\mbox{Planet Migration and Binary Companions: the case of HD $80606$b}
Y. Wu\altaffilmark{1} \& N. Murray\altaffilmark{1,2}
The exo-solar planet HD $80606b$ has a highly eccentric ($e=0.93$) and tight ($a=0.47$ AU) orbit. We study how it might arrive at such an orbit and how it has avoided being tidally circularized until now. The presence of a stellar companion to the host star suggests the possibility that the Kozai mechanism and tidal dissipation combined to %wyq: the abstract is significantly modified from here on draw the planet inward well after it formed: Kozai oscillations produce periods of extreme eccentricity in the planet orbit, and the tidal dissipation that occurs during these periods of small pericentre distances leads to gradual orbital decay. We call this migration mechanism the 'Kozai migration'. It requires that the initial planet orbit is highly inclined relative to the binary orbit. For a companion at $1000$ AU and an initial planet orbit at $5$ AU, the minimum relative inclination required is $\sim 85 \deg$. We discuss the efficiency of tidal dissipation inferred from the observations of exo-planets. Moreover, we investigate possible explanations for the velocity residual (after the motion induced by the planet is removed) observed on the host star: a second planet in the system is excluded over a large extent of semi-major axis space if Kozai migration is to work, and the tide raised on the star by HD $80606$b is likely too small in amplitude. Lastly, we discuss the relevance of Kozai migration for other planetary systems. % we discuss also the chance for this object to be a planet as opposed % to a brown-dwarf...
ms.tex
% new changes marked by %wyq2


% things to do: produce results for 1.5 M_sun and see whether that can explain everything
%		and see how the mass is constrained from spectral type		


% things I have done:  lengthen abstract
%			changes marked by %wyq


% things to do: 
%		cite Hut's equations? 
%		make all figures into double column format
%		make the angle inference more transparent so people can easily do it themselves
%	 	what happens when the binary is further apart?

%  astro-ph/0204272 From: Ray Jayawardhana <rayjay@astron.berkeley.edu>
%Title: Discovery of an Edge-On Disk in the MBM 12 Young Association
%Authors: Ray Jayawardhana, K. L. Luhman, Paola D'Alessio, and John R. Stauffer
%shows a protoplanetary disk that doesn't seem to be aligned with the 
%star plane.... neither coplanar nor parallel
% it also mentions that there are other protodisks in multiple systems

% need  $q=M_p/M_\star$ anywhere???
% for the tidal Q paper -- review what's done/known about Jupiter

% uniformity: M_p = 7.8 M_J, a_p = 0.469, e_p=0.927
% M_\star = 1.1 M_sun, M_c = 1.1 M_sun

% exoplanet.org gives M_p = 3.4 M_J ===> what to do?

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\begin{document} 
	
\title{\mbox{Planet Migration and Binary Companions: the case of HD
$80606$b}}

\lefthead{Migration \& Binarity}
\righthead{Wu \& Murray}

\author{Y. Wu\altaffilmark{1} \&  N. Murray\altaffilmark{1,2}}

\altaffiltext{1}{Canadian Institute of Theoretical Astrophysics,
	University of Toronto, 60 St. George Street, Toronto, Ontario
	M5S 3H8, Canada}
\altaffiltext{2}{Canada Research Chair in Astrophysics}

\begin{abstract}

The exo-solar planet HD $80606b$ has a highly eccentric ($e=0.93$) and
tight ($a=0.47$ AU) orbit. We study how it might arrive at such an
orbit and how it has avoided being tidally circularized until now. The
presence of a stellar companion to the host star suggests the
possibility that the Kozai mechanism and tidal dissipation combined to
%wyq: the abstract is significantly modified from here on
draw the planet inward well after it formed: Kozai oscillations
produce periods of extreme eccentricity in the planet orbit, and the
tidal dissipation that occurs during these periods of small pericentre
distances leads to gradual orbital decay. We call this migration
mechanism the 'Kozai migration'.  It requires that the initial planet
orbit is highly inclined relative to the binary orbit.  For a
companion at $1000$ AU and an initial planet orbit at $5$ AU, the
minimum relative inclination required is $\sim 85 \deg$.  We discuss
the efficiency of tidal dissipation inferred from the observations of
exo-planets.  Moreover, we investigate possible explanations for the
velocity residual (after the motion induced by the planet is removed)
observed on the host star: a second planet in the system is excluded
over a large extent of semi-major axis space if Kozai migration is to
work, and the tide raised on the star by HD $80606$b is likely too
small in amplitude. Lastly, we discuss the relevance of Kozai
migration for other planetary systems.

% we discuss also the chance for this object to be a planet as opposed
% to a brown-dwarf...
\end{abstract}

%\keywords{}

\setcounter{equation}{0}

\section{INTRODUCTION}
\label{sec:nl-intro}
% some recent article (astro-ph) talked about disk in binaries and
% their respective inclinations...

%get exoplanet's $Q$ value: physical interpretation leading to
%understanding of the planetary structure

Ongoing radial-velocity surveys have uncovered $\sim 100$ exo-solar
giant planets.\footnote{See http://cfa-www.harvard.edu/planets/ for a
constantly updated catalog.} Their orbital characteristics are
puzzling: many have eccentricities much higher than the planetary
orbits in our solar system, and a large fraction reside very close to
the host star. These orbits indicate either a formation scenario
different from that which operated in our solar system, or a migration
mechanism to bring the planets in from large distances.
 
The object HD $80606$b (Naef, Latham \& Mayor \etal, \cite{naef},
hereafter NLM, period $111.8$ days, $ m_p \sin i = 3.4
M_J$)\footnote{The host star HD $80606$ is a G5 star with [Fe/H] $=
0.43$, $m_B= 9.65$ and $m_V= 8.93$.}
% and $a=0.469$ AU according to Naef...
the focus of this article, has the highest eccentricity ($e = 0.927\pm
0.012$) and the smallest pericentre distance ($\sim 7 R_\odot$) among
all known exo-planet candidates. Is it possible to form such a system
in-situ?
% Batten catalogue from vizieR
Among the spectroscopic binaries contained in the Batten catalog
(Batten, Fletcher \& MacCarthy \cite{batten}), the only G-type binary
that has a comparably small periastron distance ($\sim 6 R_\odot$) is
HD 27935 (Griffin, Griffin \& Gunn \etal, \cite{griffin}) with $e=0.85$.
%a period of $156$ days,
%with a K5 companion, while the most eccentric binary HD
%165590 (G0+G8) has an $e=0.96$ and a period of 7400 days (Batten \etal
%\cite{batten2}). 
Duquennoy, Mayor \& Andersen \etal (\cite{Gl586}) reported a more
eccentric system, the K-dwarf binary Gl 586A, with $e= 0.975$ and a
periastron distance of $10 R_\odot$. However, doubts exist as to
whether the latter binary can be primordial. But assuming both systems
are primordial, their presence empirically suggests that HD $80606$b
might have formed as a binary companion to HD $80606$.
%The planet's
%true mass should in this case greatly exceed the projected mass to
%avoid tidal circularization of the orbit.\footnote{We assume that the
%tidal dissipation factor $Q$ does not vary with mass for sub-solar
%objects.}  
%wyq: removed the above sentence, not correct. 
This hypothesis is currently untestable, 
%wyq2: added a sentence to correspond to the referee's 2nd point
the more so since the projected mass-ratio in this system is rather
extreme.
%{\it HIPPARCOS ASTROMETRY?},
% wyq: HIPPARCOS couldn't even get parallax of the star,
%	let alone astrometric shift...
Instead we ask the following question: can HD $80606$b be formed in
the cool outer part of a protoplanetary disk and then evolve into the
current orbit via migration?

Planet-planet scattering is one such possibility, assuming HD $80606$
had or has multiple planets.
%wyq2: the range selected is a_1/a_2 = 0.769 to 0.781, while 0.769 is
%	the limit of instability
%wyq2: changed to the following after referee's point 1
Ford, Havlikova \& Rasio (\cite{ford}) integrated the evolution of
$\sim 10^3$ two-planet systems that are initially dynamically
unstable. Not a single close encounter produces a planet with $e >
0.90$.
%However, out of the $\sim 10^3$ test cases
%integrated by Ford, Havlickova \& Rasio (\cite{ford}), not a single
%close encounter produces a planet with $e > 0.90$. 
This rarity leads us to conclude that either the orbits of most
observed exo-planets are results of planet-planet scattering, or that
the HD $80606$ system is not the result of such scattering.

The leading hypothesis for planet migration involves gravitational
interactions between the planets and the gas disks out of which they
form (Goldreich \& Tremaine \cite{gt}; Lin, Bodenheimer \& Richardson
\cite{lbr}). Under certain circumstances planet-disk interactions may
also excite the orbital eccentricity of the planet (see, e.g.,
Goldreich \& Sari \cite{goldsari}).
%wyq2: started rewriting the following sentences
If so, this may explain the orbital separation and eccentricity of the
bulk of the systems discovered so far. These systems exhibit an
eccentricity distribution that is roughly flat below $e \sim 0.6$ and
distinctly drops off around $0.6$. It is believed that Jupiter-mass
planets can open up gaps in their natal gas disks with a fractional
width $\Delta a/a$ reaching up to a similar value. This coincidence
suggests that passage of an eccentric planet through the disk on
either side of the gap tends to damp the planet's orbital eccentricity
and to limit the maximum eccentricity planet-disk interaction produces
to $e\sim 0.6$. However, HD $80606b$ stands out with $e=0.93$.
%wyq2: adjusted the few sentences above to clarify (point 3 of refereee report)
%
% wyq: not sure if that's the only explanation, it can also be, e.g., the clearing
%	of lowest order resonances, if higher order resonances damp eccentricity
%		(see the following, which I deleted as I am not sure of that either...)
%	also gap width is inferred by balancing viscous torque with principal resonance
%	torque, if we use the first-order resonance torque, it's likely broader?
%There is currently no explanation for this upper
%cut-off.\footnote{However, this upper-cut off is suggestively similar
%to $e=2/3$, at which value a planet can easily clear out a wide enough
%gap in a Keplerian disk to make all principal (and most of the
%first-order) resonances impotent.}
% wyq: all principal resonances are within a(1+/- 2/3)
% 	the inward-most first-order inner Lindblad resonance at a(1-2/3)
% 	the outer-most first-order outer Lindblad resonance is at a(1+4/3)
%	if take m=1, this is negative... so minimum m is 2, 
% 	m=1, first-order ILR is called apsidal resonance
%
%Why then is the eccentricity of HD $80606$b so high?

The origin of the orbit of HD $80606$b becomes more perplexing if we
consider tidal effects.  Assuming that the planet has a tidal
dissipation efficiency similar to other known exo-planets (quality
factor $Q_p= 3\times 10^5$, Wu \cite{wu}) or to Jupiter (Goldreich \&
Soter \cite{goldreich}, Peale \& Greenberg \cite{peale}) and setting
its true mass equal to the minimum mass, its radius to a Jupiter
radius, we find that a mere $650$ Myrs ago, HD $80606$b had an
implausibly high eccentricity of $e = 0.99$.\footnote{
%wyq: added the following footnote after our discussion
% 	quantitatively: stellar size doesn't matter much (for the $Q_*$ value
%	obtained assuming a 3Myr shrinking time from 3 R_sun to 1 R_sun, and
%	to be able to synchronize out to 8 day period as in Hyades),
%	but the planet size shrinking of the same form matters a lot
%	an initial eccentricity of 0.99 can be damped to 0.97 in a few
%	Myrs
This problem is further exasperated if the planet has acquired the
same periastron distance while it was still hot and large.} The age
of the system is likely to be this old or older; 
%fitting to stellar evolutionary tracks indicates an age greater than $500$Myrs, and
%
%wyq: is the above age assuming a certain parallax? I am not sure, and removed it
the stellar projected rotational velocity is low, $v\sin i \approx 0.9
\km/\s$, typical of old G-dwarves; it is also chromospherically quiet.

Interestingly, there is a neighbor to this system, HD $80607$, a
main-sequence companion $\sim 1000$ AU away. This prompts us to
develop a theory in which the companion is responsible for the high
eccentricity and small orbit of the planet. There are two ingredients
in our theory. First, we assume that the planet was born on an orbit
of a few AU, and had an orbital plane inclined relative to the stellar
binary plane. The remote stellar companion would induce an
eccentricity oscillation in the planet's orbit via the Kozai mechanism
(Kozai\cite{kozai}; see also discussions by Holman, Touma \& Tremaine
\cite{holman}, Innanen, Zheng, \& Mikkola \etal, \cite{innanen}, and
Mazeh, Krymolowski \& Rosenfeld \cite{mazeh}).
% wyq: removed the following... 
%The initial eccentricity of the planet (before the commencement of
%Kozai oscillations) may well have been substantial, of order
%$0.2-0.5$, if the eccentricities of the other exo-solar planets are
%any guide.
%
The second component is tidal circularization. The tides operate most
effectively during episodes of high eccentricity in the Kozai
cycle. We rely on the dissipative tidal process to irreversibly draw
the planet inward. Combining the two processes, we can explain the
current high eccentricity as well as the tight orbit.  We refer to
this planet migration scenario as `Kozai migration'.

Eggleton, Kiseleva \& Hut (\cite{EKH}), Kiseleva, Eggleton \& Mikola
(\cite{kem}), and Eggleton \& Kiseleva-Eggleton (\cite{EKE}, EKE
hereafter), have developed this scenario to explain some hierarchical
triple star systems.  Blaes, Lee \& Socrates (\cite{omer}) applied an
analogous scheme (gravitational radiation being the dissipation
mechanism) to hierarchical triple black holes in galactic centers. In
this article, we adopt the formalism of EKE in an attempt to produce a
plausible life-history for HD $80606$b (\S
\ref{sec:HD80606}). The story depends on the value of the tidal
quality factor $Q$ (\S \ref{subsec:Qvalues}).  The observational
consequences of such a life-history follow in \S
\ref{subsec:secondplanet}. In \S \ref{subsec:tidalheight} we discuss
the tidal velocities induced at the stellar photosphere by the highly
eccentric planet, and the possibility of observing such tides. NLM
note the presence of substantial residuals to the Keplerian fit in the
HD $80606$ system. The tidal velocities appear to be large, but not
large enough to explain the observed velocity residuals. Lastly (\S
\ref{subsec:othersystem}), we assess the importance of Kozai migration
for other planetary systems where binary companions are also known to
exist.

\section{Kozai Migration for HD $80606$b}
\label{sec:HD80606}

% Stellar masses are both assumed to be $1.1 M_\odot$. 
The host star of HD $80606$b is known to be a member of a common
proper-motion binary with a companion (HD $80607$) that is similar to
HD $80606$ in both spectral type and metallicity . The two stars are
separated by $\sim 20''$ on the sky. Unfortunately, it is hard to
translate this into a linear dimension. The Hipparcos distances for
both stars are highly uncertain\footnote{Hipparcos parallaxes are $17
\pm 6$ mas for HD $80606$ and $10 \pm 9$ for HD $80707$. The large
error bars are presumably due to mutual contamination (see also NLM).} 
This complicates an age determination with isochrone fitting. But if
we enforce the constraint that the stars have reached the
main-sequence, isochrone fitting yields a lower limit of $\sim 65$ pc
on the distance, implying a projected binary separation of $1300$ AU
at the current epoch. On the other extreme, requiring that the stars
have not yet ascended the red giant branch yields a maximum distance
of $\sim 125$ pc, and a projected separation of $2500$ AU. As noted
above, the low $v\sin i$ and the chromospheric quietness of HD $80606$
are typical of an old G-dwarf.  In this article, we adopt a value of
$1000$ AU for the binary semi-major axis ($a_c$). The eccentricity is
taken to be $e_c = 0.5$.\footnote{We denote quantities related to the
planet and the companion star with subscripts $p$ and $c$,
respectively.} We discuss the effects on the Kozai migration when
$a_c$ is increased.
% wyq: added the above sentence.

Even at such a large distance, the companion star could significantly
perturb the planet orbit as long as the two orbital planes are
initially inclined to each other more than $39.2 \deg$.\footnote{Below
this value, the periapse argument of the planet circulates through all
angles very fast and the torque from the companion is effectively
averaged to zero. There is little variation in the planet's
inclination and eccentricity. Also see, e.g., Holman \etal
(\cite{holman}).}
%wyq2: added a footnote to explain the critical angle, referee point 4
Secular effects of the Kozai type could then occur and produce large
cyclic variations in the planet's eccentricity ($e_p$) and relative
inclination, as a result of angular momentum exchange with the
companion orbit. Effects on the companion orbit produced by the planet
can be neglected, given the extreme ratio in the two orbits' angular
momenta ($a_c \gg a_p$ and $m_c \gg m_p$).  Under this approximation,
the z-component of the planet's angular momentum, $J_z = \sqrt{G
M_\star a_p} M_p \sqrt{1-e_p^2} \cos I$, is conserved.\footnote{See
Ford, Kozinsky \& Rasio \cite{ford0} for a discussion of higher order
effects).} Here the z-axis is normal to the companion plane.  As $a_p$
is not modified by secular perturbations, the Kozai integral $H_K =
\sqrt{1-e_p^2} \cos I$ is conserved during the oscillations. Minima in
$e_p$ concur with maxima in $I$, and {\it vice versa}. Each Kozai
cycle lasts a time $\sim P_c^2/P_p$, or a few Myrs.

A much slower process, tidal circularization, gradually removes energy
from the orbit and draws the planet inward.  An initially more highly
inclined orbit (larger $I$) could reach higher eccentricity. And since
the dissipation process depends sensitively on the nearest approach
distance between HD $80606$b and its host star, such an orbit would
suffer stronger orbital decay and have shorter migration timescale.
The key issue in Kozai migration is therefore the initial value of
$I$. In \S \ref{subsec:heuristic}, we first
%first give a brief discussion on the Kozai effect (\S \ref{subsec:intuitive}), then 
estimate the minimum initial inclination required for migrating HD
$80606$b from an initial orbit of $a_p = 5$ AU and an initial
eccentricity of $e_p = 0.1$, we then confirm this estimate numerically
in \S \ref{subsec:numerical}.

%\begin{deluxetable}{ccr}
%\tabletypesize\footnotesize
\begin{table*}
%\startdata
\begin{tabular}{ccr}
\hline
Symbol & Definition & Fiducial Values \\
\hline

$M_\star, M_c, M_p$ & stellar (host \& companion) and planetary masses &
$1.1 M_\odot$, $1.1 M_\odot$, $7.80 M_J$\\

$R_\star, R_p$ & stellar and planetary radii & $R_\odot$, $R_J$ \\

$k_{2\star}, k_{2p}$ & tidal Love number & $0.028$, $0.51$ \\
%  &  (deformation under tidal or rotational perturbations) & \\

$r_{g\star}, r_{gp}$ & gyro-radius, moment of inertia $ = r_g M R^2$
 & $0.08$, $0.25$ \\

$Q_\star, Q_p$ & tidal dissipation quality factors & $10^6$,
$3\times 10^5$ \\
%& (tidal energy versus energy dissipated per tidal period) & \\

$\Omega_\star, \Omega_p$ & spin frequency & initially $20$ days \&
$10$ hours\\

%$T_\star, T_p$ & tidal dissipation time & $T = 2 Q n R^3/(G M)$ \\

$a_p$, $e_p$, $n_p$ \& $P_p$ & planet semi-major axis, eccentricity,
mean motion \& period & $n_p = \sqrt{G(M_\star + M_p)/a_p^3}$ \\

$a_c$, $e_c$, $n_c$ \& $P_c$ & parameters for the companion orbit & $a_c =
1000$ AU, $e_c = 0.5$ \\

%\enddata
\hline
\end{tabular}
\label{table:symbol}
%\end{deluxetable}
\end{table*}


\subsection{Minimum Initial Inclination: heuristic argument}
\label{subsec:heuristic}

To estimate the minimum initial inclination, we note that while the
Kozai oscillation roughly conserves $\sqrt(1-e_p^2) \cos I$ and $a_p$
during individual cycles, tidal circularization preserves orbital
angular momentum ($J= \sqrt{G M_\star } M_p \sqrt{a_p (1-e_p^2)} $)
during episodes of maximum eccentricity. The planetary spin is quickly
(pseudo-)synchronized 
%wyq2: added the following
with the orbital motion,\footnote{Pesudo-synchronization applies to an
eccentric system where the planet spin is tidally synchronized to a
rate that is in-between the orbital frequency and the angular
frequency at periapse. See, e.g., Hut (\cite{hut}).}
%wyq2: changes end here
while the star's synchronization time greatly exceeds the orbital
circularization time. Hence the planet's orbital angular momentum
cannot be absorbed into spin. Since $e_p \sim 1$ near maximum,
conservation of $J$ translates into a constant periastron distance
between maximum $e_p$ during the Kozai cycles.
%Most of the dissipation occurs near periastron. Since the planet is
%likely spinning synchronized with the angular velocity at periastron,
%we expect the tidal force to be radial and orbital angular momentum is
%conserved. 

In an orbit with a semi-major axis of $5$ AU, HD $80606$b would have
to reach an eccentricity as high as $0.993$ (at some point during the
Kozai cycle) in order to produce the currently observed
periastron. Adopting a minimum of $I \approx 40 \deg$ during the Kozai
cycles (see, e.g., Holman \etal, \cite{holman}), the Kozai integral (
$\sqrt(1-e_p^2) \cos I$) remaining constant yields $I \geq 84.8 \deg$
%\footnote{This is a minimum requirement, the actual inclination may need to 
%be slightly higher as $I$ may not reach down to $40 \deg$.}
for an initial $e_p=0.1$.  In other words, the two orbits had to be
virtually perpendicular to each other when the planet formed.
% wyq: added the following
This conclusion is independent of the companion orbital separation.

The planet in HD $80606$b is not currently undergoing Kozai
oscillations; the Kozai mechanism operated only when the planet had a
semi-major axis $a\geq 4$ AU.  To see why, note that the torque
exerted by the companion star on the planet has its value and
direction dependent on the pericentre argument of the planet
($\omega_p$). If, however, this argument also precesses under other
forces, the averaged Kozai torque is reduced.  General relativistic
effects, tidal effects, and rotational quadrapolar bulges on the
planet\footnote{Analogous but less important bulges on the star are
ignored.} are mainly responsible for these extra precession. Their
rates are summarized here (Sterne \cite{sterne}, Einstein
\cite{einstein}), with definitions for the symbols listed in Table 1.,
%\ref{table:symbol},
%Deviations from inverse-square law
\begin{mathletters}
\begin{eqnarray}
\hskip-0.5cm {d\omega_p\over dt}\Bigr|_{\rm GR} & = & 3
n_p\, {{G M_\star}\over{a_p c^2 (1-e_p^2)}},\label{eq:preGR} \\
\hskip-0.7cm {d\omega_p\over dt}\Bigr|_{\rm rot} \,\,& = & {1\over2}
n_p {k_{2}\over{(1-e_p^2)^2}}\!\left({{\Omega_p}\over{n_p}}\right)^2\!\!
{{M_\star}\over{M_p}}\!\!  \left({{R_p}\over{a_p}}\right)^5,
\label{eq:preJ2}\\
\hskip-0.7cm {d\omega_p\over dt}\Bigr|_{\rm tide} & = & {15\over2} n_p
k_{2}\! {{1+\frac{3}{2}e_p^2 + \frac{1}{8}e_p^4}\over{(1-e_p^2)^5}}
{{M_\star}\over{M_p}}\!\!  \left({{R_p}\over{a_p}}\right)^5.
\label{eq:pretide}
\end{eqnarray}
\end{mathletters}
Taking a Kozai precession rate of $\sim 0.5 n_c^2/n_p/(1-e_c^2)^{3/2}$
(Holman \etal, \cite{holman}), and assuming that the planet is
pseudo-synchronously spinning with the orbit ($e_p = 0.1$), the
relative precession rates are,
\begin{eqnarray}
%wyq: change the following numbers when scaled to 0.47 AU and 1000AU, also e_p = 0.1,
%	not 0.927
{{\dot{\omega}_p|_{\rm GR}}\over{\dot{\omega}_p|_{\rm Kozai}}}&
\approx & 5900
\left({{a_1}\over{0.47 {\rm AU}}}\right)^{-4}\!\!
\left({{a_c}\over{1000 {\rm AU}}}\right)^{3}, \nonumber\\
{{\dot{\omega}_p|_{\rm rot}}\over{\dot{\omega}_p|_{\rm Kozai}}}&
\approx & 750
\left({{a_1}\over{0.47 {\rm AU}}}\right)^{-8}\!\!
\left({{a_c}\over{1000 {\rm AU}}}\right)^{3}\!\!
\left({{R_p}\over{R_J}}\right)^{5},  \nonumber \\
{{\dot{\omega}_p|_{\rm tide}}\over{\dot{\omega}_p|_{\rm Kozai}}}&
\approx & 270
\left({{a_1}\over{0.47 {\rm AU}}}\right)^{-8}\!\!
\left({{a_c}\over{1000 {\rm AU}}}\right)^{3}\!\!
\left({{R_p}\over{R_J}}\right)^{5}\!\!\!\!.
\label{eq:precession}
\end{eqnarray}
If any of the above ratios exceeds unity, the Kozai oscillation is
destroyed. Setting the first ratio to unity, we obtain a suppression
radius $\approx 4$ AU. Currently, the Kozai oscillations are strongly
suppressed by these extra precessions. This conclusion does not affect
the minimum inclination required for Kozai migration. However, the
suppression radius does depend on the companion separation: e.g., it
is reduced to $\approx 1.5 $AU if the companion semi-major axis is as
small as $250$ AU.
%wyq2: added 'semi-major axis' above

%For the Kozai resonance to produce such an eccentricity, starting from
%a primodial eccentricity of, say, $0.1$, we require an initial
%inclination angle between the two orbital planes to be as high as $I
%\sim 84.1 \deg$. This is obtained from the Kozai constant $H_K =
%\sqrt{1 - e_p^2} \cos I $, and adopting an inclination angle of $50
%\deg$ by the time Kozai cycles stop (see Fig. \ref{fig:eggleton_short}
%and Holman \etal \cite{holman}).
%% is Holman the correct siting??
%% all these numbers acquired using estimate_kozai.math in this directory
%When adopting an initial orbit of $5$ AU, we obtain an initial
%inclination of $I \sim 84.7 \deg$. With these parameters, Kozai
%time-scale is $\sim 7 $ Myrs.  This gives rise to an eccentricity of
%$0.9917$ at $4$ AU.

\subsection{Numerical Integration}
\label{subsec:numerical}

We adopt the set of equations (eq. [11]-[17]) in EKE (also see
Eggleton, Kiseleva \& Hut \cite{EKH}) to describe the secular
evolution of the planet orbit, as well as the spin of the host star
and the planet. Any change in the companion's orbit is
ignored. Physical effects described by these equations include:
secular interactions, GR, tidal and rotational precession, and tidal
dissipation. These equations are explicitly listed in Appendix
\ref{sec:EKEeqn}. 

We integrate equations \refnew{eq:ep}-\refnew{eq:omegapi} starting
from the following initial conditions: planet orbit $a_p =5.0$ AU,
$e_p = 0.1$, relative inclination $I = 85.6\deg$ (slightly higher
%wyq: the actual number is 85.65 deg, added the 'because ...'
than the $84.8\deg$ estimate in \S \ref{subsec:heuristic} because the
actual minimum inclination is $\sim 48\deg$, not $\sim 40\deg$ as we
assumed), and periapse angle $\omega_p = 45\deg$.
%wyq: periapse angle hardly matters, in the end

Values of the system parameters are listed in Table 1.
%\ref{table:symbol}. 
The resulting orbital and spin information are presented in
Fig. \ref{fig:eggleton_short} as functions of time. Initially, the
planet undergoes Kozai oscillation with each cycle lasting $\sim 20$
Myrs. Whenever $e_p$ evolves close to maximum, a small amount of
energy (but not angular momentum) is removed from the orbit. Both
$a_p$ and $e_p$ are decreased,
% wyq: added the following, removed the original
while $a_p(1-e_p)$ remains roughly constant.  This corresponds to a
small kick to reduce $e_p$ whenever $\omega_p
\approx \pi/2$ (see Fig. 3 of Holman \etal \cite{holman}).
%wyq: changed the following sentence
%	to all values of e_p-omega_p (if initial e_p very small, circulating
%	or librating solution gives same e_max)
%	also: the actual phase space evolution remains circulatory all the time...
%So an initially circulating oscillation is slowly converted
%into a librating oscillation in the $\omega_p$-$e_p$ phase space, with
%the added complication that 
The Kozai integral slowly increases. This explains why the amplitude
of $e_p$ oscillation shrink as the planet migrates inward (top-left
panel of Fig. \ref{fig:eggleton_short}). After about $0.7$ Gyrs, the
planet has reached an orbit with $a_p \sim 2.5$ AU, and the
%wyq: changed form a_p = 4 to a_p = 2.5 AU, not much oscillation any more...
Kozai oscillation is completely destroyed by GR precession. At this
epoch, we find $e_p \sim 0.99$ and $I \sim 50 \deg$.  From this point
on, tidal circularization dominates the evolution, and the influence
of the binary companion is negligible. The planet reaches an orbit of
$e_p = 0.927$ and $a_p = 0.47$ AU $\sim 1.2$ Gyrs after its birth.
 
A few notes are in order.

We adopt a value of $Q_p = 3 \times 10^5$ for the planet, and $Q_\star
= 10^6$ for the host star.
%\footnote{Other parameters (mass, radius and
%tidal love number) also affect the tidal dissipation timescale. But we
%consider them less certain than the $Q$ value.} 
We discuss the reasons for these choices in \S
\ref{subsec:Qvalues}. Using them, the planet's spin reaches
synchronicity with the orbital frequency near periastron almost
instantly, while the star's spin is hardly affected by tides (see
lower-right plot of Fig. \ref{fig:eggleton_short}). Tidal
circularization is dominated by dissipation in the planet over that in
the star by a factor of $\sim 10$. Most of the dissipation, which
conserves orbital angular momentum, occurs when the planet has been
driven to maximum eccentricity by the companion star.
%Throughout the evolution, effects of the
%planet orbit on the stellar binary orbit are ignored. 
%With a $Q_\star = 10^6$, tidal dissipation on the star can be ignored
%relative to that on the planet, though only by a factor of $\sim 10$.
%tidal dissipation factor $Q_\star = 10^6$, although we find that tidal
%dissipation on the star is not important;
% but if Q_star = 10^5, can make some difference --> stronger dissipation

The results do not depend on the value for the longitude of the
ascending node. This is because in the quadrapole approximation for
secular interactions, the companion orbit is effectively taken to be
circular so there is no preferred axis in the binary plane. This also
explains why the vertical angular momentum of the planet orbit (Kozai
integral) is conserved -- there is a rotational invariance with
respect to the $z$ axis.

%The results do, however, depend on the periapsee angle $\omega_p$. The
%reason for this is illustrated in Fig. \ref{fig:eggleton_traj}. For a
%given initial $e_p$, the value of $\omega_p$ determines whether the
%oscillation is circulating or librating, as well as the maximum value
%of $e_p$. A smaller $|\sin (\omega_p)|$ allows the planet to reach a
%higher maximum eccentricity and consequently a shortened migration
%time-scale. For instance, varying the initial $\omega_p$ from $55\deg$
%to $65\deg$ raises the $1.1$ Gyrs cited above to $\sim 3$ Gyrs. In
%spite of these quiddities, HD $80606$b migrates to its present small
%orbit in less than a Hubble time for all values of $\omega_p$.
% wyq: ??????? doesn't depend on omega_p any more with the trajectories
%	restricted to the same e_p_max
% wyq: ??????? change all the above and remove figure 2?????
The results also do not depend on the value for the argument of
pericentre, $\omega_p$. Kozai oscillations can cause either
circulation or libration in the value of $\omega_p$. Fig. 3 of Holman
\etal (\cite{holman}) shows that the circulating solutions can reach
higher $e_p$ and therefore should lead to shorter evolutionary
timescale. However, for the initial conditions we adopted, the
circulating and librating solutions essentially coincide and give rise
to similar values for the maximum eccentricity.

%\footnote{Taking higher order
%dynamics into account will produce wandering across the separatrix and
%may dissolve this distinction. This is not investigated in the current
%article.}
%% fill in the numbers above...
%For instance, an initial $\omega_p = 0$ leads to $e_p|_{\rm max} =
%0.9967$, while that of $\omega_p = \pi/2$ gives rise to $e_p|_{\rm
%max} = 0.9936$. 

%Blaes \etal (\cite{omer}) have discussed extensively effects of
%varying different parameters in the context of triple black-holes. As
%many of our findings concur with theirs, we refer the readers to that
%article.

%wyq: removed the following
%In summary, for Kozai migration to transport the planet inward to its
%current location within a reasonable amount of time, the required
%initial conditions are: sufficiently high $I$, sufficiently small
%$a_p$, and sufficiently small $a_c$. Among these, the initial
%inclination is most important and the results depend sensitively on it.
% 
%wyq: instead, added the following
To migrate the planet from a distance larger than $5$AU, one requires
a higher minimum inclination so as to reach the same pericentre
distance at maximum eccentricity. In comparison, this inclination
hardly varies when the initial eccentricity of the planet is varied
from $0.1$ to $0.6$, this can be seen from
Fig. \ref{fig:eggleton_short}.  Lastly, if the companion is further
away than $1000$ AU, each Kozai cycle lasts longer, and it takes a
longer time for the planet to migrate to its current location. For
instance, placing the companion at $a_c = 1500$ AU, we find the planet
should now be $\sim 5 $Gyrs old.
%wyq: actually, also need to have higher initial $I$.
%

%a larger initial $a_p$ can be accomondated by a larger Kozai integral
%or a smaller $\omega_p$: e.g., $a_p = 7$ AU would require $\omega =
%33\deg$.
%to yield $t_{0927}=2.1$Gyr, $a_{0927}=0.438$ AU.
%Kozai resonance is weakened if the companion is further away than
%$1000$ AU and the corresponding time-scale increases. Moreover, this
%requires an even higher initial inclination (see the heuristic
%argument in \S \ref{subsec:heuristic}).
%But if we repeat the heuristic argument in the beginning of this
%section for $a_c = 1200$ AU (smallest $a_c$ consistent with $e_c =
%0.5$), we find that at $a_p = 5.0$AU, initial inclination has to be at
%least $86\deg$. This is confirmed by numerical experiments.
%actually, $I = 86\deg$, $t_{0927} = 1$ Gyrs, $a_{0927}=0.447$ AU
% and once can increase $\omega_p$ to $48 \deg$ 

%\begin{figure}
\begin{figure*}[t]
\centerline{\psfig{figure=eggleton_short.ps,width=0.50\hsize}}
%\epsscale{0.50}
%\plotone{f1.eps}
\caption{Evolution of the planetary orbit as a function of logarithmic time
(in years). The four panels represent (from left to right, up to
down), evolution of the planet eccentricity, planet semi-major axis
and periastron distance (in AU), inclination of the planet plane
relative to the stellar binary plane, and spin periods for both the
planet and the star (in days).  See the text for the choice of
parameters. During the Kozai cycles, maxima in eccentricity concur
with minima in inclination. At the current orbit ($e_p = 0.927$ and
$a_p = 0.47$ AU), the planet no longer undergoes Kozai oscillations as
a result of GR and tidal precession. The dashed part of each curve
depicts future evolution of the planet orbit under tidal
circularization.}
\label{fig:eggleton_short} 
\end{figure*}
%\end{figure}

%wyq: removed the following figure all together..........
%\begin{figure*}[t]
%\centerline{\psfig{figure=eggleton_traj.ps,width=0.50\hsize}}
%\caption[]{Planet trajectories in the $e_p$ - $\omega_p$ (periapsee
%angle) phase space under secular perturbations from the companion
%star. We ignore tidal dissipation and depict trajectories sharing the
%same Kozai integral of $H_K = \sqrt{1-e^2} \cos I \approx
%0.070$. Under the quadrapole approximation and in lieu of other
%%have initial parameters $e_p = 0.6$ and relative inclination $I = 85 \deg$, 
%precessions, these trajectories do not cross each other. The
%circulating trajectories can reach higher eccentricity maxima, making
%the orbit more susceptible to tidal dissipation. The latter process
%does not conserve the Kozai integral and cause the group of curves
%shown here to move upward. Other precessional effects (e.g., GR
%precession) reduce the amplitude of variations in eccentricity.}
%\label{fig:eggleton_traj} 
%\end{figure*}


\section{Discussion}
\label{sec:discussion}

\subsection{$Q$ Value}
\label{subsec:Qvalues}

A time-dependent tide raised on a celestial body, due to either its
asynchronous rotation (asynchronous tide) or its eccentric movement
(eccentricity tide), can be dissipated, leading to synchronous
spin or orbital circularization. The rate of dissipation is
conventionally described by a dimensionless quality factor $Q$, 
the ratio between the energy in the tidal bulge and the energy
dissipated per orbital period. For equilibrium tides, $Q$ is related to the
lag-angle between the tide-inducing object and the tidal bulge
$\delta$ by $\delta \approx 1/2Q$.

It is possible to adapt the $Q$ description to dynamical tides, in
which case the tidal energy refers to the energy in the (non-existent)
equilibrium tidal bulge, and $Q$ is obtained from the dissipation
averaged over a range of tidal frequencies. Back-reaction to the
orbits by the gravitational moments of the tidal waves is ignored, a
procedure strictly justifiable only when the damping time of the waves
is shorter than the orbital period. However, secular effects of the
tidal dissipation should be independent of this back-reaction.
%wyq; is the above getting a bit clearer?

Jupiter's $Q$ value has been estimated to be $10^5 \la Q \la 2\times
10^6$, with the actual value believed to be closer to the lower limit
(Goldreich \& Soter \cite{goldreich}, Peale \& Greenberg
\cite{peale}), based on the resonant configuration of the Galilean
satellites. The physical origin of this $Q$ value has remained elusive
for a few decades. As is explicitly shown in Wu (\cite{wu}),
exo-planets share similar $Q$ values ($Q_p \sim 3\times 10^5$) as
Jupiter.\footnote{This statement is independent of the $Q_\star$ value
of the host stars. The stars are typically slowly rotating so their
tidal dissipation have the effects of both circularizing and eroding
the planet orbit. This would fail to produce the observed circular
orbits, if it is more important than dissipation in the planets.}
% wyq: changed the above footnote slightly
This is striking as the exo-solar planets have different thermal
environments, formation histories, and possibly interior compositions,
than those of Jupiter. The similar $Q$ values provide an important
clue as to the underlying dissipation mechanism.  Moreover, while the
$Q$ factor for Jupiter pertains to the asynchronous tide, the one for
exo-planets concerns the eccentricity tide.  One may speculate that
the quality factors for the different Jovian-mass objects agree
because all the objects rotate fast with periods comparable to or
shorter than the tidal period.
% ($5$ hours for Jupiter, 3-5 days for the
%exo-solar planets).
%Common traits shared by these planets are that they are fully convective, and
%that they likely spin fast with periods comparable to the tidal
%periods. Close-in ($a < 0.1$ AU) planets should be tidally
%synchronized. How these relate to the dissipation is unclear.

Tidal dissipation in the host star of HD $80606$b also contributes to
circularization.  What is the tidal $Q$ factor appropriate for
solar-type stars? Field solar-type binaries are observed to be
circularized out to $\sim 12$ day orbits (Duquennoy
\& Mayor \cite{duqu}). This would imply $Q \sim 10^5$ adopting an age of $5$
Gyrs.  These binary stars have presumably been synchronized, so this
$Q$ value is affiliated with dissipation of the eccentricity tide in
rapidly spinning objects ($Q_{\rm fast}$). For the evolution of planet
orbits, the $Q$ value of relevance is that for the asynchronous tide
in slowly rotating objects to which most host stars belong ($Q_{\rm
slow}$). It is not clear that $Q_{\rm fast}$ is directly related to
$Q_{\rm slow}$.  What observational constraints could we put on
$Q_{\rm slow}$?

Short period planets can tidally spin up the star, at the cost of
planetary orbital decay. Assuming all host stars are Sun-like and all
exo-planets are Jupiter-like, we obtain a lower limit of $Q_{\rm slow}
> 10^5$ if all planets are to survive in their current orbit for
another $5$ Gyrs.
%for planet host stars based on the fact that exo-planets with orbits as
%close as $3$ days still survive today.
% this is to assume they have Jupiter mass, star is like the Sun
% and they will not crash into the star within 5 Gyrs... but of course
% it is statistics...

% for a few similarly close planets, the most massive host stars are:
% Norm's fitting gives:  tau-Boo 1.4 M_odot, Ups And 1.3 M_odot (HD 9826)
%			and HD 75289 1.3 M_odot, 
%		the latter two do not rotate fast, but their Mp sini 
%		are also 10 times smaller......
Drake \etal (\cite{drake}) considered the rotational velocities for
three planet-bearing stars ($\tau$-Boo, $\upsilon$-And, and 51
Peg). Among them, $\tau$-Boo has the closest and most massive
planet and appears to be synchronized,
%($M_p \approx 7 M_J$, $M_\star = 1.2 M_\odot$ 
%and $R_\star = 1.7 R_\odot$, age 2 Gyrs), 
while the other two have not. Taken at face value, this would imply
$Q_{\rm slow} \approx 10^4$. This contradicts our earlier statement
that $Q_{\rm slow} > 10^5$. These two results can be reconciled either
by the imminent death of the shortest period planets,\footnote{If this
possibility turns out true, tidal dissipation in stars will be held
accountable for the observed inner cutoff in the orbits of
exo-planets.} or by the possibility that $\tau$-Boo has had only its
surface convective layer synchronized with the orbit (see Goldreich \&
Nicholson \cite{goldreich2} for a similar argument on massive stars).
The latter possibility also makes it difficult to infer $Q_{\rm slow}$
using the spin data of solar-type binaries. It is interesting to
notice that among the three stars discussed by Drake \etal
(\cite{drake}), $\tau$-Boo likely has the thinnest convective
envelope.
% is this statement correct???????????????????????/
% 51 Peg is 1 M_sun and 10 Gyrs old
% tau Boo is 1.2 M_sun and 2 Gyrs old
% eps And is 1.2 M_sun and 5 Gyrs old -- ??
%Many related questions arise, e.g., the efficacy of stellar spin-down by
%magnetic braking, the coupling between stellar convective outer
%envelope and radiative interior, etc.

Unable to infer hard constraints on $Q_{\rm slow}$, we decide to adopt
$Q_\star = Q_{\rm slow} = 10^6$ in our study. With this choice,
dissipation in the planet is roughly $10$ times more important than
dissipation in the star even for a planet as massive as $M_p = 7.8
M_J$. This choice has the advantage that unless the actual $Q_\star$
is much lower, the results presented here are not qualitatively
affected.

\subsection{Constraints on possible Second Planets}
\label{subsec:secondplanet}

If there existed a second planet in the system, secular
interaction\footnote{We ignore interactions of the mean-motion type as
they are important only for a small phase space.}  between this planet
and HD $80606$b may destroy the Kozai oscillations. Kozai oscillations
require that the precession caused by a second planet acting on HD
$80606$b's orbit be no larger than that produced by the companion star
(Innanen \etal, \cite{innanen}).  We briefly investigate what
constraints this puts on the mass and orbit of the second planet.

We assume the two planets have coplanar orbits and the second planet
(denoted with subscript $s$) lies inward of the orbit of HD
$80606$b. The derivation is similar if it lies outward.
%wyq: removed the following footnote, no longer using Delta_s
%\footnote{When the second planet's orbit falls outside of HD
%$80606b$, the derivation is similar except that $\Delta_s$ is
%negligible.} 
Secular interaction between the two planets alone gives rise to
variations in their eccentricities ($e$) and periapse angles
($\omega$). Adopting a complex variable $I = e\, \exp(i \omega t)$,
and assuming that $e \ll 1$, the first-order secular evolution
equations read (Murray \& Dermott \cite{MD2000}, hereafter MD2000),
\begin{equation}
{{dI_s}\over{dt}} = i A_{ss} I_s + i A_{sp} I_p, \hskip0.2cm
{{dI_p}\over{dt}} = i A_{ps} I_s + i A_{pp} I_p,
\label{eq:linearI}
\end{equation}
where the real coefficients $A_{ij}$ are
\begin{eqnarray}
A_{ss} & = & {{\alpha^2}\over{4}} {{M_p}\over{M_\star}} n_s B_1 ,
%+ \Delta_s,
\hskip0.2cm 
A_{sp}  =  - {{\alpha^2}\over{4}} {{M_p}\over{M_\star}} n_s B_2,\nonumber \\
A_{ps} & = & - {{\alpha }\over{4}} {{M_s}\over{M_\star}} n_p B_2, \hskip0.84cm 
A_{pp}  =   {{\alpha }\over{4}} {{M_s}\over{M_\star}} n_p B_1.
\label{eq:defineAjk}
\end{eqnarray}
Here, $n_s = \sqrt{G(M_\star + M_s)/a_s^3}$, $\alpha = a_s/a_p$, and
$B_1 = b_{3/2}^{(1)}(\alpha)$, $B_2 = b_{3/2}^{(2)}(\alpha)$ with
$b_i^{(j)}$ being the usual Laplace coefficients (MD2000).
%The term $\Delta_s$ is the sum of the
%following three precession rates on the second planet (those on HD
%$80606$b are neglected)
%\begin{equation}
%\Delta_s = {{d\omega_s}\over{dt}}\Bigr|_{\rm tide} +
%{{d\omega_s}\over{dt}}\Bigr|_{\rm rot} +
%{{d\omega_s}\over{dt}}\Bigr|_{\rm GR}.
%\label{eq:defineDelta}
%\end{equation}
%They are expressed in Equations \refnew{eq:preGR} -
%\refnew{eq:pretide}. 
The general solution to eq. \refnew{eq:linearI} is composed of two
linear eigenvectors with their respective precession rates
\begin{equation}
g_{\pm} = {{(A_{ss}+A_{pp}) \pm \sqrt{(A_{ss}-A_{pp})^2 + 4
A_{sp} A_{ps}}}\over 2}.
\label{eq:solutiong}
\end{equation}
We are able to approximate the precession rate of HD $80606$b under
secular interaction as,
%Precession of the dominating component of HD $80606$b can be
%approximately expressed as
%For HD $80606b$, it precesses under secular perturbations from the
%second planet with a rate of 
\begin{eqnarray}
{{d\omega_p}\over{dt}}\Bigr|_{\rm sec} & = & 
{\rm Min}(g_+, g_-)\hskip0.5cm {\rm{if\,\, M_p > M_s}}, \nonumber \\
& & {\rm Max}(g_+, g_-)\hskip0.5cm {\rm{if\,\, M_p < M_s}}.
\label{eq:secularrate}
\end{eqnarray}

%wyq: changed the wording of the following paragraph
At birth, HD $80606$b could not have undergone Kozai oscillation if
the second planet is massive enough.  Requiring the above secular
precession rate to be smaller than the Kozai precession rate (see
eq. [\ref{eq:precession}]) when $a_p = 5$ AU, we find that the mass of
the second planet is constrained to a value that depends on its
distance to HD $80606$b. This is depicted in
Fig. \ref{fig:gaussian_2ndplanet}.

These results also strongly constrain the current state of the
planetary system. No other Jupiter-mass planet could now live between
about $0.05$ and $100$ AU.  Even earth-mass cores are excluded between
$1$ and $20$ AU. Did such objects never form?  Or were they cleared
away? We do not explore these issues here.
%We note that if the initial $e\approx0.5$ for HD $80606$b, as a result
%of planet-disk interactions for example, then clearing of Earth mass
%or smaller bodies is likely; chaotic perturbations from the massive
%planet will be very effective at removing smaller bodies.

In any case, Kozai migration is incompatible with the notion that the
claimed residual velocity in HD$80606$ (NLM) is due to a second planet
in the system.

%The above analysis applies when both planets have low eccentricities
% is this really necessary?? and have mass small compared to the host star. 
%We adopt a $a_p = 4$ AU for HD$80606b$,\footnote{This is the smallest
%orbit where Kozai is expected to be operating and is therefore the
%least demanding constraint we could set} and values as in Table
%\ref{table:symbol}, as well as assume that the spin-rate of the second
%planet is equal to its mean motion, with a tidal Love number $k_2
%\approx 0.5$ and a radius like Jupiter $R_J$. Requiring that equation
%\refnew{eq:secularrate} to be smaller than the precession rate caused
%by the Kozai effect of the stellar companion (see \S
%\ref{subsec:heuristic?})  we obtain constraints on the mass and orbit
%of the second planet. These are shown in
%Fig. \ref{fig:gaussian_2ndplanet}, onto which two other constraints
%are imposed as well.

\begin{figure*}[t]
%\begin{figure}
%\epsscale{0.50}
%\plotone{f2.eps}
\centerline{\psfig{figure=gaussian_2ndplanet.ps,width=0.50\hsize}}
\caption{The constraints on the mass (vertical axis, in unit
of $M_J$) and orbital semi-major axis (horizontal axis, in AU) of the
possible second planet placed by the Kozai migration of HD
$80606$b. The shaded region must be cleared before HD $80606$b
(assumed to be at $5$ AU) can undergo Kozai oscillations. The latter
force HD $80606$b into a variable but highly eccentric orbit with the
minimum periastron and maximum apastron delineated by the two dotted
vertical lines (see Fig. \ref{fig:eggleton_short}).  No planets should
have survived within this region. Planets lying above the slanted
dashed line would incur a radial velocity larger than $20 \m/s$ on the
host star (taking $\sin i=0.5$) and should have been detected were
they present.
%wyq: removed the following
%These results hardly vary when we include GR and tidal precessions on
%the planet orbits.  
}
\label{fig:gaussian_2ndplanet} 
%\end{figure}
\end{figure*}


\subsection{Can the Tide Raised by the Planet Account for the Residual Velocities?}
\label{subsec:tidalheight}

NLM reported residual velocities of order $\sim 40 \m/\s$ in the
spectra of HD $80606$. These are higher than the expected noise and
there is no clear trend in time. Could they be caused by anything
other than an unknown planet? We consider one option: the tidal
velocities induced by HD $80606$b on the surface of the star. The
stellar rotation is ignored in the following analysis.
%Velocity amplitude of the binary companion is of order $0.9 \km/s$.

First, we assume that the star adjusts its hydrostatic equilibrium
instantaneously according to the tidal potential of the planet at a
radial separation $D = D(t)$ (the equilibrium tide picture), and
estimate the corresponding tidal velocities. The maximum radial
displacement at the stellar surface is roughly
\begin{equation}
\xi_{\rm eq} \approx {{M_p}\over {M_\star}} \left({{R_\star}\over D}\right)^3 R_\star,
% I have ignored the potential of the tidal bulge itself, which according
% to Murray & Dermott (eq. 4.88 and around) only produces a factor of
% order unity...
\end{equation}
with the horizontal displacement being of a similar order.  Velocities
associated with these displacements are simply estimated as $v_{\rm
equi} = {{d\xi_{\rm eq}}\over{dt}}$, yielding a maximum line-of-sight
velocity $\sim 60 {\rm cm/s}$ occurring within $10$ hours of the
%%% I changed 60/\sin i to 60, \sin i should cancel out 
% what if the planet is actually a star? 
% is the tidal velocity much larger in amplitude?
% no, \sin i effect cancels out  so it doesn't matter
closest-encounter. This maximum requires conjunctions to coincide
roughly with the line of periapse, as is the case in HD $80606$.

We turn next to consider a more realistic description of the tidal
amplitudes.  The stellar response toward the time-varying potential
is decomposed into a series of eigenmode oscillations, with the mode
amplitudes maximized when the tidal forcing frequencies are in
resonance with the mode frequencies (the dynamical tide picture). If
the planetary orbit is nearly circular, the dominant forcing frequency
is $\sim 2 n_p$.
%where $m$ is the azimuthal number with $|m|
%\geq 2$. We consider the lowest order forcing $\ell = |m| = 2$. 
At an orbital period of $111$ days, this forcing resonantly excites
gravity-modes of radial order $\sim 1000$. The spatial overlap between
the tidal potential and such a high-order mode is too weak to be
interesting. However, HD $80606b$ has a highly eccentric orbit. Tidal
excitation occurs primarily near periastron and the dominant forcing
frequencies is about twice the orbital frequency near periastron, and
is related to $n_p$ as $\sim 2 n_p/ (1-e_p)^{3/2}
\sim 100 n_p$.  Gravity-modes of radial orders $\sim 30 - 80$ become
relevant and their resonant excitation give rise to much larger tidal
velocities.  A brief treatment of the dynamical tides is given in the
appendix (\S \ref{sec:dyntide}). Here we summarize our results.

\begin{figure*}[t]
\centerline{\psfig{figure=dynamic_tide.ps,width=0.50\hsize}}
%\begin{figure}
%\epsscale{0.50}
%\plotone{f3.eps}
%wyq: reworded the caption
\caption{Dynamic response of a Sun-like star  under tidal forcing from a 
planet of mass $3.9/\sin i M_J$, an orbit of period $111.82$ days and
eccentricity $0.927$. We only consider the dominant $\ell = |m| = 2$
component in the tidal forcing.  The upper panel plots mode amplitude
(in terms of surface horizontal velocity) as a function of mode period
in days. Usually one or two modes dominate the response, and in this
case it is a $n= 41$ gravity-mode. The dotted and dashed lines are the
frequency spectrum of tidal forcing over one period ($K^1_{\ell m}$,
see appendix), and the overlap integral between the tidal potential
and the eigenmode ($Q_{n\ell}$), both in arbitrary units. The forcing
spectrum peaks around $2 (1-e_p)^{-3/2} n_p$ and the overlap integral
decreases sharply when mode period (and mode radial order) increases,
roughly as $Q_{n \ell} \propto n^{-5}$. Besides from these two
functions, the tidal response also depends on the goodness of the
resonance and the mode damping rate, these are plotted in the lower
panel.  The frequency detuning $\delta \sigma = |\sigma_n -
\sigma_j|$. Here $\sigma_n$ is the frequency of a radial order $n$ mode, 
and $\sigma_j$ is the closest integer multiple of the orbital mean
motion ($\sigma_j = j n_p$ and $j$ is an even number).  The damping
rate $\gamma_n$ is estimated from radiative diffusion.  The two rows
of short vertical bars illustrate the positions of the $\sigma_j$
(upper row) and $\sigma_n$ (lower).}
% and the fractional Lagrangian pressure perturbation at the stellar surface
%(roughly also the brightness variation), 
\label{fig:dynamic_tide} 
\end{figure*}
%\end{figure}

We adopt the standard solar model (Christensen-Dalsgaard
\cite{jorgen}) to calculate the eigenmode frequencies and damping rates.
The response of the star depends sensitively on how close an orbital
multiple lies around one such eigenmodes. The maximum response occurs
when the frequency off-resonance ($\delta \sigma$) is of order or
smaller than the mode damping rate ($\gamma$), giving rise to
velocities of order $10$ to $100 \m/\s$. However, this is a rare
event. In the frequency range of interest ($150$ to $550 n_p$), there
are roughly $N_{\rm peak} \sim 200$ discreet forcing peaks, and
$N_{\rm mode} \sim 50$ eigenmodes. These modes typically have
line-widths calculated from radiative diffusion to be
$\gamma_n/\sigma_n \sim 10^{-7}$. A typical best resonance
$\delta\sigma = {\rm Min}(|\sigma_n - \sigma_j|)$ is of order $\sim
n_p/N_{\rm mode} \sim \sigma_n/10^4 \sim 10^3 \gamma_n$. So the chance
of one mode lying close to an orbital multiple with $\sigma_n
\leq \gamma_n$ is roughly $10^{-3}$. Typical tidal velocities therefore
range from $1\cm/s$ to $10 \cm/s$, well below the level of the
observed residues. One such example is shown in
Fig. \ref{fig:dynamic_tide}. The highest amplitude mode has a
fractional Lagrangian pressure perturbation $\delta p/p \sim 4\times
10^{-7}$ near the surface. Translating this into light variation, this
mode will be as hard to detect as the solar oscillation.

To showcase the sensitivity of the tidal responses, we plot in
Fig. \ref{fig:probable_tide} the marked variations in the tidal
responses when the orbital period is tuned through the observed
uncertainty range: $111.81 \pm 0.23$ days (NLM). Slight differences in
the stellar structure affect the eigenmode frequencies and would also
give rise to similar variations in the tidal responses. So the
response results are best appreciated as a probability distribution of
amplitudes, also shown in Fig. \ref{fig:probable_tide}. We find that
there is only a $\sim 10^{-3}$ chance that the tidal velocity reaches
the claimed level of residues, $\sim 40\m/\s$.

\begin{figure*}[t]
%\begin{figure}
%\epsscale{0.50}
%\plotone{f4.eps}
\centerline{\psfig{figure=probable_tide.ps,width=0.50\hsize}}
\caption{The probability distribution of the tidal response. 
The upper panel plots the horizontal velocity on the stellar surface
as a function of orbital period, with the period running through the
observational uncertainty range. Sharp spikes occur when an eigenmode
is resonantly forced ($\delta \sigma < \gamma_n$).  Other parameters
are the same as in Fig. \ref{fig:dynamic_tide}. From these data, one
finds a probability distribution for the tidal velocity to surpass a
certain value. This is depicted in the lower panel.  Typical amplitude
(probability of $0.5$) is $\sim 20 \cm/s$, with the claimed level of
residue, $\sim 40\m/\s$, occurring at a probability of $\sim 10^{-3}$.}
\label{fig:probable_tide} 
\end{figure*}
%\end{figure}

We calculate the damping rates taking only radiative diffusion into
account. Turbulent viscosity may enhance this damping. But this will
not appreciably change our conclusion: raising the damping rates by a
factor of $10$, we obtain similar results as in
Fig. \ref{fig:probable_tide} except that high tidal amplitudes are
less likely to occur.
% whether damping reduces by a factor of 10 does not seem to matter
% except a bit of the statistics at high amplitudes
% damping higher ==> smaller probability to achieve high amplitude
% damping lower ==> larger probability to achieve high amplitude

On the other hand, when we substitute the solar model for a $1.25
M_\odot$ ZAMS model (courtesy of Christensen-Dalsgaard), we found that
the typical tidal amplitudes rise by about a factor of
$10$. Equivalently, high velocity amplitudes appear roughly $10$ times
more frequent. Compared to a solar model, the $1.25 M_\odot$ model has
a thinner convection zone. Its gravity-modes have a relatively larger
surface amplitude.
%I have done the same exercise  for HD162020, 13.73 M_J, e=0.28 and p=8.30 day
% assuming 1 M_sun
% the maximum tidal amplitudes are roughly 1 m/s, about 100 times below 
% HD 80606, typical amplitude (probability=50%) is 0.03 m/s
% the probability to achieve a certain amplitude rises with stellar mass
% this is because normalized surface amplitude is much larger, by a factor
% of 10 or so

In summary, thanks to the high eccentricity of HD $80606$b, the tidal
velocities reach tantalizingly close to the current detection
thresholds, but likely fail to explain any residual velocity of order
$10 \m/\s$. However, if the residual velocity is indeed associated
with the dynamical tide, we expect it to be a coherent oscillation
throughout the orbit and have a period in the range of $0.2$ to $0.6$
days. The inclination of the orbit ($\sin i$) does not affect our
conclusions.

\subsection{Other Planetary Systems}
\label{subsec:othersystem}

\begin{figure*}[t]
%\begin{figure}
%\epsscale{0.50}
%\plotone{f5.eps}
\centerline{\psfig{figure=mass_period.ps,width=0.50\hsize}}
\caption{Period versus projected mass for all observed planet 
candidates (http://cfa-www.harvard.edu/planets/, Sep. 2002). Circles
are drawn around planets in visual binaries (Worley \& Douglass
\cite{worley}) with a projected separation $< 20\arcsec$. Besides from
the 9 binaries mentioned in Zucker \& Mazeh (\cite{zucker}, HD 142, Gl
86, HD 19994, $\epsilon$ Eri, $\tau$ Boo, HD 178911B, 16 Cyg B, HD
195019, HD 217107), there are also HD 80606, HD 213240, HD 121504, HD
46375, HD 27442, BD-103166, and $\gamma$-Cepheid.
% wyq: added gamma-cepheid
%
% Msini	       period   a(AU)    e    binarity   name
% 4.5         951      2.03     0.45   	1       HD 213240       
%		mag 6.80/12.00, sep. 19.8'' --> 17.4'', confirmed binary
% 3.41        111.78   0.439    0.927  	1       HD 80606        
%		mag 9.40/9.40, sep. 21.1''
% 0.89        64.6     0.32     0.13   	1       HD 121504       
%		mag 7.60/9.40, sep. 11.5''--> 27.2, confirmed binarity
% 1.43        423      1.18     0.02   	1       HD 27442        
%		mag 4.44/12.50, sep. 13.7''
% 0.249       3.024    0.041    0.     	1       HD 46375 
%		mag 9.0/11.0, sep. 9.4''
% 0.48        3.487    0.046    0.     	1       BD -10 3166 
%		mag 10.5/14.5, sep. 17.0'', exo-planet.org give V=10.0, so the right one
%
%The object with starry rays is HD $68988$b which Vogt \etal
%(\cite{vogt}) claimed to have a possible second object
% 1.90        6.276    0.071    0.14   	0.5     HD 68988 possibly with 2nd object,Vogt etal2002
Planets lying below the dashed line incur velocity variation higher
than $20\m/\s$. 
%wyq: removed the following sentence
% while those falling outside the plotting box have
%minimum mass approaching the current planet/brown dwarf boundary.
%
%wyq: added the following statement
There appears to be an over-abundance of close-in, massive planets
that are also in binaries. 
%
}
\label{fig:mass_period} 
%\end{figure}
\end{figure*}

Zucker \& Mazeh (\cite{zucker}) showed that planets in binaries are
statistically different from those around isolated stars in their
period-mass distribution. The former population dominates the
lower-right quadrant of Fig. \ref{fig:mass_period}.  This distinction
suggests that planet migration in binaries may be induced differently
from those around single stars, Kozai migration being one such
possibility. Among the circled planets in Fig. \ref{fig:mass_period},
we have only studied $\tau$-Boo and 16 CygB. We found that $\tau$-Boo,
the most massive close-in planet, could have gone through Kozai
migration and later circularized, while there is no compelling
evidence of Kozai migration for 16 CygB, though it could currently be
undergoing Kozai oscillations (Holman \etal (\cite{holman}). A binary
companion may assist planet migration through other means, e.g.,
exciting tidal waves in the proto-planetary disk and affecting angular
momentum transport. These deserve further study.

%wyq2: added the following paragraph following the referes's point 6
Could some of the planets around currently single stars have undergone
Kozai migration followed by the subsequent disruption of the binary? 
We consider this unlikely. The rate of Kozai migration is limited by
the tidal circularization process. For reasonable values of closest
periapse distance, the latter process takes a few $10^8$ years or
longer. By this age a typical stellar cluster has dispersed and the
%wyq2: is this true? open cluster unbound due to galactic tidal field in 10^9 yrs
%	
rate of stellar encounter has become negligible. However, it is
possible that a planet's eccentricity was excited by Kozai interaction
with the binary companion before the latter was removed from the
system. A simple estimate, however, finds that this may account for
the observed eccentricity in at most $\sim 10\%$ of the known systems.
%Supposing that the planet formation process is independent of the
%stellar binarity, and that twice as many stars were formed in binaries
%as there are in the main-sequence phase, and that the typical
%dissociation timescale is $\sim 10^7$yrs, and that only ones with
%binary plane inclined related to the planet plane $> 39.2 \deg$ can
%	(and that inclination are randomly oriented)
%effectively produce planet eccentricity, and that planets typically
%spend half their time in substantially eccentric state, we project
%that among the currently discovered planets around single stars, $\sim
%10\%$ could have had eccentricity excitation due to Kozai cycles.
%wyq2: end of changes

%wyq: removed the following paragraph, it's very vague and doesn't help
%
%The mass-period distribution of the single-star planets exhibits a
%lower bound in period that rises with mass.  If this is confirmed by a
%larger sample, it may point to an upper limit in the energy and
%angular momentum that the causal agent in planet migration can absorb.

% or lighter planet can in general be migrated further inward. 
% don't want to deliberate on this anymore, because the difference in mass
% is only a factor of 10 and it is hard to see why the relation should
% be so tight, if it exists...

% -- assuming starting at infinity, $\Delta E = G M_* M_p/a =
%10 GM_* M_J/AU $, assuming a $\Sigma \propto r^{-1/2}$ disk, and
%extending from $10 R_\odot$ to $10 AU$, we find this corresponds to a
%disk mass of $??$, roughly $??$ of the minimum solar nebula...
%% roughly fitted by (m_p/M_J)**1.5/30 \sim a/AU, or m_p/M_J/10 \sim a/AU
%Migration probably late in disk life-time -- otherwise why should
%there be a maximum dissipation a disk can receive? why shouldn't it be
%related to local environment (disk mass, metallicity, stellar
%coronae?)...


%perhaps there is an evaportaion? (mass-radius relation gives large
%radius and then Roche Lobe limits the size --> disruption, or pressure
%scale height too large at the surface to cause boiling). 
%---> checked: not important, M_J, R_J, T= T_{eq}, at 20 R_sun
%	get c_s/v_esc= 1/20, too small to be relevant

\section{Summary}
\label{sec:summary}

It is striking that a companion star as remote as $1000$ AU can force
a planet to migrate inward. Kozai migration is a long timescale
process and demands a 'clean' planetary system 
%wyq2: explained what 'clean' means
in which no other planetary object exists.
%wyq2: end of changes
Kozai migration can be responsible for the
present orbit of HD $80606$b only if the initial relative inclination
is sufficiently high ($I \geq 85 \deg$). Assuming
%wyq: reworded a bit the following content
a population of similar triple systems with randomly distributed $I$,
$\sim 7\%$ of them could have gone through Kozai migration.  So on the
strength of one object (HD $80606$b), one may speculate about the high
density of triple stellar-planetary systems yet undetected. 
%wyq: removed the following sentence, 
%	it is vague, and not discussed in the text much
%	the trend is not easy to estimate... minimum I is not affected.
%Moreover, the chance of Kozai migration diminishes with increasing binary
%separation.

%  astro-ph/0204272 From: Ray Jayawardhana <rayjay@astron.berkeley.edu>
%Title: Discovery of an Edge-On Disk in the MBM 12 Young Association
%Jayawardhana \etal (\cite{jaya}) showed a protoplanetary disk that
%doesn't seem to be aligned with the star plane.... neither coplanar
%nor parallel, it also mentions that there are other protodisks in
%multiple systems

Our prediction regarding the lack of a second planet in the system
(Fig. \ref{fig:gaussian_2ndplanet}) should be tested. In this regard,
we note the claim of the abnormally large residue velocities on the
star.  We investigated the possible association between these
velocities and surface tidal waves excited by the highly eccentric
planet. It seems unlikely that the latter can account for the former
if the star has a structure like that of our Sun. And if the system is
indeed devoid of second planet, it raises issues as to how it is
cleaned.

Zucker \& Mazeh (\cite{zucker}) pointed out that planets in binary
systems and planets around single stars do not share the same
period-mass distribution. The implications might be that binaries can
cause planet migration -- Kozai migration being one of the more
definite possibilities.

HD $80606$b's highly elongated orbit and strong tidal interaction
combine to make it a possible target for direct detection. Its current
tidal luminosity (which is likely to be converted into heat and
radiated all through the orbit) is $\sim 10^{28} \erg/\s$, or $\sim
14$ mag dimmer than the parent star, at a maximum angular separation
of $0.01\arcsec$ (assuming distance of $80$ pc). A comparable
contribution may arise if the planet evenly emits the stellar
insolation it receives during periastron passages.


\begin{acknowledgements}
Part of the research was performed while Y.W. was on a CITA-ITP
postdoctoral exchange programme funded by NSF grant PHY 99-07949.
Y.W. would like to thank Peter Eggleton for his patience, Ludmila
Kiseleva-Eggleton, Omer Blaes and Man-Hoi Lee for useful
conversations. N.M.'s research is supported by NSERC of Canada and by
the Canada Research Chair program.  This research benefited from ADS
and SIMBAD.
\end{acknowledgements}


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\begin{appendix}

\section{Equations from EKE}
\label{sec:EKEeqn}
	
We adopt the set of equations (eq. [11]-[17]) in EKE (also see
Eggleton, Kiseleva \& Hut \cite{EKH}) to describe the secular
evolution in the planet orbit, the spin of the host star and the
planet. The companion is assumed to be unaffected. Physical effects
described by these equations include: secular interactions, GR, tidal
and rotational precessions, and tidal dissipation. These equations are
explicitly listed here,
% I changed $\omega_p$ to $n_p$, but $\omega$ to $\omega_p$
\begin{eqnarray}
{1\over{e_p}}{{de_p}\over{dt}} & = & - V_\star - V_p - 5 (1-e_p^2)
S_{eq}, \label{eq:ep} \\ 
{1\over{a_p}}{{da_p}\over{dt}} & = & - 2 W_\star -
2 W_p - {{2 e_p^2}\over{1-e_p^2}} (V_\star + V_p), \label{eq:ap} \\
{{dI}\over{dt}} & = & - \sin \omega_p \left[Y_\star + Y_p + (1-e_p^2)
S_{qh}\right] + \cos\omega_p \left[ X_\star + X_p + (4 e_p^2 +1)
S_{eh}\right], \label{eq:i} \\
{{d\omega_p}\over{dt}} & = & \left[Z_\star+ Z_p + Z_{\rm GR} + (1-e_p^2)
(4 S_{ee} - S_{qq})\right] - \cos \omega_p {{\cos I}\over{\sin I}}
\left[ Y_\star + Y_p + (1-e_p^2) S_{qh}\right] \nonumber \\
& &  - 
\sin \omega_p {{\cos I}\over{\sin I}} \left[X_\star + X_p + 
(4 e_p^2 +1) S_{eh}\right], \label{eq:smallomega} \\
{{d\Omega}\over{dt}} & = & {{\cos\omega_p}\over{\sin I}}\left[ Y_\star +
Y_p + (1-e_p^2) S_{qh}\right] + {{\sin \omega_p}\over{\sin I}} \left[
X_\star + X_p + (4 e_p^2 + 1) S_{eh}\right], \label{eq:bigomega} \\
{{d \Omega_{* \gamma}}\over{dt}} & = & {{\mu h_p}\over{I_\star}} ( -
Y_\star e_\gamma + X_\star q_\gamma + W_\star h_\gamma), \label{eq:omegasi} \\
{{d \Omega_{p \gamma}}\over{dt}} & = & {{\mu h_p}\over{I_p}} ( - Y_p e_\gamma +
X_p q_\gamma + W_p h_\gamma). \label{eq:omegapi}
\end{eqnarray}
In eq. \refnew{eq:omegasi} - \refnew{eq:omegapi}, the subscript
$\gamma$ runs through the three spatial indexes, $E$, $Q$, and $H$,
with $\hat E$, $\hat Q$, $\hat H$ being the three axis defining the
orbital plane of the binary companion, and $\hat e$, $\hat q$ and
$\hat h$ defining that of the planet orbit, and $e_E = \hat e \cdot
\hat E$, etc. The inclination $I$ is taken to be that between the two
planes, with the binary plane held constant during the evolution. The
other two angles, $\omega_p$ and $\Omega$, are the argument of the
planet pericentre, and the longitude of the planet ascending node,
respectively.  The spin vector $\Omega_\star$ ($\Omega_p$) belongs to
the host star (planet), and $I_\star = r_{g\star} M_\star R_\star^2$
is the stellar moment of inertia, with $r_{g\star}$ typically called
the gyro-radius. $I_p$ is for the planet.  The planet orbit has an
angular momentum of magnitude $h_p = [G (M_\star + M_p) a_p
(1-e_p^2)]^{1/2}$, and the reduced mass of the planet-star system is
$\mu = M_\star M_p/(M_\star +M_p)$.  All other symbols are defined in
EKE. We do not repeat these definitions here.

We introduce two commonly used dimensionless numbers, $k_{2}$ (the
tidal Love number) and $Q$ (the tidal quality factor), and relate them
to the notation in EKE,
\begin{eqnarray}
A_\star = k_{2\star} R_\star^5, &  & A_p  =  k_{2p} R_p^5, \label{eq:defA}\\
{1\over{t_{F\star}}}  = {{3 k_{2\star}}\over{Q_\star}}
\left({{R_\star}\over{a_p}}\right)^5 {{M_p}\over{M_\star}}\, n_p, & & 
{1\over{t_{Fp}}}  =  {{3 k_{2p}}\over{Q_p}}
\left({{R_p}\over{a_p}}\right)^5 {{M_\star}\over{M_p}}\, n_p . \label{eq:deftF}
\end{eqnarray}



\section{Dynamical Tide}
\label{sec:dyntide}

We provide a simple description of the dynamical tide treatment. The
final results are presented in \S \ref{subsec:tidalheight}.

The period of the stellar spin is likely long compared to both the
sound crossing time and the dominant tidal forcing period. So we
assume a non-rotating star.  Following Press \& Teukolsky
(\cite{press}) and Lai (\cite{donglai}) and notations therein, we
decompose the tidal potential at a point ${\bf r} = [r,\theta,\phi]$
inside the star, due to a planet at location ${\bf D}(t) = [D(t),
\pi/2, f(t)]$, as
\begin{equation}
U ({\bf r}, t) = - G M_p \sum_{\ell, m} W_{\ell m}
{{r^{\ell}}\over{D^{\ell+1}}} Y_{\ell m} (\theta,\phi) \exp^{-i m
f(t)}.
\label{eq:tidepotential}
\end{equation}
% $f$ is the free anomaly. 
The stellar fluid responds with a displacement ${\bf \xi}({\bf r},t)$
which can be projected onto various free eigen-oscillations,
\begin{equation}
{\bf \xi}({\bf r}, t) = \sum_\alpha a_\alpha(t) {\bf \xi}_\alpha ({\bf
r}).
\label{eq:atalpha}
\end{equation}
The eigenfunctions are normalized as $\int d^3 x \rho {\bf
\xi}_\alpha \cdot {\bf \xi}_{\alpha}^* = 1$.
We denote the mode eigenfrequency as $\sigma_\alpha$ and damping rate
as $\gamma_\alpha$, and introduce ${\tilde a}_\alpha (\sigma)$ which
is related to $a_\alpha (t)$ by
\begin{equation}	
\hskip1.0cm a_\alpha (t) = \int_0^{\infty} d\sigma \exp^{-i\sigma
t} {\tilde a}_\alpha (\sigma).
\label{eq:ataa}
\end{equation}
Substituting equation \refnew{eq:atalpha} into the fluid equation of
motion, we obtain
\begin{equation}
{\tilde a}_\alpha(\sigma) = {1\over{\sigma_\alpha^2 - \sigma^2+ 2 i
\gamma_\alpha \sigma}} \sum_\ell {{G M_p}\over{D_p^{\ell+1}}}
Q_{\alpha\ell} K_{\ell m} (\sigma),
\label{eq:aalpha}
\end{equation}
where the periapse distance $D_p = a_p (1-e_p)$, $K_{\ell m}(\sigma)$
is the frequency spectrum for the $Y_{\ell m}$ component of the tidal
potential, and $Q_{\alpha \ell}$ is the spatial overlap integral
between the tidal forcing and the $\alpha$ eigenfunction,
\begin{eqnarray}
K_{\ell m}(\sigma) & = & {{W_{\ell m}}\over{2
\pi}} \int^{\infty}_{0} dt \left[{{D_p}\over{D(t)}}\right]^{\ell +1}
\exp[- im f(t) + i\sigma t],\label{eq:defineK} \\
Q_{\alpha \ell} & = & \int d^3 x\,\, \rho {\bf \xi}_\alpha^* \cdot
\nabla(r^\ell Y_{\ell m}) = \int d^3 x\,\, \rho_\alpha^{\prime *}\, r^\ell
Y_{\ell m}.
\label{eq:defineQ}
\end{eqnarray}
Here, $\rho_\alpha^\prime$ is the Eulerian density perturbation for
mode $\alpha$. The continuity equation gives $\nabla\cdot(\rho
\xi_\alpha) = - \rho_\alpha^\prime$.

We first focus on $K_{\ell m}$. For a strictly periodic orbit,
$K_{\ell m}(\sigma)$ equals a sum of delta functions with non-zero
values only at the integer multiples ($j$) of the orbital mean
motion. 
\begin{equation}
K_{\ell m}(\sigma) = \sum_{j=0}^\infty C_j(\sigma)\, \delta(\sigma - j
n_p)
\label{eq:Klminf}.
\end{equation}
To obtain $C_j$, we calculate the tidal spectrum integrated over one
orbital period,
\begin{equation}
K^1_{\ell m}(\sigma) = {{W_{\ell m}}\over{2 \pi}}
\int_0^{2\pi/n_p} dt \left[{{D_p}\over{D(t)}}\right]^{\ell +1}
\exp[-im f(t) + i\sigma t].
% I again added a negative sign before im f(t)
\label{eq:defineK0b}
\end{equation} 
Unlike $K_{\ell m}$, this quantity is a smooth function of frequency.
For $\ell = |m| = 2$, the forcing maximum lies around $\sigma \sim
(1-e_p)^{-3/2} |m| n_p$, i.e., $\sigma \sim 2 n_p$ for $e_p \ll 1$ and
$\sigma \sim 90 n_p$ for $e_p = 0.927$. This is shown in Figure
\ref{fig:dynamic_tide}. When integration in $K_{\ell m}$
is carried out over $N$ number of orbital cycles, the resulting
forcing spectrum can be approximately described by the following
log-normal distribution,
\begin{equation}
K^N_{\ell m}(\sigma) = \sum_{j=1}^{\infty} N 
\exp\left[-({\sigma\over{n_p}} - j )^2 N^2\right] \, K^1_{\ell m}(\sigma).
\label{eq:lognormal}
\end{equation}
And when $N \rightarrow \infty$, we retrieve $K_{\ell m}$ with
$C_j(\sigma) = \sqrt{\pi} n_p K_{\ell m}^1 (\sigma)$

Substituting the above expression for $K_{\ell m}(\sigma)$ into
equations \refnew{eq:atalpha}- \refnew{eq:aalpha}, we obtain the mode
amplitude,
\begin{equation}
a_\alpha (t) \approx {{G M_p}\over{D_p^{\ell + 1}}} \sqrt{\pi} n_p \,
Q_{\alpha \ell} K^1_{\ell m} (\sigma_j) \int_{(j-{1\over
2})n_p}^{(j+{1\over 2})n_p} d\sigma\, {{\delta(\sigma - j n_p) e^{- i
\sigma t}}\over{\sigma_\alpha^2 - \sigma^2 + 2 i
\gamma_\alpha \sigma}} = {{G M_p}\over{D_p^{\ell + 1}}} \sqrt{\pi} n_p\, Q_{\alpha
\ell}  K^1_{\ell m} (\sigma_j) {{ e^{- i \sigma_j
t}}\over{\sigma_\alpha^2 - \sigma_j^2 + 2 i \gamma_\alpha \sigma_j}},
\label{eq:expressat}
\end{equation}
where $\sigma_j = j n_p$ is the closest orbital multiples to the mode
frequency $\sigma_\alpha$. 

%A few scaling laws prove useful for understanding the results.

The value of the tidal overlap, $Q_{\alpha \ell}$, decays sharply with
both $\ell$ and the radial order ($n$) of the mode. For a high order
gravity-mode, most of the net contribution to $Q_{\alpha\ell}$ comes
from the upper evanescent region of the mode -- the bottom of the
surface convection zone where $\rho^\prime/\rho$ is fairly
constant. We find that $Q_{\alpha\ell}$ scales with $n$ as
\begin{equation}
Q_{\alpha\ell} \propto n^{- 5}.
\end{equation}

%As g-mode frequency $\sigma_\alpha \propto 1/n$, 

%The mode frequency is $\sigma_{n\ell} \approx \omega_0 n/\ell$, 

%The mode damping rate $\gamma_\alpha$ scales

%The frequency detuning $\delta\sigma = |\sigma- j n_p| $ scales as

%The mode that is excited to the largest surface amplitude has the
%following attributes: $\sigma_\alpha$ close to the maximum of forcing
%$2 (1-e_p)^{-3/2} n_p$; small frequency detuning and small damping
%rates, $\delta\sigma \approx \gamma_\alpha$; high spatial overlap with
%the tidal forcing -- both require small $n$.

\end{appendix}

\end{document}

Various short-hands have their definitions here,
\begin{eqnarray}
W_\alpha & = & {1\over{t_{F\alpha}}} \left[ {{1 + {{15}\over 2} e_p^2
+ {{45}\over 8} e_p^4 + {5\over{16}} e_p^6}\over{(1-e_p^2)^{13/2}}} -
{{\Omega_{\alpha h}}\over{n_p}}{{1+ 3 e_p^2 + {3\over 8}
e_p^4}\over{(1-e_p^2)^5}}\right], \label{eq:defW}\\
V_\alpha & = & {9\over{t_{F\alpha}}} \left[{{1+{{15}\over{4}} e_p^2 +
{{15}\over 8} e_p^4 + {5\over{64}} e_p^6}\over{(1-e_p^2)^{13/2}}} -
{{11 \Omega_{\alpha h}}\over{18 n_p}} {{1+{3\over 2} e_p^2 +
{1\over 8} e_p^4}\over{(1-e_p^2)^5}}\right], \label{eq:defV}\\
X_\alpha & = & - {{M_\beta A_\alpha}\over{2 \mu n_p a_p^5}}
{{\Omega_{\alpha h} \Omega_{\alpha e}}\over{2 n_p t_{F\alpha}}} 
{{1+{9\over 2} e_p^2 + {5\over 8} e_p^4}\over{(1-e_p^2)^5}}, \label{eq:defX}\\
Y_\alpha & = & - {{M_\beta A_\alpha}\over{2 \mu n_p a_p^5}}
{{\Omega_{\alpha h} \Omega_{\alpha q}}\over{(1-e_p^2)^2}} +
{{\Omega_{\alpha e}}\over{2 n_p t_{F\alpha}}} {{1+{3\over 2}
e_p^2 + {1\over 8} e_p^4}\over{(1-e_p^2)^5}}, \label{eq:defY}\\
Z_\alpha & = & {{M_\beta A_\alpha}\over{2 \mu n_p a_p^5}} \left[ {{2
\Omega_{\alpha h}^2 - \Omega_{\alpha e}^2 - \Omega_{\alpha
q}^2}\over{2 (1-e_p^2)^2}} + 15 {{G M_\beta}\over{a_p^3}} {{1+{3\over
2} e_p^2 + {1\over 8} e_p^4}\over{(1-e_p^2)^5}}
\right], \label{eq:defZ} \\
Z_{\rm GR} & = & {{3 G (M_\star + M_p) n_p }\over{a_p c^2 (1-e_p^2)}}, \label{eq:defZGR}\\
S_{ee} & = & {\cal C} (1- 3 \sin^2 I \sin^2 \omega_p), \label{eq:defSee}\\
S_{eq} & = & - 3 {\cal C} \sin^2 I \sin \omega_p \cos \omega_p , \label{eq:defSeq}\\
S_{eh} & = & - 3 {\cal C} \sin I \sin \omega_p \cos I, \label{eq:defSeh} \\
S_{qh} & = & - 3 {\cal C} \sin I \cos \omega_p \cos I, \label{eq:defSqh}\\
S_{qq} & = & {\cal C} (1 - 3 \sin^2 I \cos^2 \omega_p),\label{eq:defSqq} \\
\cal{C} & = & {1\over{4 \sqrt{1-e_p^2} (1-e_c^2)^{3/2}}} 
{{M_c n_c^2}\over{(M_p + M_\star+ M_c) n_p}}, \label{eq:defC}
\end{eqnarray}
where the subscript $\alpha$($\beta$) is to be taken as $\star$($p$)
or $p$ ($\star$). Various symbols are defined as in EKE. In
particular, we have
\begin{eqnarray}
%I_\alpha & = & r_{g\alpha} M_\alpha R_\alpha^2, \label{eq:defI}\\
A_\alpha & = & k_{2\alpha} R_\alpha^5, \label{eq:defA}\\
{1\over{t_{F\alpha}}} & = & {{3 k_{2\alpha}}\over{Q_\alpha}}
\left({{R_\alpha}\over{a_p}}\right)^5 {{M_\beta}\over{M_\alpha}}, \label{eq:definetF}\\
%\mu & = & {{M_\star M_p}\over{M_\star +M_p}}, \label{eq:definemu}\\
\Omega_{\alpha e} & = & \Omega_{\alpha E} e_E + \Omega_{\alpha Q} e_Q + 
\Omega_{\alpha H} e_H, \label{eq:spin1}\\
\Omega_{\alpha h} & = & \Omega_{\alpha E} h_E + \Omega_{\alpha Q} h_Q + 
\Omega_{\alpha H} h_H, \label{eq:spin2}\\
\Omega_{\alpha q} & = & \Omega_{\alpha E} q_E + \Omega_{\alpha Q} q_Q + 
\Omega_{\alpha H} q_H, \label{eq:spin3}\\
e_E & = & {\hat e} \cdot {\hat E} = 
\cos \Omega \cos \omega_p - \sin \Omega \sin \omega_p \cos I, \label{eq:dir1} \\
e_Q & = & {\hat e} \cdot {\hat Q} = 
\sin \Omega \cos \omega_p + \cos \Omega \sin \omega_p \cos I, \label{eq:dir2}\\
e_H & = & {\hat e} \cdot {\hat H} = \sin I \sin \omega_p, \label{eq:dir3}\\
q_E & = & {\hat q} \cdot {\hat E} = 
- \cos \Omega \sin \omega_p - \sin \Omega \cos \omega_p \cos I, \label{eq:dir4}\\
q_Q & = & {\hat q} \cdot {\hat Q} = 
-\sin \Omega \sin \omega_p + \cos \Omega \cos \omega_p \cos I , \label{eq:dir5}\\
q_H & = & {\hat q} \cdot {\hat H} = \sin I \cos \omega_p, \label{eq:dir6}\\
h_E & = & {\hat h} \cdot {\hat E} = \sin I \sin \Omega, \label{eq:dir7}\\
h_Q & = & {\hat h} \cdot {\hat Q} = - \sin I \cos\Omega, \label{eq:dir8}\\
h_H & = & {\hat h} \cdot {\hat H} = \cos I. \label{eq:dir9}
\end{eqnarray}
The dimensionless number $k_{2\alpha}$ is the apsidal constant.


\begin{table*}
\tabletypesize\small
\begin{tabular}{ccr}
\hline
Symbol & Definition & Fiducial Values \\
\hline

$M_\star, M_c, M_p$ & stellar (host \& companion) and planetary masses &
$1.1 M_\odot$, $1.1 M_\odot$, $7.80 M_J$\\

$R_\star, R_p$ & stellar and planetary radii & $R_\odot$, $R_J$ \\

$k_{2\star}, k_{2p}$ & tidal Love number & $0.028$, $0.51$ \\
%  &  (deformation under tidal or rotational perturbations) & \\

$r_{g\star}, r_{gp}$ & gyro-radius, moment of inertia $ = r_g M R^2$
 & $0.08$, $0.25$ \\

$Q_\star, Q_p$ & tidal dissipation quality factors & $10^6$,
$3\times 10^5$ \\
%& (tidal energy versus energy dissipated per tidal period) & \\

$\Omega_\star, \Omega_p$ & spin frequency & initially $20$ days \&
$10$ hours\\

%$T_\star, T_p$ & tidal dissipation time & $T = 2 Q n R^3/(G M)$ \\

$a_p$, $e_p$, $n_p$ \& $P_p$ & planet semi-major axis, eccentricity,
mean motion \& period & $n_p = \sqrt{G(M_\star + M_p)/a_p^3}$ \\

$a_c$, $e_c$, $n_c$ \& $P_c$ & parameters for the companion orbit & $a_c =
1000$ AU, $e_c = 0.5$ \\

\hline
\end{tabular}
\label{table:symbol2}
\end{table*}


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