------------------------------------------------------------------------ From: Andreas Eckart eckart@ph1.uni-koeln.de To: Heino Falcke Subject: paper \documentstyle[12pt,aasms4,psfig]{article} %\documentstyle[aas2pp4]{article} %--------------MSxxxxx %\textwidth 7.3in %\hoffset=-0.4in % This centers the extra wide page %\voffset=-0.4in % This is because the printer I use prints too low... \def\baselinestretch{0.96} %\def\baselinestretch{0.96} %\hyphenation{com-pu-scripts} \def\,{\thinspace} \def \kms{km\,s$^{-1}$} \newcommand{\solm}{M$_{\odot}$} \newcommand{\solarb}{L$_{\odot}$} \newcommand{\solar}{L$_{\odot}$\ } \newcommand{\solars}{L$_{\odot}$\ } \newcommand{\etal}{et al.\ } \newcommand{\rf}{\par\noindent\hangindent 15pt {}} \newcommand{\apjj}[2]{ApJ, #1, #2.} \newcommand{\apjjs}[2]{ApJS, #1, #2.} \newcommand{\apjjl}[2]{ApJ, #1, #2.} \newcommand{\asa}[2]{A\&A, #1, #2.} \newcommand{\asal}[2]{A\&A, #1, #2.} \newcommand{\arasa}[2]{ARA\&A, #1, #2.} \newcommand{\asas}[2]{A\&AS, #1, #2.} \newcommand{\anrev}[2]{ARA\&A, #1, #2.} \newcommand{\asar}[2]{A\&A Rev., #1, #2.} \newcommand{\ppasp}[2]{PASP, #1, #2.} \newcommand{\appss}[2]{Ap\&SS, #1, #2.} \newcommand{\mn}[2]{MNRAS, #1, #2.} \newcommand{\ajj}[2]{AJ, #1, #2.} \newcommand{\irmm}{Internat. Journ. Infrared and Millimeter Waves\ } \newcommand{\vol}[1]{1} \newcommand{\nhico}{$\frac{N_{H_2}}{I_{CO}}$} \newcommand{\mhtwo}{$M_{H_2}$ } \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 ($60^o1$) 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 ($60^o1.0$). 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