infinity of two 
colliding stars.

\bfig
\C
\resizebox{0.7\hsize}{!}{\includegraphics
	{diag_motiv/coll_time.eps}}
\caption{Definition of the {collision} time
\label{fig.coll_time}}
\EC
\efig

\noindent
Computed for an average distance of closest approach $\bar{r}_{\rm min}=
\frac{2}{3}r_{\star}$, this time is
\begin{equation}
n_{\star}V(t_{\rm coll})=1= n_{\star}\,\Sigma\, v_{\rm rel}\, t_{\rm coll}.
\end{equation}
\noindent
And so,
\begin{equation}
t_{\rm coll}=\frac{m_{\star}}{\rho_{\star}\Sigma\sigma_{\rm rel}},
\end{equation}
\noindent
with
\begin{equation}
\Sigma=\pi \bar{r}_{\rm min}^2 \bigg( 1+ \frac{2Gm_{\star}}
{\bar{r}_{\rm min}\sigma_{\rm rel}^2}\bigg);
\end{equation}
$\sigma_{\rm rel}^2=2\sigma_{\star}^2$ is the stellar velocity dispersion and
$\Sigma$ a {collision}al cross section with gravitational focusing

\section{Intention of this thesis}

This work comprises several aspects of theoretical stellar dynamics in
clusters, whether globular clusters or galactic nuclei, both
analytically and numerically.  The relevance of computer simulations
in current astrophysics is well-known and clear, for it is the only
way we have to establish (at least in tendency) what we should expect
from an analytic point of view. Also, questions for which we are not
able to develop an exact mathematical model are tackled thanks to
these methods. Astronomy is a science that, at the beginning, was
purely observational. We have rarely the chance to access direct
measurements of what we would like to analyse. In our discipline,
creating models that enable us to evaluate the physical system exposed
to study -although in a way that is more or less idealised according
to the method to which we resort- is crucial. The work presented here
is consecrated to two different aspects that involve massive black
holes in globular clusters and galactic nuclei.

As we have seen, the formation of most super-massive black holes
(SMBHs) cannot be explained from accretion of stars on to seed black
holes in nuclei, since the process is unlikely to be fast enough. It
has been argued that galactic nuclei in their formation process
produce a dense core consisting of a star-gas system or a cluster of
compact stellar remnants. In both cases the system may ultimately
collapse to a SMBH. Dense super-massive star-gas composite objects
have been regarded as a transient progenitor of a SMBH. Provided that
the central object (a star cluster, \sms~ or a mixture of both) becomes
smaller than a certain critical radius, it will collapse in a
dynamical time scale due to an instability caused by Post-Newtonian
relativistic corrections of hydrostatic equilibrium. However, it is
not clear at all whether and how this final unstable state can be
reached. Due to the complex physical nature of the interstellar
matter, star formation and stellar interactions, this is a rather
complicated question. The conditions under which a supermassive object
can form in a spherical, isolated, star-forming and collapsing gas
cloud -rotating or not- has never been exhaustively studied. In
parallel with my work on stellar clusters containing an SMBH, together
with an intense analytical study and description of such an \sms~
({\bf chapter \ref{ch.sms}}), I worked on its numerical implementation into
the anisotropic model. For that aim, I take into account the transfer
of radiation in a spherically symmetric moving medium allowing for the
contributions which are of the order of the flow velocity divided by
the velocity of light, the thermal energy equation and the turbulent
energy equation. The interaction between stars (collisions) and star
gas (drag force between stars and gas, loss-cone etc) and
post-Newtonian corrections are also considered. How these \smss~ could
power the quasar activity by star accretion and energy flows is one of
the main questions that could be answered thanks to this method that
is in a very advanced state of development, even though not yet fully
implemented into the numerical program. Here we give an exhaustive
analytical description of such a physical configuration in {\bf chapter
\ref{ch.th_model}}. I have made also a semi-analytical study of the
influence of accretion of stars on to an \sms~ harboured at the centre
of the galactic nucleus (see {\bf chapter \ref{ch.semi_analytical}}).

For the analysis of star clusters with a central BH, I employed an
anisotropic model that solves numerically moment equations of the full
Fokker-Planck equation with Boltzmann-Vlasov terms on the left- and
interaction (collisional) terms on the right-hand side of the
equations. The cluster is modelled like a self-gravitating, conducting
gas sphere. In this method, all quantities of interest are accessible
as smooth functions of the radius and time. In {\bf chapter
\ref{ch.th_model}} there is a detailed description of the approach.

This enables a detailed study of clean-cut aspects of the dynamics
without any noise that particle methods suffer from. This model allows
us to study the most important physical processes that are present in
the evolution of a spherical cluster, like self-gravity, two-body
relaxation, stellar evolution, collisions, binary stars etc and,
undoubtedly, the interaction with a central BH and the role of a mass
spectrum. I have performed calculations to follow the joint evolution
of a spherical star cluster with a central BH making feasible
anisotropy in order to check for the reliability of the method. I
include here the study of the growth of the central BH due to star
accretion at its tidal disruption radius thanks to a diffusion model
to treat loss-cone physics. The core collapse is studied in detail in
a self-consistent manner, as well as the post-collapse evolution of
the surrounding stellar cluster. The results are in good agreement
with classical literature about this subject ({\bf chapter
\ref{ch.diff_model}}). The current new version of the program enables
the analysis of the effects of an a (discretised) stellar mass
spectrum and stellar evolution. Using it, I present also more
realistic models of dense clusters and give a description of mass
segregation in these systems with and without a central BH ({\bf chapter
\ref{ch.mass_seg}}).

%%%%%%% Bibliografia del cap{\'\i}tol
\newpage
\thispagestyle{empty}
\begin{spacing}{0.95}
\cleardoublepage
\bibliographystyle{apalike}
\section{Literature of chapter \thechapter}
\bibliography{aamnem99,biblio_tesi}
\end{spacing}
\newpage

odisseu.tex
\thispagestyle{empty}
\cleardoublepage
\begin{center}
\begin{minipage}{13cm}
\begin{spacing}{0.95}
\inicialfada W{\sf \bf hen he was in this plight, Ino daughter of Cadmus, also
called Leucothea, saw him. She had formerly been a mere mortal, but
had been since raised to the rank of a marine goddess. Seeing in what
great distress Ulysses now was, she had compassion upon him, and,
rising like a sea-gull from the waves, took her seat upon the raft.
"My poor good man," said she, "why is Poseidon so furiously angry with
you? He is giving you a great deal of trouble, but for all his bluster
he will not kill you. You seem to be a sensible person, do then as I
bid you; strip, leave your raft to drive before the wind, and swim to
the Phaecian coast where better luck awaits you. And here, take my
veil and put it round your chest; it is enchanted, and you can come to
no harm so long as you wear it. As soon as you touch land take it off,
throw it back as far as you can into the sea, and then go away again."
With these words she took off her veil and gave it him. Then she dived
down again like a sea-gull and vanished beneath the dark blue waters.
But Odysseus did not know what to think. "Alas," he said to himself in
his dismay, "this is only some one or other of the gods who is luring
me to ruin by advising me to will quit my raft. At any rate I will not
do so at present, for the land where she said I should be quit of all
troubles seemed to be still a good way off. I know what I will do- I
am sure it will be best- no matter what happens I will stick to the
raft as long as her timbers hold together, but when the sea breaks her
up I will swim for it; I do not see how I can do any better than
this.}
\vskip0.075cm
\flushright{\bf \em {Odyssey, Homer (Book V)}}
\end{spacing}
\end{minipage}
\end{center}

% Odysseus

% It seems that Odysseus mistrusts the 
% Why should he leave his raft, put on a veil

% Anem a fer-ho abans en Catal�, que en lleng�es b�rbares no es pot
% pensar
\vspace{0.1cm}
\begin{spacing}{0.95}
%\small

Odysseus clearly distrusted the benevolence of the goddess. And his
resolution was quite clear: neither to accept the advice nor the
favours of superhuman origin until tragedy be imminent. While the ship
remained whole, he would not dare leave it... despite the clear help
that (whichever) gods offered. Perhaps because Leucothea had once been
mortal?  This did not seem to be the reason, in my view.  Odysseus,
one of the first western heroes of whom we have written evidence, did
not rely at all on the gods who favoured him. \\

Alone, wet, cold, in a feeble boat that could not guarantee him the
necessary cover against the wrath of another god, in the middle of the
night, battling the fearsome waves that carried him directly to the
reefs, Odysseus looks at the piece of sail that the goddess gave him
with which he will have to reach the cost, still far away.  Perhaps
because he is a hero, he arrives there where many do not. Those who,
desperate, would blindly have heeded the advice of gods. He, on the
contrary, proceeds {\em rationally}. Odysseus will not pay attention
to the advice of gods until his chances be clearly doomed, until his
ship disintegrates, taken by the rage of the sea, and he has nothing
to lose. Then, and only then, he will abandon the chunk of wood in
which he finds himself at the mercy of the storm, and he would launch
himself into the water with the hope that the madness of the
seagull-goddess be true. 

Naturally, as the reader can guess, either because he is familiar with
Homer's work or sufficiently savvy to be irrational and see it,
Odysseus reaches the shore.\\

I read this passage a couple of years ago, when I was still studying
in Val{\`e}ncia. It impressed me to see that a mythological greek
hero, maybe {\em the} mythological greek hero could think in such a
rational way and only leave his last shelter when all other
possibilities were ruled out. I interpreted it like a knowing wink of
Homer to us, poor mortals. When I started my PhD I felt a bit like 
Odysseus. Only a couple of months ago I decided to trust the veil and
now, safe in the cost, I resolved to include this text here.





%Odisseu desconfia clarament de la benevol�ncia de la deesa.  I la seua
%resoluci� �s ben clara: No seguir� els consells ni favors de
%proced�ncia sobrehumana fins que la trag�dia siga imminent. Mentre la
%barca estiga sencera, no gosar� de deixar-la... malgrat el clar ajut
%que (qualques) d�us li ofereixen. Potser perqu� na Leucothea havia
%estat abans una mortal? No sembla �sser aquesta la ra�, al meu parer.
%Odisseu, un dels primers herois occidentals de qui tenim const�ncia
%escrita, no es refia gens dels d�us que estan al seu favor. A soles,
%mullat, amb fred, a sobre d'una pobra barca que no pot garantir-li
%l'aixopluc necessari contra la ira d'un altre d�u, en meitat de la
%nit, lluitant contra les ones apaivadores que l'enduen directament als
%arrecifes??, Odisseu mira el boc� de tela que la deesa li ha donat amb
%qu� haur� d'assolir la costa, llunyana encara. Potser perqu� �s un
%heroi arriba, all� on molts, potser, desesperats, haurien fet cas
%d'una forma cega d'all� que els d�us li havien aconsellat, arriba,
%dic, a procedir {\em racionalment}. Odisseu no far� cas del consell de
%les entitats divines fins que la seua sort no siga clarament fatal,
%fins que la barca no es disolga, presa de la r�bia de la mar i no
%tinga res a perdre. Aleshores, i nom�s aleshores, abandonar� el tros
%de fusta en qu� es troba, a merc� de la tempesta, i es llencar� a
%l'aigua amb l'esperanca que les folleries de la deesa-gavina siguen
%encertades. Naturalment, con el lector, coneixedor de l'obra d'Homer o
%suficientment llest com per a ser irracional i veure-ho, Odisseu
%assoleix la platja.

\end{spacing}
%\normalsize
pagines_inicials.tex
\thispagestyle{empty}                                          
\newcommand{\barrota}{\rule{\linewidth}{1mm}}                  
\setlength{\parindent}{0mm}                                    
\setlength{\parskip}{0mm}                                      
\vspace*{\stretch{1}}                                          
\barrota                                                       
\begin{flushleft}                                              
  \Huge Pau Amaro Seoane\\[5mm]                                 
  \huge Dynamics of dense star-gas systems\\                   
BHs and their precursors                                       
\end{flushleft}                                                
\barrota                                                       
\vspace*{\stretch{2}}                                          
\begin{center}                                                 
  \Large\textsc{Heidelberg, 2004}                               
\end{center}                                                    
\newpage  
\paginablanca

\thispagestyle{empty}                                                        
\hbox{}                                                                             
\vfill                                                                       
\large Referees:\\                                                            
\bigskip
                                                                     
\hspace{3.45cm}{Prof. Dr. Rainer Spurzem 
(Th. Astroph.)}                         
\vskip.1cm                                                                    
\hspace{3.45cm}{Prof. Dr. Josef Fried
(Obs. Astroph.)}					     
\vskip.1cm                                                                    
\hspace{3.45cm}{Prof. Dr. Matthias Bartelmann
(Th. Astroph., Cosmology)}
\vskip.1cm                                                                    
\hspace{3.45cm}{Prof. Dr. H-G Dosch
(Th. Physics)}\\

{Day of oral examination:}\\

\bigskip
\hspace{3.45cm}{23rd July 2004}
\paginablanca
\thispagestyle{empty}
\hbox{}
\vfill

{\Large \bf \sf{Cl{\`a}usula de bon {\'u}s}}
\begin{quote}
{\sf Queda absolutament prohibida la utilitzaci{\'o}, recerca i
desenvolupament, de manera directa o indirecta, de qualsevol de les
aportacions cient{\'\i}fiques pr{\`o}pies de l'autor que es presenten en
aquesta mem{\`o}ria, per part de qualsevol ex{\`e}rcit del m{\'o}n o per part de
qualsevol grup armat, per a qualsevol {\'u}s militar i per a qualsevol
altre {\'u}s que atempte contra els drets humans o contra el medi ambient,
si no {\'e}s amb el perm{\'\i}s escrit de totes les persones del m{\'o}n.}
\end{quote}

{\Large \bf \sf{Good use right}} 
\begin{quote}
{\sf It is strictly prohibited to use, to investigate or to develop, in a
direct or indirect way, any of the scientific contributions of the
author contained in this work by any army or armed group in the world,
for military purposes and for any other use which is against human
rights or the environment, unless a written consent of all the persons
in the world is obtained.}
\end{quote} 

\normalsize
\paginablanca
resum.tex
\thispagestyle{empty}                                                        

\vspace{4cm}
\begin{center}
\begin{minipage}{13cm}
{\Large \bf \sf Zusammenfassung} 

\vskip0.3cm
\noindent
Diese Thesis umfasst mehrere Aspekte theoretischer Stellardynamik in
Sternhaufen, sowohl in analytischer als auch in numerischer Hinsicht.
Wir versuchen, Licht auf Ph{\"a}nomene zu werfen, welche zur Zeit in
allen Galaxietypen beobachtet werden, einschlie{\ss}lich AGNs und
Quasare, welche zu den m{\"a}chtigsten Objekten des Universums
z{\"a}hlen. Die Wechselwirkungen zwischen einem Sternsystem und einem
zentralen schwarzen Loch f{\"u}hren zu einer Menge interessanter
Ph{\"a}nomene. Die von uns verwendete Methode erm{\"o}glicht eine
Betrachtung leicht einsichtiger Aspekte ohne jegliches Rauschen,
welches die Teilchen-Methoden mit sich bringen. Wir untersuchen die
wichtigsten physikalischen Prozesse, die in der Entwicklung eines
sph{\"a}rischen Sternhaufens ablaufen, etwa Selbstanziehungskraft,
Zwei-K{\"o}rper-Relaxation etc sowie die Wechselwirkung mit
einem schwarzen Loch und die Funktion des Massenspektrums. Wir
besch{\"a}ftigen uns jedoch nicht nur mit diesem Thema alleine,
sondern auch mit einer Analyse supermassiver Sterne. Wie diese Sterne
die Aktivit{\"a}ten der Quasare durch Sternakkretion und Energiestrom
antreiben ist eine der Fragen, die hierbei aufkommen. Wir gehen auch
anderen Fragen nach, etwa jener nach der noch nicht verstandenen
Entwicklung eines solchen Objektes und seiner Wechselwirkung mit dem
ihn umgebendem Sternsystem. Dies ist ein Kernpunkt der Astrophysik, da
diese Objekte als Vorl{\"a}ufer von supermassiven schwarzen
L{\"o}chern betrachtet werden k{\"o}nnen.
\end{minipage}
\end{center}

\vskip2cm
\begin{center}
\begin{minipage}{13cm}
{\Large \bf \sf Abstract}

\vskip0.3cm
\noindent
This thesis embraces several aspects of theoretical stellar dynamics
in clusters, both analytically and numerically. We try to elucidate
the phenomena currently observed in all types of galaxies, including
AGNs and quasars, some of the most powerful objects in the
universe. The interactions between the stellar system and the central
black hole give rise to a lot of interesting phenomena. The scheme we
employ enables a study of clean-cut aspects without any noise that
particle methods suffer from. We study the most important physical
processes that are readily available in the evolution of a spherical
cluster, like self-gravity, two-body relaxation etc, the interaction with a central 
black
hole and the role of a mass spectrum. Not only embark we upon this
subject, but we set about an analysis on super-massive stars. How
these stars could power the quasar activity by star accretion and
energy flows is one of the questions that arises. We undertake other
questions, such as the uncertain evolution of such an object and its
interaction with the surrounding stellar system. This is of crucial
importance in astrophysics, for these objects could be regarded as
super-massive black holes progenitors.
\end{minipage}
\end{center}

\normalsize
%\paginablanca

semi_analytical.tex
\chapter{A semi-analytical approach to dense gas-star systems} 
\label{ch.semi_analytical}

%\vspace{-11cm}
%\begin{quote}
%\begin{flushright}
%{{\tt blabla \\
%{\vspace{0.15cm}\em autor }}
%\end{flushright}
%\end{quote}
%\vspace{11cm}

\section{Introduction}

\inicial{D}{ynamics} in galactic nuclei of dense stellar systems is
the subject to be presented in this {\mbox{chapter}} \footnote{Part of the work exposed in 
this chapter was used for \citet{AS01}, \citet{AS01b} and
\citet{AS00}} for some of the physical
configurations we introduced in former sections. It has been often
suggested that, in addition to the thermonuclear sources inside the
constituent stars, the active phenomena of high-energy emissions
occurring at galactic nuclei and quasars could be explained resorting
to the idea that a massive dark object is being harboured in the
centre of the galaxies
\citep{Magorrian98}.

We will start this discussion facing the black hole star accretion
problem in a dense stellar medium in a given time scale. Radii of
stars are extremely small ($r_{\odot}\approx 10^{10}$cm) in comparison
with the mean typical distances between stars in astrophysical stellar
systems (a typical interstellar distance in our galactic neighbourhood
is $\approx 10^{18}$cm). Even though stellar collisions occur seldom,
a single star will modify its orbit at random because of the small
pushes it receives from stars which pass close by. We can roughly
define a certain time required for the stars ``to forget their initial
orbits'' due to two-body, small-angle gravitational encounters, the
gravitational tugs of these passing-by stars; this would be one of our
timescales of interest, the {\sl relaxation time}.

Then will we follow through the initial problem introducing the
concept of {\sl super-massive star}. As a first description of it, a
{\sl super-massive star} (${\cal M} \gtrsim 10^4 \msol$, where ${\cal
M}$ is its mass) is a dense gas cloud located in the centre of dense
stellar systems which can be regarded as a general case for the black
hole accretion problem.

If the super-massive star scheme embraces the black hole accretion
problem, we could think that the most general case is to be analysed
first, since it supposedly gives an overview of the general problem.
Nevertheless, if we dare to envisage the possibility of doing it
upside-down, if we generalise the example and try to extract from it
the global concept, it reports us the satisfactory surprise that
everything turns out to be clearer after having evaluated the
particular case. The most general case becomes pointless if it yields
as a result a more difficult understanding of the subject submitted at
study.

\section{General concepts}

In this section we discuss the consequences from the astrophysical and
dynamical point of view of the presence of a black hole (or a massive
compact central object, from now onwards just BH) in a dense stellar
object in about a relaxation time. We consider the steady-state
distribution and consumption of stars orbiting a massive object at the
centre of a spherical, stellar system.

The distribution of stars is determined by the relaxation processes
associated with gravitational stellar encounters and by the
consumption of low angular momentum stars which pass within a small
distance of the central mass.  Stars whose orbits carry them within
the tidal radius $r_{\rm t}$ of the BH will be tidally disrupted. The
{\sl peribarathron} \footnote{This word fits quite well the idea of
closest approach to this ``sinking'' hole for it has the meaning of no
return.  The {\sl barathron} ($\beta\acute{\alpha}
\rho\alpha\theta\rho o\nu$) was in the ancient Greece a cliff down to
an unreacheable or unseen place where criminals were thrown} 
(distance of closest approach to the BH) is determined by the specific
orbital angular momentum {\sl L} and by the BH mass.  There are
situations in which the stars that have a radially elongated orbit and
a low angular momentum pass close by the system centre and interact
with the massive central object.  In such a situation it is
interesting to evaluate the density of those stars whose angular
momenta are limited by a superior $L_{\rm min}$; that is, the stars
which belong to a defined region in the velocity-space.

Stars at a position $r$ whose velocities are limited by a superior
limit $v_{\rm lc}(r)$ and, consequently, with an angular momentum $L <
L_{\rm min}=rv_{\rm lc}$, have orbits that will cross the tidal radius of
the central BH in their motion. They will be disrupted due to the
tidal forces and then they are lost for the stellar system. Such stars
are said to belong to the {\em loss-cone}, since they are lost for the
stellar system. The loss-cone is depleted in a crossing time $t_{\rm
  cross}=r/ \sigma$, where $\sigma$ is the 1D velocity dispersion.

The diffusion of stars into this loss-cone has been studied by
\citep{FR76} and by \citet{LS77}. We will talk about a
``critical radius'' within which stars on orbits with $r\leq r_{\rm
crit}$ diffuse. Inside they are swallowed by the BH, after being
scattered to low angular momentum {\sl loss-cone} orbits.

If there is a central point mass ${\cal M}_{\bullet}$, such that
${\cal M}_{\bullet}$ $\gg$ $m_{\star}$, then its potential well will
affect the stellar velocity field out to a distance
\begin{equation}
r_{\rm h}= G{\cal M}_{\bullet}/\sigma^2.
\end{equation}
\noindent
This expression gives us the influence radius of the central
 object. $G$ is the gravitational constant.
 
The star gets disrupted whenever the work exerted over the star by the
tidal force exceeds its own binding energy.  If we compute the work
exerted over the star by the BH we can get an expression for the tidal
radius (see Addendum A of this chapter, section \ref{add.tid_radius}),
\begin{equation}
r_{\rm t}=\Bigg[\frac{2}{3} (5-n) \frac{{\cal M}_{\bullet}}{m_{\star}}\Bigg]^{1/3} r_{\star},
\end{equation}
\noindent
where $n$ is the polytropic index of the star, $m_{\star}$ the mass of the
star and $r_{\star}$ its radius.

\section{Loss-cone phenomena}

\citet{FR76} studied how a stationary stellar density profile
around a massive star accreting BH looks like. They found that the
density profile follows a power-law within the region where the
gravity of the massive star dominates the self-gravity of the stars,
\begin{center}
\begin{equation}
\rho \propto r^{-7/4}.
\end{equation}
\end{center}
This was followed by intensive numerical studies by other authors
\citep{SM78,MS79,MS80,ST85} which are all in
agreement with the first work of Frank and Rees.

The replenishment of the loss-cone happens thanks to the small-angle
gravitational encounters in a timescale which is for the most real
stellar systems slower than the dynamical processes.

We have loss-cone effects also in the neighbourhood of a massive
central gas-formed object, an \sms, see Fig.\,(\ref{fig.sms2}). Stars with such
orbits enter the gas-formed central object and lose kinetic energy if
their density is high enough.  Nevertheless, such stars will not
disappear from the stellar system just by one crossing of the central
object, as it happens for the BH problem. In this scenario the stars
lose their energy in each crossing and their orbits come closer and
closer to the central object until they are ``trapped'' in it, their
orbits do not extend further than the massive central object radius
(confined stars). This process was described qualitatively by
\citet{Hara78}, \citet{daCosta81} and \citet{Hagio86}. Da Costa
suggested the name ``dissipation-cone''.  However, we will keep the
loss-cone term for it is the most commonly found in the related
literature. We have to take into account that the meaning is that of a
defined region of the velocity space at the position $r$, even though
there is no quick loss.

\bfig 
\C
\resizebox{0.7\hsize}{!}{\includegraphics
	{diag_semi_analytical/sms2.eps}}
\caption{Super-massive star peribarathron radius $r_{\rm p}$
\label{fig.sms2} }
\EC
\efig

In Addendum B of this chapter (section \ref{add.thetas}) we derive the
following expression for the diffusion angle (the mean deviation of a
star orbit in a dynamical time $t_{\rm dyn}$):
\begin{equation}
\theta_{\rm D} \simeq \sqrt{\frac {t_{\rm dyn}}{t_{\rm relax}}}.
\end{equation}

Now we look for a condition at a place $r>r_{\rm h}$ for a star to touch
or to cross the influence radius of the central object within a
crossing time. For this aim we look now for the amount of stars which
reaches the central influence radius with an unperturbed orbit.
Unperturbed here means that the star orbit results from the influence
of the gravitational potential and from that of the rest of the stars
and of the central object, and it is not affected by the local,
two-body, small- or big-angle gravitational encounters. We envisage
then the average part of the gravitational potential, whereas the
random component due to the individual behaviour of the stars will be
neglected.

To define the loss-cone angle we say that a star belongs to this cone
when its distance to the peribarathron (which depends on the orbit we
have, i.e., on the energy $E$ and angular momentum) is less or equal
to the tidal radius, $r_{\rm p}(E,L) \leq r_{\rm t},~ \theta \leq
\theta_{\rm lc}$.

There is a maximum $\theta$ for which the peribarathron radius is less
than or equal to the tidal radius. We define this as $\theta_{\rm lc}$
(where the subscript ``lc'' stands for loss-cone). In Addendum B
 we derive an expression for it in terms of
the influence radius of the central object.

As for the $v_{\rm lc}(r)$, we get its value using angular momentum and
energy conservation arguments in Addendum C (section
\ref{add.lc_vel}).

\begin{equation}
v_{\rm lc}(r)= \frac {r_{\rm t}}{\sqrt {r^2-r_{\rm t}^2}} \cdot
 \sqrt {2[ \phi (r_{\rm t}) - \phi (r)] + v_{\rm r}(r)^2}.
\end{equation}

\section{The critical radius}

It is interesting to evaluate a certain radius which \citet{FR76}
introduced by defining the ratio $\xi:= \theta_{\rm lc}/
\theta_{\rm D}$. When $\xi =1$, then $\theta_{\rm lc}=\theta_{\rm D}$, and this
corresponds to a ``critical radius'', $r_{\rm crit}$, if there is only
one radius with this condition. Inside the critical radius (i.e. $\xi
>1$, $\theta_{\rm lc}>\theta_{\rm D}$) stars are removed on a $t_{\rm dyn}$.
For larger radii (i.e. $\xi <1$, $\theta_{\rm lc}<\theta_{\rm D}$) we cannot
talk about a ``loss-cone'' because this $\theta_{\rm D}$ corresponds to
the variation of $\theta$ within a $t_{\rm dyn}$, and this is the
required time for the star within the loss-cone to plunge onto the BH
if the orbit is unperturbed. The angle variation happens sooner than
the required time for stars to sink into the loss-cone.  If
$\theta_{\rm D}>\theta_{\rm lc}$, loss-cone stars can get in and out of the
loss-cone faster than they could reach the central object.


\section{The loss-cone contribution to the heating rate of the gas}

Dissipation of the stellar kinetic energy of a star plunging onto the
\sms~and suffering from the drag force leads to a heating of the \sms.
Another possible consequence of the local star-gas interaction is the
formation of massive stars within the cloud due to the accretion of
ambient gas \citep{daCosta79}. This could increase the supernova rate
and be an important source of energy. Here we will assume an
equilibrium for the \sms~for the timescales of interest.

The star distribution will be affected at large radii ($r \gg {\cal
  R}$, ${\cal R}$ being the \sms~radius) by removal of stars in the
central regions of a stellar system \citep{Peebles72}.  A drift of stars
occurs in the centre of the stellar system in order to recover the
equilibrium. In the special case of having a BH at the centre of the
system, processes like tidal disruption lead to the destruction of the
star. In this arena we have an outward energy flux created by the
sinking of stars via relaxation processes (local, two-body,
small-angle gravitational encounters).

When we consider the general case of an \sms, the basic picture is the
same, but some aspects vary; the removal of stars and the inward
transport are due to a different process. The inward flux of energy is
produced not only for the local, two-body, small-angle gravitational
encounters, but also for dissipative processes. The effective sinking
for stars is now related to gas-drag energy dissipation and can
involve or not their actual physical destruction.  A star moving
through the cloud will quickly dissipate energy because of gas drag
and then it will sink into the centre of the \sms. The main difference
between a BH and an \sms~is basically that the former produces a
low-angular momentum star depletion on a crossing time, whereas the
dense gas cloud or \sms~does the same but in a dissipation time.

Since we want to analyse the effects on the dense stellar system
arising from the presence of a central gas cloud we have to
distinguish between those stars whose orbits are limited to the region
where the \sms~is located ({\sl confined stars}) and those stars in
orbits which surpass the radius of the cloud ({\sl unconfined stars}).

When we talk about confined stars, the first steps in the evolution
are determined by the energy dissipation given by the drag force that
the dense gas cloud exerts on the individual stars. Stars lose
velocity in their motion inside the cloud, they are slowed down by the
gas and therefore they cede heat to the cloud.

The slowing down of the stars makes them become a more compact
subsystem which will sink down to the centre of the cloud. The system
becomes self-gravitating and we have a cusp in the stellar
distribution. However, star-star interactions can play a decisive role
in this point, since they yield a depletion in the number of confined
stars, or we can have direct collisions between them and thus
disruption or coalescence. This could avoid that a singularity in the
core collapse crops up. We cannot exclude the exchange of mass between
individual stars and the gas as another possible way to prevent the
singularity, since this can yield the star disruption via stellar
wind, or the creation of heavier stars which become a supernova \citep{daCosta79}.

The consequent evolution of the confined system will be in part
determined by the rate at which surrounding stars outside the gas
cloud refill this confined-stars gap. The importance of the core
collapse will also be a decisive point for the evolution.

Unconfined stars move on orbits extending larger than the \sms~radius.
However, they can suffer its influence out to a radius within which
the presence of the \sms~is effective. The idea is exactly the same as
for the BH, for the \sms~is a generalisation of the former case

\section{Kinetic energy dissipation}

The drag force that the individual stars suffer when they cross the
\sms~ is given by the next equation estimated by \citet{BKS72}:

\begin{equation}
F_{\rm D}=C_{\rm D} S \rho_{\rm SMS} v_{\star}^2.
\label{eq.F_drag}
\end{equation}
\noindent
$C_{\rm D}$ is a numerical parameter of order unity, $\rho_{\rm SMS}$ is
the mean density of the gas cloud, $v_{\star}$ is the velocity of the
stars and $S$ is the cross section of the stars, $S=\pi r_{\star}^2$.  In
case of a supersonic motion of the star the force $F_{\rm D}$ can be
interpreted as caused by the ram pressure (pressure difference)
originating at a bow shock in front of the moving star. Due to the
physical shock conditions one can show in such a case $C_{\rm D} \approx 4
$ \citep{CF98}.
 
Suppose that the star crosses the \sms~from one extreme to the
opposite, i.e.  along its diameter; thus, if ${\cal R}$ is the radius
of the \sms, the stellar energy dissipated during each passage through
it is

\begin{equation}
\triangle E_{\rm D}=F_{\rm D} \cdot 2{\cal R}.
\end{equation}

The orbits of the stars within $r_{\rm h}$ will be elliptic shaped with
one focus at the \sms~centre. The semi-major orbit axis $a$ will
shrink because of the drag force, driving the orbit directly into the
\sms. The average energy dissipation rate is
\begin{equation}
- \frac {dE}{dt} = \frac {\triangle E_{\rm D}}{T} = \frac {2C_{\rm D} \rho_{\rm SMS} \pi
 r_{\star}^2 G{\cal M}}{ \pi \sqrt {4a^3/G{\cal M}}},
\end{equation}
\noindent
where $T$ is the period.

In the BH accretion problem the individual stars are doomed to sink in
the hole if they belong to the loss-cone, whereas for the \sms~it
depends on the number of passages through the gas cloud. It means that
the non-confined star can cross the gas cloud several times until it
is trapped. To make this picture clearer we resort to the BH,
particularly when we introduced the concept of $\theta_{\rm D}$.  In
the case of \sms,
\begin{equation}
\theta _{\rm D} \approx \sqrt { \frac {n_{\rm trap} \cdot t_{\rm dyn}}{t_{\rm relax}}}.
\end{equation}
We can get the number of necessary passages for the star to be trapped if we write the 
potential difference between $r$ and $\cal R$ (for the BH problem it is $n=1$),
\begin{equation}
\triangle \phi = G{\cal M}m_{\star} \big( \frac{1}{r} - \frac{1}{\cal R} \big).
\end{equation}
\noindent
Then, the number of passages is:
\begin{equation}
n_{\rm trap}= |\triangle \phi| / \triangle E_{\rm D}.
\end{equation}

The dissipation of stellar kinetic energy defines a region in
phase-space which we will call the loss-cone too, by analogy with the
BH accretion problem. We have to recall however, that the star will be
blown up by tidal forces and converted into gas by just one passage
inside the tidal radius $r_{\rm t}$ for the case of having a BH
whereas the number of passages is $n_{\rm trap} > 1$ if we consider
the gas cloud situation.


\section{Loss-cone stars velocity field distribution function}

It stands to reason that at distances much larger than the \sms radius
($r \gg {\cal R}$) the star field velocity distribution has a
Maxwellian shape:
\begin{eqnarray}
&f(r,v)=&g(r)\cdot h(v), \nonumber \\
&f(r,v)=&N(r) \cdot \exp (-\frac {v_{\rm r}^2}
{2\sigma_{\rm r} ^2})\exp (-\frac {v_{\rm t}^2}{2\sigma_{\rm t} ^2})
\end{eqnarray}
Taking into account that $d^{3}v=2\pi v_{\rm t}dv_{\rm r}dv_{\rm t}$, the density
is
\begin{equation}
\rho(r)= \int_{0}^{\infty}f(r,v)4\pi v^2 dv =
\pi \sqrt{\frac {\pi}{2}}N(r)\sigma_{\rm r} \sigma_{\rm t}^2;
\end{equation}
\noindent
then
\begin{equation}
N(r)=\frac{\rho (r)}{(2\pi)^{3/2}\sigma_{\rm r}\sigma_{\rm t}^2}
\end{equation}
\noindent
And so get we the normalised field velocity distribution,
\begin{equation}
f(v_{\rm t},v_{\rm r})= \frac{\rho(r)}{(2\pi)^{3/2}\sigma_{\rm r}\sigma_{\rm t}^2} \exp
 (-\frac {v_{\rm r}^2}{2\sigma_{\rm r} ^2})\exp (-\frac {v_{\rm t}^2}{2\sigma_{\rm t} ^2}).
\end{equation}
In order to get the density of stars within the loss-cone we have to
compute the following integral:
\begin{equation}
\rho_{\rm lc}(r)= \int_{-v_{\rm r}|_{\rm max}}^{0} \int_{-v_{\rm lc}}^{+v_{\rm lc}} f(v_{\rm r},
v_{\theta},v_{\phi}) \mathrm{d}v_{\rm r}{d}v_{\theta}{d}v_{\phi}.
\end{equation}
\noindent
Taking into account that $\mathrm {d}v_{\theta}{d}v_{\phi}=2\pi v_{\rm t}
 {d}v_{\rm t}$ and that $f(r,v)=f(r, -v)$,
\begin{equation}
\rho_{\rm lc}(r)=4 \pi \rho(r) \int_{0}^{v_{\rm r}|_{\rm max}} \mathrm{d}v_{\rm r}
 \int_{0}^{v_{\rm lc}(r,v_{\rm r})} f(r,v_{\rm r},v_{\rm t})v_{\rm t} \mathrm{d}v_{\rm t}.
\end{equation}
In this expression, the maximal radial velocity is $v_{\rm r}|_{\rm max}=
v_{\rm escape}=\sqrt{2 \phi(r)}$ and the potential is the sum of both,
the super-massive star and the stellar system potential,
\begin{equation}
\phi(r)=\phi_{\rm SMS}(r)+\phi_{\star}(r).
\end{equation}

The integral happens to be analytical, and it yields the following
result:
\begin{equation}
\rho_{\rm lc}(r,v_{\rm r})=\rho(r) \cdot \{ \alpha - \zeta \cdot \beta \cdot \psi \},
\end{equation}

\noindent
where 

\begin{eqnarray}
&\alpha \equiv& {\rm erf} (\sqrt {\phi(r)/\sigma_{\rm r}^2}) \nonumber \\
&\beta \equiv& \exp \Bigl( - {{\cal R}^2 \over r^2-{\cal R}^2} {2\triangle
\phi \over \sigma_{\rm t}^2} \Bigr)  \nonumber \\
&\psi \equiv& {\rm erf} ( \frac {1}{\zeta} \sqrt { \phi(r)/ \sigma _{\rm r}^2} ) \\
&\zeta \equiv&  \Bigg( \frac{(r^2-{\cal R}^2)\sigma_{\rm t}^2}{(r^2-{\cal R}^2)\sigma_{\rm t}^2+{\cal R}^
2\sigma_{\rm r}^2}\Bigg)^{1/2}  \nonumber \\
&\triangle \phi \equiv& \phi({\cal R})-\phi(r). \nonumber 
\end{eqnarray}

Since we are working with a Gaussian function whose width is $\sigma$,
the contributions of velocities $v_{\rm r}>2\sigma_{\rm r}$ to the total mass
are small and therefore negligible. In the practice it means that we
can approximate the integral by $v_{\rm r}=2\sigma_{\rm r}$. In such a
situation, the loss-cone adopts an easy geometrical form: an open
cylinder in the $-v_{\rm r}$ direction with a radius $v_{\rm lc}$.
\noindent
The resulting integral yields
\begin{equation}
\frac {\rho_{\rm lc}(r)}{\rho(r)}= \Bigg[ 1- \exp \Bigg\{ - \frac{{\cal R}^2}
 {r^2-{\cal R}^2} \frac{2\sigma_{\rm r}^2 + \triangle \phi (r)}{\sigma_{\rm t}^2}\Bigg\}
 \Bigg] \cdot {\rm erf} \Bigg( \sqrt {\frac {\phi (r)}{\sigma_{\rm r}^2}} \Bigg)
\end{equation}
\noindent
We use for the stellar system a polytrope with $n=5$, with a compact
core and an extended outer envelope (named {\sl Plummer model} after
\citet{Plummer11}, since it was first used by him to fit the observed
light clusters distributions), which is commonly used for several
numerical model computations.

In this model the potential is spherical,
\begin{equation}
\phi=- \frac{G{\cal M}_{\star}}{\sqrt{r^2+r_{\rm c}^2}}
\end{equation}
\noindent
where ${\cal M}_{\star}$ is the system's total mass and $r_{\rm c}$ its core radius.

Thus,
\begin{equation}
\nabla^2 \phi= \frac{3GMr_{\rm c}^2}{(r^2+r_{\rm c}^2)^{5/2}}.
\end{equation}
Then, if we resort to the equation of Poisson, we can find out that
this system happens to have the following physical properties:
\begin{eqnarray}
&\rho_{\rm p}(r)=& \frac{3M}{4 \pi r_{\rm c}^3} \bigg(1+ \frac{r^2}{r_{\rm c}^2} \bigg)^{-5/2}, \nonumber \\
&M_{\rm p}(r)=&M \cdot \frac{r^3/r_{\rm c}^3}{(1+r^2/r_{\rm c}^2)^{3/2}}, \\
&v_{\rm p}(r)=&\sqrt{\frac{GM}{6r_{\rm c}}} \bigg( 1 + \frac{r^2}{r_{\rm c}^2} \bigg)^{-1/4}. \nonumber
\end{eqnarray}
Thus,
\begin{equation}
\phi(r)=\frac {G{\cal M}}{r} + \frac{G{\cal M}_{\star}}{\sqrt{r^2 + r_{\rm c}^2}}.
\end{equation}

\noindent
Therewith, the resulting expression is

\begin{equation}
\frac {\rho_{\rm lc}(r)}{\rho(r)}= (1-\exp {A})\cdot {\rm erf}(B),
\end{equation}
\noindent
where

\begin{eqnarray}
&A \equiv& - \frac{{\cal R}^2} {r^2-{\cal R}^2} 
~ \frac {1}{\sigma_{\rm t}^2}(C) \nonumber \\
&B \equiv&  \frac{1}{\sigma_{\rm r}} \sqrt {\frac{G{\cal M}}{r} + \frac{G{\cal M}_{\star}}
{\sqrt{r^2 + r_{\rm c}^2}}} \\
&C \equiv&  2\sigma_{\rm r}^2 + \frac{G{\cal M}}{{\cal R}} - \frac{G{\cal M}}{r}+
\frac{G{\cal M}_{\star}}{\sqrt{{\cal R}^2+r_{\rm c}^2}} - \frac{G{\cal M}_{\star}} 
{\sqrt{r^2+r_{\rm c}^2}} \nonumber.
\end{eqnarray}

The velocity vectors of all stars which belong to a given (fixed)
phase space density $f=f_{0}={\rm const}$ shape an ellipsoid whose two
tangential and one radial major axes have lenghts equal to the
velocity dispersion: $\sigma_{\theta}=\sigma_{\phi}$, and $\sigma_{\rm r}$
\citep{FR76}. If one uses $f_{0}=f(\sigma_{\rm r},\sigma_{\theta},
\sigma_{\phi})$, the surface of the ellipsoid $A= \pi \sigma_{\theta}
\sigma_{\phi} \sigma_{\rm r}$ is a measure for the available velocity
space.

A cone of angle $\theta_{\rm lc}$,
\begin{equation}
\theta_{\rm lc}:={\rm arcsin} ~(\frac{v_{\rm lc}}{\sigma_{\rm r}}),
\end{equation}
cuts out a segment of the foregoing velocity ellipsoid's $A$ surface
of corresponding fraction surface $A_{\rm lc}$,
\begin{equation}
A_{\rm lc} \approx \pi \theta_{\rm lc}^2,~ (\theta_{\rm lc}<<\pi).
\end{equation}
We can then define the ratio $\Omega':=A_{\rm lc}/A \approx \theta_{\rm lc}^2/4$,
 which is a measure for the loss-cone size. 
 
With the assumption that $v_{\rm lc}$ does not depend on $v_{\rm r}$ any more
and taking into account that with a Schwarzschild-Boltzmann
distribution we can reduct the loss-cone momenta to elementary
Gaussian error functions. The quantity

\begin{equation}
\Omega:=\frac{\rho_{\rm lc}}{\rho}
\end{equation}
yields in first order
\begin{equation}
\Omega \approx \frac {v_{\rm lc}^2}{4 \sigma_{\theta}^2} \approx \frac 
{\theta_{\rm lc}^2}{4}.
\end{equation}

This is the connection between the preceding simple picture of the
loss-cone and the definition of $\Omega$ in the velocity space: For a
Schwarzschild-Boltzmann distribution we can find out that, at first
order, $\Omega$ is of the same size as $\Omega'$.

\section{Isotropy and anisotropy in the stellar system}

We introduce now an {\sl isotropy ratio} in order to study the
different possible situations for the stellar distribution: The
tangential velocity dispersion is
$\sigma_{\rm t}^2=\sigma_{\phi}^2+\sigma_{\theta}^2$; in case of isotropy,
$\sigma_{\rm r}^2=\sigma_{\phi}^2+\sigma_{\theta}^2$, then $\sigma_{\rm t}^2=
2\sigma_{\rm r}^2.$

Now we define the ratio $R:=2\sigma_{\rm r}^2/ \sigma_{\rm t}^2$. According to
this definition, $R=1$ for the isotropic case. The corresponding
values of $R$ for radial and tangential anisotropy can be obtained
bearing in mind that $\sigma^2=\sigma_{\rm r}^2+\sigma_{\rm t} ^2=\sigma_{\rm t}^2
( R/2 +1)$, $\sigma_{\rm t}=\sigma / \sqrt{R/2+1}$ and
$\sigma_{\rm r}=\sigma_{\rm t}\cdot \sqrt{R/2}.$

We have the loss-cone star density as a function of the super-massive
star radius, ${\cal R}$.  This does not provide much information,
since in principle this radius could have any size; we do not have a
criterion for it yet. Instead, what does make sense is to express this
loss-cone star density in terms of the super-massive star stability,
which is something has been studied in detail \citep{FW86}.

From \citet{Chandra64}, instability sets in when the radius of the
star ${\cal R}$ is less than a critical radius ${\cal R}^{\rm crit}$.
He shows that if the ratio of specific heats $\gamma=C_{p}/C_{v}$
exceeds 4/3 only by a small amount, then dynamical instability will
occur if the mass contract to the radius ${\cal R}^{\rm crit}$
\begin{equation}
{\cal R}^{\rm crit}=\frac {K}{\gamma - \frac{4}{3}} \Bigg(\frac{2G{\cal M}}{c^2}
 \Bigg).
\end{equation}

Thus, we introduce the {\sl stability coefficient} $\delta:={\cal
  R}/{\cal R}^ {\rm crit}$.  We just have to substitute ${\cal R} =
\delta \cdot {\cal R}^{\rm crit}$ in the loss-cone star density
formula and vary $\delta$ instead of ${\cal R}$.

\section{Connection at the influence radius}

Since we are interested in the diffusion angle, we now derive two
expressions for it, within the influence radius of the \sms~and
outside it.  For this aim we look at the dynamical and relaxation time
at this radius.

\vskip0.2cm
\noindent
The $\sigma(r)$ varies depending on $r_{\rm h}$. 
\vskip0.2cm
\noindent
$\circ$ $r \le r_{\rm h}$: 

\noindent
In this case we assume that the gravitational potential is dominated
by the \sms.
\noindent
Therefore, the velocity dispersion is:
\begin{equation}
\sigma(r)= \sqrt{\frac{G {\cal M}}{r}};
\end{equation}
\noindent
This is just an approximation which we make here for simplicity, since
we stand on radii $r<r_{\rm h}$. To include $r\ll r_{\rm h}$ we need a better
model, which can only be obtained by numerical solution of the
equation of Poisson, and this is subject of future work.  \vskip0.2cm
\noindent
$\circ$ $r > r_{\rm h}$: 

\noindent
For radii larger than $r_{\rm h}$ we will use a Plummer model for the
velocity dispersion,
\begin{equation}
\sigma(r)= \sqrt{ \frac{G{\cal M}_{\star}}{6r_{\rm c}}} \bigg(1+\frac{r^2}{r_{\rm c}^2} \bigg)^{-1/4}
\end{equation}

However, we have to match both solutions, within and outside the
\sms~influence radius, for we have to look for a velocity dispersion
connection; otherwise we get artificial, non-physical ``jumps'' in the
plots for $r \approx r_{\rm h}$. This can be performed as follows: We add
in the velocity dispersion expression a factor $\alpha$,
\begin{equation}
\sigma(r)= \sqrt{ \frac{G{\cal M}_{\star}}{6r_{\rm c}}} \frac{\alpha}{(1+ r^2/r_{\rm c}^2)^{1/4}},
\end{equation}
and we ask now for both velocities dispersions to be equal at the
influence radius:
\begin{equation}
\sigma(r_{\rm h})|_{r<r_{\rm h}} \equiv \sigma(r_{\rm h})|_{r>r_{\rm h}}.
\end{equation}
\noindent
We can conclude that $\alpha$ has the following value:
\begin{equation}
\alpha=\sqrt{\frac{6}{r_{\rm h}}\frac{\cal M}{{\cal M}_{\star}}} \cdot (r_{\rm c}^2+r_{\rm h}^2)^{1/4}
\end{equation}
\noindent
Note that $\alpha$ is necessary because our velocity dispersion is
approximate for $r<r_{\rm h}$, but not for $r \ll r_{\rm h}.$

\noindent
Therefore, the velocity dispersion outside the influence radius is:
\begin{equation}
\sigma(r) =\sqrt{ \frac{G{\cal M}_{s}}{r_{\rm c}r_{\rm h}}} \cdot \bigg[ \frac {r_{\rm c}^2}
{r_{\rm c}^2+r^2} (r_{\rm c}^2+r_{\rm h}^2) \bigg]^{1/4}.
\end{equation}

\noindent
As regards the dynamical time, as we saw in the introduction, 
$t_{\rm dyn}={r}/{\sigma_{\rm r}(r)}$, is

\be
 t_{\rm dyn} = \left\{ \begin{array}{ll}
                             \frac {r^{3/2}}{ \sqrt{G{\cal M}}} & 
\mbox{$r~ \le r_{\rm h}$} \\

                             r \sqrt { \frac{r_{\rm h}}{G{\cal M}}} 
\cdot \bigg( \frac{r^2+r_{\rm c}^2}{r_{\rm h}^2+r_{\rm c}^2} \bigg)^{1/4}  & 
\mbox {$r~ > r_{\rm h}$}
                              \end{array}
                              \right. 
\ee

For the relaxation time we need the density of the stellar system; We
already commented that, according to \citet{FR76} or to
\citet{BW76}, the density of the spherical stellar system
around the central mass should follow a power law of the form $\rho(r)
\propto r^{-7/4}$.
\begin{equation}
\rho(r)= \rho_{\rm p}(r_{\rm h}) \cdot \bigg( \frac{r}{r_{\rm h}} \bigg)^{-7/4}
\end{equation}
\noindent
where $\rho_{\rm p}(r_{\rm h})$ is 
\begin{equation}
\rho_{P}(r_{\rm h})= \frac{3{\cal M}_{\star}}{4 \pi r_{\rm c}^3} \bigg(1+ 
\frac{r_{\rm h}^2}{r_{\rm c}^2} \bigg)^{-5/2}
\end{equation}
\noindent
Therefore,
\begin{equation}
\rho(r)= \frac{3{\cal M}_{\star}}{4 \pi r_{\rm c}^3} 
\bigg(1+ \frac{r_{\rm h}^2}{r_{\rm c}^2} 
\bigg)^{-5/2}\bigg( \frac{r}{r_{\rm h}} \bigg)^{-7/4}
\end{equation}


Thus, the final expression for the relaxation time in such a system is

\be
 t_{\rm relax} = \left\{ \begin{array}{ll}
                             \frac{3 \sqrt{\pi}r_{\rm c}^3}{4{\cal M}_{\star} 
{\rm ln}(\gamma N)} \big( \frac{G{\cal M}}{r} \big)^{3/2} \big(\frac{r}{r_{\rm h}} 
\big)^{7/4} \big(1+ \frac{r_{\rm h}^2}{r_{\rm c}^2} \big)^{5/2}  &                   
\mbox{$r~ \le r_{\rm h}$} \\

                             \frac{3 \sqrt{\pi}r_{\rm c}^3}{4{\cal M}_{\star} 
{\rm ln}(\gamma N)}
\big( \frac{G{\cal M}}{r_{\rm c}r_{\rm h}} \big)^{3/2} \big(1+\frac{r^2}{r_{\rm c}^2} 
\big)^{5/2} 
\big[ \frac{r_{\rm c}^2}{r^2+r_{\rm c}^2} (r_{\rm c}^2+r_{\rm h}^2) \big]^{3/4}  & 
\mbox {$r~ > r_{\rm h}$}
                              \end{array}
                              \right.
\ee

These expressions enable us to get the value for $\theta_{\rm D}$; we
can get $\theta_{\rm lc}$ thanks to $\Omega$, $\theta_{\rm lc} \approx
2 \sqrt{\Omega}$. Then we plot them altogether to find out at what
radius do they cross, since we are looking for the critical radius,
$r=r_{\rm crit} \iff \theta_{\rm lc}=\theta_{\rm D}$.


\section{Mass accretion rates}

The rate of stars plunging onto the central \sms~is given by two
different formul\ae, depending on whether or not there is a crossing
point for the $\theta_{\rm lc}$, $\theta_{\rm D}$- plot against the radius,
{\em a critical radius}.  If we find that the curves happen to cross,
the mass accretion rate $\mdot$ has the expression
$\mdot={\cal M}_{\star}(r_{\rm h})/t_{\rm relax}(r_{\rm h})$ because the loss-cone will
be depleted in a relaxation time, and the mass to take into account is
that which lays within the critical radius.  On the other hand, if
there is no crossing point, this means that the loss-cone in not empty
and in this situation we have to employ a rather different expression,
we have to resort to the loss-cone star density expression to get the
mass being accreted into the \sms. In this case the timescale of
interest is the dynamical time, $\mdot=\Omega(r) {\cal M}_{\star}(r)/t_{\rm
  dyn}(r)$.

It may be asserted, nevertheless, that this formula is not completely
correct because it is based on the stationary model, which supposes an
empty loss-cone in the first case and a full loss-cone in the second
case.  We have to generalise it by means of a ``diffusion'' model
(see chapter \ref{ch.diff_model}). We introduce the concept
of the {\em filling degree} K of the loss-cone as follows: Let us
conjecture that $f$ is the unperturbed velocity distribution. If the
loss-cone is empty and angular momentum diffusion is neglected, then
$f=0$ inside the loss-cone and $f$ remains unchanged elsewhere in
velocity space.  Actually, this distribution function will have a
continuous transition from nearly unperturbed values at large angular
momenta towards a partially depleted value inside the loss-cone. We
approximate this by a distribution function $f$ having a sudden jump
just at the value $L_{\rm min}=m_{\star} v_{\rm lc}$ from an unperturbed value
$f_{0}$, $f=K \cdot f_{0}, ~{\rm with}~0 \le K \le 1$.  Since we work
with the hypothesis that some kind of stationary state is to be
established in the limit $t \to \infty$, the filling degree is the 
result of calculating this limit,

\be
k_{\infty}:= \lim_{t \to \infty} K(t)= \frac {t_{\rm out}} 
{P_{\rm lc}\,t_{\rm in} \chi}=
\ee
$$\Big(\frac{ \theta_{\rm D}^2} {\theta_{\rm lc}^2}\Big) \cdot \frac{1} 
{P_{\rm lc} {\chi}},$$

\noindent
where
\be
{\chi} \equiv 1+ \frac {t_{\rm out}} {P_{\rm lc}t_{\rm in}}=1+ 
\frac{4 \theta_{\rm D}^2} 
{ P_{\rm lc}{\theta_{\rm lc}^2}}.
\ee

\noindent
Therefore,

\begin{equation}
k_{\infty}=\frac{ \nu (1+\nu)}{1+\nu(1+\nu)}.
\end{equation}
\noindent
In this expression $\nu \equiv { \theta_{\rm D}^2}/{\theta_{\rm lc}^2}.$ Then
we have to multiply the accretion rates by this filling degree
$k_{\infty}$.

The stellar mass within and outside the influence radius is
\begin{equation}
M(r,r_{\rm min})|_{r<r_{\rm h}}=
\lim_{r_{\rm min} \rightarrow 0} 4\pi \int_{r_{\rm min}}^{r} 
\rho(r'){r'}^2 dr'=
\ee
$$\lim_{r_{\rm min} \rightarrow 0}4\pi \rho(r_{\rm h})
\cdot r_{\rm h}^{7/4} \int_{r_{\rm min}}^{r} {r'~}^{1/4}dr'=$$
$$\lim_{r_{\rm min} \rightarrow 0}4\pi \rho(r_{\rm h})\cdot r_{\rm h}^{7/4} 
\bigg[ \frac{4}{5} {r'~}^{5/4} \bigg]_{r_{\rm min}}^{r}= $$
$$\frac{16\pi}{5} \rho (r_{\rm h})~r_{\rm h}^3\bigg
( \frac{r}{r_{\rm h}} \bigg)^{5/4},$$

\begin{equation}
M(r)|_{r>r_{\rm h}}=M(r_{\rm h})|_{r<r_{\rm h}} + 4\pi \int_{r_{\rm h}}^{r} \rho(r'){r'}^2dr'
\end{equation}
$$=\frac{16\pi}{5} \rho (r_{\rm h})~r_{\rm h}^3 + 4\pi \frac{{\cal M}_{\star}}{r_{\rm c}^3} 
\bigg[ \frac{r^3}{ \big( 1+ {r^2}/{r_{\rm c}^2}\big)^{3/2}}  - \frac{r_{\rm h}^3}
{\big( 1+ {r_{\rm h}^2}/{r_{\rm c}^2}\big)^{3/2}}\bigg].$$

To get the total heating rates, we just have to compute the value of
\begin{equation}
L_{\rm heat,~all} = \bigg( \frac {\mdot}{m_{\star}} \bigg)\cdot E_{\rm heat,~1\star},
\end{equation}
\noindent
where $E_{\rm heat,~1\star}$ is the heating for one star (during one crossing).

\section{Heating rates: An esteem}

In this section we analyse the interaction rate of stars with the
\sms~by varying the parameters introduced in the former sections,
namely $\delta$, the core radius $r_{\rm c}$, the total stellar mass and
the super-massive star mass itself. We suppose that the stars which
conform the stellar system are solar-type stars. For all the plots we
extend the radii down to 1.001 times the \sms~radius, because we would
run into a snag if we extended it within the \sms~radius, since the
proportions $\sigma^2 \propto 1/r$ and $\rho \propto r^{-7/4}$ would
be wrong and the loss-cone star density and therefore the loss-cone
angle has been obtained considering a Maxwell-Boltzmann distribution.
It would also be an error of the problem conception itself, because we
are studying the non-confined stars and the loss-cone, and its
definition does not make any sense for radii less than the ${\cal R}$.
This explains the first inequality in ${\cal R}<r_{\rm h}<r_{\rm c}$,
which is an exigency that we must follow unfailingly because,
otherwise, the Plummer model, which we use for the stellar system,
would not be a suitable solution for it in the case that $r_{\rm
  h}<r_{\rm c}$ is not satisfied.  However, what we demand here is not
a requirement of the physics of the problem, but a condition for the
method being employed to solve it. Situations in which $r_{\rm
  h}<{\cal R}$ or $r_{\rm h}>r_{\rm c}$ have to be solved numerically
for the equation of Poisson and the velocity distributions.

In order to estimate the heating rates for a single star crossing the
\sms~we have to plot out the mass accretion rate of this central
massive object in the case that we have no crossing point for the
loss-cone and diffusion angle curves, whereas if we have a critical
radius we will have to compute the total stellar mass and dynamical
time at this value.

In Fig.\,(\ref{fig.diff_model2}) we plot the velocity dispersion
against the radius for a $10^3\msol$ \sms.  We observe a typical
power law of $\propto r^{-1/2}$ within the influence radius -which we
represent by a vertical dashed line for all cases- because we have a
cusp on velocities in this interval of radii. We have a nearly
constant velocity for later radii which lie in the section of values
close to the core radius. Then the velocity dispersion decays. The
length of this nearly constant velocity section as well as the slope
for the decay depends on the galactic nuclei in the galaxy.

Regarding Fig.\,(\ref{fig.diff_model3}), we show the loss-cone
normalised density difference between the isotropic and radially
anisotropic cases for a $10^7\msol$ \sms. In this plot we can
observe a bigger number of stars being accreted into the \sms~for the
radially anisotropic case,since a radial orbit means a lower angular
momentum and thus it is more probable that the star sinks into the
central object, and vice-versa for a tangential orbit,
Fig.\,(\ref{fig.diff_model4}).

As regards the loss-cone and diffusion angle plots, we examine two
different cases in Figs.(\ref{fig.diff_model5}) and
(\ref{fig.diff_model6}): a $10^4\msol$ and a $10^7\msol$
\sms. One may observe that for the first one we find a critical
radius, whereas for the latter one the curves do not intersect. It is
also interesting to find out which angle is bigger and where, since
for a $\theta_{\rm lc} \gg \theta_{\rm D}$ we have an almost empty
loss-cone, because it is quickly depleted and the stars replenish it
very slowly; the opposite case, $\theta_{\rm lc} \ll
\theta_{\rm D}$, implies that the loss-cone is full.

We get a maximum $\mdot$ at about a core radius in the mass
accretion rates plot for a $10^7\msol$ \sms, because the biggest
contribution of stars being accreted into the \sms~lies at this
radius. Fig.\,(\ref{fig.diff_model_jump}) shows an irregularity at the
influence radius, because the loss-cone and diffusion angle plots show
us that the former happens to be always bigger than this one and thus
we have to apply the approximation commented in the foregoing section.

When we look at the $\mdot$ for the $10^3\msol$ and
$10^4\msol$ \smss, we obtain that for the $10^3\msol$:
\noindent
 ${\mdot}|_{\rm iso}=1.75 \times 10^{-13} ~ \msol/{\rm yr}$, 
\noindent
 ${\mdot}|_{\rm tan}=1.66 \times 10^{-13} ~ \msol/{\rm yr}$,
\noindent
 ${\mdot}|_{\rm rad}=1.97 
\times 10^{-13}~ \msol/{\rm yr}$. 
\noindent
Where the subscript ``iso'' stands for isotropy, ``rad'' for radial
anisotropy and ``tan'' for tangential anisotropy.

The case of $10^4\msol$ is similar to the last one. For the
$10^7\msol$ \sms, we will select the $\mdot$ corresponding to
the core radius, since the most important contribution is reached
there: $\mdot|_{\rm core} = 10^{-2}~ \msol/{\rm yr}.$ To get the
heating rates of these non-confined stars, we just have to compute
\begin{equation}
\dot{E}= \bigg( \frac{\mdot}{\msol} \bigg) \pi r_{\star}^2 \rho_{sms}
 v_{\star}^2 \cdot 2{\cal R}/{t_{\rm cross}},
\end{equation}
where $v_{\star}^2=G{\cal M}/{\cal R}$, $t_{\rm cross}= 2{\cal R}/v_{\star}$,
 $\rho_{\rm SMS}={\cal M}/(\frac{4 \pi}{3} {\cal R}^3).$
\noindent
For the \sms~we have supposed, as a first approximation, a constant
density.

The corresponding ${\cal R}$'s are:

\vskip0.3cm
\noindent
${\cal R}_{10^3}=2.5 \times 10^{-9}$ pc

\noindent
${\cal R}_{10^4}=8 \times 10^{-8}$ pc

\noindent
${\cal R}_{10^7}=2.5 \times 10^{-2}$ pc,

\noindent
where ${\cal R}_{10^i}$ stands for the $10^i\msol$ \sms~radius.

\vskip0.2cm
\noindent
The luminosities are:

\vskip0.3cm
\noindent
$L_{10^3}/L_{\odot} =6.2 \times 10^3$

\noindent
$L_{10^4}/L_{\odot} =1.17 \times 10^4$

\noindent
$L_{10^7}/L_{\odot} =5.6 \times 10^5,$

\noindent
idem $L_{10^i}$ $10^i\msol$ \sms~luminosity.
It is not a surprise that these luminosities are not sufficient to
support quasar luminosities, which was known before. We confirm however
the earlier result by \citet{LangbeinEtAl90} with our more detailed, but
stationary loss-cone model, that the luminosities are large enough to
prevent for some time the relativistic collapse of a \sms~in a galactic
centre; that was called the quasi-pile stage by \citet{Hara78}. 

In the case of a confined star, the heating rate can be obtained in a
much simpler way. We just have to assume that the star is on a
circular orbit at $r\approx {\cal R}$, then its velocity is
\be
v_{\rm circ}^2= \frac{G{\cal M}}{{\cal R}},
\ee
\noindent
the orbital time is
\be
t=\frac{2 \pi {\cal R}} {v_{\rm circ}}=2\pi \sqrt{
\frac{{\cal R}^3}{G{\cal M}}}.
\ee
\noindent
Regarding to the drag force, $F_{\rm D}=C_{\rm D} \pi r_{\star}^2 \rho_{g}v_{\rm
circ},$ where $\rho_{g}={\cal M}/(\frac{4\pi}{3} {\cal R}^3)$ is the
average density. The heating by one star will be
\be
E_{\rm heating}|_{\rm conf}= F_{\rm D} \cdot 2 \pi {\cal R}
\ee
\noindent
The number of stars in the~\sms is given by
\be
N_{\star}= \frac{4 \pi}{3} \rho ({\cal R}) {\cal R}^3,
\ee
\noindent
where $\rho({\cal R})= \rho (r_{\rm h}) ({{\cal R}}/{r_{\rm h}})^{-7/4}.$
Therewith, the heating by confined stars is
\be
E_{\rm heating,~conf}=N_{\star} \cdot F_{\rm D} \cdot 2 \pi {\cal R}.
\ee


\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_semi_analytical/fig1.ps}}
\caption{The velocity units are $(m/s)^{2}$ and $r$ is expressed in pc. The influence radius is 
  located at $6 \cdot 10^{-4}$pc. We show its logarithm in the plot
  with a vertical dashed line. The central velocity dispersion (the
  velocity dispersion at the influence radius) is $\sigma_{\rm
  central}=\sqrt{G{\cal M}_{\star}/( 6r_{\rm c})}=84$ km/s. ${\rm M}_{\rm SMS}$
  stands for ${\cal M}$
  \label{fig.diff_model2} }
\EC
\efig

\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_semi_analytical/fig2.ps}}
\caption{$\Omega(r):=\rho_{\rm lc}(r)/\rho(r)$ decays with the distance to the \sms. 
  We have divided the radius by the core radius in order to normalise
  it. A bigger number of stars plunge on to the \sms~for the radially
  anisotropic case
  \label{fig.diff_model3} }
\EC
\efig

\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_semi_analytical/fig3.ps}}
\caption{The same as for Fig.\,(\ref{fig.diff_model3}) 
but for the tangentially anisotropic case and a 
  lower number of stars
  \label{fig.diff_model4} }
\EC
\efig

\bfig 
\C
\resizebox{0.7\hsize}{!}{\includegraphics
	{diag_semi_analytical/fig4.ps}}
\caption{$\theta_{\rm lc}=2 \cdot \sqrt{\Omega(r)}$, $\theta_{\rm D}= \sqrt{ 
    t_{\rm dyn} / t_{\rm relax}}$.  For this mass the two curves cross
  at the critical radius. For radii smaller than that, the loss-cone
  angle is bigger than the diffusion angle, and this implies that the
  loss-cone is empty. From the $r_{\rm crit}$ onwards it is no longer
  empty, for $\theta_{\rm D}>\theta_{\rm lc}$. With the crossing point we can
  work out the accretion rate and find out the differences depending
  on whether we consider an isotropic situation for the stellar
  velocity distribution function or an anisotropic one, distinguishing
  between a radial or tangential anisotropy. If we set this case
  against the $10^3\msol$ we do not find big differences
  \label{fig.diff_model5} }
\EC
\efig

\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_semi_analytical/fig5.ps}}
\caption{For higher masses, such as $10^7\msol$, the loss-cone is empty.
  We get no critical radius
  \label{fig.diff_model6} }
\EC
\efig

\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_semi_analytical/fig6.ps}}
\caption{The ``jump'' is due to the approximation we made in order to get
  the accretion rates. The maximum is reached at a distance which
  corresponds to the core radius
  \label{fig.diff_model_jump} }
\EC
\efig


\section{Discussion of the results}

We have revisited the classical loss-cone semi-analytic theory
invented by \citet{FR76} for star accretion onto central super-massive
black holes in galactic nuclei and star clusters and extended by
\citet{daCosta81} for the case of stars on radial orbits being trapped
by star-gas interactions in a central super-massive star-gas
system. In \citet{LangbeinEtAl90} such model was included in
time-dependent, spherically symmetric models of star-gas systems in
galactic nuclei. Though highly idealized, we think that such
configurations are still worth a study to understand the physical
processes at work in the early formation phase of massive galaxies
with formation of central black holes. Notions such as the critical
radius in the classical work, where the loss-cone star accretion
becomes important and flattens out the cusp density profile turn out
naturally in our model without any ad hoc assumptions. While in this
research note we keep the stellar background system fixed and develop
the old ideas in an up-to-date form, one should include them into a
self-consistent dynamical model of relaxing star clusters with central
black hole. Recent dramatic improvements of the observational
situation and the demography of black holes in nuclei demand such
progress in dynamical modelling, which is yet surprisingly poor
\citep{RichstoneEtAl98}. It is clear that in a proper cosmological
context many complications occur for simplified modelling: the whole
system is embedded in a collisionless dark halo with disputed central
density profile, it is non-stationary due to a sequence of merger
events in hierarchical structure formation, with the possible
formation of binary or multiple black holes and perturbations of
various kinds will even cause a single black hole not to be fixed in
the centre. Despite of all that have begun our work at the point where
we think present (astro)physical understanding and modelling comes to
its limits, and this is the case for a spherical dense large $N$ star
cluster, suffering from relaxation and star accretion around a fixed
massive black hole.

\section{Addendum A: The tidal radius}
\label{add.tid_radius}

When the stars approach the BH they experience strong tidal forces
which may be violent enough to blow up the star. The star gets
disrupted whenever the work exerted over the star by the tidal force
exceeds its own binding energy, (all energies are per unit mass):
\begin{equation}
E_{\rm bind}=\alpha\,\frac{Gm_{\star}}{r_{\star}},~\alpha=\frac{3}{5-n}
\label{eq.Ebind}
\end{equation}
\noindent
where $n$ is the polytropic index \citep{Chandra42}.

\bfig
\C
\resizebox{1\hsize}{!}{\includegraphics
	{diag_semi_analytical/bind_energy.eps}}
\caption{Decomposition of the tidal forces over a star 
\label{fig.bind_energy}}
\EC
\efig

\begin{equation}
(F_{1}-F_{2})2r_{\star}=\alpha\frac{Gm_{\star}}{r_{\star}}
\label{eq.F1_F2}
\end{equation}

\begin{equation}
F_{1}=\frac{G{\cal M}_{\bullet}}{(r_{\rm t}-r_{\star})^2},
~F_{2}=\frac{G{\cal M}_{\bullet}}{(r_{\rm t}+r_{\star})^2}.
\label{eq.F1_F2_b}
\end{equation}

Considering $r_{\star}\ll r_{\rm t}$, we can approximate the expressions:
\begin{eqnarray}
&\frac{1}{(r_{\rm t}-r_{\star})^2} &\approx  \frac{1}{r_{\rm t}^2}+ \frac{2r_{\star}}{r_{\rm t}^3} \nonumber \\
&\frac{1}{(r_{\rm t}+r_{\star})^2} &\approx  \frac{1}{r_{\rm t}^2}- \frac{2r_{\star}}{r_{\rm t}^3};
\label{eq.F1_F2_c}
\end{eqnarray}
\noindent
then,

\begin{equation}
r_{\rm tidal}=\Bigg[\frac{2}{3} (5-n) 
\frac{{\cal M}_{\bullet}}{m_{\star}}\Bigg]^{1/3} r_{\star}.
\label{eq.r_tid_bind}
\end{equation}

For solar-type stars it is (considering a $n=3$ polytrope)
\begin{equation}
r_{\rm t} \simeq 1.4\times 10^{11} \bigg( \frac {{\cal M}_{\bullet}}{M _{\odot}} \bigg)^{1/3}~{\rm cm}.
\label{eq.r_tib_bind_b}
\end{equation}

\section{Addendum B: The difussion and loss-cone angles}
\label{add.thetas}

\subsection{Definition of the difussion angle $\theta _{\rm D}$}

The relaxation time is the required time for $\triangle v_{\perp}^2 / 
v_{\perp}^2 \simeq 1$ (i.e. the change in the perpendicular velocity 
component is of the same order as the perpendicular velocity component itself);
\begin{equation}
\triangle v_{\perp}^2 = n_{\rm relax} \cdot \delta v_{\perp}^2,~\triangle 
v_{\perp}^2 / v_{\perp}^2 = 1 = \frac{n_{\rm relax} \cdot \delta v_{\perp}^2}
{ v_{\perp}^2}
\label{eq.thetaD_a}
\end{equation}
\begin{equation}
t_{\rm relax}=n_{\rm relax} \cdot t_{\rm dyn} = \bigg( 
\frac {v_{\perp}^2}{\delta v_{\perp}^2} \bigg) \cdot t_{\rm dyn},
\label{eq.thetaD_b}
\end{equation}

\noindent
where $n_{\rm relax}$ is the numbers of crossings for $\triangle v_{\perp}^2 / 
v_{\perp}^2 \simeq 1$. This conforms the definition of the relaxation time, 
$\triangle v_{\perp}^2 / v_{\perp}^2 = t/t_{\rm relax}$ \citep{BT87}.

If we consider that $\theta_{\rm D}$ is very small, 

\bfig
\C
\resizebox{0.7\hsize}{!}{\includegraphics
   {diag_semi_analytical/theta.eps}}
\caption{Definition of angle $\theta$
\label{fig.theta_D} }
\EC
\efig

\begin{equation}
\sin{\theta_{\rm D}} \simeq \frac{ \delta v_{\perp}}{v} \simeq \theta_{\rm D}
;~t_{\rm relax} \simeq \frac{t_{\rm dyn}} {\theta_{\rm D}^2}
\label{eq.thetaD_c}
\end{equation}

\begin{equation}
\theta_{\rm D} \simeq \sqrt{\frac {t_{\rm dyn}}{t_{\rm relax}}}.
\label{eq.thetaD_d}
\end{equation}

\subsection{Definition of the loss-cone angle $\theta_{\rm lc}$}

We suppose that the central object with mass ${\cal M}_{\bullet}$ has
an influence radius $r_{\rm h}$. To define this radius we say that a star
will interact with the central object only when $r \le r_{\rm h}$. Then we
look for a condition at a place $r>r_{\rm h}$ for a star to touch or to
cross the influence radius of the central object within a crossing
time $t_{\rm cross}= r/ \sigma_{\rm r}$.

As we saw in the text, the condition that defines this angle is the following:
\begin{equation}
r_{\rm p}(E,L) \leq r_{\rm t},~ \theta \leq \theta_{\rm lc}.
\label{eq.thetalc_a}
\end{equation}

\bfig
\C
\resizebox{0.3\hsize}{!}{\includegraphics
	{diag_semi_analytical/theta_lc.eps}}
\caption{Definition of angle $\theta_{\rm lc}$
	\label{fig.theta_lc} }
\EC
\efig

\begin{equation}
\sin{\theta} = \frac{v_{\rm t}}{v},~\theta \ll 1;~\Longrightarrow \theta \simeq 
\frac{v_{\rm t}}{v}= \frac {L/r}{v}.
\label{eq.thetalc_b}
\end{equation}



\noindent
Where $L=rv_{\rm t}$ is the specific angular momentum.

Now we derive an expression for this angle in terms of the influence radius:

\vskip0.5cm
\noindent
$ \bullet$ {\bf $r \le r_{\rm h}$}
\vskip0.5cm

In this region the star moves under the BH potential influence, then
\begin{equation}
\sigma(r) \approx \sqrt{ \frac {G{\cal M}_{\bullet}}{r}} 
= \sqrt { \frac {G{\cal M}_{\bullet}}{R_{\rm h}}} \sqrt { \frac {R_{\rm h}}{r}} =
\sigma (R_{\rm h}) \cdot \sqrt {R_{\rm h}/r} = \sigma _{c} \cdot \sqrt {R_{\rm h}/r},
\label{eq.thetalc_c}
\end{equation}
\noindent
since $ \sigma_{c}^2 \equiv G{\cal M}_{\bullet} / R_{\rm h}.$

The typical velocity of the orbit is $<v^2> \simeq 3 \sigma^3$, where the factor 
three stands for the three directions in the space. Since $\sigma$ means the one-
dimensional dispersion,  we have to take into account the dispersion of the 
velocity in each direction. Then,

\begin{equation}
<v> \simeq \sqrt{3} \sigma_{c} \sqrt {r_{\rm h}/r}.
\label{eq.thetalc_d}
\end{equation}

Finally, we obtain the loss-cone angle,

\begin{equation}
\theta _{\rm lc} = \sqrt { \frac {2}{3} \frac {r_{\rm t}}{r}}.
\label{eq.thetalc_e}
\end{equation}

\noindent
$ \bullet$ {\bf $r \ge r_{\rm h}$}
\vskip0.5cm

We can consider that the velocity dispersion is more or less constant from 
this $r_{\rm h}$ onwards, $v \approx \sqrt{3} \sigma_{c}$, 

\begin{equation}
\theta_{\rm lc} = \frac { \sqrt {2G{\cal M}_{\bullet}r_{\rm t}}}{ \sqrt {3} r \sigma _{c}};~ 
\sigma _{c}= \sqrt {G{\cal M}_{\bullet}/r_{\rm h}}.
\label{thetalc_f}
\end{equation}

The angle  is
\begin{equation}
\theta_{\rm lc} \approx \sqrt { \frac {2}{3} r_{\rm t} r_{\rm h}} /r.
\label{eq.thetalc_g}
\end{equation}

\section{Addendum C: The loss-cone velocity}
\label{add.lc_vel}

We derive the $v_{\rm lc}(r)$ using angular momentum and energy
conservation arguments. We just have to evaluate it at a general
radius $r$ and at the tidal radius $r_{\rm t}$, where the tangential
velocity is maximal and the radial velocity cancels (see
Fig.\ref{fig.peribarath}).

\bfig 
\C
\resizebox{0.6\hsize}{!}{\includegraphics[scale=0.75,bb=0 19 350 387,clip]
	{diag_semi_analytical/peribarathron.eps}}
\caption {Definition of the {\em peribarathron} as the distance of closest 
  approximation of the star in its orbit to the BH. In this
  point the radial component of the velocity of the star cancels and
  the tangential component is maximum. In the figure ``$r_{\rm p}$'' stands
  for the peribarathron radius and ``$r_{\rm t}$'' for the {tidal radius}
        \label{fig.peribarath}
}
\EC   
\efig 


\noindent
$\bullet$ general radius:


\begin{eqnarray}
&E(r)=& \phi(r) - \frac{v_{\rm tg}(r)^2}{2} - \frac{v_{\rm r}(r)^2}{2} \nonumber \\
&L(r)=&r v_{tg}(r)
\label{eq.lc_vel_a}
\end{eqnarray}
\noindent
$\bullet$ tidal radius:

\begin{eqnarray}
&E(r_{\rm t})=& \phi(r_{\rm t}) - \frac{v_{tg}(r_{\rm t})^2}{2} \nonumber \\
&L(r_{\rm t})=&r_{\rm t} v_{tg}(r_{\rm t}),
\label{eq.lc_vel_a}
\end{eqnarray}

\noindent
from the momentum conservation and the fact that $v_{\rm r}(r_{\rm t})=0$ we get that:

\begin{equation}
v_{\rm tg}(r_{\rm t})= \frac {r}{r_{\rm t}}\cdot v_{\rm tg}(r).
\label{eq.lc_vel_b}
\end{equation}

Idem energy, and using the last result, 

\begin{equation}
\phi(r) - \frac {v_{tg}(r)^2}{2} - \frac {v_{\rm r}(r)^2}{2} =\phi (r_{\rm t}) - 
\frac{r^2}{2r_{\rm t}^2} \cdot v_{tg}(r)^2.
\label{eq.lc_vel_c} 
\end{equation}

\noindent
Then we get the tangential velocity of the stars in terms of r; namely, the 
loss-cone velocity:

\begin{equation}
v_{\rm lc}(r)= \frac {r_{\rm t}}{\sqrt {r^2-r_{\rm t}^2}} \cdot \sqrt {2[ \phi (r_{\rm t}) - 
\phi (r)] + v_{\rm r}(r)^2}.
\label{eq.lc_vel_d}
\end{equation}

Therewith, the angular momentum is

\begin{eqnarray}
&L(r_{\rm t})=&r_{\rm t}\cdot v_{\rm tg}(r)|_{\rm max}=r_{\rm t}\cdot 
\frac{r}{r_{\rm t}}v_{\rm tg}(r)= \nonumber \\
&r\cdot v_{\rm tg}(r)&=r\cdot  \frac {r_{\rm t}}{\sqrt {r^2-r_{\rm t}^2}} 
\sqrt {2\triangle \phi + v_{\rm r}(r)^2},
\label{eq.lc_vel_d2}
\end{eqnarray}

\noindent
where 

\begin{equation}
\triangle \phi \equiv \phi (r_{\rm t}) - \phi (r)=
\frac{G{\cal M}_{\bullet}}{r_{\rm t}} + \phi_{\star}(r_{\rm t})-
\frac {G{\cal M}_{\bullet}}{r}- \phi_{\star}(r)
\label{eq.lc_vel_e}
\end{equation}

If we use the fact that

\begin{equation}
r \gg r_{\rm t} \Rightarrow \frac{G{\cal M}_{\bullet}}{r_{\rm t}} 
\gg \bigg( \frac{G{\cal M}_{\bullet}}{r} +
\phi_{\star}(r) \bigg)= \phi(r)
\label{eq.lc_vel_f}
\end{equation}

\noindent
Also it is
\begin{equation}
\frac {G{\cal M}_{\bullet}}{r_{\rm t}} \gg \phi_{\star}(r_{\rm t}),
~{\rm since}~{\cal M}_{\bullet}\gg {\cal M}_{\star}(r_{\rm t})
\label{eq.lc_vel_g}
\end{equation}

\noindent
thus,

\begin{equation}
v_{\rm lc}(r) \approx \frac{r_{\rm t}}{r} 
\sqrt{ \frac{2G{\cal M}_{\bullet}}{r_{\rm t}}}.
\label{eq.lc_vel_h}
\end{equation}

If we use now the fact that

\begin{equation}
\sigma_{\rm r}(r)=\sigma_{\rm r}(r_{\rm t}) \cdot 
\bigg( \frac{r}{r_{\rm t}} \bigg)^{-1/2}=
\sqrt{ \frac{G{\cal M}_{\bullet}}{r_{\rm t}}} 
\cdot \bigg( \frac{r}{r_{\rm t}} \bigg)^{-1/2}
\label{eq.lc_vel_i}
\end{equation}

\begin{equation}
\Rightarrow \sqrt{ \frac{G{\cal M}_{\bullet}}{r_{\rm t}}} 
= \sigma_{\rm r}(r)\cdot \bigg( \frac{r}{r_{\rm t}} 
\bigg)^{-1/2}
\label{eq.lc_vel_j}
\end{equation}

\noindent
It is then in fair approximation 

\begin{equation}
v_{\rm lc}(r) \approx \frac {r_{\rm t}}{r} 
\sqrt{ \frac{2G{\cal M}_{\bullet}}{r_{\rm t}}} 
\approx \sigma_{\rm r}(r) 
\cdot \bigg( \frac{r_{\rm t}}{r} \bigg)^{1/2}.
\label{eq.lc_vel_k}
\end{equation}

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\bibliography{aamnem99,biblio_tesi}
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sms.tex

\chapter{Super-massive stars}
\label{ch.sms}

%\vspace{-11cm}
%\begin{quote}
%\begin{flushright}
%{{\tt blabla \\
%{\vspace{0.15cm}\em autor }}
%\end{flushright}
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%\vspace{11cm}


\section{On the nature and peculiarities of a super-massive gaseous object}

\inicial{A}{rising} from 1962, we find in the literature one of the first proposals
on this topic \footnote{Part of this chapter was used for
\citet{JustAS01}, \citet{ASEtAl02} and
\citet{JustAS03}}: Some $10^8$ supernov{\hskip-0.05cm}\ae~were thought to happen in
$10^6$ years; since they were also thought to be the normal galactic
source of cosmic rays, we would have found the origin for this strong
emission.  Nonetheless, this was a too strong requirement for
\citet{BB62}, who found the argument not convincing. He proposed the idea
that one supernova in a tightly packed set of neighbouring stars could
trigger explosions in these. If \citet{BB62} felt satisfied with
such an idea, F. Hoyle and W. A. Fowler could not say the same, for
they found it not convincing. To fulfil Burbidge's argument the stars
should be so close packed together that they would be {\em physically
in contact}. It was impossible for F. Hoyle and W. A. Fowler to
withstand the seduction of the simple idea that at the centres of
galaxies star-like objects exist with masses of up to $10^8\msol$
\citep{HF63}. As regards the stability and/or
existence of such an object, they ``turn a blind eye''. Actually, we
have to wait one year to find a deep study on this subject by
Chandrasekhar.

F. Hoyle and W. A. Fowler developed their argument with the argument that 
wholly convective stars could ``do the job'': Consider an ideal gas with 
radiation pressure:
\begin{equation}
P=P_{\rm gas}+P_{\rm r}=\frac{\Re}{\mu} \rho T+ \frac{a}{3}T^{4},
\end{equation}
\noindent
$P$, $\mu$, $\rho$, and $T$ being total pressure, mean molecular weight, 
density and temperature;
$\Re=k/m_u=8.315 \times 10^7 {\rm erg}/(Kg)$ is the universal gas constant,
$k=1.38 \times 10^{-16} {\rm erg}/K$ the Boltzmann constant, 
$m_u=1~{\rm amu}=1.66 \times 10^{-24}g$ the atomic gas unit  
and $a=7.565 \times 10^{-15}{\rm erg}/({\rm cm^3}{K^4})$ the radiation-density 
constant. We use here the gas constant with a dimension (energy per $K$ and
per {\em unit mass}); in thermodynamics we usually find that it is energy
per $K$ and per mole. Because of that, the molecular weight $\mu$ is
dimensionless (instead of $[ \mu ]={\rm mass}/{\rm mole}$).
Assume that $\beta = {\rm P_{\rm gas}}/{\rm P}$ is constant
throughout the whole star; thus, we can see in $1-\beta=aT^4/(3P)$ that 
$\beta = {\rm constant}$ implies the relation $T^4 \sim P$. Therefore,
 \begin{equation}
P=\bigg( \frac{3 \Re ^4}{a \mu ^4} \bigg)^{1/3} \bigg( \frac{1- \beta}
{\beta ^4} \bigg)^{1/3} \rho ^{4/3},
\end{equation}
which is a polytropic relation with a polytropic index $n=3$ for constant $\beta$.
The polytropic constant
 \begin{equation}
\mathrm {K}=\bigg( \frac{3 \Re ^4}{a \mu ^4} \bigg)^{1/3} \bigg( \frac{1- \beta}
{\beta ^4} \bigg)^{1/3}, 
\label{eq.pol_const}
\end{equation}
\noindent
is a free parameter since we can choose $\beta$ in the interval $[0,1]$. 

Eddington's quartic equation is
\begin{equation}
\frac{1- \beta}{\mu ^4 \beta ^4} = 3.02 \times 10^{-3} \bigg( \frac {M}
{\msol}\bigg)^2.
\end{equation}
\noindent
The left-hand side of the equality becomes smaller in the interval $[0,1]$
when $M$ increases. In other words, the radiation pressure becomes more 
important the larger the stellar mass is. This means that \smss~are dominated
by the radiation pressure. As a consequence, the adiabatic gradient 
$\nabla _{\rm ad}$ is reduced, (particularly $\nabla _{\rm ad} \to 1/4$ 
when $\beta \to 0$, see e.g. \citealt{KW94}) and the star becomes
convective with $\nabla=\nabla _{\rm ad}$. This adiabatic structure requires
constant specific entropy $s$. 
Thanks to the first law of thermodynamics, we have that
\begin{equation}
s= \frac{4aT^3}{3\rho}.
\end{equation}
\noindent
Constant specific entropy means $\rho \sim T^3$ and from the pressure equation
we can see that $P \sim \rho$; therefore $P \sim \rho^{4/3}$, which again means 
that \smss~are polytropes with $n=3$.

The \smss~are star polytropes with a free $\mathrm {K}$, which implies that their
mass $M$ can be chosen {\em arbitrarily}. For each mass, ${1- \beta}/({\mu ^4 \beta ^4})$
can be obtained and then, Eq.\,(\ref{eq.pol_const}) gives us the corresponding value of $\mathrm {K}$.
But if the mass is given, there still exist an infinite number of models for different
radius (in spite of the fact that $\mathrm {K}$ is already determined by $M$) 
since $\mathrm {K}$ is independent of $R$.
With a specific entropy being constant, we have that the adiabatic polytropic constant
$\gamma$ is roughly the local adiabatic index $\Gamma=[d({\rm ln}P)/d({\rm ln}\rho)]_{s}$.

The \sms~evolves radiating energy and entropy and, in the case of contemplating rotation, 
losses also angular momentum via mass shedding. Since the pressure is dominated by 
radiation, the luminosity of an \sms~is close to the Eddington limit,
\begin{equation}
{\cal L}=L_{\rm Edd}=\frac{4\pi Gc{\cal M}m_{\rm p}}{\sigma_{\rm Th}}=
3.4\times 10^4  \Big(\frac{\cal M}{\msol} \Big) L_{\odot},
\label{eq:sms_Ledd}
\end{equation}
\noindent
where ${\cal L}$ is the luminosity of the \sms~, $m_{\rm p}$ the proton mass and 
$\sigma_{\rm Th}$ the Thomson cross section.


\subsection{Nuclear energy source}

The most important nuclear energy source for \smss~is hydrogen
burning. We can distinguish between two regimes: For temperatures less
than $T \equiv 5 \times 10^8K$, hydrogen burns on the standard
$\beta$-limited CNO cycle \citep{Fowler65}. For temperatures greater than
the above-mentioned one, \citet{WW81} proved that leak
out of the $\beta$-limited CNO cycle can be very important, because of
the $^{15}O(\alpha,\gamma)^{19}Ne$ reaction. When we are out of the
cycle, we have a flow consisting of radiative proton captures and
positron decays which builds up toward the iron peak: this is the
so-called rp-process. \citet{WW81} proved that when
this process operates, it has 200 or 300 times the energy generation
rate of the $\beta$-limited CNO cycle. Nevertheless, for masses in
excess of a few $10^5\msol$, nuclear burning is irrelevant prior
to the gravitational instability when we compare the nuclear energy
generation rate with the photon luminosity (see
e.g. \citealt{Fowler66,ZN71}), and electron-positron
annihilation are not important \citep{NS01}; the creation of
electron-positron pairs can be neglected for $M \gtrsim 10^4\msol$
(i.e., $T \leq T_{\rm crit} \lesssim 2.5 \times 10^9 K$) \citep{BA84}.


\subsection{Instabilities of radiation-dominated stars}

The bulk of the \smss~is expected to be convective, thus the entropy
is roughly constant. For stars in which the specific entropy is nearly
constant, the adiabatic polytropic constant is approximately the local
adiabatic index \citep{Chandra39}.  The evolution of these objects is
a rather complicated subject, for these stars have always a pressure
averaged adiabatic index of by $4/3$. In order to keep hydrostatic
equilibrium, the central temperature must increase as the star mass
rises up. The ratio of gas pressure $P_{\rm g}$ to the total pressure
$P= P_{\rm r}+P_{\rm gas}$ (where $P_{\rm r}$ is the radiation
pressure) for an index $n=3$ polytrope is
\begin{equation}
\beta=\frac {P_{\rm g}}{P} \approx \frac {4.3}{\mu} \Bigg( 
\frac{\cal M}{\msol} \Bigg)^{-1/2},
\end{equation}
\noindent
where $\cal M$ is the mass of the \sms. For stars with a huge mass like these
objects, the gas pressure is a minor perturbation on the total pressure of 
the star, $P_{\rm g}$ can only be a minute fraction of the total pressure. 

The condition for hydrostatic equilibrium can be expressed as a variational
principle \citep{Chandra64,HF63}; the 
total energy of the star, which is a function of $\cal M$, entropy and 
central density is an extremum. We can thus get the equilibrium mass for a 
given entropy and central density by extremising this total energy. If we 
employ a low enough central density, the equilibrium energy will be almost 
zero, because of the $\gamma \sim 4/3$. The \sms~will quasi-statically contract,
radiating away entropy and increasing the central density while the equilibrium
energy decreases. Gravity will have a minimum due to general relativistic 
corrections, so that beyond a certain $\rho_{\rm crit}$, energy must be supplied
in order to get equilibrium. The \sms~radiates quasi-statically its entropy away
and shrinks to the point where the central density reaches the $\rho_{\rm crit}$
and instability sets in. At this point, the equilibrium energy is stationary, and
we have instability, for a second derivative of the total energy of the system 
is zero. Thus, the \sms~becomes dynamically unstable and collapse sets in.

The nature of the \smss~is such that small effects that are normally 
negligible in the study of stellar evolution
must here be included to determine the stability properties.

\begin{itemize} \addtolength{\itemsep}{-0.5\baselineskip}
\item general relativity
\item electron-positron pair formation
\item rotation
\item dissociation of heavy nuclei
\end{itemize}

In the framework of general relativity, dynamical instability happens 
before a gaseous mass shrinks to the limiting radius compatible with 
hydrostatic equilibrium. A star whose structure is determined almost 
completely by Newtonian gravitation turns out to be unstable because 
of general relativity. The instability in radiation dominated stars 
stems from the fact that $\gamma$ is nearly $4/3$; they ``are trembling
on the verge of instability'' \citep{Fowler64}. This was first 
suggested by \citet{Chandra64} and \citet{Feynman63} and first applied 
to stars by \citet{HF63}.

From \citet{Chandra64}, (post-Newtonian) instability sets in when the radius of 
the star $\cal R$ is less than a critical radius ${\cal R}^{\rm crit}$. 
He shows that if the ratio of specific heats $\gamma=C_{\rm p}/C_{\rm v}$ exceeds 
$4/3$ only by a small amount, then dynamical instability will 
occur if the mass contract to the radius
\begin{equation}
{\cal R}^{\rm crit}=\frac {\cal K}{\gamma - \frac{4}{3}} \Bigg(\frac{2G\cal{M
}}{c^2} \Bigg),
\end{equation}
where $\cal K$ is a constant depending on the density distribution in the 
configuration and the term between brackets is the Schwarzschild radius 
of the \sms. Chandrasekhar evaluates the $\cal K$ value for the homogeneous 
sphere of constant energy density and the polytropes of indices 
$n=1,~2,~3$.
The ratio of specific heats $\gamma$ can be shown to be in the limit 
of high entropy \citep{Chandra39},
\begin{equation}
\gamma \approx \frac{4}{3}+\frac {\beta}{6} + {\cal O}( \beta ^2),
\end{equation}
\noindent
to first order in $\beta$. This means that the radiation-dominated stars 
have a $\gamma$ close to the value $4/3$ and then they are on the limit 
of instability.
With the last formula and $n=3$ we can approximate the ${\cal R}^
{\rm crit}$ by
\begin{equation}
{\cal R}^{\rm crit} \approx \frac {6.8}{\beta} \Bigg(\frac{2G{\cal M}}
{c^2} \Bigg).
\end{equation}
The Schwarzschild radius of the \sms~ is:
\begin{equation}
\frac{2G{\cal M}}{c^2}=10^{-5} \Bigg( \frac {\cal M}{10^{8}\msol} 
\Bigg) {\rm pc},
\end{equation}
Then, with the $\beta$ formula, we can find out that
\begin{equation}
{\cal R}^{\rm crit}=0.16 \mu \Bigg( \frac 
{{\cal M}}{10^8\msol} \Bigg)^{3/2} {\rm pc}.
\end{equation}
On the other hand, the critical central density for a radiation- dominated,
high entropy, index $n=3$ polytrope is
\begin{equation}
\rho_{\rm crit} = 2 \times 10^{18} \bigg( \frac{1}{2\mu} \bigg)^3
\bigg( \frac{\cal M}{\msol} \bigg)^{-7/2} {\rm g/cm}^3,
\end{equation}
\noindent
\citep{FW86}. For a higher central density, the nonlinear effects
of gravity have a destabilising influence to radial perturbations. The
critical central temperature is
\begin{equation}
T_{\rm crit}=2.5 \times 10^{13} \bigg( \frac{\cal M}{\msol}\bigg)^{-1}K,
\end{equation}
\noindent
and the corresponding equilibrium energy is
\begin{equation}
E_{\rm crit}=-3.6 \times 10^{54} {\rm erg},
\end{equation}
\noindent
which is independent of ${\cal M}$.

\noindent

\section{Fencing in the existence zone of an \sms}

It is interesting to examine the time-scales of \smss~in order to know
whether or not they are on hydrostatic equilibrium, and so to have a
general overview of their possible evolution. The time scale associated with hydrodynamic processes in
radiation-dominated stars is
\begin{equation}
{\tau_{\rm hyd}} \equiv \frac {1}{\sqrt{24\pi G \rho}} \approx 2 \times 10^{-6}
\bigg( \frac{\cal M}{\msol} \bigg)^{7/4} {\rm s}.
\end{equation}
In order to see a hydrostatic evolution phase, this time scale must be
shorter than the modified Kelvin-Helmholtz time, also called ``cooling
time'' or ``contracting time'' for it is the time-scale which gives us
an approximated idea of how long can live the the star from its
gravitational energy. In the life of a star there are phases in which
this is the main or even the only stellar energy source, and so we say
that ``the star evolves on a $\tau_{\rm KH}$''. This time is given by
\begin{equation}
{\tau_{\rm KH}} \approx \frac {|E_{\rm min}|}{L_{\rm Edd}} \approx 3 \times
10^{16} \bigg( \frac {\cal M}{\msol} \bigg)^{-1} {\rm s},
\end{equation}
where $L_{ \rm Edd}$ is the Eddington luminosity and $E_{\rm min}$ is
the value of the total energy of the star at the relativistic or
electron-positron pair instability point \citep{ST83}. We find that
\begin{equation}
{\tau_{\rm hyd}} < {\tau_{\rm KH}} \iff {\cal M} \lesssim 10^8\msol
\end{equation} 
For instance, for a $10^8 \msol$ \sms, $\tau_{\rm dyn} \sim 6.33$
yrs and $\tau_{\rm KH} \sim 9.5$ yrs. For times which are comparable
to the time-scale we have to take into account the cooling of the
\sms, since it can determine its evolution. For short times compared
to the time-scale ($\tau \ll \tau_{\rm KH}$) we can forget about the
cooling when we want to study the state and/or the evolution of the
\sms. We have to evaluate which process is more effective (which
process has a shorter time-scale), either the cooling, the accretion
of mass onto the central object or the heating of the \sms~due to
un/confined stars. In such a case, the evolution will not be
determined by the cooling, but by the faster processes.

In the case that $\tau_{\rm KH}> \tau_{\rm hyd}$, the cooling is
slower than the setting of the hydrodynamical equilibrium and so we
get a quasi- hydrostatic \sms. At this point there are two completely
different aspects which should be studied in detail. The first of them
is the state of equilibrium. We have a hydrostatic equilibrium, we
consider that we have no cooling and $P_{\rm rad} \gg P_{\rm g}$. The
Schwarzschild criterion for dynamical stability says that a star with
$\nabla_{\rm rad} > \nabla_{\rm ad}$ (the late being = $4/3$, $n=3$
polytrope) will be unstable, supposing a homogeneous chemical
composition of the star, so that $\nabla_{\mu}=0$ and the small
perturbations will increase until we have a fully convective star. If
the equilibrium is stable, we have a contracting \sms, but we are
close to instability; on the other hand, if it is not stable after the
post-Newtonian instability, we have a free-fall collapse, because the
radiation pressure grows slower than the gravity force when it
contracts. The second aspect with which we have to deal is the
quasistationary evolution, which is dominated now by $\tau_{\rm KH}$
and by the slow cooling. Ditto, we have a slow cooling and contracting
\sms~provided that we get such a state of equilibrium after having
analysed the foregoing questions. To complete the study we have to
know whether the stationary state is stable and how can we describe
it. 

We can illustrate this situation thanks to the gravitational and total
energy ($E_{\rm gr}$, $E_{\rm tot}$), which are

\begin{eqnarray}
E_{\rm gr} &=& -\frac{3}{2}\frac{G{\cal M}^2}{\cal R} = -1.3\cdot 10^{57} \,{\cal M}_8^2 \,{\cal R}_{\rm pc}^{-1} {\rm erg}  \\
E_{\rm tot} &=& \frac{3}{2}\beta E_g = -1.5\cdot 10^{54} \,{\cal M}_8^{3/2} \,{\cal R}_{\rm pc}^{-1} {\rm erg}.
\label{eq:sms_E_gr_E_tot}
\end{eqnarray}

\noindent
In these equations, ${\cal R}_{\rm pc}$ is the radius of the \sms~ in pc and
${\cal M}_{8}$ its mass in units of $10^8\msol$. We have also used the
fact that

\begin{equation}
\beta = \frac{P_g}{P_g+P_{rad}} = 7.8\cdot 10^{-4} {\cal M}_8^{-1/2}
\label{eq:sms_beta_units_m8}
\end{equation}

As we already mentioned, the contraction rate of the \sms is
determined by the Eddington luminosity which, expressed in 
${\cal M}_{8}$ (see eq. \ref{eq:sms_Ledd}), is

\begin{equation}
{\cal L}_8 = 1.3\cdot 10^{46} {\cal M}_8 \frac{\rm erg}{\rm s}\
\label{eq:sms_L_edd_units_m8}
\end{equation}

\noindent
Thus, we have a contraction time (Kelvin-Helmholtz time)
\begin{equation}
\tau_{\rm KH} = \frac{-E_{tot}}{L_{Ed}} = 3.7 M_8^{1/2} R_{pc}^{-1} {\rm yr}
\end{equation}

To determine whether a \sms~ can exist or not (from a point of view of
the hydrostatic equilibrium), we just have to check for its radius in
a ${\cal M}$-${\cal R}$ plane. We can define the radius ${\cal R}_{\rm
ff}$ at the place where $\tau_{\rm KH}$ is equal to the free-fall
time. In fig.\,(\ref{fig.sms__radii_timescales}) the time-scale
$\tau_{\rm KH}$ in which an \sms evolves is shown, from right to left.
This gives us an up-limit to the limits in the plane for the \sms. As
for the low-limit, we have to take into account the condition imposed
by the post-Newtonian unstability ${\cal R}_{\rm pn}$ or go even
``lower'' and look for lower masses when nuclear burning at the centre
sets in, ${\cal R}_{\rm nb}$. The area where such an object can exist
(is stable) starts at ${\cal R}_{\rm ff}$ .When hydrogen burning at
the centre sets in at ${\cal R}_{\rm nb}$, or when Post-Newtonian
unstabilities triumph over thermal pressure stabilisation, at ${\cal
R}_{\rm pn}$, the \sms becomes unstable. For masses $\lesssim 5\,\cdot
10^4 \msol$, the electron-positron pair instability occurs. The friction 
time-scale is also plotted ($\tau_{\rm fric}=10^5,\,10^3,\,10^1$ yr
from right to left) for a typical star with $v_{\star}=1500$ km/s
(assuming that 1\% of the orbit crosses the inner part of the
\sms). For radii smaller than the coupling radius ${\cal R}_{\rm
coupl}$ such stars are effectively decelerated. Some typical values of
$\tau_{\rm fric}$ for solar type stars are also plotted.

\bfig
\C
\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_sms/sms_radii_timescales.eps}}
\caption{Characteristic
 radii and time-scales for an \sms. The Schwarzschild radius ${\cal
 R}_{\rm Schw}$ is far below the ``allowed'' region of an \sms. The
 evolutionary tracks are horizontal lines from right to left. The
 contraction time $\tau_{\rm KH}$ is also displayed in order to show
 that without re-heating the life time of the \sms~is at $2\cdot
 10^4$ yr before nuclear burning or the post-Newtonian collapse sets
 in (see text)
\label{fig.sms__radii_timescales}}
\EC
\efig

\section{Possible origin of a BH: sequence of \smss}

It may be enlightening to distinguish two competing processes to have
a first look of the evolution of this physical problem: The
contraction (or collapse with re-bounce) of the mixed core with stars
that are trapped by friction within the gas and slow down additionally
to build a new highly condensed stellar core. This core may re-heat
the gas or decouple to build a new core.  A core-halo interaction is
also possible. The loss-cone stars of the surrounding stellar system
heat the core and feed it by means of new trapped stars. This core
collapse can be conked for a while until the loss-cone is ``empty''
(not rigorously speaking, but for the practice we can assume it is
empty) or the core becomes too massive. 

The friction force of the stars moving through the \sms~has a
dynamical and a geometrical component. For \smss~more compact than the
coupling radius ${\cal R}_{\rm coupl}$, green line in
Fig.\,(\ref{fig.sms__radii_timescales}), the typical friction
time-scale is shorter than the contraction time of the \sms. Thus,
stars are effectively decelerated by dynamical friction. As a result,
loss-cone stars will be segregated to the centre building a highly
concentrated sub-system, which may essentially change the evolution of
the \sms by enhanced interactions.

The result may be a sequence of \sms~-stars cores contained one inside
the other until the relatively low mass innermost \sms~ initiates
hydrogen burning or collapses to a BH ``seed''. In
Fig.\,(\ref{fig.sms_seq}) we give an intuitive scheme for this
scenario.

\bfig 
\C
\resizebox{0.8\hsize}{!}{\includegraphics
	{diag_sms/sms_cores.eps}}
\caption{Sequence of \sms -stars cores: {\sms $_{2}$}\,$\in$\,
{\sms $_{1}$}\,$\in$\,{\sms $_{0}$}. The non-confined stars which
belong to the loss-cone sink on to the central object after a number
of crossings, which we call the ``trap'' number. The innermost
\sms~may collapse to a BH seed
  \label{fig.sms_seq} }
\EC
\efig

\section{Stabilisation theory}

\subsection{The role of rotation}

Rotation is likely to play a decisive role in the quasistationary
evolution of an \sms, as well as in its final collapse. Since
\smss~are on the verge of stability, rotation can prolongate their
equilibrium evolution.

The gravitational instability sets in when $\Gamma_{\rm SMS}$ yields a
value less than a critical value, which is roughly
\begin{equation}
\Gamma_{\rm SMS}< \Gamma_{\rm crit} \approx \frac{2}{3} \frac{2-5\eta}{1-2\eta}
+1.12\frac{{\cal R}_{\rm Schw}}{\cal R},
\end{equation}
\noindent
where ${\cal R}_{\rm Schw}=2G{\cal M}/c^2$ is the Schwarzschild radius
of the star and $\eta=T/|W|$ the ratio of the rotational energy to the
gravitational potential energy. For $\eta=0$, the last equation
becomes $ \Gamma_{\rm crit}= {4}/{3}+ 1.12{{\cal R}_{\rm Schw}}/{\cal
R}$ (\citealt{MTW73} and references therein).  The adiabatic index
grows thus as the radius of the star becomes smaller. Rotation has a
stabilising effect and can hold the collapse if
\begin{equation}
\eta>\eta_{\rm crit} \approx \frac{1}{2}\frac{4-3(\Gamma_{\rm SMS}-1.12
{{\cal R}_{\rm Schw}}/{\cal R})}{5-3(\Gamma_{\rm SMS}-1.12
{{\cal R}_{\rm Schw}}/{\cal R})}
\end{equation}
\citet{BS99a,BS99b} study the effects of rotation on a
\sms~and follow the evolution of these objects up to the dynamical
instability and collapse.  They find out that ratios like ${\cal
R}/{{\cal R}_{\rm Schw}}$, $\cal R/ \cal M$, $T/|W|$ and $Jc/(G{\cal
M}^2)$ ($T$ is the rotational kinetic energy, $W$ the gravitational
potential energy and $J$ the angular momentum) for a maximally and
rigidly rotating $n=3$ polytrope at the onset of radial instability
are universal numbers, ``key ratios'', as they denominate them, which
are independent of the mass, spin, radius or even the history of the
star. They are thus universal constants:
\begin{equation}
\bigg( \frac{T}{|W|}\bigg)_{\rm crit} \approx 0.009,
~\bigg( \frac{Jc}{G {\cal M}^2}\bigg)_{\rm crit} \approx 0.97,
~\bigg( \frac{{\cal R}_{\rm p}}{{\cal R}_{\rm Schw}}\bigg)_{\rm crit} \approx 214,
\end{equation} 
\noindent
with the polar radius ${\cal R}_{\rm p}\approx 2 {\cal R}_{e}/3$ (${\cal R}_{e}$ stands
for the equatorial radius). This deformation could be responsible for the reduction
of the luminosity by about 36\% below the usual Eddington luminosity 
from a non-rotating \sms~ \citep{BS99a,BS99b}:
\begin{equation}
L_{\rm rot}=0.639L_{\rm Edd}.
\end{equation}
Since luminosity determines the timescale (i.e. the mean life) of
these objects, rotation seems to be a factor whose importance should
be seriously taken into account.

They perform a fully relativistic, numerical calculation and compare
it with an analytical treatment, which is in good agreement. They
claim that cooling by photon radiation drives the evolution, while the
rotating \sms~loses mass, angular momentum and entropy. Gas pressure
may stabilise \smss~in the absence of angular momentum
\citep{ZN71,ST83}, but Baumgarte and Shapiro in their paper maintain
that even a small degree of rotation could dominate gas pressure.
They analyse also the final collapse of rotating \smss, which might
not necessary be a BH.  They reckon that the collapse may be
inhibited, in the form of a disc and/or bar, or possible fragmenting
into several blobs.

The study of the importance of a dark matter background on the
stability of an \sms~ has also been done, with the conclusion that
although it has stabilising effects, it can be neglected compared with
the effects of rotation \citep{BK98}.

Nevertheless, all the work being performed as regards the stability of
\smss~ in the case that such objects would rotate, and other topics,
like the equilibrium and stability of \smss~ in binary systems
 assume that the system is an isolated one; namely,
{\em the accretion and heating are assumed to be negligible}.



\subsection{Stabilisation by fluctuations}
The density, temperature and velocity fluctuations excited by the stellar
component in the gas \citep{JustEtAl86,DeissEtAl90} do not only
decelerate the stars by dynamical friction, but there is also a stabilising
effect on the gas component, which can be described by an effective
$\overline{\beta}$.
The diffusion of photons is much faster than the dynamical time scale in the
\sms. Since the temperature is determined by the radiation field, the
perturabations are isothermal as long as the matter is optically
thick. For scales smaller than the mean free path
$\lambda=3.3\,10^{-6} M_8^{-1}R_{\rm pc}^3\,\rho_c/\rho$ pc, the
density fluctuations are adiabatic leading to an enhanced mean thermal
pressure. The basic corrections to the unperturbed parameter $\beta_0$
are
\be
\overline{\beta} = \beta_0(1+K\frac{\langle\rho_1^2\rangle}{\rho_0^2})
   + \beta_{\rm turb} > \beta_0
\ee
where $\beta_{\rm turb}$ gives the ratio of turbulent to radiation
pressure and $\langle\rho_=^2\rangle$ is the unperturbed density. The
scaling constant $K$ and the fluctuations' density
$\langle\rho_1^2\rangle$ must be determined by some fluctuation
theory.  Using a quasilinear approximation of a mode analysis the
perturbations excited by the stars moving through the gas will be
estimated to determine the magnitude of the corrections due to the
turbulent velocity field and the adiabatic density fluctuations on the
stability parameter $\beta_0$. An increase of the effective
$\overline{\beta}$ will shift the line for the Post-Newtonian
instability to higher masses.

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tesi.tex

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th_model.tex
\chapter{The theoretical model}
\label{ch.th_model}

%\vspace{-11cm}
%\begin{quote}
%\begin{flushright}
%{{\tt blabla \\
%{\vspace{0.15cm}\em autor }}
%\end{flushright}
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%\vspace{11cm}


\section{Introduction}

\inicial{F}{undamentals} and principles of the anisotropic 
gaseous model are presented in this chapter\footnote{At the beginnig
of each chapter I will give references of published work related to
the contents of the chapter. Typically, I will firstly mention the
refereed article (if any) and, secondly, contributions to conferences
({\em proceedings}). As for the chapter under discussion, some
sections were used for \citet{AS01a}, \citet{AS04} and
\citet{SpurzemEtAl03}} so as to give the description of the numerical
approach employed to model stellar clusters, as we shall see in next
chapters. We start with a brief description of the mathematical basis
which can be regarded as a short summary of the description of kinetic
theory in chapter eight of
\citet{BT87}. The reader interested in an in-depth study is addressed
to it. On the other hand, we stress on the peculiarities of the
approximation we make use of, the so-called {local approximation}, so
as to be aware of the restrictions of our theory and emphasise them.
Later, we will give the set of equations describing the system, the
interaction terms to be taken into consideration to represent
realistic models, like mass exchange by mass loss, heating, loss-cone,
exchange of kinetic energy between the stellar system and the
interstellar gas via drag forces, etc, to be included in the
right-hand terms of this set. Ensuingly we shall make a thoughtfully
depiction of the gaseous component in the cluster and their own
interaction terms.

\section{The {Fokker-Planck} equation}

The state of a system of $N$ particles with velocities 
${\bf v} =({\bf v}_i)$ and positions
${\bf x} =({\bf x}_i)$ ($i=1,\ldots N$) 
is represented by a point in $6N$-dimensional
$\Gamma$-space of positional and velocity variables. The
$N$-particle distribution $f^{(N)}({\bf x},{\bf v}, t)$ is defined as 
the probability to find the system in the volume element 
$d^3\!{\bf v}d^3\!{\bf x}$ around ${\bf x}$ and ${\bf v}$ at time $t$. 
$f^{(N)}$ is normalised as $ 1 = \int f^{(N)} d^3\!{\bf v}d^3\!{\bf x} $
The evolution of the system is a trajectory in $\Gamma$-space; the
evolution of a continuous subspace of systems at $t=t_0$ (initial conditions) 
can be seen as a flow in $\Gamma$-space, which due to the absence of
any dissipation is incompressible, and thus described by a
continuity equation in $\Gamma$-space, which is Liouville's equation:
\begin{equation}
\frac{\partial f^{(N)}}{\partial t} + \sum_{i=1}^N \Bigl\{
\frac{\partial}{\partial {\bf x}_i}
\Bigl[ f^{(N)} {d{\bf x}_i\over dt} \Bigr] + \frac{\partial}{\partial{\bf v}_i}
\Bigl[ f^{(N)} {d{\bf v}_i\over dt} \Bigr] \Bigr\} = 0  .
\end{equation}
Using the definition $d{\bf x}_i/dt = {\bf v}_i$ and 
$\partial{\bf v}_i/\partial{\bf x_i} = 0$, since ${\bf v}_i$ and
${\bf x}_i$ are independent coordinates, and if the forces are
conservative, $d{\bf v}_i/dt = -\partial\Phi_i/\partial{\bf x}_i$,
where $\Phi_i$ is the potential at the position of particle $i$ due
to the other particles, we can simplify last equation to
\begin{equation}
\frac{\partial{f^{(N)}}}{\partial{t}} + \sum_{i=1}^N \Bigl[ {\bf v}_i \cdot
 \frac{\partial{f^{(N)}}}{{\partial{\bf x}_i}} - \frac{\partial{\Phi_i}}{{\partial{\bf x}_i}} \cdot 
 \frac{\partial{f^{(N)}}}{{\partial{\bf v}_i}} \Bigr] = 0 . 
\end{equation}
Here it has been utilised that $d{\bf v}_i/dt$ does not depend on
${\bf v}_i$ itself, since the potential only depends on the
spatial coordinates. 

Now we introduce the one-particle distribution function 
$f^{(1)}({\bf x},{\bf v}, t)$ as 
\begin{equation}
f^{(1)}({\bf x},{\bf v}, t) = \int f^{(N)}({\bf x}_i,{\bf v}_i,t) 
d^3\!{\bf x_2}\ldots d^3\!{\bf x_N}d^3\!{\bf v_2}\ldots d^3\!{\bf v_N}  ,
\end{equation}
the two-particle distribution function
\begin{equation}
f^{(2)}({\bf x}_1,{\bf x}_2,{\bf v}_1.{\bf v}_2,t) = 
  \int f^{(N)}({\bf x}_i,{\bf v}_i,t)
  d^3\!{\bf x}_3\ldots d^3\!{\bf x}_Nd^3\!{\bf v}_3\ldots d^3\!{\bf v}_N \ 
\end{equation}
and the two-particle correlation function $g$ by
\begin{equation}
g({\bf x}_1,{\bf x}_2,{\bf v}_1.{\bf v}_2,t) = 
  f^{(2)}({\bf x}_1,{\bf x}_2,{\bf v}_1.{\bf v}_2,t) -
  f^{(1)}({\bf x}_1,{\bf v}_1,t)
  f^{(1)}({\bf x}_2,{\bf v}_2,t)
\end{equation}
$g$ measures the excess probability of finding a particle at 
${\bf x}_1$, ${\bf v}_1$ due to the presence of another particle
at ${\bf x}_2$, ${\bf v}_2$.
Since $f^{(1)}$ is normalised to
unity, one has $\rho({\bf x}_2,t) = mN \int f^{(1)}({\bf x}_2,{\bf v}_2,t)d{\bf v}_2$,
where $\rho$ is a mean mass density, and $m$ the individual stellar mass.
Assuming that $f^{(N)}$ is symmetric with respect to exchange of
particles (i.e. all particles are indistinguishable), and observing that 
$\Phi_1$ is also symmetric with respect to exchanges of the particles $2,\ldots, 
N$, one arrives at
\begin{equation}
\frac{\partial f^{(1)}}{\partial {t}} + {\bf v}_1 \cdot \frac{\partial f^{(1)}}
{\partial {\bf x}_1} -
{N\!-\!1\over N}\frac{\partial {\Phi_1}}{\partial {{\bf x}_1}}\cdot 
\frac{\partial {f^{(1)}}}{\partial {{\bf v}_1}} = \nonumber 
\end{equation}
\begin{equation}
-Gm(N\!-\!1) \int \frac{\partial}{\partial {\bf v}_1} \Bigl(
 g({\bf x}_1,{\bf x}_2,{\bf v}_1.{\bf v}_2,t) \Bigr)
 \frac{\partial {}}{\partial{{\bf x}_1}} \Bigl(\vert{\bf x}_1-
{\bf x}_2\vert^{-1}\Bigr)d^3\!{\bf x_2}d^3\!{\bf v_2} .
\end{equation}
Now one substitutes $f^{(1)}$ by the more common phase space density
$f = f^{(1)}/N$ and drops for simplicity all subscripts ``1''. It
follows
\begin{equation}
\frac{\partial{f}}{\partial{t}} + {\bf v} \cdot \frac{\partial{f}}
{\partial{{\bf x}}} -{N\!-\!1\over N} \frac{\partial{\Phi}}{\partial{{\bf x}}}
\cdot \frac{\partial{f}}{\partial{{\bf v}}} = \bigg(\frac{\delta f}{\delta t} 
\bigg)_{\rm enc}
\end{equation}

\noindent
If the average particle distance $\bar{d}=1/n^{1/3}$ is bigger than
the impact parameter $p_{90}$ related to a $90^o$ deflection,
most of the scattering is due to small angle encounters, which change
velocity and position of the particle only weakly. So, if we assume that 
all correlations stem from gravitational two-body scatterings of 
particles, not from higher order correlations, we have that the {Fokker-Planck}
equation is

\begin{equation}
\bigg(\frac{\delta f}{\delta t} 
\bigg)_{\rm enc}=
  -\sum_{i=1}^3 \Biggl[ \frac{\partial{}}{\partial{x_i}}\Bigl(
        f({\bf x},{\bf v})
        D(\Delta x_i)\Bigr) 
      + \frac{\partial{}}{\partial{v_i}}\Bigl(
        f({\bf x},{\bf v})
       D(\Delta v_i)\Bigr) \Biggr] 
\nonumber
\end{equation}
\begin{equation}
    + {1\over 2} \sum_{i,j=1}^3 \Biggl[ 
       {\partial^2\over\partial x_i\partial x_j}\Bigl(
        f({\bf x},{\bf v})
       D(\Delta x_i\Delta x_j)\Bigr)  
     + {\partial^2\over\partial v_i\partial v_j}\Bigl(
        f({\bf x},{\bf v})
       D(\Delta v_i\Delta v_j)\Bigr) 
\nonumber
\end{equation}
\begin{equation}
    +  {\partial^2\over\partial x_i\partial v_j}\Bigl(
        f({\bf x},{\bf v})
       D(\Delta x_i\Delta v_j)\Bigr)
     + {\partial^2\over\partial v_i\partial x_j}\Bigl(
        f({\bf x},{\bf v})
       D(\Delta v_i\Delta x_j)\Bigr)
\Biggr] \
\end{equation}
Here the convenient notation of diffusion coefficients has been
introduced, which contain the integration over the velocity
and position changes, as e.g.:
\begin{equation}
D(\Delta v_i) := \int \Delta v_i 
     \Psi({\bf x},{\bf v},{ \Delta \bf x},{ \Delta \bf v})
     d^3\!{\ \Delta \bf  x}d^3\!{\Delta \bf v},
\end{equation}
where $\Psi({\bf x},{\bf v}, {\Delta \bf x},{\Delta \bf v})$ $d^3\!
{\Delta \bf x}d^3\!{\Delta \bf v}dt$
is defined
as the probability for a star with position ${\bf x}$ and velocity 
${\bf v}$
to be scattered into a new phase space volume element 
$d^3\!{\Delta \bf x}$ $d^3\!{\Delta \bf v}$ 
located around ${\bf x}+{\Delta \bf x}$,
${\bf v}+{\Delta \bf v}$ during the time interval $dt$.

\section{The {local approximation}}

There are two alternative ways of further simplification. One is the
orbit average, which uses that any distribution function, being a
steady state solution of the collisionless Boltzmann equation, can be
expressed as a function of the constants of motion of an individual
particle (Jeans' theorem). For the sake of simplicity, it is assumed
that all orbits in the system are regular, as it is the case for
example in a spherically symmetric potential; thus the distribution
function $f$ now only depends on maximally three independent integrals
of motion (strong Jeans' theorem). Let us transform the {Fokker-Planck}
equation to a new set of variables, which comprise the constants of
motion instead of the velocities $v_i$.  Since in a spherically
symmetric system the distribution only depends on energy and the
modulus of the angular momentum vector $J$, the number of independent
coordinates in this example can be reduced from six to two, and all
terms in the transformed equation (8) containing derivatives to other
variables than $E$ and $J$ vanish (in particular those containing
derivatives to the spatial coordinates $x_i$).  Integrating the
remaining parts of the {Fokker-Planck} equation over the spatial
coordinates is called orbit averaging, because in our present example
(a spherical system) it would be an integration over accessible
coordinate space for given $E$ and $J$ (which is a spherical shell
between $r_{\rm min}(E,J)$ and $r_{\rm max}(E,J)$, the minimum and
maximum radius for stars with energy $E$ and angular momentum $J$).
Such volume integration is, since $f$ does not depend any more on
$x_i$ carried over to the diffusion coefficients $D$, which become
orbit-averaged diffusion coefficients.

Orbit-averaged {Fokker-Planck} models treat very well the diffusion of
orbits according to the changes of their constants of motion, taking
into account the potential and the orbital structure of the system in
a self-consistent way. However, they are not free of any problems or
approximations.  They require checks and tests, for example by
comparisons with other methods, like the one described in the
following.


We treat relaxation like the addition of a big non-correlated number
of two-body encounters. Close encounters are rare and thus we admit
that each encounter produces a very small deflection angle. Thence,
relaxation can be regarded as a diffusion process \footnote{Anyhow, it
  has been argued that rare deflections with a large angle may play a
  important role in the vicinity of a BH \citep{LT80}.}.

A typical two-body encounter in a large stellar system takes place in
a volume whose linear dimensions are small compared to other typical
radii of the system (total system dimension, or scaling radii of
changes in density or velocity dispersion). Consequently, it is
assumed that an encounter only changes the velocity, not the position
of a particle.  Thenceforth, encounters do not produce any changes
${\Delta \bf x}$, so all related terms in the Fokker-Planck equation
vanish. However, the local approximation goes even further and assumes
that the entire cumulative effect of all encounters on a test particle
can approximately be calculated as if the particle were surrounded by
a very big homogeneous system with the local distribution function
(density, velocity dispersions) everywhere. We are left with a
Fokker-Planck equation containing only derivatives with respect to the
velocity variables, but still depending on the spatial coordinates (a
local Fokker-Planck equation). 

In practical astrophysical applications, the diffusion coefficients
occurring in the {Fokker-Planck} equation are not directly calculated, 
containing the probability $\Psi$ for a velocity change
${\Delta \bf v}$ from an initial velocity ${\bf v}$. Since $D(\Delta
v_i)$, and $D(\Delta v_i\Delta v_j)$ are of the dimension velocity
(change) per time unit, and squared velocity (change) per time unit,
respectively, one calculates such velocity changes in a more direct
way, considering a test star moving in a homogeneous sea of field
stars. Let the test star have a velocity ${\bf v}$ and consider an
encounter with a field star of velocity ${\bf v}_f$. The result of the
encounter (i.e. velocity changes $\Delta v_i$ of the test star) is
completely determined by the impact parameter $p$ and the relative
velocity at infinity $v_{\rm rel} = \vert{\bf v} - {\bf v}_f\vert $;
thus by an integration of the type
\begin{equation}
<\!\Delta {\dot v}_i\!>_p = 2\pi\int (\Delta v_i) 
 \,v_{\rm rel}\, n_f p dp ,
\end{equation}
the rate of change of the test star 
velocity due to encounters with $v_{\rm rel}$, in field of stars with
particle density $n_f$,
averaged over all relevant impact parameters is computed. The
integration is normally carried out from $p_0$ (impact
parameter for $90^o$ deflection) until $R$, which is some
maximum linear dimension of the system under consideration. Such
integration generates
in subsequent equations the so-called Coulomb logarithm $\ln \Lambda $;
we will argue later that it can be well approximated by 
$\ln (0.11 N)$, where $N$ is the total particle number.
The diffusion
coefficient finally is
\begin{equation}
D(\Delta v_i) = \int <\!\Delta {\dot v}_i\!>_p 
                 f({\bf v}_f) d^3\!{\bf v}_f, 
\end{equation}
where $f({\bf v}_f)$ is the velocity distribution of the field stars.
In an equal mass system, $f({\bf v}_f)$ should be equal to the distribution
function of the test stars occurring in the
{Fokker-Planck} equation for self-consistency.
In case of a multi-mass system, however, $f({\bf v}_f)$ could be
different from the test-star distribution, if the diffusion coefficient
arising from encounters between two different species of stars is
to be calculated.
The diffusion coefficients are (for an exact procedure see 
Appendix 8.A of \citealt{BT87}):
\begin{eqnarray}
&D(\Delta v_i) =& 4\pi G^2 m_f \ln\Lambda \frac{\partial{}}{\partial {v_i}}h({\bf v}) \nonumber \\
&D(\Delta v_iv_j) =& 4\pi G^2 m_f \ln\Lambda \frac{{\partial^2}}{ \partial v_i \partial v_j} g({\bf v})  
\end{eqnarray}
with the Rosenbluth potentials \citet{RosenbluthEtAl57}
\begin{eqnarray}
& h({\bf v}) =& (m+m_f)
 \int {f({\bf v}_f)\over\vert{\bf v}-{\bf v}_f\vert}
 d^3\!{\bf v}_f  \nonumber \\
& g({\bf v}) =& m_f \int f({\bf v}_f) \vert{\bf v}-{\bf v}_f\vert
 d^3\!{\bf v}_f \ . 
\end{eqnarray}
With these results we can finally write down the local {Fokker-Planck}
equation in its standard form 
for the Cartesian coordinate system of the $v_i$:
\begin{equation}
\bigg(\frac{\delta f}{\delta t} 
\bigg)_{\rm enc}= -4\pi G^2 m_f \ln\Lambda \Biggl[
     \sum_{i=1}^3 \frac{\partial{}}{\partial {v_i}}\Bigl(
        f({\bf v}) \frac{\partial{h}}{\partial {v_i}} \Bigr)
    + {1\over 2} \sum_{i,j=1}^3
       \frac{{\partial^2}}{\partial v_i\partial v_j}\Bigl(
        f({\bf v})
       {\partial^2 g \over\partial v_i\partial v_j}
       \Bigr)\Biggr]  
\end{equation}
Note that in \citet{RosenbluthEtAl57} the above equation
is given in a covariant notation, which allows for a straightforward
transformation into other curvilinear coordinate systems.

Before going ahead the question is raised, why such approximation
can be reasonable, regarding the long-range gravitational force,
and the impossibility to shield gravitational forces as in the
case of Coulomb forces in a plasma by opposite charges. The key is
that logarithmic intervals in impact parameter $p$ contribute equally
to the mean square velocity change of a test particle,
provided $p\gg p_0$
(see e.g. \citealt{Spitzer87}, chapter 2.1). Imagine that
on one hand side the lower limit of impact parameters ($p_0$, the
$90^o$ deflection angle impact parameter) is small compared to
the mean interparticle distance $d$. Let on the other hand side
$D$ be a typical radius connected with a change in density or
velocity dispersions (e.g. the scale height in a disc of a galaxy),
and $R$ be the maximum total dimension of the system. Just to
be specific let us assume $D=100d$, and $R=100D$. In that case
the volume of the spherical shell with radius between $D$ and $R$
is $10^6$ times larger than the volume of the shell defined by
the radii $d$ and $D$. Nevertheless the contribution of both shells
to diffusion coefficients or the relaxation time is approximately
equal. This is a heuristic illustration why the {local approximation}
is not so bad; the reason is with other words that there are a lot
more encounters with particles in the outer, larger shell, but the
effect is exactly compensated by the larger deflection angle for
encounters happening with particles from the inner shell.
If we are in the core or in the plane of a galactic disc the density
would fall off further out, so the actual error will be smaller than
outlined in the above example. By the same reasoning one can 
see, however, that the {local approximation} for a particle in a
low-density region, which suffers from relaxation by a nearby
density concentration, is prone to failure. 

These rough handy examples should illustrate that under certain
conditions the {local approximation} is not a priori bad. 
On the other hand, it is obvious from
our above arguments, that if we are interested in relaxation effects
on particles in a low-density environment, whose orbit occasionally
passes distant, high-density regions, the {local approximation} could
be completely wrong. One might think here for example of stars on
radially elongated orbits in the halo of globular clusters or
of stars, globular clusters, or other objects
as massive black holes, on spherical orbits in the galactic
halo, passing the galactic disc. In these situations an orbit-averaged
treatment seems much more appropriate. 

\section{A numerical anisotropic model}

In this section we introduce the fundamentals of the numerical
method we use to model our system. We give a brief description of the
mathematical basis of it and the physical idea behind it. The system
is treated as a continuum, which is only adequate for a large number
of stars and in well populated regions of the phase space. We consider
here spherical symmetry and single-mass stars. We handle relaxation in
the Fokker-Planck approximation, i.e. like a diffusive process
determined by local conditions. We make also use of the hydrodynamical
approximation; that is to say, only local moments of the velocity
dispersion are considered, not the full orbital structure. In
particular, the effect of the two-body relaxation can be modelled by a
local heat flux equation with an appropriately tailored conductivity.
Neither binaries nor stellar evolution are included at the presented
work. As for the hypothesis concerning the BH, see section
({\ref{sec.diff_model_th}}). 

For our description we use polar coordinates, $r$ $\theta $, $\phi$.
The vector ${\bf v} = (v_i), i=r,\theta,\phi $ denotes the velocity in
a local Cartesian coordinate system at the spatial point
$r,\theta,\phi$. For succinctness, we shall employ the notation
$u=v_{\rm r}$, $v=v_\theta$, $w=v_\phi$. The distribution function $f$, is a
function of $r$, $t$, $u$, $v^2+w^2$ only due to spherical symmetry,
and is normalised according to

\be
\rho(r,t) = \int f(r,u,v^2+w^2,t) du\,dv\,dw.  
\label{eq.rho(r,t)}
\ee

\noindent
Here $\rho(r,t)$ is the mass density; if $m_{\star}$ denotes the
stellar mass, we get the particle density $n=\rho/m_{\star}$.  The
Euler-Lagrange equations of motion corresponding to the Lagrange
function

\be
{\cal L} = {1\over 2}\bigl({\dot r}^2 + r^2{\dot\theta}^2 +
         r^2 \sin^2\!\!\theta\, {\dot\phi}^2\bigr) - \Phi(r,t)
\label{eq.lagrange}
\ee

\noindent
are the following
\begin{eqnarray}
&{\dot u} =& - \frac{\partial \Phi}{\partial r} 
+ {v^2\!+\!w^2\over r} \nonumber \\
&{\dot v}  =& - {uv\over r} + {w^2\over r\tan\theta} \\
\label{eq.u_v_w_dot}
&{\dot w}  =& - {uw\over r} - {vw\over r\tan\theta} \nonumber 
\end{eqnarray}

\noindent
And so we get a complete local Fokker-Planck equation, 

\be
\frac{\partial f}{\partial t}+v_{\rm r}\frac{\partial f}{\partial r}+\dot{v_{\rm r}}
\frac{\partial f}{\partial v_{\rm r}}+\dot{v_{\theta}}\frac{\partial f}{\partial 
v_{\theta}}+\dot{v_{\varphi}}\frac{\partial f}{\partial v_{\varphi}}=\Bigg
( \frac{\delta f}{\delta t} \Bigg)_{FP}
\label{eq.FP}
\ee

In our model we do not solve the equation directly; we use a so-called
{\em momenta process}. The momenta of the velocity distribution
function $f$ are defined as follows 

\be
<i,j,k>:=\int^{+\infty}_{-\infty} v^{i}_{\rm r} v^{j}_{\theta} v^{k}_{\phi} 
\,f(r, v_{\rm r}, v_{\theta},v_{\phi},t)\,dv_{\rm r}dv_{\theta}dv_{\phi};
\label{eq.ijk}
\ee

\noindent
We define now the following moments of the velocity distribution function,

\begin{eqnarray}
& <0,0,0> := & \rho = \int f dudvdw \nonumber \\
& <1,0,0> := & {u} = \int uf dudvdw \nonumber \\
& <2,0,0> := & p_{\rm r} + \rho {u}^2 = \int u^2 f dudvdw \nonumber \\
& <0,2,0> := & p_\theta = \int v^2 f dudvdw
\label{eq.vel_momenta} \\
& <0,0,2> := & p_\phi = \int w^2 f dudvdw \nonumber \\
& <3,0,0> := & F_{\rm r} + 3{u}p_{\rm r} + {u}^3 = \int u^3 f dudvdw \nonumber \\
& <1,2,0> := & F_\theta + {u}p_\theta = \int uv^2 f dudvdw \nonumber \\
& <1,0,2> := & F_\phi + {u}p_\phi = \int uw^2 f dudvdw, \nonumber
\end{eqnarray}

\noindent
where $\rho$ is the density of stars, $u$ is the bulk velocity,
$v_{\rm r}$ and $v_{\rm t}$ are the radial and tangential flux velocities,
$p_{\rm r}$ and $p_{\rm t}$ are the radial and tangential pressures, $F_{\rm r}$
is the radial and $F_{\rm t}$ the tangential kinetic energy flux
\citep{LS91}. Note that the definitions of $p_i$ and $F_i$ are such
that they are proportional to the random motion of the stars. Due to
spherical symmetry, we have $p_{\theta} = p_{\phi }=: p_{\rm t}$ and
$F_{\theta} = F_{\phi} =: F_{\rm t}/2$. By $p_{\rm r} = \rho\sigma_{\rm r}^2$ and
$p{_{\rm t}} = \rho\sigma_{\rm t}^2$ the random velocity dispersions are given,
which are closely related to observable properties in stellar
clusters.

\noindent
$F = (F_{\rm r} + F_{\rm t})/2$ is a radial flux of random kinetic energy. In the
notion of gas dynamics it is just an energy flux. Whereas for the
$\theta-$ and $\phi-$ components in the set of Eqs.
(\ref{eq.vel_momenta}) are equal in spherical symmetry, for the $r$
and $t$- quantities this is not true. In stellar clusters the
relaxation time is larger than the dynamical time and so any possible
difference between $p_{\rm r}$ and $p_{\rm t}$ may {\em survive} many dynamical
times. We shall denote such differences anisotropy. Let us define the
following velocities of energy transport:

\begin{eqnarray}
&v_{\rm r} & = {F_{\rm r} \over 3 p_{\rm r}} + u, 
\label {eq.vel_energy_trans}\\
&v_{\rm t} & = {F_{\rm t} \over 2 p_{\rm t}} + u. \nonumber
\end{eqnarray}

\noindent
In case of {\it weak} isotropy ($p_{\rm r}$=$p_{\rm t}$) $2F_{\rm r}$ = $3F_{\rm t}$, and thus
$v_{\rm r}$ = $v_{\rm t}$, i.e. the (radial) transport velocities of radial and
tangential random kinetic energy are equal.

The Fokker-Planck equation (\ref{eq.FP}) is multiplicated with various
powers of the velocity components $u$, $v$, $w$. We get so up to
second order a set of moment equations: A mass equation, a continuity
equation, an Euler equation (force) and radial and tangential energy
equations. The system of equations is closed by a phenomenological
heat flux equation for the flux of radial and tangential RMS ({\em
  root mean square}) kinetic energy, both in radial direction. The
concept is physically similar to that of \citet{LBE80}. The set of
equations is

\begin{eqnarray*}
\frac{\partial{\rho}}{\partial t} + \frac{1}{r^2}\frac{\partial}
{\partial r} (r^2u\,\rho)= 0
\end{eqnarray*}

\begin{eqnarray*}
\frac{\partial u}{\partial t}+u\,\frac{\partial u}{\partial r} + 
{GM_{\rm r}\over r^2} +
{1\over\rho}\frac{\partial p_{\rm r}}{\partial r} + 2\,\frac{p_{\rm r} - p_{\rm t}}{\rho\, r} 
= 0
\end{eqnarray*}

\begin{eqnarray}
\lefteqn{
\frac{\partial{p_{\rm r}}}{\partial {t}} + \frac{1}{r^2} \frac{\partial}
{\partial r} 
(r^2 u \,p_{\rm r})+2 \,p_{\rm r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 F_{\rm r}){}} \label{eq.set_of_eqs} \\
& & {} -
\frac{2F_{\rm t}}{r} = -\frac{4}{5} 
\frac{(2p_{\rm r}-p_{\rm t})}
{\lambda_A t_{\rm relax}} \nonumber  
\end{eqnarray}

\begin{eqnarray*}
\lefteqn{
\frac{\partial{p_{\rm t}}}{\partial {t}} + \frac{1}{r^2} \frac{\partial}
{\partial r} (r^2 u \,p_{\rm t})+2 \,\frac{p_{\rm t}\,u}{r}+\frac{1}{2 r^2} \frac{\partial}{\partial r}
(r^2F_{\rm t}){}} \\
& & {} +\frac{F_{\rm t}}{r}
= \frac{2}{5} 
\frac{(2p_{\rm r}-p_{\rm t})}{\lambda_A 
t_{\rm relax}}, 
\end{eqnarray*}

\noindent
where $\lambda_A$ is a numerical constant related to the time-scale of
collisional anisotropy decay. The value chosen for it has been
discussed in comparison with direct simulations performed with the
$N$--body code \citep{GS94}. The authors find that $\lambda_A=0.1$
is the physically realistic value inside the half-mass radius for all
cases of $N$, provided that close encounters and binary activity do
not carry out an important role in the system, what is, on the other
hand, inherent to systems with a big number of particles, as this is.

\noindent
With the definition of the mass $M_{\rm r}$ contained in a sphere of radius
$r$

\be
\frac{\partial M_{\rm r}}{\partial r} = 4 \pi r^2 \rho,
\label{eq.Mr}
\ee

\noindent
the set of Eqs.\,(\ref {eq.set_of_eqs}) is equivalent to gas-dynamical
equations coupled with the equation of Poisson. To close it we need an
independent relation, for moment equations of order $n$ contain
moments of order $n\,+\,1$. For this intent we use the heat conduction
closure, a phenomenological approach obtained in an analogous way to
gas dynamics. It was used for the first time by \cite{LBE80} but
restricted to isotropy. In this approximation one assumes that heat
transport is proportional to the temperature gradient,

\be
F = -\kappa \frac{\partial T}{\partial r} = -\Lambda \frac{\partial 
\sigma ^2}{\partial r}  
\label{eq.temp_grad}
\ee

\noindent
That is the reason why such models are usually also called {\em
  conducting gas sphere models}. 

It has been argued that for the classical approach
$\Lambda\propto\bar{\lambda}^2/\tau$, one has to choose the Jeans'
length $\lambda_J^2 = \sigma ^2/(4\pi G\rho)$ and the standard
Chandrasekhar local relaxation time $t_{\rm relax}\propto \sigma
^3/\rho$ \citep{LBE80}, where $\bar{\lambda}$ is the mean free path
and $\tau$ the collisional time. In this context we obtain a
conductivity $\Lambda\propto \rho/ \sigma$. We shall consider this as
a working hypothesis. For the anisotropic model we use a mean velocity
dispersion $\sigma^2 = (\sr^2 + 2\st^2)/3$ for the temperature
gradient and assume $v_{\rm r} = v_{\rm t}$ \citep{BS86}.
\noindent
Forasmuch as, the equations we need to close our model are

\begin{eqnarray}
{v_{\rm r} - u + \frac{\lambda}{4\pi \,G\rho\,t_{\rm relax}} 
\frac{\partial \sigma^2}{\partial r} = 0} 
\label{eq.closing_eqs} \\
v_{\rm r} = v_{\rm t}.  \nonumber 
\end{eqnarray}

\section{Interaction terms and implementation of the gaseous component}

The set of Eqs.\,(\ref{eq.set_of_eqs}) describe the stellar cluster without
taking into consideration the different interaction processes which
may play a role in these systems. Heating, cooling processes, mass
exchange between stars and gas by stellar mass loss or disruptive
stellar collisions, exchange of kinetic internal energy between the
stellar system and the interstellar gas via drag forces, loss-cone
processes etc are not included in the equations. Here we will give a
description for some interesting phenomena that crop up in the systems
we want to study. Chapter \ref{ch.diff_model} is partly devoted to a
profound description of a loss-cone diffusion model.

In parallel with the description of the interaction terms for the
stellar component of a cluster, here we will describe the inclusion of
an~ \sms in the anisotropic numerical model. For that aim, one should
take into account the transfer of radiation in a spherically symmetric
moving medium allowing for the contributions which are of the order of
the flow velocity divided by the velocity of light, the thermal energy
equation and the turbulent energy equation. The equations will be
given also ``in their logarithmic form'' (i.e. in our equations we work
with $\ln x$ instead of $x$), ready to include in the gaseous model.

The interaction terms for the {\em gaseous component} \footnote{One
should here not be confused by the terminology. When we talk about the
{\em gaseous model}, we mean the numerical anisotropic model we use to
model stellar systems; it is called like that because it is based on
moments of the Boltzmann equation with relaxation. The stellar cluster
is modelled like a self-gravitating, conducting {\em gas} sphere, and
was first presented in \citet{LS91} and
\citet{GS94}. On the other hand, when we refer to the {\em gaseous 
component}, we mean the physical gas in the cluster, which may conform an 
\sms .} are also described and post-Newtonian corrections are also 
considered. How these \smss could power the quasar activity by star
accretion and energy flows is one of the main questions that could be
answered thanks to this method. Therefore we have found of paramount
importance to give the set of equations describing this situation.

\subsection{The star component}

We now introduce the interaction terms to be added to right hand of
the star component equations. This and the next section benefit from
the paper by \citet{LangbeinEtAl90}.


\subsubsection{Equation of continuity}
\citet{LangbeinEtAl90} derive the interaction terms to be
added to the basic equations of the gaseous model. According to them,
the star continuity equation is no longer
\begin{equation}
\frac {\partial \rho_{\star}}{\partial {\rm t}}+ \frac{1}{{\rm r}^2} \frac{\partial}
{\partial {\rm r}}({\rm r^2} \rho_{\star} {\rm u_{\star}})=0,
\end{equation}
\noindent
but
\begin{equation}
\frac {\partial \rho_{\star}}{\partial {\rm t}}+ \frac{1}{{\rm r}^2} \frac{\partial}
{\partial {\rm r}}({\rm r^2} \rho_{\star} {\rm u_{\star}})=\bigg( \frac{\delta \rho_{\star}}
{\delta {\rm t}} \bigg)_{\rm coll}+\bigg( \frac{\delta \rho_{\star}}
{\delta {\rm t}} \bigg)_{\rm lc};
\end{equation}
\noindent
where the right-hand term reflects the time variation of the star's density due to 
stars interactions (i.e. due to the calculation of the mean rate of gas production by 
stars {collision}s) and loss-cone (stars plunging onto the central object). 

\noindent
If $f(v_{\rm rel})$ is the stellar distribution of relative velocities,
then the mean rate of gas production by stellar {collision}s is
\begin{equation}
\bigg( \frac{\delta \rho_{\star}}{\delta {\rm t}} \bigg)_{\rm coll}=
-\int_{|v_{\rm rel|}>\sigma_{\rm coll}} \frac{\rho_{\star}f_{\rm c}(v_{\rm 
rel})}{t_{\rm coll}} f(v_{\rm rel})d^3v_{\rm rel}
\end{equation}
In the calculation of equation (14)  $f(v_{\rm rel})$ is
a Schwarzschild-Boltzmann distribution,
\begin{equation}
f(v_{\rm rel})=\frac{1}{2 \pi^{3/2} \sigma_{\rm r} \sigma_{\rm t}^2} \cdot {\rm exp}
\Bigg( -\frac{(v_{\rm rel,r}-u_{\star})^2}{4 \sigma_{\rm r}^2}-\frac{v_{\rm rel,t}^2}
{2 \sigma_{\rm t}^2}  \Bigg)
\end{equation}

\noindent
As regards $f_{\rm c}$, it is the relative fraction of mass liberated per stellar
{collision} into the gaseous medium. Under certain assumptions given in the 
initial work of \citet{SS66}, we can calculate it as an average
over all impact parameters resulting in $r_{\rm min}<2r_{\star}$ and as a function
of the relative velocity at infinity of the two colliding stars, $v_{\rm rel}$.
\citet{LangbeinEtAl90} approximate their result by

\[ f_{\rm c}(v_{\rm rel}) = \left\{ \begin{array}{ll}
  \big(1+q_{\rm coll} \sqrt{\sigma_{\rm coll}/v_{\rm rel}} \big)^{-1}& 
\mbox{$v_{\rm rel} > \sigma_{\rm coll}$}\\
  0 & \mbox{$v_{\rm rel} < \sigma_{\rm coll}$},
  \end{array} 
   \right. \]
\noindent
with $q_{\rm coll}=100$. So, we have that

\[ f_{\rm c}(v_{\rm rel}) = \left\{ \begin{array}{ll}
  0.01 & \mbox{$\sigma_{\rm coll}=v_{\rm rel}$}\\
  0 & \mbox{$\sigma_{\rm coll}>v_{\rm rel}$},
  \end{array} 
   \right. \]

The first interaction term is 
\begin{equation}
\bigg( \frac{\delta \rho_{\star}}{\delta {\rm t}} \bigg)_{\rm coll}=- \rho_{\star} 
\frac{\rm f_{\rm c}}{\rm t_{\rm coll}} \bigg[ 1-{\rm erf} \bigg( \frac{\sigma_{\rm coll}}
{\sqrt {6}\sigma_{\rm r}} \bigg) \bigg]\bigg[ 1-{\rm erf} \bigg( \frac
{\sigma_{\rm coll}}{\sqrt {6}\sigma_{\rm t}} \bigg) \bigg]^2 
\end{equation}
\noindent
which, for simplification, we re-call like this
\begin{equation}
\bigg( \frac{\delta \rho_{\star}}{\delta {\rm t}} \bigg)_{\rm coll}\equiv -\rho_{\star} 
{\rm X_{\rm coll}}.
\end{equation}
\noindent

Since the evolution of the system that we are studying can be regarded as
stationary, we introduce for each equation the ``logarithmic variables'' in 
order to study the evolution at long-term. In
the other hand, if the system happens to have quick changes, we should use
the ``non-logarithmic'' version of the equations. For this reason we will write
at the end of each subsection the equation in terms of the logarithmic variables.

In the case of the equation of continuity, we develop it and divide it by $\rho_{\star}$
because we are looking for the logarithm of the stars density, $\partial \ln \rho_{\star}
/\partial t=(1/\rho_{\star})\partial \rho_{\star}/ \partial t$. 
The result is:

\begin{equation}
\frac{\partial \ln \rho_{\star}}{\partial t} + \frac{\partial u_{\star}}{\partial r}+
u_{\star} \frac{\partial \ln \rho_{\star}}{\partial r}+ \frac{2u_{\star}}{r}= \frac{1}
{\rho_{\star}}\bigg( \frac{\delta \rho_{\star}}{\delta {\rm t}} \bigg)_{\rm coll}+
\frac{1}{\rho_{\star}}\bigg( \frac {\delta \rho_{\star}} {\delta {\rm t}} \bigg)_{\rm lc}
\end{equation}

\subsubsection{Momentum balance}

The second equation in Eq.\,(\ref{eq.set_of_eqs}) has the following
star interaction terms:

\begin{equation}
\frac{\partial u_{\star}}{\partial t}+u_{\star} \frac{\partial u_{\star}}{\partial r} + {GM_{\rm r}\over r^2} +
{1\over\rho_{\star}}\frac{\partial p_{\rm r}}{\partial r} + 2{p_{\rm r} - p_{\rm t}\over\rho_{\star} r} = 
\bigg( \frac{\delta u_{\star}}{\delta t}\bigg)_{\rm drag}
\label{eq.ustar_non_log}  
\end{equation}
\noindent
The interaction term is due to the decelerating force at which stars that move
inside the gas are subject to. As we shall see, an estimate for the force
is given by Eq.\,(\ref{eq.F_drag}). Explicitely, it is
\begin{equation}
\bigg( \frac{\delta u_{\star}}{\delta t}\bigg)_{\rm drag} = 
- X_{\rm drag}\frac{1}{\rho_{\star}} 
(u_{\star}-u_{\rm g})
\end{equation}
\noindent
where we have introduced the following definition:
\begin{equation}
X_{\rm drag} \equiv -C_{D} \frac{\pi r_{\star}^2}{m_{\star}}\rho_{\star} \rho_{\rm g} \sigma_{\rm tot},
\end{equation}
\noindent
with $\sigma_{\rm tot}^2=\sigma_{\rm r}^2+\sigma_{\rm t}^2+(u_{\star}-u_{\rm g})^2$
\vskip0.3cm
To the end of the calculation of the logarithmic variable version of the equation, we
multiply Eq.\,(\ref{eq.ustar_non_log}) by $\rho_{\star} r/p_{\rm r}$:
\begin{equation}
\frac{\rho_{\star} r}{p_{\rm r}} \bigg( \frac{\partial u_{\star}}{\partial t}+ u_{\star} \bigg)+
\frac{GM_{\rm r}}{rp_{\rm r}}\rho_{\star}+\frac{\partial \ln p_{\rm r}}
{\partial \ln r}+2(1-\frac{p_{\rm t}}{p_{\rm r}})=-X_{\rm drag} \frac{r}{p_{\rm r}}(u_{\star}-u_{\rm g})
\end{equation}

\subsubsection{Radial energy equation}

As regards the last but one equation, the interaction terms are:
\begin{equation}
\frac{\partial{p_{\rm r}}}{\partial {t}} + \frac{1}{r^2} \frac{\partial}{\partial r} 
(r^2 u_{\star} p_{\rm r})+2 p_{\rm r} \frac{\partial u_{\star}}{\partial r}+\frac{4}{5} \frac{(2p_{\rm r}-p_{\rm t})}
{t_{\rm relax}} +  \frac{1}{r^2} \frac{\partial}{\partial r} 
(r^2 F_{\rm r})- \frac{2F_{\rm t}}{r}= \nonumber
\end{equation}
\begin{equation}
\bigg( \frac{\delta p_{\rm r}}{\delta t}\bigg)_{\rm drag}+\bigg( \frac{\delta p_{\rm r}}
{\delta t}\bigg)_{\rm coll},
\label{eqn:pr} 
\end{equation}
\noindent
where
\begin{equation}
\bigg( \frac{\delta p_{\rm r}}{\delta t}\bigg)_{\rm drag}=-2X_{\rm drag} \sigma_{\rm r}^2,~
\bigg( \frac{\delta p_{\rm r}}{\delta t}\bigg)_{\rm coll}=-X_{\rm coll} 
\rho_{\star} \tilde{\sigma_{\rm r}}^2 \epsilon. 
\end{equation}
\noindent
In order to determine $\epsilon$ we introduce the ratio $k$ of kinetic
energy of the remaining mass after the encounter over its initial
value (before the encounter); $k$ is a measure of the inelasticity of
the {collision}: for $k=1$ we have the minimal inelasticity, just the
kinetic energy of the liberated mass fraction is dissipated, whereas
if $k<1$ a surplus amount of stellar kinetic energy is dissipated
during the {collision} (tidal interactions and excitation of stellar
oscillations). If we calculate the energy loss in the stellar system
per unit volume as a function of $k$ we obtain
\begin{equation}
\epsilon=f_{\rm c}^{-1}[1-k(1-f_{\rm c})].
\end{equation}

We divide by $p_{r }$ so that we get the logarithmic variable version of the equation.
We make also the following substitution:
\begin{eqnarray}
&F_{\rm r}=&3p_{\rm r}v_{\rm r} \nonumber \\
&F_{\rm t}=&2p_{\rm t}v_{\rm t}
\end{eqnarray}
\noindent
The resulting equation is
\begin{eqnarray}
\frac{\partial \ln p_{\rm r}}{\partial t}+(u_{\star}+3v_{\rm r}) \frac{\partial \ln p_{\rm r}}{\partial r}
+3 \bigg( \frac{\partial u_{\star}}{\partial r}+\frac{\partial v_{\rm r}}{\partial r} \bigg)+
\frac{2}{r}\bigg( u_{\star}+3v_{\rm r}-2v_{\rm t} \frac{p_{\rm t}}{p_{\rm r}} \bigg)+ \nonumber \\
\frac{4}{5}\frac{ 2-\frac{p_{\rm t}} {p_{\rm r}}}{t_{\rm relax}}= \frac{1}{p_{\rm r}} 
\bigg( \frac{\delta p_{\rm r}}{\delta t}\bigg)_{\rm drag}+\frac{1}{p_{\rm r}}
\bigg( \frac{\delta p_{\rm r}} {\delta t}\bigg)_{\rm coll}
\end{eqnarray}

\subsubsection{Tangential energy equation}

To conclude the set of equations of the star component
with the interaction terms, we have the following equation:

\begin{equation}
\frac{\partial{p_{\rm t}}}{\partial {t}} + \frac{1}{r^2} \frac{\partial}{\partial r} 
(r^2 u_{\star} p_{\rm t})+2 \frac{p_{\rm t}u_{\star}}{r}-\frac{4}{5} \frac{(2p_{\rm r}-p_{\rm t})}{t_{\rm relax}}+
 \frac{1}{r^2} \frac{\partial}{\partial r}(r^2F_{\rm t})+\frac{2F_{\rm t}}{r}= \nonumber
\end{equation}
\begin{equation}
\bigg( \frac{\delta p_{\rm t}}{\delta t}\bigg)_{\rm drag}+\bigg( \frac{\delta p_{\rm t}}
{\delta t}\bigg)_{\rm coll}, 
\end{equation}
\noindent
where
\begin{equation}
\bigg( \frac{\delta p_{\rm t}}{\delta t}\bigg)_{\rm drag}=-2X_{\rm drag} \sigma_{\rm t}^2,~
\bigg( \frac{\delta p_{\rm t}}{\delta t}\bigg)_{\rm coll}=-X_{\rm coll} 
\rho_{\star} \tilde{\sigma_{\rm t}}^2 \epsilon. 
\end{equation}
\noindent

We follow the same path like in the last case and so we get the following logarithmic
variable equation:

\begin{eqnarray}
\frac{\partial \ln p_{\rm t}}{\partial t}+(u_{\star}+2v_{\rm t}) \frac{\partial \ln p_{\rm t}}{\partial r}
+\frac{\partial}{\partial r}(u_{\star}+2v_{\rm t})+ \frac{4}{r}(u_{\star}+2v_{\rm t})- \\
\frac{4}{5} \frac{2\frac{p_{\rm r}}{p_{\rm t}}-1}{t_{\rm relax}}= \frac{1}{p_{\rm t}} 
\bigg( \frac{\delta p_{\rm t}}{\delta t}\bigg)_{\rm drag}+
\frac{1}{p_{\rm t}} \bigg( \frac{\delta p_{\rm t}} {\delta t}\bigg)_{\rm coll} \nonumber
\end{eqnarray}

\subsection{The gaseous component}

In this section we give the set of equations corresponding to the
gaseous component as for their right hand interaction terms.

\subsubsection{Equation of continuity}

For the \sms~the equation of continuity looks as follows:

\begin{equation}
\frac {\partial \rho_{\rm g}}{\partial { t}}+ \frac{1}{{\rm r}^2} \frac{\partial}
{\partial { r}}({\rm r^2} \rho_{\rm g} { u_{\rm g}})=\bigg( \frac{\delta \rho_{\rm g}}
{\delta { t}} \bigg)_{\rm coll}
\end{equation}
\noindent
where, for the mass conservation, we have that, obviously,
\begin{equation}
\bigg( \frac{\delta \rho_{\rm g}}{\delta { t}} \bigg)_{\rm coll}=
-\bigg( \frac{\delta \rho_{\star}}{\delta { t}} \bigg)_{\rm coll}
\end{equation}

We follow the same procedure as for the star continuity equation to get the 
equation in terms of the logarithmic variables:

\begin{equation}
\frac{\partial \ln \rho_{\rm g}}{\partial t} + \frac{\partial u_{\rm g}}{\partial r}+
u_{\rm g} \frac{\partial \ln \rho_{\rm g}}{\partial r}+ \frac{2u_{\rm g}}{r}= \frac{1}
{\rho_{\rm g}}\bigg( \frac{\delta \rho_{\rm g}}{\delta {\rm t}} \bigg)_{\rm coll}
\end{equation}

\noindent
The interaction term is in this case

\begin{equation}
\frac{1}{\rho_{\rm g}}\bigg( \frac{\delta \rho_{\rm g}}{\delta {\rm t}} \bigg)=
\frac{1}{\rho_{\rm g}}\bigg( -\frac{\delta \rho_{\star}}{\delta {\rm t}} \bigg)
= -\frac{\rho_{\star}}{\rho_{\rm g}}X_{\rm coll}
\end{equation}

\subsubsection{Momentum balance}

We modify equation number (2.9) of \citet{LangbeinEtAl90} in the following way:
\begin{equation}
\frac{\partial (\rho_{\rm g}u_{\rm g})}{\partial t}=u_{\rm g}\frac{\partial \rho_{\rm g}}{\partial t}+
\rho_{\rm g} \frac{\partial u_{\rm g}}{\partial t};
\end{equation}
\noindent
we substitute this equality in their equation, divide by $\rho_{\rm g}$ ($u_{\rm g}$ 
is the variable in our code) and make use of the equation of continuity for the 
gas component. Thus, we get the following expression:
\begin{equation}
\frac{\partial u_{\rm g}}{\partial t}+u_{\rm g} \frac{\partial u_{\rm g}}{\partial r} + {GM_{\rm r}\over r^2} +
{1\over\rho_{\rm g}}\frac{\partial p_{\rm r}}{\partial r} -\frac{4 \pi}{c} \kappa_{\rm ext} H= 
\bigg( \frac{\delta u_{\rm g}}{\delta t}\bigg)_{\rm coll}  
\end{equation}
\noindent
To get the interaction term we use the mass and momentum conservation:
\begin{equation}
\bigg( \frac{\delta \rho_{\rm g}}{\delta t}\bigg)_{\rm coll}+
\bigg( \frac{\delta \rho_{\star}}{\delta t}\bigg)_{\rm coll}=0 \nonumber
\end{equation}
\begin{equation}
\bigg( \frac{\delta (\rho_{\rm g} u_{\rm g})}{\delta t}\bigg)_{\rm coll}+
\bigg( \frac{\delta (\rho_{\star} u_{\star})}{\delta t}\bigg)_{\rm coll}=0.
\end{equation}
\noindent
We know that
\begin{equation}
\bigg( \frac{\delta u_{\star}}{\delta t}\bigg)_{\rm coll}=0,
\end{equation}
\noindent
thus,
\begin{equation}
\bigg( \frac{\delta (\rho_{\rm g} u_{\rm g})}{\delta t}\bigg)_{\rm coll}=u_{\star}\rho_{\star}X_{\rm coll}=
\rho_{\rm g} \bigg( \frac{\delta u_{\rm g}}{\delta t}\bigg)_{\rm coll} + u_{\rm g} X_{\rm coll} \rho_{\star}.
\end{equation}
\noindent
Therefore, the resulting interaction term is
\begin{equation}
\bigg( \frac{\delta u_{\rm g}}{\delta t}\bigg)_{\rm coll}= \frac{\rho_{\star}}{\rho_{\rm g}} 
X_{\rm coll}(u_{\star}-u_{\rm g})
\end{equation}
In the case of the stellar system
\begin{equation}
F= \frac{1}{2}(F_{\rm r}+F_{\rm t})=\frac{5}{2}\rho_{\star}v_{\star}
\end{equation}
By analogy, we now introduce $F_{\rm rad}$ in this way
\begin{equation}
\frac{F_{\rm rad}}{4\pi}=H=\frac{5}{2}p_{\rm g}v_{\rm g},
\end{equation}
where $v_{\rm g}$ is per gas particle.
\begin{equation}
v_{\rm g}=\frac{2}{5} \frac{H}{p_{\rm g}}
\end{equation}

As means to write the equation in its ``logarithmic variable version'', we multiply
the equation by $\rho_{\rm g} r /p_{\rm g}$, as we did for the corresponding momentum balance
star equation and replace $H$ by $5/2p_{\rm g}v_{\rm g}$, 
\begin{eqnarray}
\frac{\rho_{\rm g}r}{p_{\rm g}} \bigg( \frac{\partial u_{\rm g}}{\partial t}+u_{\rm g}
\frac{\partial u_{\rm g}}{\partial r} \bigg)+\frac{GM_{\rm r}}{rp_{\rm g}}\rho_{\rm g}+
\frac{\partial \ln p_{\rm g}}{\partial \ln r}-\frac{5}{2} \frac{\kappa_{\rm ext}}{c}
\rho_{\rm g}r v_{\rm g}= \nonumber \\
\frac{r}{p_{\rm g}} \rho_{\star}X_{\rm coll} (u_{\star}-u_{\rm g})
\end{eqnarray}

\subsubsection{Radiation transfer}

We extend here and improve the work done by \citet{LangbeinEtAl90} by resorting to a 
more detailed description of the radiation transfer \citep{Castor72}.

Consider a radiation field; we place a surface element $d \sigma$ with a 
surface normal ${\bf n}$, see Fig.\,(\ref{fig.rad_sh}); the radiation energy which passes
through $d \sigma$ per unit time at angle $\theta$ to ${\bf n}$ within a 
small range of solid angle $d \omega$ given by the directional angles 
$\theta$ and $\phi$ is 
\begin{equation}
dE=I_{\nu}(\theta, \phi) \mu d\nu d \sigma d \omega
\end{equation}
\noindent
where $\mu = \cos \theta$.

The radiation intensity $I_{\nu}(\theta, \phi)$ is defined as the amount of energy that
passes through a surface normal to the direction ($\theta$,$\phi$) per unit solid angle
(1 steradian) and unit frequency range (1 Hz) in one second. The intensity of the total
radiation is given by integrating over all frequencies,
\begin{equation}
I=\int_{0}^{\infty} I_{\nu}d \nu
\end{equation}

\begin{figure}
\begin{center}
 \leavevmode
 \epsfxsize=6cm
 \epsfysize=7cm
 \epsffile{diag_th_model/rad2.ps}
 \caption{Radiation intensity definition}
 \label{fig.rad_sh}
\end{center}
\end{figure}
\noindent

The three radiation moments (the moments of order zero, one and two) are defined by:
\begin{eqnarray}
&J=&\int_{0}^{\infty} J_{\nu} d{\nu}=\int_{0}^{\infty} d{\nu} \frac{1}{2} \int_{-1}^{+1} 
I_{\nu} d{\mu} \nonumber \\
&
H=&\int_{0}^{\infty} H_{\nu} d{\nu}=\int_{0}^{\infty} d{\nu} \frac{1}{2} \int_{-1}^{+1} 
I_{\nu} \mu d{\mu}  \\
&K=&\int_{0}^{\infty} K_{\nu} d{\nu}=\int_{0}^{\infty} d{\nu} \frac{1}{2} \int_{-1}^{+1} 
I_{\nu} \mu^2 d{\mu}, \nonumber 
\end{eqnarray}
\noindent
The moment of order zero is related to the density of energy of the field of radiation 
$E_{\rm rad}$, the moment of order one to the flux of radiation $F_{\rm rad}$ and the
moment of order one to the radiation pressure $p_{\rm rad}$,
\begin{eqnarray}
&E_{\rm rad}=& \frac{4\pi}{c}J \nonumber \\
&F_{\rm rad}=&4\pi H \\
&p_{\rm rad}=&\frac{4 \pi}{c}K \nonumber
\end{eqnarray}
The transfer of
radiation in a spherically symmetric moving medium is considered
taking into account the contributions which are of the order of the flow velocity
divided by the velocity of light; we include also the variation from the centre up 
to the atmosphere of the Eddington factor $f_{\rm Edd}=K/J$, where $K$ and $J$ are the radiation 
moments; $f_{\rm Edd}$ is obtained from a numerical solution of the
equation of radiation transfer in spherical geometry \citet{Yorke80}. 
We get the radiation transfer equations by re-writing the frequency-integrated moment 
equations from \citet{Castor72}: 

\begin{eqnarray}
{\frac{1}{c} \frac{\partial J}{\partial t}+ \frac{\partial H}{\partial r}+
\frac{2 H}{r}- \frac{J(3f_{\rm Edd}-1)}{cr} u_{\rm g}- \frac {J(1+f_{\rm Edd})}{c}
\frac {\partial \ln \rho_{\rm g}}{\partial t}=} \nonumber \\
\kappa_{\rm abs}(B-J)
\label{eq.J_nonlog}
\end{eqnarray}

\begin{eqnarray}
\frac{1}{c} \frac{\partial H}{\partial t}+\frac{\partial (J f_{\rm Edd})}{\partial r}
+ \frac{J(3f_{\rm Edd}-1)}{r}-\frac{2u_{\rm g}}{cr} H-\frac{2}{c}\frac{\partial 
\ln \rho_{\rm g}}{\partial t} H= \nonumber \\
-\kappa_{\rm ext} \rho_{\rm g} H
\label{eq.H_nonlog}
\end{eqnarray}

\noindent
In the equations $\kappa_{\rm abs}$ and $\kappa_{\rm ext}$ are the absorption and extinction
coefficients per unit mass 

\begin{equation}
\kappa_{\rm abs}=\frac{\rho_{\rm g} \Lambda(T)}{B},~
\kappa_{\rm ext}=\rho_{\rm g}(\kappa_{\rm abs}+\kappa_{\rm scatt}),
\end{equation}

\noindent
$\Lambda(T)$ is the cooling function, $B$ the Planck function and $\kappa_{\rm scatt}$ the
scattering coefficient per unit mass. We have made use of $\partial M_{\rm r}/ \partial 
r=4 \pi^2 \rho$, $f_{\rm Edd}=K/J$, 
and the Kirchhoff's law, $B_{\nu}=j_{\nu}/\kappa_{\nu}$ ($j_{\nu}$ is the emission 
coefficient), so that the right-hand terms in \citet{Castor72} are the corresponding given here.

We now look for the logarithmic variable version of both equations;
for this aim, we divide Eq.\,(\ref{eq.J_nonlog}) by $J$ and multiply Eq.\,(\ref{eq.H_nonlog}) by
$2c/(5p_{\rm g}v_{\rm g})$,
\begin{eqnarray}
\frac{1}{c}\frac{\partial \ln J}{\partial t}+\frac{5}{2J}\frac{\partial}{\partial r} 
(v_{\rm g}p_{\rm g})+\frac{5}{Jr}p_{\rm g}v_{\rm g}-\frac{3f_{\rm Edd}-1}{cr}u_{\rm g} 
-(1+f_{\rm Edd}) \nonumber \\
\frac{1}{c}\frac{\partial \ln \rho_{\rm g}} {\partial t}= 
\frac{\kappa_{\rm abs}}{J} (B-J)
\end{eqnarray}

\begin{eqnarray}
\frac{\partial \ln v_{\rm g}}{\partial t}+\frac{\partial \ln p_{\rm g}}{\partial t}+
\frac{2c}{5}\frac{1}{p_{\rm g}v_{\rm g}}\frac{\partial (J f_{\rm Edd})}{\partial r}+
\frac{2c}{5}\frac{3f_{\rm Edd}-1}{rp_{\rm g}v_{\rm g}}J-\frac{2u_{\rm g}}{r} \nonumber \\
-2\frac{\partial \ln \rho_{\rm g}}{\partial t}= -c \kappa_{\rm ext}\rho_{\rm g} 
\end{eqnarray}
\noindent
Where we have substituted $H=5p_{\rm g}v_{\rm g}/2$.

\subsubsection{Thermal energy conservation}

It is enlightening to construct from Eqs.\,(\ref{eq.set_of_eqs}) an equation for
the energy per volume unit $e=(p_{\rm r}+2p_{\rm t})/2$ which, in the
case of an isotropic gas ($p_{\rm r}=p_{\rm t}$) is $e=3p/2$. For this
aim we take, for instance, equation (\ref{eqn:pr}) and in the term
$2p_{\rm r}\partial u_{\star}/\partial r$ we include now a source for
radiation pressure, $2(p_{\rm r}+p_{\rm rad})\partial
u_{\star}/\partial r$ and we divide everything by $e$ so that we get
the logarithmic variables. The resulting equation is

\begin{equation}
\frac{\partial \ln e}{\partial t}+(u_{\rm g}+3v_{\rm g}) \frac{\partial \ln e}{\partial r}
+\frac{2}{r} (u_{\rm g}+v_{\rm g})=\frac{1}{e} \bigg( \frac{\delta e}{\delta t} \bigg)_{\rm drag}+
\frac{1}{e}\bigg( \frac{\delta e}{\delta t} \bigg)_{\rm coll}
\end{equation}

\noindent
The interaction terms for this equation are
\begin{equation}
\bigg( \frac{\delta e}{\delta t} \bigg)_{\rm drag}=X_{\rm drag}(\sigma_{\rm r}^2+\sigma_{\rm t}^2
+(u_{\star}-u_{\rm g})^2)
\end{equation}
\begin{equation}
\bigg( \frac{\delta e}{\delta t} \bigg)_{\rm coll}=\frac{1}{2}X_{\rm coll} \rho_{\star}
((\sigma_{\rm r}+\sigma_{\rm t})^2 \epsilon+(u_{\star}-u_{\rm g})^2 -\xi \sigma_{\rm coll}^2)
\end{equation}
\subsubsection{Mass conservation}

The mass conservation is guaranteed by

\begin{equation}
\frac{3}{4\pi}\frac{\partial M_{\rm r}}{\partial r^3}=\rho_{\star}+\rho_{\rm g}
\end{equation}

\section{A mathematical view of our approach}

\newcommand{\Ngrid}{\ensuremath{N_{\rm r}}}
\newcommand{\Neq}{\ensuremath{N_{\rm eq}}}
%\newcommand{\qid}[1]{\ensuremath{_{#1}}}% index for various quantites (rho, u, ...)
%\newcommand{\rid}[1]{\ensuremath{^{(#1)}}}% index for various elements on the radial mesh
\newcommand{\qid}[1]{\ensuremath{^{(#1)}}}% index for various quantites (rho, u, ...)
\newcommand{\rid}[1]{\ensuremath{_{#1}}}% index for various elements on the radial mesh
\newcommand{\xprev}{y}% x a t time t-delta_t
\newcommand{\myH}[1]{\tilde{#1}}% notation for lambda*x(t) + (1-lambda)*x(t-delta_t)
% \newcommand{\GrandVect}[1]{\underline{\underline{\cal #1}}} del Marc, pero no m'agrada
\newcommand{\GrandVect}[1]{{{\cal #1}^*}} 
\newcommand{\VecNeq}[1]{{\cal #1}}
\newcommand{\MatNeq}[1]{{\mathfrak #1}}
\newcommand{\itid}[1]{\ensuremath{_{[ #1]}}}% index for iteration during Newton-Raphson scheme

%\newcommand{\DiagMatrix}{\begin{pspicture}(0,0)(0.65,0.8)
%\psframe[fillstyle=vlines,hatchangle=0,hatchcolor=black,hatchsep=1.4pt,hatchwidth=0.5pt](0,0)(0.8,0.8)
%\end{pspicture}}

%\newcommand{\AfterDiagMatrix}{\begin{pspicture}(0,0)(0.65,0.8)
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%\psframe[fillstyle=vlines,hatchangle=-45,hatchcolor=black,hatchsep=2.8pt,hatchwidth=0.3pt](0,0)(0.8,0.8)
%\end{pspicture}}
\newcommand{\DMatrix}{\ensuremath{{\blacksquare}}}%_{\ \,}}}}
\newcommand{\CMatrix}{\ensuremath{{\square_{-}}}}
\newcommand{\EMatrix}{\ensuremath{{\square_{+}}}}
\newcommand{\CEMatrix}{\ensuremath{{\square_{\pm}}}}
\newcommand{\DerivMatrix}{\ensuremath{{\partial\GrandVect{F}}/{\partial \GrandVect{X}}}}
\newcommand{\DerivMatrixDisp}{\ensuremath{\frac{\partial\GrandVect{F}}{\partial \GrandVect{X}}}}

\bfig 
\C
\resizebox{0.65\hsize}{!}{\includegraphics[clip]{diag_th_model/rad_shells.eps}}
\caption {Representation of the logarithmic radial mesh used in the code. 
With $v$, $p$ we represent both, the radial and tangential components of the
velocity and pressure.
        \label{fig.rad_shells}
}   
\EC
\efig 

In this subsection, we explain briefly how the gaseous model is solved
numerically. We concentrate on aspects of the method not exposed in
previous work. This description is therefore complementary to
Sec.~2.2 of \citet{GS94}. The algorithm used is a partially implicit
Newton-Raphson-Henyey iterative scheme (\citealt{HenyeyEtAl59}, see
also \citealt{KW94}, Sec.~11.2).

Putting aside the bounding conditions, the set of equations to be
solved are Eqs.~\ref{eq.set_of_eqs} to \ref{eq.closing_eqs}. In
practice, however, the equations are rewritten using the logarithm of
all positive quantities as dependant functions. As explained in
\citet{GS94}, this greatly improves energy conservation. Formally, one
may write this system as follows
\be
\begin{split}
\frac{\partial x\qid{i}}{\partial t} + f\qid{i}\left( 
\left\{x\qid{j},\frac{\partial x\qid{j}}{\partial r}\right\}_{j=1}^{\Neq} 
\right) &=0 \mbox{\ \ for\ } i=1\dotso 4\\
f\qid{i}\left( 
\left\{x\qid{j},\frac{\partial x\qid{j}}{\partial r}\right\}_{j=1}^{\Neq} 
\right) &=0 \mbox{\ \ for\ } i=5\dotso \Neq
\end{split}
\label{eq:difeq_set}
\ee
where the $x\qid{i}$ are the local quantities defining the state of the cluster, i.e.
\be
\begin{split}
\underline{x} &\equiv \left\{x\qid{1}, x\qid{2},\dotso x\qid{\Neq}\right\} \\
& \equiv 
\left\{ \log\rho, u, \log p_{\rm r}, \log p_{\rm t}, \log M_{\rm r}, v_{\rm r}-u, v_{\rm t}-u \right\},
\end{split}
\ee
with $\Neq=7$ in the present application.

To be solved numerically, this set of coupled partial differential
equations have to be discretised according to time and radius. Let us
first consider time stepping. Let $\Delta t$ be the time step. Assume
we know the solution $\underline{x}(t-\Delta t)$ at time $t-\Delta t$
and want to compute $\underline{x}(t)$. For the sake of numerical
stability, a partially implicit scheme is used. We adopt the shorthand
notations $x\qid{i} \equiv x\qid{i}(t)$ and $\xprev\qid{i} \equiv x\qid{i}(t-\Delta t)$. Time
derivation is replaced by finite differences, 
\be
\frac{\partial x\qid{i}}{\partial t} \rightarrow \Delta t^{-1}(x\qid{i}-\xprev\qid{i}).
\label{eq:time_discr}
\ee
In the terms $f\qid{i}$, we replace the $x\qid{j}$ by $\myH{x}\qid{j}$
which are values intermediate between $\xprev\qid{j}$ and $x\qid{j}$,
$\myH{x}\qid{j} =
\zeta x\qid{j} + (1-\zeta)\xprev\qid{j}$, with $\zeta= 0.55$ for
stability purpose
\citep{GS94}. 

Spatial discretisation is done by defining all quantities (at a given
time) on a radial mesh, $\{r\rid{1}, r\rid{2},\dotso r\rid{N_{\rm
r}}\}$ with $r\rid{1}=0$ and $r\rid{\Ngrid}=r_{\rm max}$. A staggered
mesh is implemented. While values of $r$, $u$, $v_{\rm t}$, $v_{\rm r}$ 
and $M_{\rm r}$
are defined at the boundaries of the mesh cells, $\rho$, $p_{\rm t}$ and
$p_{\rm r}$ are defined at the centre of each cell, see
Fig.\,\ref{fig.rad_shells}. When the value of a ``boundary'' quantity
is needed at the centre of a cell, or vice-versa, one resort to simple
averaging, i.e. $\hat{b}\rid{k} = 0.5(b\rid{k-1}+b\rid{k})$,
$\hat{c}\rid{k} = 0.5(c\rid{k}+c\rid{k+1})$, if $b$ and $c$ are is
border- and centre-defined quantities, and $\hat{b}$, $\hat{c}$ their
centre- and border-interpolations, respectively. For all runs
presented here, $\Ngrid=300$ and the points $r\rid{2}$ to $r_{\rm
max}$ are logarithmically equidistant with $r_{\rm max}=10^4\,{\rm
pc}$ and $r\rid{2}\simeq 1.7\times 10^{-6}\,{\rm pc}$. Let us adopt
the notation $x\qid{j}\rid{k}$ for the value of $x\qid{j}$ at position
$r\rid{k}$ (or $\hat{r}\rid{k}$) and $\Delta r\rid{k}\equiv
r\rid{k}-r\rid{k-1}$. Then, radial derivatives in the terms $f\qid{i}$
are approximated by finite differences,
\be
\frac{\partial x\qid{j}}{\partial r}
\rightarrow 
\frac{\myH{{x}}\qid{j}\rid{k}-\myH{{x}}\qid{j}\rid{k-1}}{\Delta r\rid{k}}
\label{eq:rad_discr1}
\ee
if the derivative has to be evaluated at a point where $x\rid{k}$ is
defined (centre or border of a cell), or
\be
\frac{\partial x\qid{j}}{\partial r}
\rightarrow 
\frac{\hat{\myH{{x}}}\qid{j}\rid{k}-\hat{\myH{{x}}}\qid{j}\rid{k-1}}{\Delta r\rid{k}} =
\frac{\myH{x}\qid{j}\rid{k+1}-\myH{x}\qid{j}\rid{k-1}}{2\Delta r\rid{k}}
\label{eq:rad_discr2}
\ee
otherwise. As an exception we use upstream differencing in
$\partial{u}/\partial{r}$ for the second equation in set
\ref{eq.set_of_eqs}, i.e. the difference quotient is displaced by half
a mesh point upstream to improve stability.

By making the substitutions for ${\partial x\qid{j}}/{\partial t}$ and
${\partial x\qid{j}}/{\partial r}$ in the set of differential
equations
\ref{eq:difeq_set}, one obtains, at each mesh point $r\rid{k}$, a set
of $\Neq$ non-linear algebraic equations linking the new values to be
determined, $\underline{x}\rid{k-1}$ and $\underline{x}\rid{k}$, to
the ``old'' ones, $\underline{\xprev}\rid{k-1}$ and
$\underline{\xprev}\rid{k}$, which are known,
\be
\begin{split}
 {\cal F}\qid{i}\rid{k}\left(\underline{x}\rid{k-1},\underline{x}\rid{k} |
\underline{\xprev}\rid{k-1},\underline{\xprev}\rid{k}\right)=0\\
 \ i=1\dotso \Neq, \ k=1\dotso \Ngrid.
\end{split}
\ee

Note that the structure of the equations is the same at all mesh
points, except $k=1$ and $k=\Ngrid$. In particular, terms $k-1$ do
not appear in ${\cal F}\qid{i}\rid{1}$. Also, one has to keep in mind that
only the $\underline{x}\rid{k-1}$ and $\underline{x}\rid{k}$ are
unknown; the $\underline{\xprev}\rid{k-1}$ and
$\underline{\xprev}\rid{k}$ play the role of fixed parameters in these
equation (as do the $\Delta r\rid{k}$). If one defines a $(\Neq\times
\Ngrid)$-dimension vector $\GrandVect{X}$ whose component
$\Neq(k-1)+i$ is $x\qid{i}\rid{k}$, one can write the system of $\Neq\times
\Ngrid$ equations as $\GrandVect{F}(\GrandVect{X})=0$, i.e.
\be
\GrandVect{F}(\GrandVect{X}) \equiv \left( \begin{array}{c}
{\cal F}\qid{1}\rid{1} \\
{\cal F}\qid{2}\rid{1} \\
\vdots \\
{\cal F}\qid{\Neq}\rid{1} \\
{\cal F}\qid{1}\rid{2} \\
\vdots \\
{\cal F}\qid{\Neq}\rid{2} \\
\vdots \\
{\cal F}\qid{1}\rid{\Ngrid} \\
\vdots \\
{\cal F}\qid{\Neq}\rid{\Ngrid} \\
\end{array}\right) =  \left( \begin{array}{c}
0 \\
\vdots \\
0 \\
\end{array}\right)
\mbox{\ with\ }\GrandVect{X} \equiv \left( \begin{array}{c}
x\qid{1}\rid{1} \\
x\qid{2}\rid{1} \\
\vdots \\
x\qid{\Neq}\rid{1} \\
x\qid{1}\rid{2} \\
\vdots \\
x\qid{\Neq}\rid{2} \\
\vdots \\
x\qid{1}\rid{\Ngrid} \\
\vdots \\
x\qid{\Neq}\rid{\Ngrid} \\
\end{array}\right).
\label{eq:discreq_set}
\ee

The system is solved iteratively using Newton-Raphson scheme. If
$\GrandVect{X}\itid{m}$ is the approximation to the solution of
Eq.~\ref{eq:discreq_set} after iteration $m$, with
$\GrandVect{F}\itid{m} \equiv \GrandVect{F}(\GrandVect{X}\itid{m})\ne 0$, 
the solution is refined through the relation

\be
\GrandVect{X}\itid{m+1} = \GrandVect{X}\itid{m} - 
\left(\DerivMatrixDisp\right)^{-1} 
\GrandVect{F}\itid{m}
\label{eq:NewtRaphsonStep}
\ee

where $(\DerivMatrix)^{-1}$ is
the inverse of the matrix of derivatives. The latter, of dimension
$(\Neq\, \Ngrid)\times(\Neq\, \Ngrid)$, has the following structure

\be
\setcounter{MaxMatrixCols}{20}
\frac{\partial \GrandVect{F}}{\partial \GrandVect{X}} =
%\left(\,\,
\begin{pmatrix}
\DMatrix & \EMatrix \\
\CMatrix & \DMatrix & \EMatrix \\
 & \CMatrix & \DMatrix & \EMatrix \\
 & & \ddots & \ddots\\
 & & \CMatrix\rid{k} & \DMatrix\rid{k} & \EMatrix\rid{k} \\
 & & & \ddots & \ddots\\
% & & & &\ddots & \ddots\\
 & & & \CMatrix & \DMatrix & \EMatrix \\
 & & & & \CMatrix & \DMatrix \\
\end{pmatrix}.
%\,\,\,\,\right).
\ee
\setcounter{MaxMatrixCols}{10}
In this diagram, each square is a $\Neq\times \Neq$ sub-matrix. For $2\le k
\le \Ngrid-1$, lines ${\Neq}k-6$ to ${\Neq}k$ of $\DerivMatrix$ are composed
of a group of 3 such ${\Neq}\times {\Neq}$ matrices, $\CMatrix\rid{k},
\DMatrix\rid{k}, \EMatrix\rid{k}$ that span columns ${\Neq}k-13$ to 
${\Neq}k+{\Neq}$, while the rest is composed of zeros,

\be
\begin{aligned}
\DMatrix\rid{k} &= \begin{pmatrix}
\frac{\partial {\cal F}\qid{1}\rid{k}}{\partial x\qid{1}\rid{k}} & \frac{\partial {\cal F}\qid{1}\rid{k}}{\partial x\qid{2}\rid{k}} & \cdots & \frac{\partial {\cal F}\qid{1}\rid{k}}{\partial x\qid{{\Neq}}\rid{k}} \\
\vdots & & & \vdots \\
\frac{\partial {\cal F}\qid{{\Neq}}\rid{k}}{\partial x\qid{1}\rid{k}} & \frac{\partial {\cal F}\qid{{\Neq}}\rid{k}}{\partial x\qid{2}\rid{k}} & \cdots & \frac{\partial {\cal F}\qid{{\Neq}}\rid{k}}{\partial x\qid{{\Neq}}\rid{k}} \\
\end{pmatrix}\\
%
\CEMatrix\rid{k} &= \begin{pmatrix}
\frac{\partial {\cal F}\qid{1}\rid{k}}{\partial x\qid{1}\rid{k\pm 1}} & \frac{\partial {\cal F}\qid{1}\rid{k}}{\partial x\qid{2}\rid{k\pm 1}} & \cdots & \frac{\partial {\cal F}\qid{1}\rid{k}}{\partial x\qid{{\Neq}}\rid{k\pm 1}} \\
\vdots & & & \vdots \\
\frac{\partial {\cal F}\qid{{\Neq}}\rid{k}}{\partial x\qid{1}\rid{k\pm 1}} & \frac{\partial {\cal F}\qid{{\Neq}}\rid{k}}{\partial x\qid{2}\rid{k\pm 1}} & \cdots & \frac{\partial {\cal F}\qid{{\Neq}}\rid{k}}{\partial x\qid{{\Neq}}\rid{k\pm 1}} \\
\end{pmatrix}.\\
%
\end{aligned}
\ee

\noindent
We can see this more explicitely,

$$
\setcounter{MaxMatrixCols}{20}
\frac{\partial \GrandVect{F}}{\partial \GrandVect{X}} =
\begin{pmatrix}
\frac{\partial {\cal F}^{1}\rid{1}}{\partial x^{1}\rid{1}} & \cdots & \frac{\partial {\cal F}^{1}\rid{1}}{\partial x^{N_{\rm eq}}\rid{1}} & \frac{\partial {\cal F}^{1}\rid{1}}{\partial x^{1}\rid{2}} & \cdots & \frac{\partial {\cal F}^{1}\rid{1}}{\partial x^{N_{\rm eq}}\rid{2}} & 0 & \hdotsfor{4} & 0\\
\vdots & & & & & \vdots &\vdots & & & & &\vdots \\
\frac{\partial {\cal F}^{N_{\rm eq}}\rid{1}}{\partial x^{1}\rid{1}} & \cdots & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{1}}{\partial x^{N_{\rm eq}}\rid{1}} & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{1}}{\partial x^{1}\rid{2}} & \cdots & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{1}}{\partial x^{N_{\rm eq}}\rid{2}} & 0 & \hdotsfor{4} & 0\\
\frac{\partial {\cal F}^{1}\rid{2}}{\partial x^{1}\rid{1}} & \cdots & \frac{\partial {\cal F}^{1}\rid{2}}{\partial x^{N_{\rm eq}}\rid{1}} & \frac{\partial {\cal F}^{1}\rid{2}}{\partial x^{1}\rid{2}} & \cdots & \frac{\partial {\cal F}^{1}\rid{2}}{\partial x^{N_{\rm eq}}\rid{2}} & \frac{\partial {\cal F}^{1}\rid{2}}{\partial x^{1}\rid{3}} & \cdots & \frac{\partial {\cal F}^{1}\rid{2}}{\partial x^{N_{\rm eq}}\rid{3}} & 0 & \hdotsfor{1} & 0\\
\vdots & & & & &  &  & & \vdots & \vdots & & \vdots \\
\frac{\partial {\cal F}^{N_{\rm eq}}\rid{2}}{\partial x^{1}\rid{1}} & \cdots & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{2}}{\partial x^{N_{\rm eq}}\rid{1}} & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{2}}{\partial x^{1}\rid{2}} & \cdots & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{2}}{\partial x^{N_{\rm eq}}\rid{2}} & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{2}}{\partial x^{1}\rid{3}} & \cdots & \frac{\partial {\cal F}^{1}\rid{2}}{\partial x^{N_{\rm eq}}\rid{3}} & 0 & \hdotsfor{1} & 0\\
\hdotsfor{12} \\
\hdotsfor{12} \\
0 & \hdotsfor{4} & 0 & \frac{\partial {\cal F}^{1}\rid{N_{\rm r}}}{\partial x^{1}\rid{N_{\rm r}-1}} & \cdots & \frac{\partial {\cal F}^{1}\rid{N_{\rm r}}}{\partial x^{N_{\rm eq}}\rid{N_{\rm r}-1}} & \frac{\partial {\cal F}^{1}\rid{N_{\rm r}}}{\partial x^{1}\rid{N_{\rm r}}} & \cdots & \frac{\partial {\cal F}^{1}\rid{N_{\rm r}}}{\partial x^{N_{\rm eq}}\rid{N_{\rm r}}} \\
\vdots & & & & & \vdots & \vdots & & & & & \vdots \\
0 & \hdotsfor{4} & 0 & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{N_{\rm r}}}{\partial x^{1}\rid{N_{\rm r}-1}} & \cdots & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{N_{\rm r}}}{\partial x^{N_{\rm eq}}\rid{N_{\rm r}-1}} & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{N_{\rm r}}}{\partial x^{1}\rid{N_{\rm r}}} & \cdots & \frac{\partial {\cal F}^{N_{\rm eq}}\rid{N_{\rm r}}}{\partial x^{N_{\rm eq}}\rid{N_{\rm r}}} \\
\end{pmatrix}
$$
\setcounter{MaxMatrixCols}{10}

\newcommand{\DiagMatrix}{\begin{pspicture}(0,0)(0.65,0.8)
\psframe[fillstyle=vlines,hatchangle=0,hatchcolor=black,hatchsep=1.4pt,hatchwidth=0.5pt](0,0)(0.8,0.8)
\end{pspicture}}

\newcommand{\AfterDiagMatrix}{\begin{pspicture}(0,0)(0.65,0.8)
\psframe[fillstyle=vlines,hatchangle=45,hatchcolor=black,hatchsep=2.8pt,hatchwidth=0.3pt](0,0)(0.8,0.8)
\end{pspicture}}

\newcommand{\BeforeDiagMatrix}{\begin{pspicture}(0,0)(0.65,0.8)
\psframe[fillstyle=vlines,hatchangle=-45,hatchcolor=black,hatchsep=2.8pt,hatchwidth=0.3pt](0,0)(0.8,0.8)
\end{pspicture}}

\noindent
More schematically, to have an overview of the structure,

$$
\setcounter{MaxMatrixCols}{20}
\frac{\partial \GrandVect{F}}{\partial \GrandVect{X}} =
\left(\,\,
\begin{matrix}
\DiagMatrix & \AfterDiagMatrix \\
\BeforeDiagMatrix & \DiagMatrix & \AfterDiagMatrix \\
 & \BeforeDiagMatrix & \DiagMatrix & \AfterDiagMatrix \\
 & & \BeforeDiagMatrix & \DiagMatrix & \AfterDiagMatrix \\
 & & & \ddots & \ddots\\
 & & & &\ddots & \ddots\\
 & & & & \BeforeDiagMatrix & \DiagMatrix & \AfterDiagMatrix \\
 & & & & & \BeforeDiagMatrix & \DiagMatrix \\
\end{matrix}
\,\,\,\,\right)
$$
\setcounter{MaxMatrixCols}{10}

The Heyney method is a way to take advantage of the special structure
of matrix $\DerivMatrix$ to solve system (\ref{eq:NewtRaphsonStep})
efficiently, with a number of operation scaling like ${\cal O}(N_{\rm
r})$ rather than ${\cal O}(\Ngrid^3)$ as would be the case if one
uses a general-purpose matrix inversion scheme\footnote{Memory usage
is also reduced, scaling like ${\cal O}(\Ngrid)$ rather than ${\cal
O}(\Ngrid^2)$.}. Setting
$\GrandVect{B}\equiv-\GrandVect{F}\itid{m}$ and $\GrandVect{W}\equiv
\GrandVect{X}\itid{m+1}-\GrandVect{X}\itid{m}$,
Eq.\,(\ref{eq:NewtRaphsonStep}) is equivalent to
\be
  \left(\DerivMatrixDisp\right) \GrandVect{W} = \GrandVect{B}
\ee
with $\GrandVect{W}$ the unknown vector. We further decompose vectors
$\GrandVect{W}$ and $\GrandVect{B}$ into {\Neq}--dimensional sub-vectors,
each one representing the values at a given mesh points,
\be
\GrandVect{W} = \begin{pmatrix}
\VecNeq{W}\rid{1} \\
\VecNeq{W}\rid{2} \\
\vdots \\
\VecNeq{W}\rid{k} \\
\vdots \\
\VecNeq{W}\rid{\Ngrid} \\
\end{pmatrix}.
\label{eq:MtimesWequalB}
\ee
Then, the system (\ref{eq:MtimesWequalB}) can be written as a set of
coupled {\Neq}--dimensional vector equations,
\be
\addtolength{\arraycolsep}{-3.5pt}
\begin{array}{lclclcl}
%\begin{alignedat}{4}
                                            &   &\DMatrix\rid{1}\VecNeq{W}\rid{1}& + &\EMatrix\rid{1}\VecNeq{W}\rid{2}  & = &\VecNeq{B}\rid{1}\\
\CMatrix\rid{k}\VecNeq{W}\rid{k-1}          & + &\DMatrix\rid{k}\VecNeq{W}\rid{k}& + &\EMatrix\rid{k}\VecNeq{W}\rid{k+1}& = &\VecNeq{B}\rid{k}\\% \qquad 2\le k \le \Ngrid-1 \\
\CMatrix\rid{\Ngrid}\VecNeq{W}\rid{\Ngrid-1}& + &\DMatrix\rid{\Ngrid}\VecNeq{W}\rid{\Ngrid}&&                           & = &\VecNeq{B}\rid{\Ngrid}.
%\end{alignedat}
\end{array}
\addtolength{\arraycolsep}{3.5pt}
\ee
The algorithm operates in two passes. First, going from $k=1$ to
$\Ngrid$, one defines recursively a sequence of {\Neq}--vectors
$\VecNeq{V}\rid{k}$ and $({\Neq}\times {\Neq})$--matrices
$\MatNeq{M}\rid{k}$ through
\be
\begin{aligned}
\VecNeq{V}\rid{1} &= \left(\DMatrix\rid{1}\right)^{-1} \VecNeq{B}\rid{1}\\
\MatNeq{M}\rid{1} &= \left(\DMatrix\rid{1}\right)^{-1} \EMatrix\rid{1} \\
\VecNeq{V}\rid{k} &= \left(\DMatrix\rid{k}-\CMatrix\rid{k}\MatNeq{M}\rid{k-1}\right)^{-1} \left(\VecNeq{B}\rid{k}-\CMatrix\rid{k}\VecNeq{V}\rid{k-1}\right)\\
\MatNeq{M}\rid{k} &= \left(\DMatrix\rid{k}-\CMatrix\rid{k}\MatNeq{M}\rid{k-1}\right)^{-1} \EMatrix\rid{k}\qquad 2\le k \le \Ngrid.
\end{aligned}
\ee
$\MatNeq{M}\rid{\Ngrid}$ is not defined. In the second pass, the
values of the unknown $\VecNeq{V}\rid{k}$ are computed, climbing back
from $k=\Ngrid$ to $1$, with 
\be
\begin{aligned}
 \VecNeq{W}\rid{\Ngrid} &= \VecNeq{V}\rid{\Ngrid}\\
 \VecNeq{W}\rid{k} &= \VecNeq{V}\rid{k} - \MatNeq{M}\rid{k}\VecNeq{W}\rid{k+1}\qquad 1\le k \le \Ngrid-1.
\end{aligned}
\ee
Note that, with this algorithm, only $({\Neq}\times {\Neq})$ matrices
have to be inverted. We use Gauss elimination for this purpose because
this venerable technique proves to be robust enough to properly deal
with the kind of badly conditioned matrices that often appear in this
application.

The initial model for the Newton-Raphson algorithm is given by the
structure of the cluster at the previous time,
$\GrandVect{X}\itid{0}(t)=\GrandVect{X}(t-\Delta t)$ One iterates
until the following convergence criteria are met. Let us set $\delta
x\qid{i}\rid{k} \equiv
\left.x\qid{i}\rid{k}\right|\itid{m+1}-\left.x\qid{i}\rid{k}\right|\itid{m}$. Then,
the condition for logarithmic quantities is
\be
 \max_{i=1...\Neq} \frac{1}{\Ngrid} \sum_{k=1...\Ngrid} 
 \left(\delta x\qid{i}\rid{k}\right)^2  < \varepsilon_1,
\ee
with $\varepsilon_1=10^{-6}$. For velocities ($u$, $v_{\rm r}-u$,
$v_{\rm t}-u$), one checks
\be
\max_{i=1...\Neq} \frac{1}{\Ngrid} \sum_{k=1...\Ngrid} 
\left(\frac{ \delta x\qid{i}\rid{k} }{ x\qid{i}\rid{k}+\varepsilon_1 w\rid{k} }\right)^2  < \varepsilon_2,
\ee
with $\varepsilon_2=10^{-3}$ and $w\rid{k}=r\rid{k}(4\pi
G\rho\rid{k})^{1/2}$. Generally, two iterations are sufficient to
reach convergence.



%%%%%%% Bibliografia del cap�tol
\newpage
\thispagestyle{empty}
\begin{spacing}{0.95}
\cleardoublepage
\bibliographystyle{apalike}
\section{Literature of chapter \thechapter}
\bibliography{aamnem99,biblio_tesi}
\end{spacing}
\newpage






























diffusion_model.bbl
\begin{thebibliography}{}

\bibitem[{Aarseth}, 1999a]{Aarseth99a}
{Aarseth}, S.~J. (1999a).
\newblock {From NBODY1 to NBODY6: The Growth of an Industry}.
\newblock {\em PASP}, 111:1333--1346.

\bibitem[{Aarseth}, 1999b]{Aarseth99b}
{Aarseth}, S.~J. (1999b).
\newblock {Star Cluster Simulations: the State of the Art}.
\newblock {\em Celestial Mechanics and Dynamical Astronomy}, 73:127--137.

\bibitem[{Amaro-Seoane} et~al., 2004]{ASEtAl04}
{Amaro-Seoane}, P., {Freitag}, M., and {Spurzem}, R. (2004).
\newblock {Accretion of stars on to a massive black hole: A realistic diffusion
  model and numerical studies}.
\newblock {\em mn}.

\bibitem[{Bahcall} and {Wolf}, 1976]{BW76}
{Bahcall}, J.~N. and {Wolf}, R.~A. (1976).
\newblock Star distribution around a massive black hole in a globular cluster.
\newblock {\em ApJ}, 209:214--232.

\bibitem[{Bettwieser}, 1983]{Bettwieser83}
{Bettwieser}, E. (1983).
\newblock {A numerical method for the study of the gravothermal instability in
  star clusters}.
\newblock {\em MNRAS}, 203:811--831.

\bibitem[{Bettwieser} and {Spurzem}, 1986]{BS86}
{Bettwieser}, E. and {Spurzem}, R. (1986).
\newblock {Anisotropy in stellar dynamics}.
\newblock {\em A\&A}, 161:102--112.

\bibitem[{Bettwieser} and {Sugimoto}, 1984]{BS84}
{Bettwieser}, E. and {Sugimoto}, D. (1984).
\newblock Post-collapse evolution and gravothermal oscillation of globular
  clusters.
\newblock {\em MNRAS}, 208:493--509.

\bibitem[{Bettwieser} and {Sugimoto}, 1985]{BS85}
{Bettwieser}, E. and {Sugimoto}, D. (1985).
\newblock Gravitational n-body problem and the validity of concepts of the gas
  model.
\newblock {\em MNRAS}, 212:189--195.

\bibitem[{Binney} and {Tremaine}, 1987]{BT87}
{Binney}, J. and {Tremaine}, S. (1987).
\newblock {\em Galactic Dynamics}.
\newblock Princeton University Press.

\bibitem[{Chatterjee} et~al., 2002]{ChatterjeeEtAl02}
{Chatterjee}, P., {Hernquist}, L., and {Loeb}, A. (2002).
\newblock {Dynamics of a Massive Black Hole at the Center of a Dense Stellar
  System}.
\newblock {\em ApJ}, 572:371--381.

\bibitem[{Cohn} and {Kulsrud}, 1978]{CK78}
{Cohn}, H. and {Kulsrud}, R.~M. (1978).
\newblock The stellar distribution around a black hole - numerical integration
  of the fokker-planck equation.
\newblock {\em ApJ}, 226:1087--1108.

\bibitem[{da Costa}, 1981]{daCosta81}
{da Costa}, L.~N. (1981).
\newblock {Star distribution in the presence of a massive central gas cloud}.
\newblock {\em MNRAS}, 195:869--880.

\bibitem[{David} et~al., 1987a]{DDC87a}
{David}, L.~P., {Durisen}, R.~H., and {Cohn}, H.~N. (1987a).
\newblock The evolution of active galactic nuclei. i - models without stellar
  evolution.
\newblock {\em ApJ}, 313:556--575.

\bibitem[{David} et~al., 1987b]{DDC87b}
{David}, L.~P., {Durisen}, R.~H., and {Cohn}, H.~N. (1987b).
\newblock The evolution of active galactic nuclei. {II} - models with stellar
  evolution.
\newblock {\em ApJ}, 316:505--516.

\bibitem[{Dehnen}, 1993]{Dehnen93}
{Dehnen}, W. (1993).
\newblock {A Family of Potential-Density Pairs for Spherical Galaxies and
  Bulges}.
\newblock {\em MNRAS}, 265:250.

\bibitem[{Deiters} and {Spurzem}, 2001]{DS01}
{Deiters}, S. and {Spurzem}, R. (2001).
\newblock {Multi-mass Gaseous Models of Globular Clusters with Stellar
  Evolution}.
\newblock In {\em ASP Conf. Ser. 228: Dynamics of Star Clusters and the Milky
  Way}, page 416.

\bibitem[{Dokuchaev} and {Ozernoi}, 1977]{DO77a}
{Dokuchaev}, V.~I. and {Ozernoi}, L.~M. (1977).
\newblock Distribution of stars in the vicinity of a massive black hole.
\newblock {\em Sov. Astron. Lett.}, 3:112--114.

\bibitem[{Dorband} et~al., 2003]{DHM03}
{Dorband}, E.~N., {Hemsendorf}, M., and {Merritt}, D. (2003).
\newblock {Systolic and hyper-systolic algorithms for the gravitational N-body
  problem, with an application to Brownian motion}.
\newblock {\em Journal of Computational Physics}, 185:484--511.

\bibitem[{Duncan} and {Shapiro}, 1982]{DS82}
{Duncan}, M.~J. and {Shapiro}, S.~L. (1982).
\newblock Star clusters containing massive, central black holes. {IV} -
  galactic tidal fields.
\newblock {\em ApJ}, 253:921--938.

\bibitem[{Frank} and {Rees}, 1976]{FR76}
{Frank}, J. and {Rees}, M.~J. (1976).
\newblock Effects of massive central black holes on dense stellar systems.
\newblock {\em MNRAS}, 176:633--647.

\bibitem[{Fregeau} et~al., 2003]{FregeauEtAl03}
{Fregeau}, J.~M., {G{\" u}rkan}, M.~A., {Joshi}, K.~J., and {Rasio}, F.~A.
  (2003).
\newblock {Monte Carlo Simulations of Globular Cluster Evolution. III.
  Primordial Binary Interactions}.
\newblock {\em ApJ}, 593:772--787.

\bibitem[{Freitag}, 2003a]{Freitag03b}
{Freitag}, M. (2003a).
\newblock {Captures of stars by a massive black hole: Investigations in
  numerical stellar dynamics}.
\newblock to appear in the proceedings of ``The Astrophysics of Gravitational
  Wave Sources'', astro-ph/0306064.

\bibitem[{Freitag}, 2003b]{Freitag03}
{Freitag}, M. (2003b).
\newblock {Gravitational waves from stars orbiting the Sagittarius A* black
  hole}.
\newblock {\em ApJ Lett.}, 583:21--24.

\bibitem[{Freitag} and {Benz}, 2001]{FB01a}
{Freitag}, M. and {Benz}, W. (2001).
\newblock A new monte carlo code for star cluster simulations: I.~relaxation.
\newblock {\em A\&A}, 375:711--738.

\bibitem[{Freitag} and {Benz}, 2002]{FB02b}
{Freitag}, M. and {Benz}, W. (2002).
\newblock A new monte carlo code for star cluster simulations: {II}.~central
  black hole and stellar collisions.
\newblock {\em A\&A}, 394:345--374.

\bibitem[{Freitag} and {Benz}, 2004]{FB04}
{Freitag}, M. and {Benz}, W. (2004).
\newblock A comprehensive set of simulations of high-velocity collisions
  between main sequence stars.
\newblock submitted to MNRAS, astro-ph/0403621.

\bibitem[{Ghez} et~al., 2003]{GhezEtAl03}
{Ghez}, A.~M., {Duch{\^ e}ne}, G., {Matthews}, K., {Hornstein}, S.~D.,
  {Tanner}, A., {Larkin}, J., {Morris}, M., {Becklin}, E.~E., {Salim}, S.,
  {Kremenek}, T., {Thompson}, D., {Soifer}, B.~T., {Neugebauer}, G., and
  {McLean}, I. (2003).
\newblock {The First Measurement of Spectral Lines in a Short-Period Star Bound
  to the Galaxy's Central Black Hole: A Paradox of Youth}.
\newblock {\em ApJ Lett.}, 586:L127--L131.

\bibitem[{Giersz}, 1998]{Giersz98}
{Giersz}, M. (1998).
\newblock Monte carlo simulations of star clusters - i. first results.
\newblock {\em MNRAS}, 298:1239--1248.

\bibitem[{Giersz} and {Heggie}, 1994]{GH94}
{Giersz}, M. and {Heggie}, D.~C. (1994).
\newblock {Statistics of N-Body Simulations - Part Two - Equal Masses after
  Core Collapse}.
\newblock {\em MNRAS}, 270:298.

\bibitem[{Giersz} and {Spurzem}, 1994]{GS94}
{Giersz}, M. and {Spurzem}, R. (1994).
\newblock Comparing direct $n$-body integration with anisotropic gaseous models
  of star clusters.
\newblock {\em MNRAS}, 269:241.

\bibitem[{Giersz} and {Spurzem}, 2000]{GS00}
{Giersz}, M. and {Spurzem}, R. (2000).
\newblock A stochastic monte carlo approach to model real star cluster
  evolution - {II}. self-consistent models and primordial binaries.
\newblock {\em MNRAS}, 317:581.

\bibitem[{Giersz} and {Spurzem}, 2003]{GS03}
{Giersz}, M. and {Spurzem}, R. (2003).
\newblock {A stochastic Monte Carlo approach to modelling real star cluster
  evolution - III. Direct integration of three- and four-body interactions}.
\newblock {\em MNRAS}, 343:781--795.

\bibitem[{Goodman}, 1984]{Goodman84}
{Goodman}, J. (1984).
\newblock {Homologous evolution of stellar systems after core collapse}.
\newblock {\em ApJ}, 280:298--312.

\bibitem[{Gurevich}, 1964]{Gurevich64}
{Gurevich}, A. (1964).
\newblock Instability of the disturbed zone in the vicinity of a charged body
  in plasma.
\newblock {\em Geomag. Aeronom.}, 4:247--255.

\bibitem[{H{\' e}non}, 1965]{Henon65}
{H{\' e}non}, M. (1965).
\newblock {Sur l'{\' e}volution dynamique des amas globulaires. II. Amas
  isol\&eacute}.
\newblock {\em Annales d'Astrophysique}, 28:62.

\bibitem[{Hachisu} et~al., 1978]{HachisuEtAl78}
{Hachisu}, I., {Nakada}, Y., {Nomoto}, K., and {Sugimoto}, D. (1978).
\newblock {Post-collapse evolution of a gaseous cluster model}.
\newblock {\em Prog. Theor. Phys. 60, 393}, 393:60.

\bibitem[{Heggie}, 1984]{Heggie84}
{Heggie}, D.~C. (1984).
\newblock {Post-collapse evolution of a gaseous cluster model}.
\newblock {\em MNRAS}, 206:179--195.

\bibitem[{Heggie} and {Mathieu}, 1986]{HM86}
{Heggie}, D.~C. and {Mathieu}, R.~D. (1986).
\newblock Standardised units and time scales.
\newblock In {Hut}, P. and {McMillan}, S. L.~W., editors, {\em The Use of
  Supercomputers in Stellar Dynamics}, page 233. Springer-Verlag.

\bibitem[{Heggie} and {Ramamani}, 1989]{HR89}
{Heggie}, D.~C. and {Ramamani}, N. (1989).
\newblock {Evolution of star clusters after core collapse}.
\newblock 237:757--783.

\bibitem[{H{\'e}non}, 1971]{Henon71a}
{H{\'e}non}, M. (1971).
\newblock Monte carlo models of star clusters.
\newblock {\em Ap\&SS}, 13:284--299.

\bibitem[{Joshi} et~al., 2001]{JoshiEtAl01}
{Joshi}, K.~J., {Nave}, C.~P., and {Rasio}, F.~A. (2001).
\newblock {Monte Carlo Simulations of Globular Cluster Evolution. II. Mass
  Spectra, Stellar Evolution, and Lifetimes in the Galaxy}.
\newblock {\em ApJ}, 550:691--702.

\bibitem[{Lightman} and {Shapiro}, 1977]{LS77}
{Lightman}, A.~P. and {Shapiro}, S.~L. (1977).
\newblock The distribution and consumption rate of stars around a massive,
  collapsed object.
\newblock {\em ApJ}, 211:244--262.

\bibitem[{Lin} and {Tremaine}, 1980]{LT80}
{Lin}, D. N.~C. and {Tremaine}, S. (1980).
\newblock A reinvestigation of the standard model for the dynamics of a massive
  black hole in a globular cluster.
\newblock {\em ApJ}, 242:789--798.

\bibitem[{Louis} and {Spurzem}, 1991]{LS91}
{Louis}, P.~D. and {Spurzem}, R. (1991).
\newblock Anisotropic gaseous models for the evolution of star clusters.
\newblock {\em MNRAS}, 251:408--426.

\bibitem[{Lynden-Bell} and {Eggleton}, 1980]{LBE80}
{Lynden-Bell}, D. and {Eggleton}, P.~P. (1980).
\newblock On the consequences of the gravothermal catastrophe.
\newblock {\em MNRAS}, 191:483--498.

\bibitem[{Magorrian} and {Tremaine}, 1999]{MT99}
{Magorrian}, J. and {Tremaine}, S. (1999).
\newblock Rates of tidal disruption of stars by massive central black holes.
\newblock {\em MNRAS}, 309:447--460.

\bibitem[{Marchant} and {Shapiro}, 1979]{MS79}
{Marchant}, A.~B. and {Shapiro}, S.~L. (1979).
\newblock Star clusters containing massive, central black holes. {II} -
  self-consistent potentials.
\newblock {\em ApJ}, 234:317--328.

\bibitem[{Marchant} and {Shapiro}, 1980]{MS80}
{Marchant}, A.~B. and {Shapiro}, S.~L. (1980).
\newblock Star clusters containing massive, central black holes. {III} -
  evolution calculations.
\newblock {\em ApJ}, 239:685--704.

\bibitem[{McMillan} et~al., 1981]{McMLC81}
{McMillan}, S. L.~W., {Lightman}, A.~P., and {Cohn}, H. (1981).
\newblock Luminosity evolution of quasars and active galaxies - theoretical
  models of the evolving mass supply rate.
\newblock {\em ApJ}, 251:436--445.

\bibitem[{Milosavljevi{\' c}} and {Merritt}, 2003]{MM03}
{Milosavljevi{\' c}}, M. and {Merritt}, D. (2003).
\newblock {Long-Term Evolution of Massive Black Hole Binaries}.
\newblock {\em ApJ}, 596:860--878.

\bibitem[{Miralda-Escud{\'e}} and {Gould}, 2000]{MEG00}
{Miralda-Escud{\'e}}, J. and {Gould}, A. (2000).
\newblock A cluster of black holes at the galactic center.
\newblock {\em ApJ}, 545:847--853.

\bibitem[{Morris}, 1993]{Morris93}
{Morris}, M. (1993).
\newblock {Massive star formation near the Galactic center and the fate of the
  stellar remnants}.
\newblock {\em ApJ}, 408:496--506.

\bibitem[{Murphy} et~al., 1991]{MCD91}
{Murphy}, B.~W., {Cohn}, H.~N., and {Durisen}, R.~H. (1991).
\newblock Dynamical and luminosity evolution of active galactic nuclei - models
  with a mass spectrum.
\newblock {\em ApJ}, 370:60--77.

\bibitem[{Ozernoi} and {Reinhardt}, 1978]{OR78}
{Ozernoi}, L.~M. and {Reinhardt}, M. (1978).
\newblock {Massive black holes in nuclei of galaxies and quasars}.
\newblock {\em A\&AS}, 59:171--192.

\bibitem[{Peebles}, 1972]{Peebles72}
{Peebles}, P. J.~E. (1972).
\newblock Star distribution near a collapsed object.
\newblock {\em ApJ}, 178:371--376.

\bibitem[{Pfahl} and {Loeb}, 2003]{PL03}
{Pfahl}, E. and {Loeb}, A. (2003).
\newblock {Probing the Spacetime Around Sgr A* with Radio Pulsars}.
\newblock {\em ArXiv Astrophysics e-prints}.

\bibitem[{Sch{\" o}del} et~al., 2003]{SchoedelEtAl03}
{Sch{\" o}del}, R., {Ott}, T., {Genzel}, R., {Eckart}, A., {Mouawad}, N., and
  {Alexander}, T. (2003).
\newblock {Stellar Dynamics in the Central Arcsecond of Our Galaxy}.
\newblock {\em ApJ}, 596:1015--1034.

\bibitem[{Shapiro}, 1977]{Shapiro77}
{Shapiro}, S.~L. (1977).
\newblock The dissolution of globular clusters containing massive black holes.
\newblock {\em ApJ}, 217:281--286.

\bibitem[{Shapiro} and {Lightman}, 1976]{SL76}
{Shapiro}, S.~L. and {Lightman}, A.~P. (1976).
\newblock The distribution of stars around a massive black hole.
\newblock {\em Nat}, 262:743--745.

\bibitem[{Sigurdsson}, 2003]{Sigurdsson03}
{Sigurdsson}, S. (2003).
\newblock {The loss cone: past, present and future}.
\newblock {\em Classical and Quantum Gravity}, 20:45.

\bibitem[{Spitzer}, 1987]{Spitzer87}
{Spitzer}, L. (1987).
\newblock {\em Dynamical evolution of globular clusters}.
\newblock Princeton University Press.

\bibitem[{Spurzem}, 1991]{Spurzem91}
{Spurzem}, R. (1991).
\newblock {Gravothermal instability of anisotropic self-gravitating gas spheres
  - Singular equilibrium solution}.
\newblock {\em MNRAS}, 252:177--189.

\bibitem[{Spurzem}, 1992]{Spurzem92}
{Spurzem}, R. (1992).
\newblock Evolution of stars and gas in galactic nuclei.
\newblock In {\em Reviews of Modern Astronomy}, volume~5, pages 161--173.

\bibitem[{Spurzem}, 1999]{Spurzem99}
{Spurzem}, R. (1999).
\newblock Direct $n$-body simulations.
\newblock {\em Journal of Computational and Applied Mathematics}, 109:407--432.

\bibitem[{Spurzem} and {Aarseth}, 1996]{SA96}
{Spurzem}, R. and {Aarseth}, S.~J. (1996).
\newblock Direct collisional simulation of 100000 particles past core collapse.
\newblock {\em MNRAS}, 282:19.

\bibitem[{Spurzem} et~al., 2003]{SpurzemEtAl03}
{Spurzem}, R., {Berczik}, P., {Hensler}, G., {Theis}, C., {Amaro-Seoane}, P.,
  {Freitag}, M., and {Just}, A. (2003).
\newblock {Physical Processes in Star-Gas Systems}.
\newblock {\em ArXiv Astrophysics e-prints}.

\bibitem[{Spurzem} and {Takahashi}, 1995]{ST95}
{Spurzem}, R. and {Takahashi}, K. (1995).
\newblock Comparison between fokker-planck and gaseous models of star clusters
  in the multi-mass case revisited.
\newblock {\em MNRAS}, 272:772--784.

\bibitem[{Syer} and {Ulmer}, 1999]{SU99}
{Syer}, D. and {Ulmer}, A. (1999).
\newblock Tidal disruption rates of stars in observed galaxies.
\newblock {\em MNRAS}, 306:35--42.

\bibitem[{Takahashi}, 1995]{Takahashi95}
{Takahashi}, K. (1995).
\newblock Fokker-planck models of star clusters with anisotropic velocity
  distributions i. pre-collapse evolution.
\newblock {\em PASJ}, 47:561--573.

\bibitem[{Takahashi}, 1996]{Takahashi96}
{Takahashi}, K. (1996).
\newblock Fokker-planck models of star clusters with anisotropic velocity
  distributions {II}. post-collapse evolution.
\newblock {\em PASJ}, 48:691--700.

\bibitem[{Takahashi}, 1997]{Takahashi97}
{Takahashi}, K. (1997).
\newblock Fokker-planck models of star clusters with anisotropic velocity
  distributions {III}. multi-mass clusters.
\newblock {\em PASJ}, 49:547--560.

\bibitem[{Tremaine} et~al., 1994]{Tremaine94}
{Tremaine}, S., {Richstone}, D.~O., {Byun}, Y., {Dressler}, A., {Faber}, S.~M.,
  {Grillmair}, C., {Kormendy}, J., and {Lauer}, T.~R. (1994).
\newblock {A family of models for spherical stellar systems}.
\newblock {\em AJ}, 107:634--644.

\end{thebibliography}

mass_seg.bbl
\begin{thebibliography}{}

\bibitem[{Amaro-Seoane} and {Spurzem}, 2003]{AS03}
{Amaro-Seoane}, P. and {Spurzem} (2003).
\newblock {The anisotropic gaseous model 'Spedi' : A tool to study the
  formation and growth of black holes in galactic nuclei}.
\newblock {\em In "The formation and early evolution of galaxies", workshop
  held in Irsee, Germany, \footnote{Available at~{\tt
  http://www.mpia-hd.mpg.de/GALAXIES/CADIS/irsee2003/PROCEEDINGS/Amaro-Seoane.%
pdf}}}.

\bibitem[{Cohn}, 1980]{Cohn80}
{Cohn}, H. (1980).
\newblock Late core collapse in star clusters and the gravothermal instability.
\newblock {\em ApJ}, 242:765--771.

\bibitem[{Drukier} et~al., 1999]{DCLY99}
{Drukier}, G.~A., {Cohn}, H.~N., {Lugger}, P.~M., and {Yong}, H. (1999).
\newblock Anisotropic fokker-planck models for the evolution of globular star
  clusters: The core-halo connection.
\newblock {\em ApJ}, 518:233--245.

\bibitem[{Elson} et~al., 1987]{ElsonEtAl87}
{Elson}, R., {Hut}, P., and {Inagaki}, S. (1987).
\newblock {Dynamical evolution of globular clusters}.
\newblock {\em araa}, 25:565--601.

\bibitem[{G{\" u}rkan} et~al., 2004]{GFR04}
{G{\" u}rkan}, M.~A., {Freitag}, M., and {Rasio}, F.~A. (2004).
\newblock {Formation of Massive Black Holes in Dense Star Clusters. I. Mass
  Segregation and Core Collapse}.
\newblock {\em apj}, 604:632--652.

\bibitem[{Giersz} and {Heggie}, 1994]{GH94b}
{Giersz}, M. and {Heggie}, D.~C. (1994).
\newblock Statistics of n-body simulations - part two - equal masses after core
  collapse.
\newblock {\em MNRAS}, 270:298.

\bibitem[{Giersz} and {Heggie}, 1996]{GH96}
{Giersz}, M. and {Heggie}, D.~C. (1996).
\newblock Statistics of $n$-body simulations - {III}. unequal masses.
\newblock {\em MNRAS}, 279:1037--1056.

\bibitem[{H{\'{e}}non}, 1973]{Henon73}
{H{\'{e}}non}, M. (1973).
\newblock Collisional dynamics of spherical stellar systems.
\newblock In {Martinet}, L. and {Mayor}, M., editors, {\em Dynamical structure
  and evolution of stellar systems, Lectures of the 3rd Advanced Course of the
  Swiss Society for Astronomy and Astrophysics (SSAA)}, pages 183--260.

\bibitem[{H{\'e}non}, 1975]{Henon75}
{H{\'e}non}, M. (1975).
\newblock Two recent developments concerning the monte carlo method.
\newblock In {Hayli}, A., editor, {\em IAU Symp. 69: Dynamics of Stellar
  Systems}, page 133.

\bibitem[{Hillenbrand} and {Hartmann}, 1998]{HH98}
{Hillenbrand}, L.~A. and {Hartmann}, L.~W. (1998).
\newblock {A Preliminary Study of the Orion Nebula Cluster Structure and
  Dynamics}.
\newblock {\em apj}, 492:540.

\bibitem[{Inagaki} and {Saslaw}, 1985]{IS85}
{Inagaki}, S. and {Saslaw}, W.~C. (1985).
\newblock Equipartition in multicomponent gravitational systems.
\newblock {\em ApJ}, 292:339--347.

\bibitem[{Inagaki} and {Wiyanto}, 1984]{IW84}
{Inagaki}, S. and {Wiyanto}, P. (1984).
\newblock On equipartition of kinetic energies in two-component star clusters.
\newblock {\em PASJ}, 36:391--402.

\bibitem[{Joshi} et~al., 2000]{JRPZ00}
{Joshi}, K.~J., {Rasio}, F.~A., and {Portegies Zwart}, S. (2000).
\newblock Monte carlo simulations of globular cluster evolution. i. method and
  test calculations.
\newblock {\em ApJ}, 540:969--982.

\bibitem[{Khalisi}, 2002]{Khalisi02}
{Khalisi}, E. (2002).
\newblock Equipartition and mass segregation: Simulations of star clusters with
  two-mass components.
\newblock {\em Doctoral dissertation. Ruperto-Carola University of Heidelberg,
  Germany \footnote{Idem {\tt http://www.ub.uni-heidelberg.de/archiv/3096}}},
  349.

\bibitem[{Kim} et~al., 1998]{KLG98}
{Kim}, S.~S., {Lee}, H.~M., and {Goodman}, J. (1998).
\newblock {Two-Component Fokker-Planck Models for the Evolution of Isolated
  Globular Clusters}.
\newblock {\em apj}, 495:786.

\bibitem[{Kroupa}, 2001]{Kroupa01}
{Kroupa}, P. (2001).
\newblock {On the variation of the initial mass function}.
\newblock {\em mnras}, 322:231--246.

\bibitem[{Kroupa} et~al., 1993]{KroupaEtAl93}
{Kroupa}, P., {Tout}, C.~A., and {Gilmore}, G. (1993).
\newblock {The distribution of low-mass stars in the Galactic disc}.
\newblock {\em mnras}, 262:545--587.

\bibitem[{Lightman} and {Fall}, 1978]{LF78}
{Lightman}, A.~P. and {Fall}, S.~M. (1978).
\newblock {An approximate theory for the core collapse of two-component
  gravitating systems}.
\newblock {\em apj}, 221:567--579.

\bibitem[{Makino}, 1996]{Makino96}
{Makino}, J. (1996).
\newblock Postcollapse evolution of globular clusters.
\newblock {\em ApJ}, 471:796.

\bibitem[{Marchant} and {Shapiro}, 1980]{MS80}
{Marchant}, A.~B. and {Shapiro}, S.~L. (1980).
\newblock Star clusters containing massive, central black holes. {III} -
  evolution calculations.
\newblock {\em ApJ}, 239:685--704.

\bibitem[{McCaughrean} and {Stauffer}, 1994]{McMS94}
{McCaughrean}, M.~J. and {Stauffer}, J.~R. (1994).
\newblock High resolution near-infrared imaging of the trapezium: A stellar
  census.
\newblock {\em AJ}, 108:1382--1397.

\bibitem[{Portegies Zwart} and {McMillan}, 2000]{PZMM00}
{Portegies Zwart}, S.~F. and {McMillan}, S.~L.~W. (2000).
\newblock {Black Hole Mergers in the Universe}.
\newblock {\em apjl}, 528:L17--L20.

\bibitem[{Quinlan}, 1996]{Quinlan96}
{Quinlan}, G.~D. (1996).
\newblock The time-scale for core collapse in spherical star clusters.
\newblock {\em New Astronomy}, 1:255--270.

\bibitem[{Raboud} and {Mermilliod}, 1998]{RM98}
{Raboud}, D. and {Mermilliod}, J.-C. (1998).
\newblock {Evolution of mass segregation in open clusters: some observational
  evidences}.
\newblock {\em aa}, 333:897--909.

\bibitem[{Spitzer} and {Shull}, 1975]{SS75a}
{Spitzer}, L. and {Shull}, J.~M. (1975).
\newblock Random gravitational encounters and the evolution of spherical
  systems. {VI} - plummer's model.
\newblock {\em ApJ}, 200:339--342.

\bibitem[{Spitzer}, 1969]{Spitzer69}
{Spitzer}, L.~J. (1969).
\newblock Equipartition and the formation of compact nuclei in spherical
  stellar systems.
\newblock {\em ApJ Lett.}, 158:139.

\bibitem[{Spitzer} and {Hart}, 1971]{SH71b}
{Spitzer}, L.~J. and {Hart}, M.~H. (1971).
\newblock Random gravitational encounters and the evolution of spherical
  systems. {II}. models.
\newblock {\em ApJ}, 166:483.

\bibitem[{Spurzem} and {Aarseth}, 1996]{SA96}
{Spurzem}, R. and {Aarseth}, S.~J. (1996).
\newblock Direct collisional simulation of 100000 particles past core collapse.
\newblock {\em MNRAS}, 282:19.

\bibitem[{Stodo{\l}kiewicz}, 1982]{Stodol82}
{Stodo{\l}kiewicz}, J.~S. (1982).
\newblock Dynamical evolution of globular clusters. i.
\newblock {\em Acta Astron.}, 32:63--91.

\bibitem[{Takahashi}, 1993]{Takahashi93}
{Takahashi}, K. (1993).
\newblock A new powerful method for solving the orbit-averaged fokker-planck
  equation regarding stellar dynamics.
\newblock {\em PASJ}, 45:233--242.

\bibitem[{Takahashi}, 1995]{Takahashi95}
{Takahashi}, K. (1995).
\newblock Fokker-planck models of star clusters with anisotropic velocity
  distributions i. pre-collapse evolution.
\newblock {\em PASJ}, 47:561--573.

\bibitem[{Watters} et~al., 2000]{WJR00}
{Watters}, W.~A., {Joshi}, K.~J., and {Rasio}, F.~A. (2000).
\newblock Thermal and dynamical equilibrium in two-component star clusters.
\newblock {\em ApJ}, 539:331--341.

\end{thebibliography}

motiv.bbl
\begin{thebibliography}{}

\bibitem[{Amaro-Seoane} and {Spurzem}, 2001]{AS01}
{Amaro-Seoane}, P. and {Spurzem}, R. (2001).
\newblock {The loss-cone problem in dense nuclei}.
\newblock {\em MNRAS}, 327:995--1003.

\bibitem[{Arons} et~al., 1975]{AKO75}
{Arons}, J., {Kulsrud}, R.~M., and {Ostriker}, J.~P. (1975).
\newblock {A multiple pulsar model for quasi-stellar objects and active
  galactic nuclei}.
\newblock {\em apj}, 198:687--705.

\bibitem[{Bahcall} and {Ostriker}, 1975]{BO75}
{Bahcall}, J.~N. and {Ostriker}, J.~P. (1975).
\newblock {Massive black holes in globular clusters}.
\newblock {\em nat}, 256:23.

\bibitem[{Baumgardt} et~al., 2003a]{BaumgardtEtAl03}
{Baumgardt}, H., {Hut}, P., {Makino}, J., {McMillan}, S., and {Portegies
  Zwart}, S. (2003a).
\newblock {On the Central Structure of M15}.
\newblock {\em ApJ Lett.}, 582:L21--L24.

\bibitem[{Baumgardt} et~al., 2003b]{BaumgardtEtAl03b}
{Baumgardt}, H., {Makino}, J., {Hut}, P., {McMillan}, S., and {Portegies
  Zwart}, S. (2003b).
\newblock {A Dynamical Model for the Globular Cluster G1}.
\newblock {\em ApJ Lett.}, 589:L25--L28.

\bibitem[{Begelman} and {Rees}, 1978]{BR78}
{Begelman}, M.~C. and {Rees}, M.~J. (1978).
\newblock The fate of dense stellar systems.
\newblock {\em MNRAS}, 185:847--860.

\bibitem[{Bond} et~al., 1984]{BA84}
{Bond}, J.~R., {Arnett}, W.~D., and {Carr}, B.~J. (1984).
\newblock {The evolution and fate of Very Massive Objects}.
\newblock {\em apj}, 280:825--847.

\bibitem[{Burkert}, 2000]{Burkert00}
{Burkert}, A. (2000).
\newblock {The Structure and Evolution of Weakly Self-interacting Cold Dark
  Matter Halos}.
\newblock {\em ApJ Lett.}, 534:143--146.

\bibitem[{Chandrasekhar}, 1942]{Chandra42}
{Chandrasekhar}, S. (1942).
\newblock {Principles of stellar dynamics}.
\newblock {\em Physical Sciences Data}.

\bibitem[{Chandrasekhar}, 1964]{Chandra64}
{Chandrasekhar}, S. (1964).
\newblock {The Dynamical Instability of Gaseous Masses Approaching the
  Schwarzschild Limit in General Relativity.}
\newblock {\em apj}, 140:417.

\bibitem[{Colgate}, 1967]{Colgate67}
{Colgate}, S.~A. (1967).
\newblock Stellar coalescence and the multiple supernova interpretation of
  quasi-stellar sources.
\newblock {\em ApJ}, 150:163.

\bibitem[{Diaferio} et~al., 1999]{DiaferioEtAl99}
{Diaferio}, A., {Kauffmann}, G., {Colberg}, J.~M., and {White}, S.~D.~M.
  (1999).
\newblock {Clustering of galaxies in a hierarchical universe - III. Mock
  redshift surveys}.
\newblock {\em mnras}, 307:537--552.

\bibitem[{Duncan} and {Shapiro}, 1982]{DS82}
{Duncan}, M.~J. and {Shapiro}, S.~L. (1982).
\newblock Star clusters containing massive, central black holes. {IV} -
  galactic tidal fields.
\newblock {\em ApJ}, 253:921--938.

\bibitem[{Duncan} and {Shapiro}, 1983]{DS83}
{Duncan}, M.~J. and {Shapiro}, S.~L. (1983).
\newblock Monte carlo simulations of the evolution of galactic nuclei
  containing massive, central black holes.
\newblock {\em ApJ}, 268:565--581.

\bibitem[{Ebisuzaki} et~al., 2001]{EbisuzakiEtAl01}
{Ebisuzaki}, T., {Makino}, J., {Tsuru}, T.~G., {Funato}, Y., {Portegies Zwart},
  S., {Hut}, P., {McMillan}, S., {Matsushita}, S., {Matsumoto}, H., and
  {Kawabe}, R. (2001).
\newblock {Missing Link Found? The ``Runaway'' Path to Supermassive Black
  Holes}.
\newblock {\em ApJ Lett.}, 562:L19--L22.

\bibitem[{Ferrarese} et~al., 2001]{FPPMWJ01}
{Ferrarese}, L., {Pogge}, R.~W., {Peterson}, B.~M., {Merritt}, D., {Wandel},
  A., and {Joseph}, C.~L. (2001).
\newblock {Supermassive Black Holes in Active Galactic Nuclei. I. The
  Consistency of Black Hole Masses in Quiescent and Active Galaxies}.
\newblock {\em ApJ Lett.}, 555:79--82.

\bibitem[{Frank} et~al., 2002]{FKR02}
{Frank}, J., {King}, A., and {Raine}, D.~J. (2002).
\newblock {\em {Accretion Power in Astrophysics: Third Edition}}.
\newblock Accretion Power in Astrophysics: Third Edition, by Juhan Frank,
  Andrew King, and Derek J.~Raine.~Cambridge University Press, 2002, 398 pp.

\bibitem[{Freitag} and {Benz}, 2002]{FB02b}
{Freitag}, M. and {Benz}, W. (2002).
\newblock A new monte carlo code for star cluster simulations: {II}.~central
  black hole and stellar collisions.
\newblock {\em A\&A}, 394:345--374.

\bibitem[{Fuller} et~al., 1986]{FW86}
{Fuller}, G.~M., {Woosley}, S.~E., and {Weaver}, T.~A. (1986).
\newblock {The evolution of radiation-dominated stars. I - Nonrotating
  supermassive stars}.
\newblock {\em apj}, 307:675--686.

\bibitem[{Gebhardt} et~al., 2001]{GebhardtEtAl01}
{Gebhardt}, K., {Lauer}, T.~R., {Kormendy}, J., {Pinkney}, J., {Bower}, G.,
  {Green}, R., {Gull}, T., {Hutchings}, J.~B., {Kaiser}, M.~E., {Nelson},
  C.~H., {Richstone}, D., and {Weistrop}, D. (2001).
\newblock M33: A galaxy with no supermassive black hole.
\newblock {\em AJ}, 122:2469--2476.

\bibitem[{Gebhardt} et~al., 2002]{GRH02}
{Gebhardt}, K., {Rich}, R.~M., and {Ho}, L.~C. (2002).
\newblock {A $20000 M_\odot$ Black Hole in the Stellar Cluster G1}.
\newblock {\em ApJ Lett.}, 578:41--45.

\bibitem[{Gerssen} et~al., 2002]{GerssenEtAl02}
{Gerssen}, J., {van der Marel}, R.~P., {Gebhardt}, K., {Guhathakurta}, P.,
  {Peterson}, R.~C., and {Pryor}, C. (2002).
\newblock {Hubble Space Telescope Evidence for an Intermediate-Mass Black Hole
  in the Globular Cluster M15. II. Kinematic Analysis and Dynamical Modeling}.
\newblock {\em AJ}, 124:3270--3288.

\bibitem[{Gerssen} et~al., 2003]{GerssenEtAl03}
{Gerssen}, J., {van der Marel}, R.~P., {Gebhardt}, K., {Guhathakurta}, P.,
  {Peterson}, R.~C., and {Pryor}, C. (2003).
\newblock {Addendum: Hubble Space Telescope Evidence for an Intermediate-Mass
  Black Hole in the Globular Cluster M15. II. Kinematic Analysis and Dynamical
  Modeling}.
\newblock {\em AJ}, 125:376--377.

\bibitem[{Gezari} et~al., 2003]{GezariEtal03}
{Gezari}, S., {Halpern}, J.~P., {Komossa}, S., {Grupe}, D., and {Leighly},
  K.~M. (2003).
\newblock {Follow-Up Hubble Space Telescope/Space Telescope Imaging
  Spectroscopy of Three Candidate Tidal Disruption Events}.
\newblock {\em apj}, 592:42--51.

\bibitem[{Ghez} et~al., 2003]{GhezEtAl03}
{Ghez}, A.~M., {Duch{\^ e}ne}, G., {Matthews}, K., {Hornstein}, S.~D.,
  {Tanner}, A., {Larkin}, J., {Morris}, M., {Becklin}, E.~E., {Salim}, S.,
  {Kremenek}, T., {Thompson}, D., {Soifer}, B.~T., {Neugebauer}, G., and
  {McLean}, I. (2003).
\newblock {The First Measurement of Spectral Lines in a Short-Period Star Bound
  to the Galaxy's Central Black Hole: A Paradox of Youth}.
\newblock {\em ApJ Lett.}, 586:L127--L131.

\bibitem[{Ginzburg} and {Ozernoy}, 1964]{GO64}
{Ginzburg}, V. and {Ozernoy}, L.~M. (1964).
\newblock {\em Sov. Phys. JETP}, 20:489.

\bibitem[{Goldreich} and {Weber}, 1980]{GW80}
{Goldreich}, P. and {Weber}, S.~V. (1980).
\newblock {Homologously collapsing stellar cores}.
\newblock {\em apj}, 238:991--997.

\bibitem[{Hara}, 1978]{Hara78}
{Hara}, T. (1978).
\newblock Evolution of a super-massive star in a dense stellar system.
\newblock {\em Progress of Th. Phys.}, 60:711--723.

\bibitem[{Heckman}, 1980]{Heckman80}
{Heckman}, T.~M. (1980).
\newblock {An optical and radio survey of the nuclei of bright galaxies -
  Activity in normal galactic nuclei}.
\newblock {\em aap}, 87:152--164.

\bibitem[{Hemsendorf} et~al., 2001]{HSS01}
{Hemsendorf}, M., {Sigurdsson}, S., and {Spurzem}, R. (2001).
\newblock {Binary black holes in galactic centres}.
\newblock {\em EAS Publications Series, Volume 1.~Active galactic nuclei in
  their cosmic environment.~JENAM 99, Toulouse, France, September 7-9,
  1999.~Ed.~by B.~Rocca-Volmerange and H.~Sol.~Les Ulis: EDP Sciences, ISBN
  2-86883-563-5, 2001, p.~173}, 1:173.

\bibitem[{Herrnstein} et~al., 1999]{Herrnstein99}
{Herrnstein}, J.~R., {Moran}, J.~M., {Greenhill}, L.~J., {Diamond}, P.~J.,
  {Inoue}, M., {Nakai}, N., {Miyoshi}, M., {Henkel}, C., and {Riess}, A.
  (1999).
\newblock A geometric distance to the galaxy ngc 4258 from orbital motions in a
  nuclear gas disk.
\newblock {\em Nat}, 400:539--541.

\bibitem[{Hills}, 1975]{Hills75}
{Hills}, J.~G. (1975).
\newblock {Possible power source of Seyfert galaxies and QSOs}.
\newblock {\em Nat}, 254:295--298.

\bibitem[{Hoyle} and {Fowler}, 1963]{HF63}
{Hoyle}, F. and {Fowler}, W.~A. (1963).
\newblock {On the nature of strong radio sources}.
\newblock {\em mnras}, 125:169.

\bibitem[{Jarrett} et~al., 2003]{JarretEtAl03}
{Jarrett}, T.~H., {Chester}, T., {Cutri}, R., {Schneider}, S.~E., and {Huchra},
  J.~P. (2003).
\newblock {The 2MASS Large Galaxy Atlas}.
\newblock {\em aj}, 125:525--554.

\bibitem[{Kauffmann} et~al., 1999a]{KauffmanEtAl99a}
{Kauffmann}, G., {Colberg}, J.~M., {Diaferio}, A., and {White}, S.~D.~M.
  (1999a).
\newblock {Clustering of galaxies in a hierarchical universe - I. Methods and
  results at z=0}.
\newblock {\em mnras}, 303:188--206.

\bibitem[{Kauffmann} et~al., 1999b]{KauffmanEtAl99b}
{Kauffmann}, G., {Colberg}, J.~M., {Diaferio}, A., and {White}, S.~D.~M.
  (1999b).
\newblock {Clustering of galaxies in a hierarchical universe - I. Methods and
  results at z=0}.
\newblock {\em mnras}, 303:188--206.

\bibitem[{Kormendy}, 2003]{Kormendy03}
{Kormendy}, J. (2003).
\newblock {The Stellar-Dynamical Search for Supermassive Black Holes in
  Galactic Nuclei}.
\newblock In {Ho}, L., editor, {\em ``Coevolution of Black Holes and
  Galaxies'', Carnegie Observatories, Pasadena}.
\newblock astro-ph/0306353.

\bibitem[{Krolik}, 1999]{Krolik99}
{Krolik}, J.~H. (1999).
\newblock {\em Active galactic nuclei: from the central black hole to the
  galactic environment}.
\newblock Princeton University Press.

\bibitem[{Langbein} et~al., 1990]{LangbeinEtAl90}
{Langbein}, T., {Fricke}, K.~J., {Spurzem}, R., and {Yorke}, H.~W. (1990).
\newblock {Interactions between stars and gas in galactic nuclei}.
\newblock {\em aa}, 227:333--341.

\bibitem[{Larson}, 1970]{Larson70a}
{Larson}, R.~B. (1970).
\newblock A method for computing the evolution of star clusters.
\newblock {\em MNRAS}, 147:323.

\bibitem[{Lee}, 1995]{Lee95}
{Lee}, H.~M. (1995).
\newblock Evolution of galactic nuclei with 10-m\_ black holes.
\newblock {\em MNRAS}, 272:605--617.

\bibitem[{Lynden-Bell}, 1967]{LyndenBell67}
{Lynden-Bell}, D. (1967).
\newblock Statistical mechanics of violent relaxation in stellar systems.
\newblock {\em MNRAS}, 136:101.

\bibitem[{Lynden-Bell}, 1969]{LB69}
{Lynden-Bell}, D. (1969).
\newblock {Galactic Nuclei as Collapsed Old Quasars}.
\newblock {\em nat}, 223:690.

\bibitem[{Lynden-Bell} and {Rees}, 1971a]{LR71}
{Lynden-Bell}, D. and {Rees}, M.~J. (1971a).
\newblock {On quasars, dust and the galactic centre}.
\newblock {\em MNRAS}, 152:461.

\bibitem[{Lynden-Bell} and {Rees}, 1971b]{LBR71}
{Lynden-Bell}, D. and {Rees}, M.~J. (1971b).
\newblock {On quasars, dust and the galactic centre}.
\newblock {\em mnras}, 152:461.

\bibitem[{Makino}, 1997]{Makino97}
{Makino}, J. (1997).
\newblock {Merging of Galaxies with Central Black Holes. II. Evolution of the
  Black Hole Binary and the Structure of the Core}.
\newblock {\em apj}, 478:58.

\bibitem[{Makino} and {Ebisuzaki}, 1996]{ME96}
{Makino}, J. and {Ebisuzaki}, T. (1996).
\newblock {Merging of Galaxies with Central Black Holes. I. Hierarchical
  Mergings of Equal-Mass Galaxies}.
\newblock {\em apj}, 465:527.

\bibitem[{Makino} and {Taiji}, 1998]{MT98}
{Makino}, J. and {Taiji}, M. (1998).
\newblock {\em {Scientific simulations with special-purpose computers : The
  GRAPE systems}}.
\newblock Scientific simulations with special-purpose computers : The GRAPE
  systems /by Junichiro Makino \& Makoto Taiji.~Chichester ; Toronto : John
  Wiley \& Sons, c1998.

\bibitem[{McCray}, 1979]{McCray79}
{McCray}, R. (1979).
\newblock {\em {Spherical accretion onto supermassive black holes}}, pages
  227--239.
\newblock Active galactic nuclei.~(A79-50785 22-90) Cambridge, Cambridge
  University Press, 1979, p.~227-239.~ NSF-supported research.

\bibitem[{Merritt} et~al., 2001]{MFJ01}
{Merritt}, D., {Ferrarese}, L., and {Joseph}, C.~L. (2001).
\newblock {No Supermassive Black Hole in M33?}
\newblock {\em Science}, 293:1116--1119.

\bibitem[{Merritt} and {Poon, M. Y.}, 2003]{MP03}
{Merritt}, D. and {Poon, M. Y.} (2003).
\newblock {Chaotic Loss Cones, Black Hole Fueling and the M-Sigma Relation}.
\newblock {\em ArXiv Astrophysics e-prints}.

\bibitem[{Michell}, 1784]{Michell1784}
{Michell}, J. (1784).
\newblock {On the Means of Discovering the Distance, Magnitude, {\& }c. of the
  Fixed Stars, in Consequence of the Diminution of the Velocity of Their Light,
  in Case Such a Diminution Should be Found to Take Place in any of Them, and
  Such Other Data Should be Procured from Observations, as Would be Farther
  Necessary for That Purpose. By the Rev. John Michell, B. D. F. R. S. In a
  Letter to Henry Cavendish, Esq. F. R. S. and A. S.}
\newblock {\em Philosophical Transactions Series I}, 74:35--57.

\bibitem[{Miller} and {Colbert}, 2003]{MC03}
{Miller}, M.~C. and {Colbert}, E.~J.~M. (2003).
\newblock {Intermediate-Mass Black Holes}.
\newblock preprint, astro-ph/0308402.

\bibitem[{Milosavljevi{\'{c}}} and {Merrit}, 2001]{MM01}
{Milosavljevi{\'{c}}}, M. and {Merrit} (2001).
\newblock {Formation of Galactic Nuclei}.
\newblock {\em ApJ}, 563:34--62.

\bibitem[{Miyoshi} et~al., 1995]{Miyoshi95}
{Miyoshi}, M., {Moran}, J., {Herrnstein}, J., {Greenhill}, L., {Nakai}, N.,
  {Diamond}, P., and {Inoue}, M. (1995).
\newblock Evidence for a black-hole from high rotation velocities in a
  sub-parsec region of ngc4258.
\newblock {\em Nat}, 373:127.

\bibitem[{Moran} et~al., 1999]{MoranEtAl99}
{Moran}, J.~M., {Greenhill}, L.~J., and {Herrnstein}, J.~R. (1999).
\newblock {Observational Evidence for Massive Black Holes in the Centers of
  Active Galaxies}.
\newblock {\em Journal of Astrophysics and Astronomy}, 20:165.

\bibitem[{Murphy} et~al., 1991]{MCD91}
{Murphy}, B.~W., {Cohn}, H.~N., and {Durisen}, R.~H. (1991).
\newblock Dynamical and luminosity evolution of active galactic nuclei - models
  with a mass spectrum.
\newblock {\em ApJ}, 370:60--77.

\bibitem[{Perley} et~al., 1984]{PerleyEtAl84}
{Perley}, R.~A., {Dreher}, J.~W., and {Cowan}, J.~J. (1984).
\newblock {The jet and filaments in Cygnus A}.
\newblock {\em apjl}, 285:L35--L38.

\bibitem[{Pian} et~al., 1997]{PianEtAl97}
{Pian}, E., {Urry}, C.~M., {Treves}, A., {Maraschi}, L., {Penton}, S., {Shull},
  J.~M., {Pesce}, J.~E., {Grandi}, P., {Kii}, T., {Kollgaard}, R.~I.,
  {Madejski}, G., {Marshall}, H.~L., {Wamsteker}, W., {Celotti}, A.,
  {Courvoisier}, T.~J.-L., {Falomo}, R., {Fink}, H.~H., {George}, I.~M., and
  {Ghisellini}, G. (1997).
\newblock {Multiwavelength Monitoring of the BL Lacertae Object PKS 2155-304 in
  1994 May. II. The IUE Campaign}.
\newblock {\em apj}, 486:784.

\bibitem[{Pinkney} et~al., 2003]{PinkneyEtAl03}
{Pinkney}, J., {Gebhardt}, K., {Bender}, R., {Bower}, G., {Dressler}, A.,
  {Faber}, S.~M., {Filippenko}, A.~V., {Green}, R., {Ho}, L.~C., {Kormendy},
  J., {Lauer}, T.~R., {Magorrian}, J., {Richstone}, D., and {Tremaine}, S.
  (2003).
\newblock {Kinematics of 10 Early-Type Galaxies from Hubble Space Telescope and
  Ground-based Spectroscopy}.
\newblock {\em ApJ}, 596:903--929.

\bibitem[{Portegies Zwart} and {McMillan}, 2002]{PZMcM02}
{Portegies Zwart}, S.~F. and {McMillan}, S. L.~W. (2002).
\newblock {The Runaway Growth of Intermediate-Mass Black Holes in Dense Star
  Clusters}.
\newblock {\em ApJ}, 576:899--907.

\bibitem[{Quinlan} and {Shapiro}, 1987]{QS87}
{Quinlan}, G.~D. and {Shapiro}, S.~L. (1987).
\newblock The collapse of dense star clusters to supermassive black holes -
  binaries and gravitational radiation.
\newblock {\em ApJ}, 321:199--210.

\bibitem[{Quinlan} and {Shapiro}, 1990]{QS90}
{Quinlan}, G.~D. and {Shapiro}, S.~L. (1990).
\newblock The dynamical evolution of dense star clusters in galactic nuclei.
\newblock {\em ApJ}, 356:483--500.

\bibitem[{Rasio} et~al., 2003]{RFG03}
{Rasio}, F.~A., {Freitag}, M., and {G{\"u}rkan}, M.~A. (2003).
\newblock {Formation of Massive Black Holes in Dense Star Clusters}.
\newblock In {Ho}, L., editor, {\em ``Coevolution of Black Holes and
  Galaxies'', Carnegie Observatories, Pasadena}.
\newblock astro-ph/0304038.

\bibitem[{Rauch}, 1995]{Rauch95}
{Rauch}, K.~P. (1995).
\newblock Dynamical evolution of star clusters around a rotating black hole
  with an accretion disc.
\newblock {\em MNRAS}, 275:628--640.

\bibitem[{Rauch}, 1999]{Rauch99}
{Rauch}, K.~P. (1999).
\newblock Collisional stellar dynamics around massive black holes in active
  galactic nuclei.
\newblock {\em ApJ}, 514:725--745.

\bibitem[{Rees}, 1984]{Rees84}
{Rees}, M.~J. (1984).
\newblock "black hole models for active galactic nuclei".
\newblock {\em ARA\&A}, 22:471--506.

\bibitem[{Rees}, 1990]{Rees90}
{Rees}, M.~J. (1990).
\newblock 'dead quasars' in nearby galaxies?
\newblock {\em Science}, 247:817--823.

\bibitem[{Salpeter}, 1964]{Salpeter64}
{Salpeter}, E.~E. (1964).
\newblock {Accretion of Interstellar Matter by Massive Objects.}
\newblock {\em apj}, 140:796--800.

\bibitem[{Sanders}, 1970]{Sanders70b}
{Sanders}, R.~H. (1970).
\newblock The effects of stellar collisions in dense stellar systems.
\newblock {\em ApJ}, 162:791.

\bibitem[{Sch{\" o}del} et~al., 2003]{SchoedelEtAl03}
{Sch{\" o}del}, R., {Ott}, T., {Genzel}, R., {Eckart}, A., {Mouawad}, N., and
  {Alexander}, T. (2003).
\newblock {Stellar Dynamics in the Central Arcsecond of Our Galaxy}.
\newblock {\em ApJ}, 596:1015--1034.

\bibitem[{Schaffer}, 1979]{Schaffer79}
{Schaffer}, S. (1979).
\newblock {Michell, John and Black Holes}.
\newblock {\em Journal for the History of Astronomy}, 10:42.

\bibitem[{So{\l}tan}, 1982]{Soltan82}
{So{\l}tan}, A. (1982).
\newblock {Masses of quasars}.
\newblock {\em MNRAS}, 200:115--122.

\bibitem[{Spitzer} and {Saslaw}, 1966]{SS66}
{Spitzer}, L., J. and {Saslaw}, W.~C. (1966).
\newblock On the evolution of galactic nuclei.
\newblock {\em ApJ}, 143:400.

\bibitem[{Spitzer}, 1987]{Spitzer87}
{Spitzer}, L. (1987).
\newblock {\em Dynamical evolution of globular clusters}.
\newblock Princeton University Press.

\bibitem[{Spitzer} and {Stone}, 1967]{SS67}
{Spitzer}, L.~J. and {Stone}, M.~E. (1967).
\newblock On the evolution of galactic nuclei. {II}.
\newblock {\em ApJ}, 147:519.

\bibitem[{Stoner} and {Ptak}, 1984]{SP84}
{Stoner}, R.~E. and {Ptak}, R.~L. (1984).
\newblock {Evidence for Supermassive Stars in Three Seyfert-I Nuclei from
  Variability in IUE Spectra}.
\newblock {\em baas}, 16:988.

\bibitem[{Terlevich}, 1989]{Terlevich89}
{Terlevich}, R. (1989).
\newblock {Active galactic nuclei without black-holes}.
\newblock In {\em Evolutionary Phenomena in Galaxies}, pages 149--158.

\bibitem[{Thorne}, 1995]{Thorne95}
{Thorne}, K.~S. (1995).
\newblock {Gravitational Radiation}.
\newblock In {\em Seventeeth Texas Symposium on Relativistic Astrophysics and
  Cosmology}, page 127.

\bibitem[{van der Marel} et~al., 2002]{vdMEtAl02}
{van der Marel}, R.~P., {Gerssen}, J., {Guhathakurta}, P., {Peterson}, R.~C.,
  and {Gebhardt}, K. (2002).
\newblock {Hubble Space Telescope Evidence for an Intermediate-Mass Black Hole
  in the Globular Cluster M15. I. STIS Spectroscopy and WFPC2 Photometry}.
\newblock {\em AJ}, 124:3255--3269.

\bibitem[{Vil'Koviskii}, 1978]{Vilk78}
{Vil'Koviskii}, {\' E}.~Y. (1978).
\newblock {Interaction of galaxies with central objects, and the activity of
  galaxy nuclei}.
\newblock {\em Soviet Astronomy}, 22:385.

\bibitem[{Woosley} and {Weaver}, 1982]{WW82}
{Woosley}, S.~E. and {Weaver}, T.~A. (1982).
\newblock {Theoretical Models for Supernovae}.
\newblock In {\em NATO ASIC Proc. 90: Supernovae: A Survey of Current
  Research}, page~79.

\bibitem[{Yu} and {Tremaine}, 2002]{YT02}
{Yu}, Q. and {Tremaine}, S. (2002).
\newblock {Observational constraints on growth of massive black holes}.
\newblock {\em MNRAS}, 335:965--976.

\bibitem[{Zeldovich} and {Novikov}, 1971]{ZN71}
{Zeldovich}, Y.~B. and {Novikov}, I.~D. (1971).
\newblock {\em Relativistic astrophysics. Vol.1: Stars and relativity}.
\newblock University of Chicago Press.

\bibitem[{Zel'Dovich} and {Podurets}, 1966]{ZP66}
{Zel'Dovich}, Y.~B. and {Podurets}, M.~A. (1966).
\newblock {The Evolution of a System of Gravitationally Interacting Point
  Masses}.
\newblock {\em Soviet Astronomy}, 9:742.

\end{thebibliography}

semi_analytical.bbl
\begin{thebibliography}{}

\bibitem[{Amaro-Seoane} and {Spurzem}, 2000]{AS00}
{Amaro-Seoane}, P. and {Spurzem}, R. (2000).
\newblock {The Loss-Cone Problem in Dense Nuclei Revisited}.
\newblock In {\em Astronomische Gesellschaft Meeting Abstracts}.

\bibitem[{Amaro-Seoane} and {Spurzem}, 2001a]{AS01}
{Amaro-Seoane}, P. and {Spurzem}, R. (2001a).
\newblock {The loss-cone problem in dense nuclei}.
\newblock {\em MNRAS}, 327:995--1003.

\bibitem[{Amaro-Seoane} and {Spurzem}, 2001b]{AS01b}
{Amaro-Seoane}, P. and {Spurzem}, R. (2001b).
\newblock {The Loss-Cone Star Contribution to the Heating Rate of a
  Supermassive Star}.
\newblock In {\em Stellar Dynamics: from Classic to Modern}, page 444.

\bibitem[{Bahcall} and {Wolf}, 1976]{BW76}
{Bahcall}, J.~N. and {Wolf}, R.~A. (1976).
\newblock Star distribution around a massive black hole in a globular cluster.
\newblock {\em ApJ}, 209:214--232.

\bibitem[{Binney} and {Tremaine}, 1987]{BT87}
{Binney}, J. and {Tremaine}, S. (1987).
\newblock {\em Galactic Dynamics}.
\newblock Princeton University Press.

\bibitem[{Bisnovatyi-Kogan} and {Syunyaev}, 1972]{BKS72}
{Bisnovatyi-Kogan}, G.~S. and {Syunyaev}, R.~A. (1972).
\newblock {Quasars and the Nuclei of Galaxies: A Single Object or a Star
  Cluster?}
\newblock {\em Soviet Astronomy}, 16:204.

\bibitem[{Chandrasekhar}, 1942]{Chandra42}
{Chandrasekhar}, S. (1942).
\newblock {Principles of stellar dynamics}.
\newblock {\em Physical Sciences Data}.

\bibitem[{Chandrasekhar}, 1964]{Chandra64}
{Chandrasekhar}, S. (1964).
\newblock {The Dynamical Instability of Gaseous Masses Approaching the
  Schwarzschild Limit in General Relativity.}
\newblock {\em apj}, 140:417.

\bibitem[{Courant} and {Friederichs}, 1998]{CF98}
{Courant}, R. and {Friederichs}, K.~O. (1998).
\newblock Supersonic flow and shock waves.
\newblock In Heidelberg, S.-V.~B., editor, {\em Applied Mathematical
  Sciencies}, volume~21.

\bibitem[{da Costa}, 1979]{daCosta79}
{da Costa}, L.~N. (1979).
\newblock {Thesis. Harvard University, American Doctoral Dissertations}.

\bibitem[{da Costa}, 1981]{daCosta81}
{da Costa}, L.~N. (1981).
\newblock {Star distribution in the presence of a massive central gas cloud}.
\newblock {\em MNRAS}, 195:869--880.

\bibitem[{Frank} and {Rees}, 1976]{FR76}
{Frank}, J. and {Rees}, M.~J. (1976).
\newblock Effects of massive central black holes on dense stellar systems.
\newblock {\em MNRAS}, 176:633--647.

\bibitem[{Fuller} et~al., 1986]{FW86}
{Fuller}, G.~M., {Woosley}, S.~E., and {Weaver}, T.~A. (1986).
\newblock {The evolution of radiation-dominated stars. I - Nonrotating
  supermassive stars}.
\newblock {\em apj}, 307:675--686.

\bibitem[{Hagio}, 1986]{Hagio86}
{Hagio}, F. (1986).
\newblock {Effects of viscous drag on a slowly rotating supermassive star due
  to an embedded stellar system}.
\newblock {\em pasj}, 38:111--120.

\bibitem[{Hara}, 1978]{Hara78}
{Hara}, T. (1978).
\newblock Evolution of a super-massive star in a dense stellar system.
\newblock {\em Progress of Th. Phys.}, 60:711--723.

\bibitem[{Langbein} et~al., 1990]{LangbeinEtAl90}
{Langbein}, T., {Fricke}, K.~J., {Spurzem}, R., and {Yorke}, H.~W. (1990).
\newblock {Interactions between stars and gas in galactic nuclei}.
\newblock {\em aa}, 227:333--341.

\bibitem[{Lightman} and {Shapiro}, 1977]{LS77}
{Lightman}, A.~P. and {Shapiro}, S.~L. (1977).
\newblock The distribution and consumption rate of stars around a massive,
  collapsed object.
\newblock {\em ApJ}, 211:244--262.

\bibitem[{Magorrian} et~al., 1998]{Magorrian98}
{Magorrian}, J., {Tremaine}, S., {Richstone}, D., {Bender}, R., {Bower}, G.,
  {Dressler}, A., {Faber}, S.~M., {Gebhardt}, K., {Green}, R., {Grillmair}, C.,
  {Kormendy}, J., and {Lauer}, T. (1998).
\newblock {The Demography of Massive Dark Objects in Galaxy Centers}.
\newblock {\em AJ}, 115:2285--2305.

\bibitem[{Marchant} and {Shapiro}, 1979]{MS79}
{Marchant}, A.~B. and {Shapiro}, S.~L. (1979).
\newblock Star clusters containing massive, central black holes. {II} -
  self-consistent potentials.
\newblock {\em ApJ}, 234:317--328.

\bibitem[{Marchant} and {Shapiro}, 1980]{MS80}
{Marchant}, A.~B. and {Shapiro}, S.~L. (1980).
\newblock Star clusters containing massive, central black holes. {III} -
  evolution calculations.
\newblock {\em ApJ}, 239:685--704.

\bibitem[{Peebles}, 1972]{Peebles72}
{Peebles}, P. J.~E. (1972).
\newblock Star distribution near a collapsed object.
\newblock {\em ApJ}, 178:371--376.

\bibitem[{Plummer}, 1911]{Plummer11}
{Plummer}, H.~C. (1911).
\newblock On the problem of distribution in globular star clusters.
\newblock {\em MNRAS}, 71:460.

\bibitem[{Richstone} et~al., 1998]{RichstoneEtAl98}
{Richstone}, D., {Ajhar}, E.~A., {Bender}, R., {Bower}, G., {Dressler}, A.,
  {Faber}, S.~M., {Filippenko}, A.~V., {Gebhardt}, K., {Green}, R., {Ho},
  L.~C., {Kormendy}, J., {Lauer}, T.~R., {Magorrian}, J., and {Tremaine}, S.
  (1998).
\newblock {Supermassive black holes and the evolution of galaxies.}
\newblock {\em nat}, 395:A14+.

\bibitem[{Shapiro} and {Marchant}, 1978]{SM78}
{Shapiro}, S.~L. and {Marchant}, A.~B. (1978).
\newblock Star clusters containing massive, central black holes - monte carlo
  simulations in two-dimensional phase space.
\newblock {\em ApJ}, 225:603--624.

\bibitem[{Shapiro} and {Teukolsky}, 1985]{ST85}
{Shapiro}, S.~L. and {Teukolsky}, S.~A. (1985).
\newblock The collapse of dense star clusters to supermassive black holes - the
  origin of quasars and agns.
\newblock {\em ApJ Lett.}, 292:41--44.

\end{thebibliography}

sms.bbl
\begin{thebibliography}{}

\bibitem[{Amaro-Seoane} et~al., 2002]{ASEtAl02}
{Amaro-Seoane}, P., {Spurzem}, R., and {Just}, A. (2002).
\newblock {Super-Massive Stars: Dense Star-Gas Systems}.
\newblock In {\em Lighthouses of the Universe: The Most Luminous Celestial
  Objects and Their Use for Cosmology Proceedings of the MPA/ESO/, p. 376},
  page 376.

\bibitem[{Baumgarte} and {Shapiro}, 1999a]{BS99b}
{Baumgarte}, T.~W. and {Shapiro}, S.~L. (1999a).
\newblock {Evolution of Rotating Supermassive Stars to the Onset of Collapse}.
\newblock {\em apj}, 526:941--952.

\bibitem[{Baumgarte} and {Shapiro}, 1999b]{BS99a}
{Baumgarte}, T.~W. and {Shapiro}, S.~L. (1999b).
\newblock {Luminosity versus Rotation in a Supermassive Star}.
\newblock {\em apj}, 526:937--940.

\bibitem[{Bisnovatyi-Kogan}, 1998]{BK98}
{Bisnovatyi-Kogan}, G.~S. (1998).
\newblock {Stability of Rotating Supermassive Stars in the Presence of a Dark
  Matter Background}.
\newblock {\em apj}, 497:559.

\bibitem[{Bond} et~al., 1984]{BA84}
{Bond}, J.~R., {Arnett}, W.~D., and {Carr}, B.~J. (1984).
\newblock {The evolution and fate of Very Massive Objects}.
\newblock {\em apj}, 280:825--847.

\bibitem[{Burbidge} and {Burbidge}, 1962]{BB62}
{Burbidge}, E.~M. and {Burbidge}, G.~R. (1962).
\newblock {\em nat}, 194:367.

\bibitem[{Chandrasekhar}, 1939]{Chandra39}
{Chandrasekhar}, S. (1939).
\newblock {\em {An introduction to the study of stellar structure}}.
\newblock Chicago, Ill., The University of Chicago press [1939].

\bibitem[{Chandrasekhar}, 1964]{Chandra64}
{Chandrasekhar}, S. (1964).
\newblock {The Dynamical Instability of Gaseous Masses Approaching the
  Schwarzschild Limit in General Relativity.}
\newblock {\em apj}, 140:417.

\bibitem[{Deiss} et~al., 1990]{DeissEtAl90}
{Deiss}, B.~M., {Just}, A., and {Kegel}, W.~H. (1990).
\newblock {Fluctuations of the interstellar medium induced by the stellar
  system - Mechanical heating and dynamical friction}.
\newblock {\em aa}, 240:123--136.

\bibitem[{Feynman}, 1963]{Feynman63}
{Feynman}, R.~P. (1963).
\newblock {\em {Feynman lectures on physics - Volume 1}}.
\newblock Reading, Ma.: Addison-Wesley, 1963, edited by Feynman, Richard P.;
  Leighton, Robert B.; Sands, Matthew.

\bibitem[{Fowler}, 1964]{Fowler64}
{Fowler}, W.~A. (1964).
\newblock {Massive Stars, Relativistic Polytropes, and Gravitational Radiation
  }.
\newblock {\em Reviews of Modern Physics}, 36:545--554.

\bibitem[{Fowler}, 1965]{Fowler65}
{Fowler}, W.~A. (1965).
\newblock {Massive Stars, Relativistic Polytropes, and Gravitational
  Radiation}.
\newblock In {\em Quasi-Stellar Sources and Gravitational Collapse}, page~51.

\bibitem[{Fowler}, 1966]{Fowler66}
{Fowler}, W.~A. (1966).
\newblock {The Stability of Supermassive Stars}.
\newblock {\em apj}, 144:180.

\bibitem[{Fuller} et~al., 1986]{FW86}
{Fuller}, G.~M., {Woosley}, S.~E., and {Weaver}, T.~A. (1986).
\newblock {The evolution of radiation-dominated stars. I - Nonrotating
  supermassive stars}.
\newblock {\em apj}, 307:675--686.

\bibitem[{Hoyle} and {Fowler}, 1963]{HF63}
{Hoyle}, F. and {Fowler}, W.~A. (1963).
\newblock {On the nature of strong radio sources}.
\newblock {\em mnras}, 125:169.

\bibitem[{Just} and {Amaro-Seoane}, 2001]{JustAS01}
{Just}, A. and {Amaro-Seoane}, P. (2001).
\newblock {Gas-star-interaction in Dense Galactic Nuclei}.
\newblock In {\em Astronomische Gesellschaft Meeting Abstracts}, page 155.

\bibitem[{Just} and {Amaro-Seoane}, 2003]{JustAS03}
{Just}, A. and {Amaro-Seoane}, P. (2003).
\newblock {Stability and Evolution of Supermassive Stars (SMS)}.
\newblock {\em EAS Publications Series, Volume 10, 2003, Galactic and Stellar
  Dynamics, Proceedings of JENAM 2002, held in Porto, Portugal, 3-6 September,
  2002.~Edited by C.~M.~Boily, P.~Pastsis, S.~Portegies Zwart, R.~Spurzem and
  C.~Theis, pp.127.}, 10:127.

\bibitem[{Just} et~al., 1986]{JustEtAl86}
{Just}, A., {Kegel}, W.~H., and {Deiss}, B.~M. (1986).
\newblock {Dynamical friction between the ISM and the system of stars}.
\newblock {\em aa}, 164:337--341.

\bibitem[{Kippenhahn} and {Weigert}, 1994]{KW94}
{Kippenhahn}, R. and {Weigert}, A. (1994).
\newblock {\em Stellar Structure and Evolution}.
\newblock Springer-Verlag Berlin Heidelberg.

\bibitem[{Misner} et~al., 1973]{MTW73}
{Misner}, C.~W., {Thorne}, K.~S., and {Wheeler}, J.~A. (1973).
\newblock {\em {Gravitation}}.
\newblock San Francisco: W.H.~Freeman and Co., 1973.

\bibitem[{New} and {Shapiro}, 2001]{NS01}
{New}, K.~C.~B. and {Shapiro}, S.~L. (2001).
\newblock {Evolution of Differentially Rotating Supermassive Stars to the Onset
  of Bar Instability}.
\newblock {\em apj}, 548:439--446.

\bibitem[{Shapiro} and {Teukolsky}, 1983]{ST83}
{Shapiro}, S.~L. and {Teukolsky}, S.~A. (1983).
\newblock {\em Black holes, white dwarfs, and neutron stars: The physics of
  compact objects}.
\newblock Wiley-Interscience.

\bibitem[{Wallace} and {Woosley}, 1981]{WW81}
{Wallace}, R.~K. and {Woosley}, S.~E. (1981).
\newblock {Explosive hydrogen burning}.
\newblock {\em apjs}, 45:389--420.

\bibitem[{Zeldovich} and {Novikov}, 1971]{ZN71}
{Zeldovich}, Y.~B. and {Novikov}, I.~D. (1971).
\newblock {\em Relativistic astrophysics. Vol.1: Stars and relativity}.
\newblock University of Chicago Press.

\end{thebibliography}

tesi.bbl
\begin{thebibliography}{}

\bibitem[{Aarseth}, 1999a]{Aarseth99a}
{Aarseth}, S.~J. (1999a).
\newblock {From NBODY1 to NBODY6: The Growth of an Industry}.
\newblock {\em PASP}, 111:1333--1346.

\bibitem[{Aarseth}, 1999b]{Aarseth99b}
{Aarseth}, S.~J. (1999b).
\newblock {Star Cluster Simulations: the State of the Art}.
\newblock {\em Celestial Mechanics and Dynamical Astronomy}, 73:127--137.

\bibitem[{Amaro-Seoane} et~al., 2004]{ASEtAl04}
{Amaro-Seoane}, P., {Freitag}, M., and {Spurzem}, R. (2004).
\newblock {Accretion of stars on to a massive black hole: a realistic diffusion
  model and numerical studies}.
\newblock {\em mn}, page 224.

\bibitem[{Amaro-Seoane} and {Spurzem}, 2003]{AS03}
{Amaro-Seoane}, P. and {Spurzem} (2003).
\newblock {The anisotropic gaseous model 'Spedi' : A tool to study the
  formation and growth of black holes in galactic nuclei}.
\newblock {\em In "The formation and early evolution of galaxies", workshop
  held in Irsee, Germany, \footnote{Available at~{\tt
  http://www.mpia-hd.mpg.de/GALAXIES/CADIS/irsee2003/PROCEEDINGS/Amaro-Seoane.%
pdf}}}.

\bibitem[{Amaro-Seoane} and {Spurzem}, 2000]{AS00}
{Amaro-Seoane}, P. and {Spurzem}, R. (2000).
\newblock {The Loss-Cone Problem in Dense Nuclei Revisited}.
\newblock In {\em Astronomische Gesellschaft Meeting Abstracts}.

\bibitem[{Amaro-Seoane} and {Spurzem}, 2001a]{AS01a}
{Amaro-Seoane}, P. and {Spurzem}, R. (2001a).
\newblock {Gas in the Central Regions of AGN: The Interstellar Medium and
  Supermassive Gaseous Objects}.
\newblock In {\em ASP Conf. Ser. 249: The Central Kiloparsec of Starbursts and
  AGN: The La Palma Connection}, page 731.

\bibitem[{Amaro-Seoane} and {Spurzem}, 2001b]{AS01}
{Amaro-Seoane}, P. and {Spurzem}, R. (2001b).
\newblock {The loss-cone problem in dense nuclei}.
\newblock {\em MNRAS}, 327:995--1003.

\bibitem[{Amaro-Seoane} and {Spurzem}, 2001c]{AS01b}
{Amaro-Seoane}, P. and {Spurzem}, R. (2001c).
\newblock {The Loss-Cone Star Contribution to the Heating Rate of a
  Supermassive Star}.
\newblock In {\em Stellar Dynamics: from Classic to Modern}, page 444.

\bibitem[{Amaro-Seoane} and {Spurzem}, 2004]{AS04}
{Amaro-Seoane}, P. and {Spurzem}, R. (2004).
\newblock {Dense Gas-Star Systems: Evolution of Supermassive Stars}.
\newblock In {\em In "Carnegie Observatories Astrophysics Series, Vol. 1:
  Coevolution of Black Holes and Galaxies," ed. L. C. Ho, Pasadena, USA:
  Carnegie Observatories, 2003}.

\bibitem[{Amaro-Seoane} et~al., 2002]{ASEtAl02}
{Amaro-Seoane}, P., {Spurzem}, R., and {Just}, A. (2002).
\newblock {Super-Massive Stars: Dense Star-Gas Systems}.
\newblock In {\em Lighthouses of the Universe: The Most Luminous Celestial
  Objects and Their Use for Cosmology Proceedings of the MPA/ESO/, p. 376},
  page 376.

\bibitem[{Arons} et~al., 1975]{AKO75}
{Arons}, J., {Kulsrud}, R.~M., and {Ostriker}, J.~P. (1975).
\newblock {A multiple pulsar model for quasi-stellar objects and active
  galactic nuclei}.
\newblock {\em apj}, 198:687--705.

\bibitem[{Bahcall} and {Ostriker}, 1975]{BO75}
{Bahcall}, J.~N. and {Ostriker}, J.~P. (1975).
\newblock {Massive black holes in globular clusters}.
\newblock {\em nat}, 256:23.

\bibitem[{Bahcall} and {Wolf}, 1976]{BW76}
{Bahcall}, J.~N. and {Wolf}, R.~A. (1976).
\newblock Star distribution around a massive black hole in a globular cluster.
\newblock {\em ApJ}, 209:214--232.

\bibitem[{Baumgardt} et~al., 2003a]{BaumgardtEtAl03}
{Baumgardt}, H., {Hut}, P., {Makino}, J., {McMillan}, S., and {Portegies
  Zwart}, S. (2003a).
\newblock {On the Central Structure of M15}.
\newblock {\em ApJ Lett.}, 582:L21--L24.

\bibitem[{Baumgardt} et~al., 2003b]{BaumgardtEtAl03b}
{Baumgardt}, H., {Makino}, J., {Hut}, P., {McMillan}, S., and {Portegies
  Zwart}, S. (2003b).
\newblock {A Dynamical Model for the Globular Cluster G1}.
\newblock {\em ApJ Lett.}, 589:L25--L28.

\bibitem[{Baumgarte} and {Shapiro}, 1999a]{BS99b}
{Baumgarte}, T.~W. and {Shapiro}, S.~L. (1999a).
\newblock {Evolution of Rotating Supermassive Stars to the Onset of Collapse}.
\newblock {\em apj}, 526:941--952.

\bibitem[{Baumgarte} and {Shapiro}, 1999b]{BS99a}
{Baumgarte}, T.~W. and {Shapiro}, S.~L. (1999b).
\newblock {Luminosity versus Rotation in a Supermassive Star}.
\newblock {\em apj}, 526:937--940.

\bibitem[{Begelman} and {Rees}, 1978]{BR78}
{Begelman}, M.~C. and {Rees}, M.~J. (1978).
\newblock The fate of dense stellar systems.
\newblock {\em MNRAS}, 185:847--860.

\bibitem[{Bettwieser}, 1983]{Bettwieser83}
{Bettwieser}, E. (1983).
\newblock {A numerical method for the study of the gravothermal instability in
  star clusters}.
\newblock {\em MNRAS}, 203:811--831.

\bibitem[{Bettwieser} and {Spurzem}, 1986]{BS86}
{Bettwieser}, E. and {Spurzem}, R. (1986).
\newblock {Anisotropy in stellar dynamics}.
\newblock {\em A\&A}, 161:102--112.

\bibitem[{Bettwieser} and {Sugimoto}, 1984]{BS84}
{Bettwieser}, E. and {Sugimoto}, D. (1984).
\newblock Post-collapse evolution and gravothermal oscillation of globular
  clusters.
\newblock {\em MNRAS}, 208:493--509.

\bibitem[{Bettwieser} and {Sugimoto}, 1985]{BS85}
{Bettwieser}, E. and {Sugimoto}, D. (1985).
\newblock Gravitational n-body problem and the validity of concepts of the gas
  model.
\newblock {\em MNRAS}, 212:189--195.

\bibitem[{Binney} and {Tremaine}, 1987]{BT87}
{Binney}, J. and {Tremaine}, S. (1987).
\newblock {\em Galactic Dynamics}.
\newblock Princeton University Press.

\bibitem[{Bisnovatyi-Kogan}, 1998]{BK98}
{Bisnovatyi-Kogan}, G.~S. (1998).
\newblock {Stability of Rotating Supermassive Stars in the Presence of a Dark
  Matter Background}.
\newblock {\em apj}, 497:559.

\bibitem[{Bisnovatyi-Kogan} and {Syunyaev}, 1972]{BKS72}
{Bisnovatyi-Kogan}, G.~S. and {Syunyaev}, R.~A. (1972).
\newblock {Quasars and the Nuclei of Galaxies: A Single Object or a Star
  Cluster?}
\newblock {\em Soviet Astronomy}, 16:204.

\bibitem[{Bond} et~al., 1984]{BA84}
{Bond}, J.~R., {Arnett}, W.~D., and {Carr}, B.~J. (1984).
\newblock {The evolution and fate of Very Massive Objects}.
\newblock {\em apj}, 280:825--847.

\bibitem[{Burbidge} and {Burbidge}, 1962]{BB62}
{Burbidge}, E.~M. and {Burbidge}, G.~R. (1962).
\newblock {\em nat}, 194:367.

\bibitem[{Burkert}, 2000]{Burkert00}
{Burkert}, A. (2000).
\newblock {The Structure and Evolution of Weakly Self-interacting Cold Dark
  Matter Halos}.
\newblock {\em ApJ Lett.}, 534:143--146.

\bibitem[{Castor}, 1972]{Castor72}
{Castor}, J.~I. (1972).
\newblock {Radiative Transfer in Spherically Symmetric Flows}.
\newblock {\em apj}, 178:779--792.

\bibitem[{Chandrasekhar}, 1939]{Chandra39}
{Chandrasekhar}, S. (1939).
\newblock {\em {An introduction to the study of stellar structure}}.
\newblock Chicago, Ill., The University of Chicago press [1939].

\bibitem[{Chandrasekhar}, 1942]{Chandra42}
{Chandrasekhar}, S. (1942).
\newblock {Principles of stellar dynamics}.
\newblock {\em Physical Sciences Data}.

\bibitem[{Chandrasekhar}, 1964]{Chandra64}
{Chandrasekhar}, S. (1964).
\newblock {The Dynamical Instability of Gaseous Masses Approaching the
  Schwarzschild Limit in General Relativity.}
\newblock {\em apj}, 140:417.

\bibitem[{Chatterjee} et~al., 2002]{ChatterjeeEtAl02}
{Chatterjee}, P., {Hernquist}, L., and {Loeb}, A. (2002).
\newblock {Dynamics of a Massive Black Hole at the Center of a Dense Stellar
  System}.
\newblock {\em ApJ}, 572:371--381.

\bibitem[{Cohn}, 1980]{Cohn80}
{Cohn}, H. (1980).
\newblock Late core collapse in star clusters and the gravothermal instability.
\newblock {\em ApJ}, 242:765--771.

\bibitem[{Cohn} and {Kulsrud}, 1978]{CK78}
{Cohn}, H. and {Kulsrud}, R.~M. (1978).
\newblock The stellar distribution around a black hole - numerical integration
  of the fokker-planck equation.
\newblock {\em ApJ}, 226:1087--1108.

\bibitem[{Colgate}, 1967]{Colgate67}
{Colgate}, S.~A. (1967).
\newblock Stellar coalescence and the multiple supernova interpretation of
  quasi-stellar sources.
\newblock {\em ApJ}, 150:163.

\bibitem[{Courant} and {Friederichs}, 1998]{CF98}
{Courant}, R. and {Friederichs}, K.~O. (1998).
\newblock Supersonic flow and shock waves.
\newblock In Heidelberg, S.-V.~B., editor, {\em Applied Mathematical
  Sciencies}, volume~21.

\bibitem[{da Costa}, 1979]{daCosta79}
{da Costa}, L.~N. (1979).
\newblock {Thesis. Harvard University, American Doctoral Dissertations}.

\bibitem[{da Costa}, 1981]{daCosta81}
{da Costa}, L.~N. (1981).
\newblock {Star distribution in the presence of a massive central gas cloud}.
\newblock {\em MNRAS}, 195:869--880.

\bibitem[{David} et~al., 1987a]{DDC87a}
{David}, L.~P., {Durisen}, R.~H., and {Cohn}, H.~N. (1987a).
\newblock The evolution of active galactic nuclei. i - models without stellar
  evolution.
\newblock {\em ApJ}, 313:556--575.

\bibitem[{David} et~al., 1987b]{DDC87b}
{David}, L.~P., {Durisen}, R.~H., and {Cohn}, H.~N. (1987b).
\newblock The evolution of active galactic nuclei. {II} - models with stellar
  evolution.
\newblock {\em ApJ}, 316:505--516.

\bibitem[{Dehnen}, 1993]{Dehnen93}
{Dehnen}, W. (1993).
\newblock {A Family of Potential-Density Pairs for Spherical Galaxies and
  Bulges}.
\newblock {\em MNRAS}, 265:250.

\bibitem[{Deiss} et~al., 1990]{DeissEtAl90}
{Deiss}, B.~M., {Just}, A., and {Kegel}, W.~H. (1990).
\newblock {Fluctuations of the interstellar medium induced by the stellar
  system - Mechanical heating and dynamical friction}.
\newblock {\em aa}, 240:123--136.

\bibitem[{Deiters} and {Spurzem}, 2001]{DS01}
{Deiters}, S. and {Spurzem}, R. (2001).
\newblock {Multi-mass Gaseous Models of Globular Clusters with Stellar
  Evolution}.
\newblock In {\em ASP Conf. Ser. 228: Dynamics of Star Clusters and the Milky
  Way}, page 416.

\bibitem[{Diaferio} et~al., 1999]{DiaferioEtAl99}
{Diaferio}, A., {Kauffmann}, G., {Colberg}, J.~M., and {White}, S.~D.~M.
  (1999).
\newblock {Clustering of galaxies in a hierarchical universe - III. Mock
  redshift surveys}.
\newblock {\em mnras}, 307:537--552.

\bibitem[{Dokuchaev} and {Ozernoi}, 1977]{DO77a}
{Dokuchaev}, V.~I. and {Ozernoi}, L.~M. (1977).
\newblock Distribution of stars in the vicinity of a massive black hole.
\newblock {\em Sov. Astron. Lett.}, 3:112--114.

\bibitem[{Dorband} et~al., 2003]{DHM03}
{Dorband}, E.~N., {Hemsendorf}, M., and {Merritt}, D. (2003).
\newblock {Systolic and hyper-systolic algorithms for the gravitational N-body
  problem, with an application to Brownian motion}.
\newblock {\em Journal of Computational Physics}, 185:484--511.

\bibitem[{Drukier} et~al., 1999]{DCLY99}
{Drukier}, G.~A., {Cohn}, H.~N., {Lugger}, P.~M., and {Yong}, H. (1999).
\newblock Anisotropic fokker-planck models for the evolution of globular star
  clusters: The core-halo connection.
\newblock {\em ApJ}, 518:233--245.

\bibitem[{Duncan} and {Shapiro}, 1982]{DS82}
{Duncan}, M.~J. and {Shapiro}, S.~L. (1982).
\newblock Star clusters containing massive, central black holes. {IV} -
  galactic tidal fields.
\newblock {\em ApJ}, 253:921--938.

\bibitem[{Duncan} and {Shapiro}, 1983]{DS83}
{Duncan}, M.~J. and {Shapiro}, S.~L. (1983).
\newblock Monte carlo simulations of the evolution of galactic nuclei
  containing massive, central black holes.
\newblock {\em ApJ}, 268:565--581.

\bibitem[{Ebisuzaki} et~al., 2001]{EbisuzakiEtAl01}
{Ebisuzaki}, T., {Makino}, J., {Tsuru}, T.~G., {Funato}, Y., {Portegies Zwart},
  S., {Hut}, P., {McMillan}, S., {Matsushita}, S., {Matsumoto}, H., and
  {Kawabe}, R. (2001).
\newblock {Missing Link Found? The ``Runaway'' Path to Supermassive Black
  Holes}.
\newblock {\em ApJ Lett.}, 562:L19--L22.

\bibitem[{Elson} et~al., 1987]{ElsonEtAl87}
{Elson}, R., {Hut}, P., and {Inagaki}, S. (1987).
\newblock {Dynamical evolution of globular clusters}.
\newblock {\em araa}, 25:565--601.

\bibitem[{Ferrarese} et~al., 2001]{FPPMWJ01}
{Ferrarese}, L., {Pogge}, R.~W., {Peterson}, B.~M., {Merritt}, D., {Wandel},
  A., and {Joseph}, C.~L. (2001).
\newblock {Supermassive Black Holes in Active Galactic Nuclei. I. The
  Consistency of Black Hole Masses in Quiescent and Active Galaxies}.
\newblock {\em ApJ Lett.}, 555:79--82.

\bibitem[{Feynman}, 1963]{Feynman63}
{Feynman}, R.~P. (1963).
\newblock {\em {Feynman lectures on physics - Volume 1}}.
\newblock Reading, Ma.: Addison-Wesley, 1963, edited by Feynman, Richard P.;
  Leighton, Robert B.; Sands, Matthew.

\bibitem[{Fowler}, 1964]{Fowler64}
{Fowler}, W.~A. (1964).
\newblock {Massive Stars, Relativistic Polytropes, and Gravitational Radiation
  }.
\newblock {\em Reviews of Modern Physics}, 36:545--554.

\bibitem[{Fowler}, 1965]{Fowler65}
{Fowler}, W.~A. (1965).
\newblock {Massive Stars, Relativistic Polytropes, and Gravitational
  Radiation}.
\newblock In {\em Quasi-Stellar Sources and Gravitational Collapse}, page~51.

\bibitem[{Fowler}, 1966]{Fowler66}
{Fowler}, W.~A. (1966).
\newblock {The Stability of Supermassive Stars}.
\newblock {\em apj}, 144:180.

\bibitem[{Frank} et~al., 2002]{FKR02}
{Frank}, J., {King}, A., and {Raine}, D.~J. (2002).
\newblock {\em {Accretion Power in Astrophysics: Third Edition}}.
\newblock Accretion Power in Astrophysics: Third Edition, by Juhan Frank,
  Andrew King, and Derek J.~Raine.~Cambridge University Press, 2002, 398 pp.

\bibitem[{Frank} and {Rees}, 1976]{FR76}
{Frank}, J. and {Rees}, M.~J. (1976).
\newblock Effects of massive central black holes on dense stellar systems.
\newblock {\em MNRAS}, 176:633--647.

\bibitem[{Fregeau} et~al., 2003]{FregeauEtAl03}
{Fregeau}, J.~M., {G{\" u}rkan}, M.~A., {Joshi}, K.~J., and {Rasio}, F.~A.
  (2003).
\newblock {Monte Carlo Simulations of Globular Cluster Evolution. III.
  Primordial Binary Interactions}.
\newblock {\em ApJ}, 593:772--787.

\bibitem[{Freitag}, 2003a]{Freitag03b}
{Freitag}, M. (2003a).
\newblock {Captures of stars by a massive black hole: Investigations in
  numerical stellar dynamics}.
\newblock to appear in the proceedings of ``The Astrophysics of Gravitational
  Wave Sources'', astro-ph/0306064.

\bibitem[{Freitag}, 2003b]{Freitag03}
{Freitag}, M. (2003b).
\newblock {Gravitational waves from stars orbiting the Sagittarius A* black
  hole}.
\newblock {\em ApJ Lett.}, 583:21--24.

\bibitem[{Freitag} and {Benz}, 2001]{FB01a}
{Freitag}, M. and {Benz}, W. (2001).
\newblock A new monte carlo code for star cluster simulations: I.~relaxation.
\newblock {\em A\&A}, 375:711--738.

\bibitem[{Freitag} and {Benz}, 2002]{FB02b}
{Freitag}, M. and {Benz}, W. (2002).
\newblock A new monte carlo code for star cluster simulations: {II}.~central
  black hole and stellar collisions.
\newblock {\em A\&A}, 394:345--374.

\bibitem[{Freitag} and {Benz}, 2004]{FB04}
{Freitag}, M. and {Benz}, W. (2004).
\newblock A comprehensive set of simulations of high-velocity collisions
  between main sequence stars.
\newblock submitted to MNRAS, astro-ph/0403621.

\bibitem[{Fuller} et~al., 1986]{FW86}
{Fuller}, G.~M., {Woosley}, S.~E., and {Weaver}, T.~A. (1986).
\newblock {The evolution of radiation-dominated stars. I - Nonrotating
  supermassive stars}.
\newblock {\em apj}, 307:675--686.

\bibitem[{G{\" u}rkan} et~al., 2004]{GFR04}
{G{\" u}rkan}, M.~A., {Freitag}, M., and {Rasio}, F.~A. (2004).
\newblock {Formation of Massive Black Holes in Dense Star Clusters. I. Mass
  Segregation and Core Collapse}.
\newblock {\em apj}, 604:632--652.

\bibitem[{Gebhardt} et~al., 2001]{GebhardtEtAl01}
{Gebhardt}, K., {Lauer}, T.~R., {Kormendy}, J., {Pinkney}, J., {Bower}, G.,
  {Green}, R., {Gull}, T., {Hutchings}, J.~B., {Kaiser}, M.~E., {Nelson},
  C.~H., {Richstone}, D., and {Weistrop}, D. (2001).
\newblock M33: A galaxy with no supermassive black hole.
\newblock {\em AJ}, 122:2469--2476.

\bibitem[{Gebhardt} et~al., 2002]{GRH02}
{Gebhardt}, K., {Rich}, R.~M., and {Ho}, L.~C. (2002).
\newblock {A $20000 M_\odot$ Black Hole in the Stellar Cluster G1}.
\newblock {\em ApJ Lett.}, 578:41--45.

\bibitem[{Gerssen} et~al., 2002]{GerssenEtAl02}
{Gerssen}, J., {van der Marel}, R.~P., {Gebhardt}, K., {Guhathakurta}, P.,
  {Peterson}, R.~C., and {Pryor}, C. (2002).
\newblock {Hubble Space Telescope Evidence for an Intermediate-Mass Black Hole
  in the Globular Cluster M15. II. Kinematic Analysis and Dynamical Modeling}.
\newblock {\em AJ}, 124:3270--3288.

\bibitem[{Gerssen} et~al., 2003]{GerssenEtAl03}
{Gerssen}, J., {van der Marel}, R.~P., {Gebhardt}, K., {Guhathakurta}, P.,
  {Peterson}, R.~C., and {Pryor}, C. (2003).
\newblock {Addendum: Hubble Space Telescope Evidence for an Intermediate-Mass
  Black Hole in the Globular Cluster M15. II. Kinematic Analysis and Dynamical
  Modeling}.
\newblock {\em AJ}, 125:376--377.

\bibitem[{Gezari} et~al., 2003]{GezariEtal03}
{Gezari}, S., {Halpern}, J.~P., {Komossa}, S., {Grupe}, D., and {Leighly},
  K.~M. (2003).
\newblock {Follow-Up Hubble Space Telescope/Space Telescope Imaging
  Spectroscopy of Three Candidate Tidal Disruption Events}.
\newblock {\em apj}, 592:42--51.

\bibitem[{Ghez} et~al., 2003]{GhezEtAl03}
{Ghez}, A.~M., {Duch{\^ e}ne}, G., {Matthews}, K., {Hornstein}, S.~D.,
  {Tanner}, A., {Larkin}, J., {Morris}, M., {Becklin}, E.~E., {Salim}, S.,
  {Kremenek}, T., {Thompson}, D., {Soifer}, B.~T., {Neugebauer}, G., and
  {McLean}, I. (2003).
\newblock {The First Measurement of Spectral Lines in a Short-Period Star Bound
  to the Galaxy's Central Black Hole: A Paradox of Youth}.
\newblock {\em ApJ Lett.}, 586:L127--L131.

\bibitem[{Giersz}, 1998]{Giersz98}
{Giersz}, M. (1998).
\newblock Monte carlo simulations of star clusters - i. first results.
\newblock {\em MNRAS}, 298:1239--1248.

\bibitem[{Giersz} and {Heggie}, 1994a]{GH94}
{Giersz}, M. and {Heggie}, D.~C. (1994a).
\newblock {Statistics of N-Body Simulations - Part Two - Equal Masses after
  Core Collapse}.
\newblock {\em MNRAS}, 270:298.

\bibitem[{Giersz} and {Heggie}, 1994b]{GH94b}
{Giersz}, M. and {Heggie}, D.~C. (1994b).
\newblock Statistics of n-body simulations - part two - equal masses after core
  collapse.
\newblock {\em MNRAS}, 270:298.

\bibitem[{Giersz} and {Heggie}, 1996]{GH96}
{Giersz}, M. and {Heggie}, D.~C. (1996).
\newblock Statistics of $n$-body simulations - {III}. unequal masses.
\newblock {\em MNRAS}, 279:1037--1056.

\bibitem[{Giersz} and {Spurzem}, 1994]{GS94}
{Giersz}, M. and {Spurzem}, R. (1994).
\newblock Comparing direct $n$-body integration with anisotropic gaseous models
  of star clusters.
\newblock {\em MNRAS}, 269:241.

\bibitem[{Giersz} and {Spurzem}, 2000]{GS00}
{Giersz}, M. and {Spurzem}, R. (2000).
\newblock A stochastic monte carlo approach to model real star cluster
  evolution - {II}. self-consistent models and primordial binaries.
\newblock {\em MNRAS}, 317:581.

\bibitem[{Giersz} and {Spurzem}, 2003]{GS03}
{Giersz}, M. and {Spurzem}, R. (2003).
\newblock {A stochastic Monte Carlo approach to modelling real star cluster
  evolution - III. Direct integration of three- and four-body interactions}.
\newblock {\em MNRAS}, 343:781--795.

\bibitem[{Ginzburg} and {Ozernoy}, 1964]{GO64}
{Ginzburg}, V. and {Ozernoy}, L.~M. (1964).
\newblock {\em Sov. Phys. JETP}, 20:489.

\bibitem[{Goldreich} and {Weber}, 1980]{GW80}
{Goldreich}, P. and {Weber}, S.~V. (1980).
\newblock {Homologously collapsing stellar cores}.
\newblock {\em apj}, 238:991--997.

\bibitem[{Goodman}, 1984]{Goodman84}
{Goodman}, J. (1984).
\newblock {Homologous evolution of stellar systems after core collapse}.
\newblock {\em ApJ}, 280:298--312.

\bibitem[{Gurevich}, 1964]{Gurevich64}
{Gurevich}, A. (1964).
\newblock Instability of the disturbed zone in the vicinity of a charged body
  in plasma.
\newblock {\em Geomag. Aeronom.}, 4:247--255.

\bibitem[{H{\' e}non}, 1965]{Henon65}
{H{\' e}non}, M. (1965).
\newblock {Sur l'{\' e}volution dynamique des amas globulaires. II. Amas
  isol\&eacute}.
\newblock {\em Annales d'Astrophysique}, 28:62.

\bibitem[{Hachisu} et~al., 1978]{HachisuEtAl78}
{Hachisu}, I., {Nakada}, Y., {Nomoto}, K., and {Sugimoto}, D. (1978).
\newblock {Post-collapse evolution of a gaseous cluster model}.
\newblock {\em Prog. Theor. Phys. 60, 393}, 393:60.

\bibitem[{Hagio}, 1986]{Hagio86}
{Hagio}, F. (1986).
\newblock {Effects of viscous drag on a slowly rotating supermassive star due
  to an embedded stellar system}.
\newblock {\em pasj}, 38:111--120.

\bibitem[{Hara}, 1978]{Hara78}
{Hara}, T. (1978).
\newblock Evolution of a super-massive star in a dense stellar system.
\newblock {\em Progress of Th. Phys.}, 60:711--723.

\bibitem[{Heckman}, 1980]{Heckman80}
{Heckman}, T.~M. (1980).
\newblock {An optical and radio survey of the nuclei of bright galaxies -
  Activity in normal galactic nuclei}.
\newblock {\em aap}, 87:152--164.

\bibitem[{Heggie}, 1984]{Heggie84}
{Heggie}, D.~C. (1984).
\newblock {Post-collapse evolution of a gaseous cluster model}.
\newblock {\em MNRAS}, 206:179--195.

\bibitem[{Heggie} and {Mathieu}, 1986]{HM86}
{Heggie}, D.~C. and {Mathieu}, R.~D. (1986).
\newblock Standardised units and time scales.
\newblock In {Hut}, P. and {McMillan}, S. L.~W., editors, {\em The Use of
  Supercomputers in Stellar Dynamics}, page 233. Springer-Verlag.

\bibitem[{Heggie} and {Ramamani}, 1989]{HR89}
{Heggie}, D.~C. and {Ramamani}, N. (1989).
\newblock {Evolution of star clusters after core collapse}.
\newblock 237:757--783.

\bibitem[{Hemsendorf} et~al., 2001]{HSS01}
{Hemsendorf}, M., {Sigurdsson}, S., and {Spurzem}, R. (2001).
\newblock {Binary black holes in galactic centres}.
\newblock {\em EAS Publications Series, Volume 1.~Active galactic nuclei in
  their cosmic environment.~JENAM 99, Toulouse, France, September 7-9,
  1999.~Ed.~by B.~Rocca-Volmerange and H.~Sol.~Les Ulis: EDP Sciences, ISBN
  2-86883-563-5, 2001, p.~173}, 1:173.

\bibitem[{H{\'e}non}, 1971]{Henon71a}
{H{\'e}non}, M. (1971).
\newblock Monte carlo models of star clusters.
\newblock {\em Ap\&SS}, 13:284--299.

\bibitem[{H{\'{e}}non}, 1973]{Henon73}
{H{\'{e}}non}, M. (1973).
\newblock Collisional dynamics of spherical stellar systems.
\newblock In {Martinet}, L. and {Mayor}, M., editors, {\em Dynamical structure
  and evolution of stellar systems, Lectures of the 3rd Advanced Course of the
  Swiss Society for Astronomy and Astrophysics (SSAA)}, pages 183--260.

\bibitem[{H{\'e}non}, 1975]{Henon75}
{H{\'e}non}, M. (1975).
\newblock Two recent developments concerning the monte carlo method.
\newblock In {Hayli}, A., editor, {\em IAU Symp. 69: Dynamics of Stellar
  Systems}, page 133.

\bibitem[{Henyey} et~al., 1959]{HenyeyEtAl59}
{Henyey}, L.~G., {Wilets}, L., {B{\" o}hm}, K.~H., {Lelevier}, R., and {Levee},
  R.~D. (1959).
\newblock {A Method for Atomic Computation of Stellar Evolution.}
\newblock {\em ApJ}, 129:628.

\bibitem[{Herrnstein} et~al., 1999]{Herrnstein99}
{Herrnstein}, J.~R., {Moran}, J.~M., {Greenhill}, L.~J., {Diamond}, P.~J.,
  {Inoue}, M., {Nakai}, N., {Miyoshi}, M., {Henkel}, C., and {Riess}, A.
  (1999).
\newblock A geometric distance to the galaxy ngc 4258 from orbital motions in a
  nuclear gas disk.
\newblock {\em Nat}, 400:539--541.

\bibitem[{Hillenbrand} and {Hartmann}, 1998]{HH98}
{Hillenbrand}, L.~A. and {Hartmann}, L.~W. (1998).
\newblock {A Preliminary Study of the Orion Nebula Cluster Structure and
  Dynamics}.
\newblock {\em apj}, 492:540.

\bibitem[{Hills}, 1975]{Hills75}
{Hills}, J.~G. (1975).
\newblock {Possible power source of Seyfert galaxies and QSOs}.
\newblock {\em Nat}, 254:295--298.

\bibitem[{Hoyle} and {Fowler}, 1963]{HF63}
{Hoyle}, F. and {Fowler}, W.~A. (1963).
\newblock {On the nature of strong radio sources}.
\newblock {\em mnras}, 125:169.

\bibitem[{Inagaki} and {Saslaw}, 1985]{IS85}
{Inagaki}, S. and {Saslaw}, W.~C. (1985).
\newblock Equipartition in multicomponent gravitational systems.
\newblock {\em ApJ}, 292:339--347.

\bibitem[{Inagaki} and {Wiyanto}, 1984]{IW84}
{Inagaki}, S. and {Wiyanto}, P. (1984).
\newblock On equipartition of kinetic energies in two-component star clusters.
\newblock {\em PASJ}, 36:391--402.

\bibitem[{Jarrett} et~al., 2003]{JarretEtAl03}
{Jarrett}, T.~H., {Chester}, T., {Cutri}, R., {Schneider}, S.~E., and {Huchra},
  J.~P. (2003).
\newblock {The 2MASS Large Galaxy Atlas}.
\newblock {\em aj}, 125:525--554.

\bibitem[{Joshi} et~al., 2001]{JoshiEtAl01}
{Joshi}, K.~J., {Nave}, C.~P., and {Rasio}, F.~A. (2001).
\newblock {Monte Carlo Simulations of Globular Cluster Evolution. II. Mass
  Spectra, Stellar Evolution, and Lifetimes in the Galaxy}.
\newblock {\em ApJ}, 550:691--702.

\bibitem[{Joshi} et~al., 2000]{JRPZ00}
{Joshi}, K.~J., {Rasio}, F.~A., and {Portegies Zwart}, S. (2000).
\newblock Monte carlo simulations of globular cluster evolution. i. method and
  test calculations.
\newblock {\em ApJ}, 540:969--982.

\bibitem[{Just} and {Amaro-Seoane}, 2001]{JustAS01}
{Just}, A. and {Amaro-Seoane}, P. (2001).
\newblock {Gas-star-interaction in Dense Galactic Nuclei}.
\newblock In {\em Astronomische Gesellschaft Meeting Abstracts}, page 155.

\bibitem[{Just} and {Amaro-Seoane}, 2003]{JustAS03}
{Just}, A. and {Amaro-Seoane}, P. (2003).
\newblock {Stability and Evolution of Supermassive Stars (SMS)}.
\newblock {\em EAS Publications Series, Volume 10, 2003, Galactic and Stellar
  Dynamics, Proceedings of JENAM 2002, held in Porto, Portugal, 3-6 September,
  2002.~Edited by C.~M.~Boily, P.~Pastsis, S.~Portegies Zwart, R.~Spurzem and
  C.~Theis, pp.127.}, 10:127.

\bibitem[{Just} et~al., 1986]{JustEtAl86}
{Just}, A., {Kegel}, W.~H., and {Deiss}, B.~M. (1986).
\newblock {Dynamical friction between the ISM and the system of stars}.
\newblock {\em aa}, 164:337--341.

\bibitem[{Kauffmann} et~al., 1999a]{KauffmanEtAl99a}
{Kauffmann}, G., {Colberg}, J.~M., {Diaferio}, A., and {White}, S.~D.~M.
  (1999a).
\newblock {Clustering of galaxies in a hierarchical universe - I. Methods and
  results at z=0}.
\newblock {\em mnras}, 303:188--206.

\bibitem[{Kauffmann} et~al., 1999b]{KauffmanEtAl99b}
{Kauffmann}, G., {Colberg}, J.~M., {Diaferio}, A., and {White}, S.~D.~M.
  (1999b).
\newblock {Clustering of galaxies in a hierarchical universe - I. Methods and
  results at z=0}.
\newblock {\em mnras}, 303:188--206.

\bibitem[{Khalisi}, 2002]{Khalisi02}
{Khalisi}, E. (2002).
\newblock Equipartition and mass segregation: Simulations of star clusters with
  two-mass components.
\newblock {\em Doctoral dissertation. Ruperto-Carola University of Heidelberg,
  Germany \footnote{Idem {\tt http://www.ub.uni-heidelberg.de/archiv/3096}}},
  349.

\bibitem[{Kim} et~al., 1998]{KLG98}
{Kim}, S.~S., {Lee}, H.~M., and {Goodman}, J. (1998).
\newblock {Two-Component Fokker-Planck Models for the Evolution of Isolated
  Globular Clusters}.
\newblock {\em apj}, 495:786.

\bibitem[{Kippenhahn} and {Weigert}, 1994]{KW94}
{Kippenhahn}, R. and {Weigert}, A. (1994).
\newblock {\em Stellar Structure and Evolution}.
\newblock Springer-Verlag Berlin Heidelberg.

\bibitem[{Kormendy}, 2003]{Kormendy03}
{Kormendy}, J. (2003).
\newblock {The Stellar-Dynamical Search for Supermassive Black Holes in
  Galactic Nuclei}.
\newblock In {Ho}, L., editor, {\em ``Coevolution of Black Holes and
  Galaxies'', Carnegie Observatories, Pasadena}.
\newblock astro-ph/0306353.

\bibitem[{Krolik}, 1999]{Krolik99}
{Krolik}, J.~H. (1999).
\newblock {\em Active galactic nuclei: from the central black hole to the
  galactic environment}.
\newblock Princeton University Press.

\bibitem[{Kroupa}, 2001]{Kroupa01}
{Kroupa}, P. (2001).
\newblock {On the variation of the initial mass function}.
\newblock {\em mnras}, 322:231--246.

\bibitem[{Kroupa} et~al., 1993]{KroupaEtAl93}
{Kroupa}, P., {Tout}, C.~A., and {Gilmore}, G. (1993).
\newblock {The distribution of low-mass stars in the Galactic disc}.
\newblock {\em mnras}, 262:545--587.

\bibitem[{Langbein} et~al., 1990]{LangbeinEtAl90}
{Langbein}, T., {Fricke}, K.~J., {Spurzem}, R., and {Yorke}, H.~W. (1990).
\newblock {Interactions between stars and gas in galactic nuclei}.
\newblock {\em aa}, 227:333--341.

\bibitem[{Larson}, 1970]{Larson70a}
{Larson}, R.~B. (1970).
\newblock A method for computing the evolution of star clusters.
\newblock {\em MNRAS}, 147:323.

\bibitem[{Lee}, 1995]{Lee95}
{Lee}, H.~M. (1995).
\newblock Evolution of galactic nuclei with 10-m\_ black holes.
\newblock {\em MNRAS}, 272:605--617.

\bibitem[{Lightman} and {Fall}, 1978]{LF78}
{Lightman}, A.~P. and {Fall}, S.~M. (1978).
\newblock {An approximate theory for the core collapse of two-component
  gravitating systems}.
\newblock {\em apj}, 221:567--579.

\bibitem[{Lightman} and {Shapiro}, 1977]{LS77}
{Lightman}, A.~P. and {Shapiro}, S.~L. (1977).
\newblock The distribution and consumption rate of stars around a massive,
  collapsed object.
\newblock {\em ApJ}, 211:244--262.

\bibitem[{Lin} and {Tremaine}, 1980]{LT80}
{Lin}, D. N.~C. and {Tremaine}, S. (1980).
\newblock A reinvestigation of the standard model for the dynamics of a massive
  black hole in a globular cluster.
\newblock {\em ApJ}, 242:789--798.

\bibitem[{Louis} and {Spurzem}, 1991]{LS91}
{Louis}, P.~D. and {Spurzem}, R. (1991).
\newblock Anisotropic gaseous models for the evolution of star clusters.
\newblock {\em MNRAS}, 251:408--426.

\bibitem[{Lynden-Bell}, 1967]{LyndenBell67}
{Lynden-Bell}, D. (1967).
\newblock Statistical mechanics of violent relaxation in stellar systems.
\newblock {\em MNRAS}, 136:101.

\bibitem[{Lynden-Bell}, 1969]{LB69}
{Lynden-Bell}, D. (1969).
\newblock {Galactic Nuclei as Collapsed Old Quasars}.
\newblock {\em nat}, 223:690.

\bibitem[{Lynden-Bell} and {Eggleton}, 1980]{LBE80}
{Lynden-Bell}, D. and {Eggleton}, P.~P. (1980).
\newblock On the consequences of the gravothermal catastrophe.
\newblock {\em MNRAS}, 191:483--498.

\bibitem[{Lynden-Bell} and {Rees}, 1971a]{LR71}
{Lynden-Bell}, D. and {Rees}, M.~J. (1971a).
\newblock {On quasars, dust and the galactic centre}.
\newblock {\em MNRAS}, 152:461.

\bibitem[{Lynden-Bell} and {Rees}, 1971b]{LBR71}
{Lynden-Bell}, D. and {Rees}, M.~J. (1971b).
\newblock {On quasars, dust and the galactic centre}.
\newblock {\em mnras}, 152:461.

\bibitem[{Magorrian} and {Tremaine}, 1999]{MT99}
{Magorrian}, J. and {Tremaine}, S. (1999).
\newblock Rates of tidal disruption of stars by massive central black holes.
\newblock {\em MNRAS}, 309:447--460.

\bibitem[{Magorrian} et~al., 1998]{Magorrian98}
{Magorrian}, J., {Tremaine}, S., {Richstone}, D., {Bender}, R., {Bower}, G.,
  {Dressler}, A., {Faber}, S.~M., {Gebhardt}, K., {Green}, R., {Grillmair}, C.,
  {Kormendy}, J., and {Lauer}, T. (1998).
\newblock {The Demography of Massive Dark Objects in Galaxy Centers}.
\newblock {\em AJ}, 115:2285--2305.

\bibitem[{Makino}, 1996]{Makino96}
{Makino}, J. (1996).
\newblock Postcollapse evolution of globular clusters.
\newblock {\em ApJ}, 471:796.

\bibitem[{Makino}, 1997]{Makino97}
{Makino}, J. (1997).
\newblock {Merging of Galaxies with Central Black Holes. II. Evolution of the
  Black Hole Binary and the Structure of the Core}.
\newblock {\em apj}, 478:58.

\bibitem[{Makino} and {Ebisuzaki}, 1996]{ME96}
{Makino}, J. and {Ebisuzaki}, T. (1996).
\newblock {Merging of Galaxies with Central Black Holes. I. Hierarchical
  Mergings of Equal-Mass Galaxies}.
\newblock {\em apj}, 465:527.

\bibitem[{Makino} and {Taiji}, 1998]{MT98}
{Makino}, J. and {Taiji}, M. (1998).
\newblock {\em {Scientific simulations with special-purpose computers : The
  GRAPE systems}}.
\newblock Scientific simulations with special-purpose computers : The GRAPE
  systems /by Junichiro Makino \& Makoto Taiji.~Chichester ; Toronto : John
  Wiley \& Sons, c1998.

\bibitem[{Marchant} and {Shapiro}, 1979]{MS79}
{Marchant}, A.~B. and {Shapiro}, S.~L. (1979).
\newblock Star clusters containing massive, central black holes. {II} -
  self-consistent potentials.
\newblock {\em ApJ}, 234:317--328.

\bibitem[{Marchant} and {Shapiro}, 1980]{MS80}
{Marchant}, A.~B. and {Shapiro}, S.~L. (1980).
\newblock Star clusters containing massive, central black holes. {III} -
  evolution calculations.
\newblock {\em ApJ}, 239:685--704.

\bibitem[{McCaughrean} and {Stauffer}, 1994]{McMS94}
{McCaughrean}, M.~J. and {Stauffer}, J.~R. (1994).
\newblock High resolution near-infrared imaging of the trapezium: A stellar
  census.
\newblock {\em AJ}, 108:1382--1397.

\bibitem[{McCray}, 1979]{McCray79}
{McCray}, R. (1979).
\newblock {\em {Spherical accretion onto supermassive black holes}}, pages
  227--239.
\newblock Active galactic nuclei.~(A79-50785 22-90) Cambridge, Cambridge
  University Press, 1979, p.~227-239.~ NSF-supported research.

\bibitem[{McMillan} et~al., 1981]{McMLC81}
{McMillan}, S. L.~W., {Lightman}, A.~P., and {Cohn}, H. (1981).
\newblock Luminosity evolution of quasars and active galaxies - theoretical
  models of the evolving mass supply rate.
\newblock {\em ApJ}, 251:436--445.

\bibitem[{Merritt} et~al., 2001]{MFJ01}
{Merritt}, D., {Ferrarese}, L., and {Joseph}, C.~L. (2001).
\newblock {No Supermassive Black Hole in M33?}
\newblock {\em Science}, 293:1116--1119.

\bibitem[{Merritt} and {Poon, M. Y.}, 2003]{MP03}
{Merritt}, D. and {Poon, M. Y.} (2003).
\newblock {Chaotic Loss Cones, Black Hole Fueling and the M-Sigma Relation}.
\newblock {\em ArXiv Astrophysics e-prints}.

\bibitem[{Michell}, 1784]{Michell1784}
{Michell}, J. (1784).
\newblock {On the Means of Discovering the Distance, Magnitude, {\& }c. of the
  Fixed Stars, in Consequence of the Diminution of the Velocity of Their Light,
  in Case Such a Diminution Should be Found to Take Place in any of Them, and
  Such Other Data Should be Procured from Observations, as Would be Farther
  Necessary for That Purpose. By the Rev. John Michell, B. D. F. R. S. In a
  Letter to Henry Cavendish, Esq. F. R. S. and A. S.}
\newblock {\em Philosophical Transactions Series I}, 74:35--57.

\bibitem[{Miller} and {Colbert}, 2003]{MC03}
{Miller}, M.~C. and {Colbert}, E.~J.~M. (2003).
\newblock {Intermediate-Mass Black Holes}.
\newblock preprint, astro-ph/0308402.

\bibitem[{Milosavljevi{\' c}} and {Merritt}, 2003]{MM03}
{Milosavljevi{\' c}}, M. and {Merritt}, D. (2003).
\newblock {Long-Term Evolution of Massive Black Hole Binaries}.
\newblock {\em ApJ}, 596:860--878.

\bibitem[{Milosavljevi{\'{c}}} and {Merrit}, 2001]{MM01}
{Milosavljevi{\'{c}}}, M. and {Merrit} (2001).
\newblock {Formation of Galactic Nuclei}.
\newblock {\em ApJ}, 563:34--62.

\bibitem[{Miralda-Escud{\'e}} and {Gould}, 2000]{MEG00}
{Miralda-Escud{\'e}}, J. and {Gould}, A. (2000).
\newblock A cluster of black holes at the galactic center.
\newblock {\em ApJ}, 545:847--853.

\bibitem[{Misner} et~al., 1973]{MTW73}
{Misner}, C.~W., {Thorne}, K.~S., and {Wheeler}, J.~A. (1973).
\newblock {\em {Gravitation}}.
\newblock San Francisco: W.H.~Freeman and Co., 1973.

\bibitem[{Miyoshi} et~al., 1995]{Miyoshi95}
{Miyoshi}, M., {Moran}, J., {Herrnstein}, J., {Greenhill}, L., {Nakai}, N.,
  {Diamond}, P., and {Inoue}, M. (1995).
\newblock Evidence for a black-hole from high rotation velocities in a
  sub-parsec region of ngc4258.
\newblock {\em Nat}, 373:127.

\bibitem[{Moran} et~al., 1999]{MoranEtAl99}
{Moran}, J.~M., {Greenhill}, L.~J., and {Herrnstein}, J.~R. (1999).
\newblock {Observational Evidence for Massive Black Holes in the Centers of
  Active Galaxies}.
\newblock {\em Journal of Astrophysics and Astronomy}, 20:165.

\bibitem[{Morris}, 1993]{Morris93}
{Morris}, M. (1993).
\newblock {Massive star formation near the Galactic center and the fate of the
  stellar remnants}.
\newblock {\em ApJ}, 408:496--506.

\bibitem[{Murphy} et~al., 1991]{MCD91}
{Murphy}, B.~W., {Cohn}, H.~N., and {Durisen}, R.~H. (1991).
\newblock {Dynamical and luminosity evolution of active galactic nuclei -
  Models with a mass spectrum}.
\newblock {\em apj}, 370:60--77.

\bibitem[{New} and {Shapiro}, 2001]{NS01}
{New}, K.~C.~B. and {Shapiro}, S.~L. (2001).
\newblock {Evolution of Differentially Rotating Supermassive Stars to the Onset
  of Bar Instability}.
\newblock {\em apj}, 548:439--446.

\bibitem[{Ozernoi} and {Reinhardt}, 1978]{OR78}
{Ozernoi}, L.~M. and {Reinhardt}, M. (1978).
\newblock {Massive black holes in nuclei of galaxies and quasars}.
\newblock {\em A\&AS}, 59:171--192.

\bibitem[{Peebles}, 1972]{Peebles72}
{Peebles}, P. J.~E. (1972).
\newblock Star distribution near a collapsed object.
\newblock {\em ApJ}, 178:371--376.

\bibitem[{Perley} et~al., 1984]{PerleyEtAl84}
{Perley}, R.~A., {Dreher}, J.~W., and {Cowan}, J.~J. (1984).
\newblock {The jet and filaments in Cygnus A}.
\newblock {\em apjl}, 285:L35--L38.

\bibitem[{Pfahl} and {Loeb}, 2003]{PL03}
{Pfahl}, E. and {Loeb}, A. (2003).
\newblock {Probing the Spacetime Around Sgr A* with Radio Pulsars}.
\newblock {\em ArXiv Astrophysics e-prints}.

\bibitem[{Pian} et~al., 1997]{PianEtAl97}
{Pian}, E., {Urry}, C.~M., {Treves}, A., {Maraschi}, L., {Penton}, S., {Shull},
  J.~M., {Pesce}, J.~E., {Grandi}, P., {Kii}, T., {Kollgaard}, R.~I.,
  {Madejski}, G., {Marshall}, H.~L., {Wamsteker}, W., {Celotti}, A.,
  {Courvoisier}, T.~J.-L., {Falomo}, R., {Fink}, H.~H., {George}, I.~M., and
  {Ghisellini}, G. (1997).
\newblock {Multiwavelength Monitoring of the BL Lacertae Object PKS 2155-304 in
  1994 May. II. The IUE Campaign}.
\newblock {\em apj}, 486:784.

\bibitem[{Pinkney} et~al., 2003]{PinkneyEtAl03}
{Pinkney}, J., {Gebhardt}, K., {Bender}, R., {Bower}, G., {Dressler}, A.,
  {Faber}, S.~M., {Filippenko}, A.~V., {Green}, R., {Ho}, L.~C., {Kormendy},
  J., {Lauer}, T.~R., {Magorrian}, J., {Richstone}, D., and {Tremaine}, S.
  (2003).
\newblock {Kinematics of 10 Early-Type Galaxies from Hubble Space Telescope and
  Ground-based Spectroscopy}.
\newblock {\em ApJ}, 596:903--929.

\bibitem[{Plummer}, 1911]{Plummer11}
{Plummer}, H.~C. (1911).
\newblock On the problem of distribution in globular star clusters.
\newblock {\em MNRAS}, 71:460.

\bibitem[{Portegies Zwart} and {McMillan}, 2000]{PZMM00}
{Portegies Zwart}, S.~F. and {McMillan}, S.~L.~W. (2000).
\newblock {Black Hole Mergers in the Universe}.
\newblock {\em apjl}, 528:L17--L20.

\bibitem[{Portegies Zwart} and {McMillan}, 2002]{PZMcM02}
{Portegies Zwart}, S.~F. and {McMillan}, S. L.~W. (2002).
\newblock {The Runaway Growth of Intermediate-Mass Black Holes in Dense Star
  Clusters}.
\newblock {\em ApJ}, 576:899--907.

\bibitem[{Quinlan}, 1996]{Quinlan96}
{Quinlan}, G.~D. (1996).
\newblock The time-scale for core collapse in spherical star clusters.
\newblock {\em New Astronomy}, 1:255--270.

\bibitem[{Quinlan} and {Shapiro}, 1987]{QS87}
{Quinlan}, G.~D. and {Shapiro}, S.~L. (1987).
\newblock The collapse of dense star clusters to supermassive black holes -
  binaries and gravitational radiation.
\newblock {\em ApJ}, 321:199--210.

\bibitem[{Quinlan} and {Shapiro}, 1990]{QS90}
{Quinlan}, G.~D. and {Shapiro}, S.~L. (1990).
\newblock The dynamical evolution of dense star clusters in galactic nuclei.
\newblock {\em ApJ}, 356:483--500.

\bibitem[{Raboud} and {Mermilliod}, 1998]{RM98}
{Raboud}, D. and {Mermilliod}, J.-C. (1998).
\newblock {Evolution of mass segregation in open clusters: some observational
  evidences}.
\newblock {\em aa}, 333:897--909.

\bibitem[{Rasio} et~al., 2003]{RFG03}
{Rasio}, F.~A., {Freitag}, M., and {G{\"u}rkan}, M.~A. (2003).
\newblock {Formation of Massive Black Holes in Dense Star Clusters}.
\newblock In {Ho}, L., editor, {\em ``Coevolution of Black Holes and
  Galaxies'', Carnegie Observatories, Pasadena}.
\newblock astro-ph/0304038.

\bibitem[{Rauch}, 1995]{Rauch95}
{Rauch}, K.~P. (1995).
\newblock Dynamical evolution of star clusters around a rotating black hole
  with an accretion disc.
\newblock {\em MNRAS}, 275:628--640.

\bibitem[{Rauch}, 1999]{Rauch99}
{Rauch}, K.~P. (1999).
\newblock Collisional stellar dynamics around massive black holes in active
  galactic nuclei.
\newblock {\em ApJ}, 514:725--745.

\bibitem[{Rees}, 1984]{Rees84}
{Rees}, M.~J. (1984).
\newblock "black hole models for active galactic nuclei".
\newblock {\em ARA\&A}, 22:471--506.

\bibitem[{Rees}, 1990]{Rees90}
{Rees}, M.~J. (1990).
\newblock 'dead quasars' in nearby galaxies?
\newblock {\em Science}, 247:817--823.

\bibitem[{Richstone} et~al., 1998]{RichstoneEtAl98}
{Richstone}, D., {Ajhar}, E.~A., {Bender}, R., {Bower}, G., {Dressler}, A.,
  {Faber}, S.~M., {Filippenko}, A.~V., {Gebhardt}, K., {Green}, R., {Ho},
  L.~C., {Kormendy}, J., {Lauer}, T.~R., {Magorrian}, J., and {Tremaine}, S.
  (1998).
\newblock {Supermassive black holes and the evolution of galaxies.}
\newblock {\em nat}, 395:A14+.

\bibitem[{Rosenbluth} et~al., 1957]{RosenbluthEtAl57}
{Rosenbluth}, M.~N., {MacDonald}, W.~M., and {Judd}, D.~L. (1957).
\newblock {\em Phys. Rev. 2nd Ser.}, 107:1.

\bibitem[{Salpeter}, 1964]{Salpeter64}
{Salpeter}, E.~E. (1964).
\newblock {Accretion of Interstellar Matter by Massive Objects.}
\newblock {\em apj}, 140:796--800.

\bibitem[{Sanders}, 1970]{Sanders70b}
{Sanders}, R.~H. (1970).
\newblock The effects of stellar collisions in dense stellar systems.
\newblock {\em ApJ}, 162:791.

\bibitem[{Sch{\" o}del} et~al., 2003]{SchoedelEtAl03}
{Sch{\" o}del}, R., {Ott}, T., {Genzel}, R., {Eckart}, A., {Mouawad}, N., and
  {Alexander}, T. (2003).
\newblock {Stellar Dynamics in the Central Arcsecond of Our Galaxy}.
\newblock {\em ApJ}, 596:1015--1034.

\bibitem[{Schaffer}, 1979]{Schaffer79}
{Schaffer}, S. (1979).
\newblock {Michell, John and Black Holes}.
\newblock {\em Journal for the History of Astronomy}, 10:42.

\bibitem[{Shapiro}, 1977]{Shapiro77}
{Shapiro}, S.~L. (1977).
\newblock The dissolution of globular clusters containing massive black holes.
\newblock {\em ApJ}, 217:281--286.

\bibitem[{Shapiro} and {Lightman}, 1976]{SL76}
{Shapiro}, S.~L. and {Lightman}, A.~P. (1976).
\newblock The distribution of stars around a massive black hole.
\newblock {\em Nat}, 262:743--745.

\bibitem[{Shapiro} and {Marchant}, 1978]{SM78}
{Shapiro}, S.~L. and {Marchant}, A.~B. (1978).
\newblock Star clusters containing massive, central black holes - monte carlo
  simulations in two-dimensional phase space.
\newblock {\em ApJ}, 225:603--624.

\bibitem[{Shapiro} and {Teukolsky}, 1983]{ST83}
{Shapiro}, S.~L. and {Teukolsky}, S.~A. (1983).
\newblock {\em Black holes, white dwarfs, and neutron stars: The physics of
  compact objects}.
\newblock Wiley-Interscience.

\bibitem[{Shapiro} and {Teukolsky}, 1985]{ST85}
{Shapiro}, S.~L. and {Teukolsky}, S.~A. (1985).
\newblock The collapse of dense star clusters to supermassive black holes - the
  origin of quasars and agns.
\newblock {\em ApJ Lett.}, 292:41--44.

\bibitem[{Sigurdsson}, 2003]{Sigurdsson03}
{Sigurdsson}, S. (2003).
\newblock {The loss cone: past, present and future}.
\newblock {\em Classical and Quantum Gravity}, 20:45.

\bibitem[{So{\l}tan}, 1982]{Soltan82}
{So{\l}tan}, A. (1982).
\newblock {Masses of quasars}.
\newblock {\em MNRAS}, 200:115--122.

\bibitem[{Spitzer} and {Saslaw}, 1966]{SS66}
{Spitzer}, L., J. and {Saslaw}, W.~C. (1966).
\newblock On the evolution of galactic nuclei.
\newblock {\em ApJ}, 143:400.

\bibitem[{Spitzer}, 1987]{Spitzer87}
{Spitzer}, L. (1987).
\newblock {\em Dynamical evolution of globular clusters}.
\newblock Princeton University Press.

\bibitem[{Spitzer} and {Shull}, 1975]{SS75a}
{Spitzer}, L. and {Shull}, J.~M. (1975).
\newblock Random gravitational encounters and the evolution of spherical
  systems. {VI} - plummer's model.
\newblock {\em ApJ}, 200:339--342.

\bibitem[{Spitzer}, 1969]{Spitzer69}
{Spitzer}, L.~J. (1969).
\newblock Equipartition and the formation of compact nuclei in spherical
  stellar systems.
\newblock {\em ApJ Lett.}, 158:139.

\bibitem[{Spitzer} and {Hart}, 1971]{SH71b}
{Spitzer}, L.~J. and {Hart}, M.~H. (1971).
\newblock Random gravitational encounters and the evolution of spherical
  systems. {II}. models.
\newblock {\em ApJ}, 166:483.

\bibitem[{Spitzer} and {Stone}, 1967]{SS67}
{Spitzer}, L.~J. and {Stone}, M.~E. (1967).
\newblock On the evolution of galactic nuclei. {II}.
\newblock {\em ApJ}, 147:519.

\bibitem[{Spurzem}, 1991]{Spurzem91}
{Spurzem}, R. (1991).
\newblock {Gravothermal instability of anisotropic self-gravitating gas spheres
  - Singular equilibrium solution}.
\newblock {\em MNRAS}, 252:177--189.

\bibitem[{Spurzem}, 1992]{Spurzem92}
{Spurzem}, R. (1992).
\newblock Evolution of stars and gas in galactic nuclei.
\newblock In {\em Reviews of Modern Astronomy}, volume~5, pages 161--173.

\bibitem[{Spurzem}, 1999]{Spurzem99}
{Spurzem}, R. (1999).
\newblock Direct $n$-body simulations.
\newblock {\em Journal of Computational and Applied Mathematics}, 109:407--432.

\bibitem[{Spurzem} and {Aarseth}, 1996]{SA96}
{Spurzem}, R. and {Aarseth}, S.~J. (1996).
\newblock Direct collisional simulation of 100000 particles past core collapse.
\newblock {\em MNRAS}, 282:19.

\bibitem[{Spurzem} et~al., 2003]{SpurzemEtAl03}
{Spurzem}, R., {Berczik}, P., {Hensler}, G., {Theis}, C., {Amaro-Seoane}, P.,
  {Freitag}, M., and {Just}, A. (2003).
\newblock {Physical Processes in Star-Gas Systems}.
\newblock {\em ArXiv Astrophysics e-prints}.

\bibitem[{Spurzem} and {Takahashi}, 1995]{ST95}
{Spurzem}, R. and {Takahashi}, K. (1995).
\newblock Comparison between fokker-planck and gaseous models of star clusters
  in the multi-mass case revisited.
\newblock {\em MNRAS}, 272:772--784.

\bibitem[{Stodo{\l}kiewicz}, 1982]{Stodol82}
{Stodo{\l}kiewicz}, J.~S. (1982).
\newblock Dynamical evolution of globular clusters. i.
\newblock {\em Acta Astron.}, 32:63--91.

\bibitem[{Stoner} and {Ptak}, 1984]{SP84}
{Stoner}, R.~E. and {Ptak}, R.~L. (1984).
\newblock {Evidence for Supermassive Stars in Three Seyfert-I Nuclei from
  Variability in IUE Spectra}.
\newblock {\em baas}, 16:988.

\bibitem[{Syer} and {Ulmer}, 1999]{SU99}
{Syer}, D. and {Ulmer}, A. (1999).
\newblock Tidal disruption rates of stars in observed galaxies.
\newblock {\em MNRAS}, 306:35--42.

\bibitem[{Takahashi}, 1993]{Takahashi93}
{Takahashi}, K. (1993).
\newblock A new powerful method for solving the orbit-averaged fokker-planck
  equation regarding stellar dynamics.
\newblock {\em PASJ}, 45:233--242.

\bibitem[{Takahashi}, 1995]{Takahashi95}
{Takahashi}, K. (1995).
\newblock Fokker-planck models of star clusters with anisotropic velocity
  distributions i. pre-collapse evolution.
\newblock {\em PASJ}, 47:561--573.

\bibitem[{Takahashi}, 1996]{Takahashi96}
{Takahashi}, K. (1996).
\newblock Fokker-planck models of star clusters with anisotropic velocity
  distributions {II}. post-collapse evolution.
\newblock {\em PASJ}, 48:691--700.

\bibitem[{Takahashi}, 1997]{Takahashi97}
{Takahashi}, K. (1997).
\newblock Fokker-planck models of star clusters with anisotropic velocity
  distributions {III}. multi-mass clusters.
\newblock {\em PASJ}, 49:547--560.

\bibitem[{Terlevich}, 1989]{Terlevich89}
{Terlevich}, R. (1989).
\newblock {Active galactic nuclei without black-holes}.
\newblock In {\em Evolutionary Phenomena in Galaxies}, pages 149--158.

\bibitem[{Thorne}, 1995]{Thorne95}
{Thorne}, K.~S. (1995).
\newblock {Gravitational Radiation}.
\newblock In {\em Seventeeth Texas Symposium on Relativistic Astrophysics and
  Cosmology}, page 127.

\bibitem[{Tremaine} et~al., 1994]{Tremaine94}
{Tremaine}, S., {Richstone}, D.~O., {Byun}, Y., {Dressler}, A., {Faber}, S.~M.,
  {Grillmair}, C., {Kormendy}, J., and {Lauer}, T.~R. (1994).
\newblock {A family of models for spherical stellar systems}.
\newblock {\em AJ}, 107:634--644.

\bibitem[{van der Marel} et~al., 2002]{vdMEtAl02}
{van der Marel}, R.~P., {Gerssen}, J., {Guhathakurta}, P., {Peterson}, R.~C.,
  and {Gebhardt}, K. (2002).
\newblock {Hubble Space Telescope Evidence for an Intermediate-Mass Black Hole
  in the Globular Cluster M15. I. STIS Spectroscopy and WFPC2 Photometry}.
\newblock {\em AJ}, 124:3255--3269.

\bibitem[{Vil'Koviskii}, 1978]{Vilk78}
{Vil'Koviskii}, {\' E}.~Y. (1978).
\newblock {Interaction of galaxies with central objects, and the activity of
  galaxy nuclei}.
\newblock {\em Soviet Astronomy}, 22:385.

\bibitem[{Wallace} and {Woosley}, 1981]{WW81}
{Wallace}, R.~K. and {Woosley}, S.~E. (1981).
\newblock {Explosive hydrogen burning}.
\newblock {\em apjs}, 45:389--420.

\bibitem[{Watters} et~al., 2000]{WJR00}
{Watters}, W.~A., {Joshi}, K.~J., and {Rasio}, F.~A. (2000).
\newblock Thermal and dynamical equilibrium in two-component star clusters.
\newblock {\em ApJ}, 539:331--341.

\bibitem[{Woosley} and {Weaver}, 1982]{WW82}
{Woosley}, S.~E. and {Weaver}, T.~A. (1982).
\newblock {Theoretical Models for Supernovae}.
\newblock In {\em NATO ASIC Proc. 90: Supernovae: A Survey of Current
  Research}, page~79.

\bibitem[{Yorke}, 1980]{Yorke80}
{Yorke}, H.~W. (1980).
\newblock {Numerical solution of the equation of radiation transfer in
  spherical geometry}.
\newblock {\em aa}, 86:286--294.

\bibitem[{Yu} and {Tremaine}, 2002]{YT02}
{Yu}, Q. and {Tremaine}, S. (2002).
\newblock {Observational constraints on growth of massive black holes}.
\newblock {\em MNRAS}, 335:965--976.

\bibitem[{Zeldovich} and {Novikov}, 1971]{ZN71}
{Zeldovich}, Y.~B. and {Novikov}, I.~D. (1971).
\newblock {\em Relativistic astrophysics. Vol.1: Stars and relativity}.
\newblock University of Chicago Press.

\bibitem[{Zel'Dovich} and {Podurets}, 1966]{ZP66}
{Zel'Dovich}, Y.~B. and {Podurets}, M.~A. (1966).
\newblock {The Evolution of a System of Gravitationally Interacting Point
  Masses}.
\newblock {\em Soviet Astronomy}, 9:742.

\end{thebibliography}

th_model.bbl
\begin{thebibliography}{}

\bibitem[{Amaro-Seoane} and {Spurzem}, 2001]{AS01a}
{Amaro-Seoane}, P. and {Spurzem}, R. (2001).
\newblock {Gas in the Central Regions of AGN: The Interstellar Medium and
  Supermassive Gaseous Objects}.
\newblock In {\em ASP Conf. Ser. 249: The Central Kiloparsec of Starbursts and
  AGN: The La Palma Connection}, page 731.

\bibitem[{Amaro-Seoane} and {Spurzem}, 2004]{AS04}
{Amaro-Seoane}, P. and {Spurzem}, R. (2004).
\newblock {Dense Gas-Star Systems: Evolution of Supermassive Stars}.
\newblock In {\em In "Carnegie Observatories Astrophysics Series, Vol. 1:
  Coevolution of Black Holes and Galaxies," ed. L. C. Ho, Pasadena, USA:
  Carnegie Observatories, 2003}.

\bibitem[{Bettwieser} and {Spurzem}, 1986]{BS86}
{Bettwieser}, E. and {Spurzem}, R. (1986).
\newblock {Anisotropy in stellar dynamics}.
\newblock {\em A\&A}, 161:102--112.

\bibitem[{Binney} and {Tremaine}, 1987]{BT87}
{Binney}, J. and {Tremaine}, S. (1987).
\newblock {\em Galactic Dynamics}.
\newblock Princeton University Press.

\bibitem[{Castor}, 1972]{Castor72}
{Castor}, J.~I. (1972).
\newblock {Radiative Transfer in Spherically Symmetric Flows}.
\newblock {\em apj}, 178:779--792.

\bibitem[{Giersz} and {Spurzem}, 1994]{GS94}
{Giersz}, M. and {Spurzem}, R. (1994).
\newblock Comparing direct $n$-body integration with anisotropic gaseous models
  of star clusters.
\newblock {\em MNRAS}, 269:241.

\bibitem[{Henyey} et~al., 1959]{HenyeyEtAl59}
{Henyey}, L.~G., {Wilets}, L., {B{\" o}hm}, K.~H., {Lelevier}, R., and {Levee},
  R.~D. (1959).
\newblock {A Method for Atomic Computation of Stellar Evolution.}
\newblock {\em ApJ}, 129:628.

\bibitem[{Kippenhahn} and {Weigert}, 1994]{KW94}
{Kippenhahn}, R. and {Weigert}, A. (1994).
\newblock {\em Stellar Structure and Evolution}.
\newblock Springer-Verlag Berlin Heidelberg.

\bibitem[{Langbein} et~al., 1990]{LangbeinEtAl90}
{Langbein}, T., {Fricke}, K.~J., {Spurzem}, R., and {Yorke}, H.~W. (1990).
\newblock {Interactions between stars and gas in galactic nuclei}.
\newblock {\em aa}, 227:333--341.

\bibitem[{Lin} and {Tremaine}, 1980]{LT80}
{Lin}, D. N.~C. and {Tremaine}, S. (1980).
\newblock A reinvestigation of the standard model for the dynamics of a massive
  black hole in a globular cluster.
\newblock {\em ApJ}, 242:789--798.

\bibitem[{Louis} and {Spurzem}, 1991]{LS91}
{Louis}, P.~D. and {Spurzem}, R. (1991).
\newblock Anisotropic gaseous models for the evolution of star clusters.
\newblock {\em MNRAS}, 251:408--426.

\bibitem[{Lynden-Bell} and {Eggleton}, 1980]{LBE80}
{Lynden-Bell}, D. and {Eggleton}, P.~P. (1980).
\newblock On the consequences of the gravothermal catastrophe.
\newblock {\em MNRAS}, 191:483--498.

\bibitem[{Rosenbluth} et~al., 1957]{RosenbluthEtAl57}
{Rosenbluth}, M.~N., {MacDonald}, W.~M., and {Judd}, D.~L. (1957).
\newblock {\em Phys. Rev. 2nd Ser.}, 107:1.

\bibitem[{Spitzer} and {Saslaw}, 1966]{SS66}
{Spitzer}, L., J. and {Saslaw}, W.~C. (1966).
\newblock On the evolution of galactic nuclei.
\newblock {\em ApJ}, 143:400.

\bibitem[{Spitzer}, 1987]{Spitzer87}
{Spitzer}, L. (1987).
\newblock {\em Dynamical evolution of globular clusters}.
\newblock Princeton University Press.

\bibitem[{Spurzem} et~al., 2003]{SpurzemEtAl03}
{Spurzem}, R., {Berczik}, P., {Hensler}, G., {Theis}, C., {Amaro-Seoane}, P.,
  {Freitag}, M., and {Just}, A. (2003).
\newblock {Physical Processes in Star-Gas Systems}.
\newblock {\em ArXiv Astrophysics e-prints}.

\bibitem[{Yorke}, 1980]{Yorke80}
{Yorke}, H.~W. (1980).
\newblock {Numerical solution of the equation of radiation transfer in
  spherical geometry}.
\newblock {\em aa}, 86:286--294.

\end{thebibliography}
