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%astro-ph/0807.2239
%http://www.journals.uchicago.edu/doi/ref/10.1086/591734

%\documentclass[12pt,preprint]{aastex}
\documentclass{emulateapj}

\newcommand{\msun}{M_{\sun}}

\begin{document}

\title{Origin of the S Stars in the Galactic Center}

\author{Ulf L\"ockmann, Holger Baumgardt, and Pavel Kroupa}
\affil{Argelander-Institut f\"ur Astronomie, Universit\"at Bonn, Auf dem 
H\"ugel 71, 53121 Bonn, Germany}
\email{uloeck@astro.uni-bonn.de}


\begin{abstract}
Over the last 15 years, around a hundred very young stars have been 
observed in the central parsec of our Galaxy.
While the presence of young stars forming one or two stellar disks at 
$\approx 0.1\,$pc from the supermassive black
hole (SMBH) can be understood through star formation in accretion disks,
the origin of the S stars observed a factor of 10 closer to the SMBH has 
remained a major puzzle.
Here we show the S stars to be a natural consequence of dynamical 
interaction of two stellar disks at larger radii.
Due to precession and Kozai interaction,
individual stars achieve extremely high eccentricities at random 
orientation.
Stellar binaries on such eccentric orbits are disrupted due to close 
passages near the SMBH, leaving behind a single S star on a much tighter 
orbit.
The remaining star may be ejected from the vicinity of the SMBH,
thus simultaneously providing an explanation for the observed 
hypervelocity stars in the Milky Way halo.
\end{abstract}

\keywords{black hole physics --- Galaxy: center --- methods: $n$-body 
simulations --- stellar dynamics}



\section{Introduction}

The central parsec of our Galaxy harbors different groups of young stars 
orbiting the central SMBH.
These include one or two stellar disks at $\approx 0.1\,$pc from Sgr 
A$^*$ made up of 5--6\,Myr old stars \citep{pau06},
and a number of stars on highly eccentric orbits even closer, the 
so-called S stars \citep{eg97}.
With orbital periods as small as 15 years, the S stars provide powerful 
constraints on the mass and size of the central dark object \citep{ghe05}.
While the origin of the stellar disk(s) observed in the Galactic center 
can be understood \citep{lb03,gs03,nc05,ger01,mp03},
the S stars show a puzzling combination of interesting features 
\citep{eg97}: They are very young ($< 10$\,Myr), with less than 0.01\,pc 
a factor of 10 closer to the SMBH than the closest young population 
known (the stellar disks), and move around Sgr A$^*$ on randomly 
oriented eccentric orbits.


A number of mechanisms to create the S stars have been proposed, but so 
far none of them was able to account for all their properties \citep[see 
also review by][]{a05}:
It has been suggested that the S stars are not young but rejuvenated 
stars or exotic objects,
but this is
in conflict with their spectral properties, showing they are ordinary 
main-sequence B-type stars \citep{ghe05,eis05,mge08}.
In situ formation is excluded by the inexplicably high pressure required 
to overcome the strong tidal forces, and will also lead to a stellar 
disk \citep{ghe05},
which cannot account for the random orientation and high eccentricity of 
the orbits within the stars' lifetime \citep{gh07}.
The latter argument also argues against the suggested scenario of an 
infalling cluster of young stars \citep{ger01},
which would moreover require a very high mass \citep{pmg03}, or an 
intermediate-mass black hole in its center for it to survive the tidal 
force of Sgr A$^*$ \citep{hm03}.
Exchange capture with compact remnants \citep{al04} only works with a 
large cluster of black holes and fails to explain the youth and high 
initial eccentricities.
The Lense-Thirring effect may account for the innermost star 
\citep{lb03}, S2, being dragged out of the disk plane,
but cannot explain the orbital orientation of the longer-period S stars, 
nor the proximity of the orbit itself.
Interaction of stars in a central cluster with a fossil accretion disk 
can lead to high eccentricities and orbital decay \citep{sk05} if the 
disk mass is sufficiently high, but fails to explain the observed bias 
toward young B-type stars. Furthermore, such an accretion disk is 
expected to fragment and form stars and should thus have been observed.

Tidal disruption of binaries falling into the Galactic center on highly 
eccentric orbits has been shown to be an effective mechanism to create S 
stars,
but until now required very massive star clusters
passing close to Sgr A$^*$ \citep{gq03}
or the presence of a large enough population of compact massive 
perturbers in combination with many young stars on eccentric orbits 
\citep{pha06}.

In the following we show how the dynamical interaction of two stellar 
disks at larger radii, as they are observed in the Galactic center, 
naturally leads to the creation of S stars. Section \ref{sec:model} 
describes the setup of our numerical simulations. In \S\ 
\ref{sec:high-ecc}, we present the outcome of these simulations, showing 
how the disk stars can eventually gain very high eccentricities. Section 
\ref{sec:disrupt} explains how binary stars on such eccentric orbits 
around the SMBH get disrupted, leaving behind an S star and possibly a 
hypervelocity star. In \S\ \ref{sec:discuss} we discuss our results, 
showing how the mechanism suggested here explains all the observed 
properties of the S stars.



\section{Modeling disk interaction}
\label{sec:model}

\begin{figure}
%\epsscale{1.15}
\plotone{f1}
\caption{
\label{fig:discevol1}
Aitoff projection of normal vectors of the orbits of stars situated in 
two interacting disks.
The separation of two points in the projected plot indicates the 
relative inclination of the two corresponding stellar orbits,
and thus the disk thickness.
The red squares indicate the initial state of two nearly flat disks 
inclined at an angle of 130$^{\circ}$.
The orientation of the disks changes with time due to precession.
By 5\,Myr (circles), the massive disk shows a slight warping (bottom).
The less massive disk (top to left) precesses faster and is affected 
more strongly by warping,
especially towards the inner edge (green and yellow circles), where it 
loses its disk shape.}
\end{figure}

An extensive analysis of young massive stars in the innermost parsec of 
our Galaxy shows that
almost all stars reside in two $6 \pm 2$\,Myr old rotating 
disks---except for the S stars in the central arcsecond \citep{pau06}.
While the clockwise disk is very distinct, there is some debate over the 
disk structure of the counterclockwise system \citep{lu06}.
However, \citet{caa08} have shown that a single stellar disk cannot 
explain the high inclinations and eccentricities observed.
Using direct $N$-body calculations as described below, we find that the 
distribution of eccentricities and inclinations of the observed young 
stars' orbits is consistent with an initial configuration of two 
relatively flat disks, since the less massive disk is destroyed over 
time due to dynamical interaction (see \S\ \ref{sec:high-ecc}).

To investigate the stellar dynamical evolution of two interacting 
stellar disks,
we have computed a number of $N$-body integrations. The parameters of 
our model are
chosen so as to best fit the observations by \citet{pau06} and are as 
follows:
Our model consists of a $3.5 \times 10^6\,\msun$ SMBH and two flat 
circular stellar disks, mutually inclined at an angle of $130^{\circ}$,
with a surface density profile that scales with distance as $r^{-2.5}$. 
The disks have well-defined initial extents
of $0.05-0.5\,$pc and $0.07-0.5\,$pc, respectively.

We assume that the lower mass limit of the stars observed by 
\citet{pau06} is $20\,\msun$. For our models, we assume a disk age of 
5\,Myr, and
hence an upper initial mass limit of $42.5\,\msun$ for the observed disk 
stars \citep{hpt00,fba06}. Using the initial mass function
${\rm d}N/{\rm d}m \propto m^{-1.35}$ as derived from the $K$-band 
luminosity function by \citet{pau06} with mass limits 1 and
120\,$\msun$, this leads to a total initial stellar mass of 
$10,380\,\msun$ to explain the 73 observed O stars. Allowing for
30\% binaries and another 10\% for stars disrupted by the SMBH or 
unobserved, and using a mass ratio of 2\,:\,1 for the disks, we have
initial stellar masses of $9,900$ and $4,950\,\msun$, respectively. As a 
consequence, our disks initially consist of 535 and 267 multimass
stars (and binaries), respectively.
The above parameters have proven to result in
a distribution of eccentricities and semimajor axes as well as an 
inclination between the two disks after 5 Myr which best match
the observations.

We used our \textsc{bhint} code \citep{lb08} for the integration, which 
has been developed specifically to calculate the dynamics of
stars orbiting a SMBH, and includes post-Newtonian treatment up to order 
2.5 to account for the effects of general relativity.
Furthermore, we have added the \textsc{SSE} package by \citet{hpt00} to 
account for the effects of stellar mass loss.

The stars and binaries are modeled as point masses, and all binaries are 
assumed to be equal-mass.
Tidal disruption is not considered; however, we assume that stars on 
very tight orbits ($a<80$\,AU) are swallowed by the SMBH.
We switch on the post-Newtonian terms for the central motion around the 
SMBH in our \textsc{bhint} integrator \citep{lb08}
for eccentricities $e>0.9$, and switch it off when $e$ falls below 0.8. 
A fully post-Newtonian calculation shows that in the regime
considered in this Letter, relativistic effects are negligible for 
orbits with lower eccentricities within the considered timescale.

We finally note that the detailed choice of parameters seems not to be 
too important. A number of models with variations in total mass,
mass function, and disk age lead to comparable results.



\section{Achieving very high eccentricities}
\label{sec:high-ecc}

Due to mutual torques, the two stellar disks described above precess 
about each other, with a frequency proportional to the other disk's mass:
The precession frequency of a star orbiting a SMBH of mass $M_{\rm 
SMBH}$ at radius $R$ at an inclination $\beta$ relative to a narrow disk 
of mass $M_{\rm disk}$ at a radius $R_{\rm disk}$ can be approximated as 
\citep{nay05}
\begin{equation}
\omega_p = -\frac{3}{4}\frac{M_{\rm disk}}{M_{\rm 
SMBH}}\cos{\beta}\sqrt{\frac{GM_{\rm SMBH}}{R^3}}\frac{R^3R_{\rm 
disk}^2}{\left(R^2+R_{\rm disk}^2\right)^{5/2}}.
\end{equation}

As can be seen, the
precession frequency depends also on the distance of a star from the 
central SMBH \citep{nay05},
and since both disks have a finite extent, stars at different central 
distances precess with different frequencies,
thus warping the originally flat disks. In particular, the innermost 
stars quickly get inclined with respect to their disk of origin,
and so start precessing about both disks. This leads to further 
thickening and deformation of the disks as well as large inclinations.

Figure \ref{fig:discevol1} depicts the evolution of the stellar disks' 
shape and thickness.
Being exposed to the larger torque, the less massive disk is warped much 
faster,
explaining why the observed thickness of the less massive disk
in the center of our Galaxy is substantially higher, making its present 
existence
controversial \citep{lu06}.

Apart from precession, all stars are subject to the Kozai mechanism 
\citep{k62}: Stars on highly inclined orbits relative to an axisymmetric 
perturbation periodically obtain small inclinations, and simultaneously 
gain high eccentricities.
At their pericenter, stars on orbits with eccentricity $e>0.99$ pass the 
central SMBH within $\sim 100$\,AU, less than the S stars' pericenter 
distances.
The effects of general relativity causes eccentric orbits to precess, 
which may damp the Kozai mechanism and prevent further growth in 
eccentricity \citep{htt97}.
However, our simulations show that in the presence of very massive, 
nearby disks, relativistic effects do not prevent eccentricities as high 
as $e=0.999$.

Figure \ref{fig:eccent} shows the eccentricity evolution of some stars 
in our simulation, and the total fraction of stars
that have reached eccentric orbits within a given time.
A significant fraction of the stars obtain very high eccentricities.
In contrast, a single stellar disk around a SMBH evolves only very 
slowly: In a corresponding integration,
no star obtained an orbital eccentricity above 0.8 within a comparable 
time span. This is mainly due to the lack of an external torque,
and the lack of large inclinations required for the Kozai mechanism to 
become effective.

\begin{figure}
\begin{center}

%\epsscale{1.1}
\plotone{f2}
\caption{
\label{fig:eccent}
Eccentricity evolution of
stellar orbits.
The top panel shows as an example the time evolution of orbital 
eccentricity of three stars with short Kozai periods.
The cyclic nature of the Kozai mechanism, leading to extremely high 
eccentricities, is evident.
The solid and dotted lines in the bottom panel depict the fraction of 
stars having reached eccentricities higher than 0.9 and 0.99, 
respectively, as a function of time.
Any star on a Kozai oscillation keeps a high eccentricity for a short 
time only, but as most stars have a Kozai period longer than the 
computed time,
these numbers are increasing until the end of the integration.
}
\end{center}
\end{figure}



\section{Disruption of stellar binaries}
\label{sec:disrupt}

As most stars form in binaries \citep{gkgb07},
it is natural to assume that a significant fraction of stars in the 
stellar disks are followed by a companion.
The orbits of these binaries around the SMBH are subject to the same 
Kozai mechanism as single stars, but a very close pericenter passage
may break up the binary and leave one star on a tightly bound orbit 
around the SMBH.

In a set of three-body integrations of a stellar binary orbiting a SMBH, 
we have investigated the distribution of binary disruption remnants.
As can be seen in Figure \ref{fig:eccent}, the eccentricity evolution 
due to the Kozai effect is smooth.

We calculated a set of models including a SMBH, an intermediate-mass 
black hole (IMBH) to drive the Kozai, and a stellar binary on an 
initially circular orbit at high inclination with respect to the IMBH's 
orbit in order to test the impact of a Kozai oscillation on the binary's 
inner orbital parameters.
Figure \ref{fig:binkozai} shows the time evolution of a 0.1\,AU binary's 
both outer and inner eccentricity and semimajor axis. It can be seen 
that until binary disruption, only the outer eccentricity is changing.
In particular, the binary does not widen.
Only during the last few pericenter passages does the inner eccentricity 
grow to a value of $e=0.6$, but this does not significantly change the 
resulting semimajor axis of the bound member, nor the velocity of the 
ejected star.
Computations with wider binaries (1\,AU) yield similar results.

\begin{figure}
%\epsscale{1.08}
\plotone{f3}
\caption{
Evolution of orbital parameters of a stellar binary undergoing Kozai 
resonance. The long-dashed and dotted lines show the outer and inner 
eccentricity of the binary, respectively, while the solid and 
short-dashed lines show the outer and inner semimajor axis, respectively.
It is seen that the binary's inner orbital parameters stay constant 
until binary disruption, while the outer eccentricity grows due to the 
Kozai mechanism.
\label{fig:binkozai}
}
\end{figure}

We thus conclude that any tight binary undergoing Kozai resonance 
reaches its point of disruption basically unperturbed.
We have therefore computed a set of models where
the pericenter distance of a binary orbiting a SMBH is equal to the 
disruption distance (Fig.\ \ref{fig:disrupt}).
We find that the disruption remnants of binaries with separations larger 
than a few AU
orbit the central SMBH on similar trajectories as the initial binary, 
and so can still be observed as disk stars.
However, the disruption of a tight binary (with separation of 1\,AU or 
less) forces one star onto a very tight and eccentric S-star-type orbit, 
while the other star gains a significant amount of energy and may even 
be ejected from the Galaxy, thus becoming a progenitor of the 
hypervelocity stars observed in the Galactic halo 
\citep[e.g.,][]{b07a,gps05}.
For example, disruption of binaries with a separation of 0.2\,AU leaves 
one star on an orbit with semimajor axis consistent with those of the 
observed S stars, while the other star is ejected from the Galaxy with a 
velocity of up to 1000\,km\,s$^{-1}$ in more than half of our integrations.

\begin{figure}
%\epsscale{1.08}
\plotone{f4}
\caption{
\label{fig:disrupt}
Orbital parameters about the SMBH of the (more strongly) bound member 
star of a disrupted equal-mass binary as a function of initial binary 
separation.
While the disruption of wide binaries leaves the individual stars on 
orbits similar to the initial trajectory,
the bound star of a disrupted tight binary ends up on a very tight orbit 
around the SMBH.
The solid line depicts the initial semimajor axis before disruption.
The shaded area indicates the semimajor axes of the S stars in the 
Galactic center.
}
\end{figure}

Binary disruption has been shown to be an effective mechanism to create 
the S stars \citep{gq03}.
One way to create young close-passage binaries is by massive perturbers 
like star clusters or molecular clouds at distances of a few pc from the 
Galactic center \citep{pha06}.
Our simulations show that these binaries are also a natural consequence 
of the interaction of the two stellar disks:
Using a binary fraction of 30\%, 18 of the B-type binaries achieve 
pericenter distances below their tidal radius, assuming a binary 
separation of 0.1\,AU.
Given that not much is known about the binary fraction in the Galactic 
center, this is fully consistent with the observed number of 15 S stars.



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

The mechanism described above naturally explains the reported mysteries 
of the S stars:
The disruption of stellar binaries as tight as 0.1--1\,AU can leave a 
single star orbiting the central SMBH
a factor of 10 closer than the original binary. This is just the 
distance ratio of S stars and stellar disks observed
in the center of our Galaxy.

The orientation of the disks rapidly changes due to precession. 
Furthermore, the change of orientation of a binary's orbit
is a natural consequence of the Kozai effect. Once a binary is disrupted 
and leaves a single star on a much closer orbit, this orbit is
decoupled from the originating stellar disk's motion, as both the 
precession frequency and the Kozai oscillation period are functions of 
central distance,
explaining the widely varying orbital orientations observed.

The eccentricities created for the tightly bound stars after tidal 
disruption are as high as 0.99. Resonant relaxation \citep{rt96} and 
Kozai interactions
between the S stars are expected to turn these extremely high 
eccentricities into an isotropic distribution on a short timescale,
just as is observed for the S stars.

While all of the S stars observed so far are main-sequence B-type stars, 
observations of the stellar disks in the Galactic center
mostly revealed OB giants and Wolf-Rayet stars. But even for the 
unusually flat mass function suggested \citep{pau06},
one would expect a factor of 5 more B-type main-sequence stars in the 
disks than O stars. Most of these stars have not been observed yet,
mainly due to crowding of the observed region, limited spatial 
resolution, and strong nebular emission \citep{pau06}.
Our computation assuming equal-mass binaries predicts only one S star 
above 20\,$\msun$ per 18 B-type S stars. The lack of observed O-type S 
stars is therefore not statistically significant.
The number of expected massive S stars would drop further for random 
pairing of binary component masses, since disruption of a high mass 
ratio binary will only moderately change the orbit of the massive member.
Furthermore, the massive O stars might not form in as tight binaries as 
are required to produce S stars.
Hence, it is legitimate to assume that both S stars and disk stars 
formed in the same environment,
thus providing a solution to the ``paradox of youth''.

In this Letter, we studied the evolution of two isolated stellar disks 
orbiting a SMBH. In a forthcoming publication, we will analyze the 
effects of other contributions of a perturbing potential, such as a 
stellar cusp \citep[e.g.,][]{sea+07}, a massive circumnuclear disk 
\citep{cssy05}, an intermediate-mass black hole and/or a star cluster 
\citep[probably IRS 13;][]{mpsr04}.
Preliminary results indicate, e.g., that the presence of a cusp of 
stellar black holes \citep[e.g.,][]{fak06} does not prevent the 
dynamical formation of S stars as described in this Letter.






\acknowledgments

We thank Ladislav {\v S}ubr for helpful discussions. This work was 
supported by the German Research Foundation (DFG) through the priority 
program 1177 ``Witnesses of Cosmic History: Formation and Evolution of 
Black Holes, Galaxies and Their Environment.''

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