Way. Near-infrared imaging and =
spectroscopic observations allow an
accurate determination of =
the orbit of the star. Given S2's short
with =
current technology. We show that perturbations due to an IBH =
in
as large or even larger than the GR shift. An IBH =
will also induce
degree per period. We apply observational =
orbital fitting techniques to
will =
occur in =
2018.
black hole physics -- stellar =
dynamics - methods: $N$-body simulations -- Galaxy: =
centre
supermassive black hole (MBH). =
Near-infrared observations of the
stars. These can =
be used to derive fundamental parameters like the
general relativity (GR) \citep{rubilar2001,zucker2006}. =
Since the
migrated to their =
current location. Among the in-situ models are the
\citep{madigan09}. The two main models suggested =
for the transport of
scenario =
\citep{gq03,perets07,perets09}. The properties and =
efficiency
of these models are discussed in =
\citet{PG10}.
0.0034$ \citep{gillS2}. =
Adopting a distance to the Galactic
0.585\mpc$, =
$r_a =3D 9.419\mpc$ . For this set of orbital
\citep{Gillessen2009}.
\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)}
hole. For the =
Galactic centre black hole, $R_S \approx
\citep{gillS2}.
semi-major axis.
\left(\frac{\mbh}{4.3\times10^6\msun}=
\right) =
\left(\frac{8.28\kpc}{R_0}\right)
which=
amounts to $\sim 0.83\mas$ for star S2.
within S2's orbit, consisting of stars and stellar =
remnants. Assuming
\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 =
distributed
mass within S2's orbit is $\simgreat =
10^{-3}\mbh$.
Disentangling these two sources of precession =
will be difficult, requiring
measurements of the radial =
velocity of S2 near periapse =
passage
\citep[e.g.][]{saha10}.
Non-sph=
erically-symmetric perturbations, if present, can affect =
not
just the periapse angle but also the angular momentum of =
S2's orbit,
resulting in changes in $e$ and in the orientation =
of its orbital
plane. Potentially the largest source of =
such perturbations is a
second massive black hole orbiting =
somewhere near S2. In fact, the
orbits of the S-stars =
are consistent with the long-term presence of an
IBH in their =
midst, with mass $\sim 10^3\msun$ \citep{mgm09,GM2009}.
While =
an IBH would also contribute to the periapse advance of S2, =
its
key 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 the
observable effects =
of an IBH on the orbit of star S2 over a time-scale
of 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 with
perturbations 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-star
cluster. The =
initial conditions for the stars are derived from the
orbital =
elements given by \citet{Gillessen2009}. For star S2, we =
take
the improved elements from \citet{gillS2}. Of the =
28 stars for which
those 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, =
for
which we determined positions and velocities at 2008 AD =
from the
classical 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 =
four
different 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 =
angular
momentum vector (the same set as in \citet{GM2009}), =
for a total of
960 sets of initial conditions. The IBH =
begins from orbital periapsis
in all cases. The initial =
value of the mean anomaly is likely to be
unimportant in all =
cases for which the orbital period of the binary is
much =
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 the
integration, and in principle the choice of the =
initial position might
have an effect on the interaction with =
S2. However, at a distance of
$30\mpc$ the IBH is completely =
outside S2's orbit for $e \simless
0.7$, and there is =
essentially no detectable signature, as shown
below. The only =
set of simulations for which the initial mean anomaly
of the =
IBH might be important is the $a=3D10\mpc$, which corresponds =
to
an orbital period of about $43\yr$ for the IBH. We perform =
an extra
set of simulations for $a=3D10\mpc$ starting the IBH =
at apoapsis rather
than 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 algorithmic
regularisation method that =
is able to reproduce the motion of tight
binaries for long =
periods of time with extremely high precision. The
code =
combines the use of the chain structure, introduced originally =
by
\cite{ma93}, with a new time transformation to avoid =
singularities and
achieve high precision for arbitrary mass =
ratios. Note that we
self-consistently follow not just =
the interactions of S2 with the MBH
and IBH, but all other =
interactions as well, including star-star
interactions. =
The integration interval was 50 years, during which
time =
S2 performs three full orbits.
The AR-CHAIN =
code includes relativistic corrections to the
accelerations up =
to 2.5 post-Newtonian order for all interactions
involving the =
MBH particle. General relativistic advance of =
the
periapse, 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 term
arising from the =
emission of gravitational waves is accounted for by
the 2.5PN =
term; this term is potentially important for the IBH, =
for
which 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 decay
time-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 =
adopted
values of the initial eccentricity. This =
time-scale is always much
longer than our integration interval =
of $\sim 50$ yr. In addition, it
is 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 =
to
associate our initial parameters for the IBH/MBH binary =
with the
parameters at some much earlier time, e.g. the epoch =
preceding
formation of the S-stars. In the latter runs, =
the orbit of the IBH at
some much earlier time would have been =
larger and/or more eccentric.
The maximum relative variation =
of the binary semi-major axis in the $N$-body
integrations is =
$\Delta a / a \sim 10^{-3}$ while the absolute variation
of =
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 $\varpi$ evolves
due to in-plane =
precession. The two remaining angles that define =
the
orientation of the orbit, $i$, the inclination, and =
$\Omega$, the
position angle of the ascending node, are fully =
conserved in the
relativistic 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 =
are
conserved at the 1PN level, and we expect very small =
deviations due to
higher order PN corrections in the =
integrations \citep{soffel89}. In
the limit of small =
star-to-black hole mass ratio, the semi-major axis
and =
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 vectors
between the star and the MBH, $M$ is the =
total mass, and $c$ is the speed
of =
light.
The presence of the other S-stars =
introduces small deviations from
spherical symmetry in the =
gravitational potential but the effect on
the 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 S2
in our =
simulations,
namely the semi-major axis, eccentricity, =
inclination, and position
angle of the ascending node, can be =
attributed to perturbations from
the =
IBH.
Figure~\ref{fig:change} summarises the =
changes in the orbital elements
of S2 found in the $N$-body =
integrations. Plotted are the variations
over one =
revolution averaged over the twelve different orientations =
of
the 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 =
the
variations due to GR. The shift in the periapse =
corresponds to an
observable angle $\Delta \Theta_p =3D \Delta =
\Theta_a
\left(1-e\right)/\left(1+e\right)$, where $\Delta =
\Theta_a$ is defined
in Eq.~\ref{eq:wein}. Note that the =
variations in the inclination $i$
and position angle $\Omega$ =
of the ascending node reach values close
to 1 degree for the =
most massive IBHs considered. This is of the same
order as the =
current observational accuracy ($\sim 0.7\,\rm deg$). In
50 =
years, this value will drop to about $0.4\,\rm deg$, assuming =
there
are no technological =
improvements.
While precession induced by the =
PN terms is restricted to the orbital
plane, an IBH induces =
more general changes in the orbital elements,
including =
changes in the direction of the orbital angular =
momentum
vector. 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 =
averagingover the 12 different IBH =
orientations.
The changes in the orbital plane =
of S2 are larger for more massive
IBHs and can reach values of =
$\sim 1$ degree for $q =3D
10^{-3}$. Out-of-plane motion is =
also affected by the size of the
MBH/IBH binary orbit such =
that changes are largest for $a =
\simless
10\mpc$.
The three-body =
system composed of MBH, IBH and S2 is reminiscent of a
Kozai =
triple \citep{kozai62}. However, the changes that we observe =
in
the orbital elements are not generally attributable to the =
Kozai
mechanism. Kozai oscillations can be induced if =
(i) the MBH - S2 -
IBH system can be regarded as a =
hierarchical triple, with
well-separated orbital periods for =
the inner and outer orbit; (ii) the
period predicted for the =
Kozai oscillations is shorter than that of
any other =
precessional period, in this case, GR precession; (iii) =
the
outer orbit is largely inclined with respect to the inner =
orbit. The
first 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 =
outer
binary, $\eout$ is the eccentricity of the outer orbit =
and
$\mathcal{K}$ is a numerical coefficient which depends =
only on the
initial values of the relative inclination angle =
$\alpha$, the inner
binary eccentricity $\einn$ and the inner =
binary argument of periapsis
$\varpi$. The maximum =
eccentricity $e_{\rm max}$ attained by the
inner 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 =
hole
binary mass ratio for the assumed values of the =
eccentricity and for
two allowed separations. We find =
that $T_K$ is always longer then
S2's GR precession timescale =
of $2.5\times10^4\yr$ for $a=3D30\mpc$. For
the 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 =
the
applicability of the Kozai mechanism to a small subset of =
our
simulations, in contrast with the results shown =
in
Figure~\ref{fig:change} and in the following section. =
We conclude that
the Kozai mechanism is not the dominant =
effect producing changes in
orbital $e$ and $i$ in our =
simulations. In those cases where it is
potentially =
relevant, the eccentricity is predicted to oscillate
between =
$e_{\rm min} =3D 0.24 - 0.74$ 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 sensitive
to =
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 IBH
becomes larger than the relativistic =
shift. This suggests a variation
in the orbital elements which =
is potentially observable with =
current
instrumentation.
However, the =
observability of variations in the orbit of S2 depends =
on
several factors. In the following section we thoroughly =
examine all
such factors and use orbital fitting to determine =
whether an IBH is
detectable via on-going monitoring of the =
S-cluster. Theoretically,
it would be possible to use =
other S-stars to investigate the effects
of a hypothetical =
IBH. Given that the shifts in the apparent
location of =
periapsis and apoapsis depend only on the eccentricity and
not =
on the semi-major axis of the stellar orbit, it would =
seem
appropriate to consider all stars with $e \simgreat 0.8$ =
for such an
analysis. =46rom an observational point of =
view, however, S2 is the
only star in the sample which is =
bright enough and not affected by
confusion to allow for =
meaningful tests of the gravitational
potential. We therefore =
limit our study to star =
S2.
\section{Orbita=
l fitting}
In this section we extract observational-like data =
from the simulated
orbital traces of S2. We assume that eight =
or nine astrometric epochs
can be obtained each year over the =
course of 50 years. The eight or
nine yearly epochs are not =
evenly distributed but are spread over
seven months only, thus =
taking into account the fact that the Galactic
Centre is =
accessible with NIR observations only for part of the
year. At =
the chosen epochs, the original, simulated positions =
are
perturbed by an astrometric error, assumed to be =
distributed in a
Gaussian fashion. We use a value of =
$300\,\mu$as per coordinate, which
is a conservative =
assumption for S2 \citep{Fritz2010}. The =
statistical
uncertainty of the measured S2 positions is =
smaller than the assumed
value. However, unrecognised =
confusion events with fainter stars are
an additional, =
important error source, such that we consider our =
value
realistic. Furthermore, we assume that the radial =
velocity of S2 is
determined at two epochs per year. Here, we =
adopt a Gaussian error of
$15\kms$. This value is typically =
reached with current NIR
medium-resolution =
spectrographs.
In this way we obtain 960 =
simulated data sets that are reasonably
close to what one =
would obtain by simply continuing the monitoring of
orbits in =
the Galactic Centre with existing instruments. The fact =
that
within the next 50 years new facilities with improved =
angular
resolution will become available (such as the NIR =
interferometers
ASTRA \citep{astra2008} and GRAVITY =
\citep{eisen08}, or the extremely
large telescopes TMT and =
E-ELT) means that an IBH probably will be
detectable more =
easily than what we derive here. There is a second
reason why =
we consider our procedure conservative. Observers may adopt
a =
sampling strategy for the orbits other than a simple constant =
rate,
as we assume here. Given the fact that the most =
constraining part of
an orbit is the periapse passage, and the =
event is predictable, an
intensification of the observations =
around the periapse passage
provides 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 13
parameters describing =
the orbits and the potential: The six orbital
elements =
$(a,\,e,\,i,\,\Omega,\,\varpi,\,t_P)$ and seven =
parameters
describing 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 the
case when the data sets =
are considered as mock observations. Then
these quantities =
have to be treated as free fit parameters, since they
need to =
be determined from the same orbital data from which =
the
presence of an IBH shall be judged. Currently, nearly all =
constraints
on these parameters actually come from the =
S2-orbit which we consider
here. In future, some of the =
parameters describing the MBH might be
determined =
independently from the orbit of other S-stars. We neglect
this =
here and keep all seven parameters completely free. This =
is
conservative, because additional constraints would make the =
presence
of an IBH more easily =
detectable.
Using a purely Keplerian point-mass =
model is inadequate for all our
data sets. The relativistic =
precession is too large during the 50
years of evolution and =
therefore we include the first order PN
correction 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 the
relativistic =
equations. In contrast, none of these data sets can =
be
described by a purely Keplerian point-mass potential. =
Still, the
relativistic potential only yields a perfect =
fit for some of the data
sets. For others, the IBH perturbs =
the dynamics too strongly, and
would therefore be detectable. =
Given that our sampling in radial
velocity is rather =
sparse, the inclusion of special relativistic
corrections 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 by
comparing the obtained =
parameters with those used as input for the
simulations. =
Instead, we use the reduced $\chi^2$ for each of the 960
fits =
to decide whether a fit is acceptable or not. We obtain =
values
between 0.88 and 357. In addition, we examine the =
residuals of each
fit by eye, dividing them into two =
categories: A set of fits for which
the residuals do not =
visually show obvious correlations and a set for
which it is =
apparent that the chosen gravitational potential model is
not =
adequate. Figure~\ref{fig:residuals} shows typical examples of =
the
residuals from three =
simulations.
Each of the simulated data sets =
has $418\pm 1$ astrometric data points
and $142\pm 1$ radial =
velocity measurements. The exact numbers vary due
to the =
random variations of the sampling pattern. Given that our =
fits
have 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 or
not. Clearly, the reduced $\chi^2$ can be used =
as discriminator. The
largest reduced $\chi^2$ corresponding =
to a fit classified as
'acceptable' is 1.36, the smallest =
reduced $\chi^2$ corresponding to a
fit classified as 'not =
acceptable' is 1.12. The optimum cut is at a
reduced $\chi^2$ =
of 1.22, yielding as many good fits above as bad fits
below =
the threshold.
The total number of fits misclassified by =
this
cut is 41. The total number of bad fits is 409 and =
correspondingly 551
fits have a reduced $\chi^2$ below the =
threshold. Hence, the IBH
would be detectable from the data in =
$\approx$43\% of the cases. We
find a dependence of the =
fraction of bad fits on the assumed mass for
the 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\%, and
66\%, =
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 effects
such as confusion =
events, recognised or unrecognised. Hence, the
actual $\chi^2$ =
might be larger than in the simulations and the =
simple
Gaussian statistics will not apply. We are nevertheless =
confident that
by observing the astrometric (and photometric) =
residuals as a function
of time the effects of an IBH can be =
disentangled from confusion
events. The patterns of the =
residuals contain more information than a
single $\chi^2$ =
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 the
first row shows that the fits on average get =
worse when the mass of
the assumed IBH is increased, as =
expected. The right panel in the
first row shows that a value =
for the semi major axis of the IBH
around $\approx 1\mpc$ =
leads on average to the worst =
fits.
\begin{figure}
\begin{center}
<=
div> =
\includegraphics[width=3D6cm]{figure8.ps}
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. =
}
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 of
the eccentricity of the =
IBH orbit (second row of
Figure~\ref{fig:dist2}, left panel). =
The right panel in the second
row shows that the =
periapsis distance $p=3Da\,(1-e)$ of the IBH is a
less good =
predictor for the fit than the semi major axis. The =
reduced
$\chi^2$ does not correlate with the finite set of =
orbital
orientations probed, as shown in the third row, left =
panel of
Figure~\ref{fig:dist2}. Here, $\alpha$ represents the =
angle between
the angular momentum vectors of S2 and the IBH =
at the start of the
simulations. This incidentally argues =
against the possibility that the
observed deviations are due =
to Kozai oscillations, since the mechanism
requires large =
relative inclinations to operate. However, fits with
the =
same sets of parameters $(q,\,e,\,a)$ but different $\alpha$ =
can
have different values of $\chi^2$. This means that =
the knowledge of
$(q,\,e,\,a)$ is insufficient to predict the =
reduced $\chi^2$ but
information on the sky position is =
necessary. Finally, the right panel
in the third row of =
Figure~\ref{fig:dist2} shows that the minimum 3D
distance =
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 not
independent of the semi-major =
axis.
The initial =
parameters adopted for the black hole binary are not =
sampled
homogeneously. This is obvious for the first three =
panels in
Figure~\ref{fig:dist2} showing the reduced $\chi^2$ =
as a function of
the binary parameters $(q,\,e,\,a)$. But it =
also holds (and is less
obvious) for the plot investigating =
the minimum distance between S2
and 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 case
of an IBH initially at =
the apoapsis of its orbit. This can be
attributed to the fact =
that S2's apoapsis, where the star spends most
of its time, is =
about $10\mpc$.
Finally, we also investigated =
the minimum time required for the IBH
to become detectable. =
For this purpose, we repeated the orbital fits
for 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 =
S2
happens after 10 years, the second after 26 years and the =
third after
42 years. Figure~\ref{fig:time} shows that the =
reduced $\chi^2$ starts
to increase beyond our threshold of =
1.22 after the second periapsis
passage for fits that show a =
large reduced $\chi^2$ after 50 simulated
years (red/dashed, =
blue/long-dashed and green/solid curves). Only for
fits that =
after 50 simulated years have a reduced $\chi^2 \simless 3$
is =
the threshold passed after the third periapse passage =
(black
curve). The discrete nature of periapse passages also =
is the reason
why the blue/long-dashed and green/solid curves =
level out after about
45 years.
This =
is interesting in comparison with the current status of =
the
observations. Since by now only one periapse passage of S2 =
has been
observed in 2002, one would not expect to have =
detected an IBH from
the actual S2 data so far. This is =
particularly true since the assumed
level of accuracy was not =
reached during the first years of the
observations (1992 - =
2002), and radial velocity information is only
available after =
2002 (with the exception of one point in 2000). Hence,
the =
first real chance to detect an IBH will be after the next =
S2
periapse passage, which will happen in =
2018.
\section{Other sources of =
orbital =
torque}
\label{sec:torque}
Here we =
consider other perturbations that could induce changes
in S2's =
orbital angular momentum, potentially complicating the
signal =
from an IBH.
We find that almost all such alternative =
perturbations are small
compared 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 the
orbital angles ($\varpi,\Omega,i$) =
of S2 with an accuracy of
about 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 an
additional in-plane precession term as well as a =
precession of the
orbital plane. Defining $i^\prime$ and =
$\Omega^\prime$ as
the 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 =
the
Schwarzschild 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 the
runs 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 the
distributed mass is appreciably =
non-spherical, the resultant torques
could 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 =
could
be stratified on triaxial ellipsoids at all =
radii.
A homogeneous, non-rotating triaxial bar induces =
changes in the
inclination 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 =
torques
from 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 remnants
is also a potential source of =
torque.
``Vector resonant relaxation'' produces changes in =
the
direction 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 number
of such stars within S2's =
orbit.
Writing $N=3DM_\star/m$, with $M_\star$ the distributed =
mass
within 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-dragging
unless 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 =
massive
than $\sim 1000\msun$ at a distance of $\sim 1-5\mpc$ =
is potentially
discoverable at the next periapse passage of =
S2.
The presence of an IBH companion to the MBH =
might produce an
observable feature in the radio observations =
of SgrA*. In particular,
some combinations of the binary =
parameters considered here might
produce a reflex motion of =
the MBH which is larger than the currently
available limits on =
the proper motion of SgrA*. Such combinations of
parameters =
could therefore be considered unlikely. \citet{r09} =
report
a proper motion of $\left(7.2 \pm 8.5\right)\kms$ in =
the plane of the
Galaxy and of $\left(-0.4\pm0.9\right)\kms$ =
in the direction
perpendicular to the plane. Since it appears =
unlikely that the motion
of the MBH lies primarily in the =
Galactic plane, we adopt the value of
the perpendicular =
component of the velocity as our fiducial value. =
In
Figure~\ref{fig:vbh} we compare the 3D root mean square =
velocity of
the 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 is
generally smaller than the $3\sigma$ limit derived from =
radio
observations of SgrA*. Only for $q > 10^{-3}$ and =
$a=3D0.3\mpc$ does the
simulated motion exceed the limit. Of =
course, if the motion induced by
the IBH were to lie primarily =
in the Galactic plane, it would be very
hard to detect via =
radio observations, even for large values of =
$q$.
\section*{Acknowledgments}
DM acknowledges support from the National Science Foundation =
under
grants no. AST 08-07910, 08-21141 and by the National =
Aeronautics and
Space Administration under grant no. =
NNX-07AH15G.
%\bibliographystyle{mn2e}
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