Issue 
A&A
Volume 657, January 2022



Article Number  L12  
Number of page(s)  17  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/202142465  
Published online  19 January 2022 
Letter to the Editor
Mass distribution in the Galactic Center based on interferometric astrometry of multiple stellar orbits
^{1}
Max Planck Institute for extraterrestrial Physics, Giessenbachstraße 1, 85748 Garching, Germany
^{2}
LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France
^{3}
Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
^{4}
1st Institute of Physics, University of Cologne, Zülpicher Straße 77, 50937 Cologne, Germany
^{5}
Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
^{6}
Universidade de Lisboa – Faculdade de Ciências, Campo Grande, 1749016 Lisboa, Portugal
^{7}
Faculdade de Engenharia, Universidade do Porto, rua Dr. Roberto Frias, 4200465 Porto, Portugal
^{8}
European Southern Observatory, KarlSchwarzschildStraße 2, 85748 Garching, Germany
^{9}
European Southern Observatory, Casilla, 19001 Santiago 19, Chile
^{10}
Sterrewacht Leiden, Leiden University, Postbus 9513, 2300 RA Leiden, The Netherlands
^{11}
Departments of Physics and Astronomy, Le Conte Hall, University of California, Berkeley, CA 94720, USA
^{12}
CENTRA – Centro de Astrofísica e Gravitação, IST, Universidade de Lisboa, 1049001 Lisboa, Portugal
^{13}
Department of Astrophysical & Planetary Sciences, JILA, Duane Physics Bldg., 2000 Colorado Ave, University of Colorado, Boulder, CO 80309, USA
^{14}
CERN, 1 Esplanade des Particules, Genève 23, 1211, Switzerland
^{15}
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA
^{16}
Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany
^{17}
Gemini Observatory/NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, HI 96720, USA
Received:
17
October
2021
Accepted:
18
November
2021
Stars orbiting the compact radio source Sgr A* in the Galactic Center serve as precision probes of the gravitational field around the closest massive black hole. In addition to adaptive opticsassisted astrometry (with NACO/VLT) and spectroscopy (with SINFONI/VLT, NIRC2/Keck and GNIRS/Gemini) over three decades, we have obtained 30–100 μas astrometry since 2017 with the fourtelescope interferometric beam combiner GRAVITY/VLTI, capable of reaching a sensitivity of m_{K} = 20 when combining data from one night. We present the simultaneous detection of several stars within the diffraction limit of a single telescope, illustrating the power of interferometry in the field. The new data for the stars S2, S29, S38, and S55 yield significant accelerations between March and July 2021, as these stars pass the pericenters of their orbits between 2018 and 2023. This allows for a highprecision determination of the gravitational potential around Sgr A*. Our data are in excellent agreement with general relativity orbits around a single central point mass, M_{•} = 4.30 × 10^{6} M_{⊙}, with a precision of about ±0.25%. We improve the significance of our detection of the Schwarzschild precession in the S2 orbit to 7σ. Assuming plausible density profiles, the extended mass component inside the S2 apocenter (≈0.23″ or 2.4 × 10^{4} R_{S}) must be ≲3000 M_{⊙} (1σ), or ≲0.1% of M_{•}. Adding the enclosed mass determinations from 13 stars orbiting Sgr A* at larger radii, the innermost radius at which the excess mass beyond Sgr A* is tentatively seen is r ≈ 2.5″ ≥ 10× the apocenter of S2. This is in full harmony with the stellar mass distribution (including stellarmass black holes) obtained from the spatially resolved luminosity function.
Key words: black hole physics / instrumentation: interferometers / Galaxy: center
© GRAVITY Collaboration 2022
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Open Access funding provided by Max Planck Society.
1. Introduction
The GRAVITY instrument on the Very Large Telescope Interferometer (VLTI) has made it possible to monitor the positions of stars within 0.1″ from Sgr A*, the massive black hole (MBH) at the Galactic Center (GC), with a precision of ≈50μas (GRAVITY Collaboration 2017). The GRAVITY data taken in 2017–2019 together with the adaptive optics (AO) and Speckle data sets obtained since 1992 (at ESO telescopes), or since 1995 at the Keck telescopes have delivered exquisite coverage of the 16year highly elliptical orbit of the star S2, which passed its most recent pericenter in May 2018. Besides the direct determinations of the mass of Sgr A* (M_{•}) and the distance to the GC (R_{0}), these interferometric data have provided strong evidence for general relativistic effects caused by the central MBH on the orbit of S2, namely, the gravitational redshift and the prograde relativistic precession (GRAVITY Collaboration 2018a, 2019, 2020, 2021; Do et al. 2019).
Due to its short period and brightness, S2 is the most prominent star in the GC, but ever higher quality, highresolution imaging and spectroscopy of the nuclear star cluster over almost three decades have delivered orbit determinations for some 50 stars (Schödel et al. 2002, 2003; Ghez et al. 2003, 2008; Eisenhauer et al. 2005; Gillessen et al. 2009, 2017; Meyer et al. 2012; Boehle et al. 2016). The motions of these stars show that the gravitational potential is dominated by a compact, central mass of ≈4.3 × 10^{6} M_{⊙} that is concentrated within S2’s (3D) pericenter distance of 14 mas (or 120 AU) and 1400 times the event horizon radius R_{S} of a Schwarzschild (nonrotating) MBH for a distance of 8.28 kpc (GRAVITY Collaboration 2019, 2021).
S2 passes its pericenter with a mildly relativistic orbital speed of 7700 km s^{−1} (β = v/c = 0.026). Based on the monitoring of the star’s radial velocity and motion on the sky from data taken prior to and up to two months after pericenter, GRAVITY Collaboration (2018a) were able to detect the first postNewtonian effects of general relativity (GR), namely: the gravitational redshift, along with the transverse Doppler effect of special relativity. The combined effect for S2 shows up as a 200 km s^{−1} residual centered on the pericenter time, relative to the Keplerian orbit with the same parameters. GRAVITY Collaboration (2019) improved the statistical robustness of the detection of the gravitational redshift to 20σ. Do et al. (2019) confirmed these findings from a second, independent data set mainly from the Keck telescope. While the redshift occurs solely in the wavelength space, the superior astrometry of GRAVITY sets much tighter constraints on the orbital geometry, mass, and distance, all the while decreasing the uncertainty on the redshift parameter more than three times relative to data sets from single telescopes.
The precession due to the Schwarzschild metric is predicted to lead to a prograde rotation of the orbital ellipse in its plane of Δω = 12.1″ per revolution for S2, corresponding to a shift in the milliarcsec regime of the orbital trace on sky; hence using interferometry is particularly advantageous in this case. GRAVITY Collaboration (2020) detected the Schwarzschild precession at the 5σ level. The uncertainties on the amount of precession can then be interpreted as limits on how much extended mass (leading to retrograde precession) might be present within the S2 orbit.
Here, we expand our analysis by two more years, up to 2021.6. We combine GRAVITY data from four stars, alongside with the previous AO data. Section 2 presents the new data and Sect. 3 describes our analysis. In Sect. 4, we show the combined fits, improving the accuracy of the measured postNewtonian parameters of the central black hole and the limits on the extended mass (Sect. 5). In combination with earlier measurements of stars with larger apocenters, we study the mass distribution out to ≈3″. Section 6 summarizes our conclusions.
2. Observations
Interferometric astrometry with GRAVITY has several distinct advantages over singletelescope, AO imaging (Fig. 1):
Fig. 1.
Progress in stellar astrometric imaging in the GC, from seeing limited observations (≈0.4 − 0.5″ FWHM, top left) to AO imaging on 8 − 10 m class telescopes (50 − 60 mas FWHM, ≈300 − 500 μas astrometry (topright), showing a LucyRichardson deconvolved and beamrestored AO image), to current interferometric imagery with GRAVITY/VLTI (bottom panels, 2 × 4 mas FWHM resolution, 30 − 100 μas astrometry). Bottom right: central 0.2″ region centered on Sgr A*(IR) on May 30, 2021 (see Appendix A.2, GRAVITY Collaboration 2022). Over 2021, the central GRAVITY field of view was dominated by four stars (S29, S55, S62, and S300) with Kband magnitudes between 16.7 and 19.2, in addition to variable emission from Sgr A*(IR) itself. The dashed circle indicates the fieldofview of GRAVITY, defined by the Gaussian acceptance profile of the pickup fibres in the instrument. Comparing in the bottom left panel three such images taken on March 30 (red), May 30 (green), and July 26 (blue), the orbital motions of all four stars are easily seen, topped by the ≈8740 km s^{−1} velocity of S29 at its pericenter on 2021.41 (R_{peri} ≈ 100 AU). 
First, the higher angular resolution yields an order of magnitude better astrometric precision for isolated sources.
Secondly, for crowded environments, such as the central arcsecond that has a surface density > 100 stars per square arcsecond to K < 17 (and more for fainter limits, Genzel et al. 2003; Baumgardt et al. 2018; Waisberg et al. 2018), interferometric data are much less affected (by a factor of several hundred) by confusion noise. In the context of the GC cluster imaging, this issue was recognized early on (Ghez et al. 2003, 2008; Gillessen et al. 2017; Do et al. 2019; GRAVITY Collaboration 2020). For “orbit crossings” of modest duration for individual brighter stars, this often means that data over a duration of a few years are affected. The situation is worse at the pericenter passage of S2 (2002, 2018), when the star and the variable source Sgr A* are in the same diffraction beam of an 8 − 10 m class telescope (Ghez et al. 2008). For 2021/2022, our data show explicitly that in addition to Sgr A*, several stars are present in the central beam (see also GRAVITY Collaboration 2022), making singletelescope astrometry even more uncertain or unusable.
Third, close to Sgr A*, astrometry with interferometry reduces to fitting the phases and visibilities with a multiple point source model in a single pointing of the interferometric fiber, which is straightforward, once the optical aberrations across the fiber field of view are corrected for (GRAVITY Collaboration 2020, 2021). Interferometric measurements beyond the fibre field of view require double pointings, which, thanks to GRAVITY’s metrology system, can be related astrometrically to each other (Appendix A.1.2). The GRAVITY positions are directly referring to Sgr A*, since it is visible in each exposure and since it is one of the point sources in the multisource model. In contrast, AO astrometry relies on establishing a global reference frame by means of stars visible both at radio wavelengths (where Sgr A* is visible as well) and in the nearinfrared. A few such SiO maser stars exist in the GC (Reid et al. 2007) and they reside at larger separations (≈20″). Hence, to establish an AObased reference frame, it is necessary to correct for the distortions of the imaging system (Plewa et al. 2015; Sakai et al. 2019; Jia et al. 2019).
In addition to S2 (now moving away from Sgr A* since its 2018.38 pericenter), we used the stars S29 (K = 17.6, pericenter 2021.41), S38 (K = 18.3, 2022.70), and S55 (K = 17.2, 2021.7). The latter star was labeled S0102 in Meyer et al. (2012). These authors determined its orbital period to be 12 years.

For S2, we included 128 NACO astrometric data points, 92 SINFONI, 3 Keck (2000–2002), 2 NACO (2003) and 4 GNIRS / GEMINI spectra, and 82 GRAVITY astrometric measurements. Compared to GRAVITY Collaboration (2020), we added the 4 GNIRS spectra, and 17 GRAVITY epochs. These data cover the time span of 1992.22021.6.

For S29, we included 94 NACO, 17 SINFONI, 2 GNIRS, and 21 GRAVITY measurements, covering 2002.32021.6.

For S38, we included 110 NACO, 8 SINFONI, 1 Keck, and 6 GRAVITY measurements, covering 2004.22021.6.

For S55, we included 42 NACO, 2 SINFONI, and 18 GRAVITY measurements, covering 2004.52021.6.

We also analyzed the NACO and SINFONI 2002–2019 data for another 13 stars with K ≈ 12 − 16 that have sufficient data to infer a mass (and their orbital parameters), with R_{0} fixed to 8277 pc (the best fit value, Sect. 4).
We report the full list of observations in Appendix D.1.
3. Analysis
For a singlestar fit, we typically fit for 14 parameters: six parameters describing the initial osculating Kepler orbit (a, e, i, ω, Ω, t_{0}), R_{0} and M_{•}, along with five coordinates describing the onsky position and the 3D velocity of the mass (relative to the AO spectroscopic/imaging frame), and a dimensionless parameter encoding the nonKeplerian effect we are testing for. For the gravitational redshift, we used f_{gr}, which is 0 for Newtonian orbits and 1 for GRorbits. In GRAVITY Collaboration (2018a) we found f_{gr} = 0.90 ± 0.17; and in GRAVITY Collaboration (2019), we found f_{gr} = 1.04 ± 0.05. Do et al. (2019) reported f_{gr} = 0.88 ± 0.17.
For the Schwarzschild precession, we use the firstorder postNewtonian expansion for a massless test particle (Will 1993) and add a factor f_{SP} in the equation of motion in front of the 1PN terms, where f_{SP} = 0 corresponds to Keplerian motion and f_{SP} = 1 to GR. In GRAVITY Collaboration (2020), we found f_{SP} = 1.10 ± 0.19.
Similarly, we parameterize an extended mass distribution by including a parameter f_{Pl} in the normalization of the profile. Following Gillessen et al. (2017) and GRAVITY Collaboration (2020), we assume a Plummer (1911) profile
with scale length, a_{Pl}, and total mass of f_{Pl}M_{•}. We use a_{Pl} = 1.27a_{apo}(S2)=0.3″ (Mouawad et al. 2005). The enclosed mass within R is . We fit for the fraction of M_{•} that is in the extended mass, f_{Pl}.
Following GRAVITY Collaboration (2018a, 2019, 2020), we find the bestfit values by fitting simultaneously all parameters, including prior constraints. Throughout our study, we used an outlier robust fitting (Sect. 3.2 in GRAVITY Collaboration 2020). The inferred uncertainties are affected and partially dominated by systematics, especially when combining data from three or more measurement techniques. Our parameterization keeps correlations between f_{SP} or f_{Pl} and M_{•} or R_{0} small, but the two parameters of interest show some degeneracy with the coordinate system offsets.
To check the formal fit errors, we carried out a (MetropolisHastings) Markovchain Monte Carlo (MCMC) analysis. Using 100 000 realizations, we found the distributions and parameter correlations of the respective dimensionless parameter, f_{SP} or f_{Pl}, with the other parameters and test whether they are well described by Gaussian distributions. For more details, see GRAVITY Collaboration (2018a, 2019, 2020, 2021) and Appendix A.
In Gillessen et al. (2009, 2017), we consistently found, based on the AO data, that the basic parameters describing the gravitational potential (M_{•} and R_{0}) and the extended mass (e.g., f_{Pl} for an assumed Plummer distribution) are best constrained by the S2 data. Including other stars only moderately improved the fitting quality and uncertainties. This is because of the superior number and quality of the S2 data compared to those of the other stars. Since the higher resolution GRAVITY astrometry became available, S2 has completely dominated our knowledge about the central potential. Another reason is that it is only for S2 that we have data at or near pericenter, which are the most sensitive component of the data to the mass distribution, as the explicit analysis in Gillessen et al. (2017) and Heißel et al. (2022) shows.
This situation changes with the data set used here. We now have GRAVITY data of four stars with comparable pericenters at 12 mas (S29), 14 mas (S2), 26 mas (S38), and 29 mas (S55). Naturally, we need to fit for 4 × 6 orbital parameters (a_{i}, e_{i}, i_{i}, ω_{i}, Ω_{i}, t_{0, i}) in addition to the NACO/SINFONI zero points (x_{0}, y_{0}, vx_{0}, vy_{0}, vz_{0}), M_{•} and R_{0}, as well as f_{SP} and/or f_{Pl}. However, the inclusion of nearpericenter GRAVITY data of S29, S38, and S55 lessens the parameter correlations and uncertainties (Fig. 2); this is also because the orbits are oriented almost perpendicular to each other in at least one of the Euler angles. Furthermore, the orbits of the four stars probe the precession over a wider parameter range in semimajor axis and eccentricity.
Fig. 2.
Summary of the 1992–2021 astrometric data and best fitting orbits for the stars S2, S29, S38, and S55. The black cross marks the position of the compact radio source Sgr A* and mass center. The directions of projected orbital motions are marked by arrows. 
A comparison of the results obtained with different fitting schemes and codes of the consortium (of MPE, Univ. Cologne and LESIA) underlines the importance of two aspects of the data analysis: First, AObased astrometry of stars near pericenter is subject to confusion, with the emission from Sgr A* and other neighboring stars contributing to the centroid. These data now can be discarded in favor of betterresolution interferometric data. Second, the NACOframe zero point and drift on the one hand, and the pro or retrograde precession on the other hand, are degenerate. To reduce this degeneracy, we can use three sources of additional information: (i) NACO astrometry of flares, (ii) the prior from the construction of the AO reference frame (Plewa et al. 2015), or (iii) the orbits of further stars that have sufficient phase coverage to constrain the zero points. In GRAVITY Collaboration (2020), we combined (i) and (ii). Avoiding the potentially confused flare positions (i), here we use the combination of (ii) and the orbits of 13 further stars (iii). We derive x_{0} = 0.57 ± 0.15 mas, y_{0} = −0.06 ± 0.25 mas (both epoch 2010.35), vx_{0} = 63 ± 7 μas yr^{−1}, vy_{0} = 33 ± 2 μas yr^{−1}, which are consistent with the earlier estimates, but with smaller uncertainties.
4. Schwarzschild precession for S2
Repeating the analysis of GRAVITY Collaboration (2020) (S2 alone but with the updated zero points) and solving for the Schwarzschild precession parameter we find f_{SP} = 0.85 ± 0.16 (). This is naturally very similar to our previous results^{1}, but the new data have decreased the 1σ uncertainty from ±0.19 to ±0.16.
Next, we fit with the four star (S2, S29, S38, S55) data, and find f_{SP} = 0.997 ± 0.144, with (Fig. 2). Figure 3 shows the residuals of the best fit and from the corresponding Newtonian (f_{SP} = 0) orbit. The combination of the nearpericenter GRAVITY data of four stars improves the constraints on the common parameters. The contributions raising come from the NACO data of S29, S38, and S55 covering the outer parts of their orbits more affected by confusion.
Fig. 3.
Illustration of the fit results for f_{SP}. We show the key figures for S2. Two left panels: the residuals in RA (top) and Dec (bottom) between the GRAVITY (cyan filled circles, and 1σ uncertainties) and NACO data (blue filled circles) and the best GR fit (red curve, f_{SP} = 1, Rømer effect, plus special relativity, plus gravitational redshift and Schwarzschild precession), relative to the same orbit for f_{SP} = 0 (Kepler/Newton, plus Rømer effect, plus special relativity, plus redshift). The orbital elements for nonKeplerian orbits (i.e., with f_{SP} ≠ 0) are interpreted as osculating parameters at apocenter time, 2010.35. NACO and GRAVITY data are averages of several epochs. The grey bars denote averages over all NACO data near apocenter (2004–2013). Top right: the same for the residual orbital angle on the sky δϕ = ϕ(f_{SP} = 1)−ϕ(f_{SP} = 0). Bottom right: zoom into the 2005/2021 part of the orbit, plotted in the mass rest frame. The earlier orbital trace does not coincide with the current one due to the Schwarzschild precession. 
Applying MCMC analysis we find the most likely values of f_{SP} = 0.85 ± 0.18 (only S2) and f_{SP} = 0.99 ± 0.15 (S2, S29, S38, S55). Figure B.1 shows the full set of parameter correlations, including the wellknown degeneracy between M_{•} and R_{0} (Ghez et al. 2008; Boehle et al. 2016; Gillessen et al. 2009, 2017). All of the 32 parameters of the fourstar fit are well constrained.
As discussed by GRAVITY Collaboration (2020), the impact of the high eccentricity of the S2 orbit (e = 0.88) is that most of the precession happens in a short timeframe around pericenter. Due to the geometry of the orbit most of the precession shows up in the RAcoordinate, and the change in ω after pericenter appears as a kink in the RAresiduals. The data are obviously in excellent agreement with GR. Compared to GRAVITY Collaboration (2020), the significance of this agreement has improved from 5 to 7σ, from the combination of adding two more years of GRAVITY data to the S2 data set and the expansion to a fourstar fit. Table B.1 gives the best fit orbit parameters, zero points, M_{•}, and R_{0}.
As of 2021, S2 is sufficiently far away from pericenter, such that the Schwarzschild precession can now be seen as a ≈0.6 mas shift between the data sets in RA (and less so in Dec) between two consecutive passages of the star on the apocenterside of the orbit. This effect is obvious when comparing the 2021 GRAVITY data to the 2005 NACO data, exactly one period prior (Fig. 3 right). This comparison illustrates that the Schwarzschild precession dominates the entire orbit and that there is no detectable retrograde (Newtonian) precession due to an extended mass component (see Heißel et al. 2022).
5. Limits on extended mass
In the following, we fix f_{SP} = 1 at its GR value and allow now for an extended mass component parameterized by f_{Pl}. We find f_{Pl} = (2.7 ± 3.5) × 10^{−3} from a single S2 fit and f_{Pl} = (−3.8 ± 2.4) × 10^{−3} for the fourstar fit, in accordance with the findings of Heißel et al. (2022). The latter 1σ error is consistent, albeit three to four times smaller than that of Gillessen et al. (2017) and 1.7 times smaller than that of GRAVITY Collaboration (2020), corresponding to 2400 M_{⊙} within the apocenter of S2. In Fig. 4, we include the 3σ uncertainty as a conservative upper limit, indicating that the extended mass cannot exceed 7500 M_{⊙}. As in Gillessen et al. (2017) and GRAVITY Collaboration (2020), we find again that varying a_{Pl} or replacing the Plummer distribution by a suitable power law changes this result trivially. Furthermore, we get a weaker limit by a factor of ≈2 when omitting the NACO astrometry, using only GRAVITY & SINFONI data.
Fig. 4.
Summary of the central mass distribution in the GC, within 20× the apocenter radius of the S2 orbit, ≈5″. The filled blue circle is the central mass of 4.30 × 10^{6} M_{⊙} (for R_{0} = 8277 pc, and its 1σ uncertainty of 12 000 M_{⊙}), which our fourstar fitting has established to lie within the 1200R_{S} pericenter of S29. The black arrow denotes the 3σ upper limit of M_{ext}(≤0.3″) of the sum of M_{•} and any extended Plummer mass of assumed scale radius 0.3″. The two open blue circles and two red filled squares show averages of enclosed masses within the semimajor axes of other Sstars and clockwise disk stars. The magenta dashed line is the sum of M_{•} and the extended stellar mass distribution from the literature (e.g., Genzel et al. 2010; Alexander 2017; Schödel et al. 2018; Baumgardt et al. 2018). All data are in excellent agreement with a point mass (the MBH Sgr A*) and a star cluster with a powerlaw density slope γ ≈ 1.6 (Genzel et al. 2003; GallegoCano et al. 2018), consisting of mainsequence stars and a small contribution of giants. No extra component from dark matter or an intermediate mass black hole > 10^{3} M_{⊙} (GRAVITY Collaboration 2020) is required or compatible with the data. 
The impact of an extended mass is naturally largest near apocenter of an orbit (e.g. Heißel et al. 2022). Figure C.2 shows the result of adding various amounts of extended mass on top of the bestfit residuals with a point mass only. Our data are just commensurate with an additional f_{Pl} = 0.25% of M_{•}, but a larger mass is excluded by both the nearperi and nearapocenter data. The apparent sensitivity of the nearpericenter data in Fig. C.2 is the result of referring the residuals to the osculating Keplerian orbit at apocenter in 2010.35, such that the accumulating retrograde precession enters the nearpericenter data.
A second, independent measurement of the dynamically inferred mass distribution comes from fitting for the central mass using 13 individual stellar orbits with a = 0.1″ to 3.8″ (Gillessen et al. 2017), with R_{0} and zeropoints fixed to the best fitting values of the fourstar fit (Fig. C.1). We then averaged the results in four groups of four stars with 0.11″ < a < 0.22″, five stars with 0.27″ < a < 0.4″, three stars with 0.55″ < a < 1.6″, and two stars with 1.6″ < a < 3.8″. Most of the stars in the first two groups are classical “Sstars” (mostly earlytype B stars) with typically large eccentricities (e.g., Ghez et al. 2008; Gillessen et al. 2017), while most of the stars in the outer groups are O and B stars in the clockwise disk (Paumard et al. 2006; Bartko et al. 2009; Lu et al. 2009), with modest eccentricities. The stars in the first two groups indicate that the mass is consistent with M_{•} to within 0.3–0.6%. There is no indication for an extended mass larger than ≈25 000 M_{⊙} within 2 a_{apo}(S2)≈0.5″. The outer stars suggests an extended mass of 15 000 M_{⊙} (and conservatively a 3σ limit of 50 000 M_{⊙}) within 5 − 10 a_{apo}(S2).
Figure 4 summarizes the mass distribution within 5″ (≈20× the apocenter of S2). These estimates and limits are in excellent agreement with the distribution of stars (and stellar mass black holes and neutron stars) contained in this inner region around Sgr A*, as estimated from models and simulations (Alexander 2017; Baumgardt et al. 2018), or from observations of faint stars and diffuse stellar light (Fig. 4, Genzel et al. 2010; GallegoCano et al. 2018; Schödel et al. 2018; Habibi et al. 2019).
In summary, several precise (O(0.1–0.3%), 1σ) determinations show that the mass distribution in the GC within 5″ ≈ 5 × 10^{5}R_{S} of Sgr A* is dominated by a central, compact mass. This mass is definitely enclosed within the pericenter of S29 (12 mas, ≈1200R_{S}). Taking the gas motions at ≈3 − 5 R_{S} (GRAVITY Collaboration 2018b) and the mmsize of Sgr A* (Doeleman et al. 2008; Johnson et al. 2017; Issaoun et al. 2019) into account, the data are in excellent agreement with the MBH paradigm.
6. Conclusions
Here, we present GRAVITY data obtained at the VLTI in 2021. Within the central 20 mas, we observe the motions of four stars between March and July, illustrating the power of the interferometric, high spatial resolution. Using the novel astrometry of the stars S2, S29, S38, and S55, along with new radial velocities obtained with GNIRS, we update our orbital analysis.
The star S2 has now returned to the part of its 16year orbit for which good NACO AOassisted positions were obtained during its previous passage. A direct comparison of the positions confirms that the orientation of the orbital ellipse has indeed shifted in its plane by the 12.1′ expected from the prograde Schwarzschild precession induced by the gravitational field of the MBH, as reported in GRAVITY Collaboration (2020).
At K = 14.1, S2 is comparably bright. With its increased distance from Sgr A* in 2021, we are now able to map with GRAVITY the immediate vicinity of the MBH to significantly fainter objects. This provided accurate positions for S29, S38, and S55. These stars have previously measured NACO positions when they were further away from Sgr A*. Combining these with the GRAVITY data improves the orbital parameters of the three stars substantially. In particular, S29 is on a deeply plunging (e = 0.97) orbit with a period of ≈90 years and pericenter passage in 2021.41, with a space velocity of ≈8740 km s^{−1} at only 100 AU from Sgr A*.
S2, S29, S38, and S55 orbit in the same gravitational potential, and combining their astrometry and radial velocity data improves the accuracy of the determination of the properties of the central MBH. This leads to a 14% measurement precision of the Schwarzschild precession, which is in full agreement with the prediction of GR. The best fit further yields R_{0} = (8277 ± 9) pc and M_{•} = (4.297 ± 0.012) × 10^{6} M_{⊙} (statistical errors, see GRAVITY Collaboration 2021 for a discussion of the systematics that are ≈30 pc for R_{0} and ≈40 000 M_{⊙} for M_{•}).
Any smooth extended mass distribution would lead to a retrograde precession of the S2 orbit relative to the relativistically precessing one and we can thus place a limit on a hypothetical mass distribution. The measurement errors leave room for at most ≲3000 M_{⊙} in extended mass out to 230 mas. We included 13 stars further out with earlier measurements to trace the mass as a function of radius. The data are fully consistent with a single point mass, and only at r ≳ 2.5″ does the enclosed mass tentatively exceed M_{•}, which is consistent with the theoretically expected stellar mass distribution. Inside the 100 AU pericenter of S29, the orbits of Sgr A* flares (GRAVITY Collaboration 2018b), together with the coincidence of mass location and light centroid (Plewa et al. 2015; Reid & Brunthaler 2020) constrain the mass distribution further, excluding, for example, dark matter spikes, as proposed by BecerraVergara et al. (2020, 2021), and the presence of an intermediate mass black hole as well (GRAVITY Collaboration 2020).
Our multiepoch GRAVITY data also confirm that at any time, there are likely a few stars that are sufficiently close to Sgr A* on the sky to systematically influence its position derived with AOassisted imaging on single telescopes. In addition, in 2022, two stars will pass the pericenters of their orbits at less than 100 mas distance (S38 and S42). The upgrade of GRAVITY to GRAVITY+ (Eisenhauer 2021) will push the sensitivity limit to K > 20, which may reveal more stars with even smaller orbits. The 39 m ELT equipped with MICADO (Davies et al. 2021) and HARMONI (Thatte et al. 2021) might be the prime choice for obtaining radial velocities of such stars. Yet, GRAVITY+ will beat the ELT’s angular resolution by a factor three, allowing continued < 50 μas astrometry and going even deeper than what we have demonstrated so far (GRAVITY Collaboration 2022).
Following strictly GRAVITY Collaboration (2020), i.e., using the S2 data plus the flare positions of Sgr A* with the zero point priors of Plewa et al. (2015) yields f_{SP} = 1.23 ± 0.14 ().
Acknowledgments
We are very grateful to our funding agencies (MPG, ERC, CNRS [PNCG, PNGRAM], DFG, BMBF, Paris Observatory [CS, PhyFOG], Observatoire des Sciences de l’Univers de Grenoble, and the Fundação para a Ciência e Tecnologia), to ESO and the Paranal staff, and to the many scientific and technical staff members in our institutions, who helped to make NACO, SINFONI, and GRAVITY a reality. S.G. acknowledges the support from ERC starting grant No. 306311. F.E. and O.P. acknowledge the support from ERC synergy grant No. 610058. A.A., A.F., P.G. and V.C. were supported by Fundação para a Ciência e a Tecnologia, with grants reference SFRH/BSAB/142940/2018, UIDB/00099/2020 and PTDC/FISAST/7002/2020. Based on observations collected at the European Southern Observatory under the ESO programmes listed in Appendix D. The GNIRS spectra were obtained at the international Gemini Observatory, a program of NSF’s NOIRLab, managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation (NSF) on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). This work was enabled by observations made from the Gemini North telescope, located within the Maunakea Science Reserve and adjacent to the summit of Maunakea. We are grateful for the privilege of observing the Universe from a place that is unique in both its astronomical quality and its cultural significance. We thank our anonymous referee for very useful comments.
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Appendix A: Experimental techniques
A.1. GRAVITY: Determining astrometric separations
The full width half maximum (FWHM) of the interferometric field of view (IFOV) of GRAVITY is 70 mas. In consequence, not all stars discussed in this paper are observable simultaneously. The star S2 has moved too far away from Sgr A* compared to 2018 to be observable simultaneously with Sgr A*, while the stars S55 and S29 (and others, see GRAVITY Collaboration 2022) are always observed alongside Sgr A*. Depending on the separation, there are two methods for determining the positions of the stars relative to Sgr A*: singlebeam and dualbeam astrometry. Singlebeam positions are extracted from pointings in which more than one source is present in the IFOV. The distances between the stars are extracted by fitting a multisource model to the visibility amplitude and closure phase, each of which are measured at ≈ ten spectral channels for six baselines. This yields positions of the sources with respect to each other. Since Sgr A* is visible in all our central frames, for those pointings the relative positions also are the absolute ones, that is, with respect to the mass center. If the stars are not observable in a single IFOV, we need to observe them separately and apply the dualbeam technique. For the case of two isolated stars, one interferometrically calibrates the first source with the second. The first source serves as a phase reference relative to which offsets of the second source can be measured.
A.1.1. Singlebeam astrometry
If a star is in the same IFOV as Sgr A* (of particular interest here in 2021 are S29 and S55), we determine the relative separation between the star and Sgr A* by interferometric model fitting to the visibility amplitude and closure phases in the Sgr A* pointing. This methodology is unchanged with regard to the way separations were determined in GRAVITY Collaboration (2021). We thus take into account the effect of phase aberrations as well as bandwidth smearing (GRAVITY Collaboration 2020).
A.1.2. Dualbeam astrometry
If the stars are separated by more than the IFOV of GRAVITY, we measure the separation between the two sources by using one of them as the phase reference for the other target, namely, by calibrating the complex visibilities of the object of interest with the ones of the phase reference. We use S2 as the phase reference, which in its interferometric observables is consistent with a single point source. The separation between any star and S2 is determined by two vectors:

the vector by which the IFOV was moved between S2 and the star. This vector is measured by the metrology system of GRAVITY monitoring the internal optical path differences

the phase center offset in the S2calibrated star observation. It is determined by fitting the visibility phase. Also for the dualbeam analysis, we account for the effect of phase aberrations and bandwidth smearing when calculating the model visibility phase.
The separation is affected by inaccuracies and systematic uncertainties of the metrology. Such telescopebased errors are inherent to the dualbeam part of the measurement.
Typically, we find more stars than just one in the IFOV. We thus need to take into account the signatures induced by the additional stars in the dualbeam measurement. This occurs, for example, for the Sgr A* pointings (where S29, S55, S62, and S300 are present), but also for the S38 pointing (with S60 and S63 being present in the IFOV). We thus fit the interstellar separations and the phase center offset simultaneously in order to take into account their degeneracies. However, the separation vectors are mostly sensitive to the visibility amplitude and the closure phase information from the visibility phase, while the phase center offsets mostly acts as an additional term in the visibility phase. In this way we can relate all positions to our calibrator source, S2. Hence, we can relate the positions also to Sgr A* by subtracting the startoS2 and S2toSgr A* separations.
Telescopebased errors cancel out in the closure phase and therefore the relative positions of the sources are not affected by phase errors, but the visibility phases carry the information of how different IFOVs are located with respect to one another. We find that by fitting the closure phases and the visibility phases with equal weights, we minimize the effect of the telescopebased errors, while still being sensitive to the phase information.
In order to average out the phase errors, we calibrate all N frames of a given pointing with all M available S2 frames individually. For each of the N × M resulting data sets, we determine the phase center position and average the resulting phase center locations. This calibration uncertainty adds a systematic uncertainty of 60 μas, divided by the square root of the number of available calibrations.
We further improve the accuracy of our phase center measurement by determining the best fit fringetracker and science target separation by fitting the S2 observations with a drifting point source model. This takes into account our imperfect knowledge of the separation prior to the observation. Here, we follow the concepts set out in GRAVITY Collaboration (2020).
A.2. GRAVITY: Deep imaging
To obtain deep, highresolution images of the GC, we developed a new imaging code called GRAVITYRESOLVE or G^{R} which is drawn from RESOLVE (Arras et al. 2021), a Bayesian imaging algorithm formulated in the framework of information field theory (Enßlin 2019) that is customtailored to GRAVITY observations of the GC. Here, we briefly outline the main ideas, (see GRAVITY Collaboration (2022) for a detailed description.
With a Bayesian forward modeling approach, we can address data sparsity and account for various instrumental effects that render the relation between image and measurement more complicated than the simple Fourier transform of the vanCittert Zernike theorem. To this end, the algorithm formulates a prior model which permits to draw random samples, processes them with the instrumental response function, and evaluates the likelihood to compare the predicted visibilities with the actual measurement. This approach can handle the noninvertible measurement equation and allows us to work with nonlinear quantities such as closure phases. The exploration of the posterior distribution is done with metric Gaussian variational inference (Knollmüller & Enßlin 2019), and infers the mean image with respect to the posterior jointly with an uncertainty estimate. There are already some imaging tools available for optical/nearIR interferometry that implement a forwardmodeling approach such as MIRA (Thiébaut et al. 2008) or SQUEEZE (Baron et al. 2010). Our code differs from those with regard to the details of the measurement equation, the prior model, and how the maximization and exploration of the posterior are performed.
In the measurement equation, we implemented all instrumental effects relevant for GRAVITY: coupling efficiency, aberration corrections (GRAVITY Collaboration 2021), averaging
over finite sized wavelength channels (also known as bandwidth smearing), and the practice in optical and nearIR interferometry to construct the complex visibility as the coherent flux over a baseline divided by the total flux of each of the two telescopes. The latter signifies that the visibility amplitude can be unity at most, but coherence loss can degrade the observed visibility from the theoretical expectation. This we account for by a selfcalibration approach where we infer a time and baselinedependent calibration factor jointly with the image.
An appropriate prior model is essential to address the data sparsity inherent to optical/nearIR interferometry, and we specifically tailor it to the GC observations. There, we see Sgr A* as a point source in addition to some relatively bright stars whose approximate positions are known from orbit predictions. For those objects, we directly infer the position and brightness using a Gaussian and a lognormal prior respectively. The variability and polarization of Sgr A* is accounted for by allowing for an independent flux value in each frame and polarization state observed. In the actual image itself, we expect to see few faint, yet unknown, point sources and thus impose the individual pixels to be independent with their brightness following an Inverse Gamma distribution. We note that all sources other than Sgr A*, that is all nonvariable sources, could in principle also be attributed to the image. However, modeling them as additional point sources improves convergence and mitigates pixelization errors.
A.3. GNIRS: Determining radial velocities
In 2021 we had four successful observations with the longslit spectrograph GNIRS using the AO system ALTAIR at the Gemini observatory. We used the long slit in the KBand with the 10.44 l/mm grating. The slit was positioned such that we observed S2 and S29 simultaneously. To calibrate the data we used the daytime calibration from the day after the observation, which contains a set of dark frames to determine a bad pixel mask, flat frames, and a wavelength calibration. Additionally, a telluric star was observed right after the observation. To determine the velocity of the stars we used template fitting with a high SNR S2 spectrum, in the same way as we extracted the SINFONI velocities (see GRAVITY Collaboration 2018a). We were able to detect a velocity for S2 in all four observing nights. As S29 is significantly fainter than S2 we needed excellent conditions to get a detection, which was only possible in two of the four nights.
Appendix B: Fit details
In Table B.1, we give the bestfit parameters of the fourstar fitdetermining f_{SP}, comparing also with similar fits from the literature. Figure B.1 gives the full posterior of the fourstar fit in the form of a corner plot.
Bestfit orbit parameters of the fourstar fit determining f_{SP}.
Fig. B.1.
Corner plot of the full 32dimensional parameter space of the fourstar fit, consisting of (in order) the central mass M_{•}, the offsets x_{0}, y_{0}, the distance R_{0}, the velocity offsets vx_{0}, vy_{0}, vz_{0}, the precession parameter f_{SP}, followed by four times six orbital elements for each of the four stars used, in the order a, e, i, Ω, ω, t_{peri} for S2, S29, S38, S55. 
Appendix C: Additional figures
In Figure C.1 we show the orbital data of additional Sstars that were auxiliary in this work. Figure C.2 illustrates that our S2 data are compatible at most with an extended mass component of around 0.1% enclosed within the S2orbit.
Fig. C.1.
Visualization of the orbital (astrometric) data used in determining the mass distribution in the GC. The panels group the stars according to the semimajor axes of their orbits, as indicated in the top left in each panel. For reference, we show in each panel the orbits from the fourstar fit (S2: red line, S29 violet, S38 black, S55 blue), Left: Orbital data for S9, S13, S18 and S21. Middle: S4, S12, S31 and S42. Right: S66, S67, S87, S96 and S97. These data are complemented by multiepoch spectroscopy for the orbital fitting. 
Fig. C.2.
Same data and RA/Decresidual plots as the left panels in Figure 3, with the dashed red curve denoting the f_{SP} = 1 GR curve for the best fitting orbit and mass. In addition, we show orbital models with the same central mass, distance and orbital parameters but now adding an extended mass component assumed to have a Plummer shape (Gillessen et al. 2017) showing the impact of adding a Plummer mass of M_{ext} within the 0.25″ apocenter radius of S2. Black, orange and blue solid curves show the changes expected if this extended Plummer mass is 0.1, 0.3 and 0.6% of M_{•} (4.4 × 10^{3}, 8.9 × 10^{3} and 1.78 × 10^{4}M_{⊙} within the apocenter of S2, R_{apo} = 0.24″). Formal fitting shows that such an extended mass greater than about ≈0.1% of M_{•} is incompatible with the data. 
Appendix D: List of observations
In Tables D.1, D.2, and D.3 we list the observations used in this work. We note that most of the raw data are available from the observatories’ archives, most easily found via the program ID numbers provided here. The derived astrometric and spectroscopic data set can be made available upon personal request and after discussing the terms of a collaboration on a casebycase basis.
GRAVITY observations used in this work.
Radial velocity observations used in this work.
Astrometric observations used in this work.
All Tables
All Figures
Fig. 1.
Progress in stellar astrometric imaging in the GC, from seeing limited observations (≈0.4 − 0.5″ FWHM, top left) to AO imaging on 8 − 10 m class telescopes (50 − 60 mas FWHM, ≈300 − 500 μas astrometry (topright), showing a LucyRichardson deconvolved and beamrestored AO image), to current interferometric imagery with GRAVITY/VLTI (bottom panels, 2 × 4 mas FWHM resolution, 30 − 100 μas astrometry). Bottom right: central 0.2″ region centered on Sgr A*(IR) on May 30, 2021 (see Appendix A.2, GRAVITY Collaboration 2022). Over 2021, the central GRAVITY field of view was dominated by four stars (S29, S55, S62, and S300) with Kband magnitudes between 16.7 and 19.2, in addition to variable emission from Sgr A*(IR) itself. The dashed circle indicates the fieldofview of GRAVITY, defined by the Gaussian acceptance profile of the pickup fibres in the instrument. Comparing in the bottom left panel three such images taken on March 30 (red), May 30 (green), and July 26 (blue), the orbital motions of all four stars are easily seen, topped by the ≈8740 km s^{−1} velocity of S29 at its pericenter on 2021.41 (R_{peri} ≈ 100 AU). 

In the text 
Fig. 2.
Summary of the 1992–2021 astrometric data and best fitting orbits for the stars S2, S29, S38, and S55. The black cross marks the position of the compact radio source Sgr A* and mass center. The directions of projected orbital motions are marked by arrows. 

In the text 
Fig. 3.
Illustration of the fit results for f_{SP}. We show the key figures for S2. Two left panels: the residuals in RA (top) and Dec (bottom) between the GRAVITY (cyan filled circles, and 1σ uncertainties) and NACO data (blue filled circles) and the best GR fit (red curve, f_{SP} = 1, Rømer effect, plus special relativity, plus gravitational redshift and Schwarzschild precession), relative to the same orbit for f_{SP} = 0 (Kepler/Newton, plus Rømer effect, plus special relativity, plus redshift). The orbital elements for nonKeplerian orbits (i.e., with f_{SP} ≠ 0) are interpreted as osculating parameters at apocenter time, 2010.35. NACO and GRAVITY data are averages of several epochs. The grey bars denote averages over all NACO data near apocenter (2004–2013). Top right: the same for the residual orbital angle on the sky δϕ = ϕ(f_{SP} = 1)−ϕ(f_{SP} = 0). Bottom right: zoom into the 2005/2021 part of the orbit, plotted in the mass rest frame. The earlier orbital trace does not coincide with the current one due to the Schwarzschild precession. 

In the text 
Fig. 4.
Summary of the central mass distribution in the GC, within 20× the apocenter radius of the S2 orbit, ≈5″. The filled blue circle is the central mass of 4.30 × 10^{6} M_{⊙} (for R_{0} = 8277 pc, and its 1σ uncertainty of 12 000 M_{⊙}), which our fourstar fitting has established to lie within the 1200R_{S} pericenter of S29. The black arrow denotes the 3σ upper limit of M_{ext}(≤0.3″) of the sum of M_{•} and any extended Plummer mass of assumed scale radius 0.3″. The two open blue circles and two red filled squares show averages of enclosed masses within the semimajor axes of other Sstars and clockwise disk stars. The magenta dashed line is the sum of M_{•} and the extended stellar mass distribution from the literature (e.g., Genzel et al. 2010; Alexander 2017; Schödel et al. 2018; Baumgardt et al. 2018). All data are in excellent agreement with a point mass (the MBH Sgr A*) and a star cluster with a powerlaw density slope γ ≈ 1.6 (Genzel et al. 2003; GallegoCano et al. 2018), consisting of mainsequence stars and a small contribution of giants. No extra component from dark matter or an intermediate mass black hole > 10^{3} M_{⊙} (GRAVITY Collaboration 2020) is required or compatible with the data. 

In the text 
Fig. B.1.
Corner plot of the full 32dimensional parameter space of the fourstar fit, consisting of (in order) the central mass M_{•}, the offsets x_{0}, y_{0}, the distance R_{0}, the velocity offsets vx_{0}, vy_{0}, vz_{0}, the precession parameter f_{SP}, followed by four times six orbital elements for each of the four stars used, in the order a, e, i, Ω, ω, t_{peri} for S2, S29, S38, S55. 

In the text 
Fig. C.1.
Visualization of the orbital (astrometric) data used in determining the mass distribution in the GC. The panels group the stars according to the semimajor axes of their orbits, as indicated in the top left in each panel. For reference, we show in each panel the orbits from the fourstar fit (S2: red line, S29 violet, S38 black, S55 blue), Left: Orbital data for S9, S13, S18 and S21. Middle: S4, S12, S31 and S42. Right: S66, S67, S87, S96 and S97. These data are complemented by multiepoch spectroscopy for the orbital fitting. 

In the text 
Fig. C.2.
Same data and RA/Decresidual plots as the left panels in Figure 3, with the dashed red curve denoting the f_{SP} = 1 GR curve for the best fitting orbit and mass. In addition, we show orbital models with the same central mass, distance and orbital parameters but now adding an extended mass component assumed to have a Plummer shape (Gillessen et al. 2017) showing the impact of adding a Plummer mass of M_{ext} within the 0.25″ apocenter radius of S2. Black, orange and blue solid curves show the changes expected if this extended Plummer mass is 0.1, 0.3 and 0.6% of M_{•} (4.4 × 10^{3}, 8.9 × 10^{3} and 1.78 × 10^{4}M_{⊙} within the apocenter of S2, R_{apo} = 0.24″). Formal fitting shows that such an extended mass greater than about ≈0.1% of M_{•} is incompatible with the data. 

In the text 
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