------------------------------------------------------------------------
From: Andreas Eckart eckart@ph1.uni-koeln.de
To: Heino Falcke
Subject: paper
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\lefthead{Eckart et al.}
\righthead{Stellar Orbits Near Sgr~A*}
\begin{document}
\title{Stellar Orbits Near Sagittarius~A*}
\author{A. Eckart}
\affil{I.Physikalisches Institut, Universit\"at zu K\"oln,
Z\"ulpicher Str.77, 50937 K\"oln, Germany}
\and
\author{R. Genzel, T. Ott, R. Sch\"odel}
\affil{Max-Planck-Institut f\"ur extraterrestrische Physik (MPE),
D-85740 Garching, Germany}
\received{}
\begin{abstract}
\noindent
The SHARP/NTT stellar proper motion data now cover an interval from
1992 to 2000 and allow us to determine orbital accelerations
for some of the most central stars.
We confirm the stellar acceleration measurements
obtained by Ghez et al. (2000) with NIRC at the Keck telescope.
Our analysis differs in 3 main points from that of Ghez et al.:
1) We combine the high precision
but shorter time scale NIRC/Keck data
with the lower precision but longer time scale
SHARP/NTT data set;
2) We statistically correct the observed accelerations for geometrical
projection effects;
3) We exclude star S8 from the analysis of the amount and position of
the central mass.
\\
\\
From the combined SHARP/NTT and NIRC/Keck data sets we
show that the stars S2, and most likely S1 and S8 as well,
are on bound, fairly inclined ($60^o*1$) that just happen to fly by the central position on
almost linear trajectories.
In the following we argue
that the sources S1, S2, and S8 are likely located within a
spherical volume of radius 15$\pm$3~$mpc$ (0.4''$\pm$0.1'').
{\it Estimate from velocities and positions:}
In the following $R$ denotes the projection of the 3-dimensional
separation $r$ of the star from the center
and $V_{star}$ the projection of the 3-dimensional
velocity $v_{star}$ of the star.
An estimate of the size can be obtained
by comparing the proper motion velocities $V_{star}$
to the 3-dimensional velocity dispersion
obtained from the parameterized functional form
of the true number density distribution $n(r)$
and the true radial and tangential velocity dispersions
$\sigma_r(r)$ and $\sigma_t(r)$ of the best anisotropic
Jeans model (Genzel et al. 2000).
Those quantities are integrated along the line of
sight at the projected separation $R$ from Sgr~A* of
the individual stars (formulae 5 and 6 in Genzel et al. 2000).
This way we use the projected position and velocity information
for each of the stars simultaneously.
Both $V_{star}$ and $R_{star}$ are lower limits to
$v_{star}$ and $r_{star}$.
For the probability of a star to exhibit a proper motion velocity $V$
in excess of $V_{star}$ we can therefore write
\begin{equation}
\label{eq01a}
P(V>V_{star},R) \ge P(V>V_{star},r) \ge P(V>v_{star},r)~~.
\end{equation}
For a given velocity dispersion
$\sigma^2(r) = \sigma_r^2(r) + 2 \sigma_t^2(r)$
the probability $P(V>V_{star},r)$ can be calculated via
\begin{equation}
\label{eq01b}
P(V>V_{star},r) = 1- P(V \le V_{star},r) =
1 - 1/\sigma^2(r)\int_0^{V_{star}} v\exp(-v^2/(2\sigma^2(r))) dv~~.
\end{equation}
The velocity dispersion $\sigma(r)$ and therefore also the
probability $P(V>V_{star},r)$ decrease with increasing radii $r$.
For fast stars it becomes increasingly unlikely that they belong to
statistical samples at correspondingly larger radii.
Therefore, we interprete $P(V>V_{star},r)$ as a measure of how likely
it is that the star belongs to a sample of stars
at that radius $r$ or larger.
Using $R$ instead of $r$ we can calculate
$P(V>V_{star},R)$ as an upper limit of this probability
$P(V>V_{star},r)$.
The mean probability of the three stars S1, S2, and S8 to belong to
samples of stars at the corresponding radii
$R_{S1}$, $R_{S2}$, and $R_{S8}$
- or larger - is only about $P_{init}=$33\%.
This implies that the mean probability of these three sources
to belong to samples of stars at their true 3-dimensional separations
$r_{S1}$, $r_{S2}$, and $r_{S8}$ - or larger - is even less than that.
The value of $P_{init}$ drops by a factor of 2 (i.e. to the FWHM
value of that probability) at a mean radius of $r=13.7~mpc$
and by a factor of
three at a radius of $r=14.8~mpc$.
The probability of the stars to belong to samples at even larger radii
lies well below 10\%.
We therefore adopted a value of about 15~$mpc$ (0.4``).
as a reasonable estimate of the radius of the volume that contains
all three stars.
{\it A safe lower bound} to the size estimate of the volume
described above is given by the
upper limit of the projected separation of the stars from SgrA$^*$.
Of the three stars S8 has the largest projected separation from the
center (see Tab.~\ref{t02}).
Therefore we adopt 12~$mpc$ (0.3'') as a lower bound to the radius of the
volume containing the three stars. This limit compares
favorably with the size estimate derived above.
\subsection{
\label{sec3.2}
Enclosed mass estimates from accelerations}
\normalsize
\subsubsection{
\label{sec3.2.1}
Correction of accelerations for projection effects}
\normalsize
For a star at a projected separation $R$ from
the center and a total enclosed mass M
one can calculate
the projected, observable acceleration $a_{obs}$ via
\begin{equation}
\label{eq02}
a_{obs} = G M cos^3(\theta) R^{-2}~~~.
\end{equation}
Here $\theta$ is the angle between the radius vector to the star and the
plane of the sky containing the central mass.
Plotting the lower limits $M cos^3(\theta)$ of the enclosed mass
as a function of the
projected radius is equivalent to the assumption that the stars
are in exactly the same plane of the sky as the central dark mass
at the position of Sgr~A$^*$.
This assumption is not justified and the
approach does not answer the question of whether or not the
observed projected accelerations are in agreement with the value
and compactness of the enclosed mass derived at larger radii with
different methods (Genzel et al. 2000, Ghez et al. 1998).
A more realistic approach needs to correct for
geometrical projection effects.
%Without information on the inclination of the orbits, the
%acceleration provides only a lower limit on the enclosed mass.
A statistical estimate of $M$ can be derived by using median values.
As a consistent error estimate we use the median error defined as the
median of the deviations of the individual estimates from their median.
The quantity $(cos \theta)^{-1}$ increases monotonically with the distance
from the plane of the sky in which Sgr~A$^*$ is located and its median
can be calculated under the assumption of a stellar density
distribution $n(r)$.
Using median values and a volume derived for a ensemble of
stars make this method of correcting for geometrical effects
much less susceptible to extreme correction values that occur
for instance in the case of stars with large physical separations and
small projected separations from SgrA* (i.e. $\theta$ approaching $\pi$/2).
Contributions from those values would become dominant in case of
a calculation of an expectation value for $(cos \theta)^{-1}$ using
$n(r)$ values as a weights.
\subsubsection{
\label{sec3.2.2}
Validity of the approach}
\normalsize
In order to verify that the above described method results in acceptable
statistically corrected enclosed mass estimates we performed simulations.
The results of the de-projection procedure
- presented in the $R-log(M)$-plane -
show that the distribution of the de-projected mass estimate tends
to under-estimate $M$ for flat stellar distributions, whether the
central mass is assumed to be point-like or extended, but is tightly peaked
around the true value of $M$ for steep cusp-like distributions.
We assumed a sphere with radius $r=15~mpc$ and a stellar
number density $n(r)$ surrounding the dark mass of
3$\times$10$^6$\solm.
For each star (we used several 1000) at a separation $r$ from
the center and a total enclosed mass M we calculated the projected radius
R and the projected, observable acceleration $a_{obs}$ via
equation~(\ref{eq02}).
In Fig.~\ref{fig05}a,b,c we show the density of
data points in the R-logM-plane for combinations of
a central point mass (BH) or an
extended central mass distribution
and a constant or cusp-like
stellar number density distribution.
Almost all data points underestimate the enclosed mass
$M$ and the value of that
estimate drops dramatically towards smaller values of R.
The density increase of data points towards larger projected distances from
the center is due to the fact that the projection effects decrease for
stars towards the projected edge of the limited volume.
In Fig.~\ref{fig05}d,e,f we correct both the projected
radius R and the upper limit of
the mass using the formalism outlined above in section \ref{sec3.2.1}.
A higher density of points is located along the correct value of the enclosed
mass and the remaining estimates are almost equally distributed
above and below that value.
The available number density counts provide some evidence for
an increased volume density of stars towards the center
(Genzel et al. 2000, Alexander 1999, see also Alexander \& Sternberg 1999).
In Fig.~\ref{fig05}b and Fig.~\ref{fig05}e we have calculated the
expected projected and statistically corrected mass estimates using
a $r^{-7/4}$ stellar density law as an extreme case.
The corrected mass estimates spread almost symmetrically
about the expected value.
At any projected radius the stellar number density along the line of sight
is now biased towards the plane of the sky that contains SgrA*.
This results in enclosed mass estimates that are less affected by
the geometrical projection.
For about 70\% of all stars in Fig.~\ref{fig05}b the projected
enclosed mass estimate accounts for more than 70\% of the true mass value.
This demonstrates that for steep cusps the majority of the projected
mass estimates will be much closer to the true value than in the case
of a constant density distribution.
For a $r^{-7/4}$ density law we would expect at least for two stars
a mass estimate of at least $\sim$70\% of the enclosed mass.
This is only barely fulfilled by S8 and S2. This shows, that a
larger number of stars with significant curvature will greatly
improve our knowledge on the presence and nature of a central stellar cusp.
\subsubsection{
\label{sec3.2.3}
Application to the measured data}
\normalsize
In Fig.~\ref{fig06}a we show projected mass estimates derived from the
observed accelerations as a function of the projected radius listed in
Tab.~\ref{t01}.
In addition we indicate the mass distribution obtained
from stellar and gas dynamics (for $R=8.0$ kpc; see caption of
Fig.~\ref{fig06} and Genzel et al. 2000, 1997, 1996,
Eckart \& Genzel 1996, 1997, Ghez et al. 2000,1998).
As expected from the simulations presented in Fig.~\ref{fig05}
the estimates obtained from S1 and S2 fall well below the
enclosed mass estimate of 3$\times$10$^6$\solm ~derived
previously (see references above).
For S8, however, (see discussion in section ~\ref{sec4.2.2})
and especially for S7, S10, S11
(not shown in Fig.~\ref{fig06}a) - for which only {\it upper}
limits of the acceleration could be obtained - the mass estimates are
well above 3$\times$10$^6$\solm.
For S7, S10, and S11 these {\it upper} limits range between
1.1$\times$$10^7$\solm and 1.6$\times$$10^7$\solm.
In Fig.~\ref{fig06}b we show the mass estimates
for S1, S2, and S8 as derived from the observed
accelerations and corrected for projection effects following the method
outlined in sections ~\ref{sec3.2.1} and ~\ref{sec3.2.2}.
The correction factors obtained for different volume sizes are listed in
Tab.~\ref{t04}. The derived corrected radii and mass estimates are given
in Tab.~\ref{t05}.
The values cover a mass range of 2.9 to 7.2$\times$10$^6$\solm~ over
separations from Sgr~A$^*$ between 8 and 15~$mpc$.
We compare the data to enclosed mass estimates
as a function of separation from Sgr~A$^*$
obtained assuming (physically not realistic)
Plummer like density distributions
(see discussion in Genzel et al. 1996, 2000)
with a core radius $r_c$ and a mass density $\rho(0)$
at the very center of the distribution.
We chose the exponent $\alpha$=5, since this corresponds to
the steepest currently observed drop in cluster mass density.
For the stars S1 and S2 which are currently closest in projection to
SgrA$^*$ the mean value and error of the enclosed mass corrected for
a volume radius of about $15~mpc$ is
$M_{acc}=(5 \pm 3) \times 10^6$\solm.
This value is fully consistent with an enclosed mass distribution that
is flat down to radii of about 8~$mpc$ with a value of
$3 \times 10^6$\solm~and a lower limit to the
mass density of $3.7 \times 10^{12}$\solm$~pc^{-3}$
for a core radius of $r_c=5.8~mpc$ as previously derived from the proper
motion data (Genzel et al. 2000).
As is apparent from Fig.~\ref{fig06}b this is the smallest range of true
(not projected) separations from SgrA* for which a mass estimate
corrected for projection effects has been derived so far.
The fact that $M_{acc}$ lies systematically above the enclosed mass obtained
at larger radii can very likely be attributed to the fact that
the volume size has been estimated correctly but the stars
are systematically
closer to Sgr~A* along the line of sight than the median distance
at the given projected radius (see Fig.~\ref{fig04}).
Alternatively, the estimate of the volume radius in which
the two stars S1 and S2
are located is too large - which will result in the same
effect and therefore in a correction
that systematically over estimates the enclosed mass.
Orbit calculations assuming a $3\times10^6$\solm~point mass
yield separations from the SgrA* plane of the sky of $6-7~mpc$ for S1 and S2.
This indicates that for a volume radius of $15~mpc$
the described effect is in fact relevant.
Assuming a compact enclosed mass of $3 \times 10^6$\solm~the range of
derived mass estimates can also be used to qualitatively judge its
compactness.
For comparison we plotted in Fig.~\ref{fig06}b
the calculated Plummer like enclosed mass distributions for central
mass density $\rho(0)$ and core radius values $r_c$
of $10^{13}$\solm$pc^{-3}$ and $4.2~mpc$
and $10^{14}$\solm$pc^{-3}$ and $1.9~mpc$, respectively.
This comparison demonstrates that
\\
a) the estimates of the central enclosed mass and compactness derived from
acceleration measurements for stars S1 and S2 are fully consistent
with previously determined values (Genzel et al. 2000) and that
\\
b) under the assumption of a compact $3 \times 10^6$\solm ~central
dark mass the current acceleration data allow central
mass densities of $>10^{13}$\solm$pc^{-3}$ and core radii of $<4~mpc$.
The star S2 currently (2000) is at a projected distance of
only about $60~mas$ from the center.
This is 4 times smaller than the minimum
radius reached by the Jeans modeling (Genzel et al. 2000).
If the orbit of S2 remains consistent with a
compact mass of 3.0$\times$10$^6$\solm ~~the mass density
is at least 64 times higher than the value based on the Jeans modeling
i.e. 2.4$\times$10$^{14}$\solm~pc$^{-3}$.
In this case the collapse life time of a hypothetical cluster of dark
mass would shrink to only a few 10$^6$ years (Maoz 1998).
\section{
\label{sec4}
STELLAR ORBITS CLOSE TO Sgr~A$^*$}
\normalsize
In the following we discuss possible Keplerian orbits for
the three early type S-sources
S1, S2, and S8 as well as two late type stars: S18 and star No.25 in
Tab.1 by Genzel et al. (2000).
In the following section \ref{sec4.1}
we first describe the algorithm we use to constrain the stellar orbits.
In sections \ref{sec4.2} and \ref{sec4.3} we then apply it to the combined
SHARP/NTT and NIRC/Keck data sets of the three high velocity stars S1, S2, and
S8 and the two late type stars, respectively.
\subsection{
\label{sec4.1}
Orbit calculations}
\normalsize
A complete global fit has to include the measurement
errors of the relative positions and velocities
given in Tab.\ref{t02} and Tab.\ref{t03}, the uncertainties
in the position of the central mass as well as its amount.
In order to get a first insight into the stellar orbits we first
restrict ourself to the case of a compact mass of
3$\times$10$^6$\solm~and and a location of it that coincides with
the nominal position of SgrA*.
The influences of the uncertainties of these quantities on the
3-dimensional orbits will be discussed in section \ref{sec4.3}.
We have chosen to present the results of our simulations in the
$v_{z}$-$s_{z}$-plane rather than the semi-major axes and eccentricity
plane since
this representation is closer to the observations.
Progress in diffraction limited near-infrared
spectroscopy now allows ongoing experiments to determine the line of
sight velocity of the central stars.
Calculated semi-major axes and eccentricities of the resulting
orbits are listed in Tab.\ref{t06}.
For two late type stars at the projected separations of about 0.5'' and 1''
from SgrA* radial and proper motion velocities are known.
For these stars only the positions
along the line of sight are undetermined.
For the stars S1, S2, and S8 the line of
sight velocities $v_{z}$ and positions $s_{z}$
are currently unknown.
We considered orbits for the ranges
of $-3500~<~v_{z}~<~3500~km~s^{-1}$ and $0~<~|s_{z}|~<~40~mpc$.
These intervals correspond to more than 5 times the central
velocity dispersion and about twice the radius of the Sgr~A$^*$ cluster
and include all possible bound orbits.
To judge the quality of the orbital fits we calculated
reduced $\chi^2$ values via
\begin{equation}
\label{eq04}
\chi^2=
\frac{1}{m-n}
\sum{\frac{(|{\bf x(t_i)-c(t_i)}|)^2}{\sigma^2}}~~.
\end{equation}
Here {$\bf x(t_i)$} and {$\bf c(t_i)$} are the measured and calculated
position vectors and $\sigma$ the measurement uncertainties
as a function of time, $m$ is the number of observed data points
and $n$ the number of free parameters. We have used $n=2$ since
$v_{z}$ and $s_{z}$ are undetermined
and currently the dominant source of uncertainty (see \ref{sec4.3}).
The pairs $v_{z}$ and $s_{z}$ and $-v_{z}$ and $-s_{z}$ result
in the same projected orbits and $\chi^2$ values.
Using the orbital data point that is closest to
our reference position (see Tab.\ref{t02}) we synchronized the
densely sampled calculated orbit with the measurements.
In Fig.\ref{fig07} we show diagrams for the type for simulations
described above.
For the stars S1, S2, and S8 the resulting $\chi^2$ values are shown in the
$v_{z}$-$s_{z}$-plane in Fig.~\ref{fig08}, ~\ref{fig10}, and ~\ref{fig12}.
In those diagrams we can in general distinguish between three areas
labeled A, B, and C in Fig.\ref{fig07}:
\begin{itemize}
\item[A]
At small separations from Sgr~A$^*$ ($s_z<5~mpc$) the calculated
orbits have acceleration values which are well above what is measured.
The corresponding $\chi^2_A$ values are highest.
These orbits can clearly be excluded.
\item[B]
At large separations from Sgr~A$^*$ ($s_z>10~mpc$) or large line of
sight velocities ($|v_z| > 2000~km/s$) the orbits result in
linear trajectories over the time interval from 1992 to 2000.
The accelerations are too small.
These orbital solutions can be excluded as well.
Large $\chi^2_B$ values in that region are due to the
measurement uncertainties as well as a mismatch with respect to a
straight line. This mismatch is due to the curvature in the
measured orbital section.
\item[C]
Finally, there is an area in the $v_{z}$-$s_{z}$-plane in which the $\chi^2$
values are lowest and correspond to acceptable orbital solutions
with curvatures similar to what is measured.
These minimum fit errors $\chi^2_C$ are only dominated by the
scatter in the data.
The difference between $\chi^2_B$ and $\chi^2_C$ is
a measure of the true $\chi^2$ deviation of the measured
curved orbital section from a simple linear trajectory - not
contaminated by the scatter in the data.
\end{itemize}
In Fig.~\ref{fig07} we show the results of
orbit calculations applied to simulated data for stars similar to S2.
Compared to the available measurements these data have a similar
sampling but are noise free with respect to the calculated orbits
from which they have been drawn.
The calculations show that the shape of the
$\chi^2$ minima depends on the orbital section
for which measurements are available.
Towards larger velocities and line of sight separations from SgrA*,
i.e. lower orbital curvatures, it becomes increasingly difficult
to distinguish between possible orbital solutions.
The location of the minimum $\chi^2$ values are smeared out towards
this region.
For a less curved section the line of sight separation
can be higher to result in a similarly curved orbit section
over the same amount of time and hence minimum $\chi^2$
values at a higher velocity.
Despite of this effect the simulations also show that
a common intersection (marked with a filled circle in Fig.~\ref{fig07})
of the regions of minimum $\chi^2$ values
remains at the correct $v_{z}$- and $s_{z}$-values with
which the stellar orbits are launched at the corresponding epoch
assumed for the simulations - excepting of course the ambiguity in
the sign of those quantities (see section \ref{sec1.1}).
How deep and close the absolute minimum of the $\chi^2$ values is
with respect to this location depends on the resolution (sampling in
the $v_{z}$-$s_{z}$-plane) of the calculation, the signal to noise,
and sampling of the observations.
To get a clear measure of the true $\chi^2_*$ deviation of the measured
curved orbital section from a simple linear trajectory - not
contaminated by the scatter in the data -
we corrected for both the SHARP/NTT and the NIRC/Keck data
the $\chi^2$ values by the corresponding minimum $\chi^2_C$ values
(see before).
\begin{equation}
\label{eq040}
\chi_*^2= \chi^2 - \chi^2_C
\end{equation}
We then combined both data sets in a maximum
likelihood (ML) analysis via:
\begin{equation}
\label{eq000}
log(ML) = - \chi_{SHARP/NTT}^2/2 - \chi_{NIRC/Keck}^2/2.
\end{equation}
The results are shown on the right hand site panels of
Fig.~\ref{fig08} ,~\ref{fig10}, and~\ref{fig12} and discussed in the following
section.
\subsection{
\label{sec4.2}
The central high velocity stars}
\normalsize
\noindent
In the previous section \ref{sec4.1} we presented a general discussion
of the procedure we use to match the data with Keplerian orbits.
We now discuss detailed results for the individual stars
obtained from the $\chi^2$ fits in the $v_{z}$-$s_{z}$-planes
and present characteristic orbits.
We show that the high velocity stars S2, and most likely S1 and S8 as well
are on bound, inclined ($60^o**1.0$).
The orbital calculations reveal a well defined single $\chi^2$ minimum.
For star S1 about 80\% of the $log(ML)$ values within the
1~$\sigma$ contour in Fig.~\ref{fig08}d correspond to
eccentricities of $e\le 1.0$ and the
separation from the SgrA* plane of the sky is about $\sim$7~$mpc$.
For star S2 {\it all} $log(ML)$ values within the 1~$\sigma$ contours in
Fig.~\ref{fig10} b) and d) are consistent with bound orbits, i.e. e$<$1.0.
Here both data sets (SHARP/NTT and NIRC/Keck) indicated
separations from the SgrA* plane of the sky of $\sim$6~$mpc$
and a line of sight velocity in the range of $\pm$500$km/s$.
For both data sets and sources characteristic orbital solutions are shown in
Fig.~\ref{fig09} and Fig.~\ref{fig11}.
For both stars the eccentricities are most likely in the range of
$0.4\le e < 1.0$.
For larger and smaller values of
$s_z$ the eccentricities and half axes become correspondingly larger and
smaller (Tab.~\ref{t06}).
For S2 the orbital elements listed in Tab.\ref{t06} are defined best.
For S1 about 20\% of the orbital fits obtained from the possible
$v_{z}$-$s_{z}$ points (Fig.\ref{fig08}d) result in large semi-major
axes and high eccentricities.
\noindent
{\bf S8:}
The $v_{z}$-$s_{z}$-planes are shown in Fig.~\ref{fig12} and
for both data sets we show characteristic Keplerian orbits
in Fig.~\ref{fig13} (see also Tab.~\ref{t06}).
From Fig.~\ref{fig12}, however, it is evident that there is
a clear mismatch between the measured curvature values
and those indicated by the $\chi^2$ minima.
These minima show that pure Keplerian orbits
result in a curvature of 0.6 to 0.8 $mas~yr^{-2}$
rather than about 3 $mas~yr^{-2}$ as obtained by the NTT and Keck
proper motion experiments (see Tab.\ref{t01}).
This mismatch corresponds to a 3-4$\sigma$ deviation
from the measured value. The correspondence would be better for central masses
above 3$\times$10$^{6}$\solm. However, already the lower enclosed
mass limit derived from the accelerations of star S8
represents a 3-4$\sigma$ deviation
from the values obtained via Jeans modeling and other mass estimations based on
proper motions and Doppler velocities (see Tab.\ref{t05} and Fig.\ref{fig06}
in this paper and Tab.5 and Fig.17 by Genzel et al. 2000).
\\
The orbital elements for S8 listed in Tab.\ref{t06} correspond to
the best fits to the data shown in Fig.\ref{fig01} in this paper and
and Fig.\ref{fig01} in Ghez et al. (2000) and reproduce the observed
time averaged positions and velocities but not the curvatures
(see discussion in section \ref{sec4.2.2}).
Orbits with large curvatures can clearly be excluded.
The most probable eccentricities are just below $e\sim1.0$.
The proper motion velocity is too large for orbits with apoastron
positions, i.e. regions of higher curvature closer to the present
location of the star. This is a strong indication for the fact
that the observed amount of curvature is not solely due to orbital motion.
If this result is confirmed by further measurements
consequences are that S8 cannot be used to
pinpoint the location of SgrA* (see section \ref{sec2.2}).
{\bf Influence of the position and amount of the central mass:}
The errors of the orbital elements in Tab.\ref{t06} have been
derived from the uncertainties of the 3D-positions and velocities
listed below.
Since the possible range of $v_{z}$ and $s_{z}$ that results
from our fit is large compared to
the measurement uncertainties of the proper motion velocities
and positions (see Tab.\ref{t02})
the resulting uncertainties on the orbital elements are much
smaller and well covered by their errors listed in Tab.\ref{t06}.
\\
The $\pm$30~$mas$ (Menten et al. 1998) uncertainty
of the position of SgrA* - which we assume to be
associated with the central mass - is comparable to the uncertainty
of the line of sight separation $s_{z}$.
It amounts, however, to only less than about 1/8 of the 3-D
separation of the stars from SgrA*.
%
A simultaneous variation of the amount and position of the central mass
within the $\pm$30~$mas$ and the (2.6-3.3)$\times$10$^6$\solm~ intervals
shows that the orbital elements in Tab.\ref{t06} represent a solution
at the global $\chi^2$ minimum of the orbital fits to the measured data.
%
We find that such a variation of the position and amount
of the central mass causes changes in the eccentricities and the
semi-major axes that are well covered by the errors of the orbital
elements given in Tab.\ref{t06}.
Therefore the main result - that the central stars are on
fairly inclined and eccentric orbits - is independent of the
variation of the involved quantities within their errors.
\subsubsection{
\label{sec4.2.2}
What causes the acceleration of S8~?}
\normalsize
\noindent
In the previous section we have shown that the orbital curvature observed
by both proper motion experiments is too large for being solely due to
Keplerian motion.
We now discuss a variety of reasons that could explain the
observed acceleration of the star S8.
{\it Stellar scattering:}
The curvature of S8's orbit corresponds to a deviation
from a straight line by an angle $\psi$.
If this is caused by a scattering star
of mass $m$, then its distance $r_s$ from S8 is given by
\begin{equation}
\label{eqs1}
r_s \sim 2G (m_{S8}+m)/(v_{\infty}^2 \psi)
\end{equation}
(Binney \& Tremaine 1994), where
$v_{\infty}$ is the relative velocity at infinity between the two stars.
The probability of such a scattering event occurring during the time
$\Delta t$ of the monitoring campaign is
\begin{equation}
\label{eqs2}
P = \pi r_s^2 n v_{\infty} \Delta t
\approx 4 \pi G^2 n (m_{S8}+m)^2 \Delta t/(v_{\infty}^3 \psi^2),
\end{equation}
where $n$ is the stellar number density.
Here $v_{\infty}\sim$$\sigma_{central}$$\sim$500 km/s
corresponds to the velocity
difference of both stars at large separations.
The mass of S8 is assumed to be
$m_{S8}\sim15-20$\solm~(Eckart, Genzel, Ott 1999, Genzel et al. 1998).
This implies $m'$=$(m_{S8}+m)\sim$20\solm~ for $m$$\le$1\solm.
With the central stellar mass density given by (Genzel et al. 1998, 2000)
we assume that the stellar number density is of the order of
$n$$\sim$10$^6$ pc$^{-3}$.
From the acceleration values in Tab.\ref{t01} we derive an observed
scattering angle $\psi$ of the order of 20 degrees for S8.
Equation \ref{eqs2} can then be written as
\begin{equation}
\label{eqs3}
P \approx 5 \times 10^{-8} \times
n[10^6 pc^{-3}] (m'[20M_{\odot}])^2 (v_{\infty}[500 km/s])^{-3}
\end{equation}
This shows that even if the stellar number density is higher by a
few orders of magnitude due to a stellar cusp or if $v_{\infty}$
varies by a few 100~km/s the scattering probability is always very low.
{\it Flux density of neighboring stars:}
A K=15-17 background or foreground star close to the current line of
sight toward S8 could also be responsible for a positional shift
that gives rise to the observed apparent acceleration.
However, S8 has moved by about 160~$mas$ over the past 8 years.
At a wavelength of 2$\mu$m this corresponds
to the angular resolving power of the NTT and about 3 times the
resolving power of the Keck telescope.
Such a star near S8 has not yet been reported
but - if present and not strongly variable - should be detected soon.
If the S8 acceleration
is due to such a star the S8 trajectory should straighten again
in the near future.
{\it Alternatives: }
If other observational biases (e.g. misalignments in position
or position angle) were relevant one would expect even larger
variations in proper motions at increasing projected separations
from the center.
These variations are not observed in both independent data sets.
Also a systematical underestimation of the enclosed mass from proper
motions and radial velocities is not likely. See detailed discussions in
Genzel et al. (2000).
A lensing event can also be excluded as a straightforward explanation
for the observed acceleration. For stars as bright as S8 such events
are very unlikely and result in a flux density increase over a
period of approximately 1 year (Alexander \& Loeb 2001,
Alexander \& Sternberg 1999).
Within less than about 0.5 magnitudes S8 was constant in
flux density over the past 8 year.
{\it As a conclusion} the acceleration of S8 that has been detected
in both the SHARP/NTT and the NIRC/Keck experiment is either due to
a flux contamination of an unrelated object along the same line of
sight or due to a rare scattering event in the dense environment
of the central stellar cluster.
\subsubsection{
\label{sec4.2.3}
Other central early type stars}
\normalsize
\noindent
For the remaining early type stars of the central Sgr~A* cluster
positions and proper motions are known
(Genzel et al. 1997,
Ghez et al. 1998,
Ghez et al. 2000,
Genzel et al. 2000).
Orbit calculations show that stars in the
Sgr~A* cluster with line of sight separations from the
center of $s_z$$<$30~$mpc$ and line of sight velocities
$v_z$ smaller than 2 to 3 times the velocity dispersion of the
central arcsecond will be on bound orbits around the black hole.
A more detailed analysis, however, still awaits a detection of their
orbital curvature and/or their radial velocity.
\subsection{
\label{sec4.3}
Late type stars at small projected separations}
\normalsize
There are two stars with prominent CO band head absorption that are located at
small projected separations from SgrA* and for which the full 3-dimensional
velocity information is available.
The corresponding $v_{z}$-$s_{z}$-planes and characteristic orbital
solutions are shown in Fig.~\ref{fig14}.
It cannot be fully excluded that these stars are at small
physical distances to the center. Our orbital analysis, however, shows that
the current data suggest a likely location outside
the central 0.3~pc diameter section of the Galactic Center stellar cluster
which is dominated by the early type He-stars. In the following we discuss the
results of our orbital analysis for both stars.
{\it No.25 - 0.43''E; 0.96''S of SgrA*:} Based on R$\sim$5000
VLT ISAAC observations Eckart, Ott,\& Genzel (1999) report
the presence of a late type star with strong 2.3$\mu$m CO band head absorption
about 1'' south of the center.
We identify this object with the K=12.4 proper motion star No.25 in Tab.1
of Genzel et al. (2000) and star S1-5 in Tab.1 by Ghez et al. (1998).
This star is located at a projected separation of
1.05'' about 0.43''E and 0.96''S of SgrA*. This star is approximately 0.3
magnitudes brighter than the overall southern part of the SgrA* cluster
(containing S9, S10, S11, and a few K$\ge$16 stars just E of S10 and S11).
In a 0.3''-0.5'' seeing under which the VLT data (Eckart, Ott, Genzel 1999)
were taken the flux density contribution of this star in a 0.6'' NS slit
is comparable to that of the southern part of the SgrA* cluster.
A fainter star almost exactly 1.1''S of SgrA*
can be excluded as a possible identification of the late type star, since its
brightness is about 0.3 magnitudes fainter than the individual stars S11 or
S10 and hence almost a full magnitude fainter than the total of the
southern part of the SgrA* cluster.
The wavelength calibration of the R$\sim$5000 VLT ISAAC data - as well as
a comparison to the spectrum of the late type star IRS14SW (see Tab.1 by
Genzel et al. 2000) that fell into the NS oriented slit and was acquired
simultaneously - indicate a line of sight velocity of the star
0.43''E and 0.96''S of SgrA* of -80$\pm$40 km/s.
The SHARP/NTT proper motion data of this object including the results
of the 1999 and 2000 observing run are shown in Fig.\ref{fig02}
and listed in Tab.\ref{t03}.
A comparison of the radial and proper motion velocities indicates that
this star is on a predominantly tangential orbit in the plane of the sky.
In Fig.\ref{fig14} we investigate the possible orbital solutions that
lead to bound orbits. The thick (red) continuous and
dashed lines mark line of
sight separations from SgrA* for which the eccentricities $e<1$.
About 60\% of the possible current line of sight separations are located
beyond the radius within which the He-stars dominate the emission.
About 30\% even lie
beyond the core radius of the central stellar cluster of $\sim0.3~pc$.
If the line of sight separations $s_z$ are of the order of 150 to 200~mpc
the eccentricities are smaller than unity and the
semi-major axes of the orbits will be of the same order as $s_z$.
These numbers are lower limits only, since they are derived for simple
Keplerian orbits under the assumption of a dominant central mass of
$3\times10^6$\solm. For orbits with eccentricities (as calculated
for the simple Keplerian case in Fig.\ref{fig14}) closer to $e=1.0$ and
values for $s_z$$>$$200~mpc$ the orbits will have large (several degrees)
Newtonian periastron shifts.
The stars reach true physical separations from the center
of well beyond 1.0~pc for which the mass of the
stellar cluster starts to dominate. Under
these conditions bound stellar orbits with line of sight separations
larger than what is indicated by the thick (red) lines are possible.
\\
\\
{\it S18 - 0.04''W; 0.45''S of SgrA*:}
This star is listed as S18 in Tab.1 of Genzel et al. (2000) but not
contained in the corresponding list of Ghez et al. (1998).
Its most recent SHARP/NTT proper motion data are listed in Tab.~\ref{t03}.
Based on deep CO(2-0)
line absorption Gezari et al. (2000) identify this object as an early
K-giant which is blue shifted with respect to the Galactic Center
stellar cluster at about -300 km/s. Of all late type stars in the central
stellar cluster S18 has the smallest angular separation ($<$0.5'')
from SgrA* reported to date.
A comparison of the radial and proper motion velocities indicates that
this star could be on a predominantly radial orbit.
In Fig.\ref{fig14} we investigate the possible
orbital solutions that lead to bound
orbits. The thick (red) continuous and
dashed lines mark line of sight separations from
SgrA* for which the eccentricities $e<1$.
About 50\% of the possible current line of sight locations are located beyond
the radius of the He-stars cluster and reach out to the core radius of
the central stellar cluster. For the reasons mentioned above
bound stellar orbits with line of sight separations
larger than what is indicated by the red lines are possible.
\\
\\
Highly eccentric orbits like those labeled with '$I$' in Fig.\ref{fig14}
bring both late type stars physically too close to the position of SgrA*.
These orbits can be excluded because
for a black hole mass of $3\times$10$^6$\solm
~the tidal disruption radius for a giant is
\begin{equation}
\label{eq7}
R_t \sim 1.2mas \times (R_{*}/10^{12}cm) \times (M_{*}/ M_{\odot})^{1/3}
\end{equation}
(e.g. Frank \& Rees 1976, Binney \& Tremaine 1994), where
$R_{*}$ and $M_{*}$ are the giant's radius and mass. For orbits with
semi-major axes of $a \sim 20mpc \sim 0.5''$ and eccentricities of $e >$0.94
every giant will be destroyed on its periastron passage. The coupling between
the orbit and the tides raised on the star will cause deviations from
a point mass behavior even at separations larger than $R_t$.
Correspondingly this would allow only wider orbits for giants.
Along the same line of arguments equation \ref{eq7} also provides additional
evidence that the central high velocity stars (the S-stars) are O-stars rather
than late type giants.
This identification, however, still awaits spectroscopic confirmation.
\section{
\label{sec5}
SUMMARY AND CONCLUSIONS}
\normalsize
The combination of the high precision but shorter time scale NIRC/Keck
data with the lower precision but longer time scale SHARP/NTT data set
allows us to have a first insight into the nature of
{\it individual} stellar orbits
as close to the massive black hole at the center of the Milky Way
as currently possible.
We have shown that a statistical correction for geometrical projection
effects allows us to derive an enclosed mass estimate from the
observed accelerations of stars S1 and S2 of
$M_{acc}=(5 \pm 3) \times 10^6$\solm.
This value is fully consistent with an enclosed mass that
is flat down to radii of about 8~mpc with a value of about
$3 \times 10^6$\solm~and mass density of $3.7 \times 10^{12}$\solm$~pc^{-3}$
for a core radius of $r_c=5.8~mpc$ as derived from the proper
motion data (Genzel et al. 2000).
Our most recent data - compared to and combined with
published data on proper motions and accelerations
(Ghez et al. 2000,
Genzel et al. 2000,
Eckart et al. 2000,
Ghez et al. 1999,
Ghez et al. 1998) -
show that S2 - and most likely S1 and S8 as well - are on
orbits around a central, dark, and massive object coincident with
the position of the radio source SgrA*.
The stars are on bound fairly inclined ($60^o*