Monitoring stellar orbits around the Massive Black Hole in the Galactic Center
S. Gillessen, F. Eisenhauer, S. Trippe, T. Alexander, R. Genzel, F. Martins, T. Ott
aa r X i v : . [ a s t r o - ph ] O c t Draft version October 22, 2018
Preprint typeset using L A TEX style emulateapj v. 03/07/07
MONITORING STELLAR ORBITS AROUND THE MASSIVE BLACK HOLE IN THE GALACTIC CENTER
S. Gillessen , F. Eisenhauer , S. Trippe , T. Alexander , R. Genzel , F. Martins , T. Ott Draft version October 22, 2018
ABSTRACTWe present the results of 16 years of monitoring stellar orbits around the massive black hole incenter of the Milky Way using high resolution near-infrared techniques. This work refines our previousanalysis mainly by greatly improving the definition of the coordinate system, which reaches a long-term astrometric accuracy of ≈ µ as, and by investigating in detail the individual systematicerror contributions. The combination of a long time baseline and the excellent astrometric accuracyof adaptive optics data allow us to determine orbits of 28 stars, including the star S2, which hascompleted a full revolution since our monitoring began. Our main results are: all stellar orbits are fitextremely well by a single point mass potential to within the astrometric uncertainties, which are now ≈ × better than in previous studies. The central object mass is (4 . ± . | stat ± . | R ) × M ⊙ where the fractional statistical error of 1 .
5% is nearly independent from R and the main uncertaintyis due to the uncertainty in R . Our current best estimate for the distance to the Galactic Center is R = 8 . ± .
35 kpc. The dominant errors in this value is systematic. The mass scales with distanceas (3 . ± . × ( R / . M ⊙ . The orientations of orbital angular momenta for stars in thecentral arcsecond are random. We identify six of the stars with orbital solutions as late type stars, andsix early-type stars as members of the clockwise rotating disk system, as was previously proposed. Weconstrain the extended dark mass enclosed between the pericenter and apocenter of S2 at less than0 . Subject headings: blackhole physics — astrometry — Galaxy: center — infrared: stars INTRODUCTION
Observations of Keplerian stellar orbits in the Galac-tic Center (GC) that revolve in the gravitational poten-tial created by a highly concentrated mass of roughly4 × M ⊙ (Sch¨odel et al. 2002; Eisenhauer et al. 2005;Ghez et al. 2003, 2005) currently constitute the bestproof for the existence of an astrophysical massive blackhole. In this experiment the stars in the innermost arc-second (the so-called S-stars) of our galaxy are used astest particles to probe the potential in which they move.Unlike gas, the motion of stars is determined solely bygravitational forces. Since the beginning of the obser-vations in 1992 one of the stars, called S2, has nowcompleted one full orbit. Its orbit (Sch¨odel et al. 2002;Ghez et al. 2003) has a period of 15 years. Since 2002 thenumber of reasonably well-determined orbits has grownfrom one to 28; in total we currently monitor 109 stars,see Figure 1 .Due to the high interstellar extinction of ≈
30 magni-tudes in the optical towards the GC the measurementshave to be performed in the near infrared (NIR), wherethe extinction amounts to only ≈ Max-Planck-Institut f¨ur Extraterrestrische Physik, 85748Garching, Germany Physics Department, University of California, Berkeley, CA94720, USA Faculty of Physics, Weizmann Institute of Science, POB 26,Rehovot 76100, Israel; William Z. and Eda Bess Novick CareerDevelopment Chair GRAAL-CNRS, Universit´e Montpellier II, Place Eug`eneBataillon, 34095 Montpellier, France This work is based on observations collected between 1992 and2008 at the European Southern Observatory, both on Paranal andLa Silla, Chile. imaging at ESO’s NTT in La Silla, a 4m-telescope, and in1995 at the Keck telescope, a 10m-telescope. Since 1999(Keck: Ghez et al. (2001)) and 2002 (VLT: Sch¨odel et al.(2002)) the combination of 8m/10m-class telescopes andadaptive optics (AO) has been routinely used for deep(H ≈
19) diffraction limited (FWHM 40-100 mas) imag-ing and spectroscopy.The GC is a uniquely accessible laboratory for explor-ing the interactions between a massive black hole (MBH)and its stellar environment. By tracking the orbits ofstars close to the MBH one can gather information onthe gravitational potential in which they move. Of primeinterest is the value of R , the distance to the GC, as itis one of the fundamental quantities in models for ourGalaxy. Equally interesting is the nature of the massresponsible for the strong gravitational forces observed.While the measured mass makes a compelling case for aMBH, the exact form of the potential encodes answers tomany interesting questions. Clearly, testing general rela-tivity for such a heavy object is among the goals; the firststep would be to detect the Schwarzschild precession ofthe pericenters of some orbits. A measurable deviationfrom a point mass potential would give access to a possi-ble cluster of dark objects around the MBH, testing manytheoretical ideas, such as mass segregation or the conceptof a loss-cone. Another focus of interest are the proper-ties of the stellar orbits. The distributions of the orbitalelements may have conserved valuable information aboutthe formation scenario of the respective stars. This ad-dresses for example the so-called ’paradox of youth’ forthe stars in the central arcsecond (Ghez et al. 2003) orthe puzzling existence of a large number of O-stars andWolf-Rayet stars in the GC (Paumard et al. 2006).This paper is the continuation of our long-term work on Gillessen et al. S112 S106S105S104S103S102S101 S100S99 S111S110S109 S107S108S98S93S97S96 S95 S70S56 S48 S94S92 S91 S90 S89S88S87 S86S85 S84S39S29S81 S83S82 S55S61 S11 S73S79 S78 S72S68 S71S75S76S77S80S63 S74S69 S65S40S4S57S23S14 S44 S42/S41S58S25 S50S51S33S32S60 S62 S43S21S18 S17/SgrA*(?)S59 S53 S28S27S12 S64 S31 S36 S37S5S26S49 S47S6S7 S46S34S35 S8 S9S22 S10S1S2/S13
S67 S19S54 S24S45S20 S66S52 S30 S38
Fig. 1.—
Finding chart of the S-star cluster. This figure is based on a natural guide star adaptive optics image obtained as part ofthis study, using NACO at UT4 (Yepun) of the VLT on July 20, 2007 in the H-band. The original image with a FWHM of ≈
75 maswas deconvolved with the Lucy-Richardson algorithm and beam restored with a Gaussian beam with FWHM = 2 pix=26 . m H = 19 . m K = 17 .
7) are detected at the 5 σ level. Only stars that are unambiguously identified inseveral images have designated names, ranging from S1 to S112. Blue labels indicate early-type stars, red labels late-type stars. Stars withunknown spectral type are labelled in black. At the position of Sgr A* some light is seen, which could be either due to Sgr A* itself or dueto a faint, so far unrecognized star being confused with Sgr A*. stellar motions in the vicinity of Sgr A*. We reanalyzedall data available to our team from 16 years. The basicsteps of the analysis are: • Obtain high quality, astrometrically unbiased mapsof the S-stars. Obtain high quality spectra for thesestars. • Extract pixel positions from the maps and radialvelocities from the spectra. • Transform the pixel positions to a common astro-metric coordinate system; transform the radial ve-locities to the local standard of rest (LSR). For theastrometric data several steps are needed: – Relate the fainter S-stars positions to those ofthe brighter S-stars (Speckle data only). – Relate the S-stars positions to a set of selectedreference stars. – Relate the reference stars to a set of SiOmaser stars, of which the positions relative toSgr A* are known with good accuracy fromradio (VLA) observations (Reid et al. 2007). • Fit the data with a model for the potential andgather in that way orbital parameters as well asinformation about the potential.We organize this paper according to these steps. DATA BASE
The present work relies on data obtained over manyyears with different instruments. In this section webriefly describe the different data sets.
SHARP
The first high resolution imaging data of the GC regionwere obtained in 1992 with the SHARP camera built atMPE (Hofmann et al. 1992; Eckart et al. 1994). SHARPwas used by MPE scientists until 2002 at ESO’s 3.5mtellar orbits in the Galactic Center 3NTT in Chile. The data led to the detection of highproper motions close to Sgr A* (Eckart & Genzel 1996).The camera was operating in speckle mode with exposuretimes of 0 . . . NACO
The first AO imaging data available to us of the GC re-gion was obtained in 2002 with the Naos-Conica (NACO)system mounted at the fourth unit telescope Yepun ofthe VLT (Lenzen et al. 1998; Rousset et al. 1998). Com-pared to the SHARP data the NACO data are superiordue to the larger telescope aperture (8 . . . • In order to measure the positions of the SiO maserstars IRS9, IRS10EE, IRS12N, IRS15NE, IRS17,IRS19NW, IRS28 and SiO-15 (Reid et al. 2007),we used the 27 mas/pix image scale both in H-and K-band in all years since 2002. The data are described in Reid et al. (2007); Trippe et al.(in prep.) and summarized in table 2. The typ-ical single detector integration time was two sec-onds, such that the bright IR sources present inthe r ≈ ′′ field covered did not get saturated.Mostly, we used a dither pattern of four positionsthat guaranteed that the central ten arcsecondsare imaged in each pointing position. The num-ber of useful maser positions per image varied be-tween six and eight. IRS19NW was not in the im-ages in 2002, 2003 and 2006; SiO-15 was not cov-ered in 2003. Due to their brightness IRS17 andIRS9 were in the non-linear regime of detector inthe observations from June 12, 2004 and thus ex-cluded for that epoch. Since the NACO camerawhen operated in the 27 mas/pix mode exhibits no-table geometric image distortions we constructedde-distorted mosaics from the individual images byapplying a distortion correction, involving rebin-ning of the measured flux distribution to a newpixel grid. The procedure is described in detail inTrippe et al. (in prep.) and relies on comparing dis-tances between stars present in the different point-ings. The distortion model used is ~p = ~p ′ (1 − β ~p ′ )with β ≈ × − where ~p and ~p ′ denote true anddistorted pixel positions with respect to some ori-gin in the image that also is determined from thedata. See also fig. 6. We did not apply deconvolu-tion techniques on these images. • The positions of the S-stars were determinedmostly from images obtained in the 13 mas/pix im-age scale. (Only when no image in the 13 mas/pixscale was available sufficiently close in time, weused also images obtained in the 27 mas/pix scale.)A typical data set contains two hours of data. Thesingle detector integration time was mostly around15 seconds, and the field of view was moved after afew integrations successively to four positions suchthat the central four arcseconds are present in allframes. The data are summarized in table 1 anda complete list of the data sets used is given inthe table in appendix B. The reduction followedthe usual steps of sky subtraction and flat-fielding.Manually selected high quality frames were com-bined to a single ssa map per epoch since the opti-cal distortions are small enough to be neglected inthe 13 mas/pix scale (Trippe et al. in prep.) for theframe combination. A distortion model of the sametype as for the 27 mas/pix scale images was con-structed for each epoch; however the best-fittingmodel parameters varied more than expected be-tween the different epochs. We concluded that wewere not able to solve for the distortion parame-ters with our observations. Hence, we did not ap-ply distortion models to the 13 mas/pix data butused higher order transformations when relatingpixel positions to astrometric positions (see fig. 5).In order to separate sources we moderately decon-volved the central five arcseconds of these mapswith the Lucy-Richardson algorithm. The latterused a point spread function constructed from themap itself obtained by applying the starfinder code(Diolaiti et al. 2000). In order to estimate the de- Gillessen et al.
TABLE 1Summary of the yearly number of epochs for which weobtained S-star images and the yearly mean number ofS-star positions determined per epoch.
Year Instrument S − star positionsepoch TABLE 2Summary of the number of available maser star mosaicimages, number of maser stars present in each frame andthe respective FWHM of the point spread function in theimages.
Date convolution error we divided each 13 mas/pix dataset into two and obtained two coadded maps, eachwith half of the integration. Both maps were thendeconvolved the same way as the full coadd.
SINFONI
Spectroscopy enables one to determine radial velocitiesof stars if the positions of known atomic or molecularlines can be measured in the stellar spectra. The GC isbest exploited with integral field spectroscopy as one isinterested in the radial velocities of all stars for whichone can hope to determine orbits, i.e. all stars in thecentral arcsecond. In the NIR the K-band (2 . − . µ m)is best suited since it contains the hydrogen line Bracket- γ at 2 . µ m. This line is present in absorption forB type stars, the most common spectral type for the S-stars (Eisenhauer et al. 2005). For late-type stars theCO band heads at 2 . µ m, 2 . µ m, 2 . µ m and2 . µ m are also covered by the K-band.Since July 2004 we regularly monitored the GCwith the AO assisted field spectrometer SINFONI(Eisenhauer et al. 2003; Bonnet et al. 2004). The in-strument is mounted at the Cassegrain focus of ESO’sUT-4 (Yepun) and offers several operation modes con-cerning pixel scale and wavelength coverage. For the TABLE 3Summary of SINFONI data used for this work. Theexposure time is the effective shutter-open time on S2,for other stars the actual exposure time might bedifferent since the observations were mosaicing aroundSgr A*. The FWHM was determined from a median imageof the respective cube on the unconfused star S8.
Date Band t exp on S2 FWHM >
250 5October 2-6 2005 H+K 120 74 22March 16 2006 H+K 110 76 27April 21 2006 H+K 10 100 6August 16/17 2006 H+K 100 88 18March 26 2007 H+K 20 86 10July 18-23 2007 H+K 133 78 15September 3/4 2007 H+K 70 81 15April 4-9 2008 H+K 200 65 40June 4 2008 H+K 10 84 3
GC we operated SINFONI mostly in the AO scale, map-ping 0.8” × ×
32 spatial pixels. We used theK-band grating and the combined H+K grating of SIN-FONI, with spectral resolutions of 4000 and 1500 respec-tively. For most of the data sets, the single exposuretime per frame was 10 minutes; a few data sets also used5 minute exposures. We chose various mosaicking pat-terns inside the central arcsecond for the different runs;mainly with the aim to have a good compromise betweenmonitoring the activity of Sgr A* and building up inte-gration on the S-stars. For stars at somewhat larger radii( r > × m R =14 .
65. As a result the performance of the AO stronglydepends on the seeing conditions. Therefore the qualityof our SINFONI data is variable over the data set. For atypical run, one can detect Bracket- γ absorption of early-type stars as faint as m K = 15 . m K = 16 .
0. A summary of ourdata is given in Table 3.We applied the standard data reduction for SINFONIdata, including detector calibrations (such as bad pixelcorrections, flat-fielding and distortion corrections) andcube reconstruction. The wavelength scale was cali-brated by means of emission line lamps and finetuned onthe atmospheric OH lines. The remaining uncertainty ofthe wavelength scale corresponds to typically .
10 km/s.We did not trust the SINFONI cubes for their astro-metric precision, they were used only for their spectraldimension. Nevertheless it is easy to identify stars in thecubes.tellar orbits in the Galactic Center 5
Other
Beyond the data sets described so far, we included afew more data points which we describe briefly in thissection. • Positions from public Gemini data for 2000:
In addition to our observations we included imagesfrom the Galactic Center Demonstration ScienceData Set obtained in 2000 with the 8-m-telescopeGemini North on Mauna Kea, Hawaii, using theAO system Hokupa’a in combination with the NIRcamera Quirc. These images were processed by theGemini team and released to be used freely. Wetreated this data in the same way as the SHARPdata. • Published radial velocities of S2 in 2002:
Thefirst radial measurements of S2 were obtained byGhez et al. (2003). We included the two publishedradial velocities since they extend the sampled timerange by one year and clearly contribute signifi-cantly in fixing the epoch of pericenter passage t P for S2. • Radial velocities from longsplit spectroscopywith NACO in 2003:
We used NACO in itsspectroscopic mode to measure the radial veloc-ity of S2 in 2003. The data are described inEisenhauer et al. (2003). • Radial velocities from integral field spec-troscopy with SPIFFI in 2003:
SPIFFI is theintegral field spectrometer inside SINFONI. Weused it without AO in 2003 as guest instrumentat ESO-VLT UT-4 (Yepun) under superb atmo-spheric conditions and obtained cubes from whichradial velocities for 18 stars (namely S1, S2, S4, S8,S10, S12, S17, S19, S25, S27, S30, S35, S65, S67,S72 S76, S83, S95, S96). The data are described inEisenhauer et al. (2003). ANALYSIS OF ASTROMETRIC DATA
This section describes in some technical detail the as-trometric calibration of our data. The first step is to mea-sure the positions of stars on the astrometric maps. Next,these positions of stars on the detector have to be trans-formed into a common astrometric reference frame. Thisprocedure ultimately relies on measurements of eight SiOmaser stars of which positions can be determined bothin the radio and in NIR images. However, a direct com-parison of the central arcsecond and the maser stars onone and the same image is impractical for two reasons:a) The exposure times necessary to obtain sufficientlydeep images for the S-stars saturates the detector at thepositions of the maser stars. b) The field of view of the13 mas/pix pixel scale is too small to show enough maserstars. Therefore we need to crosscalibrate the S-stars im-ages with the maser star images. This is done by a set ofselected reference stars (fig. 2), which are present bothin the S-star images and the maser star images. For theSHARP data, even an additional step of cross-calibrationis taken. We selected reference stars with 1” ≤ r ≤ m K ≈ . Extraction of pixel positions
All pixel positions were obtained by two-dimensionalGaussian fits in the images. The fits yielded both thepositions and estimates for the statistical error of thepositions (section 3.4.3). For each epoch for which wehave useful S-stars data we extracted pixel positions forthe S-stars and for the reference stars.
SHARP
Only star images that are not visually distorted (e.g.due to a confusion event) were used from the SHARPdata. • Reference stars : We obtained the reference stars’positions from the four single-pointing maps fromeach epoch. Due to the limited field of view, ineach frame only a subset of the reference stars ispresent. • Brighter S-stars : For the brighter S-stars (e.g.S2, S1, S8, S10, S30, S35) typically all four differ-ent pointing positions could be used. The astro-metric position of each star was determined fromthe corresponding four pixel positions using the as-trometric average position (see section 3.4.3). • Fainter S-stars : In order to detect faint S-starswe used the fifth coadded map which can be trustedastrometrically only for the innermost arcsecond.The limiting magnitude for a non-confused sourcewas typically m K ≈ .
8. We determined the pixelpositions of the weaker S-stars as well as the onesof the brighter S-stars. The latter served as refer-ence for relating the fainter stars to the astrometriccoordinate system (see section 3.4.3).Since we had two different deconvolutions at hand, weextracted pixel positions from both sets of images. Thus,up to eight (= two deconvolutions × four pointings) pixelpositions were obtained per star and epoch. NACO
For the NACO data, we used both the 27 mas/pix dataand the 13 mas/pix data. • SiO maser stars : Positions for the SiO maserstars were obtained by Gaussian fits to the stars’images in the 27 mas/pix mosaics. The SiO maserstars were unconfused in all mosaics. • Reference stars : The positions of the referencestars were measured both on the 27 mas/pix mo-saics as well as on the 13 mas/pix maps (Table 1),since they serve as cross-calibration between thetwo sets. They were selected to be unconfused,thus essentially it was possible to use all referencestars visible on any given frame. • S-stars : For isolated S-stars, the positions wereobtained from a simple Gaussian fit to the manu-ally identified stars in the maps. Due to the highersampling rate with NACO confusion events canbe tracked much better in the AO data than inthe SHARP data. Therefore it was reasonable toalso measure positions when stars are partly over-lapping. In such a case, a simultaneous, multiple Gillessen et al.Gaussian fit to the individual peaks was used, re-sulting of course in larger statistical uncertaintiesof the obtained positions.
Fig. 2.—
The open symbols mark the sample of 91 referencestars which are used to define the astrometric frame for the S-stars.The underlying image was obtained on April 3, 2007 in H-band,deconvolved and beam-restored with a beam of 2 pix. North is up,East is left. The field is 9.3” × Relating the reference stars to the SiO maser stars
The goal of this step is to obtain linear models forthe motions of the reference stars, i.e. to express theirvelocities and positions with linear functions x ( t ) , y ( t )in terms of astrometric coordinates. These models thendefine a common reference frame that is calibrated inposition and velocity such that radio Sgr A* should be atrest at the origin of the system. Such a coordinate systemallows one to test if the center of mass obtained fromorbital fitting coincides with the compact radio source .In the following we present two ways to obtain the de-sired calibration. They differ in the way in which thepositions and velocities for the reference stars are deter-mined: either all maser star images are tied to the re-spective maser positions (multi-epoch cross calibration);or only one maser star image is used to tie to the ra-dio maser positions, and the other maser star images arematched to that by an additional step of cross calibrationthat only involves infrared data (single epoch cross cali-bration). It turns out that both ways have their specificadvantages and disadvantages in terms of position andvelocity calibration of the resulting coordinate systems.We finally constructed a third coordinate system com-bining the advantages and rejecting the disadvantages. Multi-epoch cross calibration with all maser starimages Systematic problems of the coordinate system could be ab-sorbed into the orbital fitting by allowing the center of mass tohave an offset from 0/0 and a non-zero velocity, at the cost of notbeing able to test the coincidence of the center of mass with radioSgr A*.
Using the results from Reid et al. (2007) we calculatedthe expected radio maser positions for the given obser-vation epochs. The different maser images contained be-tween seven and nine SiO masers of which we used sixto eight since we excluded IRS7 due to its brightness of m K ≈ .
5. By allowing for a linear transformation oftype ~x = ~x + M.~p between the astrometric positions ~x and the pixel positions ~p in the respective image we de-termined a transformation by which any detector posi-tion can be converted into astrometric coordinates. Notethat the use of a linear transformation is justified sincethe IR images were distortion-corrected mosaics. Therms of the 1D-residuals of the SiO masers (thus apply-ing the transformation to the SiO masers’ pixel positionsand comparing the result with the expected radio po-sitions) was 2 .
28 mas. Correspondingly we expect thatfrom our 11 images a coordinate system can be definedto at most an 1D-accuracy of 2 . / √
11 mas ≈ . .
45 mas.The next step of refinement was to compare all mea-sured positions in one mosaic with the positions expectedfrom the fits, effectively checking how well a given im-age fits to the other 10 images. A visual inspection ofmaps of residual vectors showed that the residuals arenot randomly distributed but unveiled some systematicshift and rotation for each image. Since each image iscompared with 10 other images, any systematic prob-lem in the given image is most likely to come from thatimage and not from a combined effect of the others. In-deed, the interpretation of the observed systematic effectis straightforward, it means that each individual mosaicis not registered perfectly with respect to the sample av-erage, i.e. the transformation for the respective imageis slightly wrong. This systematic error is naturally ex-plained by measurement errors of the positions of theSiO maser stars in the respective image. Such an er-ror translates into an error of the parameters of the lin-ear transformation used to tie the astrometric frame tothe pixel positions in the mosaic and shows up as a sys-tematic effect in the residuals of the independent set ofreference stars. Thus, we were able to determine bet-ter transformation parameters by adding to the originallinear transformation the linear transformation that min-imizes the residuals of the reference star sample, yieldinga corrected linear transformation. We applied it to thedata and obtained the final linear motion models for thereference stars. The rms of the 1D-residuals now was0 .
55 mas. This step changed the position of the origintellar orbits in the Galactic Center 7by (∆ α, ∆ δ ) = ( − . , .
05) mas and the velocity of thesystem by less than 4 µ as/yr, these quantities being themean differences of the respective quantities for the ref-erence stars before and after the refinement. Hence, therefinement effectively did not change the coordinate sys-tem calibration. We call the coordinate system so definedthe ‘maser system’ in the following.The position of the origin of the maser system and itsvelocity are uncertain due to two effects: a) the non-zero errors of the SiO maser stars’ radio positions andvelocities and b) the IR positions of the SiO maser starsshow some residuals to the best fitting linear motion,indicative of residual image distortions and of measure-ment errors in the pixel positions in the IR images. Thepropagation of the statistical errors into the definitionof the coordinate system was addressed using a Monte-Carlo technique. We varied the input to the transfor-mations according to the measured errors and residuals.We created 10 realizations of transformations, assuminga Gaussian distribution of the simulated values aroundthe original values. The standard deviation of the posi-tions obtained for Sgr A* estimates the positional uncer-tainty of the maser system under the assumption of un-correlated measurement errors. We obtained (∆ α, ∆ δ ) =(0 . , .
77) mas. Similarly, the standard deviation of thevelocities obtained for Sgr A* estimates the uncertaintyof the maser system’s velocity under the same assump-tion. We obtained (∆ v α , ∆ v δ ) = (0 . , .
55) mas/yr.However, in our data the assumption that the errorsfrom the 11 maser images are uncorrelated is not fulfilled.We rather observe a typical residual per SiO maser starfor all epochs when comparing the transformed, mea-sured positions with the predicted radio positions. Pos-sible reasons are: first, the linear motion models obtainedfor the SiO masers could be inaccurate due to some un-known some unknown systematic problem of the radiopositions. Secondly, the radio positions could not be ap-plicable to the IR positions, for instance if the maseremission would originate from far away of the stellar sur-face. Thirdly, the correlation could arise due to someunaccounted systematics in the infrared frames, such asuncorrected distortion. We fitted the residuals of eachstar with linear functions and obtained in that way es-timates for the mean position and mean velocity uncer-tainty for each star. Then we calculated the mean devi-ation (over the SiO maser stars which are <
15” awayfrom Sgr A*) of these linear motion model parametersas estimates for the positional and velocity uncertaintyof the maser system given the correlations in our data.With our initial transformation we obtained (∆ α, ∆ δ ) =(0 . ± . , . ± .
43) mas and (∆ v α , ∆ v δ ) = (0 . ± . , . ± .
24) mas/yr. After the refinement wegot (∆ α, ∆ δ ) = (0 . ± . , . ± .
58) mas and(∆ v α , ∆ v δ ) = (0 . ± . , . ± .
33) mas/yr. Finallywe conservatively adopt for the uncertainties of the masersystem (∆ α, ∆ δ ) = (1 . , .
5) mas and ∆ v α = ∆ v δ =0 . Single-epoch cross calibration with one maser starimage
The sensitivity of the reference star velocities to er-rors in the SiO maser velocities can be avoided by anadditional step of cross-calibration: We can measure thepositions of the reference stars in all maser images withrespect to a much larger sample of stars in these images.This cluster of stars is assumed to be non-rotating andnot moving with respect to Sgr A*. The cluster is tied tothe astrometric frame for just one epoch, as given by theradio positions of the SiO masers, which can be calcu-lated for the chosen epoch from Reid et al. (2007). Forall other epochs it is assumed that the mean cluster isstationary in time. Hence, the velocity calibration relieson the statistical argument that for a sufficiently largesample of cluster stars the mean velocity of the cluster isexpected to become very small. For a typical velocity of v for a cluster star and N stars, the error of this meanshould be of order v/ √ N .Effectively the few maser stars are only used once inthis scheme. Any error in their radio positions, radio ve-locities or NIR detector positions will therefore translateinto a positional offset, but not into a systematic velocityof the coordinate system. The latter is instead connectedto the validity of the assumption that the cluster meanis stationary. In order to ensure the best estimate of thevelocity calibration we adopted the following procedure:1. We selected the maser mosaic from 12 May 2005,which was chosen since it is of good quality androughly corresponds to the middle of the range intime covered with NACO. Building upon the workdone by Trippe et al. (in prep.) we selected an en-semble of stars in that mosaic of which the positionscan be measured with high reliability: Take allstars that have a peak flux of more than 25 countswhich at the given noise level of 1.9 counts selectshigh-significance stars. In a second step many starsget excluded again: All stars with more than 700counts (they could be saturated in other frameswith longer single detector integration times) andall stars that have a potential source (peak with 5counts) within 10 pixels. Furthermore a Gaussianfit was required to yield a FWHM < .
05 pix andthe fitted position must coincide with the positionobtained using DAOPHOT FIND. This yielded asample of 433 stars. We determined the astromet-ric positions at the given epoch of the 433 stars bymeans of a linear transformation that was deter-mined from the eight maser stars.2. We then determined preliminary astrometric posi-tions for a much larger sample of 6037 stars in all11 mosaics, by tying their pixel positions at all 11epochs to the astrometric positions of the 433 starsat the reference epoch with a linear transformation.Note that not taking into account proper motionsin that step makes the velocity calibration indepen-dent from the radio measurements. The error dueto the omission of the proper motions is minimizedby using 433 stars instead of few masers for thecross-calibration.3. We fitted linear motion models to all 6037 stars.After the fit we determined the residuals of all star Gillessen et al.positions in all mosaics. By inspecting the residu-als of any mosaic as function of position, we wereable to map the residual image distortions in thegiven mosaic. These residual image distortions canarise due to imperfect registration of the individ-ual exposures to the respective mosaic, or due toan error in the distortion correction applied to theindividual frames.4. For each star in each mosaic we determined an es-timate of the residual image distortion by calculat-ing the mean of the residuals of the stars in thevicinity ( r < ′′ ) of the chosen star. The radiuswas chosen such that a suitable number of starswas present in the area from which the correctionwas determined and the area was sufficiently lo-cal. A value of r <
2” was a good compromise,yielding 30-50 stars typically. The estimate forthe residual image distortion was then subtractedfrom the given star; the typical values applied were∆ x = 0 . ± .
20 mas and ∆ y = 0 . ± .
24 mas.5. In a second fit we used the corrected astrometricpositions in order to obtain updated linear motionsmodels for the 6037 stars.6. Then we defined the final cluster: it consists of allstars which were present in all 11 mosaics, withradii between 2” and 15” and which are not knownearly-type stars. These criteria yielded a clustersample size of 2147 stars.7. We determined the cluster mean velocity, yielding( − . , .
00) mas/yr, and subtracted that valuefrom all velocities of the 6037 stars. This then isthe final, velocity-calibrated list of linear motionsfrom which the reference star sample is extracted.The mean radius of the 2147 stars of the clustersample is 9.89”, the root mean square (rms) speedof the stars in the sample is 157 km/s ≈ .
15 mas/yr(for R = 8 kpc).We call the coordinate system defined in this way the‘cluster system’. Since we expect the mean of the clus-ter to show a net motion of order 157 / √ / s =3 . / s = 0 .
09 mas/yr, we estimate the uncer-tainty of the velocity calibration to be of the samesize. We checked this number more thoroughly bymeans of a Monte Carlo simulation: We dividedthe cluster into nine radial bins with boundaries[2 , , , . , , , , , ,
15] mas (selected such that ineach bin roughly the same number of stars is present).For each bin we have determined the rms velocityyielding 3 . , . , . , . , . , . , . , . , . .
06 mas/yr; even a bit bet-ter than the simple estimate. Hence, if the assumptionof isotropy is correct, the cluster system should allow fora better calibration of the reference star velocities than with the maser system. The assumption could be wrong,for example if a net streaming motion were present inthe GC cluster.The statistical positional uncertainty of the origin ofthe cluster system was estimated by the same meansas for the maser system. We obtained (∆ α, ∆ δ ) =(0 . , .
51) mas. In addition to these uncertainties, theresiduals of the SiO masers also need to be considered,for the epoch at hand the mean deviation is (∆ α, ∆ δ ) =(1 . , .
12) mas. The uncertainties here are greater thanthe respective numbers for the maser system due to thefact that the position of Sgr A* in the cluster systemis effectively measured only on one frame while in themaser system it is measured in several and the residualsare not fully correlated.
The final, combined coordinate system
The maser system has a smaller systematic error in itsposition calibration, while the cluster system is superiorwith respect to the velocity calibration. Hence, by com-bining the two we were able to construct a system thatcombines both advantages. The idea simply is to correcteither the velocity calibration of the maser system suchthat it agrees with the one from the cluster system or tocorrect the origin of the cluster system such that it co-incides with the origin of the maser system (taking intoaccount that the systems refer to two different epochs).Note that this implicitly uses the fact that the second, re-fining transformation of the maser system did not changeits calibration properties.We used the sample of reference stars to compare thetwo systems. The mean positional offset between the twolists of positions for the epoch of the cluster system was ~p CSys − ~p MSys = (cid:18) − . . (cid:19) ± (cid:18) . . (cid:19) mas . (1)Here, ‘Csys’ denotes the cluster system, ‘MSys’ the masersystem. The errors are the standard deviation of thesample of differences. We also calculated the differencesof the reference star velocities, as given by the two lin-ear motion models obtained for each reference star. Weobtained ~v CSys − ~v MSys = (cid:18) − . . (cid:19) ± (cid:18) . . (cid:19) masyr , (2)where again the errors are the standard deviation of thesample of differences.This means that the two coordinate systems differ sig-nificantly in position and velocity calibration in a sys-tematic way. It should be noted that only the differencebetween the two coordinate systems is that well defined;for the question how well each of the coordinate systemsrelates to Sgr A*, the larger, systematic errors of sec-tions 3.2.1 and 3.2.2 need to be considered. It is exactlythe fact the difference between the coordinate systemsis well defined that allowed us to combine the two co-ordinate systems and to gain accuracy in the combinedsystem that way. Also note that the size of the offsetsoccurring here are consistent with the combined uncer-tainties of the two coordinate system; much larger offsetswould have meant that the coordinate systems would beinconsistent with each other.Finally we chose the method which corrects the clustersystem by a positional offset. The positional differencetellar orbits in the Galactic Center 9from equation (1) was subtracted from all positions of thecluster stars (and thus also from the reference stars thatare a subset of the cluster). This combined coordinatesystem has the same prior as the cluster system, namelythat the cluster is at rest with respect to Sgr A*. Thelinear motion models so obtained were then used for thefurther analysis. Relating the S-stars to the reference stars
We constructed the transformation from pixel posi-tions on the detector to astrometric positions by meansof the reference stars. For each given image, we calcu-lated the expected astrometric positions of the referencestars using the linear motions models as obtained in sec-tion 3.2.3. Given the pixel positions of the reference starsin the respective image, we related the two sets of posi-tions by means of a cubic transformation (20 parameters)of type x sky = p + p x + p y + p x + p xy + p y + p x + p x y + p xy + p y y sky = q + q x + q y + q x + q xy + q y + q x + q x y + q xy + q y . (3)Once the transformations are known, it is straight-forward to apply them to the pixel positions of the S-stars.The parameters p i , q i were found by demanding thatthe transformation should map the two lists of positionsoptimally in a χ sense. Since the problem is linear, theparameter set can be found with a pseudo-inverse matrix(we always used at least 50 stars, thus 100 coordinates,for 20 parameters). The procedure also allows for anoutlier rejection. For this purpose we applied the trans-formation to reference stars themselves and calculatedthe residuals to the expected astrometric positions. Byonly keeping reference star positions which are not moreoff than 15 mas from the expected position we cleanedour sample. This excluded in total 19 of the 7189 refer-ence star positions. For the cleaned set we redeterminedthe linear motion model for each star under the side con-dition that the refinement would not change the meanposition or the mean velocity of the sample of referencestars, thus avoiding a change of the origin of the coor-dinate system and a change of its velocity. Comparedto previous work the number of reference stars used isroughly a factor eight larger. This reduced the statisti-cal uncertainty of this calibration step to a very smalllevel .For the SHARP data we had to use some additionalsteps for relating the S-star positions to the referencestars, since for a given epoch we used two deconvolu-tions for which we had four single-pointing frames andone combined map respectively. We used the pixel posi-tions of the reference stars in the two times four single-pointing images together with the predicted astrometricpositions of the reference stars to set up eight transfor-mations of the type given in equation 3. Not all refer-ence stars are present in all pointings, but in all cases Actually some of the reference stars relatively close to Sgr A*were also considered as S-stars for which we tried to determineorbits. Indeed, four of those stars showed significant accelerations.However, we did not exclude them from the sample of referencestars. Therefore, an additional, obvious step of refinement wouldbe to allow for quadratic motion models for the reference stars. their number exceeded 50, such that the transformationparameters were well determined. With these transfor-mations we calculated the astrometric positions of thebrighter S-stars detected in the eight frames and usedthe average astrometric position in the end. The stan-dard deviation of the eight astrometric positions was in-cluded in the error estimate. For the fainter S-stars weused the coadded maps. For the two coadded maps (twodeconvolutions) per epoch we set up two times four fullfirst order transformations relating pixel positions of thebrighter S-stars in each coadded map to the respectivepixel positions in the four single-pointing frames. Withthese transformations we determined the pixel positionsof the fainter S-stars which they would have had in thesingle-pointing frames. These fictitious pixel positionswere then transformed with the cubic transformation ofthe respective single-pointing frame into astrometric po-sitions. The average of the latter was used in the end, thestandard deviation was included in the error estimate.
Estimation of astrometric errors
The goal of this section is to understand the errors ofthe astrometric data. This includes both statistical andsystematic error terms. The statistical error is due to theuncertainty of the measured pixel positions. Among thesystematic error terms are the influence of the coordinatesystem, residual image distortions, transformation errorsand unrecognized confusion.
Offset and velocity of the coordinate system
The accuracy in 2D-position (∆ x , ∆ y ) and 2D-velocity(∆ v x , ∆ v y ) of the combined coordinate system is givenby the numbers in sections 3.2.1 and 3.2.2. In the thirddimension, we don’t use any priors for ∆ z , since we wishto determine R from our data.For ∆ v z we use the prior that Sgr A* is not mov-ing radially, based both on theoretical arguments andon radio and NIR measurements. Even if Sgr A* is dy-namically relaxed in the stellar cluster surrounding it,some random Brownian motion due to the interactionwith the surrounding stars is expected. Merritt et al.(2007) calculated this number and concluded that themotion should be ≈ . v l = 18 ± v b = − . ± . R = 8 kpc). The significance of thefact that v l = 0 is disputed, and furthermore it is notclear, whether it is truly due to a peculiar motion ofSgr A* or due to a difference between the global and localmeasures of the angular rotation rate of the Milky Way(Reid & Brunthaler 2004). Clearly, the motion of Sgr A*perpendicular to the galactic plane is very small as ex-pected. In the third dimension, the velocity of Sgr A*can only be determined indirectly by radial velocity mea-surements of the stellar cluster surrounding it. Using asample of 85 late-ytpe stars Figer et al. (2003) found thatthe mean radial velocity of the cluster is consistent with0: v z = − ±
11 km/s. Trippe et al. (in prep.) useda larger sample of 664 late-type stars and found con-sistently v z = 4 . ± . U ≈ . v z = 0 ± x = 0 .
95 mas ≈ . y = 2 .
35 mas ≈ . v x = 0 .
06 mas / yr ≈ . / yr∆ v y = 0 .
06 mas / yr ≈ . / yr∆ v z = 5 km / s . (4) Rotation and pumping of the coordinate system
Potentially, there are two more degrees of freedom,which could affect the reliability of the chosen coordi-nate system, namely rotation and pumping. An artifi-cial rotation can be introduced if the selected stars bychance preferentially move on tangential tracks with apreferred sense of rotation. Similar, artificial pumpingcan occur: suppose that by chance all selected stars moveon perfect radial trajectories and that stars further outmove faster than stars closer to Sgr A*. Such a pattern,which would be somewhat similar to the Hubble flow ofgalaxies, would yield under the set of transformationsa time-dependent plate scale and otherwise stationarystars. Both effects can affect the selection of the refer-ence star sample and (less important) the selection ofcluster stars.The chosen coordinate system relies on the assumptionthat the cluster does not show any net motion (see sec-tion 3.2.2), net rotation or net pumping. The selection ofa finite number of cluster stars however limits the accu-racy with which these conditions can be satisfied. Given2147 stars with a RMS velocity of ≈
157 km/s and a typ-ical distance of 10” we expect that any selection leads toa pumping or rotation effect of the order of 9 µ as/yr/”.Due to the errors in the SiO maser positions, the masersystem can show artificial pumping or rotation. Similarto what was done in sections 3.2.1 and 3.2.2 we simulatedin a Monte Carlo fashion the error propagation. From10 realizations of the transformations, assuming the ob-served errors of the SiO maser positions in the NIR andradio, we created perturbed sets of reference stars. Thestandard deviation of the pumping and rotation motion( v r /r and v t /r respectively) over these sets then estimatethe stability of the maser system. We obtained v r /r | MSys = 37 µ as / yr / ′′ ,v t /r | MSys = 33 µ as / yr / ′′ . (5)The cluster system (and therefore also the combinedsystem) can be checked against the maser system. Bycalculating the difference in velocity for each referencestar and subtracting from those the difference of the twocoordinate system velocities we obtained a vector field ofresidual velocities, which is well described by: v r /r | CSy − v r /r | MSy = (32 ± | stat ± | sys ) µ as / yr / ′′ v t /r | CSy − v t /r | MSy = (6 ± | stat ± | sys ) µ as / yr / ′′ (6)The combined size of the effects from equations 5 and 6estimate the error made when using the assumption thatthe combined coordinate system is non-rotating and non-pumping. At 1” these effects can sum up over 15 years to at most 0 . ≈ . Statistical errors of the pixel positions
This paragraph deals with the uncertainties of the stel-lar positions on a given image; the unit of this error termas measured is therefore pixels. The error which is mosteasily accessible is the formal fit error of the Gaussian fitto a source. However, in deconvolved and beam-restoredimages it might be a bad estimator for the positional un-certainties. Therefore we compared additionally differentdeconvolutions of the same image for each epoch in orderto get a more robust estimate.
100 200 500 1000 2000 50001.000.500.200.100.05 Flux H ADU L D de v H m a s L Fig. 3.—
The statistical errors of the pixel positions for theNACO K-band data as a function of arbitrary detector units offlux. The thin lines show the respective error model for each epoch;the thick dashed line is the mean for the data. The mean has afloor at 99 µ as, the median (not shown) at 84 µ as. For the SHARP data we used up to eight (= two decon-volutions × four pointings) pixel positions. The standarddeviation of the astrometric positions was included in theerror estimate for the statistical position error. For starswhich were present only in one frame, the typical errorof the epoch was used instead.For NACO we split up each data set into two parts anddeconvolved both co-added images with the same pointspread function as the co-added image of the completedata set (see Section 2.2). We determined the pixel posi-tions of the reference and S-stars in the two deconvolvedframes and applied a pure shift between the two listsof pixel positions such that the average pixel positionis the same for both. The remaining difference betweenrespective positions of one star estimates the statisticaluncertainty for that star. The error estimates obtainedthis way were a strong function of the stellar brightness.Therefore we described the error estimates as a functionof flux for each epoch (see Figure 3) using a simple em-pirical model of the form ax − n + b . The mean floor ¯ b overall data sets is 99 µ as, while for lower fluxes the error in-creases up to 2 mas. We used the empirical descriptionof each image to assign an error to all stellar positionsobtained from that frame. Finally we checked whetherthe formal fit error of the positions was greater than theestimate from the empirical error model. In such a casewe used the formal fit error instead. Figure 4 showsthe final distribution of statistical errors for the NACOtellar orbits in the Galactic Center 11data. It is effectively the mean error model folded withthe brightness distribution of the S-stars. peak ž H mas L N po s i t i on s Fig. 4.—
The measured distribution of the statistical errors ofthe pixel positions for the NACO data. The characteristic statis-tical error (defined as the peak of the distribution) is 108 µ as, thesystematic error terms have to be added to this to come to a fairestimate of the true uncertainty. For the SHARP data we obtained a broad distributionof the statistical pixel position errors with no clear max-imum and a tail to 2 mas. The median error is 360 µ as,the mean error 760 µ as in the SHARP data. Residual image distortions
A main source of error at the sub-milliarsecond level isimage distortions. We estimated this error term by com-paring distances of stars in different pointing positionswith a dither offset of 7” (see Figure 5). If we had usedonly the raw positions and linear transformations, the re-sulting mean 1D position error would be as large as 1 masfor the 13 mas/pix NACO data. By applying a distortionmodel (see section 2.2) plus a linear transformation thiserror can be reduced to 600 µ as. Allowing for a cubictransformation onto a common grid yields an error of240 µ as only. This justifies our choice to use a high ordertransformation rather than to de-distort the 13 mas/pixNACO images. The numbers obtained in this way areactually the combined error of the statistical and trans-formation uncertainties with the residual image distor-tions. Subtracting the former we conclude that residualimage distortions have a contribution of 210 µ as to theerror budget of each individual astrometric data point.We thus added this value in squares to all other errorterms, effectively acting as a lower bound for the astro-metric errors.We applied the same analysis to the 27 mas/pix NACOdata which had a dither offset of 14” (see Figure 6). Theraw differences showed a skewed distribution, indicatingthe presence of image distortions. The rms of this dis-tribution is 2 . . . - - D dist @ pix D p @ % D raw frames Σ = - - D dist @ pix D p @ % D distortion corrected Σ = - - D dist @ pix D p @ % D transformation Σ =
Fig. 5.—
Determination of residual image distortions for theNACO H-band data from September 8, 2007, 13 mas/pix. The his-tograms show the differences of detector distances for a set of bonafide stars as measured in the four pointing positions with a ditheroffset of 7”. Left: Using the raw frames. Middle: After applicationof a distortion model, Right: After transforming the raw positionswith a cubic transformation onto a common grid. The correspond-ing 1D coordinate errors are determined from Gaussian fits to thedistributions and are quoted at the top of each panel. - - D dist @ pix D p @ % D raw framesrms = - - D dist @ pix D p @ % D distortion corrected Σ = - - D dist @ pix D p @ % D transformationrms = Fig. 6.—
Determination of residual image distortions for theNACO K-band data from March 16, 2007, 27 mas/pix. The his-tograms show the differences of detector distances for a set of bonafide stars as measured in the four pointing positions with a ditheroffset of 14”. Left: Using the raw frames. Middle: After appli-cation of a distortion model, Right: After transforming the rawpositions with a cubic transformation onto a common grid. Thecorresponding 1D coordinate errors are quoted at the top of eachpanel, in the middle panel the value is determined from a Gaussianfit, for the other two the rms is quoted due to the non-Gaussianityof the distributions. residual image distortions of 0 . . Transformation errors
It is important to notice that any error in derivingthe motions for the reference stars only translates into aglobal uncertainty of the coordinate system (which couldshow up as an offset of the center of mass from 0/0 ora net motion of the coordinate system). It will howevernot affect the accuracy of individual data points in thissystem. Only the selection of reference stars and trans-formation errors contribute to the errors of the individualdata points. We estimated them by performing all coor-dinate transformations not only once but also with sub-sets of the available reference stars. The standard devia-tion of the sample of obtained astrometric positions wasthen included in the astrometric error estimate. The typ-ical uncertainty introduced by the transformations wasquite small, namely 23 µ as for the NACO data. Thisis consistent with the fact that ≈
100 stars have beenused of which each can be determined with an accuracy2 Gillessen et al.of ≈ µ as. For the SHARP data we found a valueof 100 µ as, again consistent with the characteristic singleposition error of ≈ Differential effects in the field of view
At the sub-mas level, there is a multitude of differentialeffects over the field of view that can influence astromet-ric positions. The most prominent ones are relativisticlight deflection in the gravitational field of the sun, lightaberration due to Earth’s motion or refraction in theatmosphere. Since our analysis is based on relative as-trometry, the absolute magnitudes of the effects do notmatter. Only the differential effects over the field of viewcan contribute to the positional uncertainties. • Over a field of view of 20” the differential effectsof aberration can be described by a global changeof image scale (Lindegren & Bastian 2006). Sincewe fit the image scale for each epoch separately,the differential aberration is absorbed into the lin-ear terms of the transformation and thus is notaffecting the astrometry. The size of the effect fora small field of view with a diameter f amountsto f × v/c × cos Ψ where Ψ is the angle betweenthe observation direction and the apex point. For f ≈ ′′ and v ≈
30 km/s this yields ≈ • The light deflection can be approximated by4 mas × cot Ψ / µ as as long as Ψ > . ◦ , which is guaran-teed for all our data. • From the usual refraction formula R = 44 ′′ tan z (for a standard pressure of 740 mbar at Paranal)we find a differential effect of 4 − − Unrecognized confusion
One important contribution to the position errors isthe fact that stars can be confused and that sometimesthe confusion is not recognized. This problem is moresevere for the SHARP data than for the NACO data dueto the lower resolution. Of course we excluded positionsfor which we know that they are confused. However,unrecognized confusion cannot be dealt with by princi-ple. We therefore simply accept that these events hap-pen. This means in turn, that we expect to find a re-duced χ > Gravitational lensing
Gravitational lensing might affect the measured posi-tions. A quantitative analysis shows that the effects arevery small except in unusual, exceptional geometric con-figurations. For a star at a distance z ≪ R sufficientlyfar behind Sgr A* the angle of deflection as measuredfrom Earth is θ = zR GM MBH c b , (7)where b is the impact parameter. For the GC, this evalu-ates to θ ≈ µ as × z/b , indicative of a very small astro-metric effect unless z/b ≫ z ≈ b ≈ ≈
16 AU. In our data set, none ofthe stars get close to the regime that gravitational lens-ing actually becomes important. Therefore, we neglectedthe effect.
Comparison of error estimates with noise
We were able to check how well our error estimatesagree with the intrinsic noise of the data. For this pur-pose we fitted all measured positions of the referencestars with simple quadratic functions. After exclusion of3 σ -outliers, we have calculated the reduced χ for eachreference star. The mean reduced χ for the NACO datais 2 . ± .
7, while for the SHARP data we obtained valuesbetween 0.5 and 2.0 with a mean of 1.0. peak ž H mas L N po s i t i on s Reference Stars SHARP peak ž H mas L N po s i t i on s Reference Stars NACOpeak ž H mas L N po s i t i on s S - Stars SHARP peak ž H mas L N po s i t i on s S - Stars NACO
Fig. 7.—
Final distribution of total astrometric errors for ourdata. Left column: SHARP data, right column: NACO data.Top row: reference stars, bottom row: S-stars. The curves showempirical fits to the histograms in order to determine the respectivecharacteristic error as the peak of the distribution.
Since our data set consists of two subsets (SHARP andNACO), each covering roughly the same amount of time,the relative weight of the two subsets matters. Given thatwe seem to underestimate the errors for NACO a bit,while the SHARP errors seem consistent with the noisein the data, we decided to apply a global rescaling factorof r = 1 .
42 to all NACO data points. This procedureadjusts the relative weight between the two subsets. Stillwe expect a reduced χ > µ as, in the SHARP data itamounts to 760 µ as. For the S-stars, the histogram of theNACO errors has a peak also around 325 µ as and a tailtowards larger errors, essentially telling us that for brightS-stars the astrometry is as good as one could hope for(since it is equally good as for the reference stars). Thetail is due to the fact that many of the S-stars are faint(hence the statistical error is severe) and probably alsounrecognized confusion events affect the statistical er-ror since confusion can alter the shapes of the imagesof faint stars. In the SHARP data, the typical S-starserror is 2 mas and the lower end of the distribution at ≈ S2 in 2002
Our data set covers the pericenter passages of severalstars. Particularly important to our analysis is the one ofthe star S2. The star is one of the brightest in the sam-ple and we observed a full orbit (see Figure 13). In 2002S2 passed its pericenter, thus changing quickly in veloc-ity throughout a period of a few months. These dataare particularly useful for constraining the potential ofthe MBH. However, as we will now discuss, the photom-etry of the star near pericenter-passage is puzzling andmay indicate that the positional information is affectedby a possible confusion event with another star. Fig-ure 8 shows a K-band PSF-photometrically determinedlight curve for the star (Rank 2007). It is clear that S2was brighter in 2002 than in the following years. Thereare several reasons why a star could change its apparentbrightness.
S2S82003 2004 2005 2006 200714.114.213.914.13.813.714.314.414.514.6 year m a g H K L Fig. 8.—
The K-band magnitude of S2 as function of time inthe NACO data, determined by means of PSF photomery (blackdata). For comparison the star S8 is shown (red data).
1. In 2002, S2 was positionally nearly coincident withSgr A* and thus confused with the NIR counter-part of the MBH. Typically, Sgr A* is fainter than m K = 17 and thus the extra-light from Sgr A*in quiescence is not sufficient to explain the ob-served increase in brightness of S2. However,Sgr A* is known to exhibit flares that can reacha brightness level that could account for the ob-served increase in brightness (Genzel et al. 2003a; Trippe et al. 2007). In that case we would expectto see intra-night variability of S2 in the 2002 data.Assuming conservatively that we can determine therelative flux of S2 to ∆ m K = 0 . m K ≈
14) we esti-mate that we would have noticed any variations inSgr A* that exceed m K ≈ .
5. Since we did notobserve any intra-night variability we exclude thatflares from Sgr A* significantly contributed to theincreased brightness of S2 in 2002.2. Intrinsic variability of S2 might explain the ob-served light curve. However, it is unlikely to bethe correct explanation, since it would be a big co-incidence that the brightening happens during thepericenter passage. Also an eclipsing binary seemsunlikely given the slow variation.3. The star could change its properties duringthe pericenter passage. While tidal heating(Alexander 2005) cannot plausibly change thetemperature of a star within a few months, the in-teraction of S2 with some ambient medium does notseem ruled out. Such an encounter would primar-ily change the surface temperature of the star andtherefore would act nearly instantaneously. Effec-tively the light curve would then be a direct traceof the density of the surrounding gas encounteredalong the orbital path of S2. However, energeti-cally, this scenario seems unlikely: Given the max-imum velocity of S2 at pericenter ( v ≈ r = 11 R ⊙ , Martins et al.(2008)) and assuming that the kinetic energy ofthe gas that hits the geometric cross section ofthe star is converted to radiation, one can estimatethe number density n necessary to produce the ob-served brightness increase of ∆ m K ≈ .
5. We ob-tained n ≈ cm − , which is unrealistically high,and so we do not favor this scenario.4. Loeb (2004) proposed that the stellar winds ofearly-type stars passing their pericenters close tothe MBH could alter the accretion flow ontoSgr A*. Such an event would produce a changein the brightness of Sgr A* on the timescale ofmonths, compatible with Figure 8. However,Martins et al. (2008) showed that the mass lossrate of S2 is too low for this mechanism to work.5. The extinction could be locally smaller than theaverage value. For instance, Sgr A* could removedust in the interstellar medium in its vicinity. Thishypothesis can be tested in the future by observingother S-stars passing close to Sgr A* during thepericenters of their orbits.6. The brightness of S2 could be affected by dust inthe accretion flow onto the MBH. The dust wouldbe heated by S2 and account for the excess bright-ness, a proposal that was used by Genzel et al.(2003b) to explain the MIR excess of S2/Sgr A*.7. The star could be confused with another star. IfS2 had been located very close to another star inprojection, the true nature of this encounter could4 Gillessen et al.remain undiscovered, but the observed brightnessof S2 would be increased.Of the three viable explanations (5 to 7), the first wouldnot lead to astrometric biases, the others however woulddisplace S2 artificially. Given the importance of the 2002data, we decided not to discard it completely but to es-timate the astrometric error assuming a confusion event,given the measured increase in brightness. - - - - - - - - (cid:144) - - - - e xc l uded : e lli p t i c i t y o f s po t > % H pix L m ag K centroid deflection H pix L Fig. 9.—
Simulation of a confusion event. The contour lines showby which amount a m K = 14 source is displaced if it is confusedwith a second source that has certain magnitude and that is locatedin a given distance. The units are pixels, the simulation assumedsimple Gaussian point spread functions that are sampled as it isthe case for the NACO detector in K-band. The area to the topright can be excluded since a relatively bright source a few pixelsapart from the primary would produce an elongated shaped image(which is not observed for S2 in 2002). The line denotes the limitat which the major axis is 30% larger than the minor axis. Thehorizontal lines indicate the brightnesses that a secondary sourcewould have needed to push the S2 brightness up by the observedamount for the observed magnitudes at the dates indicated. Foreach date a mean deflection can be read from this plot. That valueis used as astrometric error for S2 at the given date. For this purpose we simulated confusion events. We as-sumed simple Gaussian point spread functions and sam-pled them as they are sampled by the 13 mas/pix scaleof the NACO camera in K-band. By polluting a pri-mary source with a fainter secondary source we gener-ated a confused stellar image. This was then fit by atwo-dimensional Gaussian and the displacement from theposition of the primary source was determined. We var-ied brightness ratio and distance between the two sourcessystematically, yielding a displacement map (Figure 9).This map allows the determination of the possible rangeof displacements if the brightness of the secondary sourceis known. The range can be constrained further, since abright secondary source in a few pixels distance will leadto very eccentric images that would be easily detected inthe data. We excluded all points that would lead to astellar image of which the major axis is more than 30%larger than the minor axis. Thus, from the measuredS2 fluxes, the known, unconfused brightness of S2 andthe roundness of the S2 images, we were able to con-strain the astrometric bias due to confusion. For each date we looked up in figure 9 the possible range of as-trometric displacements given the observed brightness ofS2, essentially determining the profile along a horizon-tal line in the plot. The mean of this distribution wasthen considered as an additional 2D error to be addedto the respective astrometric errors for that date. Thesuch obtained error terms ranged between 2 .
37 mas and3 .
76 mas - - - - H '' L D e c H '' L Fig. 10.—
The 2002 data of S2. The grey symbols show themeasured positions, the errors are as obtained from the standardanalysis and are not yet enlarged by the procedure described insection 3.5. The black dots are the positions predicted for theobservation dates using an orbit fit obtained from all data otherthan 2002. The blue shaded areas indicate the uncertainties in thepredicted positions resulting from the uncertainties of the orbitalelements and of the potential, taking into account parameter cor-relations. The little ellipse close to the origin denotes the positionof the fitted mass and the uncertainty in it. This plot shows thatthe S2 positions are dragged for most of the data by ≈
10 mas tothe NE; they are not biased towards Sgr A*.
We checked whether the residuals of the 2002 data, rel-ative to an orbit fit to the data other than 2002, showsome systematic trend (figure 10) and found that in-deed all points appear to be shifted systematically by10 mas ≈ m K ≈ . m K ≈ . − . − . ≈
40 mas nearly parallel to S2. Itis extremely unlikely that we have missed such an event.From this analysis, it is clear that the weight of the2002 data will influence the resulting orbit fits, sincethese points will systematically change the orbit figureat its pericenter. At the same time we have no plausibletellar orbits in the Galactic Center 15explanation for the increase in brightness and the sys-tematic residuals in the 2002 data; in particular a confu-sion event seems unlikely. Thus, it is clear that using the2002 data will affect the results, but we cannot decidewhether it biases towards the correct solution or awayfrom it. Therefore we use in the following two options:a) we include the 2002 data with the increased error bars;b) we completely disregard the 2002 data of S2. ANALYSIS OF SPECTROSCOPIC DATA
Most of the radial velocities were obtained withSINFONI. For the few non-SINFONI data we usedthe already published values (Ghez et al. 2003;Eisenhauer et al. 2003).From the SINFONI cubes we determined spectra bymanually selecting on- and off-pixels for each S-star andcalculating the mean of the on-pixels minus the mean ofthe off-pixels. The spectra were then used to determinethe radial velocities of the respective stars at the givenepoch. We only used spectra in which we were able to vi-sually identify the stellar absorption lines without doubt.The most prominent features are the Br- γ line for early-type stars and the CO band heads for late-type stars.Both line profiles are non-trivial, possibly biasing theresult when using a simple Gaussian profile to fit the line.The bias can be avoided by crosscorrelating the spectrawith a template and determining the maximum of thecrosscorrelation.For the CO band heads we used a template spec-trum from Kleinmann & Hall (1986). We used the well-established tool ‘fxcor’ which is part of NOAO-package iniraf. We identified the following stars as late-type stars:S10, S17, S21, S24, S25, S27, S30, S32, S34, S35, S38,S45, S68, S70, S73, S76, S84, S85, S88, S89, S111.Also for the early-type stars one might be worried thatradial velocity measurements are biased due to a com-plex line profile. In particular, Br- γ might be affected bynearby He lines. We tested this for the bright star S2,by generating a template from our 2004 - 2006 data :we estimated for all S2-spectra the velocities by simpleGaussian fits to the Br- γ line. We then Doppler-shiftedall spectra to the 0-velocity (using the iraf task ’dopcor’)and coadded them (using the iraf task ’scombine’). Thisresulted in a first template for S2. With this templatewe crosscorrelated all individual S2-spectra in the wave-length range 2 . − . µ m (using the iraf task ’fxcor’)and obtained better estimates for the velocities. Withthese new velocities we reassembled the template spec-trum. We stopped after this first iteration since the ve-locity differences had already converged to a mean de-viation of 0 . γ line for the other early-type stars. We identifiedthe following stars as early-type: S1, S2, S4, S5, S6, S7,S8, S9, S11, S12, S13, S14, S18, S19, S20, S22, S26, S31,S33, S37, S52, S54, S65, S66, S67, S71, S72, S83, S86, The combined S2 spectrum created in this context was alsothe basis for the work of Martins et al. (2008).
S87, S92, S93, S95, S96, S97.
He−I 2.1126 He−I 2.1137 He−I 2.1500 Br−g 2.16612He−I 2.1846He−I 2.1613 / 2.1615
Fig. 11.—
The combined S2 spectrum from the 2004 - 2006SINFONI data, used as velocity template.
Before the measured velocities can be used in a fit theyhave to get referred to a common reference frame. Themost suitable choice is the LSR. We used standard toolsto determine the corrections which for our data only de-pend on the observing date and the source location. Theobservatory’s position on Earth does not matter at thelevel of 15 km/s accuracy, since it leads to a correction < . Radial velocity errors
All radial velocities crucially depend on an exact wave-length calibration. The errors in the radial velocitieswere estimated from the following terms: • The formal fit error . For radial velocities whichwere obtained from a cross correlation with a tem-plate spectrum, the formal fit error is given by thefit error of the peak in the cross correlation, whichis calculated routinely with the cross correlationroutine. For the data for which we fitted a simpleline profile to the spectrum the formal fit error isalso an output of the fit routine. The magnitudeof this error depends on the spectral type and theSNR in the spectrum. For a bright late-type star,e.g. S35 with m K ≈ .
3, the formal fit error canbe as small as 10 km/s, for a bright early-type star,e.g. S2 with m K ≈ .
0, a typical value is 30 km/s. • Accuracy of wavelength calibration for Br- γ .We used the non-sky-subtracted data cubes in or-der to determine the positions of atmospheric OH-lines. Comparing those to the nominal positionsallowed us to estimate the accuracy of the wave-length calibration in the range of Br- γ and the He-lines around 2 . µ m. The rms of the OH-line posi-tions around their nominal positions yielded errorsin the order of 2 − • Accuracy of wavelength calibration for COband heads . Since there are no OH emission linesat wavelengths longer than 2 . µ m, we used atmo-spheric absorption features in the non-atmosphere-divided spectra of the respective standard stars inorder to asses the accuracy of the wavelength cali-bration at the wavelengths of the CO band heads.This was possible since our standard stars wereearly-type stars (spectral type around B5) that donot show spectral features at the region of interest.We divided the region from 2 . µ m to 2 . µ m into6 Gillessen et al.short windows of ∆ λ = 0 . µ m and cross corre-lated each with a respective theoretical spectrumof the atmosphere. The typical resulting deviationwas measured to be 10 km/s. The accuracy of theprocedure is limited however by the accuracy bywhich the individual deviations can be measured,which yielded a value of 10 km/s, too. So probablythe calibration is even more accurate than 10 km/sand consistent with what is found for the accuracyof the calibration for the shorter wavelengths. • Uncertainty of the underlying spectrum . TheGC region is highly confused. Therefore we did notuse an automated procedure to extract the spec-tra from the data cubes but selected the respectivesignal and off pixels manually. Since there is noclear prescription for what the optimum way forthat procedure is, we extracted each spectrum sev-eral times. This allowed us to estimate the errordue to the selection of signal and off pixels. Whilefor bright stars ( m K ≈
14) this error term is below10 km/s, it becomes dominant for fainter stars. Foran early-type star of m K ≈ . ORBITAL FITTING
The aim of the orbital fitting is to infer the orbits ofthe individual stars as well as information on the grav-itational potential. A Keplerian orbit can be describedby the six parameters semi major axis a , eccentricity e ,inclination i , angle of the line of nodes Ω, angle fromascending node to pericenter ω and the time of the peri-center passage t P . If the orbit is only approximatelyKeplerian, these parameters should be interpreted as theosculating orbital parameters. The parameters describ-ing a simple point mass potential are the distance to theGC, R , the mass of the central object, M MBH , its po-sition and velocity. Note that the potential might alsobe more complicated, for example due to an extendedmass component or due to the corrections arising fromthe Schwarzschild metric. These parameters can be in-ferred from our data by orbital fitting.After 16 years of high-precision astrometry of the in-nermost stars in our galaxy and a few years of Doppler-based radial velocity measurements the accuracy of theavailable data has reached a level at which one mighthope to detect deviations from the Keplerian orbits onwhich the stars apparently move due to the existence ofthe MBH at the dynamical center of the Milky Way. Suchdeviations may be due to relativistic effects or are the ef-fect of an extended mass component possibly residingin the vicinity of the MBH. Both cases are scientificallyhighly interesting. In order to analyze these effects weimplemented a general orbital fitting routine that per-mits the fitting of orbits in an arbitrary potential andthat can take into account also relativistic effects.For a 1 /r potential it is well-known that the solutionsof the equations of motion of test particles are (Kepler)ellipses. Assuming such a potential, orbits can be fit-ted by adjusting the orbital elements, since there is a straightforward prescription for the calculation of the po-sition and velocity vectors at any given time from theorbital elements. However, a more general approach isneeded if an arbitrary potential determines the dynam-ics. Then the trajectory has to be determined numeri-cally. The problem can be described by the initial condi-tions of each test particle plus the parameters describingthe potential. For each set of parameters a χ with re-spect to the measured data can be calculated. One seeksthe parameter values which minimize the χ . This isa computationally demanding problem as at each stepof the high-dimensional minimization the equations ofmotion are solved numerically. We chose the high-leveltool Mathematica (Wolfram Research 2005) for the im-plementation and tested it thoroughly, e.g. by comparingresults with results obtained from the former routine thatexplicitly uses ellipse-shaped orbits and that was usedfor the work of Eisenhauer et al. (2005). Some featuresof the new routine are: • The NIR flares of SgrA* are believed to appearat the position of the center of mass for the orbits(Genzel et al. 2003a). When a flare occurs it there-fore is reasonable to take the measured position ofthe flare into account and to identify it with thecenter of mass. This can be achieved by lettingthis measurement contribute to the χ of the fit.In total we measured 22 times a position of Sgr A*(at various brightness levels, typically at m K ≈ • We implemented four relativistic effects:a) the geometric retardation due to the finite speedof light, also called the Roemer effect. This in-volves numerically solving the retardation equation t obs = t em − z ( t em ) /c , where z is the coordinatealong the line of sight, in order to know the positionand velocity of the star at the time of emission.b) the relativistic Doppler formula, giving rise tothe so-called transverse Doppler effect, affectingonly the radial velocities.c) the gravitational redshift due to the potentialof the central point mass, altering the conversionof line positions to radial velocities. Zucker et al.(2006) show that effects a) - c) might become vis-ible in the radial velocity measurements during aclose periastron passage of a star.d) the first general relativistic correction to theNewtonian potential as given by the Schwarzschildmetric: V ( r ) = − GM MBH /r + GM MBH l /c r where l is the orbital angular momentum of thestar.Within the fitting routine all four effects can beturned on or off, or the strength of the effect canbe used as a fit parameter where 0 means the ef-fect is not present and 1 corresponds to the case inwhich the effect is as strong as expected from thetheory. • We allow for additional mass components in the po-tential, described by an arbitrary number of addi-tional parameters, all of which can be either treatedtellar orbits in the Galactic Center 17as fixed or as free fit parameters. The additionalmass components can be given either as a term inthe potential or as a function describing the densityas function of the spatial coordinates. In the lattercase the routine determines the potential from themass distribution by solving the Poisson equation ∇ V ( r ) = 4 πGρ ( r ). Here, one encounters eithera case in which a closed solution for V ( r ) can befound or it might happen that for each set of pa-rameters for which χ is calculated during the fitthe Poisson equation has to be solved numerically. • For some of the parameters of the problem therecould exist independent measurements which onemight want to take into account during the fit. Anexample is the position of the central mass. Weused radio measurements of Sgr A* to determinethe coordinate system and thus we expect the cen-tral point mass to reside in the origin of the chosencoordinate system. We therefore implemented theuse of priors for any of the parameters, which canbe done straightforwardly by including them intothe calculation of χ . • Instead of fitting the semi-major axis, we fit theperiastron distance p . This has the advantage thatwe can allow values of e <
0, effectively exchangingthe role of major and minor semi axis. By using p the parameter space is compact and the fittingroutine can smoothly pass e = 0.We followed the usual approach when calculating the sta-tistical fit errors (Press 1992). For the given best fitsolution at a certain set of values { p i } for the parame-ters we determine the Hessian matrix from the curvatureof the χ -surface: ∂ χ /∂p i ∂p j . The formal fit errorsare the diagonal elements of the inverse of that matrix.Note that still these are only formal, statistical fit errors.Possible systematic errors come in addition to them. Pa-rameter correlations are taken into account by the matrixinversion. All orbital elements for a given star are corre-lated with each other and with the potential parameters.However, the other matrix elements describing correla-tions between orbital elements of different stars can beset to 0. This reflects the test particle approach in whichone star can only influence the fit result for another starvia its influence on the potential. We explicitly use thetest particle approach also when calculating χ for morethan one star. It allows one to use several CPUs in paral-lel since the contributions to χ from the individual starsare independent. RESULTS
In order to predict the motion of a star in a given gravi-tational potential one has to know six phase space coordi-nates, e.g. its position and velocity at a given time. Sincethe radial position is not measurable for any of the S-starsand only for a few the radial velocity is measured, oneneeds additional dynamical quantities. As such one canuse accelerations, either in the proper motion or in theradial velocity. Also higher order derivatives (e.g. da/dt )of the astrometric data can be used as additional dynam-ical measurables. If more than six dynamical quantitiesare measures, the star can be used to retrieve informationabout the potential. This section is organized as follows: First, we checkby polynomial fits (going up to third order), for whichstars we can expect to find orbital solutions and whichstars can contribute in the determination of the potential.Then we determine the potential, yielding also the orbitsof the stars used in this step. Finally, we determine theorbits of the remaining stars in the given potential.
Polynomial fits
For stars for which a significant part of the orbit is sam-pled, the astrometric data cannot be described by poly-nomial fits anymore. Most prominently, in our data setthis is S2 of which our astrometric measurements covermore than one complete revolution. For all other starswe report the polynomial fits to the astrometric data inthe table in appendix C. We also give there polynomialfits to the radial velocity data of those stars for whichwe were able to determine orbits. The order of the poly-nomials in all cases was chosen such that the highestorder term still differed significantly (at the 5- σ level)from 0. Significances were calculated after rescaling theerrors such that the reduced χ of the respective fit was1, which is a conservative approach.Astrometrically, we found significant da/dt (requiringat least a 5- σ level) values for the stars S1, S4, S12, S13,S14, S17 and S31. Significant astrometric accelerations(at the 5- σ level or above) were found in addition forS5, S6, S8, S9, S18, S19, S21, S23, S24, S27, S28, S29,S33, S38, S39, S40, S48, S58, S66, S67, S71, S83, S87and S111, where we checked that the acceleration vectoractually points towards Sgr A*.We measured changes in the radial velocity for S1, S2,S4, S8, S13, S17, S19 and S24 (all > σ , except S24 with4 . σ ). Additionally, we were able to determine radialvelocities for S5, S6, S7, S9, S10, S11, S12, S14, S18,S20, S21, S22, S25, S26, S27, S29, S30, S31, S32, S33,S34, S35, S37, S38, S45, S52, S54, S65, S66, S67, S70,S71, S72, S73, S76, S83, S84, S85, S86, S87, S88, S89,S92, S93, S95, S96, S97 and S111.Summarizing, we expect • that the S2 data will dominate the problem of de-termining the gravitational potential; • that S1, S4, S8, S12, S13, S14, S17, S19, S24 andS31 can be used additionally to constrain the po-tential further; • that we can find orbits in addition for S5, S6, S9,S18, S21, S27, S29, S33, S38, S66, S67, S71, S83,S87 and S111.The data for the stars for which we found orbital solu-tions is presented in figures 12 and 13, see also table C. Mass of and distance to Sgr A*
Here and in the following we report always the fit re-sults including the (downweighted) 2002 data of S2 andexcluding it. The coordinate system priors were used asgiven in equation 4. The fit errors reported are rescaledsuch that the reduced χ = 1. Note that these errorsinclude the formal fit errors, taking into account param-eter correlations between the parameters reported here8 Gillessen et al. S1S4 S8S9 S12S17S14 S13 S21S33 S19S31 S24S27 S29S96 S67S5 S6S83S87 S97S66S111 S71S18S38 - - - - - t H yr L D ec H " L S1 S4S8 S9S12S17S14 S13 S38 S19S33 S21S29 S24S27 S31 S96S67 S5 S6S83S87S97S66 S111S71S18 - - - - - t H yr L R . A . H " L S1S4S8S17S13S19S243. 4. 5. 6. 7. 8. - - - - - t H yr L v r a d H m (cid:144) s L Fig. 12.—
The orbital data for the S-stars other than S2, the data of which is given in figure 13. Left: The measured declinations asfunction of time for the stars for which we were able to determine orbits together with the orbital solution. Middle: The same plot for rightascenscion. Right: The measured radial velocity for those stars for which we were able to measure changes in the radial velocity togetherwith the orbital solutions. The radial velocities for the other stars are given in table C. and the respective orbital elements determined simulta-neously. The systematic uncertainty due to the coordi-nate system is included here as well, since these param-eters were varied during the fits, too. The importance ofthis was pointed out also by Nikiforov (2008). R and mass from S2 data only First, we used the S2 data only to determine a Kep-lerian gravitational potential (see figure 13). Using thepriors as obtained in equation 4, the fits yield the num-bers in the first and second row of table 4. The twovalues for R differ by more than what the errors sug-gest; indicating that the 2002 data influences R . Thisconfirms the presumption from section 3.5. We exploitedthis further in figure 14. Assigning the 2002 data higherweights (smaller errors) pushes the distance estimate up,smaller weights lower it.Mass and distance are strongly correlated parameters,see Figure 15. The scaling of mass with R in our dataset is a power law with M MBH ∼ R . For a purelyastrometric data set one would have an exponent of 3and a complete degeneracy; the fact that the exponentis < R effectivelychanges the conversion from measured angles (in mas)to physical lengths (in pc), i.e. changing R changesthe semi major axis. Since the orbital period is welldetermined in our data, the mass has to change in orderto fulfill Kepler’s third law.The strong dependency means that the uncertainties for mass and distance are coupled. Fixing the dis-tance yields a very small fractional error on the massof ∆ M MBH ≈ . M MBH . This shows that the error ofthe fitted mass is completely dominated by the uncer-tainty in the distance. Once the distance is known, themass immediately follows from the scaling relation M MBH = (3 . ± . | stat ± . | R ) × M ⊙ (cid:18) R (cid:19) . (incl . ,M MBH = (4 . ± . | stat ± . | R ) × M ⊙ (cid:18) R (cid:19) . (excl . , (8)where the error due to R corresponds to the fit errorreported in table 4. Position of the central point mass
By construction the position of the radio source Sgr A*in our coordinate system is located at the origin. Since itis clear that Sgr A* is the MBH candidate we used thisfact when applying the priors of equation 4. However,our data actually allows us to test this hypothesis. Byleaving the position and proper motion of the mass com-pletely free, we can check how well the position of themass coincides with Sgr A*. Using the S2 data only, no2D priors but the prior in v z from equation 4 we obtainedthe numbers presented in the third and fourth row of ta-ble 4. We note that the mass is located within ≈ ≈ - - - H " L D ec H " L - - - - - - - v r ad H k m (cid:144) s L Fig. 13.—
Top: The S2 orbital data plotted in the combinedcoordinate system and fitted with a Keplerian model in which thevelocity of the central point mass and its position were free fit pa-rameters. The non-zero velocity of the central point mass is thereason why the orbit figure does not close exactly in the overlapregion 1992/2008 close to apocenter. The fitted position of thecentral point mass is indicated by the elongated dot inside the or-bit near the origin; its shape is determined from the uncertaintyin the position and the fitted velocity, which leads to the elonga-tion. Bottom: The measured radial velocities of S2 and the radialvelocity as calculated from the orbit fit.
We also report the S2-only fits when not using anycoordinate system priors at all (rows 5 and 6 in table 4).This enlarges the errors on R and M MBH substantially,the fit values however are not significantly different fromthe respective fits in which the v z -prior was applied. Notapplying the v z -prior also shows a large uncertainty on v z of ≈
50 km/s; this parameter also is degenerate with R . R H k p c L R from S2 data Fig. 14.—
Fitted value of R for various scaling factors of theS2 2002 data, using a fit with the coordinate system priors. Thefactor by which the 2002 astrometric errors of the S2 data is scaledup strongly influences the distance. The mean factor determinedin Figure 9 is ≈
7, corresponding to R ≈ . ´ M Ÿ H R (cid:144) L H incl. 2002 L ´ M Ÿ H R (cid:144) L H excl. 2002 L R H kpc L M M B H H M Ÿ L Fig. 15.—
Contour plot of χ as function of R and central pointmass. The two parameters are strongly correlated. The contoursare generated from the S2 data including the 2002 data; fittingat each point all other parameters both of the potential and theorbital elements. The black dots indicate the position and errors ofthe best fit values of the mass for the respective distance; the blueline is a power law fit to these points; the corresponding function isgiven in the upper row of the text box. The central point is chosenat the best fitting distance. The red points and the red dashedline are the respective data and fit for the S2 data excluding the2002 data; the fit is reported in the lower row of the text box.The contour levels are drawn at confidence levels corresponding to1 σ, σ, σ, σ, σ . From the numbers it seems that the fit excluding the2002 data agrees better with the expectations for the co-ordinate system (equation 4) than the fit including it.The latter is marginally consistent with the priors, whilethe former is fully consistent. This means that the 2002data not only affects R (which we want to measure andthus cannot use to judge the result) but also the positionand velocity of the mass for which we have an indepen-dent measurement via the coordinate system definition.This argument points towards rejecting the 2002 data.0 Gillessen et al. Position of the IR counterpart of Sgr A*
At 22 epochs we have identified a source in the NACOdata between 2003 and 2008 that might be associatedwith Sgr A*. In some cases, e.g. when a bright flare oc-curred, the identification seems unproblematic. In othercases, one cannot be sure that the emission is not dueto an unrecognized star at or very close to the positionof Sgr A*; an example is Figure 1. Due to this proba-bly very frequent confusion we expect that the measuredpositions are very noisy and we decided not to includethem into the orbital fits. However, we checked whetherthe measured positions are compatible with the orbitalfits. Fitting a linear motion model to the Sgr A* datawe obtained α [mas] = (1 . ± .
8) + (0 . ± . × ( t [yr] − . δ [mas] = (2 . ± . − (0 . ± . × ( t [yr] − . χ of 1. Thevelocity errors are approximately a factor 5 larger thanthe priors from equation 4, justifying our choice not toincorporate this data into the orbital fits. Given the un-certainties, the position of the IR counterpart of Sgr A* isconsistent with the position of the central point mass. In-terestingly, that data seems to prefer a position of Sgr A*marginally North of the expected position, which is alsothe case for the orbit fits which include the 2002 data ofS2. This weakens again the conclusion from section 6.2.2that the 2002 data should be rejected. R and mass from a combined orbit fit Given the large uncertainties due to the 2002 data ofS2, we decided to obtain more information about thepotential by using a combined orbit fit and the coordi-nate system priors. For comparison, we also excludedS2 completely. We used the stars S1, S2, S8, S12, S13,S14. We selected these stars from the sample that cancontribute to the potential (section 6.1) since for them alarge fraction of the respective orbit is covered. We didnot select S4 and S17 as they suffered confusion in theSHARP data. S19 was omitted because its time base isquite short still (the star was not detected before 2003).Since S24 would only contribute marginally to the po-tential, it was left out, too. Finally, we did not selectS31, since the nearby sources S59 and S60 were confusedwith S31 in the earlier NACO data. Not surprising, thefinal sample contains the same stars as Eisenhauer et al.(2005) had reported orbits for.In order to balance the relative weights of the starsused, we had fitted the five additional stars first alone,leaving also the potential free (but applying the priors).While the such obtained fits were not of interest per se,they still provided a smooth, unbiased model for eachstar. Hence, we used the resulting reduced χ valuesto rescale the astrometric and radial velocity errors suchthat all stars yielded a value of 1. The scaling factorsapplied ranged from 1.20 to 2.33, the latter value beingextreme and occurring for S13, which perhaps sufferedfrom confusion in the SHARP data and of which thedata in 2006/2007 was affected by confusion with S2.Our procedure guaranteed that such a star with a highastrometric noise would not contribute overly much tothe combined χ . We obtained the results given in rows7, 8 and 9 of table 4: These numbers agree with each other within the uncertainties. The combined fit includ-ing the S2 2002 data also agrees with the correspondingS2-only fit. This is not true for the combined fit exclud-ing the S2 2002 data, which is hardly compatible withthe respective S2-only fit. A possible reason is that theS2 data before 2002 is only relying on the SHARP mea-surements, which not only have larger formal errors butalso is more affected by unrecognized confusion eventsthan the NACO data.By fitting the combined data at various, fixed valuesof R we obtain again the scaling of mass and distance: M MBH = (3 . ± . | stat ± . | R ) × M ⊙ (cid:18) R (cid:19) . (incl . ,M MBH = (4 . ± . | stat ± . | R ) × M ⊙ (cid:18) R (cid:19) . (excl . ,M MBH = (3 . ± . | stat ± . | R ) × M ⊙ (cid:18) R (cid:19) . (excl . S2) , (10) Other systematic errors for R Beyond what was considered before, the physicalmodel for the potential is another source of uncertainty.For example using a relativistic model instead of a Ke-plerian orbit model increased the distance by ∆ R =0 .
18 kpc (0 .
09 kpc) when including (excluding) the 2002data. This is consistent with the formal error on R .Since we do not detect explicitly relativistic effects, westay with Keplerian orbits and consider the shift of thevalue as an uncertainty for R . Fitting a Plummermodel (as in Section 6.3) instead of a point mass poten-tial increases the distance by a similar value: 0 .
14 kpc(0 .
03 kpc) when including (excluding) the 2002 data.The additional degree of freedom in this fit increased theformal uncertainty by 0 .
11 kpc added in squares. Finally,we adopted for the uncertainties of the potential an errorof ∆ R = 0 .
25 kpc.An additional, systematic error is whether the use ofpriors (equation 4) is correct. In order to address this, werepeated the combined orbit fits without the 2D priors.We obtained the numbers in rows 10 and 11 of table 4.The influence of the priors on the value of R is relativelysmall (compare rows 7 and 8 with 10 and 11 in table 4).We adopt for this source of uncertainty ∆ R = 0 .
10 kpc.Furthermore, rows 7, 8 and 9 of table 4 show that theuncertainty of the weights of the 2002 data from S2 in acombined fit alters R by ∆ R = 0 .
13 kpc. DeselectingS2 from the fits changes the result by ∆ R = 0 .
07 kpc.Finally, we assign ∆ R = 0 .
15 kpc for the uncertaintiesrelated to the selection of data.Adding up the uncertainties yields that the uncertaintyof the distance to GC is still rather large with ∆ total R =0 .
35 kpc. Table 5 summarizes the error terms for R . Final estimate for R and mass We finally adopt the potential from the combined fitincluding the S2 2002 data, the difference to the one ex-cluding that data is negligible given the formal fit errors(section 6.2.4). This potential will be used in section 6.4tellar orbits in the Galactic Center 21
TABLE 4Results for the central potential from orbital fitting, from either S2 data only (rows 1 - 6) or a combined fit using inaddition S1, S8, S12, S13, S14 (rows 7 - 12). In rows 9 and 12, the combined fit was done without S2. The third columnindicates whether the 2002 data from S2 was used or not; the fourth column informs about which of the priors fromequation 4 have been used.
Fit S2 priors R M MBH α δ v α v δ v z M ⊙ ) (mas) (mas) ( µ as/yr) ( µ as/yr) (km/s)1 S2 only yes 2D, v z . ± .
33 4 . ± .
35 0 . ± .
64 2 . ± . − ±
87 119 ±
78 0 . ± .
22 S2 only no 2D, v z . ± .
43 3 . ± .
35 0 . ± . − . ± . − ±
91 103 ± − . ± .
23 S2 only yes v z . ± .
38 4 . ± .
41 0 . ± .
73 2 . ± .
94 76 ±
131 231 ±
107 0 . ± .
14 S2 only no v z . ± .
45 3 . ± .
36 0 . ± . − . ± . − ±
137 154 ± − . ± .
35 S2 only yes none 8 . ± .
53 4 . ± .
75 0 . ± .
71 2 . ± .
89 74 ±
127 220 ±
107 29 ±
366 S2 only no none 6 . ± .
91 2 . ± .
74 0 . ± .
75 -2.00 ± . − ±
148 162 ± − ±
447 comb. yes 2D, v z . ± .
17 4 . ± .
22 0 . ± .
63 2 . ± . − ±
71 100 ±
68 0 . ± .
08 comb. no 2D, v z . ± .
18 4 . ± .
22 1 . ± .
58 1 . ± . − ±
73 86 ±
71 0 . ± .
19 w/o S2 - 2D, v z . ± .
29 4 . ± .
49 1 . ± .
99 2 . ± . − ± − ± − . ± .
110 comb. yes v z . ± .
16 4 . ± .
21 0 . ± .
65 2 . ± .
61 51 ±
106 211 ± − . ± .
111 comb. no v z . ± .
20 4 . ± .
26 1 . ± .
81 1 . ± .
83 0 ±
133 164 ± − . ± .
312 w/o S2 - v z . ± .
31 4 . ± .
55 6 . ± . . ± . − ± − ± − . ± . TABLE 5Systematic errors for the distance to the GC, R . Error source ∆ R (kpc)Fit error including position and velocityuncertainty of coordinate system 0.17Assumed potential 0.25Using priors or not 0.10Selection of data 0.15Total 0.35 to determine the orbits of the other stars for which weexpect to find an orbital solution. Hence, we find R = 8 . ± . | stat ± . | sys kpc . (11)It should be noted that this value is consistent withinthe errors with values published earlier (Eisenhauer et al.2003, 2005). The improvement of our current work is themore rigorous treatment of the systematic errors. Alsoit is worth noting that adding more stars did not changethe distance much over the equivalent S2-only fit. Forthe mass we adopt M MBH = (3 . ± . | stat ± . | R , stat ± . | R , sys ) × M ⊙ (cid:18) R (cid:19) . = (4 . ± . × M ⊙ for R = 8 .
33 kpc . (12) Testing for an extended mass component
While Newtonian physics seems to describe the S-starsystem reasonably well, one actually expects to detectdeviations from purely Keplerian orbits with accurateenough astrometric and spectroscopic data. There aretwo main reasons for this: • The relativistic effects as described in Sec-tion 5 lead to deviations (Rubilar & Eckart 2001;Weinberg et al. 2005; Gillessen et al. 2006). Notethat for S2 the pericenter advances by 0 . ◦ perorbital revolution, not far from the precision of theorbit orientation in Table 7. • In addition to the MBH a substantial amount ofmass might reside in form of a cluster of dark stellar remnants around the MBH (Morris 1993;Miralda-Escud´e & Gould 2000; Muno et al. 2005;Mouawad et al. 2005; Hopman & Alexander 2006).This will also lead to a non-Keplerian orbit, withthe pericenter precessing in retrograde fashion.Given our current data base S2 is the only star for whichone can hope to find a deviation from a Keplerian orbit.Fitting a relativistic orbit to the S2 data yields a similar χ (158.5 compared to 158.7 for the Keplerian fit, bothwith 114 degrees of freedom). Allowing for an extendedmass component in addition does not change χ much,typically we found χ ≈ . ρ ( r ) described by ρ ( r ) = ρ . (13)More realistic is a power law model ρ ( r ) = ρ (cid:18) rr (cid:19) α . (14)The power law model is motivated by the findings ofGenzel et al. (2003b) who show that the stellar num-ber counts display such a density profile, which is alsoexpected on theoretical grounds (Bahcall & Wolf 1977;Young 1980). The parameters ρ and α are a character-istic density at the given radius and the power law index.We assumed for the following α = − . α = − .
75 and α = − . ρ ( r ) = 3 µM MBH πr (cid:18) r r (cid:19) − / . (15)The free parameters of the Plummer model are thecore radius r core and the mass parameter µ , which cor-responds to the ratio of total extended mass versusmass of the central point mass. This model allows aconvenient analytical description of the null hypothe-sis - no stellar cusp - and roughly describes the surfacelight density distribution around Sgr A* (Scoville et al.2003; Sch¨odel et al. 2007). We adopt a core radius of2 Gillessen et al. TABLE 6Results from S2 fits including an extended masscomponent. The parameter η describes the ratio ofextended mass to the central point mass. The extendedmass is accounted for in a spherical shell from thepericenter distance of S2 to the apocenter distance. Thetable shows the results for various potentials Fit R (kpc) η incl. 2002 data of S2 ρ = const . ± .
25 0 . ± . α = − . . ± .
26 0 . ± . α = − .
75 8 . ± .
26 0 . ± . α = − . . ± .
27 0 . ± . . ± .
26 0 . ± . ρ = const . ± .
33 0 . ± . α = − . . ± .
34 0 . ± . α = − .
75 8 . ± .
34 0 . ± . α = − . . ± .
35 0 . ± . . ± .
33 0 . ± . r core = 15 mpc, which matches the observed light pro-file (Mouawad et al. 2005).We fitted the S2 data for all three mass models andincluded in all cases the relativistic effects. The coordi-nate system priors were applied (equation 4) and an ad-ditional prior was set on the R = 8 . ± .
29 kpc fromthe combined fit that excluded S2 completely (row 9, ta-ble 4). Any such fit can only test for mass inside the S2orbit; therefore we express the results in terms of massenclosed between S2’s apocenter ( r = 0 . ′′ = 8 . r = 0 . ′′ = 0 .
58 mpc) relative to themass of the MBH and call this parameter η : η M MBH = 4 π Z apoperi dr r Z dm n ( r, m ) (16)The results are shown in Table 6 from which we obtain η S2 = 0 . ± . | stat ± . | mod (incl . η S2 = 0 . ± . | stat ± . | mod (excl . . (17)The statistical fit error includes the uncertainties due tothe coordinate system definition. The result correspondsto a 1- σ upper limit of η ≤ .
040 (0 . η ≤ .
066 (0 . η (Feldman & Cousins 1998). The (small) uncertainty in η due to the model uncertainty has been excluded for thecalculation of the upper limit since Table 6 shows that itaffects rather the amplitude of η than its significance.So the basic result of this study, improving measure-ment uncertainties by a factor of six over Sch¨odel et al.(2002); Ghez et al. (2005); Eisenhauer et al. (2005), isthat a single point mass potential is (still) the best de-scription of the data. Any deviations are smaller than afew percent of the point mass, within the orbits of thecentral S-star cluster. Stars with orbits
Assuming the potential from section 6.2.6 we were ableto determine orbits for the stars listed in section 6.1.During these fits, each star was considered separatelyand the potential was fixed. This yielded a total of 26 measured orbits as expected from section 6.1. An illus-tration of the (inner) stellar orbits is shown in Figure 16,the orbital elements for all 26 stars for which we foundorbits are summarized in Table 7. For the calculationof the errors quoted, all measurement errors (astrometryand radial velocities) were rescaled such that the reduced χ = 1. Furthermore, the uncertainties of the potentialwere included.As a double-check, we ran Markov-Chain Monte Carlo(MCMC) simulations (Tegmark et al. 2004) in order toasses the probability density distribution of the orbitalelements in the six dimensional parameter space. Sucha chain efficiently samples high-dimensional parameterspaces. The algorithm is simple:1. Choose a reasonable starting point in the parame-ter space and calculate χ for that point.2. Draw a random jump in the parameter space withthe typical jump distance simultaneously for eachparameter being the respective 1- σ uncertainty di-vided by the square root of the number of parame-ters (hence the mean jump distance corresponds toa 1- σ jump). The uncertainties are obtained fromthe Hessian matrix at the given point in parameterspace.3. Calculate χ n for the new point.4. If χ n < χ accept the new point, else accept thenew point with a probability of exp( − ( χ n − χ ) / S2S1S4S8 S9 S12 S13S14 S17 S21S24 S31S33S27 S29S5S6 S19S18S380.4 0.2 0. - - - - H " L D ec H " L Fig. 16.—
The stellar orbits of the stars in the central arcsecond for which we were able to determine orbits. In this illustrative figure,the coordinate system was chosen such that Sgr A* is at rest.
Among the stars with orbital solution, six stars arelate type (S17, S21, S24, S27, S38 and S111). It is worthnoting that for the first time we determine here the or-bits of late-type stars in close orbits around Sgr A*. Inparticular S17, S21 and S38 have small semi major axesof a ≈ . ′′ . The late-type star S111 is marginally un-bound to the MBH, a result of its large radial velocity( −
740 km/s) at r = 1 . ′′ which brings its total velocityup to a value ≈ σ above the local escape velocity.Furthermore we determined (preliminary) orbits forS96 (IRS16C) and S97 (IRS16SW), showing marginal ac-celerations (2 . σ and 3 . σ respectively). These stars areof special interest, since they were proposed to mem-ber of a clockwise rotating disk of stars (Paumard et al.2006). Similarly, we could not detect an accelerationfor S95 (IRS16 NW). This excludes the star from beinga member of the counter-clockwise disk (Paumard et al.2006), since in that case it should show an accelerationof ≈ µ as/yr , while we can place a safe upper limit of a < µ as/yr . H ° L W H ° L S1 0.44 0.46 0.48 0.50112114116118120 e Ω H ° L S1 15202530354045500.360.380.400.420.440.460.48 t Peri H yr - L a H '' L S20
Fig. 17.—
Examples from the Markov-Chain Monte Carlo sim-ulations. Each panel shows a 2D cut through the six dimensionalphase space of the orbital elements for the respective star. Left:Example of two well constrained and nearly uncorrelated param-eters. Middle: Example for two correlated parameters, which arenonetheless well constrained. Right: Example of badly constrainedparameters, showing a non-compact configuration in parameterspace. DISCUSSION
TABLE 7Orbital parameters of those S-stars, for which we were able to determine orbits. The parameters were determined in thepotential as obtained in section 6.2.6, the errors quoted in this table are the formal fit errors after rescaling them suchthat the reduced χ = 1 and including the uncertainties from the potential. The last three columns give the spectraltype (’e’ for early-type stars, ’l’ for late-type stars), the K-band magnitude and the global rescaling factor for thatstar. S111 formally has a negative semi major axis, indicative for a hyperbolic orbit with e > . We also cite the orbitalsolutions for the stars S96 and S97 which showed only marginal accelerations, see section 7.3.1. Star a [”] e i [ ◦ ] Ω [ ◦ ] ω [ ◦ ] t P [yr-2000] T [yr] Sp m K rS1 0 . ± .
028 0 . ± .
028 120 . ± .
46 341 . ± .
51 115 . ± . . ± .
27 132 ±
11 e 14.7 1.49S2 0 . ± .
001 0 . ± .
003 135 . ± .
47 225 . ± .
84 63 . ± .
84 2 . ± .
01 15 . ± .
11 e 14.0 1.22S4 0 . ± .
019 0 . ± .
022 77 . ± .
32 258 . ± .
30 316 . ± . − . ± . . ± . . ± .
042 0 . ± .
017 143 . ± . ±
10 236 . ± . − . ± . . ± . . ± .
153 0 . ± .
026 86 . ± .
59 83 . ± .
69 129 . ± . ±
21 105 ±
34 e 15.4 1.45S8 0 . ± .
004 0 . ± .
014 74 . ± .
73 315 . ± .
50 345 . ± . − . ± . . ± . . ± .
052 0 . ± .
020 81 . ± .
70 147 . ± .
44 225 . ± . − . ± . ± . . ± .
008 0 . ± .
003 31 . ± .
76 240 . ± . . ± . − . ± .
03 62 . ± . . ± .
012 0 . ± .
023 25 . ± . . ± . . ± . . ± .
09 59 . ± . . ± .
010 0 . ± .
006 99 . ± . . ± .
70 339 . ± . . ± .
06 47 . ± . . ± .
004 0 . ± .
015 96 . ± .
18 188 . ± .
32 31945 ± . − . ± . . ± . . ± .
080 0 . ± .
052 116 . ± . . ± . . ± . − . ± . ±
16 e 16.7 2.34S19 0 . ± .
064 0 . ± .
062 73 . ± .
61 342 . ± . . ± . . ± .
22 260 ±
31 e 16.0 2.31S21 0 . ± .
041 0 . ± .
028 54 . ± . . ± . . ± . . ± . . ± . . ± .
178 0 . ± .
010 106 . ± .
93 4 . ± . . ± . . ± . ±
73 l 15.6 1.78S27 0 . ± .
078 0 . ± .
006 92 . ± .
73 191 . ± .
92 308 . ± . . ± . ±
18 l 15.6 1.79S29 0 . ± .
335 0 . ± .
048 122 ±
11 157 . ± . . ± . ±
18 91 ±
79 e 16.7 1.92S31 0 . ± .
044 0 . ± .
007 153 . ± . ±
11 314 ±
10 13 . ± . . ± . . ± .
088 0 . ± .
039 42 . ± . . ± . . ± . − . ± . ±
21 e 16.0 2.02S38 0 . ± .
041 0 . ± .
041 166 ±
22 286 ±
68 203 ±
68 3 . ± . . ± . . ± .
126 0 . ± .
039 135 . ± . . ± . ± . − ±
23 486 ±
41 e 14.8 1.15S67 1 . ± .
102 0 . ± .
041 139 . ± . . ± . . ± . − ±
16 419 ±
19 e 12.1 1.53S71 1 . ± .
765 0 . ± .
075 76 . ± . . ± . . ± . − ±
251 399 ±
283 e 16.1 2.44S83 2 . ± .
234 0 . ± .
096 123 . ± . . ± . . ± . ±
25 1700 ±
205 e 13.6 1.23S87 1 . ± .
161 0 . ± .
036 142 . ± . . ± . . ± . − ±
38 516 ±
44 e 13.6 0.94S111 − . ± . . ± .
094 103 . ± . . ± . ± − ± − l 13.8 0.94S96 1 . ± .
209 0 . ± .
054 126 . ± . . ± .
93 231 . ± . − ±
34 701 ±
81 e 10.0 1.40S97 2 . ± .
844 0 . ± .
308 114 . ± . . ± .
15 38 ±
52 175 ±
88 1180 ±
688 e 10.3 1.15
The distance to the Galactic Center
Our estimate R = 8 . ± . | stat ± . | sys kpc(equation 11) is compatible with our earlier work(Eisenhauer et al. 2003, 2005). While the underlyingdata base is partially identical, this work mainly im-proved the understanding of the systematic uncertain-ties. In particular, the astrometric data during the peri-center passage of S2 is hard to understand. This is anunfortunate situation, since that data potentially is mostconstraining for the potential. During the passage thestar sampled a wide range of distances from the MBH,corresponding to a radially dependent measurement ofthe gravitational force acting on it. Probably only fu-ture measurements of either S2 or other stars passingclose to Sgr A* will allow one to answer the question,whether the confusion problem close to Sgr A* is genericor whether 2002 was a unlucky coincidence.Besides stellar orbits, there are other techniques to de-termine R . A classical one is to use the distributionof globular clusters. Bica et al. (2006) applied this tech-nique to a sample of 153 globular clusters and obtained R = 7 . ± .
3. This value is only marginally compatiblewith our result. However, the error quoted by Bica et al.(2006) corresponds to the formal fit error derived fromtheir figure 4. Therefore, one might suspect that system-atic problems owed to the method were not yet includedin the error estimate.The fact that the absolute magnitudes of red clumpstars is known and that the red clump can be identi- fied in the luminosity function obtained from the ap-parent magnitudes of stars in the galactic bulge wasused by Nishiuyama et al. (2006). These authors obtain R = 7 . ± . | stat ± . | sys kpc, where the statisticalerror is owed mainly to the uncertainty of the local redclump stars luminosities and the systematic error termsincludes uncertainties in the extinction and populationcorrections, the zero point of photometry, and the fittingof the luminosity function of the red clump stars. Thisresult is in agreement with our measurement, given theerrors of both results.The known absolute magnitudes from RR Lyraestars and Cepheids are the key to the work fromGroenewegen, Udalski & Bono (2008). Their result R = 7 . ± . | stat ± . | stat kpc is fully consistentwith our result. The statistical error here is due tothe photometric measurement errors, the zero point ofphotometry and the uncertainty of extinction correction.The systemtatic error includes the calibration of period-luminosity relations used and the selection effect, whichcould affect the result since only 39 Cepheids and 37 RRLyrae stars have been used for this statistical approach. Limits on the binarity of Sgr A*
It is interesting to see how our data limits the possi-ble existence of a second, intermediate mass black hole(IMBH) in the GC. Here, we do not aim at a rigoroustreatment of the problem (which would be beyond thescope of this paper) but limit ourselves to estimates thatappear reasonable given our findings.tellar orbits in the Galactic Center 25The first constraint comes from the fact that the centerof mass does not move fast. If the central mass were inorbit with an IMBH, the orbital reflex motion of Sgr A*might show up in our data. The upper limit on the ve-locity which we obtain from row 7 in table 4 correspondsto a line in a phase space plot of IMBH mass versusIMBH-MBH distance (Figure 18), separating configura-tions at smaller masses from systems with higher masses.From our data, we would not have been able to detectsuch an orbital motion of the MBH if the orbital pe-riod P were too short, namely much shorter than theorbital period of S2. We estimate that configurationswith P > v b = − . ± . P > yr for the IMBH-MBHsystem excludes configurations towards smaller distancesand higher masses. Dynamical stability can also be de-manded for the S-star cluster as such. Mikkola & Merritt(2008) have shown that an IMBH with a mass of10 − M MBH in a distance of 1 mpc would make the S-stars cluster unstable. It is reasonable to assume thatthis also holds for larger masses and radii at least aslarge as the S-star cluster extends ( ≈ ′′ ).Based on simulations, Gualandris & Merritt (2007)concluded that an IMBH will reach a stalling radiusthat is proportional to the mass of the IMBH: a stall =3 . µ as × M IMBH [ M ⊙ ] (for our values of mass and dis-tance). Since one does not expect an IMBH to reside ata much smaller radius, this puts another constraint onthe IMBH-MBH binary.Finally, also the S2 orbit allows us to exclude part ofthe phase space. Motivated by the findings of section 6.3and equation 17, we simply assume that no mass largerthan 0 . M MBH can be hidden inside the S2 orbit. Ac-tually, also somewhat smaller radii than the pericenterdistance r p of S2 are excluded, since this would still per-turb the orbit figure notably. We estimate that down to0 . r p no IMBH more massive than 0 . M MBH can reside.
Properties of the stellar orbits
We obtained orbits for 20 early-type stars. This rel-atively large number - Eisenhauer et al. (2005) had sixorbits, Ghez et al. (2005) seven - allows us to assess dis-tributions of orbital parameters and study the proper-ties of the stellar orbits thereby characterizing the S-starpopulation. S ! s t a r s Σ c . l . S ! s t a r s c . l . P $ ! star cluster r ad i o S g r A % c . l . r ad i o S g r A % Σ c . l . S2 excluded H M l i f e t i m e $ y r G M ! ! Mass 2 nd BH ! M ! " d i s t an c e M B H ! nd B H ! m a s " Fig. 18.—
Constraints on the binarity of Sgr A* as functionof the mass of the secondary black hole and the distance betweenthe black holes. The shaded areas are excluded due to various ar-guments. The diagonal lines assume an orbital motion of Sgr A*around the secondary and correspond to velocity limits obtainedeither from the S-stars or the motion of Sgr A* (Reid & Brunthaler2004). We estimate that only periods longer than 5 yr would leadto an observable effect, thus excluding an area towards highermasses and large distances. Demanding that the lifetime of thebinary black hole exceeds 10 yr yields another constraint (fromHansen & Milosavljevic (2003)). These authors also made similararguments for the motion of the Sgr A*, the resulting constraintis replicated in this plot (denoted as HM03). The stability of theS-star cluster puts a further constraint (Mikkola & Merritt 2008),as does the stalling radius found by Gualandris & Merritt (2007),denoted as GM07. Finally also the S2 orbit excludes some part ofthe diagram, since it apparently is Keplerian. Orientations of orbital planes
Figure 19 illustrates the orientations of the orbitalplanes for all stars from Table 7. Paumard et al. (2006)suggested that the six stars S66, S67, S83, S87, S96 andS97 (E17, E15 (S1-3), E16 (S0-15), E21, E20 (IRS16C)and E23 (IRS16SW) in their notation) are members ofthe clockwise disk. Our findings explicitly confirm this.All six stars have an angular distance to the disk be-tween 9 ◦ and 21 ◦ with a mean and standard deviationof 15 ◦ ± ◦ . This is somewhat (a factor of 2) more thanthe disk thickness of 14 ◦ ± ◦ found by Paumard et al.(2006). However, statistically the difference is not verysignificant and only the inner edge of the disk is sampledhere. All six disk stars have a semi major axis of a ≈ ′′ and a small eccentricity ( e ≈ . − .
4) in agreement withthe estimates from Paumard et al. (2006). The orbitalplane of S5 is also consistent with the disk given its dis-tance of 18 ◦ . However, the lower brightness ( m K = 15 . e > .
8) of the or-bit make it unlikely that S5 is a true disk member. Thenext closest star to the disk beyond the six disk starsand S5 is S31 with an angular distance of 27 ◦ . We alsonote that the orbital solutions for S96 and S97 derivedfrom marginal accelerations are consistent with the diskhypothesis. Therefore we are confident in these orbits,too.We used a Rayleigh test (Wilkie 1983) to check whetherthe distribution of orbital angular momenta for the 22other stars for which we found orbits is compatible with6 Gillessen et al. SunLine of sight - °- ° ° ° °- ° - ° - ° ° ° ° S1 S2S4S5S6 S8S9 S12S13 S14 S17S18S19 S21S24 S27S29 S31S33 S38S66S67 S71S83S87S96S97 S111
Fig. 19.—
Orientation of the orbital planes of those S-stars for which we were able to determine orbits. The orientation of the orbits inspace is described by the orbital angular momentum vector, corresponding to a position in this all sky plot, in which the vertical dimensioncorresponds to the inclination i of the orbit and the horizontal dimension to the longitude of the ascending node Ω. A star in a face-on,clockwise orbit relative to the line of sight, for instance, would be located at the top of the graph, while a star with an edge-on seenorbit would be located on the equator of the plot. The error ellipses correspond to the statistical 1 σ fit errors only, thus the area coveredby each is 39% of the probability density function. Stars with an ambiguous inclination have been plotted at their more likely position.The stars S66, S67, S83, S87, S96 and S97 which were suspected to be part of the clockwise stellar disk by Paumard et al. (2006) at(Ω = 99 ◦ , i = 127 ◦ ) actually are found very close to the position of the disk. The latter is marked by the thick black dot and the dashedlines, indicating a disk thickness of 14 ◦ ± ◦ , the value found by Paumard et al. (2006). The orbits of the other stars are oriented randomly. a random distribution. We found a probability of ran-domness of p = 0 .
74; meaning that the non-disk stars donot show a preferred orbit orientation. Using the projec-tion method from Cuesta-Albertos, Cuevas & Fraiman(2007) we obtained p = 1 .
0. The same statement alsoholds when testing for randomness of the subset of early-type stars.
Distribution of semi major axes
Figure 20 shows the cumulative probability distribu-tion function (pdf) for the semi major axes of stars whichhave semi major axis smaller than 0.5”, thus excludingthe stars that are identified to be members of the clock-wise disk. The statistic is limited still (15 stars makeup this sample), but nevertheless the distribution allowsus to estimate the functional behavior of the pdf n ( a ).Due to the small number of data points we did not binthe data but used a log-likelihood fit for n ( a ). We found n ( a ) ∼ a . ± . . This can be converted to a numberdensity profile as a function of radius (Alexander 2005).We obtain n ( r ) ∼ r − . ± . , consistent with the massprofile in Genzel et al. (2003b) who found ρ ( r ) ∼ r − . and with the newer work in Sch¨odel et al. (2007) whofound ρ ( r ) ∼ r − . for the innermost region of the cusp. Distribution of eccentricities
The distribution of eccentricities allows us to estimatethe velocity distribution. Figure 21 shows the cumu-lative pdf for the eccentricities of those young (early-type) stars which are not associated with the clockwisestellar disk. Using again a log-likelihood fit, we find n ( e ) ∼ e . ± . . The profile still is barely consistent with n ( e ) ∼ e , corresponding to an isotropic, thermal veloc-ity distribution (Sch¨odel et al. 2003; Alexander 2005). Fig. 20.—
The cumulative pdf for the semi major axis of theearly-type stars with a < . ′′ . The two curves correspond to thetwo ways to plot a cumulative pdf, with values ranging either from0 to (N-1)/N or from 1/N to 1. The distribution can be representedby n ( a ) ∼ a . ± . . This would be the expectation for a relaxed stellar sys-tem. However, given that the maximal lifespan for Bstars ( . yr) is much shorter than the local two bodyrelaxation (TBR) time ( ≈ yr, Alexander (2005)) onedoes not expect a thermal distribution. In this light, it isinteresting to notice that the distribution appears to bea bit steeper (i.e. peaked towards higher eccentricities)than a thermal distribution. This might be a first hinttowards the formation scenario for the S-stars. For exam-ple, it is exactly what one expects in the binary capturescenario (Perets, Hopman & Alexander 2007), in whichthe S-stars are initially captured on very eccentric or-bits ( e & . Fig. 21.—
The cumulative pdf for the eccentricities of the early-type stars which are not identified as disk members. The twocurves correspond to the two ways to plot a cumulative pdf, withvalues ranging either from 0 to (N-1)/N or from 1/N to 1. Thedistribution is only marginally compatible with n ( e ) ∼ e (dashedline), the best fit is n ( e ) ∼ e . ± . . Estimates of the extended mass component
In addition to the population of stars not yet re-solved by current instrumentation a cluster of dark ob-jects - e.g. stellar mass black holes (SBHs) as pro-posed in Morris (1993); Miralda-Escud´e & Gould (2000);Muno et al. (2005); Hopman & Alexander (2006) - isplausibly present in the GC. As shown in section 6.3the orbital data allows to test for such extended masscomponents. Here we investigate several theoretical andobservational constraints on the extended mass distribu-tion and relate these to η . We mostly assume that theextended mass distribution is due to SBHs with a mass of M ⋆ = m ⋆ M ⊙ with m ⋆ = 10 (Timmes et al. 1996), sincethis component is likely to make up most of the mass ofa potential dark cluster (Alexander 2007). Stellar number counts
Genzel et al. (2003b) and Sch¨odel et al. (2007) haveinferred a stellar density profile for the GC fromcompleteness-corrected stellar number counts. Assum-ing that the luminous objects trace the total mass, thenumber density profile that is determined reliably on the > .
01 pc scale can be extrapolated to the S2 orbit. Weobtain η = 3 . × − × (cid:16) m ⋆ (cid:17) , (18)This extrapolation is quite uncertain, since mass segrega-tion predicts that the SBHs should have a much steeperslope than the less-massive luminous stars in the cen-tral 0 .
01 pc, where the SBHs dominate the total mass(Hopman & Alexander 2006; Alexander 2007). There-fore both the mass-to-number ratio and the slope of the density profile are expected to have a significant radialdependence.
The drain limit
The drain limit is a conservative theoretical upperlimit of the number of compact objects that can existin steady-state around a MBH. It is given by the con-dition that the number of SBHs that can be packed in-side any given radius in steady state has to be smallerthan the number of SBHs scattered into the MBH overthe age of the Galaxy (Alexander & Livio 2004). Thiscan be translated into a theoretical limit for η . Close to m ⋆ = 10 and using M MBH = 4 × M ⊙ and t = 10 Gyrthe relation can be approximated by η . . × (cid:16) m ⋆ (cid:17) − . . (19)The drain limit could be violated for a non steady-statesituation. Indeed, the existence of the young star diskswith a relatively well-defined age of 6 My suggests thatstar formation in the GC is episodic. However, theamount of mass from SBHs would hardly exceed 10 M ⊙ even assuming an optimistic, top-heavy initial mass func-tion, given that the total amount of mass in the disks is ≈ M ⊙ . Dynamical modeling of the dark cluster in the GC
The expected degree of central concentration of SBHsaround the MBH can be estimated by modeling the dy-namical evolution of a system with a present-day massfunction similar to that of the GC (Alexander 2005).Monte Carlo simulations of the GC using the H´enonmethod and including also stellar collisions and tidal dis-ruptions (M. Freitag, priv. comm.; see also Freitag et al.(2006)) but neglecting star formation yield a ratherflat mass density profile of 10 ( M ⊙ / pc ) ( r/ .
01 pc) − . ,which translates to η ∼ − . Due to the statisticalnature of this method the density profile at the very cen-ter is not well determined. An alternative analytic solu-tion for the steady state distribution using a much moreidealized formulation of the mass segregation problem(Hopman & Alexander 2006) yields a similar result of η ∼ × − . However, in this method the fixed bound-ary conditions far from the MBH may artificially main-tain a high density in the center by preventing the ex-pansion of the system. Nevertheless, the fact that thesetwo different methods yield similar results also consistentwith the drain limit lends some credence to this estimate. Diffuse X-ray emission of a dark cluster
A cluster of compact objects will accrete the surround-ing gas and thus lead to X-ray emission, which for currentX-ray satellites ( ≈ ′′ ) would be barely resolved. Indeed,the X-ray source at the position of Sgr A* is slightly ex-tended (Baganoff et al. 2003). We fit the radial profileof Sgr A* as reported by Baganoff et al. (2003) by thesuperposition of a point source with a Gaussian widthof σ pt = 0 . ′′ (Baganoff et al. 2003) and an extendedcomponent with a free width σ ext . We obtain as em-pirical description for the surface brightness profile ofBaganoff et al. (2003), Figure 6: B ( r ) [cts / arcsec ] = 73 . e − r / σ +40 . e − r / σ (20)8 Gillessen et al.with σ ext = 1 . ′′ . Thus, we obtain for the extendedluminosity (assuming the same spectral index of pointlike and extended component) L X , ext = 1 . × erg / s,accounting for ≈
80% of the total X-ray luminosity.The expected X-ray luminosity of a single compact ob-ject is given by the mass accretion rate and the radiationefficiency. A simple estimate is given by assuming Bondiaccretion (Bondi 1952):˙ M B = 4 πλ ( GM ⋆ ) n e µ m p c − s ≈ g / s , (21)where λ = 1 / n e = 26 cm − the electron numberdensity, µ = 0 . m p the pro-ton mass, c s = p k T e / µ m p the speed of sound and T e = 1 . M P = πr ρ v ≈ g / s , (22)with the accretion radius r acc = 2 GM ⋆ /v ≈ × cm. Using the density from above and the Kep-lerian velocity at r = 1 ′′ one obtains consistently ˙ M P ≈ g/s ≈ ˙ M B .The radiation efficiency depends on the type of objectconsidered (Haller et al. 1996). For neutron stars 10% isassumed (Pessah & Melia 2003), since the accreted ma-terial will fall onto a hard surface and the energy re-leased can be radiated away, resulting in a luminosity of L NS ≈ erg/s. For SBHs due to the absence of a sur-face most of the emission will be thermal bremsstrahlungyielding only L ⋆ ≈ × erg/s (Haller et al. 1996).This shows that L X , ext cannot be due to SBHs, since onewould need 10 objects to explain the observed luminos-ity. In the case of neutron stars, one would need ≈ r . ′′ in order to account for the ob-served luminosity, corresponding to η ≈ .
07. However,this number exceeds the estimate of the segregated cuspmodel of Hopman & Alexander (2006) who predict only ≈
100 neutron stars there.
X-ray transients in a dark cluster
Muno et al. (2005) report an overabundance of X-raytransients in the inner parsec of the GC compared tothe overall distribution of X-ray sources. The sourcesare classified as X-ray binaries (XRBs). These authorssuggest a dynamical origin of the XRBs, namely an ex-change of type Binary + SBH → XRB + Star. The ratedensity for this reaction is γ + = n ⋆ n b Σ σ (23)where n b is the density of binaries,Σ = πa + 2 πaG ( M b + M ⋆ ) /σ (24)the exchange cross section and σ = ( GM MBH / r ) / the 1D velocity dispersion. According to Muno et al.(2005) the number of XRBs is limited by dynamicalfriction which yields a characteristic life time of τ =10 Gyr ( M ⋆ /M ⊙ ) − ( r/ pc) / . Refining this argument,we also take into account the back reaction XRB+Star → Binary + SBH and assume for simplicity equal exchangecross sections for forward and backward reaction. Botheffects together yield a rate density of γ − = 12 n n XRB Σ σ + n XRB τ , (25) where n is the number density of stars, n XRB the numberdensity of XRBs. In equilibrium one has γ + = γ − ,which allows one to solve for n XRB . After integrating n XRB over volume out to 1 pc and assuming a = 0 . n = 10 pc − ( r/ pc) − , n b = 0 . n , M b = 3 M ⊙ , M ⋆ =10 M ⊙ and n ⋆ = f ⋆ n one obtains the number of XRBsin the central parsec as N XRB = 7 × f ⋆ , where f ⋆ isthe relative number of SBHs to ordinary stars. A certainfraction f X of those will shine up as X-ray transients: f X N XRB = N X . Calculating η from this yields η = f ⋆ M ⋆ M MBH Z apoperi πr n ( r ) dr (26)= 3 . × − N X ( < f X , (27)relating the number of X-ray transients in the centralparsec with η . Using the values f X . .
01 and N X = 4(Muno et al. 2005) we obtain η ≈ . × − .A more detailed investigation by Deegan & Nayakshin(2007) shows that within r < . ≈ n ( r ) ∼ r − / into η yields η ≈ . × − , which is very similar to our estimate inthe previous paragraph. Further aspects
There are at least three other aspects of an extendedmass component in the GC which are worth mentioningbut beyond the scope of this paper. • Star formation in the presence of a dark cluster.
The process of star formation in the GC mightbe altered significantly by the presence of a sub-stantial dark component. The additional perturba-tive gravitational forces due to the SBHs might as-sist star formation since they increase the inhomo-geneities in a star forming gas cloud. On the otherhand, close encounters between individual clumpsand SBHs might result in disruption of the clumps. • Interaction of the spin of the MBH with the darkcluster.
The spin of the MBH is subject to evo-lution by several processes. While gas accre-tion and major mergers can increase the spin,the accretion of SBHs tends to decrease the spin(Hughes & Blandford 2003; Gammie et al. 2004),assuming many random infall events of isotropi-cally distributed SBHs. Furthermore there is thegeneral relativistic spin-orbit-coupling between aSBH and the MBH spin, leading to a change ofthe spin direction of the MBH but not to a changeof its modulus (Lodato & Pringle 2006). • Dark matter.
Dark matter, which is widelyaccepted in cosmology, might also show up indynamic measurements in the GC. However,Gendin & Primack (2004) show that the den-sity of the dark matter at 0 .
01 pc is ρ DM ≈ × M ⊙ / pc , which is negligible compared The factor 1 / tellar orbits in the Galactic Center 29to the theoretically predicted stellar densitythere (Hopman & Alexander 2006). See alsoVasiliev & Zelnikov (2008). Conclusions for an extended mass component
The various estimates for η all consistently point to-wards an expected value of ≈ − − − , approximatelytwo orders of magnitude smaller than what we can mea-sure with orbital dynamics today. Nevertheless, someastrophysical insights are possible.Among the most important scientific questions in theGC is the origin of the S-stars, being a population of ap-parently young stars close to the MBH (Ghez et al. 2003;Martins et al. 2008). One possible origin is that thesestars have reached their current orbits by TBR. Thenthe S-stars would have an isotropic, thermal velocity dis-tribution, naturally explaining the observed random dis-tribution of angular momentum vectors (Figure 19). Thenumber of stars visible is by far too low to make TBRefficient enough to account for the present populationof S-stars. A hypothetical cluster of SBHs could accel-erate the process. The Chandrasekhar TBR timescale(Binney & Tremaine 1987) is given by t r ≈ . σ G h M ⋆ i n ⋆ ln Λ . (28)For a power law cusp around a MBH, the velocity disper-sion and the density are related to each other. Assumingln Λ ≈
10, a power law index of − / t r ≈ . × yr η − (cid:16) m ⋆ (cid:17) − , (29)Thus, if the S-stars formed at the same epoch as thestellar disks 6 × years ago (Paumard et al. 2006)and reached their present-day orbits by TBR, one needs η & .
033 for m ⋆ = 10 (Timmes et al. 1996). This ex-ceeds the expectations by at least two orders of mag-nitudes. If the S-stars were not born in the presentlyobserved disks, but in older, now-dispersed disks, onecan use Equation 29 with the typical age of B stars( ≈ × yr). For m ⋆ = 10 this yields η & . × − ,which could be marginally compatible with the other es-timates for η .In order to assess the expected progress we simu-lated future observations with existing instrumentationand similar sampling. Continuing the orbital monitor-ing for two more years will lower the statistical error to∆ η ≈ .
01, corresponding to t r ≈ × yr. This meanswe will soon be able to test the hypothesis that the S-stars formed in the disks and reached their current orbitsby TBR. Furthermore there is a chance to rule out anyTBR origin of the S-stars observationally in the near fu-ture, namely when η . . × − is reached. SUMMARY
We continued our long-term study of stellar orbitsaround the MBH in the Galactic Center. This workis based on our large, high quality data base which isbased on high resolution imaging and spectroscopy fromthe years 1992 to 2008. The main results are • The best current coordinate reference system usesall available IR positions of the SiO maser stars(Reid et al. 2007) for the definition of the originand assumes that the stellar cluster around Sgr A*is intrinsically at rest such that it can be used forthe calibration of the coordinate system velocity.Having more measurements of the maser sourcesboth in the radio and the IR domain we eventuallywill be able to directly tie the coordinate systemvelocity to radio Sgr A* with a sufficient precision.Then the intermediate step of cross calibration withthe stellar cluster can be dropped and the coordi-nate system definition would be independent fromthe assumption that the stellar cluster is at restwith respect to Sgr A*. • We obtained orbits for 28 stars. Eleven of thosecan contribute to the determination of the gravi-tational potential, we used up to six. For the firsttime we were able to determine orbital parametersfor six of the late-type stars in our sample. Fur-thermore, we confirm unambiguously the earlier re-port (Paumard et al. 2006) that six of the stars aremembers of the clockwise disk. • Overall, we improved measurement uncertaintiesby a factor of six over the most recent set of Galac-tic Center papers (Sch¨odel et al. 2002; Ghez et al.2005; Eisenhauer et al. 2005). A single point masspotential continues to be the best fit to these im-proved data as well. The main contribution to theerror in the mass of Sgr A* and the distance tothe Galactic Center are systematic uncertainties.While the value of the mass is driven by the dis-tance estimate, the latter is subject to many sys-tematic uncertainties that amount to 0 .
31 kpc. Thestatistical error now decreased to 0 .
17 kpc and be-came smaller than the systematic one. The mostfruitful way to overcome current limitations wouldprobably be the observation of another close peri-center passage of an S-star. Our current best valuesare: M = (3 . ± . | stat ± . | R , stat ± . | R , sys ) × M ⊙ × ( R / . = (4 . ± . × M ⊙ for R = 8 .
33 kpc R = 8 . ± . | stat ± . | sys kpc (30)It should be noted that this value is consistentwithin the errors with values published earlier(Eisenhauer et al. 2003, 2005). The improvementof our current work is the more rigorous treatmentof the systematic errors. Also it is worth notingthat adding more stars did not change the distancemuch over the equivalent S2-only fit. • We have obtained an upper limit for the mass en-closed within the S2 orbit in units of the mass ofthe MBH: η = 0 . ± . | stat ± . | model . (31)which corresponds to a 1 σ upper limit of η ≤ . APPENDIX
ACKNOWLEDGEMENTS
We would like to thank D.T. Jaffe, U Texas, for helpful discussions and U. Bastian, U Heidelberg, for access toGAIA internal notes. T. A. is supported by Minerva grant 8563, ISF grant 968/06 and a New Faculty grant by SirH. Djangoly, CBE, of London, UK. We also thank our referee, whose report helped in improving the manuscript.
A. LIST OF NACO DATA SETS
Date Band maspix
DIT NDIT maspix
DIT NDIT tellar orbits in the Galactic Center 31
B. POLYNOMIAL FITS TO THE S-STARS DATA
The following table lists the polynomial fits to the S-stars data (except S2 which is not well described by polynomialfits). For stars with a significant (at the 5- σ level) astrometric acceleration pointing towards Sgr A* we report quadraticfits. For stars with significant da/dt we report the cubic fit. Otherwise linear fits are given. Similarly, for stars forwhich detected changes in the radial velocities, we report linear fits. For stars for which we determined orbits but didnot detect changes in v vrad we report the weighted averages. Name, m K α [mas] = t [yr] for ( α, δ ) δ [mas] = t [yr] for v z v z [km / s] =S1, 14.7 ( − . ± .
4) + (19 . ± . t + (0 . ± . t ) + ( − . ± . t ) − . ± .
5) + ( − . ± . t + (1 . ± . t ) + (0 . ± . t ) . ± .
4) + ( − . ± . t + (13 . ± . t ) S4, 14.4 (269 . ± .
2) + (15 . ± . t + ( − . ± . t ) + ( − . ± . t ) . ± .
1) + ( − . ± . t + ( − . ± . t ) + (0 . ± . t ) − . ± .
3) + ( − . ± . t S5, 15.2 (352 . ± .
3) + ( − . ± . t + ( − . ± . t ) . ± .
4) + (8 . ± . t + ( − . ± . t ) ± . ± .
1) + (6 . ± . t + ( − . ± . t ) . ± .
2) + (0 . ± . t + ( − . ± . t ) ± . ± .
2) + ( − . ± . t − . ± .
3) + ( − . ± . t S8, 14.5 (336 . ± .
2) + (15 . ± . t + ( − . ± . t ) − . ± .
2) + ( − . ± . t + (0 . ± . t ) − (53 . ± .
6) + ( − . ± . t S9, 15.1 (181 . ± .
3) + (1 . ± . t + ( − . ± . t ) − . ± .
3) + ( − . ± . t + (0 . ± . t ) ± . ± .
1) + ( − . ± . t − . ± .
1) + (2 . ± . t S11, 14.3 (142 . ± .
3) + (8 . ± . t − . ± .
2) + ( − . ± . t S12, 15.5 ( − . ± .
3) + (4 . ± . t + (1 . ± . t ) + ( − . ± . t ) . ± .
3) + (29 . ± . t + ( − . ± . t ) + (0 . ± . t ) ± − . ± .
6) + (15 . ± . t + (3 . ± . t ) + ( − . ± . t ) − . ± .
0) + (44 . ± . t + ( − . ± . t ) + ( − . ± . t ) − . ± .
8) + (186 . ± . t S14, 15.7 (64 . ± .
4) + (17 . ± . t + (2 . ± . t ) + ( − . ± . t ) . ± .
2) + (14 . ± . t + (2 . ± . t ) + ( − . ± . t ) . ± . . ± .
0) + (2 . ± . t + ( − . ± . t ) + (0 . ± . t ) − . ± .
0) + (20 . ± . t + (1 . ± . t ) + ( − . ± . t ) . ± .
2) + ( − . ± . t S18, 16.7 ( − . ± .
5) + ( − . ± . t + (0 . ± . t ) − . ± .
6) + ( − . ± . t + (0 . ± . t ) − ± . ± .
1) + ( − . ± . t + ( − . ± . t ) − . ± .
9) + ( − . ± . t + (6 . ± . t ) − . ± .
4) + (10 . ± . t + (88 . ± . t ) S20, 15.7 (220 . ± .
6) + ( − . ± . t . ± .
4) + ( − . ± . t S21, 16.9 ( − . ± .
2) + ( − . ± . t + (1 . ± . t ) − . ± .
4) + (4 . ± . t + (0 . ± . t ) ± . ± .
2) + (22 . ± . t − . ± .
3) + ( − . ± . t S23, 17.8 (307 . ± .
1) + ( − . ± . t + ( − . ± . t ) − . ± .
9) + ( − . ± . t + (0 . ± . t ) S24, 15.6 ( − . ± .
3) + (6 . ± . t + (0 . ± . t ) − . ± .
3) + (10 . ± . t + (0 . ± . t ) − . ± .
5) + ( − . ± . t )S25, 15.2 ( − . ± .
3) + ( − . ± . t − . ± .
3) + (1 . ± . t S26, 15.1 (514 . ± .
2) + (6 . ± . t . ± .
2) + (0 . ± . t S27, 15.6 (146 . ± .
3) + (0 . ± . t + ( − . ± . t ) . ± .
3) + (3 . ± . t + ( − . ± . t ) − ± Name, m K α [mas] = t [yr] for ( α, δ ) δ [mas] = t [yr] for v z v z [km / s] =S28, 17.1 ( − . ± .
6) + (4 . ± . t + (0 . ± . t ) . ± .
6) + (12 . ± . t + ( − . ± . t ) S29, 16.7 ( − . ± .
2) + (1 . ± . t + (0 . ± . t ) . ± .
3) + ( − . ± . t + ( − . ± . t ) − ± − . ± .
1) + (0 . ± . t . ± .
1) + (3 . ± . t S31, 15.7 ( − . ± .
5) + (5 . ± . t + (0 . ± . t ) + (0 . ± . t ) . ± .
5) + ( − . ± . t + ( − . ± . t ) + ( − . ± . t ) − ± − . ± .
1) + ( − . ± . t − . ± .
2) + (0 . ± . t S33, 16.0 ( − . ± .
5) + ( − . ± . t + (0 . ± . t ) − . ± .
4) + (0 . ± . t + (0 . ± . t ) − ± . ± .
3) + (9 . ± . t − . ± .
3) + (3 . ± . t S35, 13.3 (540 . ± .
1) + (1 . ± . t − . ± .
1) + (3 . ± . t S36, 16.4 (276 . ± .
2) + ( − . ± . t . ± .
3) + ( − . ± . t S37, 16.1 (331 . ± .
4) + ( − . ± . t . ± .
3) + (10 . ± . t S38, 17.0 ( − . ± .
6) + ( − . ± . t + (4 . ± . t ) . ± .
7) + ( − . ± . t + ( − . ± . t ) − ± − . ± .
7) + ( − . ± . t + (1 . ± . t ) . ± .
2) + (33 . ± . t + ( − . ± . t ) S40, 17.2 (144 . ± .
5) + (3 . ± . t + ( − . ± . t ) . ± .
9) + (1 . ± . t + ( − . ± . t ) S41, 17.5 ( − . ± .
6) + ( − . ± . t − . ± .
6) + ( − . ± . t S42, 17.5 ( − . ± .
7) + ( − . ± . t − . ± .
1) + (18 . ± . t S43, 17.5 ( − . ± .
3) + (5 . ± . t − . ± .
4) + (7 . ± . t S44, 17.5 ( − . ± .
5) + ( − . ± . t − . ± .
0) + ( − . ± . t S45, 15.7 (193 . ± .
2) + ( − . ± . t − . ± .
3) + ( − . ± . t S46, 15.7 (246 . ± .
4) + (0 . ± . t − . ± .
4) + (5 . ± . t S47, 16.3 (383 . ± .
8) + ( − . ± . t . ± .
4) + (5 . ± . t S48, 16.6 (438 . ± .
5) + ( − . ± . t + ( − . ± . t ) . ± .
5) + (12 . ± . t + ( − . ± . t ) S49, 17.5 (585 . ± .
6) + (15 . ± . t . ± .
7) + (1 . ± . t S50, 17.2 ( − . ± .
3) + ( − . ± . t − . ± .
3) + (9 . ± . t S51, 17.4 ( − . ± .
3) + (7 . ± . t − . ± .
3) + (8 . ± . t S52, 17.1 (200 . ± .
5) + (2 . ± . t . ± .
3) + ( − . ± . t S53, 17.2 (323 . ± .
7) + (12 . ± . t . ± .
7) + (9 . ± . t S54, 17.5 (135 . ± .
2) + ( − . ± . t − . ± .
9) + ( − . ± . t S55, 17.5 (93 . ± .
9) + ( − . ± . t − . ± .
0) + (22 . ± . t S56, 17.0 (143 . ± .
2) + ( − . ± . t . ± .
5) + (0 . ± . t S57, 17.6 (393 . ± .
1) + ( − . ± . t − . ± .
8) + ( − . ± . t S58, 17.4 ( − . ± .
4) + (5 . ± . t + (0 . ± . t ) − . ± .
5) + (4 . ± . t + (0 . ± . t ) S59, 17.2 ( − . ± .
4) + (5 . ± . t . ± .
8) + ( − . ± . t S60, 16.3 ( − . ± .
9) + (3 . ± . t . ± .
5) + ( − . ± . t S61, 17.9 ( − . ± .
4) + ( − . ± . t − . ± .
4) + ( − . ± . t S62, 17.8 ( − . ± .
3) + ( − . ± . t . ± .
7) + (2 . ± . t tellar orbits in the Galactic Center 33 Name, m K α [mas] = t [yr] for ( α, δ ) δ [mas] = t [yr] for v z v z [km / s] =S63, 17.5 (196 . ± .
1) + ( − . ± . t − . ± .
8) + ( − . ± . t S64, 17.5 ( − . ± .
1) + ( − . ± . t . ± .
3) + (8 . ± . t S65, 13.7 ( − . ± .
1) + (2 . ± . t − . ± .
1) + ( − . ± . t S66, 14.8 ( − . ± .
2) + (12 . ± . t + (0 . ± . t ) − . ± .
2) + ( − . ± . t + (0 . ± . t ) ± . ± .
2) + ( − . ± . t + ( − . ± . t ) . ± .
2) + (1 . ± . t + ( − . ± . t ) ± . ± .
4) + (4 . ± . t . ± .
4) + (2 . ± . t S69, 16.8 ( − . ± .
3) + ( − . ± . t . ± .
5) + (0 . ± . t S70, 16.9 ( − . ± .
1) + ( − . ± . t . ± .
2) + ( − . ± . t S71, 16.1 ( − . ± .
2) + (7 . ± . t + (0 . ± . t ) − . ± .
4) + (14 . ± . t + (0 . ± . t ) − ± − . ± .
2) + (8 . ± . t − . ± .
2) + ( − . ± . t S73, 16.1 ( − . ± .
1) + ( − . ± . t − . ± .
2) + ( − . ± . t S74, 16.9 ( − . ± .
2) + ( − . ± . t − . ± .
2) + (4 . ± . t S75, 17.1 ( − . ± .
2) + (6 . ± . t − . ± .
2) + (1 . ± . t S76, 12.8 (355 . ± .
1) + ( − . ± . t − . ± .
1) + (4 . ± . t S77, 15.8 (364 . ± .
4) + (10 . ± . t − . ± .
4) + ( − . ± . t S78, 16.5 (453 . ± .
2) + ( − . ± . t − . ± .
4) + ( − . ± . t S79, 16.0 (646 . ± .
3) + (1 . ± . t − . ± .
3) + (1 . ± . t S80, 16.9 (991 . ± .
3) + ( − . ± . t − . ± .
3) + (4 . ± . t S81, 17.2 (769 . ± .
2) + ( − . ± . t − . ± .
1) + ( − . ± . t S82, 15.4 (54 . ± .
3) + ( − . ± . t . ± .
4) + ( − . ± . t S83, 13.6 ( − . ± .
1) + ( − . ± . t + (0 . ± . t ) . ± .
2) + ( − . ± . t + ( − . ± . t ) − ± − . ± .
2) + (4 . ± . t − . ± .
1) + (1 . ± . t S85, 15.6 ( − . ± .
1) + (5 . ± . t . ± .
1) + ( − . ± . t S86, 15.5 ( − . ± .
2) + (1 . ± . t . ± .
2) + ( − . ± . t S87, 13.6 ( − . ± .
1) + (10 . ± . t + (0 . ± . t ) − . ± .
2) + ( − . ± . t + (0 . ± . t ) ± − . ± .
3) + ( − . ± . t − . ± .
3) + ( − . ± . t S89, 15.3 ( − . ± .
2) + ( − . ± . t − . ± .
2) + ( − . ± . t S90, 16.1 (531 . ± .
3) + (0 . ± . t − . ± .
2) + (0 . ± . t S91, 12.2 (778 . ± .
2) + (11 . ± . t − . ± .
2) + (2 . ± . t S92, 13.0 (987 . ± .
1) + (5 . ± . t . ± .
3) + (1 . ± . t S93, 15.6 (1083 . ± .
3) + ( − . ± . t . ± .
4) + ( − . ± . t S94, 16.7 ( − . ± .
5) + ( − . ± . t . ± .
6) + (2 . ± . t S95, 10.2 (22 . ± .
2) + (6 . ± . t . ± .
2) + (0 . ± . t Name, m K α [mas] = t [yr] for ( α, δ ) δ [mas] = t [yr] for v z v z [km / s] =S96, 10.0 (1132 . ± .
2) + ( − . ± . t . ± .
2) + (7 . ± . t ± . ± .
1) + (7 . ± . t − . ± .
2) + (2 . ± . t ± − . ± .
1) + ( − . ± . t . ± .
2) + (2 . ± . t S99, 16.9 ( − . ± .
2) + ( − . ± . t . ± .
2) + (1 . ± . t S100, 15.4 ( − . ± .
2) + ( − . ± . t . ± .
3) + ( − . ± . t S101, 17.4 ( − . ± .
3) + (3 . ± . t . ± .
5) + (7 . ± . t S102, 17.6 ( − . ± .
4) + ( − . ± . t . ± .
3) + (7 . ± . t S103, 18.3 ( − . ± .
7) + (10 . ± . t . ± .
6) + ( − . ± . t S104, 17.6 ( − . ± .
7) + (10 . ± . t . ± .
6) + ( − . ± . t S105, 16.5 ( − . ± .
2) + (3 . ± . t . ± .
2) + ( − . ± . t S106, 17.1 ( − . ± .
2) + (1 . ± . t . ± .
4) + (2 . ± . t S107, 14.8 ( − . ± .
2) + ( − . ± . t − . ± .
2) + (5 . ± . t S108, 17.0 ( − . ± .
3) + (3 . ± . t − . ± .
3) + (1 . ± . t S109, 17.3 ( − . ± .
2) + (5 . ± . t − . ± .
4) + ( − . ± . t S110, 16.9 ( − . ± .
3) + ( − . ± . t − . ± .
2) + ( − . ± . t S111, 13.8 ( − . ± .
2) + ( − . ± . t + (0 . ± . t ) − . ± .
2) + ( − . ± . t + (0 . ± . t ) − ± − . ± .
2) + (4 . ± . t . ± .
3) + (11 . ± . t tellar orbits in the Galactic Center 35tellar orbits in the Galactic Center 35