A geometric distance measurement to the Galactic Center black hole with 0.3% uncertainty

We present a 0.16% precise and 0.27% accurate determination of R0, the distance to the Galactic Center. Our measurement uses the star S2 on its 16-year orbit around the massive black hole Sgr A* that we followed astrometrically and spectroscopically for 27 years. Since 2017, we added near-infrared interferometry with the VLTI beam combiner GRAVITY, yielding a direct measurement of the separation vector between S2 and Sgr A* with an accuracy as good as 20 micro-arcsec in the best cases. S2 passed the pericenter of its highly eccentric orbit in May 2018, and we followed the passage with dense sampling throughout the year. Together with our spectroscopy, in the best cases with an error of 7 km/s, this yields a geometric distance estimate: R0 = 8178 +- 13(stat.) +- 22(sys.) pc. This work updates our previous publication in which we reported the first detection of the gravitational redshift in the S2 data. The redshift term is now detected with a significance level of 20 sigma with f_redshift = 1.04 +- 0.05.


Introduction
Measuring distances is a key challenge in astronomy. While many distance estimators rely on secondary calibration methods, the basis for the whole distance ladder is laid by a few methods. These methods all compare an angular scale on sky with a size that is known in absolute terms. Foremost is of course the parallax method. It compares an observed reflex motion on the sky, measured in angular units with the size of Earth's orbit. Recently, Gaia improved the number and quality of available parallaxes substantially (Gaia Collaboration 2018). However, Gaia works in the optical and at moderate spatial resolution and does not provide any parallaxes toward the crowded and highly dust-obscured center of the Milky Way. The extinction can be overcome by observing at longer wavelengths, in the near-infrared (NIR; 1−5 µm). Very large telescopes with adaptive optics (AO), and recently, interferometry between large telescopes (Gravity Collaboration 2017), overcome the stellar crowding. This allowed us to determine the orbits of 40 stars around the central massive black hole with periods between 13 GRAVITY has been developed by a collaboration of the Max Planck Institute for Extraterrestrial Physics, LESIA of Paris Observatory/CNRS/UPMC/Univ. Paris Diderot and IPAG of Université Grenoble Alpes/CNRS, the Max Planck Institute for Astronomy, the University of Cologne, the Centro de Astrofísica e Gravitaçâo, and the European Southern Observatory. and a few thousand years ). These stars offer another direct method of determining a distance. The distance to the Galactic center (GC), R 0 , can be determined by comparing the radial velocities (measured in km/s) of these stars with their proper motions (measured in mas/yr). The measurement is direct because this can be done for individual stellar orbits, as opposed to using a sample of stars together with a dynamical model like in van de Ven et al. (2006) for the globular cluster ω Cen or in Chatzopoulos et al. (2015) for the Milky Way nuclear cluster.
Most suitable for the orbit method is the star S2 on a 16year orbit (the second shortest period known so far, Meyer et al. 2012), with a semimajor axis a ≈ 125 mas. S2 has an apparent K-band magnitude of m K ≈ 14, which is bright enough for spectroscopy. It is a massive, young main-sequence B star (Ghez et al. 2003;Martins et al. 2008;Habibi et al. 2017) that offers a few atmospheric absorption lines in the observable parts of the spectrum. Several works used S2 to measure the distance to the GC. The first measurement was in Eisenhauer et al. (2003), who reported R 0 = 7940 ± 420 pc. Eisenhauer et al. (2005) updated this value to R 0 = 7620 ± 320 pc. Ghez et al. (2008) reported R 0 = 8400 ± 400 pc, and Gillessen et al. (2009) differed slightly with R 0 = 8330 ± 350 pc. More recently, Boehle et al. (2016) measured R 0 = 7860 ± 140 ± 40 pc, and Gillessen et al. (2017) obtained R 0 = 8320 ± 70 ± 140 pc. Here and in what follows, the first error is statistical, and the second error is systematic. All these measurements rely on AO data. For general recent overviews of R 0 determinations, see Genzel et al. (2010) and Bland-Hawthorn & Gerhard (2016).
The star S2 passed the pericenter of its orbit in May 2018, an event that we followed in detail with astrometry and spectroscopy (Gravity Collaboration 2018a). The primary goal of these observational efforts was the detection of relativistic effects in the orbital motion. However, the data also allow for an unprecedentedly accurate measurement of R 0 because of the large swing in radial velocity (from +4000 to −2000 km s −1 ) and the large orbital phase that was covered in 2018. Gravity Collaboration (2018a) presented the detection of the gravitational redshift from Sgr A* in the S2 spectra. Our previous analysis included data up to end of June 2018. It addressed the question whether the gravitational redshift and Doppler terms are in agreement with the predictions of Einstein's theory of relativity. At the same time, our orbital solution also included the most precise determination of R 0 so far, R 0 = 8122 ± 31 pc, where the error is only statistical. Several authors studying the Milky Way structure have used this result (McGaugh 2018;Drimmel & Poggio 2018;Mróz et al. 2019;Eilers et al. 2019). Here, we update our value for R 0 , using data up to the end of 2018, and we apply the relativistic corrections assuming General Relativity is correct. This yields one fit parameter less. We also investigate the systematic error on R 0 from our measurement, which we did not consider in Gravity Collaboration (2018a).

Data
Gravity Collaboration (2018a) used 45 AO-based astrometric points (after down-sampling), 77 radial velocities, and 30 GRAVITY interferometric data points. The present study adds ten epochs of radial velocity measurements from late June 2018 to late September 2018, and ten epochs of GRAVITY astrometry. Furthermore, we reanalyzed our radial velocity data from SIN-FONI and the GRAVITY astrometry, implementing an improved understanding of the respective systematic effects. This also led to a slightly different data selection and different grouping of the observations.
For the SINFONI data we revisited the wavelength calibration, yielding an improved wavelength dispersion solution. Where possible, we determined the radial velocities by template fitting. The uncertainties are a combination of formal fit error, wavelength error, and the error introduced by selecting a certain extraction mask in the field of the integral field unit. For the details see Appendix A.
For the GRAVITY data we replaced the manual frame selection with an objective outlier-rejection and included the (minor) effect of atmospheric differential refraction. The data analysis includes data selection, binary fitting, correction for atmospheric refraction, outlier rejection, nightly averaging, correction for effective wavelength, adding systematic errors, and error scaling. We report the details in Appendix B.
Overall, our new data set consists of 169 AO-based astrometry points between 1992 and 2019, 91 radial velocities between 2000 and 2019, and 41 GRAVITY-based astrometry points in 2017 and 2018.
Our AO data set samples the on-sky motion of S2 at high cadence. The distance between subsequent data points is typically smaller than the size of the point spread function. Any confusion event with unrecognized faint stars thus might affect several data points, leading to correlated measurements. As in Gravity Collaboration (2018a), we therefore down-sampled the AO data set into intervals of constant arc length on the sky, and we down-weighted these AO data by a factor two in order to take the additional uncertainty due to unseen confusion events into account. Furthermore, we omitted the 2018 data where additional confusion with Sgr A* affects the data, leading to 48 AO-based astrometric data points. We also developed a different approach for the same problem, namely a noise model (see Sect. 3). This gives a second data set, in which we used all 169 AO-based astrometric points.

Analysis
We used the same techniques as in Gravity Collaboration (2018a) and Gillessen et al. (2017). The analysis essentially consists of one step: determining the best-fit orbit for the data given, and the corresponding uncertainties. We employed a χ 2minimization to determine the best-fit, and for the uncertainties, we used the standard error matrix approach, a Markov chain Monte Carlo (MCMC) sampler and a bootstrapping technique. The latter bootstraps an artificial data set by drawing from the original data separately for the AO astrometry, the radial velocities, and the GRAVITY data. In order to avoid problems that might arise because the AO points are correlated, we used the down-sampled data set for the bootstrapping.
For a different approach with the AO data, we implemented a noise model of the type presented in Plewa & Sari (2018) for the AO-based astrometry. Such a model has the advantage that it estimates the additional amount of error and the correlation length from the data themselves, avoiding any prior choices on how to treat the data. In our implementation, we exchanged the temporal correlation length of Plewa & Sari (2018) with a spatial one. The underlying reason for a correlation between different data points is confusion with unseen stars that can be described naturally by a length scale in the image plane. Because S2 has a widely varying proper motion, a temporal correlation length is less suited. This model adds two additional fit parameters: the spatial correlation length and the typical confusion amplitude, which correspond to the down-sampling and down-weighting in the other data set. We did not exclude all 2018 data for this data set, but only the epochs at which Sgr A* apparently affected the position measurements, as visible in an elongated source structure or excess flux of S2. We also analyzed a third data set excluding all AO astrometry. Perhaps somewhat surprisingly, the two years of GRAVITY data already are the much stronger constraint for the orbit compared to the past 27 years of AO imaging data.
Compared to the analysis in Gravity Collaboration (2018a), we included in the calculation of the transverse Doppler effect the apparent proper motion of Sgr A* to the southwest of (−3.151, −5.547) mas yr −1 , a reflex of the solar motion around the Milky Way center (Reid & Brunthaler 2004). This corresponds to v ≈ 250 km s −1 , while S2 at pericenter reaches an onsky motion of v S2 ≈ 7320 km s −1 . Because in the Doppler formula a term of type (v S2 + v ) 2 ≈ v 2 S2 (1 + 2v /v S2 ) occurs, the proper motion of Sgr A* leads to a small but noticeable correction. We parameterized the strength of the combined redshift and transverse Doppler effect with an artificial parameter f redshift such that f redshift = 0 corresponds to classical physics, while f redshift = 1 corresponds to the effects occurring as predicted by General Relativity. Including the proper motion of Sgr A* induces a change of ∆ f redshift = + 0.038, and a change in distance of ∆R 0 = +6 pc.

Distance R 0 to the Galactic center
If the fit has as few free parameters as possible, the estimate for R 0 is the most precise. We therefore assumed that General L10, page 2 of 10 The GRAVITY Collaboration: A geometric distance measurement to the Galactic center black hole with 0.3% uncertainty Notes. The parameters x 0 , y 0 , vx 0 , and vy 0 describe the location and motion of the mass in the coordinate system of the AO data in RA and Dec. Because GRAVITY directly measures the separation vector, we do not need to include such coordinate system offsets for the GRAVITY data. The third velocity vz 0 is the offset of the motion in the radial direction along the line of sight, the negative sign means a blueshift or a motion toward the observer. The parameters (a, e, i, Ω, ω, t peri ) are the classical orbital elements semimajor axis, eccentricity, inclination, position angle of ascending node, longitude of pericenter, and the epoch of pericenter passage. The orbital elements are defined as the osculating orbital elements at t = 2010.0, i.e., the conversion to position and velocity is done at that epoch assuming a Kepler orbit. Relativity holds and fixed the parameter f redshift = 1. We further used the Rømer delay and included the first-order correction from the Schwarzschild metric. The coordinate system parameters only apply to the AO astrometry because GRAVITY directly measures the vector S2 -Sgr A*. We list our best-fit results in Table 1 and show the best fit in Fig. 1. The error bars we report are the formal fit errors from the error matrix. The three data sets yield completely consistent parameters within the formal uncertainties. The reduced χ 2 values by construction of the errors are close to 1 (Appendices A and B).
The noise model has two additional free parameters, the noise amplitude σ = 0.83 ± 0.15 mas and the spatial correlation length λ = 21.2 ± 3.8 mas. These numbers define by how much a certain data point is expected to be off from the model, given the other data. The correlation length is on the same order of magnitude as the AO point spread function radius, and the amplitude is reasonable. Our best-fit σ corresponds to a perturbing star of m K ≈ 17 at a distance of our best-fit λ (Plewa & Sari 2018).
Using the MCMC sampler, we obtained the full 13dimensional posterior distribution. All parameters are well constrained, and Fig. 2 shows the diagrams with the strongest parameter correlations: mass versus R 0 , semimajor axis versus R 0 , and inclination versus R 0 . The most probable value agrees with the best-fit value, and the 1σ uncertainty from the posterior is 13 pc, which is fully consistent with the estimate from the error matrix. We furthermore estimated our errors by bootstrapping (and refitting each artificial data set). For this, we used the down-sampled data set because here the most important correlation between data points is removed. Figure 3 shows the resulting distribution for N = 20 000 bootstraps. The most likely value agrees with the best-fit value, and so do the error bars: R 0 = 8178 +13 −12 pc. Figure 4 shows the normalized residual (residual divided by the error) distributions for each of the three subsets of data and for the whole data set. The distributions are well behaved and reasonably close to a Gaussian with mean 0 and width 1.
The size of the R 0 error of 13/8178 ≈ 0.16% is comparable to what a simple estimate yields. R 0 is directly related to the ratio of proper motion (arc length divided by time) and radial velocity. The most constraining part of the orbit is the pericenter swing, which we followed with GRAVITY in 2017 and 2018.
-The arc length is ≈150 mas, more than 1000× larger than the median 2D error of the 41 GRAVITY points 1 . The astrometric precision is thus at the 0.01% level and does not contribute significantly to the statistical error.   2300 km s −1 . The spectroscopic precision is thus at the 0.1% level. It dominates the measurement error, and it is of the same magnitude as the actual statistical error on R 0 . We conclude that R 0 = 8178 ± 13 stat. pc. However, we still lack an estimate for the systematic error.

Systematic errors
Our estimate for R 0 is direct and as such does not depend on intermediate calibration steps. Any systematic error is directly related to how accurately we understand the instruments we use, that is, how accurate are the on-sky positions we measure and how accurate are the radial velocities. Figure 2 shows the strongest parameter correlations for R 0 from the posterior distribution of the 13-dimensional fit. They are with mass, semimajor axis, and inclination. These correlations can be understood qualitatively.
-R 0 is inversely proportional to the semimajor axis a. A biased determination of a in angular units would bias R 0 because the radial velocity data determine a in absolute units; for S2, a ≈ 1023 AU. The slope of the correlation in Fig. 2 (middle) confirms this, R 0 × a ≈ 1023 AU. The instrumental reason why a could be biased is an error in the image scale. A scale error of 1% would imply a distance error of ≈80 pc. at the best-fit R 0 , corresponding to ≈1 day pc −1 for the sensitivity to P. We note that the error we make in measuring the period due to the uncertainty in the underlying data is captured in the statistical error on R 0 . What matters here would be a systematic error in measuring time, which we can exclude at the relevant level. The mass-distance degeneracy is not a source of potential systematic error. We conclude that if the parameter degeneracies were to introduce a systematic error on R 0 , it would originate from an error in the astrometry. Furthermore, we note that the GRAV-ITY data completely dominate our astrometry (see Table 1), and that the AO + GRAVITY data sets yield the same result as the GRAVITY-only fit. This means that the uncertainty in the GRAVITY astrometry dominates the systematic error from the astrometry. In Appendix C we show that we estimate this uncertainty to be 19 pc or 0.24%.
When we used the GRAVITY astrometry, we assumed that the near-infrared (NIR) counterpart of Sgr A* is at the position of the center of mass. Gravity Collaboration (2018b) reported that the flaring emission from Sgr A* moves in a circular pattern with a radius of a few Schwarzschild radii, ≈50 µas. The flares are compact regions of transiently heated electrons that emit synchrotron light, powered probably by magnetic reconnection events (Dodds-Eden et al. 2009). They occur very close to the innermost stable circular orbit, and orbital motion of a few 10 µas has been proposed since their discovery Broderick & Loeb 2005;Hamaus et al. 2009). Observationally, the center of motion matches the position of the mass to within ≈50 µas. We used this as uncertainty on our assumption and estimated the effect on R 0 by artificially displacing the mass by that amount. This yields changes in R 0 of +8 pc, −8 pc, −6 pc, and +5 pc to the north, south, east, and west. We include 6 pc in the systematic error for the assumption that GRAVITY directly measures the separation vector between S2 and mass center.
With full coverage of the orbit, the measurement of R 0 is no longer degenerate with the offset vz 0 in radial velocity (Fig. 5, cf. Ghez et al. 2008). A general offset in the radial velocity would be absorbed fully into vz 0 , but it would not affect our measurement of R 0 . The zeroth order of the wavelength calibration is thus not a source of systematic error. The leading order could only be the first order, that is, the dispersion solution.
Our spectra are calibrated with a higher order polynomial, using multiple atmospheric lines in the same spectra as calibration points. From the residuals of our dispersion solution at these calibration points, we estimated the systematic uncertainty in the wavelength axis to 2.5 km s −1 over the range relevant for S2. Together with the mean absolute radial velocity in 2017 and 2018 (2300 km s −1 ), we obtain a systematic error of 0.11% or 9 pc.
Taken together, we thus estimate our systematic error on R 0 to be 22 pc. Our main result is R 0 = 8178 ± 13 stat. ± 22 sys. pc.
The statistical error is dominated by the measurement uncertainties of the radial velocities, and the systematic error by the GRAVITY astrometry.

Update on the gravitational redshift in S2
With the new data sets in hand, we repeated the posterior analysis of Gravity Collaboration (2018a) to determine the combined effect of gravitational redshift and transverse Doppler effect. Using an orbit model including the first-order correction due to the Schwarzschild metric and including the Rømer delay, we find f redshift = 1.047 ± 0.052 for the noise model fit and f redshift = 1.036 ± 0.052 when we use the down-sampled data set. Figure 6 shows the radial velocity residuals to the classical part of the true best-fit orbit. For this we set f redshift = 0 without refitting after fitting with f redshift = 1. We compared these residuals to the true model (i.e., with the effects turned on, f redshift = 1). We exclude that purely Newtonian physics can describe our data at a significance level of 20σ.

Distance estimate without radial velocities
Our GRAVITY measurement also provides the first direct distance measurement from orbital motion without the need for radial velocities. The key for this is the Rømer effect: The light travel time across the orbit causes astrometric points to appear slightly ahead or lagging behind the orbit, depending on whether S2 is in front of or behind Sgr A*. For a Keplerian orbit with astrometric data only and no light-time travel effect, the distance cannot be determined. The best-fit mass and distance are degenerate along a line M ∝ R 3 0 . Because the light travel time across the orbit between 2017 and 2018 (where we have GRAVITY data) is about three days and because we can detect the daily motion of S2 in the GRAVITY data, our astrometry breaks the degeneracy. Figure 7 (left) shows that this is indeed the case. The best-fit distance for this case is R 0 = 9.5 ± 1.5 kpc, consistent with our best estimate. To our knowledge, Anglada-Escudé & Torra (2006) were the first to propose this type of distance measurement, but we are not aware of an application anywhere so far.
If we were to ignore the Rømer effect for the purely astrometric data set, we would not obtain as a return a fully degenerate mass-distance relation. Instead, the fit then tries to become as small a distance as possible (Fig. 7, right), that is, in the sense of a limit, we obtain R 0 → 0. This is where the light travel time effect is minimal, as imposed by the wrong orbit model without Rømer delay. This just shows in a different way that our astrometry requires a finite speed of light and thus can estimate R 0 .

Discussion
The best estimate for R 0 from Bland-Hawthorn & Gerhard (2016) using only their set of ten independent best measurements that did not invoke Sgr A* is R 0 = 8210 ± 80 pc, in perfect agreement with our value. This means that Sgr A* is indeed at the center of the Milky Way bulge.
Our value of R 0 together with the proper motion of Sgr A* of 6.379 ± 0.026 mas yr −1 = 30.24 ± 0.12 km s −1 kpc −1 from Reid & Brunthaler (2004) Fig. 7. Posterior distributions for the data set without radial velocities. Left:usingthecorrectorbitmodel.Right:usinganorbitmodelthatneglects the Rømer effect. In this plot we allowed negative distances (and correspondingly negative masses) to avoid having a bound of the parameter space at 0, where the actual maximum of the distribution falls. and V is the peculiar solar motion toward l = 90 • . The error on Θ 0 + V is composed roughly equally of the error in the proper motion of Sgr A* and the uncertainty in R 0 . This constraint on Θ 0 + V is compatible with the recent determination from Hayes et al. (2018), who found Θ 0 + V = 253 ± 6 km s −1 from Gaia astrometry of the Sgr stream. Bland-Hawthorn & Gerhard (2016) estimate V = 11 ± 2 km s −1 , but to take into account the radial variations in the median v φ seen by Gaia Collaboration (2018), we used a total uncertainty of 4 km s −1 . Together with our estimate for Θ 0 + V this implies Θ 0 = 236.9 ± 4.2 km s −1 . From combining Gaia DR2 and APOGEE data, Eilers et al. (2019) found Θ 0 = 229 ± 6 km s −1 , where the error is the reported systematic uncertainty. Wegg et al. (2019) used Gaia DR2 and RR Lyrae stars to derive Θ 0 = 217±6 km s −1 . Using trigonometric parallaxes of high-mass star-forming regions, Reid et al. (2014) Another remarkable result is the fact that the offset in the radial velocity, vz 0 , is small and consistent with zero. The offset absorbs any possible systematic offset in the radial velocity.
-The surface gravity of S2 contributes ∆vz 0 = GM S2 /r S2 c = 1.6 km s −1 (Lindegren & Dravins 2003), where we used r S2 , the radius of S2, and M S2 , its mass, from Habibi et al. (2017). -The contribution of the Galactic potential can be approximated by ∆vz 0 = v 2 /c ln(R 0 /R S2 ), where v is the Sun's circular galactocentric speed and R S2 is the galactocentric radius of S2 (Lindegren & Dravins 2003). The approximation surely does not hold inside the sphere of influence of Sgr A* (≈3 pc), where the massive black hole dominates the potential. However, because of the logarithm in the expression, the actual effective value for R S2 does not matter strongly.
With v ≈ 230 km s −1 and R S2 = 3 pc, we obtain ∆vz 0 = 1.4 km s −1 , and when we use the apocenter distance R S2 = 0.009 pc, the number is ∆vz 0 = 2.4 km s −1 . Furthermore, an offset in vz 0 could be due to the intrinsic motion of Sgr A* with respect to the Milky Way. Reid & Brunthaler (2004) measured the motion of Sgr A* perpendicular to the Galactic plane to be 0.4 ± 0.9 km s −1 . For the third dimension, the motion along the Galactic plane, Reid et al. (2009) reported −7.2 ± 8.5 km s −1 , and the update in Reid et al. (2014) implies tighter constraints around 2−3 km s −1 . The expected "Brownian motion" of Sgr A* that is due to scattering with stars in its vicinity is even slightly smaller than these limits with 0.2 km s −1 (Chatterjee et al. 2002;Merritt et al. 2007).
The parameter vz 0 is the sum of these offsets. Our fit results and the two redshift terms yield a value of ≈−6 ± 6 km s −1 . The uncertainty on this number is larger than the fit error because of the systematic uncertainties, such as the actual value for the Galactic potential that is used for S2, but also the systematic uncertainties in the wavelength calibration. The most likely reason why the sum is small is that the summands are small. Under this hypothesis, we conclude that to within a few km/s, Sgr A* is at rest at the center of theMilkyWayandthattheLSRismovingtangentially.Thevalueis lower than might be expected from the combined effect of Galactic bar and spiral arms; however, their quantitative effect on the velocity streamlines at the solar position is not well known.
Our data very strongly constrain the angular diameter of Sgr A*. Because mass and R 0 are correlated, the constraint is stronger than what simple error propagation would yield. We find R S /R 0 = 10.022 ± 0.020 stat. ± 0.032| sys. µas. The combined uncertainty corresponds to 50 000 km at our R 0 . This sets a strong prior for the analysis of data obtained from global millimeter very long baseline interferometry that aims at resolving Sgr A* (Falcke et al. 2000;Doeleman et al. 2009).
A potential caveat of our analysis might be that the physical model of the orbit is too simple. So far, S2 did not reveal any signs of binarity. For GRAVITY, S2 is an unresolved point source (Gravity Collaboration 2017). The resolution of GRAV-ITY in GC observations is about 2.2 mas × 4.7 mas, excluding a source extension larger than or a companion farther away than ≈1 mas. Chu et al. (2018) used the radial velocity data of S2 and reported an upper limit of M companion sin i ≤ 1.6 M for periods between 1 and 150 days. Longer periods would not be stable against tidal break-up. Furthermore, the motion of either S2 or Sgr A* could be affected by as yet unknown massive objects in the GC. To some extent, such a perturbation can always be absorbed into the orbital elements (Gualandris et al. 2010), resulting in biased estimates for the parameters. According to our current knowledge, S2 is a suitable probe for R 0 . It is an ordinary massive main-sequence star of type B0-B3 (Ghez et al. 2003;Martins et al. 2008;Habibi et al. 2017). The atmospheric absorption lines we used are expected to be fair tracers of the motion of the star, together with its (unresolved) photocenter.
The value from Boehle et al. (2016), R 0 = 7.86 ± 0.14 ± 0.04 kpc, disagrees with our result. However, it comes from a combined fit of the stars S2 and S38. The S2-only result of these authors is R 0 = 8.02 ± 0.36 ± 0.04 kpc, which is completely consistent with our result. Furthermore, we note that combining different stars in the orbit fit tends to change the parameter mass and R 0 by rather large amounts  because small inconsistencies in the data sets are amplified by the fact that in the mass-R 0 plane two narrow, curved posterior distributions are combined. The statistical error of a combined fit does not catch this and could thus miss part of the true uncertainties.
Overall, we used accurate radial velocities from SINFONI and proper motions from GRAVITY of the star S2 as it orbits Sgr A* to set the absolute size of the orbit and determine the distance to the GC with unprecedented accuracy to R 0 = 8178 pc. The statistical error is only 13 pc and is dominated by the measurement errors of the radial velocities. The systematic error of 22 pc is dominated by the calibration uncertainties of the astrometry. Our analysis also demonstrates that the relative velocity of the LSR along the line of sight to Sgr A* is consistent with zero to within a few km/s, implying that Sgr A* is at rest in the GC and the LSR is moving tangentially. The addition of further SINFONI and GRAVITY data taken in 2018 also allowed us to increase the significance of the previously published measurement of the gravitational redshift caused by Sgr A* to 20σ.

Appendix A: Radial velocities from SINFONI
For the SINFONI data we improved the wavelength calibration. Explicitly, we modified our atmospheric line list that serves as reference for the wavelength calibration by excluding double lines or lines with a low signal-to-noise ratio (S/N) following the line atlas of Rousselot et al. (2000). We also improved the finetuning of the spectrum to the OH lines, leading to an improved wavelength dispersion solution. With these changes, we typically achieved a calibration error of below 2 km s −1 , measured by the residuals of the OH lines. With the improved data reduction, we re-reduced all available data since October 2004. The earlier data (two epochs in 2004 and one in 2003) were obtained during commissioning time and need a dedicated calibration procedure, which we did not repeat. We combined data from different nights when the expected velocity change was smaller than the calibration error. We omitted one measurement from 2008 with low S/N (from a single ten-minute exposure) and included one more epoch from 2009 and 2015 each and two more from 2010 and 2011 each. We split up data that previously were combined into one cube into two epochs in two occasions, in 2013 and 2015.
For spectra in which both the He-I line (2.112 µm) and Brackett-γ (2.166 µm) lines are unaffected by atmospheric residuals, we used template fitting to determine the radial velocities. For this we fit the long-time average S2 spectrum (Habibi et al. 2017) to the data. For spectra with sufficient S/N and no artifacts (e.g., from imperfect atmosphere correction), template fitting yields more accurate velocities. When either of the lines showed artifacts, we fit a double-Voigt profile to the other unaffected line.
The errors are a combination of fit error and wavelength calibration uncertainty. The fit error is obtained from the formal fit error σ, the S/N, and by varying the pixel selection. For the S/N-related error we established a relation between σ and S/N of σ ∝ S /N −0.92 . The 1 / S/N behavior is consistent with the uncertainty of a centroid fit (Fritz et al. 2010). To assess the impact of different background subtractions and extraction regions, we extracted eight spectra for each observation and determined the standard deviation of the radial velocities from the different masks. Because these three error estimates are strongly correlated, we used the largest of the three as fit error. We linearly added the wavelength calibration error to obtain a preliminary error. These preliminary errors establish the relative weight of the different radial velocities. Using these, we obtained a preliminary orbit fit, which showed that we overestimated the errors because the residuals around the best preliminary fit are on average 76.8% of the errors. Thus we rescaled the errors by that factor.
With this improvement of the SINFONI analysis, we reach an error of ≈7 km s −1 for the best data. The median error is 12.3 km s −1 , which is an improvement by 46% compared to the previous set of radial velocity data.

B.1. Data selection
We started from all observations of Sgr A* or S2 (793 exposures, each 30 × 10s = 5 min on source, i.e., a total of 66 h on source), regardless of observing conditions and instrument performance.
In 2017, S2 was still at a distance of 54−67 mas from Sgr A*, which is comparable to the photometric field of view (FWHM ≈ 65 mas), and too little flux from Sgr A* was injected into the fibres of the exposure pointing on S2 for a reliable interferometric binary signature (Perrin & Woillez 2019). We therefore only considered the observations centered on Sgr A* (261 exposures). We furthermore rejected all Sgr A* observations for which the instrument internal pupil control (Gravity Collaboration 2017) reported an error >6 cm for any of the telescopes (12 exposures) or for which the pointing of any telescope was too far from Sgr A* (83 exposures). We used a box spanning ∆RA = −45 . . . 10 mas, ∆Dec = −30 . . . 30 mas around Sgr A*, which especially avoided pointings toward the opposite side of S2. This selection keeps 166 exposures in 2017.
For 2018, we had 373 exposures on Sgr A*. Again, we rejected exposures with pupil errors >6 cm (18 exposures). Because of a newly introduced laser-metrology guiding with substantially improved pointing accuracy, we rejected exposures already when the estimated pointing error for any telescope was outside ∆RA/∆Dec = −10 . . . 10 mas around Sgr A* (35 frames). Because S2 was always closer than 23 mas to Sgr A* during our March-June 2018 observing campaigns, both sources were well within the photometric field of view. We also used the 43 exposures centered on S2 that were obtained during this period. Out of these observations, we rejected three exposures because of a pupil error > 6 cm, and five exposures because of a pointing error larger ∆RA/∆Dec = −10 . . . 10 mas. This yields a total of 355 exposures in 2018.

B.2. Binary fitting and correction for atmospheric refraction
In a second step, three independent subgroups fit the individual exposures with a binary model as described in Gravity Collaboration (2018a,b), using three different codes ("Waisberg (W)", "Pfuhl (P)", "Rodriguez-Coira (R)"). The codes differ in detail in the relative weighting of closure-phases, visibilities, and square visibilities, the free fit parameters (e.g., color of Sgr A* or flux ratio per telescope), and the numerical implementation (e.g., least-squares minimization or MCMC), but give overall consistent results for the binary separations.
We furthermore corrected each binary fit for the differential atmospheric refraction between the comparably "blue" S2 and "red" Sgr A* (see Appendix A7.4 of Gravity Collaboration 2018b). Because Sgr A* was in its faint quiescent state for most of our observations, we used the redder low-flux spectral index S v ∝ v −1.6 from Witzel et al. (2018) for the subsequent analysis. With S v ∝ v 2 for S2, and for the given effective spectral resolution of 127 nm (low-resolution mode of GRAVITY), the difference in effective wavelength between S2 and Sgr A* is ∆λ = 2.2 nm, and the resulting atmospheric differential refraction is ∆R = 45 µas nm −1 × ∆λ tan z = 99 µas tan z, where z is the zenith distance. Because we typically observed the GC close to zenith, the atmospheric differential refraction was on average only 30 µas, and often with opposite signs during a night, which resulted in a mean correction of ∆RA = −1 µas and ∆Dec = −5 µas.

B.3. Outlier rejection and nightly averaging
For each of the three sets of binary fits we determined a preliminary orbit for error scaling and outlier rejection. We rejected observations for which the residuals were outside the 80% quantile constructed in the 2D error-normalized position residual plane 2 . The final data set contains 818 (W), 795 (P), and 737 (R) binary fits, corresponding to about 400 exposures of five minutes each, that is, about 33 h on source. We combined these and derived nightly (error-weighted) mean and standard errors (with variance weights). Only in the few cases when we had fewer than ten binary-fits per night (26/27 March 2017, 28/29 March 2017, 10/11/12 July 2017) did we combine several nights to one average. The statistical 1D astrometric error of these combined nightly averages is between 10 and 110 µas.
B.4. Correction for effective wavelength, systematic error, and final error scaling In a last step we corrected the nightly average separation for the effective wavelength shift of 2.3 nm (0.1%) between the wavelength calibration with our 2800 K calibration lamp and the very red highly dust obscured S2/Sgr A* data (see Appendix A7.2 in Gravity Collaboration 2018b).
To account for the systematic error in the wavelength calibration, which we estimate to be 1/20 detector pixel, or equivalently, 2.5 nm, we added in square the corresponding scale error of 0.11%. This error in the effective wavelength translated into an astrometric error of about 10 µas for the time around peripassage, and up to ∆RA = 66 µas and ∆Dec = 33 µas for March 2017, when the S2-Sgr A* distance was largest in our observations.
Finally, to account for unknown additional errors, we scaled the GRAVITY astrometric errors by a factor 2.2 to match the residuals of a best-fitting preliminary orbit. The resulting astrometric errors around the S2 peri-passage in our data from 24 April to 27 June 2018 are ∆RA = 22−101 µas and ∆Dec = 38−112 µas, with a mean of 51 µas and 60 µas, respectively.