Stellar orbits near Sagittarius A*

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 of the Galaxy. We conﬁrm the stellar acceleration measurements obtained by Ghez et al. with NIRC at the Keck telescope. Our analysis differs in three 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 (cid:133) 60 8 , i , 80 8 (cid:134) , and eccentric (cid:133) 0 : 4 , e , 0 : 95 (cid:134) orbits around a central dark mass. The combination of both data sets results in a position of this central mass of 48 (cid:135) 54 2 24 mas E and 18 (cid:135) 42 2 61 mas S of the nominal radio position of Sgr A*. The mean statistically corrected enclosed mass derived from accelerations is M acc (cid:136) (cid:133) 5 ^ 3 (cid:134) £ 10 6 M ( with current radial separations of S1 and S2 from Sgr A* of about 8–10 mpc. This enclosed mass estimate is derived from individual stellar orbits as close to the massive black hole at the centre of the Milky Way as currently possible. Although the uncertainties are large, this estimate is fully consistent with the enclosed mass range of (cid:133) 2 : 6–3 : 3 (cid:134) £ 10 6 M ( derived by Genzel et al. from radial and/or proper motion velocities of a homogenized sample of sources. Star S8 was excluded from the analysis, since for the current proper motion velocity and radial separation from the centre we ﬁnd that the measured acceleration requires orbital motion around a compact object with a mass in excess of 3 £ 10 6 M ( . The data suggest either that this star was or is subject to a close interaction with a different object or that its position measurements are inﬂuenced by the emission of a different cluster star. Therefore we base the analysis of the enclosed mass solely on the available data for stars S1 and S2. We also discuss two late-type stars with projected separations from Sgr A* of about 0.5 and 1 arcsec. In addition to proper motions, these stars have known radial velocities. Orbit calculations indicate that such stars are very likely at larger physical distances from the centre and part of the larger scale central stellar cluster with a core radius of approximately 0.3 pc.

have been conducting a programme to study the properties of the central nuclear stellar cluster via near-infrared, high spatial resolution measurements. This programme has resulted in the very first detection of proper motions of stars that correspond to velocities of up to 1400 km s 21 in the central arcsecond in the vicinity of Sgr A* (Eckart & Genzel 1996. These results had been confirmed by Ghez et al. (1998) using NIRC on the Keck telescope. At the 1.6s to 3s level we have now detected orbital curvatures of the stars S1, S2 and S8, which confirm the recent results by Ghez et al. (2000).
Over the past 8 years the observed motions translate into a twodimensional velocity dispersion of the stars in the central arcsecond (corrected for the measurement error) of the order of 600 km s 21 . Overall, the stellar motions do not deviate strongly from isotropy, and are consistent with a spherical isothermal stellar cluster . However, a small deviation from isotropy is found for the sky-projected velocity components of the young, early-type stars. Most of the bright He I emission-line stars are on tangential orbits. This overall rotation could be a remnant of the original angular momentum pattern in the interstellar cloud from which these stars were formed. The projected radial and tangential proper motions of the fainter, fast-moving stars within <1 arcsec of SgrA* (the 'SgrA* cluster') suggest that they may largely be moving on very eccentric orbits. Speckle spectroscopy with SHARP at the NTT ) and slit spectroscopy with ISAAC at the VLT (Eckart, Ott & Genzel 1999;Figer et al. 2000) suggest that several of them are early-type stars. This is consistent with the idea that these stars are members of the earlytype cluster with small angular momentum, and therefore fell into the immediate vicinity of SgrA* Gerhard 2001; see also discussion in Ghez et al. 2000).
The detection of orbital curvature for three central stars now allows us for the first time to constrain their orbital elements and to investigate the central mass distribution at the smallest currently accessible separations from Sgr A*. In Section 1.1 we give a summary of why it is worthwhile to determine stellar orbits near the dark mass at the Galactic Centre, and give a general description of how that can best be done. In Section 2 we present the new proper motion and acceleration data, and derive the position of the central dark mass. In Section 3 we show how the acceleration measurements can be used to obtain an enclosed mass estimate. In Section 4 we then make use of the enclosed mass and its position to discuss possible orbital solutions for several of the central stars. Finally, a summary and conclusions are given in Section 5.

Stellar orbits and the central mass distribution
The point source sensitivities currently reached in near-infrared observations of the central stellar cluster are of the order of K , 16 -17 at an angular resolution of 50 -150 mas (Eckart & Genzel 1996;Genzel et al. 2000;Ghez et al. 1998Ghez et al. , 2000. These current limits are mainly a result of the imaging techniques at the diffraction limit, and are not due to the sensitivity of the instrumentation. With these values the analysis of the central mass distribution for separations of less than 0.5 arcsec (,20 mpc) from the radio point source SgrA* has to be based on detailed measurements of individual stellar orbits. We assume that the trajectories of stars can be approximated by Keplerian-shaped orbits around a central mass.
Relative Keplerian orbits are described by six orbital elements: three angles, the eccentricity e, the semimajor axis a, and a reference time T for a defined position (i.e., the periastron). Currently, for a few of the early-type, high-velocity S-sources, as well as two late-type stars that are close to the centre in projection, our detailed orbital analysis (Section 4) can be based on only five known quantities that constrain the orbits. In addition to the projected positions and proper motions, these are the orbital curvatures (for the S-sources) and the line-of-sight velocities (for the late-type stars). However, for visual binaries -as is the case for a combination of the mass at the centre of the Galaxy and any of the fast-moving stars in its vicinity -all these orbital elements can be derived from a sufficiently large section of the orbit, except for the sign of the inclination. This has to be inferred from Doppler shifts via spectroscopy. The visual binaries situation holds if the exact position of the black hole on the near-infrared images is known (see Menten et al. 1997). The orbital period and central mass can then be determined via Kepler's third law and the equation of motion, i.e., from Newton's law. Here we present a first step to reaching that goal using the combined SHARP/NTT and NIRC/Keck data sets.
The orbital analysis in the presence of a possibly partially extended central mass has to await the detection of fainter stars ðK $ 17Þ closer (, 2 mpc) to SgrA* or the availability of longperiod measurements (of the order of 100 yr or more; see Section 4.2) of the presently known high-velocity stars within the central arcsecond. The expected orbits have been discussed in the context of the Galactic Centre by Rubilar & Eckart (2001; see also Jaroszyn´ski 1998, Munyaneza, Tsiklauri & Viollier 1998and Fragile & Mathews 2000. The shortest orbital time-scales that could be observed in the near future will be of the order of only a few months. These measurements are well within reach using the new instrumentation (VLT, Keck and LBT interferometers; see Eckart et al. 2001a,b andEckart 2001).

AC C E L E R AT I O N S
From stellar orbital accelerations we can determine the position of the central mass. They can also be used to derive an additional estimate of the enclosed mass, help to constrain the stellar orbital parameters, and to show whether or not the high-velocity stars in the vicinity of SgrA* are bound. In the following we first present the results from the SHARP/NTT data and then describe how we determine the position of the dark mass from the combined SHARP/NTT and NIRC/Keck data sets.

Stellar accelerations and proper motions
In order to derive accelerations, a precise measurement of the change in proper motion velocities is required. Such a measurement can be performed only for stars that are bright, well separable from each other, and close enough to the centre to show detectable accelerations. At the Galactic Centre the stars S1, S2 and S8 fulfil these requirements, and a curvature of their trajectories has already been detected by Ghez et al. (2000). In order to estimate stellar accelerations from the SHARP/NTT data, we have used three different methods: (a) variation of proper motion velocities as a function of time, (b) parabolic fits to the time position plots, and (c) fitting Keplerian orbits to the observed data. The results of all three methods are in good agreement, and give for stars S1, S2 and S8 accelerations similar to those reported by Ghez et al. (2000). In the following we discuss properties and results of the three methods in more detail.
We prefer to estimate the stellar accelerations via the variation of proper motion velocities, since this method stays closest to the data. The only requirement is that over a period of 3 to 4 years a linear fit to the data results in reliable estimates of proper motions. The SHARP/NTT proper motion data set covers the time interval from 1992 to 2000, and allows us to derive and confirm for the stars S1, S2 and S8 curvatures of their orbits at the 1.6s to 3s level. In Fig. 1 (see also Table 1) we show the offset positions from SgrA* in right ascension (RA) and declination (Dec.) as a function of time. In the case of S1, S2 and S8 we compare them to parabolic functions (see below). Linear fits to the first and second half of the data sets for these three stars are shown by Eckart et al. (2001a,b), and clearly result in different slopes over successive time intervals. Figure 1. Positions of the stars S1, S2, S8, S7, S10 and S11 relative to the position of Sgr A* as a function of the SHARP/NTT observing epoch from 1992 to 2000. For the stars S1, S2 and S8 we show parabolic variations of the positions that correspond to the data presented in the text and in Table 1. For the other stars, for which only upper limits of the curvature were derived, we show linear fits. The declination velocity plot has been shifted by þ 300 mas. This difference in slopes divided by the time difference between the two intervals is a direct measure of the orbital curvature. We divided the data into an early ð1992 -1996Þ and a late epoch ð1997 -2000Þ. For each epoch we calculated three velocities from data in different subintervals. Individual accelerations were then estimated from quotients of all combinations of velocity and time differences between the early and late epochs. We verified the method by comparing the results to those obtained by the other two methods. We also applied it to the data shown in fig. 2 in Ghez et al. (2000), and reproduced the acceleration values quoted in their table 1 for the time-averaged epoch 1 1997.6. In Table 1 we compare the total accelerations and their 1s uncertainties for the three stars S1, S2 and S8 as derived from the SHARP/NTT and the NIRC/Keck data. The data agree to within 3.3s.
A parabolic fit is adequate as long as the acceleration is nearly constant over the observed orbit segment. This approximation is not always justified for the entire 8 years of SHARP/NTT observations analysed here. In particular, star S2 may have completed about half of its orbit during this period. Depending on the observed orbital section, a parabolic fit may therefore result in an inaccurate estimate of the acceleration. We have, however, used this method for the purpose of comparison. In Fig. 1 we show the data, together with parabolic functions that have curvatures as listed in Table 1. For stars S1 and S2, parabolic fits give similar results. For star S8, parabolic fits indicate a curvature closer to 2.3 mas yr 22 (as plotted in Fig. 1) rather than 3.3 mas yr 22 , which is, however, well within the errors of the value derived above.
Fitting Keplerian orbits to the data has the advantage that it provides the appropriate dependence of the projected positions as a function of time. It has the disadvantage that it assumes the presence of Keplerian orbits, and therefore requires knowledge of the enclosed mass and mass density. In Section 4 we show that the enclosed mass and mass density estimates derived from accelerations are in agreement with a compact 3 £ 10 6 M ( central object; we make use of this fact to fit Keplerian orbits to the data. For the stars S1 and S2 the range of Keplerian accelerations is in close agreement with our estimates from the other two methods. For star S8 the observed acceleration lies well above the prediction from Keplerian orbits. Reasons for that are discussed in detail in Section 4.2.2. Determination of proper motions: In the presence of significant orbital curvature a precise knowledge of positions and proper motion velocities for a given reference epoch is required in order to carry out orbit calculations. As reference epochs we chose the centres of the time intervals covered by the SHARP/NTT and NIRC/Keck data sets. In Table 2 we list the positions that we derived for those epochs from our own data and the data presented by Ghez et al. (2000). In both cases the near-infrared positions are measured relative to the radio position of SgrA* (Menten et al. 1997). For S1 and S2 the accelerations given in Table 1 correspond to a yearly change of the proper motion velocity of about 80 -160 km s 21 . This is of the order of both the mean measurement uncertainty and the mean differences in proper motion velocities between the SHARP/NTT and NIRC/Keck data sets (with a time difference of about 1 yr between central epochs; Eckart & Genzel 1996;Ghez et al. 1998Ghez et al. , 2000Genzel et al. 2000). In order to combine the SHARP/NTT and NIRC/Keck data sets and to obtain higher precision proper motion values for orbital calculations, we have used our acceleration values and the proper motion velocities for epoch 1995.4 by Ghez et al. (1998) to derive mean proper motion velocities for the two reference epochs 1996.5 for the SHARP/NTT data set and 1997.6 for the NIRC/Keck data set (listed in Table 3; see also Tables A1 and A2 in Appendix A).
We have also investigated the proper motion data of other stars within the SgrA* cluster. The number of further candidates that would allow us to detect curvature from the present SHARP/NTT data set is small. Star S3 apparently became fainter and disappeared -stars S4, S5 and S6 are in a crowded area, with S4 even being multiple -and S9 is too faint and is approaching S10 (see notes in the appendix of Genzel et al. 1997). In addition to S1, S2 and S8, the best candidates for detecting orbital curvature, in terms of their brightness and source crowding, are S7, S10 and S11. In Fig. 1 we show the proper position time diagrams for these three stars, and list their separations from Sgr A* and upper limits for their accelerations in Table 1.
In addition to the central early-type stars, there are two late-type stars with projected separations of only about 0.5 and 1 arcsec, listed in Table 1 as S18 and star No. 25 respectively. Those stars are discussed in detail in Section 4. In Fig. 2 we show the proper Positions relative to SgrA* in arcsec for the mean centre epochs of 1996.5 (SHARP) and 1997.6 (Keck) as derived from our own data and the data presented by Ghez et al. (2000). The assumed relative 1s errors are 10 mas (SHARP) and 5 mas (Keck; Ghez et al. 2000), respectively. For star No. 25 and S18, data are taken from Genzel et al. (2000). Keck data for S18 are taken from Gezari et al. (2000).
The velocities in km s 21 were obtained for epochs 1996.5 (SHARP) and 1997.6 (Keck) via interpolation of the measured velocities for epochs 1996.5 (SHARP) and 1995.4 (Keck) with the measured projected accelerations in RA and Dec. (see Table 1). For S18 a radial velocity of v z ¼ 2300 km s 21 is given by Gezari et al. (2000), and for No. 25 a radial velocity of 280 km s 21 is derived from ISAAC VLT data (Eckart et al. 1999; this paper). For errors see caption to Table A1. position time diagrams for these two stars that show no significant curvature.

The position of the central mass
The accelerations are consistent with recent results by Ghez et al. (2000), and imply that the three stars orbit a central, compact mass. For each of the sources S1 and S2 the acceleration values define an acceleration vector at an angle f that should point towards the central source. Here we assume that the probability for the location of the central mass is uniform in f. In Fig. 3 the stars S1 and S2 have been plotted at their time-averaged position resulting from the corresponding data sets. The measurement uncertainties define an error cone. For the presentation in Fig. 3 the dashed and dotted lines indicate error cones that correspond to a width of 2s in deviation from the nominal direction indicated by the acceleration vector. In order to determine the location of the central mass, we perform a maximum-likelihood (ML) analysis. As a ML score we use The thin contour lines in Fig. 3 indicate the locations at which log(ML) drops by 0.5 below the corresponding peak values. Here x 2 ¼ ðf 2 f 0 Þ 2 /ðDfÞ 2 , and f 0 denotes the angle of the acceleration vector, f the angle of any radial line within an error cone, and Df ¼ s the half-width of the cone. The central filled dashed circle in Fig. 3 marks the radio position of SgrA* and corresponding uncertainties of^30 mas. Using the observed curvature value and the enclosed mass range of 2.6 to 3:3 £ 10 6 M ( imposes a limit on the projected distances of S1 and S2 from SgrA*. For the SHARP/NTT data this leads to an improvement in the determination of the SgrA* position. We account for this effect in Fig. 3 by multiplying the log(ML) scores of the error cones with a Gaussian prior of the appropriate 1/e width centred on the timeaveraged positions of S1 and S2. From the projection of the 1s contour line (thin contour line E at the centre of Fig. 3) the multiplied probabilities derived from the SHARP/NTT and NIRC/Keck data result in a position of a central dark mass of 48 þ54 224 mas E and 18 þ42 261 mas S of the nominal radio position of Sgr A*. Within these limits the central mass is located at the 68.5 per cent confidence level ðDx ¼ 1:0Þ. At the 90 per cent confidence level ðDx ¼ 2:71Þ the central mass is located in an interval given by 48 þ109 248 mas E and 18 þ72 2133 mas S (thick contour line E at the centre of Fig. 3). Fig. 3 shows that at the current^30 mas uncertainty of the radio position of Sgr A* the presently available accelerations of stars S1 and S2 alone are fully consistent with the hypothesis that the radio source SgrA* is coincident with the centre of the dark mass.

T H E E N C L O S E D M A S S
Assuming Keplerian orbits, the observed projected stellar accelerations allow us only to derive lower limits to the enclosed mass. By inferring an estimate of the true physical separation of the stars from the centre, these lower limits can be statistically corrected for projection effects, and compared to previous estimates of the amount and compactness of the enclosed dark mass. In Section 3.1 we first determine the volume that contains the three stars S1, S2 and S8, describe in Section 3.2.1 how we use this information to correct for projection effects, verify the validity of that method in Section 3.2.2, and then apply it in Section 3.2.3 to the observed data.

Location of stars relative to Sgr A*
In Fig. 4 we plot the quantities that are relevant in describing the position of stars relative to the centre of the Galaxy. The line-ofsight positions of individual stars with respect to the plane of the sky containing Sgr A* are unknown. A correction for a geometrical projection of the acceleration values can, however, be carried out statistically by simply estimating the volume that contains the central stars S1, S2 and S8. In order to derive an estimate of this volume, we may use the proper motion velocities and projected positions of these stars in comparison to the velocity dispersions and number density as a function of the true (non-projected, threedimensional) radius r. The central fast-moving stars have significantly curved orbits. It is therefore unlikely that these fastmoving stars are objects on highly elliptical orbits ðe . 1Þ 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 to be located within a spherical volume of radius 15^3 mpc ð0:4^0:1 arcsecÞ.
Estimate from velocities and positions: In the following, R denotes the projection of the three-dimensional separation r of the star from the centre, and V star the projection of the threedimensional velocity v star of the star. An estimate of the size can be obtained by comparing the proper motion velocities V star to the three-dimensional velocity dispersion obtained from the parametrized functional form of the true number density distribution n(r) and the true radial and tangential velocity dispersions s r (r) and s t (r) of the best anisotropic Jeans model . 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 For a given velocity dispersion s 2 ðrÞ ¼ s 2 r ðrÞ þ 2s 2 t ðrÞ the probability PðV . V star ; rÞ can be calculated via The velocity dispersion s(r), and therefore also the probability PðV . V star ; rÞ, decrease with increasing radii r. For fast-moving Figure 3. The position of the central dark mass derived from the SHARP/NTT and NIRC/Keck acceleration data of S1 and S2. For the stars S1 and S2 the measured acceleration data correspond well to the curvature values expected from orbital motion around a central dark mass. The grey shading is proportional to the probability of the location of the central dark mass, i.e., the maximum-likelihood score given in the text. The smaller probability plot for the combined SHARP/NTT-NIRC/Keck acceleration data has been inserted into the probability plot derived from the SHARP/NTT data alone. The central filled dashed circle marks the radio position of SgrA* and corresponding uncertainties of^30 mas. The thin contour lines indicate the limit at which the log(ML )-score drops by a factor of 0.5 below its maximum. The a and d coordinates of the central mass can be read off from the projections of the thin contour lines at the 68.4 per cent and from the thick contour line at the 90 per cent level. The time-averaged positions of the SHARP/NTT and NIRC/Keck data, as well as the error cones of the acceleration data for S1 and S2, are indicated by different symbols explained in the figure legend. For the NIRC/Keck we have taken the error cone widths used in fig. 2 of Ghez et al. (2000) rather than those implied by the values in their table 1. The positional uncertainties for the SHARP and NIRC positions have been included in the grey-scale plot and the width of the cones. stars it becomes increasingly unlikely that they belong to statistical samples at correspondingly larger radii. Therefore we interpret PðV . V star ; rÞ 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 PðV . V star ; RÞ as an upper limit of this probability PðV . V star ; rÞ. The mean probability of the three stars S1, S2 and S8 belonging to samples of stars at the corresponding radii R S1 , R S2 and R S8 -or larger -is only about P init ¼ 33 per cent. This implies that the mean probability of these three sources belonging to samples of stars at their true three-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 ¼ 13:7 mpc, and by a factor of 3 at a radius of r ¼ 14:8 mpc. The probability of the stars belonging to samples at even larger radii lies well below 10 per cent. We therefore adopted a value of about 15 mpc (0.4 arcsec) as a reasonable estimate of the radius of the volume that contains all three stars.
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 centre (see Table 2). Therefore we adopt 12 mpc (0.3 arcsec) as a lower bound to the radius of the volume containing the three stars. This limit compares favourably with the size estimate derived above.

Correction of accelerations for projection effects
For a star at a projected separation R from the centre and a total enclosed mass M one can calculate the projected, observable acceleration a obs via Here u 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 (u) 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 (Ghez et al. 1998;Genzel et al. 2000). A more realistic approach needs to correct for geometrical projection effects. 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 u) 21 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 makes 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., u approaching p=2Þ. Contributions from those values would become dominant in case of a calculation of an expectation value for (cos u) 21 using n(r) values as a weights.

Validity of the approach
In order to verify that the above-described method results in acceptable statistically corrected enclosed mass estimates, we performed simulations. The results of the deprojection procedurepresented in the R 2 logðMÞ plane -show that the distribution of the deprojected mass estimate tends to underestimate 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 ¼ 15 mpc and a stellar number density n(r) surrounding the dark mass of 3 £ 10 6 M ( . For each star (we used several thousand stars) at a separation r from the centre and a total enclosed mass M we calculated the projected radius R and the projected, observable acceleration a obs via equation (4). In Figs 5(a)-(c) we show the density of data points in the R -log M 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 centre is due to the fact that the projection effects decrease for stars towards the projected edge of the limited volume. In Figs 5(d) -(f) we correct both the projected radius R and the upper limit of the mass using the formalism outlined above in Section 3.2.1. 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 centre (Alexander 1999;Genzel et al. 2000; see also Alexander & Sternberg 1999). In Figs 5(b) and (e) we have calculated the expected projected and statistically corrected mass estimates using a r 27/4 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 per cent of all stars in Fig. 5(b) the projected enclosed mass estimate accounts for more than 70 per cent 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 27/4 density law we would expect for at least two stars a mass estimate of at least , 70 per cent 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 of the presence and nature of a central stellar cusp.

Application to the measured data
In Fig. 6(a) we show projected mass estimates derived from the observed accelerations as a function of the projected radius listed in Table 1. In addition, we indicate the mass distribution obtained from stellar and gas dynamics (for R ¼ 8:0 kpc; see caption of Fig. 6 and Genzel et al. 1996, 2000, Eckart & Genzel 1996and Ghez et al. 1998, 2000.
As expected from the simulations presented in Fig. 5, the estimates obtained from S1 and S2 fall well below the enclosed mass estimate of 3 £ 10 6 M ( derived previously (see references above). For S8, however (see discussion in Section 4.2.2), and especially for S7, S10 and S11 (not shown in Fig. 6a) -for which only upper limits of the acceleration could be obtained -the mass estimates are well above 3 £ 10 6 M ( . For S7, S10 and S11 these upper limits range between 1:1 £ 10 7 and 1:6 £ 10 7 M ( . In Fig. 6(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 3.2.1 and 3.2.2. The correction factors obtained for different volume sizes are listed in Table 4. The derived corrected radii and mass estimates are given in Table 5. The values cover a mass range of 2:9 £ 10 6 to 7:2 £ 10 6 M ( 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. 1996Genzel et al. , 2000 with a core radius r c and a mass density r(0) at the very centre of the distribution. We chose the exponent a ¼ 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 ¼ ð5^3Þ £ 10 6 M ( . 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 £ 10 6 M ( and a lower limit to the mass density of 3:7 £ 10 12 M ( pc 23 for a core radius of r c ¼ 5:8 mpc as previously derived from the proper motion data .
As is apparent from Fig. 6(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. 4). 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 overestimates the enclosed mass. Orbit calculations assuming a 3 £ 10 6 M ( 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 £ 10 6 M ( , the range of derived mass estimates can also be used to qualitatively judge its compactness. For comparison, we plotted in Fig. 6(b) the calculated Plummer-like enclosed mass distributions for central mass density r(0) and core radius values r c of 10 13 M ( pc 23 and 4.2 mpc, and 10 14 M ( pc 23 and 1.9 mpc, respectively. This comparison demonstrates that (1) the estimates of the central enclosed mass and compactness derived from acceleration measurements for stars S1 and S2 are  Here we used the correction factors for a flat stellar distribution and r V ¼ 0.3 arcsec listed in Table 4. fully consistent with previously determined values , and (2) under the assumption of a compact 3 £ 10 6 M ( central dark mass the current acceleration data allow central mass densities of .10 13 M ( pc 23 and core radii of ,4 mpc. The star S2 currently (2000) is at a projected distance of only about 60 mas from the centre. This is 4 times smaller than the minimum radius reached by the Jeans modelling . If the orbit of S2 remains consistent with a compact mass of 3:0 £ 10 6 M ( , the mass density is at least 64 times higher than the value based on the Jeans modelling i.e., 2:4 £ 10 14 M ( pc 23 . In this case the collapse lifetime of a hypothetical cluster of dark mass would shrink to only a few Myr (Maoz 1998).

S T E L L A R O R B I T S C L O S E T O S G R A *
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 table 1 by Genzel et al. (2000). In Section 4.1 we first describe the algorithm we use to constrain the stellar orbits. In Sections 4.2 and 4.3 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.

Orbit calculations
A complete global fit has to include the measurement errors of the relative positions and velocities given in Tables 2 and 3, the uncertainty 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 ourselves to the case of a compact mass of 3 £ 10 6 M ( and a location of it that coincides with the nominal position of SgrA*. The influences of the uncertainties of these quantities on the threedimensional orbits will be discussed in Section 4.3.
We have chosen to present the results of our simulations in the v zs z plane rather than the semimajor axes and eccentricity plane, since the former representation is closer to the observations. Progress in diffraction-limited, near-infrared spectroscopy now allows ongoing experiments to determine the line-of-sight velocities of the central stars. Calculated semimajor axes and eccentricities of the resulting orbits are listed in Table 6. For two late-type stars at projected separations of about 0.5 and 1 arcsec 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 23500 , v z , 3500 km s 21 and 0 , js z j , 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 x 2 values via Here x(t i ) and c(t i ) are the measured and calculated position vectors, s the measurement uncertainties as a function of time, m the number of observed data points, and n the number of free parameters. We have used n ¼ 2, since v z and s z are undetermined and currently the dominant source of uncertainty (see Section 4.3). The pairs v z and s z and 2 v z and 2s z result in the same projected orbits and x 2 values. Using the orbital data point that is closest to our reference position (see Table 2), we synchronized the densely sampled calculated orbit with the measurements. In Fig. 7 we show diagrams for the type of simulations described above. For the stars S1, S2 and S8 the resulting x 2 values are shown in the v zs z plane in Figs 8, 10 and 12. In those diagrams we can in general distinguish between three areas labelled A, B and C in Fig. 7.
(A) At small separations from Sgr A* ðs z , 5 mpcÞ the calculated orbits have acceleration values which are well above what is measured. The corresponding x 2 A values are highest. These orbits can clearly be excluded.
(B) At large separations from Sgr A* ðs z . 10 mpcÞ or large line-of-sight velocities ðjv z j . 2000 km s 21 Þ 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 x 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.
(C) Finally, there is an area in the v zs z plane in which the x 2 values are lowest and correspond to acceptable orbital solutions with curvatures similar to what is measured. These minimum fit errors x 2 C are dominated only by the scatter in the data. The difference between x 2 B and x 2 C is a measure of the true x 2 deviation 2400 þ800 2300 0^500 0^1000 Top: Orbital elements for the stars S1, S2 and S8 as derived from the combined SHARP/NTT and NIRC/Keck data sets (see Table 6). We listed the orbital elements for the orbits labelled 'I' in Figs 8-13. The uncertainties correspond to the variation of orbital elements within the 1s contour of Figs 9, 11 and 13. The orbital elements for S2 are defined best. For S1 we have listed the orbital parameters for the ,80 per cent of the fit areas within the 1s contours with eccentricities below e ¼ 0:9. In the remaining ,20 per cent of the cases the quantities marked with an asterisk may become significantly larger. These orbits for S1 may even be not bound. We listed the inclination i, the sky position angle v of the periastron position, the sky position angle V of the northern most knot (transition between dashed and solid sections of the orbits in Figs 8, 10 and 12), as well as the eccentricity e and semimajor axes a of the deprojected orbits, and finally the year T of the periastron transition and the orbital period P. of the measured curved orbital section from a simple linear trajectory -not contaminated by the scatter in the data.
In Fig. 7 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 noisefree with respect to the calculated orbits from which they have been drawn. The calculations show that the shape of the x 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 x 2 values are smeared out towards this region. For a less curved section the line-of-sight separation can be higher, resulting in a similarly curved orbit section over the same amount of time, and hence minimum x 2 values at a higher velocity. Despite this effect, the simulations also show that a common intersection (marked with a filled circle in Fig. 7) of the regions of minimum x 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 1.1). How deep and close the absolute minimum of the x 2 values is with respect to this location depends on the resolution (sampling in the v zs z plane) of the calculation, the signal to noise ratio, and the sampling of the observations.
To get a clear measure of the true x 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 minimum x 2 values by the corresponding x 2 C values (see before): We then combined both data sets in an ML analysis via The results are shown on the right-hand panels of Figs 8, 10 and 12, and discussed in the following section.

The central high-velocity stars
In Section 4.1 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 x 2 fits in the v zs z planes, and present characteristic orbits. We show that the high-velocity star S2, and most likely S1 and S8 as well, are on bound, inclined ð608 , i , 808Þ, and eccentric ð0:4 # e # 1:0Þ orbits around a central, dark mass. For these three stars we list the three-dimensional positions and velocities at the time-averaged epoch of the current SHARP/NTT data set (1996.5), and the orbital elements derived from the combined SHARP/NTT and NIRC/Keck data sets in Table 6. Our analysis also demonstrates that right now it is only the combination of the two sets that actually allows a first derivation of orbital elements.
4.2.1 Shapes of the orbits S1 and S2: From the v zs z planes shown in Figs 8 and 10 two classes of orbits can clearly be excluded. These are those with high curvature, corresponding to small distances to the plane of the sky in which SgrA* is located, as well as those orbits with small curvatures (, 1 mas yr 22 ) and high eccentricities ðe . 1:0Þ. The orbital calculations reveal a well-defined single x 2 minimum. For star S1, about 80 per cent of the log(ML) values within the 1s contour in Fig. 8(d) correspond to eccentricities of e # 1:0, and the separation from the SgrA* plane of the sky is about 7 mpc. For star S2, all log(ML) values within the 1s contours in Figs 10(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 ,6 mpc and a line-of-sight velocity in the range of^500 km s 21 . For both data sets and sources, characteristic orbital solutions are shown in Figs 9 and 11. For both stars the eccentricities are most likely to be in the range of 0:4 # e , 1:0. For larger and smaller values of s z the eccentricities and half-axes become correspondingly larger and smaller (Table 6). For S2 the orbital elements listed in Table 6 are defined best. For S1 about 20 per cent of the orbital fits obtained from the possible v zs z points (Fig. 8d) result in large semimajor axes and high eccentricities.

S8:
The v zs z planes are shown in Fig. 12, and for both data sets we show characteristic Keplerian orbits in Fig. 13 (see also Table 6). From Fig. 12, however, it is evident that there is a clear mismatch between the measured curvature values and those indicated by the x 2 minima. These minima show that pure Keplerian orbits result in a curvature of 0.6 to 0.8 mas yr 22 , rather than about 3 mas yr 22 as obtained by the NTT and Keck proper motion experiments (see Table 1). This mismatch corresponds to a 3 -4s deviation from the measured value. The correspondence would be better for central masses above 3 £ 10 6 M ( . However, The grey-scale plot shows the likelihood score (see text) derived via the x 2 values from the SHARP/NTT and the NIRC/Keck data shown in panels (a) and (c). A continuous black contour line indicates the 1s limit at which the log(ML ) score drops by a factor of 0.5 below its maximum. Roman numbers label the positions of representative best-fitting orbits within this contour. The projections of the corresponding orbits on to the plane of the sky is shown in Fig. 9. The calculated accelerations a are represented by thick contour lines with labels in units of mas yr 22 . Thin contour lines represent the orbital eccentricities e. The a ¼ 0:5 mas yr 22 and e ¼ 0:5 contours are shown as dashed lines. In panels (c) and (d) we show the corresponding information for the NIRC/Keck data . While the grey scale likelihood plot is the same in panels (b) and (d) the labelled contour lines represent the eccentricities and accelerations obtained from the orbit calculations only using the SHARP/NTT and the NIRC/Keck data, respectively. already the lower enclosed mass limit derived from the accelerations of star S8 represents a 3 -4s deviation from the values obtained via Jeans modelling and other mass estimations based on proper motions and Doppler velocities (see Table 5 and Fig. 6 in this paper and table 5 and fig. 17 in Genzel et al. 2000).
The orbital elements for S8 listed in Table 6 correspond to the best fits to the data shown in Fig. 1 in this paper and fig. 1 in Ghez et al. (2000), and reproduce the observed time-averaged positions and velocities but not the curvatures (see discussion in Section 4.2.2). Orbits with large curvatures can clearly be excluded. The most probable eccentricities are just below e , 1:0. 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 of the fact that the observed amount of curvature is not solely due to orbital motion. If this result is confirmed by further measurements, then S8 cannot be used to pinpoint the location of SgrA* (see Section 2.2).
Influence of the position and amount of the central mass: The errors of the orbital elements in Table 6 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 Table 2), the resulting uncertainties on the orbital elements are much smaller and well covered by their errors listed in Table 6.
The^30 mas (Menten et al. 1997) uncertainty of the position of SgrA* -which we assume to be associated with the central massis comparable to the uncertainty of the line-of-sight separation s z . It amounts, however, to less than about 1/8 of the 3D separation of the  . Selected Keplerian orbits with velocities and separations from SgrA* for star S1 as given by the data in Tables 2 and 3. The orbit labels correspond to those in Figs 8(b) and (d). The orbits represent the range of best fits (within the 1s contour line in Fig. 8; see text) to the 1992 to 2000 SHARP/NTT data (crosses) and the 1996 to 1999 NIRC/Keck data (crosses also marked by thin lines on either site of the orbital tracks; Ghez et al. 2000). The dashed lines indicate those sections of the orbits that are -with respect to the current position of the stars -on the opposite site of the plane of the sky from SgrA* (see also caption of Table 6). The projected position of SgrA* is indicated by an asterisk. stars from SgrA*. A simultaneous variation of the amount and position of the central mass within the^30 mas and the ð2:6 -3:3Þ £ 10 6 M ( intervals shows that the orbital elements in Table 6 represent a solution at the global x 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 semimajor axes that are well covered by the errors of the orbital elements given in Table 6. 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.

What causes the acceleration of S8?
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.
Stellar scattering: The curvature of S8's orbit corresponds to a deviation from a straight line by an angle c. If this is caused by a scattering star of mass m, then its distance r s from S8 is given by r s , 2Gðm S8 þ mÞ=ðv 2 1 cÞ ð 8Þ (Binney & Tremaine 1994), where v 1 is the relative velocity at infinity between the two stars. The probability of such a scattering event occurring during the time Dt of the monitoring campaign is where n is the stellar number density.
Here v 1 , s central , 500 km s 21 corresponds to the velocity difference of both stars at large separations. The mass of S8 is assumed to be m S8 , 15 -20 M ( (Eckart et al. 1999;Genzel et al. 1998). This implies m 0 ¼ ðm S8 þ mÞ , 20 M ( for m # 1 M ( . With the central stellar mass density given by (Genzel et al. 1998 we assume that the stellar number density is of the order of n , 10 6 pc 23 . From the acceleration values in Table 1 we derive an observed scattering angle c of the order of 208 for S8. Equation (9) can then be written as 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 1 varies by a few 100 km s 21 , the scattering probability is always very low. Flux density of neighbouring stars: A K ¼ 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 mm 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 -it should be detected soon. If the S8 acceleration is due to such a star, the S8 trajectory should straighten again in the near future.
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 centre. 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 yr (Alexander & Sternberg 1999;Alexander & Loeb 2001). Within about 0.5 mag, S8 was constant in flux density over the past 8 years.
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.

Other central early-type stars
For the remaining early-type stars of the central Sgr A* cluster, positions and proper motions are known Ghez et al. 1998Ghez et al. , 2000Genzel et al. 2000). Orbit calculations show that stars in the Sgr A* cluster with line-of-sight separations from the centre 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.

Late-type stars at small projected separations
There are two stars with prominent CO band head absorption that are located at small projected separations from SgrA* and for which the full three-dimensional velocity information is available. The corresponding v zs z planes and characteristic orbital solutions are shown in Fig. 14. It cannot be fully excluded that these stars are at small physical distances to the centre. Our orbital analysis, however, shows that the current data suggest a likely location outside the central 0.3 pc diameter section of the Galactic Centre 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.
No. 25 -0.43 00 E; 0.96 00 S of SgrA*: Based on R , 5000 VLT ISAAC observations Eckart et al. (1999) report the presence of a late-type star with strong 2.3-mm CO band-head absorption about 1 arcsec south of the centre. We identify this object with the K ¼ 12:4 proper motion star No. 25 in table 1 of Genzel et al. (2000) and star S1-5 in table 1 of Ghez et al. (1998). This star is located at a projected separation of 1.05 arcsec about 0.43 arcsec E and 0.96 arcsec S of SgrA*. This star is approximately 0.3 mag brighter than the overall southern part of the SgrA* cluster (containing S9, S10 and S11, and a few K $ 16 stars just east of S10 and S11). In the seeing of 0:3 -0:5 arcsec under which the VLT data (Eckart et al. 1999) were taken the flux density contribution of this star in a 0.6-arcsec NS slit is comparable to that of the southern part of the SgrA* cluster. A fainter star almost exactly 1.1 arcsec S of SgrA* can be excluded as a possible identification of the late-type star, since its brightness is about 0.3 mag 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 , 5000 VLT ISAAC data -as well as a comparison to the spectrum of the late-type star IRS14SW (see table 1 by Genzel et al. 2000) that fell into the NS oriented slit and was acquired simultaneously -indicates a line-of-sight velocity of the star 0.43 arcsec E and 0.96 arcsec S of SgrA* of 280^40 km s 21 . The SHARP/NTT proper motion data of this object including the results of the 1999 and 2000 observing run are shown in Fig. 2 and listed in Table 3. 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. 14 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 per cent of the possible current line-of-sight separations are located beyond the radius within which the He stars dominate the emission. About 30 per cent even lie beyond the core radius of the central stellar cluster of ,0.3 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 semimajor 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 £ 10 6 M ( . For orbits with eccentricities (as calculated for the simple Keplerian case in Fig. 14) closer to e ¼ 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 centre 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. S18 -0.04 arcsec W; 0.45 arcsec S of SgrA*: This star is listed as S18 in table 1 of Genzel et al. (2000), but is not contained in the corresponding list of Ghez et al. (1998). Its most recent SHARP/NTT proper motion data are listed in Table 3. Based on deep COð2-0Þ line absorption, Gezari et al. (2000) identify this Stellar orbits near Sagittarius A* object as an early K-giant which is blueshifted with respect to the Galactic Centre stellar cluster at about 2 300 km s 21 . Of all latetype stars in the central stellar cluster, S18 has the smallest angular separation (, 0.5 arcsec) 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. 14 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 per cent 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 indicated by the red lines are possible.
Highly eccentric orbits like those labelled with 'I' in Fig. 14 bring both late-type stars physically too close to the position of SgrA*. These orbits can be excluded since for a black hole mass of 3 £ 10 6 M ( the tidal disruption radius for a giant is For orbits with semimajor axes of a , 20 mpc , 0:5 arcsec 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 behaviour even at separations larger than R t . Correspondingly, this would allow only wider orbits for giants. Along the same line of arguments, equation (11) 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.

S U M M A RY A N D C O N C L U S I O N S
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 individual stellar orbits as close to the massive black hole at the centre 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 ¼ ð5^3Þ £ 10 6 M ( . This value is fully consistent with an 43 arcsec E, 0.96 arcsec S orbit IV is close to an EW flip of orbit III, and for S18 orbit IV is close to an NS flip of orbit III. enclosed mass that is flat down to radii of about 8 mpc with a value of about 3 £ 10 6 M ( and mass density of 3:7 £ 10 12 M ( pc 23 for a core radius of r c ¼ 5:8 mpc as derived from the proper motion data . Our most recent data -compared to and combined with published data on proper motions and accelerations (Ghez et al. 1998(Ghez et al. , 1999Eckart et al. 2000;Genzel et al. 2000) -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 ð608 , i , 808Þ and eccentric ð0:4 , e , 0:95Þ orbits. In Fig. 15 we show for star S2, right ascension and declination as a function of time as predicted from our analysis of the combined data set. The analysis is in agreement with the result by Genzel et al. (2000) that the central stars are preferentially on eccentric orbits. This statement also holds for S8 -just taking into account its velocity at the current projected position -although it is not likely that the entire amount of its measured acceleration is due to orbital motion.
Properties of a possible cusp: The current kinematical data on individual central stars make it difficult to accommodate a highdensity cusp with a stellar density law as steep as nðrÞ , r 27=4 . In Section 3.2.2 we have shown that for steep cusps the majority of the projected mass estimates are much closer to the true value than in the case of, for instance, a constant-density distribution. For three stars randomly picked out of a cluster following a r 27/4 density law we would at least for two stars expect that the mass estimate derived from their accelerations accounts for , 70 per cent of the enclosed mass. With S8 and possibly S2 this is just barely the case for the Galactic Centre. The presence of a pronounced stellar cusp within the central few 100 mpc would make that area collisionly dominated, and most of the stars would be expected to be on parabolic orbits (Alexander 1999). Our orbit calculations show that the current data of both the SHARP/NTT and NIRC/Keck experiment clearly indicate that the central stars are on bound and not parabolic orbits. This weakens the case for a strong cusp, and suggests that the stellar density is not high enough to provide a collisionally dominated environment.
Due to the limited number of detected stars a minimum radius of 10 mpc (0.25 arcsec) is currently used for the determination of the mass and mass density from proper motion measurements. The a ¼ 5 Plummer-like model of a dark cluster results in a core radius of such a hypothetical cluster of r core ¼ 5:8 mpc (0.15 arcsec) and corresponding central density of 3:7 £ 10 12 M ( pc 23 . The statistical correction for geometrical projection effects presented in Section 3.2 shows that the acceleration data of the two closest stars S1 and S2 are in agreement with such a compact mass. Accepting the correction for the most likely volume that contains S1 and S2, the data would even be fully consistent with a core radius of r core ¼ 4:2 mpc (0.17 arcsec) and corresponding central density of 10 13 M ( pc 23 . If the trajectory of S2 remains consistent with an orbit around a compact mass of 3:0 £ 10 6 M ( , the lower limit for the mass density that can be derived from this stellar orbit may be as large as 2:4 £ 10 14 M ( pc 23 . In this case the collapse lifetime of a hypothetical cluster of dark mass would shrink to only a few Myr (Maoz 1998).
Two late-type stars for which proper motion and radial velocities are known, and that are at projected separations of about 0.5 and 1.0 arcsec from SgrA* are very likely to be at larger physical distances from the centre, and part of the larger scale central stellar cluster with a core radius of approximately 0.3 pc. Our analysis indicates that the strong curvature of the available orbital section of star S8 is possibly due to either a flux contamination by an unrelated object along the same line of sight or a result of a rare scattering event in the dense environment of the central stellar cluster.
The analysis of stellar orbits Genzel et al. 2000;Ghez et al. 2000; this paper) clearly supports the presence of a compact dark mass at the position of SgrA*. The VLBI maser nucleus of NGC 4258 (Greenhill et al. 1995;Myoshi et al. 1995) and the dark mass at the centre of the Milky Way are currently the best and most compelling cases for the existence of supermassive nuclear black holes (Maoz 1998).
Future developments: Future observations at higher sensitivity and angular resolution will allow us to find and track the motion of stars that are even closer, i.e., , 4 mpc, to the centre than those currently accessible. Orbital time-scales at the resolution limit of the VLT, LBT, or Keck interferometer could be in the range of a few months. A detection of a relativistic or Newtonian periastron shift would ultimately result in a direct determination of the compactness of the enclosed central mass (Rubilar & Eckart 2000; see also Fragile & Mathews 2000). Measurements of the prograde relativistic or the retrograde Newtonian orbital periastron shifts are within reach of current and upcoming instrumentation.

AC K N OW L E D G M E N T S
We are thankful to the ESO Director General and his staff for letting us bring the SHARP camera to the NTT in 1999 and 2000. This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) via grant SFB 494. We thank Reiner Hofmann and Klaus Bickert for their help with SHARP at the NTT. We are also grateful to the NTT and La Silla team for their interest and technical support of the SHARP camera.

R E F E R E N C E S
Alexander T., 1999, ApJ, 527, 835 Alexander T., Loeb A., 2001, ApJ, 551, 223 Alexander T., Sternberg A., 1999, ApJ, 520, 137 Figure 15. Prediction of position time plots for different possible orbits of the star S2. For a line-of-sight separation from the plane of the sky containing Sgr A* of s z ¼ 6 mpc the following line-of-sight velocities have been used (see Section 4 and Fig. 11): I: s v ¼ 0 km s 21 ; II: s v ¼ 2500 km s 21 ; III: s v ¼ þ500 km s 21 . The longest orbital time-scales suggested by the current data correspond to orbits with s z ¼ 7 mpc and a: s v ¼ 2500 km s 21 or b: s v ¼ 500 km s 21 . The SHARP/NTT data are shown as well. The declination velocity plot has been shifted by þ 300 mas.