 | Progress in stellar astrometric imaging in the GC, from seeing limited observations ($\approx 0.4-0.5''$ FWHM, top left) to AO imaging on $8-10\,$m class telescopes ($50-60\,$mas FWHM, $\approx 300-500\,\mu$as astrometry (top-right), showing a Lucy-Richardson deconvolved and beam-restored AO image), to current interferometric imagery with GRAVITY / VLTI (bottom panels, $2\times4$ mas FWHM resolution, $30-100\,\mu$as astrometry). Bottom right: Central $0.2''$ region centered on Sgr~A*(IR) on May 30, 2021 (see Appendix~\ref{sec:A2}, \citealt{JULIA}). Over 2021, the central GRAVITY field of view was dominated by four stars (S29, S55, S62, and S300) with K-band magnitudes between 16.7 and 19.2, in addition to variable emission from Sgr~A*(IR) itself. The dashed circle indicates the field-of-view of GRAVITY, defined by the Gaussian acceptance profile of the pick-up 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 $\approx8740\,$km/s velocity of S29 at its pericenter on 2021.41 ($R_\mathrm{peri} \approx 100\,$AU). |
 | 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. |
 | Illustration of the fit results for $f_\mathrm{SP}$. We show the key figures for S2. The two left panels show the residuals in RA (top) and Dec (bottom) between the GRAVITY (cyan filled circles, and $1\sigma$ uncertainties) and NACO data (blue filled circles) and the best GR fit (red curve, $f_\mathrm{SP}=1$, R{\o}mer effect, plus special relativity, plus gravitational redshift and Schwarzschild precession), relative to the same orbit for $f_\mathrm{SP}=0$ (Kepler/Newton, plus R{\o}mer effect, plus special relativity, plus redshift). The orbital elements for non-Keplerian orbits (i.e., with $f_\mathrm{SP} \ne 0$) 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 $\delta \phi =\phi(f_\mathrm{SP}=1)-\phi(f_\mathrm{SP}=0)$. 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. |
 | Summary of the central mass distribution in the GC, within 20$\times$ the apocenter radius of the S2 orbit, $\approx 5''$. The filled blue circle is the central mass of $4.30\times10^6 M_\odot$ (for $R_0=8277\,$pc, and its $1 \sigma$ uncertainty of $12,000 M_\odot$), which our four-star fitting has established to lie within the $1200 R_\mathrm{S}$ pericenter of S29. The black arrow denotes the $3\sigma$ upper limit of $M_\mathrm{ext}(\le0.3'')$ of the sum of $M_\bullet$ 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 semi-major axes of other S-stars and clock-wise disk stars. The magenta dashed line is the sum of $M_\bullet$ and the extended stellar mass distribution from the literature (e.g., \citealt{2010RvMP...82.3121G, 2017ARA&A..55...17A, 2018A&A...609A..27S, 2018A&A...609A..28B}). All data are in excellent agreement with a point mass (the MBH Sgr~A*) and a star cluster with a power-law density slope $\gamma\approx1.6$ \citep{2003ApJ...594..812G, 2018A&A...609A..26G}, consisting of main-sequence stars and a small contribution of giants. No extra component from dark matter or an intermediate mass black hole $>10^3 M_\odot$ \citep{2020A&A...636L...5G} is required or compatible with the data. |
 | Corner plot of the full 32-dimensional parameter space of the four-star fit, consisting of (in order) the central mass $M_\bullet$, the offsets $x_0$, $y_0$, the distance $R_0$, the velocity offsets $vx_0$, $vy_0$, $vz_0$, the precession parameter $f_\mathrm{SP}$, followed by four times six orbital elements for each of the four stars used, in the order $a$, $e$, $i$, $\Omega$, $\omega$, $t_\mathrm{peri}$ for S2, S29, S38, S55. |
 | Visualization of the orbital (astrometric) data used in determining the mass distribution in the GC. The panels group the stars according to the semi-major axes of their orbits, as indicated in the top left in each panel. For reference, we show in each panel the orbits from the four-star 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 multi-epoch spectroscopy for the orbital fitting. |
 | Same data and RA-/Dec-residual plots as the left panels in Figure~\ref{fig4}, with the dashed red curve denoting the $f_\mathrm{SP}=1$ 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 \citep{2017ApJ...837...30G} showing the impact of adding a Plummer mass of $M_\mathrm{ext}$ 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 $M_\bullet$ ($4.4\times10^3$, $8.9\times10^3$ and $1.78\times10^4 M_\odot$ within the apocenter of S2, $R_\mathrm{apo}=0.24''$). Formal fitting shows that such an extended mass greater than about $\approx 0.1$\% of $M_\bullet$ is incompatible with the data. |