The Influence of Hydrodynamical Winds on Hot Accretion Disk Solutions
Ranjeev Misra\altaffilmark{1}
The effect of a possible hydrodynamical wind on the nature of hot accretion disk solutions is studied. It is found that the advection dominated branch, in the presence of a wind, maintains the self-similar solution for the disk structure with the temperature, $\theta \propto 1/r$, optical depth $\tau \propto r^{P-1/2}$ and accretion rate $\dot m \propto r^P$. Based on global solutions cooling due to wind energy loss and advection are found to be equally important. For a wide range of viscosity and wind parameters, the temperature is about one-tenth of the virial value and $P \approx 0.9$, independent of the mass accretion rate and radius. In the context of cooling by unsaturated Comptonization of soft photons, solutions also exist where radiative cooling, advection, and wind cooling are important. In this case, wind regulated solutions are possible. Here, the radial dependence of the critical mass accretion rate above which solutions do not exist is unchanged from those solutions without winds. The wind/advection dominated solutions are locally unstable to a new type of instability called ``wind driven insbility'' where the presence of a wind causes the disk to be unstable to long wavelength perturbations of the surface density. The growth rate of this instability is inversely proportional to the ratio of the radiative cooling to the gravitational energy dissipation rates and it can grow on a timescale much longer than the viscous timescale in the disk for sufficiently small radiative cooling efficiencies.
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\shorttitle{Hot disk solutions with winds}
\shortauthors{R. Misra and R. E. Taam}


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\begin{document}

%\setcounter{footnote}{3}

\title{The Influence of Hydrodynamical Winds on Hot Accretion Disk
        Solutions}



\author{Ranjeev Misra\altaffilmark{1}} \and \author{Ronald E. Taam\altaffilmark{1}}


\altaffiltext{1}{Department of Physics and Astronomy,
Northwestern University, 2131 Sheridan Road, Evanston,IL 60208;
ranjeev@finesse.astro.nwu.edu;taam@apollo.astro.nwu.edu}



\begin{abstract}

The effect of a possible hydrodynamical wind on the nature of hot accretion 
disk solutions is studied. It is found that the advection dominated branch, 
in the presence of a wind, maintains the self-similar solution for the disk 
structure with the temperature, $\theta \propto 1/r$, optical depth $\tau 
\propto r^{P-1/2}$ and accretion rate $\dot m \propto r^P$. Based on global 
solutions cooling due to wind energy loss and advection are found to be equally 
important. For a wide range of viscosity and wind parameters, the temperature is 
about one-tenth of the virial value and $P \approx 0.9$, independent of the mass
accretion rate and radius. 

In the context of cooling by unsaturated Comptonization of soft photons, solutions
also exist where radiative cooling, advection, and wind cooling are important. In 
this case, wind regulated solutions are possible.  Here, the radial dependence of 
the critical mass accretion rate above which solutions do not exist is unchanged 
from those solutions without winds. 

The wind/advection dominated solutions are locally unstable to a new type of instability 
called ``wind driven insbility'' where the presence of a wind causes the disk 
to be unstable to long wavelength perturbations of the surface density.  The 
growth rate of this instability is inversely proportional to the ratio of the 
radiative cooling to the gravitational energy dissipation rates and it can grow 
on a timescale much longer than the viscous timescale in the disk for sufficiently 
small radiative cooling efficiencies.	
\end{abstract}

\keywords{accretion, accretion disks-black hole physics -hydrodynamics}


\section{INTRODUCTION} \label{sec: I} 

Hot accretion disks are a natural site where externally or internally 
produced soft photons can be Comptonized to higher energies by disk 
electrons. It is this property that makes hot accretion disk models attractive 
in explaining the spectra of compact X-ray binaries and active galactic nuclei.  
Alternative models to this picture have also been 
proposed involving shocks (Chakrabarti \& Titarchuk 1995), patchy or uniform
coronae (Haardt \& Maraschi 1993; Liang \& Price 1977), or spatially extended 
regions in a transition disk model (Misra, Chitinis \& Melia 1998). Detailed 
comparisons of the observed X-ray spectra with predictions of these different models 
can be used to determine the applicability of each model to a given system. In 
addition, constraints imposed by X-ray timing analysis can also be used to further 
distinguish between these models (Nowak et al 1999). Apart from these observational 
tests, theoretical studies of the stability and self-consistencies of these models 
are important for determining the circumstances under which they are physically 
realizable. 

In a seminal study, Shapiro, Lightman \& Eardley (1976) proposed a hot accretion 
disk model to explain the X-ray spectra of the black hole system Cygnus X-1. 
Here, a hot inner disk was assumed to be cooled by external soft photons 
originating from an outer cold disk.  In this model, the gravitational energy 
dissipated in the hot disk heats the protons, and due to the inefficiency of the 
Coulomb interaction rate in these environments, leads to a large temperature 
differences between the protons and electrons.  The outer cold disk was assumed 
to be transformed into a hot inner disk for radii less than a critical value due 
to a thermal instability associated with the dominance of radiation pressure 
(Lightman \& Eardley 1974). Despite the success of this model in explaining the 
X-ray observations of Cygnus X-1, there remains several outstanding theoretical 
issues. Specifically, it was not clear whether an unstable cold disk would in 
reality evolve to a hot disk (Chen \& Taam 1994) or whether the narrow transition 
between the hot and cold disk was energetically possible. Moreover, the hot accretion 
disk solution, itself, was thermally unstable.

Against this backdrop, the importance of radial advection of energy in such disks 
was noted by Ichimaru (1977). Taking this effect into account, Narayan \& Yi (1994) 
constructed a second type of hot disk solution with a self-similar structure called 
Advection Dominated Accretion Flows (ADAF). These solutions were, in general, 
radiatively inefficient (with most of the energy advected into a black hole) compared 
to the non-advective, radiatively
cooled solutions found earlier. In contrast to the radiatively cooled solutions,
ADAFs are thermally stable, and it was shown that a sharp transition between the 
hot inner regions and the cool outer regions could exist, provided an additional 
mechanism (e.g. conduction/turbulent diffusion) is assumed which transfers energy 
from the hot disk to the transition radius (Honma 1996). The topology of these 
solutions was studied by Chen et al. (1995) and Zdziarski (1998), and it was found 
that the advection dominated and radiatively cooled solutions only existed below 
a critical mass accretion rate.  Observational support for the radiatively 
inefficient ADAF solutions was provided by the X-ray spectral fitting of 
low luminosity galactic centers and quiescent X-ray Novae (Narayan et al. 1998).  

It was noted by Narayan \& Yi (1994) that steady state rotating solutions do not 
exist for an adiabatic index of the accreting flow, $\Gamma_D = 5/3$.  In this 
case the ADAF solutions become self-similar spherical Bondi accretion flows. This 
is not a general result, however, since Ogilvie (1999) has shown that time dependent 
self similar solutions for $\Gamma_D = 5/3$ do exist. If we consider only time 
independent flows, Narayan \& Yi (1994) constructed self-similar rotating solutions 
in the case of a disk with a nearly equipartition magnetic field (corresponding to 
an effective adiabatic index of $4/3$). 
Since it is not clear 
that all accretion flows will contain such fields (e.g., Balbus \& Hawley 1998), 
the applicability of these 
solutions could be limited.  Moreover, Bisnovatyi-Kogan \& Lovelace (2000) have 
recently argued that, in the presence of an equipartition field, the electrons 
in the disk will be preferentially heated compared to the protons and a two 
temperature disk may not exist. In contrast, Blandford \& Begelman (1999) have  
recently shown that a rotating self-similar flow with $\Gamma_D = 5/3$ is possible, 
in the presence of a wind, if the accretion flow rate in the disk varies with radius 
as $\dot M \propto R^P$. A strong hydrodynamic 
wind from hot disks is possible, in principle,
since the ion temperature in the disk is close to the virial value as shown in the 
early work by Takahara, Rosner, \& Kusunose (1989) and Kusunose (1991). Recently 
confirmation of these results has been obtained by Beckert (2000) for a variety of
viscosity prescriptions.  In a parallel development, Chakrabarti (1999) and Das 
(1999) have studied the possibility of outflows in the context of shock/centrifugal
barrier models for the accreting flow.  Comparisons with observations 
reveal that the X-ray spectra of such a wind driven self-similar 
flow called ADIOS (Advection dominated inflow-outflow solutions) can explain 
the observed spectra of black hole candidates in quiescence 
(Quataert \& 
Narayan 1999). 
In addition, Lin, Misra, \& 
Taam (2000) find that the spectral characteristics of high luminosity black hole 
systems suggest that winds may be important for these systems as well. 

Theoretically, the conditions under which wind dominated accreting flows exist are
unclear.  For example, Abramowicz, Lasota, \& Igumenshchev (2000) argue, based on 
the sign of the Bernoulli function and on detailed two dimensional hydrodynamical 
simulations that winds are not expected for disks characterized by low viscosities 
($\alpha < 0.1$).  Although modest outflows have been seen in simulations of disks 
characterized by high viscosities (Igumenshchev \& Abramowicz 1999; see also Stone 
et al. 1999), the relaxation of the assumption of identical adiabatic indices in the 
disk and wind (reflecting different mechanisms for driving accretion and outflow) 
can enhance the effectiveness of winds (see Chakrabarti 1999). The existence of 
winds can also be affected by the presence of convection which can modify the 
temperature profile (Narayan, Igumenshchev, \& Abramowicz 2000). In particular, they 
find that the thermal disk structure is dependent on the magnitude of the $\alpha$ 
viscosity parameter and the direction in which angular momentum is transported by 
the convection. For outward transport of angular momentum the ADAF solutions are 
recovered, whereas for inward transport both non-accreting solutions with 
lowered temperatures ($\alpha < 0.05$) and the standard ADAF solutions are found
($\alpha > 0.05$).  Thus the effect of convection on the existence of winds remains 
inconclusive especially since  it is dependent on the unknown distribution of energy 
dissipation in the vertical extent. It is clear that a strong wind could exist under
certain conditions, and it would be useful to study the effects of such a wind on 
the structure of hot accretion disks. 

In ADIOS solutions it is generally assumed that the mass flow rate has a power-law 
dependence on radius with the power law index, $P$, treated as a parameter (Blandford 
\& Begelman 1999; Beckert 2000). In principle, the 
variation of the accretion rate with radii should be obtained self-consistently 
from the surface wind velocity and the specific energy of the wind matter.  
This, in turn, depends on the disk structure, geometry and the thermal 
properties of the wind. A realistic computation of this disk/wind configuration 
is difficult due to the complexity of the problem and the unknown vertical 
distribution of the viscous dissipation in the disk. However, we show that 
significant progress can be attained if only the dependence of the 
surface velocity on the disk structure is estimated and its numerical
value is parameterized to take into account the uncertainties involved. 

In the next section, the basic equations of hot disks with winds are 
described. In \S 3 the results of the calculations are presented and 
in \S 4 the stability of the solutions is discussed. Finally, we summarize
the results and conclude in the last section. 


\section{DISK STRUCTURE EQUATIONS} \label{sec: 2} 

The standard equations for a Newtonian, keplerian hot accretion disk are 
modified to take account of mass, energy, and angular momentum loss due to 
a hydrodynamical wind. To facilitate the presentation of these equations in 
a compact form, the following dimensionless quantities are introduced: the 
ion temperature $\theta \equiv kT_i/m_p c^2$; the electron temperature $\theta_e 
\equiv k T_e/m_e c^2$; the radius $r = R/r_g$ where $r_g \equiv GM/c^2$ and $M$ is
the mass of the black hole; the accretion rate $\dot m \equiv \dot M / \dot 
M_{Edd}$ where the Eddington accretion rate is defined to be $\dot M_{Edd} =
4 \pi m_p c r_g /\sigma_T$; Thomson optical depth $\tau \equiv n \sigma_T H$, 
where $n$ and $H$ are the number density and half-height of the disk respectively; 
the speed of the wind at the disk surface $v_w \equiv V_w/c_s$ where $c_s = c 
\theta^{1/2}$ is the local isothermal sound speed of the disk and the Coulomb 
energy exchange frequency $\nu^*_c = (r_g/c) \nu_c$.
 
For a non-magnetized two-temperature disk i.e. when the ion temperature ($T_i$) is
much higher than the electron temperature ($T_e$), the local pressure is simply
$P_g = n kT_i$ and the equation for hydrostatic equilibrium can be written as
\begin{eqnarray}
(H/R)^2  & = &  \theta r
\label{eq:hysteq}
\end{eqnarray}
In the presence of a wind the accretion rate is no longer constant and hence 
mass conservation gives,
\begin{eqnarray}
d \dot M \over d R   & = &  4 \pi R m_p n V_w
\end{eqnarray}
which in dimensionless form can be re-written using equ. (\ref{eq:hysteq}) as,
\begin{eqnarray}
d \dot m \over d r   & = &  \tau v_w r^{-1/2}.
\label{eq:mdot}
\end{eqnarray}
The torque equation (e.g. Beckert 2000) is,
\begin{eqnarray}
d T \over d R   & = &  \dot M (R) {d l \over d R}
\end{eqnarray}
where $T$ is the viscous torque and where it has been assumed that the specific
angular momentum of the wind is same as that in the disk i.e. $l = \sqrt {G M R}$. 
The viscous integrated stress is described in 
terms of the $\alpha$-prescription of Shakura \& Sunyaev (1973), and it scales
as the pressure, i.e. $T = \alpha P_g H$ for a disk in hydrostatic equilibrium. 
The above equation can then be written in dimensionless form as
\begin{eqnarray}
\alpha {d (\tau \theta r^2 ) \over d r}   & = & {1\over 2} \dot m (r) r^{-1/2}. 
\label{eq:torq}
\end{eqnarray}
Note that since the accretion rate has a radial dependence (due to the presence 
of a wind) the above differential equation cannot be trivially integrated.

For a disk in thermal equilibrium, the local vertically  integrated energy dissipation
 rate  ($Q_D$) 
should be equal to the sum of the Coulomb energy loss ($Q_C$), the advective energy 
flux ($Q_A$) which is non-local radial energy transfer rate associated with the flow
 and the energy flux carried away by the wind ($Q_W$). The energy 
dissipation rate due to the viscous torque for each half of the disk is
\begin{eqnarray}
Q_D & = & {3\over 8 \pi} {\dot M_{Edd}  c^2 \over r_g^2} \;\; \alpha  \theta  
\tau   r^{-3/2}.
\label{eq:Qdis}
\end{eqnarray}
A fraction of this dissipated energy  will be transfered by the  protons to 
electrons via Coulomb interactions. The Coulomb energy exchange flux is
\begin{eqnarray}
Q_C & = & {3\over 8 \pi} {\dot M_{Edd}  c^2 \over r_g^2} \;\;   \tau \nu^*_c (\theta - 
{m_e\over m_p}\theta_e) \\
 & \approx  & {3\over 8 \pi} {\dot M_{Edd}  c^2 \over r_g^2} \;\;   \tau \nu^*_c \theta.
\label{eq:Qcoul}
\end{eqnarray}
Here, the dimensionless Coulomb frequency is given by,
\begin{eqnarray}
\nu^*_c  & = &{ r_g \over c} \nu_c  = 4.4 \times 10^{-4} \hbox {ln} \Delta \tau \theta_e^{-3/2} \theta^{-1/2} r^{-3/2}
\label{eq:nuc}
\end{eqnarray}
where ln$\Delta \approx 15$ is the Coulomb logarithm (Spitzer 1962).  

The advective energy flux is
\begin{eqnarray}
Q_A   & = & {\dot M \over 4 \pi R^2} {P_g\over n m_p} \xi_A = {1\over 4 \pi} {\dot M_{Edd}  c^2 \over r_g^2} \;\; \dot m  \theta  \xi_A   r^{-2} 
\label{eq:Qadv}
\end{eqnarray}
where $\xi_A$ the advective factor is defined to be (Chen et. al. 1995)
\begin{eqnarray}
\xi_A    & = & -( {3 r \over 2 \theta} {d \theta \over dr} - {r \over \tau} {d \tau \over dr}
+ { r \over H} {d H \over dr}).
\label{eq:xiA}
\end{eqnarray}
It has been implicitly assumed in this definition that the specific energy of the 
gas is $(3/2) k T_i$ or in other words, the adiabatic index of the accretion flow 
$\Gamma_D = 5/3$. Note that self-similar (ADAF) solutions without winds, $\theta 
\propto r^{-1}$, $H \propto r$ and $\tau \propto r^{-1/2}$, leads to $\xi_A = 0$. 
This is consistent and expected since, for $\Gamma_D = 5/3$ and in the absence of 
a wind, there are no steady self-similar (ADAF) rotating solutions and only spherical 
Bondi accretion flows are allowed (Narayan and Yi 1994).

The energy flux carried away by the wind is
\begin{eqnarray}
Q_W   & = & n V_W E_W = {1\over 8 \pi} {\dot M_{Edd}  c^2 \over r_g^2} \;\; \tau  v_w e_w   r^{-5/2} 
\label{eq:Qwin}
\end{eqnarray}
Here $E_W$ is the average energy imparted to a proton in the wind from the local 
disk and $e_w \equiv E_W/(GMm_p/2R)$ is its dimensionless form. The external 
energy required for a particle to escape to infinity is  $\approx GMm_p/2R -
(3/2)(kT_i -kT_f)$ where $T_i$ and $T_f$ are the proton temperature of the wind 
at the disk surface and at infinity respectively. If the wind is isothermal or if 
$kT_i << GMm_p/2R$ then this external energy input per particle is simply $GM/2R$ 
or the binding energy of the particle. Further, if this energy is obtained only from
the local region of the disk from where the particle originates, then $e_w \approx 
1$. Thus, $e_w < 1$ corresponds to the case where only a fraction of the required 
energy is obtained from the local region of the disk and the rest presumably is due 
to heating of the wind from a different region of the disk. The $e_w > 1$ corresponds 
to the unlikely case where more energy than required is deposited from the local 
disk to the wind. This additional energy may be radiated away by a radiatively 
cooled wind and/or appear as kinetic energy of the wind particles at infinity. The 
uncertainties involved in the energy coupling between the wind and the disk is 
factored into $e_w$ which is treated as a parameter in this work.

The equation describing conservation of energy ($Q_D = Q_C + Q_A +Q_W$ ) can be 
written in a compact form using equs. (\ref{eq:Qdis}), (\ref{eq:Qcoul}),
(\ref{eq:Qadv}) and (\ref{eq:Qwin}) as 
\begin{eqnarray}
{\nu^*_c r^{3/2} \over \alpha} + {2\over 3} {\dot m  \xi_A \over  \alpha 
\tau r^{1/2}} + {1\over 3} { v_w e_w \over \alpha  \theta r}  = 1.
\label{eq:Qtot}
\end{eqnarray}

The value of the electron temperature ($\theta_e$) which is required
to calculate the Coulomb frequency (equ. \ref{eq:nuc}), depends
on the dominant radiative cooling mechanism operating in the hot disk.
Following Zdziarski (1998) it is assumed here that the disk cools primarily
by unsaturated Comptonization of soft photons. This is motivated by the
high energy spectral characteristics of black hole binaries in the low-state
since the Comptonization process is characterized by the Compton y-parameter
$ (y \equiv 4kT_e /m_e c^2)$ max$(\tau,\tau^2) \approx 1$. Thus
\begin{eqnarray}
\theta_e \approx {y \over 4 max(\tau,\tau^2)}
\label{eq:thetae}
\end{eqnarray}
can be used to estimate the electron temperature using  $y$ as a parameter
of order unity.  Technically the electron temperature profile in the disk
should be obtained by solving the geometry dependent radiative transfer 
equations
in both vertical and radial directions. Thus it should be emphasized 
that equation (\ref{eq:thetae}) is a simplifying assumption which may be
approximately valid only under certain conditions 
(Shapiro, Lightman \& Eardley 1976).

Determining the wind velocity at the surface of the disk is complicated because
it depends on the vertical structure of the disk which in turn depends on the 
detailed variation of the  unknown viscosity with height. Furthermore, the opening 
angle for the wind flow needs to be determined using the geometry of the accreting 
flow and the pressure gradient in the wind in the radial direction. However, a 
qualitative estimate of the dependence of the wind velocity with respect to
the average temperature and gravitational potential in the disk, can be obtained by 
studying the
simpler geometry of a spherical flow from a radius $R$, temperature $T_i$ and 
potential $\approx GM/2R$. The factor of two in the potential takes into account 
the angular momentum of the disk.  The estimate obtained from such a spherical 
flow can be scaled by an unknown factor $\eta$ which would then take into account 
the uncertainties in the geometry and vertical structure of the disk,
\begin{eqnarray}
v_w = \eta v_s (\theta r)
\label{eq:v0}
\end{eqnarray}
where $v_s$ is the velocity of the wind at the surface of the sphere divided
by the isothermal sound speed ($ = \theta^{1/2} c$). Thus the motivation here is
to obtain a qualitative dependence of the disk surface wind velocity with 
temperature and radius of the disk, rather than a quantitative estimation of 
its magnitude. The standard equations of the hydrodynamical theory of stellar 
winds have been used to compute the wind velocity at the surface of the sphere. 
Using Bernoulli's equation and the fact that the Mach number at the sonic point 
is unity, the following relation for the sonic radius is obtained
\begin{eqnarray}
{r_c \over r_s} & = & {\lambda [{5 - 3 \Gamma \over 4 (\Gamma -1)}] \over {v_s^2 \over 2} + {\Gamma \over
\Gamma - 1} - \lambda}
\label{eq:v1}
\end{eqnarray}
where $r_c$ is the location of the sonic point, $r_s$ is the radius of the sphere, 
$\Gamma$ is the adiabatic index of the wind and $\lambda$ is the ratio of the 
gravitational potential at the surface to the temperature of the sphere i.e. 
$\lambda \equiv \phi/kT_i = GMm_p/(2 r_s k T_s)$. Note that the adiabatic index 
in the wind may differ from that in the disk (see Chakrabarti 1999). Conservation 
of mass and the equation of state, $P_g \propto \rho^\Gamma$ (where $\rho$ is the 
mass density) gives,
\begin{eqnarray}
v_s & = & ({\lambda \over 2})^{{1\over2}{\Gamma+1 \over \Gamma -1}}\;\; \Gamma^{-{1 \over \Gamma -1}} \;\; ({r_c\over r_s})^{{3 \Gamma - 5 \over 2 (\Gamma -1)}}
\label{eq:v2}
\end{eqnarray}
In this work, equations (\ref{eq:v1}) and (\ref{eq:v2}) are solved simultaneously to 
yield $v_s (\lambda , \Gamma)$ where $\lambda$ for an isothermal disk is $1/(2 \theta r)$.
If $v_s << \lambda$,  then 
\begin{eqnarray}
v_s (\lambda , \Gamma) & = &  ({1\over 2})^{\Gamma_2} \Gamma^{-{3 \over 2}} \lambda^2 [( 5 - 3 \Gamma)(1 - {\lambda (\Gamma - 1) \over \Gamma})]^{\Gamma_1}
\label{eq:v3}
\end{eqnarray}
where $\Gamma_1 = {3 \Gamma - 5 \over 2 (\Gamma -1)}$ and $\Gamma_2 = {7 \Gamma - 9 \over 2 (\Gamma -1)}$
. For the isothermal case ($\Gamma \rightarrow 1$), equ. (\ref{eq:v3}) reduces to
\begin{eqnarray}
v_s (\lambda , \Gamma \rightarrow 1) =  ({\lambda\over 2})^2 \hbox {exp}( 3/2  - \lambda)
\label{eq:v4}
\end{eqnarray}
There are several points which should be noted here. First, equs. (\ref{eq:v3}) 
and (\ref{eq:v4}) seem to suggest that $v_s$ has a maxima for variations with 
$\lambda$. This is an artifact of the assumption $v_s << \lambda$ and if equs. 
(\ref{eq:v1}) and (\ref{eq:v2}) are used instead (as is done in this work), then 
$v_s$ is a monotonic function of $\lambda$. Second, there is a maximum value of 
$\lambda$ beyond which there are no wind solutions or $v_w = 0$ for $\theta < 
\theta_{max} = (\Gamma - 1)/(2 \Gamma r)$. Third, inspection of equ. (\ref{eq:v3})
reveals that for $\theta \simgreat \theta_{max}$, $v_w$ is a rapidly varying 
function of $\theta$, a factor two increase in $\theta$ could lead to an increase 
of two orders of magnitude in $v_w$ (Lin, Misra, \& Taam 2000).

The equations corresponding to conservation of mass (\ref{eq:mdot}),
angular momentum (\ref{eq:torq}) and energy (\ref{eq:Qtot}) can
be solved to obtain the disk structure using the estimated values for electron
temperature (\ref{eq:thetae}) and wind velocity (\ref{eq:v0}).

\section{DISK SOLUTIONS WITH WINDS} \label{sec: 3} 
\subsection{Local Solutions} \label{sec: 3a} 

The differential equations governing the disk structure can be solved using 
appropriate boundary conditions. However, these equations can be reduced to 
algebraic ones, by treating the logarithmic derivatives of physical quantities 
as constant (i.e., independent of radii) parameters. The solutions for this set
of algebraic equations depends only on the local radius and hence are local 
solutions. Such solutions generally provide important insight on the disk 
structure and its dependence on disk parameters.

To obtain local solutions it is assumed that the logarithmic derivative of 
temperature ($\xi_T = - d \hbox {log} T / d \hbox {log} R$) and accretion rate 
($P = d \hbox {log} \dot M / d \hbox {log} R$) with respect to the logarithm 
of the radius are constants. Integrating equation
(\ref{eq:torq}) with the constant of integration chosen such the viscous
torque vanishes in the inner radius, leads to 
\begin{equation}
\label{eq:3torq}
\tau = {\dot m (r) r^{-{3\over 2}} \hbox {J}(r) \over (1 + 2 P) \alpha \theta }
\end{equation}
where J $(r) \equiv 1 - (6/r)^{P+1/2}$ takes into account the torque free
inner boundary condition at $r = 6$. The advective fraction (equ. \ref{eq:xiA})
is reduced to
\begin{equation}
\xi_A = 3 (\xi_T -1) + P
\label{eq:3xiA}
\end{equation}
where it has been assumed that J$(r) \approx 1$ is a constant. While it is
convenient to consider $\xi_T$ as a parameter, $P$ can be obtained self-consistently
by mass conservation (equ. \ref{eq:mdot}),
\begin{equation}
P  = {\tau r^{1/2} \over \dot m} \eta v_s (\theta r) = {1 \over 1 + 2 P} 
\big ({ \eta \over \alpha }\big ) \big ({v_s (\theta r) \over \theta r}\big ).
\label{eq:3mdot}
\end{equation}

Hence, the local disk solution is obtained by solving the energy equ.
(\ref{eq:Qtot}) with equs. (\ref{eq:3torq}), (\ref{eq:3xiA}) and (\ref{eq:3mdot}). 
Figure 1 shows the variation of the normalized accretion rate, $\dot m$, with 
optical depth,  $\tau$, for $r = 40$. Comparison with Figure 1 of Zdziarski (1998) 
shows that the topology of local solutions does not change with the inclusion 
of a wind. For accretion rates less than a critical value ($ \dot m < \dot 
{m}_{crit}$), there exists a radiatively cooled solution branch and an 
advection/wind dominated solution branch. No solutions exist for $ \dot m > 
\dot {m}_{crit}$. Note, that for $\Gamma = 5/3$, comparisons cannot be made 
with respect to solutions without winds since steady state solutions do not exist. 

For the advection/wind dominated solution branch, the Coulomb exchange rate can be
neglected ($Q_C \approx 0$) and equ. (\ref{eq:Qtot}) can be written for $\xi_T = 1$ 
as
\begin{eqnarray}
{3 \over 2} & = & P (1 + 2 P) \theta r + {e_W\over 2} ({\eta \over \alpha}) ({v_s (\theta r) \over \theta r} ) \nonumber \\
& = & ({\eta \over \alpha}) v_s (\theta r) \big ( 1 + {e_W \over 2 \theta r}\big )
\label{eq:3theta}
\end{eqnarray}
The above with equs. (\ref{eq:3torq}) and (\ref{eq:3mdot}) lead to self-similar
solutions for an advection/wind dominated flow: $\theta \propto 1/r$, $\tau 
\propto r^{P-1/2}$ and $\dot m \propto r^P$. Thus, with the inclusion of a wind, 
self-similar flows are maintained which is a direct consequence of the form of 
the wind velocity chosen in equ. (\ref{eq:v0}).  For example, if the parameter 
$\eta$ is a function of radius rather than a constant as assumed here, self-similar 
solutions will no longer be allowed. The solution to equ (\ref{eq:3theta}) is
shown in Figure 2, where $\theta r$ is plotted as a function of $\eta/\alpha$ 
for different values of $\Gamma$ and $E_W$. For values of $\eta/\alpha > 2$, 
$\theta r$ depends only logarithmically with $\eta/\alpha$ and is $ \approx 0.1$. 
The ratio of the wind to advective cooling $Q_W/Q_A = e_W/(2 \theta r) \approx 5$ 
making both cooling channels equally important for this solution branch. It 
is illustrative to solve for $P$ using equs. (\ref{eq:3mdot}) and (\ref{eq:3theta})
yielding 
\begin{equation}
\label{eq:3P}
 P = \Big (-1 + \sqrt{1 + 24/(e_W + 2 \theta r)} \Big )/4
\end{equation}
which implies that $P \approx 0.9$ for $e_W \approx 1$ and $\theta r \approx 0.1$.
This is shown in Figure 3 where $P$ is plotted as a function of $\eta/\alpha$.
Note that from equ. (\ref{eq:3P}) a smaller value of $e_W = 0.5$ corresponds to 
$P \approx 1.3$ and a larger value $e_W = 1.5$ corresponds to $P \approx 0.7$. Thus
P remains of order unity as long as $e_W \approx 1$ which is expected 
unless the wind is predominately heated non-locally or if the energy transferred by
the disk to the wind greatly exceeds the energy required to generate the wind.

At the critical accretion rate, $\dot {m}_{crit}$, radiative cooling becomes
comparable to the local dissipation ($Q_C \approx Q_D$). Using the self-similar
solution obtained above, $\dot {m}_{crit} \propto r^{3/10}$ which is the
same dependence obtained for hot solutions without winds (Zdziarski 1998). The
value of $\dot {m}_{crit}$ is however, generally lower than for solutions without
winds. 

In summary, for a wide range of values for the viscosity parameter ($\alpha$) and the
wind parameter ($\eta$), advection/wind dominated solutions are characterized
by $\theta \approx 0.1/r$ and $P \approx 0.9$, independent of the accretion
rate or radius.  We note that the result that the ion temperatures of wind dominated 
solutions are about an order of magnitude smaller than the no-wind ADAF solutions 
justifies the assumption of a keplerian disk.

\subsection{Global Solutions} \label{sec: 3b} 
 
It is prudent to confirm the inferences obtained from local analysis by constructing
global solutions of the disk. These global solutions involve solving the three
differential equations for conservation of mass, angular momentum and energy 
(equs. \ref{eq:mdot}, \ref{eq:torq} and \ref{eq:Qtot}) with three independent 
boundary conditions. For convenience, the boundary conditions adopted in this work 
are that the viscous torque vanishes at the inner radius ($r_i = 6$), and 
the accretion rate, $\dot {m} (r_o)$, and the temperature, $\theta (r_o)$, are 
specified at the outer radius, $r_o$. For a given $\dot m (r_o)$, $\theta (r_o)$ 
is computed from the local solution described in \S \ref{sec: 3a}.

Figure 4 illustrates the variation of the mass accretion rate with respect to 
radius for different values of $\dot m (r_o)$. Here, $\theta (r_o)$ is chosen 
from the advection/wind dominated  branch of the local solutions and hence, the 
global solution presented is also advection/wind dominated. The global solutions 
are, indeed, similar to the type inferred from local analysis with $P \approx 0.9$ 
except when $r \simgreat 6$ where as expected the flow is no longer self-similar. 
The dotted line is the critical accretion rate (beyond which hot disk solutions 
no longer exist) computed using local analysis but with values of $P$ and $\xi_T$ 
taken from the global solution.

Figure 5 presents the global solutions for the case when $\theta (r_o)$ is chosen 
from the radiatively cooled  branch of the local solutions. For $\dot m (r_o) << 
\dot {m}_{crit} (r_o)$, the accretion rate is a constant since winds are not 
important (i.e., $\theta << 1/r$). However, for $\dot m (r_o) \simless \dot {m}_{crit} 
(r_o)$, winds do become important for certain regions of the disk. In such cases
the wind regulates the flow such that $\dot m (r) \simless \dot {m}_{crit} (r)$.
These wind regulated solutions occur for a limited range of $\dot m (r_o)$ which
is due to the weak dependence of $\dot {m}_{crit}$ with radius ($\dot {m}_{crit} 
\propto r^{3/10}$).  In contrast, the accretion rate at the innermost regions of 
the disk can be significantly reduced from $\dot m(r_o)$ for the advection dominated 
branch.  For radiative cooling mechanisms other than soft photon 
Comptonization, $\dot {m}_{crit}$ may depend more steeply on radius, leading to a 
wider range of situations where a wind-regulated accretion flow would exist.


\section{STABILITY OF DISK SOLUTIONS WITH WINDS} \label{sec: 4} 

Radiatively cooled hot disks (without winds) are  known to be thermally
unstable. This result remains valid for the radiatively cooled branch described  
in \S \ref{sec: 3a}, simply because winds are not important for these solutions.
Advection dominated hot disks (without winds) are known to be thermally
stable and again this result is not altered for the advection/wind dominated
solution since the energy loss rate due to the wind increases with temperature.


Formally the criterion for thermal stability is that the derivative of the 
energy dissipation rate with respect to temperature should be smaller than the
derivative cooling terms with respect to temperature, at a constant surface
density. This leads to (Appendix: equ. \ref{eq:delQT})
\begin{equation}
\label{eq:thermstab}
S_T  \equiv  {Q_C \over 2} + 2 Q_A + \xi_{v \theta } Q_W - Q_D > 0
\end{equation}
as the criterion for thermal stability provided that the advection timescale is 
longer than the thermal timescale. Here, $\xi_{v \theta} \equiv d \hbox {log} v_s 
/ d \hbox {log} \theta>  1$. For the
wind/advection dominated solutions, $Q_D = Q_A + Q_W$ and
$Q_C << Q_D$, hence the disk will be locally stable to thermal perturbations.
For the radiatively cooled branch, 
at high accretion rates, advection and wind cooling may be sufficiently important
to stabilize the disk. 

On timescales longer than the viscous timescale, the surface density (or
optical depth) will respond to fluctuations in the disk and its temporal
behavior is governed by,
\begin{equation}
{\partial  \tau \over \partial t} = - {1 \over R}{\partial \over \partial R} ( R \tau V_r) - {\tau V_W \over H}
\label{eq:dtaudt}
\end{equation}
where $V_r$ is the radial velocity. From the torque equation (see equ. \ref{eq:torq}), it can be
seen that fluctuations in $V_r$ are linearly proportional to variations
in temperature in the long wavelength limit. On the other hand $V_W/H 
(\propto  v_w)$ is a rapidly increasing function of temperature. Thus, in 
thermal equilibrium,  if $\tau$ is inversely proportional to temperature, 
the disk will be unstable to long wavelength perturbations. In other words, 
if a decrease in $\tau$ leads to an increase in the temperature,
the mass loss rate by the wind will increase rapidly which will cause a further
decrease in $\tau$ leading to a runaway situation. 

Technically, for the local wind dominated solution described in \S \ref{sec: 3a} 
without radiative cooling ($Q_C = 0$), the temperature of the disk is independent 
of $\tau$ (equ \ref{eq:3theta}). However, even if there is only a small contribution 
due to radiative cooling (i.e. $Q_C << Q_D$), an increase in $\tau$ will cause 
$Q_C$ to increase which leads to a decrease in the temperature. Thus, wind 
dominated solutions are subject to the instability described above in the long 
wavelength limit. A linear stability analysis leads to the dispersion relation 
for $\lambda >> R$ (Appendix: equ. 
\ref{eq:iomega0})
\begin{equation}
i \omega  = {5\over 2} ({V_r \over R}) ({Q_C \over S_T}) P (\xi_{v \theta} -1).
\label{eq:iomega}
\end{equation}
For wind dominated solutions, $\xi_{v \theta} > 1$ and the disk is thermally stable ($S_T > 0$), hence
$i \omega$ is real and positive indicating that the mode is unstable. For
a radiatively cooled disk, $S_T < 0$ and this mode is stable for such solutions. 
The growth time of the wind instability ($\approx 1/\omega$) can be significantly 
longer than the viscous (or accretion) timescale ($\approx R/V_r$) if ${Q_C/S_T} 
\approx Q_C/Q_D << 1$. This implies that the disk structure can
 be described in terms of local steady state
solutions during the linear growth of the wind driven instability.
Equ. (\ref{eq:iomega}) and the inferred arguments depend
on the form of the radiative cooling mechanism (equ. \ref{eq:thetae}) assumed in
the analysis. In the general case, $Q_C \propto \tau^{\beta}$, and it can be shown that
the disk will suffer this instability provided $\beta > 1$ and if $S_T > 0$. For the
unsaturated Comptonization model (i.e. equ. \ref{eq:thetae}) $\beta = 7/2$ and
hence the disk is found to be unstable. Note however, that $\beta < 1$ corresponds
to the situation where the electron temperature is proportional to the optical
depth and varies faster than $\theta_e \propto \tau ^{2/3}$. Since for most 
physical radiative processes, $\theta_e$ is expected to be inversely proportional
to $\tau$, hot disks with winds would in general be unstable. 
The above results are based on local and linear stability 
analysis and require confirmation by global simulations.  


\section{SUMMARY AND DISCUSSIONS} \label{sec: 5} 
  
The effect of a hydrodynamical wind on a hot accretion disk has been studied for  
disks characterized by an adiabatic index, $\Gamma_D = 5/3$.  It is 
found that the topology of local ADAF solutions found earlier (for $\Gamma_D < 5/3$) 
remains the same for $\Gamma_D = 5/3$ with the inclusion of a wind. In particular, 
there exists two branches of solutions 
for accretion rates below a critical value, while there are no solutions for higher 
accretion rates. One solution corresponds to the radiative cooling branch where the
effects of the wind are negligible whereas both winds and advection are 
important for the second branch. 
 
We have shown that the local solutions for the wind/advection dominated branch may 
be described in a self-similar form with the temperature, $\theta \propto 1/r$, 
optical depth $\tau \propto r^{P-1/2}$ and accretion rate $\dot m \propto r^P$. For a 
wide range of viscosity and wind parameters, the temperature is about 10\% of the 
virial value ($\theta r \approx 0.1$) and $P \approx 0.9$, independent of the 
accretion rate and radius. These results were confirmed by global analysis for 
which self-similar solutions were obtained for radii much greater than the inner 
disk radius, thereby providing a theoretical basis for the model advanced by Blandford 
\& Begelman (1999).  Global solutions also revealed that under certain boundary 
conditions (see \S 3.2), the accretion rate $\dot m \approx \dot {m}_{crit} \propto r^{3/10}$.
For these ``critical'' solutions radiative cooling and energy loss due to the wind, 
were both important and hence they are not self-similar. 

The wind/advection dominated solutions are thermally stable, but are locally 
unstable to a new type of instability which we denote as ``wind driven 
instability.'' The presence of a wind causes the disk to be unstable to long 
wavelength perturbations in the surface density.  Since the growth rate of this 
instability is inversely proportional to the ratio of the radiative cooling to 
the gravitational energy dissipation rates, the timescale for the instability could 
be much larger than the viscous time scale in the disk for a low radiative 
cooling efficiency.
 
The accretion rate index, P, $\approx 1$ implies that, if the outer radius
of the hot disk is much larger than the inner one, most of the accreting matter
is lost in the form of a wind rather than accreted onto the star. If the compact 
object is a neutron star or white dwarf, it is this residual accretion which 
gives rise to the X-ray luminosity from the surface of the star. Loeb, Narayan \& 
Raymond (2000) point out that in such a scenario (with nearly similar outer
boundary conditions), the surface emission from the star will scale roughly as 
$\propto R_*^{P-1}$, where $R_*$ is the radius of the star. From the observed 
luminosities of soft X-ray transients and cataclysmic variables, they estimate
$P = 0.9 \pm 0.5$, which is in agreement with the theoretical results presented here.

For this radial dependence of the mass flow rate in the disk, the total energy 
dissipated in the disk scales as log $(R_o/R_i)$, where $R_o$ and $R_i$ are the 
outer and inner radii of the hot disk, rather than $R_i^{-1}$. Thus both the 
inner and outer regions of hot disk are equally important energetically. It is 
generally assumed that the hot disk is surrounded and connected to an outer cold 
disk by a narrow transition region. The energy required to make this transition 
cannot be generated locally if the transition zone is small (Misra \& Melia 1996) 
and is probably due to global transfer from the inner hot regions to the transition 
zone (Honma 1992, see also Abramowicz et al. 2000). For the wind/advection dominated 
case this may not be possible since the energy generated in the inner regions is 
only comparable to that of the outer region. In this case, the transition zone would 
then have to be extended and could significantly contribute to the luminosity of the 
source (Misra \& Melia 1996).  However, the above speculation has to be verified by 
detailed multi-dimensional simulations of the transition zone taking into
account the possibility that the hot disk may be wind dominated.

Since the radiative cooling rate (equ. \ref{eq:Qcoul}) for the self-similar solution
scales as $\propto R^{(14 P - 15)/4} \approx  R^{-1/2}$ for $P \approx 0.9$, 
most of the luminosity arises from the outer regions of the hot disk. This result,
however, depends on the radiative cooling mechanism operating in the hot disk. For 
example, if the electron temperature is chosen to be $\theta_e \propto R^{\alpha_e}$ 
instead of equ. \ref{eq:thetae}, then the radiative flux is $\propto R^{2 P - 3 - 
3 \alpha_e/2} \approx R^{- (1 + 3 \alpha_e/2)}$. Hence if $\alpha_e > 2/3$ then 
the luminosity will be dominated by the inner region. 

The terminal velocity of the wind should
be approximately equal to the sound speed at the surface of the disk. This
implies that the bulk energy per proton of the wind flow at infinity is $\approx 
0.1 GM/R$ since the temperature of the disk $\theta \approx 0.1 r$ For $P 
\approx 1$ most of the wind arises from the outermost radius and hence the total 
bulk energy flux in the wind $\approx 0.1 \dot M_{wind} GM /R_o$,
where $\dot M_{wind}$ is the total mass loss rate and $R_o$ is the outer 
radius of the hot disk. Since the radiative efficiency of such disks is $<< 0.1$, the
bulk energy of the wind flow in some circumstances may be much larger than the
radiative flux from the source. 

The viscous timescale for the self-similar solution is
\begin{equation}
t_v = {R\over V_R} \approx {R^{1/2}\over (1 + 2P) (\theta r) \alpha} ({r_g\over c}) 
\end{equation}
which numerically is $t_v \approx 250 (r_g/c) \approx 0.01$ s for $\alpha = 0.1$,
$R = 50 r_g$ and a 10 \msun black hole. Thus, a wind driven instability may 
occur on timescales $\sim 1 - 10$ s for $Q_C/Q_D \approx 0.1 - 0.01$. The instability 
may not lead to disruption of the disk, but instead give rise to large amplitude 
oscillations. For example,
the surface density could increase, accompanied by a decreasing temperature. When 
the temperature decreases sufficiently, the wind is quenched and radiative 
cooling becomes important. Since the radiatively cooled disk will be thermally
unstable, the system may return to the wind dominated state on thermal timescales, 
giving rise to cyclic behavior. Alternatively, the surface density may decrease and 
the disk may then be evacuated and finally filled again on the viscous timescale. 
In either case, the disk may evolve through epochs of high wind activities on timescale 
$\approx$ seconds. Such a disk/wind interaction may be important for explaining the 
spectral and temporal behavior of the regular bursting activity exhibited by the 
microquasar GRS 1915-105 (Taam et al 1997).  

Future work involving time dependent numerical simulations of hot disk with winds are 
needed to confirm the above speculations and to quantify the rate of mass loss and 
varying spectral characteristics during such events. Such studies may also shed light
on the relevance of wind instabilities as a mechanism for either oscillatory behavior 
or evaporative behavior in the hot inner disk regions. Besides these issues, further 
theoretical investigations of the transition from a cold to a hot wind dominated 
disk and the role of two dimensional effects along the lines suggested by Turolla \& 
Dullemond (2000) would also be illuminating. 

\acknowledgements

R.M. gratefully acknowledges support from  the Lindheimer Fellowship at 
Northwestern University. 

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\newpage

\begin{appendix}

\section{Linear stability analysis in the long wave-length approximation}

The linear response of the disk structure is calculated for perturbation
of the ion temperature $\theta \rightarrow \theta ( 1 + \delta \theta
e^{i(\omega t - kr)})$ and optical depth $\tau \rightarrow 
\tau ( 1 + \delta \tau e^{i(\omega t - kr)})$, where $\delta \theta$ and 
$\delta \tau$ are small dimensionless quantities. In the long
wavelength approximation, it is  assumed
that $kr << 1$ such that $e^{ikr} \approx 1$ which implies that the
radial derivatives of perturbed quantities can be neglected.

In this formalism, the linear response of physical
quantities (e.g., the energy flux) can be obtained directly from the
dependence of the quantity on temperature and optical depth. The
dissipative energy rate $Q_D \propto \theta \tau$ (equ. \ref{eq:Qdis})
and hence $\delta Q_D = \delta \theta + \delta \tau$. The Coulomb 
energy exchange flux $Q_C \propto \tau^{7/2} \theta^{1/2}$ (equs.
\ref{eq:Qcoul}, \ref{eq:nuc}, \ref{eq:thetae}) while the energy
flux carried away by the wind $Q_W \propto \tau v_w(\theta)$ (equ.
\ref{eq:Qwin} ). The advective energy flux (equ. \ref{eq:Qadv}),
$Q_A \propto \dot m \theta \propto \tau \theta^2$ since
$\dot m \propto \tau \theta$ (equ.
\ref{eq:torq}) for
a self-similar steady state solution with $kr << 1$. 
Note that in the long wavelength approximation
the advective factor ($\xi_A$) is invariant to perturbations. Thus
the normalized response of the total energy flux $Q_T \equiv Q_D - Q_C - Q_W - Q_A$ is,
\begin{eqnarray} 
Q_T \delta Q_T  & = & (Q_D -{1\over 2} Q_C - \xi_{v \theta} Q_W - 2 Q_A) 
\delta \theta + (Q_D - {7\over 2} Q_C - Q_W - Q_A) \delta \tau \nonumber \\ 
 &  = & -S_T \delta \theta -{5\over 2} Q_C \delta \tau 
\label{eq:delQT}
\end{eqnarray}
where $\xi_{v \theta} \equiv d \hbox {log} v_s / d \hbox {log} \theta$. On
thermal timescales $\delta \tau << \delta \theta$ 
and $- Q_T \delta Q_T / \delta \theta = S_T > 0$ is 
the criterion for thermal stability (equ. \ref{eq:thermstab}).

The time dependent form of the mass conservation equation (\ref{eq:mdot})
governs the temporal behavior of the surface density or optical depth (equ. \ref{eq:dtaudt})
\begin{equation}
({r_g \over c}) {\partial \tau \over \partial t } = {1\over r} {\partial \dot m \over \partial r} - \tau v_w r^{-3/2}
\label{eq:dtaudt1}
\end{equation}
Noting that $\dot m \propto \tau \theta$,  the linearized form of the above
equation is
\begin{equation}
i \omega \tau \delta \tau = ({c \over r_g}) ({1\over r}{\partial \dot m \over \partial r}) \delta \theta (1 - \xi_{v \theta})
\label{eq:iomega1}
\end{equation}
For timescales much longer than the thermal timescale, $\delta Q_T \approx 0$ i.e.
the system will be in thermal equilibrium. Combining equations (\ref{eq:delQT}) and
(\ref{eq:iomega1}), yields the dispersion relation in the long wavelength limit
(equ. \ref{eq:iomega})
\begin{equation}
i \omega  = {5\over 2} ({V_r \over R}) ({Q_C \over S_T}) P (\xi_{v \theta} -1 )
\label{eq:iomega0}
\end{equation}
where $V_r$ is the magnitude of the radial velocity at radius $R$.




  



\end{appendix}
\newpage
\input psbox.tex
\begin{figure*}[h]
\hspace{-1.5cm}
{\mbox{\psboxto(17cm;20cm){Fig1.ps}}}
\caption{Variation of the normalized accretion rate ($\dot m$) with optical depth ($\tau$)
for $r = 40$. Solid line: $\alpha = 0.5$; Dashed line: $\alpha = 1.0$. Other parameters are
$\eta = 1, e_W = 1, y = 1,\Gamma = 1.2$ and $\xi_T = 1$.  }
\end{figure*}

\begin{figure*}[h]
\hspace{-1.5cm}
{\mbox{\psboxto(17cm;20cm){Fig2.ps}}}
\caption{Variation of $\theta r$  with $\eta/\alpha$
for  $\Gamma = 1.01,\; e_W = 1$ (solid line), $\Gamma = 1.01, \; e_W = 0.5$ (short dashed line) and
$\Gamma = 1.2,\; e_W = 1$ (long dashed line).}
\end{figure*}

\begin{figure*}[h]
\hspace{-1.5cm}
{\mbox{\psboxto(17cm;20cm){Fig3.ps}}}
\caption{Variation of the accretion slope $P$  with $\eta/\alpha$
for  $\Gamma = 1.01,\; e_W = 1$ (solid line), $\Gamma = 1.01, \; e_W = 0.5$ (short dashed line) and
$\Gamma = 1.2,\; e_W = 1$ (long dashed line).}
\end{figure*}

\begin{figure*}[h]
\hspace{-1.5cm}
{\mbox{\psboxto(17cm;20cm){Fig4.ps}}}
\caption{Accretion rate vs. radii for different values of accretion rate at the outer boundary
($ r = 400$). The temperature at the outer boundary is chosen from the advection/wind dominated branch of
the local solutions. Dashed curve represents the local critical accretion rate. Parameters used for
the plot are $\alpha = 0.1$, $y = 1$, $\eta = 1$, $e_W = 1$ and  $\Gamma = 1.2$  }
\end{figure*}


\begin{figure*}[h]
\hspace{-1.5cm}
{\mbox{\psboxto(17cm;20cm){Fig5.ps}}}
\caption{Accretion rate vs. radii for different values of accretion rate at the outer boundary
($ r = 400$). The temperature at the outer boundary is chosen from the radiatively cooled branch of
the local solutions. Dashed curve represents the local critical accretion rate. Parameters used for
the plot are $\alpha = 0.1$, $y = 1$, $\eta = 1$, $e_W = 1$ and  $\Gamma = 1.2$  }
\end{figure*}

\end{document}



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%       %         %%%%%%        %%%%%%    %%%%%   %     %
%
%       By Jean Orloff
%
% WHAT IS IT:
% psbox is a set of machine-independent TeX macros to
% 1) allow (Encapsulated) PostScript figure inclusion in all versions
%    of TeX (Plain, LaTeX) on all machines using a PostScript printer
% 2) facilitate the communication (e-mail, ftp, ...) of all the files
%    (text, macros, figs) needed to reproduce a TeX document by grouping
%    them together into a single, TeXable file.
%
% History:
%  1.34  \readfilename=final fix for all filename scans; try \psforptips
%  1.33: corrects \psnewinput for LaTeX (still fails if fname=a{b}c)
%  1.32: corrects \psfordvialw and adds .TEX to PSBOXALL(!)
%  1.31: adds \psfordvialw(?)
%  1.30: adds \splitfile & \joinfiles for multi-file management
%  1.24: fix error handling & add \psonlyboxes
%  1.22: makes \drawingBox \global for use in Phyzzx
%  1.21: accepts %%BoundingBox: (atend)
%  1.20: tries to add \psfordvitps for the TeXPS package.
%  1.10: adds \psforoztex, error handling...
%2345678 1 2345678 2 2345678 3 2345678 4 2345678 5 2345678 6 2345678 7 23456789
%
% Checking version no to avoid multiple loadings
\def\temp{1.34}%
\let\tempp=\relax
\expandafter\ifx\csname psboxversion\endcsname\relax
  \message{PSBOX(\temp) loading}%
\else
    \ifdim\temp cm>\psboxversion cm
      \message{PSBOX(\temp) loading}%
    \else
      \message{PSBOX(\psboxversion) is already loaded: I won't load
        PSBOX(\temp)!}%
      \let\temp=\psboxversion
      \let\tempp=\endinput
    \fi
\fi
\tempp
\let\psboxversion=\temp
\catcode`\@=11
% Every macro likes a little privacy...
%
%Trying to tame the variety of \special commands for Postscript: the
%  universal internal command \PSspeci@l##1##2 takes ##1 to be the
%  filename and ##2 to be the integer scale factor*1000 (as for usual
%   TeX \scale commands)
%
\def\psfortextures{%     For TeXtures on the Macintosh
%-----------------
\def\PSspeci@l##1##2{%
\special{illustration ##1\space scaled ##2}%
}}%
\def\psfordvitops{%      For the DVItoPS converter on IBM mainframes
%----------------
\def\PSspeci@l##1##2{%
\special{dvitops: import ##1\space \the\drawingwd \the\drawinght}%
}}%
\def\psfordvips{%      For DVIPS converter on VAX, UNIX and PC's
%--------------
\def\PSspeci@l##1##2{%
%    \special{/@scaleunit 1000 def}% never read dox without trying!
\d@my=0.1bp \d@mx=\drawingwd \divide\d@mx by\d@my% BUG! for large \drawingwd
\special{PSfile=##1\space llx=\psllx\space lly=\pslly\space%
urx=\psurx\space ury=\psury\space rwi=\number\d@mx
}}}%
\def\psforoztex{%        For the OzTeX shareware on the Macintosh
%--------------
\def\PSspeci@l##1##2{%
\special{##1 \space
      ##2 1000 div dup scale
      \number-\psllx\space \number-\pslly\space translate
}}}%
\def\psfordvitps{%       From the UNIX TeXPS package, vers.>3.12
%---------------
% Convert a dimension into the number \psn@sp (in scaled points)
\def\psdimt@n@sp##1{\d@mx=##1\relax\edef\psn@sp{\number\d@mx}}
\def\PSspeci@l##1##2{%
% psfig.psr contains the def of "startTexFig": if you can locate it
% and include the correct pathname, it should work
\special{dvitps: Include0 "psfig.psr"}% contains def of "startTexFig"
\psdimt@n@sp{\drawingwd}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\drawinght}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psllx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\pslly bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psurx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psury bp}
\special{dvitps: Literal "\psn@sp\space startTexFig\space"}
\special{dvitps: Include1 "##1"}
\special{dvitps: Literal "endTexFig\space"}
}}%
\def\psfordvialw{%   Try for dvialw, a UNIX public domain
%---------------
\def\PSspeci@l##1##2{
\special{language "PostScript",
position = "bottom left",
literal "  \psllx\space \pslly\space translate
  ##2 1000 div dup scale
  -\psllx\space -\pslly\space translate",
include "##1"}
}}%
\def\psforptips{%   For MS-DOS; LUOMA@brandeis.bitnet
%---------------
\def\PSspeci@l##1##2{{
\d@mx=\psurx bp
\advance \d@mx by -\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\incm{\d@mx}
\let\tmpx\dimincm
\d@my=\psury bp
\advance \d@my by -\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\incm{\d@my}
\let\tmpy\dimincm
\d@mx=-\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\d@my=-\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\at(\d@mx;\d@my){\special{ps:##1 x=\tmpx, y=\tmpy}}
}}}%
\def\psonlyboxes{%     Draft-like behaviour if none of the others works
%---------------
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1})}}\hss}}}}
}}%
\def\psloc@lerr#1{%
\let\savedPSspeci@l=\PSspeci@l%
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1}) #1}}\hss}}}}
\let\PSspeci@l=\savedPSspeci@l% restore normal output for other figs!
}}%
%\def\psfor...  add your own!
%
% Some common defs
%
\newread\pst@mpin
\newdimen\drawinght\newdimen\drawingwd
\newdimen\psxoffset\newdimen\psyoffset
\newbox\drawingBox
\newcount\xscale \newcount\yscale \newdimen\pscm\pscm=1cm
\newdimen\d@mx \newdimen\d@my
\newdimen\pswdincr \newdimen\pshtincr
\let\ps@nnotation=\relax
{\catcode`\|=0 |catcode`|\=12 |catcode`|%=12 |catcode`~=12
|catcode`#=12 |catcode`*=14
|xdef|backslashother{\}*
|xdef|percentother{%}*
|xdef|tildeother{~}*
|xdef|sharpother{#}*
}%
% useful to display special chars in \tt; fails for \,#,%
\def\R@moveMeaningHeader#1:->{}%
\def\uncatcode#1{%
\edef#1{\expandafter\R@moveMeaningHeader\meaning#1}}%
%
\def\execute#1{#1}% NOT stupid: cs in #1 are then identified BEFORE execution
\def\psm@keother#1{\catcode`#112\relax}% borrowed from latex
\def\executeinspecs#1{%
\execute{\begingroup\let\do\psm@keother\dospecials\catcode`\^^M=9#1\endgroup}}%
\def\@mpty{}%
% \if\matchin#1#2<=> \iftrue if #1 contains #2, <=>\iffalse otherwise:
% \if\matchexpin: idem, but #1 & #2 are first fully expanded (no \if
% inside!)
% \tmpa & \tmpb contain what's before and after the occurence of #2
\def\matchexpin#1#2{
  \fi%
%\message{(#1>#2)}
  \edef\tmpb{{#2}}%
  \expandafter\makem@tchtmp\tmpb%
  \edef\tmpa{#1}\edef\tmpb{#2}%
  \expandafter\expandafter\expandafter\m@tchtmp\expandafter\tmpa\tmpb\endm@tch%
  \if\match%
}%
\def\matchin#1#2{%
  \fi%
  \makem@tchtmp{#2}%
  \m@tchtmp#1#2\endm@tch%
  \if\match%
}%
\def\makem@tchtmp#1{\def\m@tchtmp##1#1##2\endm@tch{%
  \def\tmpa{##1}\def\tmpb{##2}\let\m@tchtmp=\relax%
  \ifx\tmpb\@mpty\def\match{YN}%
  \else\def\match{YY}\fi%
}}%
% converts any dimen in cm, with 1E-4 cm precision
\def\incm#1{{\psxoffset=1cm\d@my=#1
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\number\d@mx.}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \edef\dimincm{\dimincm\number\d@mx}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\dimincm\number\d@mx}
}}%
%
%  \ReadPSize{PSfilename} reads the dimensions of a PostScript drawing
%      and stores it in \drawinght(wd)
\newif\ifNotB@undingBox
\newhelp\PShelp{Proceed: you'll have a 5cm square blank box instead of
your graphics (Jean Orloff).}%
\def\s@tsize#1 #2 #3 #4\@ndsize{
  \def\psllx{#1}\def\pslly{#2}%
  \def\psurx{#3}\def\psury{#4}%  needed by a crazyness of dvips!
  \ifx\psurx\@mpty\NotB@undingBoxtrue% this is not a valid one!
  \else
    \drawinght=#4bp\advance\drawinght by-#2bp
    \drawingwd=#3bp\advance\drawingwd by-#1bp
%  !Units related by crazy factors as bp/pt=72.27/72 should be BANNED!
  \fi
  }%
\def\sc@nBBline#1:#2\@ndBBline{\edef\p@rameter{#1}\edef\v@lue{#2}}%
\def\g@bblefirstblank#1#2:{\ifx#1 \else#1\fi#2}%
{\catcode`\%=12
\xdef\B@undingBox{%%BoundingBox}}%
%% is not a true comment in PostScript, even if % is!
\def\ReadPSize#1{
 \readfilename#1\relax
 \let\PSfilename=\lastreadfilename
 \openin\pst@mpin=#1\relax
 \ifeof\pst@mpin \errhelp=\PShelp
   \errmessage{I haven't found your postscript file (\PSfilename)}%
   \psloc@lerr{was not found}%
   \s@tsize 0 0 142 142\@ndsize
   \closein\pst@mpin
 \else
% each entry in \GlobalInputList should be unique
   \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
   \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
     \immediate\write\psbj@inaux{\lastreadfilename,}%
   \fi%
   \loop
     \executeinspecs{\catcode`\ =10\global\read\pst@mpin to\n@xtline}%
     \ifeof\pst@mpin
       \errhelp=\PShelp
       \errmessage{(\PSfilename) is not an Encapsulated PostScript File:
           I could not find any \B@undingBox: line.}%
       \edef\v@lue{0 0 142 142:}%
       \psloc@lerr{is not an EPSFile}%
       \NotB@undingBoxfalse
     \else
       \expandafter\sc@nBBline\n@xtline:\@ndBBline
       \ifx\p@rameter\B@undingBox\NotB@undingBoxfalse
         \edef\t@mp{%
           \expandafter\g@bblefirstblank\v@lue\space\space\space}%
         \expandafter\s@tsize\t@mp\@ndsize
       \else\NotB@undingBoxtrue
       \fi
     \fi
   \ifNotB@undingBox\repeat
   \closein\pst@mpin
 \fi
\message{#1}%
}%
%
% \psboxto(xdim;ydim){psfilename}: you specify the dimensions and
%    TeX uniformly scales to fit the largest one. If xdim=0pt, the
%    scale is fully determined by ydim and vice versa.
%    Notice: psboxes are a real vboxes; couldn't take hbox otherwise all
%    indentation and all cr's would be interpreted as spaces (hugh!).
%
\def\psboxto(#1;#2)#3{\vbox{
   \ReadPSize{#3}%
   \divide\drawingwd by 1000
   \divide\drawinght by 1000
   \d@mx=#1
   \ifdim\d@mx=0pt\xscale=1000
         \else \xscale=\d@mx \divide \xscale by \drawingwd\fi
   \d@my=#2
   \ifdim\d@my=0pt\yscale=1000
         \else \yscale=\d@my \divide \yscale by \drawinght\fi
   \ifnum\yscale=1000
         \else\ifnum\xscale=1000\xscale=\yscale
                    \else\ifnum\yscale<\xscale\xscale=\yscale\fi
              \fi
   \fi
   \divide\pswdincr by 1000 \multiply\pswdincr by \xscale
   \divide\pshtincr by 1000 \multiply\pshtincr by \xscale
   \divide\psxoffset by1000 \multiply\psxoffset by\xscale
   \divide\psyoffset by1000 \multiply\psyoffset by\xscale
   \global\divide\pscm by 1000
   \global\multiply\pscm by\xscale
   \multiply\drawingwd by\xscale \multiply\drawinght by\xscale
   \ifdim\d@mx=0pt\d@mx=\drawingwd\fi
   \ifdim\d@my=0pt\d@my=\drawinght\fi
   \message{scaled \the\xscale}%
 \hbox to\d@mx{\hss\vbox to\d@my{\vss
   \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{
      \kern-\psyoffset
      \PSspeci@l{\PSfilename}{\the\xscale}%
      \vss}\hss\ps@nnotation}%
   \advance\pswdincr by \drawingwd
   \advance\pshtincr by \drawinght
   \global\wd\drawingBox=\the\pswdincr
   \global\ht\drawingBox=\the\pshtincr
   \baselineskip=0pt
   \copy\drawingBox
 \vss}\hss}%
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm %should not be necessary
  \global\drawingwd=\drawingwd
  \global\drawinght=\drawinght
}}%
%
% \psboxscaled{scalefactor*1000}{PSfilename} allows to bypass the
%   rounding errors of TeX integer divisions for situations where the
%   TeX box should fit the original BoundingBox with a precision better
%   than 1/1000.
%
\def\psboxscaled#1#2{\vbox{
  \ReadPSize{#2}%
  \xscale=#1
  \message{scaled \the\xscale}%
  \advance\drawingwd by\pswdincr\advance\drawinght by\pshtincr
  \divide\pswdincr by 1000 \multiply\pswdincr by \xscale
  \divide\pshtincr by 1000 \multiply\pshtincr by \xscale
  \divide\psxoffset by1000 \multiply\psxoffset by\xscale
  \divide\psyoffset by1000 \multiply\psyoffset by\xscale
  \divide\drawingwd by1000 \multiply\drawingwd by\xscale
  \divide\drawinght by1000 \multiply\drawinght by\xscale
  \global\divide\pscm by 1000
  \global\multiply\pscm by\xscale
  \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{
     \kern-\psyoffset
     \PSspeci@l{\PSfilename}{\the\xscale}%
     \vss}\hss\ps@nnotation}%
  \advance\pswdincr by \drawingwd
  \advance\pshtincr by \drawinght
  \global\wd\drawingBox=\the\pswdincr
  \global\ht\drawingBox=\the\pshtincr
  \baselineskip=0pt
  \copy\drawingBox
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm
  \global\drawingwd=\drawingwd
  \global\drawinght=\drawinght
}}%
%
%  \psbox{PSfilename} makes a TeX box having the minimal size to
%      enclose the picture
\def\psbox#1{\psboxscaled{1000}{#1}}%
%------------------------------------------------------
%  \joinfiles file1, file2, ...n \into joinedfilename .
%     makes one file out of many
%  \splitfile joinedfilename
%     the opposite
\newif\ifn@teof\n@teoftrue
\newif\ifc@ntrolline
\newif\ifmatch
\newread\j@insplitin
\newwrite\j@insplitout
\newwrite\psbj@inaux
\immediate\openout\psbj@inaux=psbjoin.aux
\immediate\write\psbj@inaux{\string\joinfiles}%
\immediate\write\psbj@inaux{\jobname,}%
%
% INPUT REDEFINITION
%
% works if #1 is a single character
\def\toother#1{\ifcat\relax#1\else\expandafter%
  \toother@ux\meaning#1\endtoother@ux\fi}%
\def\toother@ux#1 #2#3\endtoother@ux{\def\tmp{#3}%
  \ifx\tmp\@mpty\def\tmp{#2}\let\next=\relax%
  \else\def\next{\toother@ux#2#3\endtoother@ux}\fi%
\next}%
%
% \readfilename defs:
%
\let\readfilenamehook=\relax
\def\re@d{\expandafter\re@daux}% spares typing 10 \expandafter's...
\def\re@daux{\futurelet\nextchar\stopre@dtest}%
\def\re@dnext{\xdef\lastreadfilename{\lastreadfilename\nextchar}%
  \afterassignment\re@d\let\nextchar}%
\def\stopre@d{\egroup\readfilenamehook}%
\def\stopre@dtest{%
  \ifcat\nextchar\relax\let\nextread\stopre@d
  \else
    \ifcat\nextchar\space\def\nextread{%
      \afterassignment\stopre@d\chardef\nextchar=`}%
    \else\let\nextread=\re@dnext
      \toother\nextchar
      \edef\nextchar{\tmp}%
    \fi
  \fi\nextread}%
\def\readfilename{\vbox\bgroup%
  \let\\=\backslashother \let\%=\percentother \let\~=\tildeother
  \let\#=\sharpother \xdef\lastreadfilename{}%
  \re@d}%
%
% redefines \input using \readfilename
%
\xdef\GlobalInputList{\jobname}%
\def\psnewinput{%
  \def\readfilenamehook{% each entry in \GlobalInputList should be unique
    \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
    \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
      \immediate\write\psbj@inaux{\lastreadfilename,}%
    \fi%
    \ps@ldinput\lastreadfilename\relax%
    \let\readfilenamehook=\relax%
  }\readfilename%
}%
\expandafter\ifx\csname @@input\endcsname\relax    % then Plain
  \immediate\let\ps@ldinput=\input\def\input{\psnewinput}%
\else
  \immediate\let\ps@ldinput=\@@input
  \def\@@input{\psnewinput}%
\fi%
%
\def\nowarnopenout{%
 \def\warnopenout##1##2{%
   \readfilename##2\relax
   \message{\lastreadfilename}%
   \immediate\openout##1=\lastreadfilename\relax}}%
\def\warnopenout#1#2{%
 \readfilename#2\relax
 \def\t@mp{TrashMe,psbjoin.aux,psbjoint.tex,}\uncatcode\t@mp
 \if\matchexpin{\t@mp}{\lastreadfilename,}%
 \else
   \immediate\openin\pst@mpin=\lastreadfilename\relax
   \ifeof\pst@mpin
     \else
     \errhelp{If the content of this file is so precious to you, abort (ie
press x or e) and rename it before retrying.}%
     \errmessage{I'm just about to replace your file named \lastreadfilename}%
   \fi
   \immediate\closein\pst@mpin
 \fi
 \message{\lastreadfilename}%
 \immediate\openout#1=\lastreadfilename\relax}%
% % will have an unusual catcode below; use * instead
%\vbox
{\catcode`\%=12\catcode`\*=14
\gdef\splitfile#1{*
 \readfilename#1\relax
 \immediate\openin\j@insplitin=\lastreadfilename\relax
 \ifeof\j@insplitin
   \message{! I couldn't find and split \lastreadfilename!}*
 \else
   \immediate\openout\j@insplitout=TrashMe
   \message{< Splitting \lastreadfilename\space into}*
   \loop
     \ifeof\j@insplitin
       \immediate\closein\j@insplitin\n@teoffalse
     \else
       \n@teoftrue
       \executeinspecs{\global\read\j@insplitin to\spl@tinline\expandafter
         \ch@ckbeginnewfile\spl@tinline%Beginning-Of-File-Named:%\endcheck}*
       \ifc@ntrolline
       \else
         \toks0=\expandafter{\spl@tinline}*
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \fi
   \ifn@teof\repeat
   \immediate\closeout\j@insplitout
 \fi\message{>}*
}*
\gdef\ch@ckbeginnewfile#1%Beginning-Of-File-Named:#2%#3\endcheck{*
 \def\t@mp{#1}*
 \ifx\@mpty\t@mp
   \def\t@mp{#3}*
   \ifx\@mpty\t@mp
     \global\c@ntrollinefalse
   \else
     \immediate\closeout\j@insplitout
     \warnopenout\j@insplitout{#2}*
     \global\c@ntrollinetrue
   \fi
 \else
   \global\c@ntrollinefalse
 \fi}*
\gdef\joinfiles#1\into#2{*
 \message{< Joining following files into}*
 \warnopenout\j@insplitout{#2}*
 \message{:}*
 {*
 \edef\w@##1{\immediate\write\j@insplitout{##1}}*
\w@{% This collection of files was produced with CERN psbox package}*
\w@{% To decompose and tex it:}*
\w@{%-save this with a filename CONTAINING ONLY LETTERS and a .TEX}*
\w@{% extension (say, JOINTFIL.TEX), in some uncrowded directory;}*
\w@{%-make sure you can \string\input\space psbox.tex (version>=1.3);}*
\w@{%  (else ftp cs.nyu.edu(=128.122.140.24):pub/TeX/psbox/, then get}*
\w@{%  and tex the file psboxall.tex; more info in psbREAD.ME)}*
\w@{%-tex JOINTFIL.TEX using Plain, or LaTeX, or whatever is needed by}*
\w@{%  the first file in the joining (after splitting JOINTFIL.TEX into}*
\w@{%  it's constituents, TeX will try to process it as it stands).}*
\w@{\string\input\space psbox.tex}*
\w@{\string\splitfile{\string\jobname}}*
\w@{\string\let\string\autojoin=\string\relax}*
}*
 \expandafter\tre@tfilelist#1, \endtre@t
 \immediate\closeout\j@insplitout
 \message{>}*
}*
\gdef\tre@tfilelist#1, #2\endtre@t{*
 \readfilename#1\relax
 \ifx\@mpty\lastreadfilename
 \else
   \immediate\openin\j@insplitin=\lastreadfilename\relax
   \ifeof\j@insplitin
     \errmessage{I couldn't find file \lastreadfilename}*
   \else
     \message{\lastreadfilename}*
     \immediate\write\j@insplitout{%Beginning-Of-File-Named:\lastreadfilename}*
     \executeinspecs{\global\read\j@insplitin to\oldj@ininline}*
     \loop
       \ifeof\j@insplitin\immediate\closein\j@insplitin\n@teoffalse
       \else\n@teoftrue
         \executeinspecs{\global\read\j@insplitin to\j@ininline}*
         \toks0=\expandafter{\oldj@ininline}*
         \let\oldj@ininline=\j@ininline
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \ifn@teof
     \repeat
   \immediate\closein\j@insplitin
   \fi
   \tre@tfilelist#2, \endtre@t
 \fi}*
}%
% To be put at the end of a file, for making a tar-like file containing
%   everything it used.
\def\autojoin{%
 \immediate\write\psbj@inaux{\string\into{psbjoint.tex}}%
 \immediate\closeout\psbj@inaux
 \expandafter\joinfiles\GlobalInputList\into{psbjoint.tex}%
}%
%----------------------------------------------------------------
%  Annotations & Captions etc...
%
%
% \centinsert{anybox} is just a centered \midinsert, but is included as
%    people barely use the original inserts from TeX.
%
\def\centinsert#1{\midinsert\line{\hss#1\hss}\endinsert}%
\def\psannotate#1#2{\vbox{%
  \def\ps@nnotation{#2\global\let\ps@nnotation=\relax}#1}}%
\def\pscaption#1#2{\vbox{%
   \setbox\drawingBox=#1
   \copy\drawingBox
   \vskip\baselineskip
   \vbox{\hsize=\wd\drawingBox\setbox0=\hbox{#2}%
     \ifdim\wd0>\hsize
       \noindent\unhbox0\tolerance=5000
    \else\centerline{\box0}%
    \fi
}}}%
% for compatibility with older versions, but \psfig is a bad name!
%\def\psfig#1#2#3{\pscaption{\psannotate{#1}{#2}}{#3}}
%\def\psfigurebox#1#2#3{\pscaption{\psannotate{\psbox{#1}}{#2}}{#3}}
%
% \at(#1;#2)#3 puts #3 at #1-higher and #2-right of the current
%    position without moving it (to be used in annotations).
\def\at(#1;#2)#3{\setbox0=\hbox{#3}\ht0=0pt\dp0=0pt
  \rlap{\kern#1\vbox to0pt{\kern-#2\box0\vss}}}%
%
% \gridfill(ht;wd) makes a 1cm*1cm grid of ht by wd whose lower-left
%   corner is the current point
\newdimen\gridht \newdimen\gridwd
\def\gridfill(#1;#2){%
  \setbox0=\hbox to 1\pscm
  {\vrule height1\pscm width.4pt\leaders\hrule\hfill}%
  \gridht=#1
  \divide\gridht by \ht0
  \multiply\gridht by \ht0
  \gridwd=#2
  \divide\gridwd by \wd0
  \multiply\gridwd by \wd0
  \advance \gridwd by \wd0
  \vbox to \gridht{\leaders\hbox to\gridwd{\leaders\box0\hfill}\vfill}}%
%
% Useful to measure where to put annotations
\def\fillinggrid{\at(0cm;0cm){\vbox{%
  \gridfill(\drawinght;\drawingwd)}}}%
%
% \textleftof\anybox: Sample text\endtext
%   inserts "Sample text" on the left of \anybox ie \vbox, \psbox.
%   \textrightof is the symmetric (not documented, too uggly)
% Welcome any suggestion about clean wraparound macros from
%   TeXhackers reading this
%
\def\textleftof#1:{%
  \setbox1=#1
  \setbox0=\vbox\bgroup
    \advance\hsize by -\wd1 \advance\hsize by -2em}%
\def\textrightof#1:{%
  \setbox0=#1
  \setbox1=\vbox\bgroup
    \advance\hsize by -\wd0 \advance\hsize by -2em}%
\def\endtext{%
  \egroup
  \hbox to \hsize{\valign{\vfil##\vfil\cr%
\box0\cr%
\noalign{\hss}\box1\cr}}}%
%
% \frameit{\thick}{\skip}{\anybox}
%    draws with thickness \thick a box around \anybox, leaving \skip of
%    blank around it. eg \frameit{0.5pt}{1pt}{\hbox{hello}}
% \boxit{\anybox} is a shortcut.
\def\frameit#1#2#3{\hbox{\vrule width#1\vbox{%
  \hrule height#1\vskip#2\hbox{\hskip#2\vbox{#3}\hskip#2}%
        \vskip#2\hrule height#1}\vrule width#1}}%
\def\boxit#1{\frameit{0.4pt}{0pt}{#1}}%
%
%
\catcode`\@=12 % cs containing @ are unreachable
%
% CUSTOMIZE YOUR DEFAULT DRIVER:
%    Uncomment the line corresponding to your TeX system:
%\psfortextures%     For TeXtures on the Macintosh
%\psforoztex   %     For OzTeX shareware on the Macintosh
%\psfordvitops %     For the DVItoPS converter for TeX on IBM mainframes
 \psfordvips   %     For DVIPS converter on VAX and UNIX
%\psfordvitps  %     For dvitps from TeXPS package under UNIX
%\psfordvialw  %     For dvialw, UNIX public domain
%\psonlyboxes  %     Blank Boxes (when all else fails).

astro-ph0102119.extracted_bib
\begin{thebibliography}{}

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\end{thebibliography}