`%astro-ph/1004.2703`

```
\documentclass[useAMS,usenatbib]{mn2e}\usepacka=
ge{latexsym,graphicx,natbib}\usepackage{amsmath}
```

=

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% Units\newcommand\msun{\, \rm =
M_\odot} \newcommand\rsun{\, \rm =
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% =
Names\newcommand\mbh{M_{\rm =
MBH}}\newcommand\mim{M_{\rm =
IBH}}\newcommand\mstar{m_{\rm =
*}}\newcommand\mtot{M_{\rm MBH} + M_{\rm =
IBH}}\newcommand\tpr{T_\varpi}\newcommand\tgw{T_{\rm=
GW}}\newcommand\mas{\,\rm =
mas}\newcommand\einn{e_{\rm =
inn}}\newcommand\eout{e_{\rm =
out}}\newcommand\Pinn{P_{\rm =
inn}}\newcommand\Pout{P_{\rm =
out}}

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```
\title[The Galactic Centre star S2 as dynamical =
probe]{The Galactic
```

```
Centre star S2 as a dynamical =
probe for intermediate-mass black =
holes}
```

```
\author[A. Gualandris, S. =
Gillessen and
```

```
D. Merritt]{A. =
Gualandris$^{1}$\thanks{E-mail:
```

```
=
alessiag@mpa-garching.mpg.de}, S. Gillessen$^{2}$ =
and
```

```
D. Merritt$^{3}$\footnotemark[1]\\ =
$^{1}$Max-Planck Institut
```

```
f\"{u}r Astrophysik, =
Karl-Schwarzschild-Str. 1, D-85748 Garching, =
Germany
```

```
\\ $^{2}$Max-Planck Institut f\"{u}r =
Extraterrestrische Physik,
```

```
Giessenbach-Str., =
D-85748, Garching, Germany\\ $^{3}$Department of
```

```
=
Physics and Center for Computational Relativity and =
Gravitation,
```

```
Rochester Institute of Technology, =
\\ 54 Lomb Memorial Drive,
```

```
Rochester, =
NY}
```

`\begin{document}`

`\date{}`

```
\pagerange{\pageref{firstpage=
}--\pageref{lastpage}} =
\pubyear{2010}
```

`\maketitle`

\label{firstpage}

\begin{abstract}We=
study the short-term effects of an intermediate mass black =
hole(IBH) on the orbit of star S2 (S02), the shortest-period =
star known toorbit the supermassive black hole (MBH) in the =
centre of the MilkyWay. Near-infrared imaging and =
spectroscopic observations allow anaccurate determination of =
the orbit of the star. Given S2's shortorbital period and =
large eccentricity, general relativity (GR) needsto be taken =
into account, and its effects are potentially measurablewith =
current technology. We show that perturbations due to an IBH =
inorbit around the MBH can produce a shift in the apoapsis of =
S2 that isas large or even larger than the GR shift. An IBH =
will also inducechanges in the plane of S2's orbit at a level =
as large as onedegree per period. We apply observational =
orbital fitting techniques tosimulations of the S-cluster in =
the presence of an IBH and find thatan IBH more massive than =
about $1000\msun$ at the distance of theS-stars will be =
detectable at the next periapse passage of S2, whichwill =
occur in =
2018.

\end{abstract}

\begin{keywords}black hole physics -- stellar =
dynamics - methods: $N$-body simulations -- Galaxy: =
centre\end{keywords}

\s=
ection{Introduction}The innermost arcsecond of the Milky Way =
harbours a cluster of youngmassive stars (the S-star cluster) =
in eccentric orbits around thesupermassive black hole (MBH). =
Near-infrared observations of thecluster allow a precise =
determination of the trajectories of about 20stars. These can =
be used to derive fundamental parameters like themass of the =
MBH and the distance to the Galactic centre, as well as =
toconstrain the gravitational potential and test predictions =
fromgeneral relativity (GR) \citep{rubilar2001,zucker2006}. =
Since thecollapse of a giant molecular cloud at the =
distance of the S-clusteris prevented by the tidal force of =
the MBH, it is believed that eitherthe S-stars formed in-situ =
via a non-standard process or that theyoriginated outside the =
black hole's sphere of influence and thenmigrated to their =
current location. Among the in-situ models are thetwo =
disks model \citep{lock09} and the eccentric instability =
model\citep{madigan09}. The two main models suggested =
for the transport ofthe stars are the cluster infall scenario =
\citep{gerhard01}, alsoaided by an intermediate-mass black =
hole (IBH) \citep{hm03, kim04, levin05, mgm09, =
fujii09, fujii10}, and the binary disruptionscenario =
\citep{gq03,perets07,perets09}. The properties and =
efficiencyof these models are discussed in =
\citet{PG10}.

The star with the shortest =
orbital period ($\sim 15\yr$), calledS2 \citep{schodel2002, =
ghez03}, has a semi-major axis $a =3D(0.1246\pm 0.0019)''$ =
and an eccentricity $e =3D 0.8831 \pm0.0034$ \citep{gillS2}. =
Adopting a distance to the Galacticcentre $R_0 =3D =
8.28\kpc$, $a \sim 5.0018\mpc$, while theperiapsis and =
apoapsis distances are, respectively, $r_p =3D0.585\mpc$, =
$r_a =3D 9.419\mpc$ . For this set of orbitalelements, =
GR precession is measurable with currentinstrumentation on a =
time-scale of about 10 =
years\citep{Gillessen2009}.

The =
relativistic (prograde) advance of the periapse angle is given =
inthe case of a non-rotating black hole =
by\begin{equation}\Delta\varpi =3D =
\frac{3\pi}{1-e^2} \frac{R_S}{a} =3D \frac{6\pi G}{c^2} =
\frac{\mbh}{a(1-e^2)}\end{equation}per radial =
period \citep{weinberg72},where $R_S =3D 2G\mbh/c^2$ is =
the Schwarzschild radius of the blackhole. For the =
Galactic centre black hole, $R_S \approx4.1\times10^{-7}\pc$ =
for an assumed mass of =
$\mbh=3D4.3\times10^6\msun$\citep{gillS2}.As a =
result of the orbit's precession, there is a displacement =
inthe star's apoapse position that is given =
by\begin{equation}\Delta r_a \approx a (1+e) =
\Delta\varpi \approx \frac{6\pi =
G\mbh}{c^2(1-e)}\end{equation}again per radial =
period; note that this expression is independent of =
thesemi-major axis.As seen from Earth, this shift =
corresponds to an angle on the sky =
of\begin{equation}\label{eq:wein}\Delta =
\Theta_a \sim 0.097{\mas} =
\left(\frac{1}{1-e}\right)\left(\frac{\mbh}{4.3\times10^6\msun}=
\right) =
\left(\frac{8.28\kpc}{R_0}\right)\end{equation}which=
amounts to $\sim 0.83\mas$ for star S2.

A =
complicating factor is the likely presence of a distributed =
masswithin S2's orbit, consisting of stars and stellar =
remnants. Assumingthat the mass density follows =
$r^{-\gamma}$, with $r$ the distancefrom the MBH, the =
advance, per radial period, of orbital periapse dueto the =
distributed mass is\begin{equation}\Delta\varpi =
\approx 2\pi =
\frac{M_\star(a)}{\mbh}\sqrt{1-e^2}F(\gamma)\end{equation}in the retrograde sense;here $M_\star(r)$ is the =
distributed mass enclosed within radius$r$ and $F=3D(3/2,1)$ =
for $\gamma=3D(0,1)$ \citep[e.g.][]{MAMW2010}.Setting =
$F\approx 1$, this implies, for S2, a shift on the sky =
of\begin{equation}\Delta\Theta_a \approx 0.69 =
\mathrm{mas} \left(\frac{10^3 =
M_\star(a)}{\mbh}\right).\end{equation}This is =
comparable with the relativistic precession if the =
distributedmass within S2's orbit is $\simgreat =
10^{-3}\mbh$.Disentangling these two sources of precession =
will be difficult, requiringmeasurements of the radial =
velocity of S2 near periapse =
passage\citep[e.g.][]{saha10}.

Non-sph=
erically-symmetric perturbations, if present, can affect =
notjust the periapse angle but also the angular momentum of =
S2's orbit,resulting in changes in $e$ and in the orientation =
of its orbitalplane. Potentially the largest source of =
such perturbations is asecond massive black hole orbiting =
somewhere near S2. In fact, theorbits of the S-stars =
are consistent with the long-term presence of anIBH in their =
midst, with mass $\sim 10^3\msun$ \citep{mgm09,GM2009}.While =
an IBH would also contribute to the periapse advance of S2, =
itskey signature would be a change in S2's orbital angular =
momentum.

In this work, we combine =
high-accuracy $N$-body simulations (\S2) with orbital =
fitting techniques (\S3) to investigate theobservable effects =
of an IBH on the orbit of star S2 over a time-scaleof a few =
orbital revolutions.In \S4 we discuss the other types of =
perturbation that could potentially =
induce changes in S2's orbital angular momentum and =
compare them withperturbations from an IBH.\S5 =
sums =
up.

\section{Initia=
l models and numerical methods}We consider $N$-body models =
that include a MBH, an IBH and the S-starcluster. The =
initial conditions for the stars are derived from theorbital =
elements given by \citet{Gillessen2009}. For star S2, we =
takethe improved elements from \citet{gillS2}. Of the =
28 stars for whichthose authors provide orbital elements, we =
exclude the six stars (S66,S67, S83, S87, S96, S97) which =
likely belong to the clockwise disk\citep{genzel03, =
paumard06} and star S111 which appears to be unbound.We are =
left with a sample of 21 stars with well defined orbits, =
forwhich we determined positions and velocities at 2008 AD =
from theclassical elements. The masses of the S-stars =
were set to $10\msun$\citep[e.g.][]{eisen05} except for star =
S2 for which a value of$20\msun$ was adopted =
\citep{martins08}.

The IBH is =
placed on a Keplerian orbit around the MBH. We adopt =
fourdifferent values for the mass ratio of the black hole =
binary $q \equiv\mbh/\mim =3D (1.0\times10^{-4}, =
2.5\times10^{-4}, 5.0\times10^{-4},1.0\times10^{-3})$, five =
values for the semi-major axis $a =3D (0.3, 1,3, 10, =
30)\mpc$, four values for the eccentricity $e =3D (0, 0.5, =
0.7,0.9)$ and twelve choices for the direction of the orbit's =
angularmomentum vector (the same set as in \citet{GM2009}), =
for a total of960 sets of initial conditions. The IBH =
begins from orbital periapsisin all cases. The initial =
value of the mean anomaly is likely to beunimportant in all =
cases for which the orbital period of the binary ismuch =
smaller than the integration time ($50\yr$), i.e. $a=3D0.3, =
1,3\mpc$. For $a=3D30\mpc$, only part of the IBH orbit is =
sampled by theintegration, and in principle the choice of the =
initial position mighthave an effect on the interaction with =
S2. However, at a distance of$30\mpc$ the IBH is completely =
outside S2's orbit for $e \simless0.7$, and there is =
essentially no detectable signature, as shownbelow. The only =
set of simulations for which the initial mean anomalyof the =
IBH might be important is the $a=3D10\mpc$, which corresponds =
toan orbital period of about $43\yr$ for the IBH. We perform =
an extraset of simulations for $a=3D10\mpc$ starting the IBH =
at apoapsis ratherthan periapsis and we compare the results =
in the two cases.

We advanced each $N$-body =
system in time using the AR-CHAIN code\citep{mm08}, a recent =
implementation of the algorithmicregularisation method that =
is able to reproduce the motion of tightbinaries for long =
periods of time with extremely high precision. Thecode =
combines the use of the chain structure, introduced originally =
by\cite{ma93}, with a new time transformation to avoid =
singularities andachieve high precision for arbitrary mass =
ratios. Note that weself-consistently follow not just =
the interactions of S2 with the MBHand IBH, but all other =
interactions as well, including star-starinteractions. =
The integration interval was 50 years, during whichtime =
S2 performs three full orbits.

The AR-CHAIN =
code includes relativistic corrections to theaccelerations up =
to 2.5 post-Newtonian order for all interactionsinvolving the =
MBH particle. General relativistic advance of =
theperiapse, which operates on a =
time-scale\begin{eqnarray} \tpr & =
\equiv &\left|\frac{\Delta\varpi}{2\pi P}\right|^{-1} =
=3D \frac{2 \pi c^2 \left(1-e^2\right) a^{5/2}}{3 =
\left(G \mbh \right)^{3/2}} \nonumber \\ =
& \approx & 1.3\times10^5 {\rm yr} =
\left(\frac{a}{5\mpc}\right)^{5/2} =
\left(\frac{4.3\times10^6\msun}{\mbh}\right)^{3/2}\left(1-e^2\r=
ight)\end{eqnarray}is accounted for by the 1PN and =
2PN terms. The dissipative termarising from the =
emission of gravitational waves is accounted for bythe 2.5PN =
term; this term is potentially important for the IBH, =
forwhich the associated coalescence time-scale =
is\begin{eqnarray}\label{eq:tgw} &nb=
sp;\tgw & =3D & \frac{5}{256 F(e)} \frac{c^5}{G^3} =
\frac{a^4}{\mu \left(\mtot\right)^2} =
\nonumber\\ & \approx & \frac{1.96 \times =
10^{13} \yr}{F(e)} =
\left(\frac{a}{5\mpc}\right)^4 \left(\frac{4.3\times=
10^6\msun}{\mbh}\right) \left(\frac{10^3\msun}{\mim}=
\right)\nonumber\\ & & =
\times \left(\frac{4.3\times10^6\msun}{\mtot}\right)=
\end{eqnarray}where\begin{equation}=
F(e) =3D \left(1-e^2\right)^{-7/2} \left(1 + =
\frac{73}{24}e^2 + \frac{37}{96} e^4\right) =
\nonumber\end{equation}and\begin{equation=
} \mu =3D \frac{\mbh\mim}{\mtot} \approx \mim =
\nonumber\end{equation}is the reduced mass of the =
IBH/MBH binary. The orbital decaytime-scale for the =
black hole binary is shown in Figure~\ref{fig:times}for the =
two extreme values of the mass ratio and the four =
adoptedvalues of the initial eccentricity. This =
time-scale is always muchlonger than our integration interval =
of $\sim 50$ yr. In addition, itis longer than the =
main-sequence lifetime ($\sim 10^7$ yr) of a $20$solar mass =
star for all initial configurations excepting the cases $a=3D =
0.3\mpc$ and $e \geq 0.7$. In the former runs, it is justified =
toassociate our initial parameters for the IBH/MBH binary =
with theparameters at some much earlier time, e.g. the epoch =
precedingformation of the S-stars. In the latter runs, =
the orbit of the IBH atsome much earlier time would have been =
larger and/or more eccentric.The maximum relative variation =
of the binary semi-major axis in the $N$-bodyintegrations is =
$\Delta a / a \sim 10^{-3}$ while the absolute variationof =
the eccentricity is $\Delta e \sim =
10^{-2}$.\begin{figure} \begin{center} =
\includegraphics[width=3D8cm]{figure1.eps} \en=
d{center} \caption{Time-scales associated with =
orbital evolution in our models. =
Solid lines show the GW time-scale, =
Eq.~(\ref{eq:tgw}), for a black hole binary =
with $q =3D 10^{-4}$ (top) and $q =3D 10^{-3}$ =
(bottom), for four different values of the eccentricity $e=3D0, =
0.5, 0.7, 0.9$. Dashed-dotted lines show =
the GR precession time-scale for two =
different values of the eccentricity: $e =3D 0$ (upper =
line) and $e =3D 0.9$ (lower line). The =
vertical dotted lines represent the adopted =
values for the binary initial semi-major axis. =
The filled grey region indicates the =
estimated ages of the S-stars while the =
striped area shows the radial range of S2's =
orbit.} \label{fig:times}\end{figure}

\section{IBH =
perturbations}

In the Schwarzschild metric, the =
argument of periapse $\varpi$ evolvesdue to in-plane =
precession. The two remaining angles that define =
theorientation of the orbit, $i$, the inclination, and =
$\Omega$, theposition angle of the ascending node, are fully =
conserved in therelativistic two-body problem.\footnote{We =
follow the standard practise of using the plane of =
the sky as the reference plane for defining =
$(\Omega,i)$.} The semi-major axis and eccentricity =
areconserved at the 1PN level, and we expect very small =
deviations due tohigher order PN corrections in the =
integrations \citep{soffel89}. Inthe limit of small =
star-to-black hole mass ratio, the semi-major axisand =
eccentricity in the PN approximation are given by =
\citep{soffel89}\begin{equation}a =3D =
\frac{-GM}{2\mathcal{E}} \left[ 1 + \frac{7}{2} =
\frac{\mathcal{E}}{c^2}\right]\end{equation}\begin{e=
quation}e =3D \sqrt{1 + \frac{2\mathcal{E}}{G^2 =
M^2} \left(1+ \frac{17}{2} =
\frac{\mathcal{E}}{c^2}\right) \left(\mathcal{J}^2 =
+ 2 \frac{G^2M^2}{c^2}\right) =
}\end{equation}where\begin{equation}\mathcal{E} =3D \frac{1}{2}v^2 - \frac{GM}{r} + =
\frac{3}{8}\frac{v^4}{c^2} + \frac{GM}{2rc^2} \left[3v^2 + =
\frac{GM}{r}\right]\end{equation}is the specific =
post-Newtonian energy =
and\begin{equation}\mathcal{J} =3D =
|\overrightarrow{x} \times \overrightarrow{v}|\left[ 1 + =
\frac{1}{2} \frac{v^2}{c^2} + \frac{3GM}{rc^2} =
\right]\end{equation}the specific angular =
momentum. Here, $\overrightarrow{x}$ =
and$\overrightarrow{v}$ are the relative position and =
velocity vectorsbetween the star and the MBH, $M$ is the =
total mass, and $c$ is the speedof =
light.

The presence of the other S-stars =
introduces small deviations fromspherical symmetry in the =
gravitational potential but the effect onthe orbital elements =
over these short time-scales is negligible,as we show in =
Section \ref{sec:torque}.Therefore, the variations that we =
observe in the orbital elements of star S2in our =
simulations,namely the semi-major axis, eccentricity, =
inclination, and positionangle of the ascending node, can be =
attributed to perturbations fromthe =
IBH.

Figure~\ref{fig:change} summarises the =
changes in the orbital elementsof S2 found in the $N$-body =
integrations. Plotted are the variationsover one =
revolution averaged over the twelve different orientations =
ofthe initial IBH/MBH =
orbit.\begin{figure} \begin{center}=
=
\includegraphics[width=3D4cm]{figure2a.eps} =
\includegraphics[width=3D4cm]{figure2b.eps} =
\includegraphics[width=3D4cm]{figure2c.eps} =
\includegraphics[width=3D4cm]{figure2d.eps} =
\includegraphics[width=3D4cm]{figure2e.eps} =
\includegraphics[width=3D4cm]{figure2f.eps} \e=
nd{center} \caption{Average changes in the orbital =
elements (semi-major axis, eccentricity, =
inclination, position angle of the ascending =
node, periapsis and apoapsis) of star S2 =
over one full orbit, versus the mass ratio =
of the black hole binary. Different symbols are =
for different initial semi-major axes of =
the binary. Each point is an average over =
the 12 orientations of the IBH/MBH orbital =
angular momentum vector. The dotted lines =
represent the GR shift in the periapse and =
apoapse.} \label{fig:change}\end{figure}<=
/div>The dotted lines in the periapsis and apoapsis panels indicate =
thevariations due to GR. The shift in the periapse =
corresponds to anobservable angle $\Delta \Theta_p =3D \Delta =
\Theta_a\left(1-e\right)/\left(1+e\right)$, where $\Delta =
\Theta_a$ is definedin Eq.~\ref{eq:wein}. Note that the =
variations in the inclination $i$and position angle $\Omega$ =
of the ascending node reach values closeto 1 degree for the =
most massive IBHs considered. This is of the sameorder as the =
current observational accuracy ($\sim 0.7\,\rm deg$). In50 =
years, this value will drop to about $0.4\,\rm deg$, assuming =
thereare no technological =
improvements.

While precession induced by the =
PN terms is restricted to the orbitalplane, an IBH induces =
more general changes in the orbital elements,including =
changes in the direction of the orbital angular =
momentumvector. We measure the latter via the =
angle\begin{equation}\cos \phi =3D \left( =
\frac{\overrightarrow{L_i} \cdot =
\overrightarrow{L_f}}{L_i\,L_f}\right)\,.\end{equation}\begin{figure} \begin{center} &nb=
sp; =
\includegraphics[width=3D8cm]{figure3.eps} \en=
d{center} \caption{Mean variation of the orbital =
plane of star S2 as a function of the =
binary mass ratio. For each combination of binary =
mass ratio, semi-major axis and eccentricity, the results =
are averaged over the 12 orbital =
orientations of the =
IBH.} \label{fig:phi}\end{figure}Figure~\ref{fig:phi} plots $\phi$ for all the runs, after =
averaging over the 12 different IBH =
orientations.

The changes in the orbital plane =
of S2 are larger for more massiveIBHs and can reach values of =
$\sim 1$ degree for $q =3D10^{-3}$. Out-of-plane motion is =
also affected by the size of theMBH/IBH binary orbit such =
that changes are largest for $a =
\simless10\mpc$.

The three-body =
system composed of MBH, IBH and S2 is reminiscent of aKozai =
triple \citep{kozai62}. However, the changes that we observe =
inthe orbital elements are not generally attributable to the =
Kozaimechanism. Kozai oscillations can be induced if =
(i) the MBH - S2 -IBH system can be regarded as a =
hierarchical triple, withwell-separated orbital periods for =
the inner and outer orbit; (ii) theperiod predicted for the =
Kozai oscillations is shorter than that ofany other =
precessional period, in this case, GR precession; (iii) =
theouter orbit is largely inclined with respect to the inner =
orbit. Thefirst condition is only satisfied in the runs =
with $a =3D 10, 30\mpc$.

The timescale for =
Kozai oscillations can be written as =
\citep{KH07}\begin{equation}\label{eq:tkozai}<=
div>T_K =3D \frac{4 \mathcal{K}}{3\sqrt{6}\pi} =
\frac{\Pout^2}{\Pinn}\frac{\mbh + \mstar}{\mim} =
\left(1-\eout^2\right)^{3/2}\end{equation}where =
$\Pinn$ and $\Pout$ are the period of the inner and =
outerbinary, $\eout$ is the eccentricity of the outer orbit =
and$\mathcal{K}$ is a numerical coefficient which depends =
only on theinitial values of the relative inclination angle =
$\alpha$, the innerbinary eccentricity $\einn$ and the inner =
binary argument of periapsis$\varpi$. The maximum =
eccentricity $e_{\rm max}$ attained by theinner binary is =
also a function of the initial values of $\alpha$,$\einn$ and =
$\varpi$ only. =
\begin{figure} \begin{center}=
=
\includegraphics[width=3D4cm]{figure4a.eps} =
\includegraphics[width=3D4cm]{figure4b.eps} \e=
nd{center} \caption{Timescale for Kozai =
oscillations in star S2 as a function of =
the black hole binary mass ratio for different values of =
the binary eccentricity. The left panel =
refers to simulations with $a=3D10\mpc$ =
while the right panel is for $a=3D30\mpc$. The =
horizontal lines marks the GR precession =
timescale for =
S2.} \label{fig:kozai}\end{figure}<=
div>Figure~\ref{fig:kozai} shows $T_K$ as a function of the black =
holebinary mass ratio for the assumed values of the =
eccentricity and fortwo allowed separations. We find =
that $T_K$ is always longer thenS2's GR precession timescale =
of $2.5\times10^4\yr$ for $a=3D30\mpc$. Forthe remaining case =
$a=3D10\mpc$, $T_K$ is short enough only for$q\simgreat =
2.5\times10^{-4}$ and $e\simgreat 0.7$. This restricts =
theapplicability of the Kozai mechanism to a small subset of =
oursimulations, in contrast with the results shown =
inFigure~\ref{fig:change} and in the following section. =
We conclude thatthe Kozai mechanism is not the dominant =
effect producing changes inorbital $e$ and $i$ in our =
simulations. In those cases where it ispotentially =
relevant, the eccentricity is predicted to oscillatebetween =
$e_{\rm min} =3D 0.24 - 0.74$ and $e_{\rm max} =3D 0.89 - =
0.99$over one Kozai period, depending on the values of =
$\alpha$ and$\varpi$. This implies variations of the =
order of $2 \times 10^{-5} -6\times 10^{-3}$ over the =
integration time of 50 years =
(for$a=3D10\mpc$).

The apoapsis =
shift due to perturbations from the IBH is very sensitiveto =
the binary parameters. In the case of $q \simgreat =
5\times10^{-4}$and $a \simless 3\mpc$, the shift over one =
revolution due to the IBHbecomes larger than the relativistic =
shift. This suggests a variationin the orbital elements which =
is potentially observable with =
currentinstrumentation.

However, the =
observability of variations in the orbit of S2 depends =
onseveral factors. In the following section we thoroughly =
examine allsuch factors and use orbital fitting to determine =
whether an IBH isdetectable via on-going monitoring of the =
S-cluster. Theoretically,it would be possible to use =
other S-stars to investigate the effectsof a hypothetical =
IBH. Given that the shifts in the apparentlocation of =
periapsis and apoapsis depend only on the eccentricity andnot =
on the semi-major axis of the stellar orbit, it would =
seemappropriate to consider all stars with $e \simgreat 0.8$ =
for such ananalysis. =46rom an observational point of =
view, however, S2 is theonly star in the sample which is =
bright enough and not affected byconfusion to allow for =
meaningful tests of the gravitationalpotential. We therefore =
limit our study to star =
S2.

\section{Orbita=
l fitting}In this section we extract observational-like data =
from the simulatedorbital traces of S2. We assume that eight =
or nine astrometric epochscan be obtained each year over the =
course of 50 years. The eight ornine yearly epochs are not =
evenly distributed but are spread overseven months only, thus =
taking into account the fact that the GalacticCentre is =
accessible with NIR observations only for part of theyear. At =
the chosen epochs, the original, simulated positions =
areperturbed by an astrometric error, assumed to be =
distributed in aGaussian fashion. We use a value of =
$300\,\mu$as per coordinate, whichis a conservative =
assumption for S2 \citep{Fritz2010}. The =
statisticaluncertainty of the measured S2 positions is =
smaller than the assumedvalue. However, unrecognised =
confusion events with fainter stars arean additional, =
important error source, such that we consider our =
valuerealistic. Furthermore, we assume that the radial =
velocity of S2 isdetermined at two epochs per year. Here, we =
adopt a Gaussian error of$15\kms$. This value is typically =
reached with current NIRmedium-resolution =
spectrographs.

In this way we obtain 960 =
simulated data sets that are reasonablyclose to what one =
would obtain by simply continuing the monitoring oforbits in =
the Galactic Centre with existing instruments. The fact =
thatwithin the next 50 years new facilities with improved =
angularresolution will become available (such as the NIR =
interferometersASTRA \citep{astra2008} and GRAVITY =
\citep{eisen08}, or the extremelylarge telescopes TMT and =
E-ELT) means that an IBH probably will bedetectable more =
easily than what we derive here. There is a secondreason why =
we consider our procedure conservative. Observers may adopta =
sampling strategy for the orbits other than a simple constant =
rate,as we assume here. Given the fact that the most =
constraining part ofan orbit is the periapse passage, and the =
event is predictable, anintensification of the observations =
around the periapse passageprovides an improved sampling =
pattern.

We fit each of the 960 data sets with =
the same code as used in\citet{Gillessen2009}. =46rom the =
fits, we determine the full set of 13parameters describing =
the orbits and the potential: The six orbitalelements =
$(a,\,e,\,i,\,\Omega,\,\varpi,\,t_P)$ and seven =
parametersdescribing the gravitational potential of the MBH: =
Mass, distance,on-sky position (2 parameters) and velocity (3 =
parameters) of the MBH.While in the simulations these =
quantities are known, this is not thecase when the data sets =
are considered as mock observations. Thenthese quantities =
have to be treated as free fit parameters, since theyneed to =
be determined from the same orbital data from which =
thepresence of an IBH shall be judged. Currently, nearly all =
constraintson these parameters actually come from the =
S2-orbit which we considerhere. In future, some of the =
parameters describing the MBH might bedetermined =
independently from the orbit of other S-stars. We neglectthis =
here and keep all seven parameters completely free. This =
isconservative, because additional constraints would make the =
presenceof an IBH more easily =
detectable.

Using a purely Keplerian point-mass =
model is inadequate for all ourdata sets. The relativistic =
precession is too large during the 50years of evolution and =
therefore we include the first order PNcorrection to the =
equations of motion when fitting the =
orbits\citep{Gillessen2009}. For example, all twelve =
data sets with $(q=3D1.0\times 10^{-3},\, a =3D =
30\,\mathrm{mpc},\, e=3D0.9)$ are well fit by therelativistic =
equations. In contrast, none of these data sets can =
bedescribed by a purely Keplerian point-mass potential. =
Still, therelativistic potential only yields a perfect =
fit for some of the datasets. For others, the IBH perturbs =
the dynamics too strongly, andwould therefore be detectable. =
Given that our sampling in radialvelocity is rather =
sparse, the inclusion of special relativisticcorrections to =
the radial velocities is not =
needed.

\begin{figure} \beg=
in{center} =
\includegraphics[width=3D8cm]{figure5.eps} \en=
d{center} \caption{Examples of fit residuals. Top =
row: A fit using a simulated data set with =
$(q=3D1.0 \times =
10^{-4},\,a=3D3\,\mathrm{mpc},\,e=3D0)$. Middle row: a fit =
with $(q=3D2.5 \times =
10^{-4},\,a=3D0.3\,\mathrm{mpc},\,e=3D0.9)$. =
Bottom row: a fit with $(q=3D5 \times =
10^{-4},\,a=3D1\,\mathrm{mpc},\,e=3D0)$. =
The first (top) one is classified as acceptable and has a =
reduced $\chi^2 =3D 1.15$. The other two =
are classified as not acceptable and have =
reduced $\chi^2$ values of 4.4 and 22.7 =
respectively.} \label{fig:residuals}\end{=
figure}

Note that it is not legitimate to =
assess the goodness of a fit bycomparing the obtained =
parameters with those used as input for thesimulations. =
Instead, we use the reduced $\chi^2$ for each of the 960fits =
to decide whether a fit is acceptable or not. We obtain =
valuesbetween 0.88 and 357. In addition, we examine the =
residuals of eachfit by eye, dividing them into two =
categories: A set of fits for whichthe residuals do not =
visually show obvious correlations and a set forwhich it is =
apparent that the chosen gravitational potential model isnot =
adequate. Figure~\ref{fig:residuals} shows typical examples of =
theresiduals from three =
simulations.

Each of the simulated data sets =
has $418\pm 1$ astrometric data pointsand $142\pm 1$ radial =
velocity measurements. The exact numbers vary dueto the =
random variations of the sampling pattern. Given that our =
fitshave 13 free parameters, the number of degrees of freedom =
is$n_\mathrm{dof}\approx 2\times =
418+142-13=3D965$.

\begin{figure} =
; \begin{center} =
\includegraphics[width=3D6cm]{figure6.eps} \en=
d{center} \caption{Distribution of reduced =
$\chi^2$ for the 960 fits in logarithmic =
bins. Entries are coloured according to our =
visual classification of whether the =
residuals of any given fit is acceptable =
(blue/solid) or not (red/hatched). The black, =
dashed line marks the optimum cut at 1.22 =
separating good from bad fits by the value =
of their reduced =
$\chi^2$.} \label{fig:dist}\end{figure}

Figure~\ref{fig:dist} shows the distribution of =
reduced $\chi^2$together with the flag whether a given fit is =
acceptable ornot. Clearly, the reduced $\chi^2$ can be used =
as discriminator. Thelargest reduced $\chi^2$ corresponding =
to a fit classified as'acceptable' is 1.36, the smallest =
reduced $\chi^2$ corresponding to afit classified as 'not =
acceptable' is 1.12. The optimum cut is at areduced $\chi^2$ =
of 1.22, yielding as many good fits above as bad fitsbelow =
the threshold. The total number of fits misclassified by =
thiscut is 41. The total number of bad fits is 409 and =
correspondingly 551fits have a reduced $\chi^2$ below the =
threshold. Hence, the IBHwould be detectable from the data in =
$\approx$43\% of the cases. Wefind a dependence of the =
fraction of bad fits on the assumed mass forthe IBH. For =
$q=3D10^{-4}$, $2.5\times 10^{-3}$, $5\times =
10^{-3}$,$10^{-3}$ the percentage of detectable IBHs is 15\%, =
39\%, 51\%, and66\%, =
respectively.

Having a reduced $\chi^2 \geq =
1.22$ corresponds formally to a$4.7\sigma$ detection of the =
effects of the IBH for$n_\mathrm{dof}=3D965$. In =
reality, there will be perturbing effectssuch as confusion =
events, recognised or unrecognised. Hence, theactual $\chi^2$ =
might be larger than in the simulations and the =
simpleGaussian statistics will not apply. We are nevertheless =
confident thatby observing the astrometric (and photometric) =
residuals as a functionof time the effects of an IBH can be =
disentangled from confusionevents. The patterns of the =
residuals contain more information than asingle $\chi^2$ =
value.

\begin{figure} \begi=
n{center} =
\includegraphics[width=3D8cm]{figure7.ps} \end=
{center} \caption{Distributions of reduced =
$\chi^2$ for the 960 fits as a function of =
the black hole binary initial parameters: mass, =
semi-major axis, eccentricity, periapse distance, =
orbital orientation and minimum distance to =
S2.} \label{fig:dist2}\end{figure}<=
div>In Figure~\ref{fig:dist2} we examine the dependence of the =
reduced$\chi^2$ on the black hole binary parameters. The left =
panel in thefirst row shows that the fits on average get =
worse when the mass ofthe assumed IBH is increased, as =
expected. The right panel in thefirst row shows that a value =
for the semi major axis of the IBHaround $\approx 1\mpc$ =
leads on average to the worst =
fits.\begin{figure} \begin{center}<=
div> =
\includegraphics[width=3D6cm]{figure8.ps} \end=
{center} \caption{Reduced $\chi^2$ for the 960 =
fits as a function of mass and semi major =
axis of the IBH. The plot shows the median at each =
grid point of the $\chi^2$ values. The =
thick dashed line marks our threshold of =
$\chi^2 =3D 1.22$. Fits right of the line are not =
acceptable and thus the IBH would be discoverable. =
} \label{fig:dist3}\end{figure} These two parameters correlate actually best with the =
reduced $\chi^2$, and in Figure~\ref{fig:dist3} we show =
the reduced $\chi^2$ in the $M_\mathrm{IBH}$-$a$-plane. =
The unacceptable fits occupy a well-defined region in =
this plot.

The goodness of the fits, on the =
other hand, is fairly independent ofthe eccentricity of the =
IBH orbit (second row ofFigure~\ref{fig:dist2}, left panel). =
The right panel in the secondrow shows that the =
periapsis distance $p=3Da\,(1-e)$ of the IBH is aless good =
predictor for the fit than the semi major axis. The =
reduced$\chi^2$ does not correlate with the finite set of =
orbitalorientations probed, as shown in the third row, left =
panel ofFigure~\ref{fig:dist2}. Here, $\alpha$ represents the =
angle betweenthe angular momentum vectors of S2 and the IBH =
at the start of thesimulations. This incidentally argues =
against the possibility that theobserved deviations are due =
to Kozai oscillations, since the mechanismrequires large =
relative inclinations to operate. However, fits withthe =
same sets of parameters $(q,\,e,\,a)$ but different $\alpha$ =
canhave different values of $\chi^2$. This means that =
the knowledge of$(q,\,e,\,a)$ is insufficient to predict the =
reduced $\chi^2$ butinformation on the sky position is =
necessary. Finally, the right panelin the third row of =
Figure~\ref{fig:dist2} shows that the minimum 3Ddistance =
between S2 and the IBH also correlates with the =
reduced$\chi^2$. In general it holds that the smaller the =
minimum distance,the worse the corresponding fit. Clearly, =
this parameter is notindependent of the semi-major =
axis.

The initial =
parameters adopted for the black hole binary are not =
sampledhomogeneously. This is obvious for the first three =
panels inFigure~\ref{fig:dist2} showing the reduced $\chi^2$ =
as a function ofthe binary parameters $(q,\,e,\,a)$. But it =
also holds (and is lessobvious) for the plot investigating =
the minimum distance between S2and the =
IBH.

\begin{figure} \begin{=
center} =
\includegraphics[width=3D4cm]{figure9a.eps} =
\includegraphics[width=3D4cm]{figure9b.eps} \e=
nd{center} \caption{Reduced $\chi^2$ as a function =
of black hole mass (left) and eccentricity =
(right) for two set of runs with the IBH starting =
at periapsis (dots) and apoapsis =
(circles).} \label{fig:cfr}\end{figure}A comparison of the goodness of fit for the runs starting =
with the IBH at periapsis and apoapsis is shown in =
Figure~\ref{fig:cfr}.In both cases, the semi-major axis of =
the black hole orbit is $a=3D10\mpc$. We find a modest =
worsening of the $\chi^2$ in the caseof an IBH initially at =
the apoapsis of its orbit. This can beattributed to the fact =
that S2's apoapsis, where the star spends mostof its time, is =
about $10\mpc$.

Finally, we also investigated =
the minimum time required for the IBHto become detectable. =
For this purpose, we repeated the orbital fitsfor a few cases =
assuming that the observations span 10,15,20,25, 30, 35, 40, =
45 or 50 years =
(Figure~\ref{fig:time}). \begin{figure} &n=
bsp;\begin{center} =
\includegraphics[width=3D6cm]{figure10.ps} \en=
d{center} \caption{Reduced $\chi^2$ as a function =
of the length of the observations for four =
cases: blue (long-dashed): $q=3D5 \times =
10^{-4}$, $e=3D0.9$, $a=3D1\mpc$; red (dashed): $q=3D10^{-3}$, =
$e=3D0.9$, $a=3D1\mpc$; green (solid): =
$q=3D10^{-3}$, $e=3D0$, $a=3D0.3\mpc$; black =
(dot-dashed): $q=3D10^{-3}$, $e=3D0.7$, $a=3D3\mpc$. The vertical =
lines indicate the second and third =
periapse passage covered by the =
simulations, the first one happens at $t\approx 10\yr$. =
The horizontal, dashed line is our optimum =
cut at =
1.22.} \label{fig:time}\end{figure}=
Our initial conditions are such that the first periapse passage of =
S2happens after 10 years, the second after 26 years and the =
third after42 years. Figure~\ref{fig:time} shows that the =
reduced $\chi^2$ startsto increase beyond our threshold of =
1.22 after the second periapsispassage for fits that show a =
large reduced $\chi^2$ after 50 simulatedyears (red/dashed, =
blue/long-dashed and green/solid curves). Only forfits that =
after 50 simulated years have a reduced $\chi^2 \simless 3$is =
the threshold passed after the third periapse passage =
(blackcurve). The discrete nature of periapse passages also =
is the reasonwhy the blue/long-dashed and green/solid curves =
level out after about45 years.

This =
is interesting in comparison with the current status of =
theobservations. Since by now only one periapse passage of S2 =
has beenobserved in 2002, one would not expect to have =
detected an IBH fromthe actual S2 data so far. This is =
particularly true since the assumedlevel of accuracy was not =
reached during the first years of theobservations (1992 - =
2002), and radial velocity information is onlyavailable after =
2002 (with the exception of one point in 2000). Hence,the =
first real chance to detect an IBH will be after the next =
S2periapse passage, which will happen in =
2018.

\section{Other sources of =
orbital =
torque}\label{sec:torque}

Here we =
consider other perturbations that could induce changesin S2's =
orbital angular momentum, potentially complicating thesignal =
from an IBH.We find that almost all such alternative =
perturbations are smallcompared to the torque produced by an =
IBH.

Angles quoted in this section are =
intrinsic, not astrometric.As noted above, current data =
are able to determine theorbital angles ($\varpi,\Omega,i$) =
of S2 with an accuracy ofabout one degree.Changes =
induced by an IBH per orbit of S2 are $10^{-3} =
\mathrm{deg} \simless \Delta(i,\Omega) \simless 1 =
\mathrm{deg}$(Figure~2).

{\em Spin =
of the MBH.}Frame-dragging effects from a spinning MBH =
include anadditional in-plane precession term as well as a =
precession of theorbital plane. Defining $i^\prime$ and =
$\Omega^\prime$ asthe inclination and nodal angle of S2's =
orbit with respect to the MBH's equatorial =
plane, to lowest PN order, frame dragging induces =
changes\begin{subequations}\begin{eqnarray}\Delta\varpi &=3D& -2A_J\cos i^\prime, =
\\ \Delta\Omega^\prime &=3D& =
A_J\end{eqnarray}\end{subequations}per =
revolution in the angle of periapse and the line of =
nodes,respectively, =
where\begin{subequations}\begin{eqnarray}=
A_J &=3D& \frac{4\pi\chi}{c^3} =
\left[\frac{GM_\mathrm{MBH}}{(1-e^2)a}\right]^{3/2} =
\\&\approx& 0.115' \left(1-e^2\right)^{-3/2}\chi =
\left(\frac{a}{\mathrm{mpc}}\right)^{-3/2}\end{eqnarray}<=
div>\end{subequations}and $\chi\le 1$ is the dimensionless =
spin of the MBH \citep[e.g.][]{MAMW2010}.The orbital =
inclination $i^\prime$ remains unchanged.For S2, the spin =
contribution to advance of the periapse is $\sim 1\%$ of =
theSchwarzschild contribution even for $\chi=3D1$ and so can =
be ignored.The nodal advance is $\Delta\Omega^\prime \approx =
0.002\,\chi\,\rm deg$, too small to be detectable in the =
next few decades of monitoring,and smaller than the changes =
induced by an IBH in almost all of theruns carried out here =
(Fig.~2).Effects of frame dragging are only likely to be =
important for stars much closer to SgrA$^*$ than S2 =
\citep{MAMW2010}.(Frame dragging could nevertheless be =
relevant to the orbit of an IBH.For the IBH orbit with =
smallest$a$ ($0.3$ mpc) and largest $e$ ($0.9$) considered =
here,the precession time drops to $\sim 300\chi^{-1}$ =
yr.)

{\em A stellar bar.} If the gravitational =
potential due to thedistributed mass is appreciably =
non-spherical, the resultant torquescould affect all of the =
orbital elements of S2 aside from $a$.For instance, the =
nuclear bar described by \citet{alard01} has been =
modelled as a triaxial spheroid with central density$\sim 150 =
\msun \mathrm{pc}^{-3}$ \citep{combes08}.In the most extreme =
case, the nuclear star cluster (NSC) of the Milky Way =
couldbe stratified on triaxial ellipsoids at all =
radii.A homogeneous, non-rotating triaxial bar induces =
changes in theinclination and nodal angle of a test star =
orbiting near the MBH,with characteristic time =
scale\begin{equation}T_{\Omega,i}\approx =
\frac{\sqrt{1-e^2}}{T_i-T_j} \frac{1}{P =
G\rho_b}\end{equation}\citep[][equations =
11-15]{MV10}.Here, $\rho_t$ is the density of the triaxial =
component,and $(T_x,T_y,T_z)$ are the dimensionless =
coefficients,of order unity, that define the shape of the =
triaxial component\citep{chandra69}; the torque in the =
$(i,j)$ principal plane is proportional to $T_i-T_j$, =
etc.For S2, the angular reorientation over one orbit due to =
torquesfrom a triaxial bar would be of =
order\begin{equation}\Delta\phi\approx =
0.2\mathrm{mas}\ (T_i-T_j)\left(\frac{\rho_b}{100 \msun =
\mathrm{pc}^{-3}}\right),\end{equation}undetectable =
even if $\rho_b\approx 10^5\msun \mathrm{pc}^{-3}$,the =
approximate density of the NSC at $1$ pc from =
SgrA$^*$.

{\em Resonant =
relaxation:}Discreteness in the distribution of stars and =
stellar remnantsis also a potential source of =
torque.``Vector resonant relaxation'' produces changes in =
thedirection of the orbital angular momentum of =
order \begin{equation}\Delta\phi \approx =
\frac{\sqrt{N}m}{\mbh} \end{equation}per =
radial period,where $m$ is the mass of a background star and =
$N$ is the numberof such stars within S2's =
orbit.Writing $N=3DM_\star/m$, with $M_\star$ the distributed =
masswithin S2's orbit, this =
is\begin{eqnarray}\Delta\phi &\approx& =
\left(\frac{M_\star}{\mbh}\frac{m}{\mbh}\right)^{1/2} =
\\&\approx& 0.003\ =
\mathrm{deg} \left(\frac{M_\star/\mbh}{10^{-2}}\right)^{1/=
2}\left(\frac{\mbh}{4\times 10^6m}\right)^{-1/2}. =
\end{eqnarray}Since $M_\star\simless =
10^{-2}\mbh$ \citep{gillS2},this is smaller even than the =
change due to frame-draggingunless the mean perturber mass =
$m\gg\msun$.

\secti=
on{Discussion and conclusions}We have shown that an IBH =
orbiting Sgr A$^*$ at the distance of the S-stars can =
cause observable deviations in the orbit of star S2. The =
perturbations potentially affect all the orbital =
elements, but the key signature would be a change in =
S2's angular momentum (eccentricity, orbital plane) due =
to the non-spherically-symmetric forces from the =
IBH. In particular, we find that an IBH more =
massivethan $\sim 1000\msun$ at a distance of $\sim 1-5\mpc$ =
is potentiallydiscoverable at the next periapse passage of =
S2.

The presence of an IBH companion to the MBH =
might produce anobservable feature in the radio observations =
of SgrA*. In particular,some combinations of the binary =
parameters considered here mightproduce a reflex motion of =
the MBH which is larger than the currentlyavailable limits on =
the proper motion of SgrA*. Such combinations ofparameters =
could therefore be considered unlikely. \citet{r09} =
reporta proper motion of $\left(7.2 \pm 8.5\right)\kms$ in =
the plane of theGalaxy and of $\left(-0.4\pm0.9\right)\kms$ =
in the directionperpendicular to the plane. Since it appears =
unlikely that the motionof the MBH lies primarily in the =
Galactic plane, we adopt the value ofthe perpendicular =
component of the velocity as our fiducial value. =
InFigure~\ref{fig:vbh} we compare the 3D root mean square =
velocity ofthe MBH obtained from the simulations with the =
observational $3\sigma$upper =
limit.\begin{figure} \begin{center}=
=
\includegraphics[width=3D8cm]{figure11.eps} \e=
nd{center} \caption{Root mean square velocity of =
the MBH as a function of the binary mass =
ratio. The dotted line represents the $3\sigma$ =
limit on the 3D proper motion of SgrA* =
derived from =
\citet{r09}.} \label{fig:vbh}\end{figure}=
We find that the motion of the MBH induced by the orbiting =
IBH isgenerally smaller than the $3\sigma$ limit derived from =
radioobservations of SgrA*. Only for $q > 10^{-3}$ and =
$a=3D0.3\mpc$ does thesimulated motion exceed the limit. Of =
course, if the motion induced bythe IBH were to lie primarily =
in the Galactic plane, it would be veryhard to detect via =
radio observations, even for large values of =
$q$.

```
\section*{Acknowledgments}DM acknowledges support from the National Science Foundation =
undergrants no. AST 08-07910, 08-21141 and by the National =
Aeronautics andSpace Administration under grant no. =
NNX-07AH15G.
```

%\bibliographystyle{mn2e}=
%\bibliography{biblio}

```
\begin{thebibliography}{}<=
/div>
```

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```
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