Improved GRAVITY astrometric accuracy from modeling of optical aberrations
GRAVITY Collaboration, R. Abuter, A. Amorim, M. Bauböck, J.P. Berger, H. Bonnet, W. Brandner, Y. Clénet, R. Davies, P.T. de Zeeuw, J. Dexter, Y. Dallilar, A. Drescher, A. Eckart, F. Eisenhauer, N.M. Förster Schreiber, P. Garcia, F. Gao, E. Gendron, R. Genzel, S. Gillessen, M. Habibi, X. Haubois, G. Hei?el, T. Henning, S. Hippler, M. Horrobin, A. Jiménez-Rosales, L. Jochum, L. Jocou, A. Kaufer, P. Kervella, S. Lacour, V. Lapeyrère, J.-B. Le Bouquin, P. Léna, D. Lutz, M. Nowak, T. Ott, T. Paumard, K. Perraut, G. Perrin, O. Pfuhl, S. Rabien, G. Rodríguez-Coira, J. Shangguan, T. Shimizu, S. Scheithauer, J. Stadler, O. Straub, C. Straubmeier, E. Sturm, L.J. Tacconi, F. Vincent, S. von Fellenberg, I. Waisberg, F. Widmann, E. Wieprecht, E. Wiezorrek, J. Woillez, S. Yazici, A. Young, G. Zinsınst{9}
AAstronomy & Astrophysics manuscript no. gravity-phasemaps © ESO 2021January 29, 2021
Improved GRAVITY astrometric accuracy from modeling of opticalaberrations
GRAVITY Collaboration ‹ : R. Abuter , A. Amorim , , M. Bauböck , J.P. Berger , , H. Bonnet , W. Brandner ,Y. Clénet , R. Davies , P.T. de Zeeuw , , J. Dexter , , Y. Dallilar , A. Drescher , , A. Eckart , , F. Eisenhauer ,N.M. Förster Schreiber , P. Garcia , , F. Gao , E. Gendron , R. Genzel , , S. Gillessen , M. Habibi , X. Haubois ,G. Heißel , T. Henning , S. Hippler , M. Horrobin , A. Jiménez-Rosales , L. Jochum , L. Jocou , A. Kaufer ,P. Kervella , S. Lacour , V. Lapeyrère , J.-B. Le Bouquin , P. Léna , D. Lutz , M. Nowak , , T. Ott , T. Paumard ,K. Perraut , G. Perrin , O. Pfuhl , , S. Rabien , G. Rodríguez-Coira , J. Shangguan , T. Shimizu , S. Scheithauer ,J. Stadler , O. Straub , C. Straubmeier , E. Sturm , L.J. Tacconi , F. Vincent , S. von Fellenberg , I. Waisberg , ,F. Widmann , E. Wieprecht , E. Wiezorrek , J. Woillez , S. Yazici , , A. Young , and G. Zins (A ffi liations can be found after the references) January 29, 2021
ABSTRACT
The GRAVITY instrument on the ESO VLTI pioneers the field of high-precision near-infrared interferometry by providing astrometry at the10 ´ µ as level. Measurements at such high precision crucially depend on the control of systematic e ff ects. Here, we investigate how aberrationsintroduced by small optical imperfections along the path from the telescope to the detector a ff ect the astrometry. We develop an analytical modelthat describes the impact of such aberrations on the measurement of complex visibilities. Our formalism accounts for pupil-plane and focal-planeaberrations, as well as for the interplay between static and turbulent aberrations, and successfully reproduces calibration measurements of a binarystar. The Galactic Center observations with GRAVITY in 2017 and 2018, when both Sgr A* and the star S2 were targeted in a single fiber pointing,are a ff ected by these aberrations at a level of less than 0.5 mas. Removal of these e ff ects brings the measurement in harmony with the dual beamobservations of 2019 and 2020, which are not a ff ected by these aberrations. This also resolves the small systematic discrepancies between thederived distance R to the Galactic Center reported previously. Key words. instrumentation: high angular resolution, instrumentation: interferometers, methods: data analysis, galaxy: center, galaxy: funda-mental parameters
1. Introduction
The distance to the Galactic Center (GC), R , can be measureddirectly from stellar orbits around Sgr A*, the radio source asso-ciated with the GC massive black hole (MBH) (see e.g. Genzelet al. 2010 and Bland-Hawthorn & Gerhard 2016 for a recentoverview of alternative methods). To this end, the star’s propermotion, given in angle per unit time, is compared to its radialvelocity, obtained in absolute length per units time from spectro-scopic observations. The GC distance then follows directly as ascaling parameter between the two measurements. Most suitedto measure R is S2, a massive young main sequence B-star ona 16-year orbit with semi-major axis a »
125 mas and appar-ent K-band magnitude m k »
14 (Ghez et al. 2003; Eisenhaueret al. 2005; Martins et al. 2008; Gillessen et al. 2009a, 2017;Habibi et al. 2017). During its pericenter passage in 2018, S2 wasclosely monitored in astrometry and spectroscopy (Gravity Col-laboration et al. 2018; Do et al. 2019). In particular, the GRAV- ‹ GRAVITY is developed in a collaboration by the Max PlanckInstitute for extraterrestrial Physics, LESIA of Observatoire deParis / Université PSL / CNRS / Sorbonne Université / Université de Parisand IPAG of Université Grenoble Alpes / CNRS, the Max Planck Insti-tute for Astronomy, the University of Cologne, the CENTRA - Centrode Astrofisica e Gravitação, and the European Southern Observatory.Corresponding authors: J.Stadler (email [email protected]) and F.Widmann (email [email protected]).
ITY instrument (Gravity Collaboration et al. 2017) directly mea-sured the distance between S2 and Sgr A* during the fly-by athigh angular resolution of around 30 µ as. The combination ofultra-high astrometric precision from near-infrared interferome-try and the spectroscopic precision of À
10 km / s allowed to de-termine the GC distance at the unprecedented precision of ă ff erential complex visibilities. The instru-ment’s extremely high angular resolution of » µ as and100 µ as (Gravity Collaboration et al. 2017). However, the lat-est R measurement in Gravity Collaboration et al. (2020) in-dicates a possible systematic di ff erence with earlier determina-tions (Gravity Collaboration et al. 2018, 2019). While the shiftis small, of O p q only, it is nevertheless significant due to thehigh precision of the measurement.The di ff erence in the measured GC distance coincides with achange in the observing mode. GRAVITY observes the GalacticCenter with two di ff erent methods, depending on the separationbetween Sgr A* and S2. Close to pericenter passage, i.e. in 2017and 2018, the sources are detected simultaneously in a single Article number, page 1 of 17 a r X i v : . [ a s t r o - ph . GA ] J a n & A proofs: manuscript no. gravity-phasemaps fiber pointing in the so-called single-beam mode. In later epochs,their separation exceeds the fiber’s field of view (FOV), and S2and Sgr A* are targeted individually. This is referred to as dual-beam mode.In single-beam mode, it is not possible to align the twosources with the fiber center. Hence, to further improve theGRAVITY astrometry, we conducted an analysis of how opticalaberrations a ff ect the visibility measurement across the full fieldof view. A similar concept of field-dependent errors already existin radio interferometry, where it is known as direction dependente ff ects (DDEs) (see e.g. Bhatnagar et al. (2008); Smirnov (2011);Smirnov & Tasse (2015); Tasse et al. (2018) and references therein). The DDEs can arise either at the instrument level from theantenna beam pattern or at the atmospheric level such as fromthe ionosphere. In particular for the latest generation of interfer-ometers (e.g. VLA, Meerkat, LOFAR) with a wide FOV and alarge fractional bandwidth DDEs cannot be neglected. However,to our knowledge there is no equivalent discussion in the contextof optical / near-IR interferometry.Indeed, our analysis shows that small optical imperfectionsin the beam combiner induce field-dependent phase errors thatreflect in the inferred binary separation. We developed an an-alytical model to describe this e ff ect, and verified it by appli-cation to a dedicated test-case observation. Applied to the GCobservations, the model induces a shift in the S2 relative posi-tion of order 0 . ´ . „ .
2. Formal description of static aberrations and theirimpact on visibility measurements
Static aberrations along the instrument’s optical path a ff ect themeasured visibilities by introducing a complex, field-dependentfactor for each telescope. We express this gain in its polar repre-sentation and decompose it into a phase map φ i p α q and an am-plitude map A i p α q . Here, the index i labels the telescope and α denotes positions in the image plane. Phase and amplitude mapslead to a modification of the observed complex visibilities V obs from the well-known van Cittert-Zernike theorem (c.f. Eq. 23).As we demonstrate in the following, they are given by V obs “ ş d α A i p α q A j p α q O p α q e ´ π i α ¨ b i , j { λ ` i p φ i p α q´ φ j p α qq bş d α A i p α q O p α q ş d α A j p α q O p α q , (1)where b i , j is the baseline vector between the two telescopes and O p α q denotes the intensity distribution of the observed astro-nomical object.In this section, we show how the phase- and amplitude-maps follow from optical aberrations. To this end, we start fromthe overlap integral, which determines the electromagnetic field from a single telescope arriving at the beam combiner. Subse-quently, we propagate the e ff ect of static aberrations from theoverlap integral to the measured complex visibility to arrive at arigorous derivation of Eq. (1). Finally, we account for the super-position of static and turbulent aberrations, to obtain a formalismwhich is applicable in realistic observation scenarios. Single mode fibers transport the light collected by each telescope E tel to the beam combiner instrument. The overlap integral be-tween light and the fiber mode E fib then determines the transmit-ted electric field (Neumann 1988), E p β q “ E fib p β q ˆ η “ E fib ˆ ż d ξ E tel p ξ q E ˚ fib p ξ q . (2)Here, we assume a normalized fiber mode ş d ξ | E fib p ξ q| “ ξ and β . Following the description of Perrin & Woillez(2019), the overlap integral is converted to the pupil plane bythe Parseval-Plancharel theorem, η “ ż d u F ´ “ E obj ‰ P p u q F ´ “ E ˚ fib ‰ p u q , (3)where F ´ denotes the inverse Fourier transform, i.e. transfor-mation from the image to the pupil plane, and E obj the lightemitted by the astronomical object. The latter is connected to F ´ p E tel q by multiplication with the pupil function P p u q , cor-responding to a convolution in the image plane. In the most sim-ple case of a single point source located at α , the light is de-scribed by a pure phase F ´ r E psobj s “ exp p´ π i u ¨ α q . Thepupil- and image-plane coordinates, ξ and u respectively, areFourier-conjugate to each other and chosen to be dimensionless.That is, any length scale in the pupil plane is given by λ u where λ refers to the wavelength and u “ | u | . For discussion, we convertthe dimensionless image plane coordinates ξ to the correspond-ing angular separation in UT observations. In an aberration-freescenario, the pupil function of a spherical telescope with diame-ter 2 r tel and central obscuration 2 r cent simply is˜ P p u q “ $&% u ď r cent { λ r cent ă u ď r tel { λ u ą r tel { λ . (4)Optical aberrations multiply the pupil function by a position-dependent, complex phase, and we here consider the case ofpurely static aberrations. These are characterized by an opticalpath di ff erence (OPD) d pup p u q in the pupil plane that can be ex-panded in terms of Zernike polynomials Z mn , d pup p u q “ n max ÿ n “ n ÿ m “´ n A mn Z mn p λ u { r tel q . (5)We adopt the convention that Z mn is dimensionless and the co-e ffi cient A mn corresponds to the term’s root mean square overthe unit circle. Defining the turbulence-free complex fiber modeapodised by the pupil function as Π e “ e π i d pup p u q{ λ ˜ P p u q F ´ “ E ˚ fib ‰ p u q , (6)the overlap integral reads η “ ż d u F ´ “ E obj ‰ p u q Π e p u q . (7) Article number, page 2 of 17RAVITY Collaboration: R. Abuter et al.: Improved GRAVITY astrometric accuracy from modeling of optical aberrations
The overlap integral obviously depends on the fiber profilewhich, for a perfectly aligned ideal single-mode fiber, is F ´ “ ˜ E ˚ fib ‰ “ exp ˜ ´ λ u σ ¸ . (8)GRAVITY was designed for optimal fiber injection (Pfuhl et al.2014), which is obtained for σ fib “ r tel ? { p π(cid:15) q (Wallneret al. 2002). Here, the parameter (cid:15) is of order unity and describesthe pupil shape.From comparison between model predictions and the cali-bration measurements in Sec. 3.2, we find that pupil-plane dis-tortions alone are not su ffi cient to describe the observed aber-ration pattern. We also need to account for optical errors in thefocal plane. Misalignment of the optical fiber, as well as higherorder aberrations at fiber injection, introduce a complex phase toEq. (8) and can distort the amplitude of the fiber profile.To illustrate the e ff ect of focal plane aberrations, we first con-sider the three types of misalignment depicted in Fig. 1: (A) Lat-eral misplacement of the fiber by p δ x , δy q , which in the pupilplane produces a phase slope ξ fib “ p δ x { f , δy { f q , with f beingthe focal length. (B) Fiber tilt by an angle ϕ fib “ p ϕ , ϕ q withrespect to the optical axis of the system which shifts the back-propagated fiber mode by u fib “ ϕ ¨ f { λ . And (C), a defocusor axial fiber misplacement by δ z that introduces an additionalphase curvature exp “ π i δ z λ { f u ‰ . Taking all three e ff ects intoaccount, the generalized fiber profile, projected to the pupil, is(Wallner et al. 2002) F ´ “ E ˚ fib ‰ “ F ´ “ ˜ E ˚ fib ‰ p u ´ u fib q (9) ˆ exp " ´ π i „ πδ z f p u ´ u fib q ´ ξ fib ¨ p u ´ u fib q * . By rearranging the phase term in the pupil plane, one can de-compose it into a piston, tip-tilt and defocus d pistonfib p u q “ ´ λ ˆ δ z λ f u fib ` ξ fib ˙ ¨ u fib ´ δ z f , (10) d tip ´ tiltfib p u q “ λ ˆ δ z λ f u fib ` ξ fib ˙ ¨ u , (11) d defocusfib p u q “ ´ δ z f ` λ u ´ ˘ . (12)The phase terms in Eqs. (10) to (12) thus a ff ects the overlap inte-gral in the same way as the lowest-order aberrations in d pup p u q .For the coordinate shift of the Gaussian profile, on the otherhand, there is no such correspondence, and it alters the way inwhich the optical fiber scans the pupil-plane aberrations.During GRAVITY observations, the misplacement term (A)depends on the performance of the fiber tracker but also on theuncertainty of the source position. In particular for exoplanet ob-servations, the latter can be sizable. Fiber tilt (B) is controlled bythe GRAVITY pupil tracker, and the adaptive optics calibrationis one example that impacts the defocus (C).While lateral misplacement (A) and defocus (C) describethe misplacement of a point-like fiber entrance, fiber tilt (B)accounts for the alignment of the fiber’s surface. This surfacecan exhibit irregularities beyond a simple tilt, which lead to aposition-dependent OPD in the focal plane, d foc p x q , as illus-trated in Fig. 1. Generally, aberrations from optical elements notconjugated to the pupil are field-dependent and known as Sei-del aberrations. In this context, d foc p x q arising in the focal plane d pup p u q d foc p x q o p t i c a l fi b e r δ x “ f ¨ ξ fib Af δ z C ϕ B Fig. 1.
Schematic depiction of the pupil and focal plane aberrationswhich enter the overlap integral. Both e ff ects in combination are re-quired to describe the aberration patterns observed in calibration mea-surements. The lowest-order aberrations in the pupil function are shownexplicitly, which are (A) lateral fiber misplacement, (B) fiber tilt and (C)defocus. Their e ff ect is further explained in the text. constitutes an extreme example. Still, it is possible to decom-pose the focal plane distortions into a series of Zernike polyno-mials, in analogy to Eq. (5). In this representation, axial fibero ff set (C) and fiber tilt (B) simply correspond to the lowest-order coe ffi cients, and higher-order terms amount to a general-ization of Wallner et al. (2002). Again, the phase terms intro-duced in F ´ “ ˜ E ˚ fib ‰ by higher order aberrations are degeneratewith d pup p u q , but the amplitude distortions need to be modeledexplicitly by themselves.Finally, for a single point source, located at α in the im-age plane, the overlap integral averaged over a time scale muchlonger than the source’s coherence time x ... y obj is x η ps y obj ż d u e ´ π i u ¨ α Π e p u q “ F r Π e s p α q . (13)Evaluation of the Fourier transform as function of α results ina two-dimensional complex map. We show several examples ofsuch maps in Fig. 2, assuming di ff erent Zernike coe ffi cients todetermine d pup p u q . The perfect Airy pattern, obtained in the limitof zero aberrations, exhibits zero phase in the central part anda phase jump by 180 ˝ at | α | » . λ { p r tel q . Anti-symmetricterms, such as tilt, coma and trefoil (not shown), only alter thelocation and shape of the phase jump, while defocus (not shown),astigmatism and higher order terms produce smooth phase gradi-ents. For a general choice of d pup p u q and in the absence of focal-plane aberrations, there is a saddle point where the phase mapsaverage to zero, but significant phase shifts are encountered atlarger radii.Focal-plane aberrations break the radial symmetry of thefiber profile. Still, if the perturbations are small enough, thephase maps show a saddle point, but its value di ff ers from zeroand its location may be shifted. In any case, the transmitted am-plitude is deformed and / or misplaced from the perfect Airy case.Pupil-plane aberrations typically widen the amplitude, whileimage-plane aberrations have the opposite e ff ect. They lead toa widening of the fiber in the pupil plane and correspondinglyto a narrower image-plane profile. The exact scaling relation forthe position of the Airy ring remains true only approximately inthe presence of higher-order aberrations such that maps at twodi ff erent wavelengths, λ and λ , can be related by x η ps y obj p α , λ q » x η ps y obj ˆ α λ λ , λ ˙ . (14) Article number, page 3 of 17 & A proofs: manuscript no. gravity-phasemaps
Fig. 2.
Example phase screens (top) and amplitude maps (bottom) in the image plane induced by low-order Zernike aberrations in the pupil planeat a wavelength of λ “ . µ m. From left to right the considered aberrations are: perfect Airy pattern, vertical tilt of 0 . µ m RMS, verticalastigmatism of 0 . µ m RMS, vertical coma of 0 . µ m RMS, and combination of astigmatism, coma and trefoil (with RMS 0 . µ m, 0 . µ m, and0 . µ m, respectively). The rightmost panel also considers an additional fiber tilt with 0 . µ m RMS. The overlap integral defines the electromagnetic wave transmit-ted to the beam combiner from each of the four telescopes. Af-ter pairwise beam combination, the complex visibilities are ob-tained from the inference pattern I i , j , I i , j “ ż d β A | E i p β q ` E j p β q| E obj (15) “ A | η i | E obj ` A | η j | E obj ` (cid:60) A η i η ˚ j E obj , (16)where i and j denote the telescopes involved in the measurementand I is the intensity. The complex pupil function enters each ofthese terms. Focusing on the single-telescope component first,we find from Eq. (7) A | η i | E obj “ ż d α F r Π e , i b Π e , i s p α q O p α q“ ż d α | F r Π e , i s p α q| O p α q , (17)where the b -operator denotes auto-correlation, and O p α q “ ˇˇ E obj p α q ˇˇ is the brightness distribution of the observed astro-nomical object which obeys A F ´ “ E obj ‰ p u q F ´ “ E obj ‰ ˚ p u q E obj “ F ´ r O p α qs p u ´ u q . (18) Similarly, the inference term is given by A η i η ˚ j E obj “ ż d α F r Π e , i b Π e , j s p α q O p α q e ´ π i α ¨ b i , j { λ “ ż d α F r Π e , i s p α q F r Π e , j s ˚ p α q O p α q e ´ π i α ¨ b i , j { λ , (19)where b i , j is the baseline vector.All optical aberrations discussed previously are encoded inthe back-projected apodized pupil, which is a complex field-dependent function. Expressing the pupil function in its polarrepresentation, F r Π e , i s “ A i p α q e i φ i p α q , (20)we refer to A i as the telescope-dependent "amplitude map" andto φ i as the "phase map". Note that these quantities are closelyrelated to the photometric and the interferometric lobes, L i p α q “ A i p α q and L i , j p α q “ A i p α q e i φ i p α q A j p α q e ´ i φ j p α q , (21)respectively.From the measured inference pattern, the complex visibili-ties are obtained as V obs p b i , j { λ q “ A η i η ˚ j E obj N cA | η i | E obj A | η j | E obj . (22)By contrast, in an ideal, aberration-free setting, the van-Cittert-Zernike theorem relates the complex visibilities to the object’sbrightness distribution V mod p b i , j { λ q “ ş d α O p α q e ´ π i α ¨ b i , j { λ ş d α O p α q . (23) Article number, page 4 of 17RAVITY Collaboration: R. Abuter et al.: Improved GRAVITY astrometric accuracy from modeling of optical aberrations
Comparison of Eq. (22) and Eq. (23) readily suggests that staticaberrations at fiber injection distort both the measured visibilityphases and amplitudes. We thus need to adapt the interferometricequation accordingly. To make this e ff ect even more explicit, wefirst consider the case of a single, unresolved object at position α , V obsps p b i , j { λ q “ L i , j p α q a L i p α q L j p α q e ´ π i α ¨ b i , j { λ . (24)In the aberration-free case, the phase and amplitude maps of ei-ther telescope are given by the perfect Airy pattern shown inthe very left panel of Fig. 2, and φ i { j p α q equals zero or 2 π .The presence of static aberrations introduces a phase shift by φ i p α q ´ φ j p α q . For an interferometric binary with positions α , α and flux ratio f bin the measured visibility becomes V obsbin “ L i , j p α q e ´ π i α ¨ b i , j { λ ` f bin L i , j p α q e ´ π i α ¨ b i , j { λ a r L i p α q ` f bin L i p α qs r L j p α q ` f bin L j p α qs . (25)Finally, for a generic extended object with an intensity distribu-tion O p α q the van-Cittert- Zernike theorem generalizes to theexpression stated at the beginning of this section, in Eq. (1) V obs “ ş d α L i , j p α q O p α q e ´ π i v ec α ¨ b i , j { λ bş d α L i p α q O p α q ş d α L j p α q O p α q . Single point sources typically are observed at the fiber cen-ter, where fiber injection is highest and the phase distortions areclose to zero. In situations where a very precise alignment is notpossible, like for example in exoplanet observations, the visibil-ities can pick up some small contribution from the phase maps.For binaries with a separation comparable to the fiber width, aconfiguration in which the phase and amplitude maps are irrel-evant cannot be obtained in principle. In this case, the e ff ect ofstatic aberrations needs to be modeled and corrected for in thedata analysis. To this point, we have not considered the e ff ect of time varyingphase aberrations. These are introduced by atmospheric turbu-lence or time-varying imperfections in the optical system suchas tip-tilt jitter from the adaptive optics. Their e ff ect is to mul-tiply the static pupil function by another, time dependent phase P e “ Π e e i φ turb p u , t q . (26)To see how time-dependent aberrations a ff ect the visibility mea-surement, we briefly recap the arguments of Perrin & Woillez(2019). Assuming that the detector integration time by far ex-ceeds the coherence time of phase fluctuations, the long-timeaverage x ... y turb over the telescope lobes is x L i p α qy turb “ A | F r P e , i s p α q| E ““ F ” p Π e , i b Π e , i q p u q e ´ D φ p u q ı , (27) @ L i , j p α q D turb “ x F r P e , i s p α qy turb @ F “ P e , j ‰ p α q D ˚ turb “ F “` Π e , i b Π e , j ˘ p u q e ´ σ φ ‰ , (28)where D φ p u q is the structure function of the turbulent phase(Roddier 1981), which saturates to 2 σ φ on large scales. Two as-sumptions underlie these expressions, first that the fluctuations are stationary and second that the baseline between the tele-scopes is long enough for the respective apertures to becomeuncorrelated. As in Perrin & Woillez (2019), we assume both tobe fulfilled.In the case of GRAVITY observations, atmospheric phasevariations across the telescope apertures are corrected by theadaptive optics system and the turbulent aberrations are domi-nated by tip-tilt jitter. Thus, the turbulent phase is φ turbi “ π t i p t q ¨ u , (29)where the two directions of t i p t q are independent and follow aGaussian distribution with zero mean and variance σ t . The struc-ture function then becomes D t p u q “ p πσ t u q , and the photo-metric lobe is given by x L i p α qy turb “ | F r Π e , i s p α q| f exp ˆ ´ α σ t ˙ , (30)where f denotes convolution. In case of the interferometric lobe,we further assume that the jitter is uncorrelated between tele-scopes which yields @ L i , j p α q D turb “ ˆ F r Π e , i s f e ´ α σ t ˙ ˚ ˆ F r Π e , j s f e ´ α σ t ˙ . (31)These turbulent lobes replace the static expressions of the pre-vious sections in the prediction of the observed visibility, i.e. inEq. (1), Eq. (24) and Eq. (25). The tip-tilt jitter acts like a Gaus-sian convolution kernel on the static maps, which is applied tothe amplitude map squared in case of the photometric lobe butto the full complex map in the case of the interferometric lobe.
3. Measurement and characterization of aberrationsfor the GRAVITY beam combiner
GRAVITY observes the Galactic Center in its so-called dual-field mode, which requires the presence of a bright reference tar-get (IRS 16C) within 2” of the actual science targets, Sgr A* andS2. The field at each telescope is split, and reference and sciencesource are separately injected into the fringe tracking (FT) andscience channel (SC) fibers. Short detector integration times onthe FT allow for the optical path delay to be constantly adjustedfor atmospheric turbulence in order to maintain a high fringecontrast. The science channel then measures a di ff erential visi-bility phase with respect to the fringe tracker on each baseline.Phase and amplitude maps are inherently single-field e ff ectsin the sense that they individually a ff ect the fringe tracker andthe science channel for each telescope separately. Based on theoptical layout of the fiber coupler (Pfuhl et al. 2014), there is noreason to expect equal aberrations on the SC and FT. However,the fringe tracking object is a bright, unresolved source which isactively tracked by the fiber center in closed loop, such that thephase distortions introduced from static aberrations are small.Moreover, any possible phase distortion from the fringe trackercancels in the analysis of closure phases or induces a global shiftwithout a ff ecting the binary separation in the analysis of visibil-ity phases. However, a description of the SC phase and amplitudemaps is essential to robustly measure a binary separation in thescience channel.Here we report on measurements with the GRAVITY Cali-bration Unit (Blind et al. 2014) and on our subsequent analysis Article number, page 5 of 17 & A proofs: manuscript no. gravity-phasemaps
Fig. 3.
Examples of the scanning pattern applied in the Calibration Unitmeasurements. SC aberration maps where obtained with a slow modu-lation frequency (left). For the corresponding FT measurement, a fasterscanning was used, and the right panel only shows a single iteration ofin- and out-spiral. to extract SC phase and amplitude maps. We then fit the static-aberration model from Sec. 2.1 to those maps in order to demon-strate its validity and to obtain compressed representation of theaberrations in form of a small number of Zernike coe ffi cients. The GRAVITY Calibration Unit, which we use for the mea-surement of static aberrations, is directly attached to the beamcombiner and creates the light of an artificial science and fringetracker star. By modulating the voltage on GRAVITY’s position-ing mirror, the position of that star relative to the fiber can bechanged. We scan the FOV out to „
70 mas in a pattern of in-and out-spiral, which is applied simultaneously to the FT and SCon one single telescope at a time, see Fig. 3.In normal observation mode, GRAVITY controls the di ff er-ential OPD between science channel and fringe tracker by itslaser metrology and the common path to the telescopes by fringetracking. During the phase map calibration measurement, how-ever, fringe tracking is not possible because the fringes are lostat the margins of the scanning region. Instead, the common pathfrom the telescope to the instrument drifts in time. Thus the de-termination of the aberration pattern from the absolute FT andSC phase requires a drift correction. On the FT, the short detec-tor integration time with maximum sampling frequency of 1 kHzallows one to resolve fast modulation of the source position andthe full FOV can be scanned within „
15 s. Over this short timespan, the drift is well described by a constant velocity, which wefit and subtract from the data. On the SC, in contrast, the min-imum detector integration time is 0 .
13 s and a full scan of theFOV takes 2 ´ ff erential, drift-freeSC-FT phase. The pure science channel aberrations then followfrom knowledge of the absolute fringe tracker phase.The data are reduced by the standard GRAVITY pipeline andwe obtain the correlated flux in six FT spectral channels (rangingfrom 1 . ´ . µ m) and in medium resolution for the SC (233wavelength bins in the range 1 . ´ . µ m). With the chosensetup, where the source position is varied on only one of the twobeams forming a baseline, the measured correlated flux is givenby B η ps i p α q ´ η ps j p q ¯ ˚ F obj “ A i p α q e i φ i p α q A j p q e ´ i φ j p q . (32) Fig. 4.
Science channel phase maps reconstructed by the procedure ofSec. 3.1 from the Calibration Unit measurement on 03 / /
20 for all fourGRAVITY beams.
Thus, the measurement directly scans the phase and amplitudemaps on the modulated channel. Potential o ff sets in the accom-panying non-modulated beam, φ j p q ‰
0, can only cause aglobal phase shift, which we fit and remove in the subsequentanalysis. Finally, we consider the amplitude maps normalized totheir maximum value, such that A j p q has no impact on our re-sult.In summary we apply the following analysis steps to obtainthe FT and the di ff erential SC-FT phase and amplitude maps.1. We fit and subtract a linear time drift from the phases mea-sured in each spectral channel and on each baseline.2. Phases and amplitudes are binned on a spatial grid with res-olution 1 mas and averaged over all periods of in- and out-spiral available.3. The image plane coordinates do not align perfectly with theamplitude maximum, i.e. the source position for which thecoupling to the fiber is most e ffi cient. We correct for thise ff ect by fitting a Gaussian profile and shifting the coordinateorigin to its maximum.4. Interpolation over the gridded data gives one phase and am-plitude map per spectral channel and baseline.5. All spectral channels are combined into a single map at refer-ence wavelength λ “ . µ m, by applying the approximatecoordinate scaling from Eq. (14). Here, we verified that theindividual maps are consistent over the full spectral range.Cross-validation with simulated maps shows that the errorintroduced by the approximate scaling relation is small, apartfrom the very margins of the map. It further cancels betweenchannels above and below λ to a very good degree.6. From consideration of all baselines, three maps are availablefor each telescope. We again verify their consistency and av-erage them into a single phase and amplitude map.This method results in a FT and a di ff erential SC-FT map foreach telescope. Subtracting the former from the latter, we finally Article number, page 6 of 17RAVITY Collaboration: R. Abuter et al.: Improved GRAVITY astrometric accuracy from modeling of optical aberrations arrive at the desired SC phase map, which is shown in Fig. 4.The amplitude map on the SC, on the other hand, is measureddirectly.The Calibration Unit measurement was performed twicewith a four month break, in late-2019 and early-2020, and we usethe data to construct two independent sets of maps. These agreevery well in the qualitative features and structures displayed. Onthe quantitative level the maps display moderate di ff erences ofthe order of „ ˝ , which are smaller at the center and increasetowards the map’s margins. Analyzing the Calibration Unit measurement as described in theprevious subsection, we obtain the phase and amplitude mapson a grid discretizing the image plane. We use this result to in-fer the underlying pupil-plane and fiber aberrations, d pup p u q and d foc p u q , in their Zernike representation. To this end, we devel-oped a simulation tool that creates complex maps of image-planedistortions from a set of Zernike coe ffi cients according to Eq. (5),Eq. (6) and Eq. (13).For the fit we consider the two Calibration Unit measure-ments from 2019 and 2020 separately and combine the phase andamplitude maps for each telescope into a complex map. We thenminimize the square absolute di ff erence to the model predictionsummed over all pixels with respect to the input coe ffi cients. Dueto the nature of the approximate coordinate scaling (step 5 of theanalysis pipeline), at a map’s edge only the smallest wavelengthscontribute. We limit the radius to which the data is considered inthe fit to α max ˆ λ low { λ high . With α max being the size of the fullmap and λ low and λ high the wavelength of the lowest and highestchannel, respectively. This choice ensures equal participation ofall channels in the fit.The optical layout of observations with the Calibration Unithas some important di ff erences with the on-sky situation, forwhich the phase maps will be applied later. Namely, the lack ofa central obscuration and an enlarged outer stop r GCU “ . { r tel “ . { λ , ˜ σ fib “ (cid:15)λ { ` r tel ? ln 2 ˘ , i.e. d foc p α q “ n max ÿ n “ m ÿ m “´ n B mn Z mn p α { ˜ σ fib q . (33)Of the di ff erent types of maps constructed, the fringe trackerprovides the cleanest system and thus gives an important bench-mark point for the agreement between model and data. We thususe the FT-maps to determine the order n max to which Zernikepolynomials in the pupil- and focal-plane aberrations are con-sidered. Successively increasing the fit order, we find that pupil-plane aberrations with n max “ n max “ Fig. 5.
Science channel phase maps obtained from fits to the di ff erentialSC-FT maps, measured on 03 / /
20 for all four GRAVITY beams.
Fig. 6.
Phase residuals of the fit to the di ff erential SC-FT map measuredon 03 / /
20 for all four GRAVITY beams. Only the data within thedashed circle is considered in the fit; at larger radii the cancellation ofwavelength-dependent scaling errors is not guaranteed. accounts for the overall amplitude scaling between measured andpredicted maps, such that each fit constrains at least 34 degreesof freedom. The phase RMS achieved for the fringe tracker fitsis of order „ ˝ for all beams and data sets; extrapolation of thefit result to the full map radius yields an RMS of a few degrees.In principle, it is possible to directly fit the SC maps by thesame procedure employed for the FT. However, by further re- Article number, page 7 of 17 & A proofs: manuscript no. gravity-phasemaps fining the analysis we can remove additional systematic e ff ectsfrom the SC maps. Creating the maps, we corrected for misalign-ment of the image plane coordinates with the amplitude maxi-mum (step 3 in the analysis pipeline). This shift, however, is notguaranteed to be identical on SC and FT, and as a result there canbe a small o ff set between the FT phase entering the di ff erentialSC-FT measurement. To describe this e ff ect, we fit a di ff eren-tial map, predicted from two sets of Zernike coe ffi cients, to theSC-FT maps. The latter of this two sets of parameters is largelyfixed to the previously obtained FT coe ffi cients, and only the tip-tilt terms are allowed to vary. The SC parameters, on the otherhand, are all free, such that the fit eventually determines the de-sired SC maps and the o ff set between the two channels.From the best-fit coe ffi cients of the di ff erential SC-FT fit,which we summarize in App. A, we reconstruct a complex SCmap. Its phase is displayed in Fig. 5. As expected, the structureagrees very well with the maps obtained by direct evaluation ofthe Calibration Unit measurement in Fig. 4. Residuals betweenmeasured and fitted SC-FT map, shown in Fig. 6, are low overthe full radius considered for the fit. We obtain a best-fit RMSof 1 ˝ ´ ˝ for most beams and data sets and two slightly worseresults with RMS „ ˝ and „ ˝ . Going to larger radii, thedisagreement between fit and data starts to increase. This canbe caused either by wavelength-dependent errors or by higher-order aberrations, beyond those considered for the fit. Indeed,in optimizing n max , we noted that every increase improved the extrapolation to large separations. However, at such large o ff -axis distances, fiber damping becomes very significant, resultingin a poor signal-to-noise ratio. Thus, we consider the Zernikedecomposition up to 6th order su ffi cient for our applications.
4. Application to GRAVITY observations
Static, field-dependent aberrations a ff ect the visibility measure-ment whenever the size of an observed object is comparable tothe fiber FOV. Here, we apply the formalism developed in Sec. 2alongside the characterization of aberrations from Sec. 3 to ob-servations of two di ff erent binary systems. First, as a proof ofconcept, we consider a test-case binary observed with the Aux-iliary Telescopes (ATs), where the system’s position in the FOVwas systematically varied and thus screened over the phase andamplitude maps. Second, we apply the aberration-correction toGC observations with the UTs from 2017 and 2018. Duringthose epochs, close to pericenter passage, S2 and Sgr A* whereobserved simultaneously in a single fiber pointing.The data considered in either analysis consists of visibilityamplitudes, squared visibilities and closure phases with a rela-tive weighting of (1:1:2). To infer the sources’ separation, we fita binary model based on Eq. (25), which we extend to accountfor the e ff ect of finite spectral resolution and for a homogeneousbackground with flux ratio f bkg relative to the first binary com-ponent, V obsbin p b i , j { λ q “ ˜ A i p α q ˜ A j p α q V λ rp b i , j ¨ α ´ d i , j p α qq , ν s ` ˜ A i p α q ˜ A j p α q V λ rp b i , j ¨ α ´ d i , j p α qq , ν s bś x “ i , j “ ˜ L x p α q V λ p , ν q ` f bin ˜ L x p α q V λ p , ν q ` f bkg V λ ` , ν bkg ˘‰ . (34)Phase distortions enter this expression via the OPD correction d i , j “ p ˜ φ i ´ ˜ φ j q ˆ λ { π . Further, the point-source visibility av-eraged over a spectral channel is V λ p d , ν q “ ż d λ P p λ q ˆ λ . µ m ˙ ´ ´ ν e ´ π i d { λ . (35)The spectral bandpass P p λ q is given by a top hat function. Thesource positions α and α , the flux ratios f bin and f bkg as wellas the spectral index of the central component ( ν ) and the back-ground flux ( ν bkg ) are free fit parameters, while the companion’sspectral slope is fixed to ν “ A i { j , ˜ φ i { j and ˜ L i { j in Eq. (34) refer to the phase maps,amplitude maps and the photometric lobes as they are encoun-tered in on-sky observations. Those have two important di ff er-ences with the Calibration Unit measurement. Firstly, while thepupil-plane representation of the aberrations is the same for bothsettings, the presence of a central obscuration and the smallerouter stop a ff ects the realization of the maps in the image plane.This is conveniently captured by using the Zernike coe ffi cientsfound in Sec. 3.2 to create a new set of maps with adjustedpupil configuration. Secondly, the maps are subject to turbulentsmoothing according to Eqs. (30) and (30). The test-case observations, carried out with the ATs in astro-metric configuration, targeted HIP 41426, a binary with K-band magnitude m K » .
393 at R . A . “ .
75 h, Dec . “´ . ˘
400 mas on AT2. At each o ff set, ten frameswith a 6 s integration time were taken. The setup is illustratedin Fig. 7, which shows both binary components relative to thefiber profile on all four telescopes. The shift was applied alongthe x-axis in the frame of the GRAVITY pupil, whose rotationwith respect to the field results in a diagonal movement on thesky.We use the Zernike coe ffi cients obtained for the SC inSec. 3.2 to produce phase and amplitude maps tailored to ob-servations with the ATs. In this case, the pupil, c.f. Eq. (4), isdefined by r tel “ .
82 m { r cent “ .
14 m {
2. After beamcollimation, ATs and UTs illuminate the same section on theGRAVITY mirrors, such that the pupil-plane phase screen cansimply be scaled to the AT radius, i.e. r tel “ .
82 m { φ i { j “ A i { j “ ff sets, the signal-to-noise ratio on AT2is poor due to large fiber damping and we consequently discardthese data. The remaining pointings are shown in Fig. 7, and thecorresponding separation, measured from a binary fit to the dataaccording to Eq. (34), is given in Fig. 8.The AT binary test-case clearly validates our aberration cor-rections. Di ff erent configurations yield consistent results only ifphase and amplitude maps are considered in the analysis. Includ- Article number, page 8 of 17RAVITY Collaboration: R. Abuter et al.: Improved GRAVITY astrometric accuracy from modeling of optical aberrations O ff s e t [ m a s ] AT 1 AT 2
200 0 200Offset [mas]2000200 O ff s e t [ m a s ] AT 3
200 0 200Offset [mas]
AT 4
Fig. 7.
Illustration of the AT binary test observations, showing the posi-tion of the two binary components (circles and diamonds, respectively)relative to the fiber profile (gray shading). Color gradients are chosenin accordance with Fig. 8. For this test, the fiber position was varied onAT2 only, but kept fixed on the other three telescopes. D E C [ m a s ] F i b e r O ff s e t [ p i x ] Fig. 8.
Binary separation inferred for a varying fiber o ff set on AT2 with(right panel) and without (left panel) application of the phase and am-plitude maps. Each data point shows the average over two polarizationstates, and the range of o ff sets corresponds to ˘
200 mas, approximately. ing the correct aberration model in the analysis clearly shifts theresult and reduces the scatter. Even more importantly, however,the separation found in the no-map analysis systematically de-pends on the fiber position; it is largest for positive fiber-o ff setsand smallest for o ff sets in the negative direction. With applica-tion of the aberration-correction, this systematic is largely re-moved.We consider the binary test-case observations primarily asa proof of concept and therefore forgo a full analysis of the Fig. 9.
The orbit of S2 relative to the phase maps as applied for theGC analysis (measurement from 03 / / σ t “
10 mas). Dots indicatethe position of S2 on 2017.2, 2017.6, 2018.2 and 2018.7, respectively,while the cross marks Sgr A*. measurement’s systematic error as carried out for the GC. Suchuncertainties arise from the accuracy to which the phase mapscan be determined and from the uncertainty of the atmosphericsmoothing kernel. Further, there can be minor di ff erences in thephase and amplitude maps between AT und UT observations,and our treatment is optimized to the UT scenario.As the shift in its central value indicates, the binary separa-tion is large enough that even at perfect fiber pointing at leastone source lies in a region of the FOV where aberration-inducedphase errors are significant. Accurate astrometry thus is not aquestion of precise fiber alignment but is only possible with aconsistent treatment of the pupil-plane distortions in the analy-sis. Having verified our approach to correct for aberration-inducedsystematic errors, we also apply it to Galactic Center observa-tions with GRAVITY. During 2017 and 2018, i.e. close to peri-center passage, S2 and Sgr A* where observed simultaneouslyin a single fiber pointing. In particular during 2017, when theo ff -axis distance of S2 was larger, the aberration correction im-proves the inferred binary separation. In 2019, in contrast, theSgr A*-S2 separation exceeds the single telescope beam size ofabout 60 mas, and GRAVITY observes both sources separatelyin so called dual-beam mode. Their separation is then obtainedby calibrating Sgr A* with S2 and fitting a point source modelto its visibilities (see Gravity Collaboration et al. (2020) for de-tails). In this configuration, each source can be well aligned withthe fiber center, such that field-dependent aberrations do not im-pact the measurement.To derive the aberration-induced shift of the S2 position, weexamine a subset of the GRAVITY data used in Gravity Collab-oration et al. (2019). In particular, we apply stricter quality cuts Article number, page 9 of 17 & A proofs: manuscript no. gravity-phasemaps and demand a high signal-to-noise ratio. Phase and amplitudemaps are generated from the coe ffi cients obtained in Sec. 3.2by accounting for the specific geometry of UT-observations, i.e. r tel “ . { r cent “ .
96 m {
2. The residual turbulent tip-tilt is between 10 mas and 15 mas per axis (Perrin & Woillez2019). In total, we consider four di ff erent realizations of theaberration maps which are given by the independent analysis ofthe two calibration measurements in 2019 and 2020 each con-volved with the minimum and maximum smoothing assumption.A representative example for the phase maps applied in the GCanalysis is shown in Fig. 9 in relation to the orbit of S2.Our main result, the di ff erence in S2 position with andwithout aberration-corrections averaged per month, is shown inFig. 10. As expected, the correction is largest in early-2017 andsmallest around peri-center passage in May 2018. Further, themean corrections per epoch obtained with the four di ff erent re-alizations of the aberration maps are consistent over the full ob-servational period.As the orbit of S2 smoothly scans over the phase and am-plitude maps (see Fig. 9), we also expect a smooth variation inthe position-correction. Indeed, the time-dependence in Fig. 10is well described by a second-order polynomial fit ∆ R . A . “ ` ´ . τ ` . τ ` . ˘ mas , (36) ∆ Dec . “ ` . τ ´ . τ ´ . ˘ mas , (37)where τ “ t { years ´ . ff ect to the finalcorrection. To this end, we take the position error of the original,un-corrected data point from which we draw 100 realizationsand shift the aberration maps by it. We then derive the correc-tion from each realization independently and use their scatter toestimate the statistical error of the S2 position correction. The re-sulting mean statistical uncertainty per epoch is small, between10 µ as and 30 µ as, but we nevertheless also account for it in theorbit fitting.A further check is to ask the question, what correction makesthe 2017 and 2018 GRAVITY positions optimally match to therest of the S2 data. To this end, we included a scaling factor f corr in the correction we apply, such that f corr “ f corr “ f corr “ . ˘ .
06, i.e. identical to the correction we have de-rived purely from calibration data. This gives an independentconfirmation of our concept and the resulting aberration correc-tion: Our correction yields the most consistent S2 orbit.The aberration correction presented here constitutes a furtherrefinement of the analysis in Gravity Collaboration et al. (2020).There, we applied the measured aberration maps as shown inFig. (4) directly, rather than the fitted decomposition in terms ofpupil-plane Zernike polynomials. To account for the wideningof the maps, which occurs when projecting from the enlargedstop on the Calibration Unit to the telescope pupil, in addition misalignment between mass and IR-emission 12 pcwavelength calibration of SINFONI 9 pcGRAVITY astrometry 29 pcbaseline accuracy 4 pcwavelength accuracy 9 pcmodel & data selection 9 pcatmospheric di ff erential dispersion 5 pcaberration-correction 23 pcmetrology correction 10 pc Table 1.
Contribution to the systematic errors a ff ecting the measure-ment of R , for details see Gravity Collaboration et al. (2019). Addingall contributions quadratically, we find a total systematic uncertainty of33 pc. to the e ff ect of turbulence, we applied a smoothing kernel of σ t “ p ˘ q mas. The resulting best-estimate for the correc-tion is depicted in Fig. 10 as dashed line. Both methods giveconsistent results, a ffi rming the robustness of the approach. Theonly sizable deviation is in 2017.2, when S2 was observed at aseparation comparable to the maximum radius for which we ob-tained the calibration measurement (see Fig. 9). This case showsthe strength of the Zernike decomposition, which allows for awell-defined extrapolation.
5. Results
In the following we evaluate the e ff ect of the aberration correc-tion on the S2 orbit. The data used is similar to Gravity Collab-oration et al. (2020) and described in detail in Appendix B. Weemploy the same fitting procedure as in Gravity Collaborationet al. (2020), using a 13-parameter, Post-Newtonian orbit model.Six of those parameters describe the Kepler orbit ( a , e , i , ω , Ω , t peri ), and another six describe the reference frame relative to theAO spectroscopy and assumed Local Standard of Rest (LSR)correction, ( x , y , R , x , y , z ). Here, R is the distance to theGC, the prime focus of this work, and M ‚ the central mass. Thebest-fit parameters are given in Tab. 2.For determining the systematic uncertainty, we follow the ap-proach in Gravity Collaboration et al. (2019) of varying our as-sumptions and tracing the associated changes in R . Comparedto our earlier work, we also include the uncertainty due to theaberration correction, as given by the gray band in Fig. 10. Theindividual contributions are given in Tab. 1. It turns out that theaberration correction is the dominant contributor to the system-atic error. The total systematic uncertainty is 33 pc when addingthe contributions quadratically.Our best estimate of the Galactic Center distance thus is R “ ˘ | stat . ˘ | sys . pc . (38) Our previous determinations of the GC distance in Gravity Col-laboration et al. (2018), Gravity Collaboration et al. (2019) andGravity Collaboration et al. (2020) were biased by the field-dependent aberrations. Taking them into account brings all our
Article number, page 10 of 17RAVITY Collaboration: R. Abuter et al.: Improved GRAVITY astrometric accuracy from modeling of optical aberrations C o rr e c t i o n R A [ m a s ] t =102019 data, t =152020 data, t =102020 data, t =152017.2 2017.4 2017.6 2017.8 2018.0 2018.2 2018.4 2018.6Time [yrs]0.00.51.01.5 C o rr e c t i o n D e c [ m a s ] Fig. 10.
The di ff erence in S2 position obtained from an analysis with and without application of the aberration corrections. Colored dots indicatethe epoch-wise mean for di ff erent realizations of the phase and amplitude maps, gray dots the results for individual observations. From these, wedetermine a mean position-correction as function of time with a corresponding upper and lower limit as indicated by the black solid line and thegray band. The thin dashed line, finally, represents the correction applied in Gravity Collaboration et al. (2019). G ill e ss en e t a l . a G ill e ss en e t a l . b G ill e ss en e t a l . G R AV I T Y c o ll . G R AV I T Y c o ll . G R AV I T Y c o ll . a G R AV I T Y c o ll . b G he z e t a l . B oeh l e e t a l . D o e t a l . Year R [ k p c ] Fig. 11.
Measurements of the Galactic Center distance over time with a focus on studies of the S2 orbit. Blue points show results obtained with theSINFONI, NACO and GRAVITY data with (dark blue) and without (light blue) application of the aberration corrections. Gray R determinationsare based on data from the Keck observatory. For comparison, we show in black results based on the statistical parallax of the nuclear starcluster (Chatzopoulos et al. 2015) and from modeling the Milky Way dynamics based on observations of molecular masers (Reid et al. 2019).Bland-Hawthorn & Gerhard (2016), finally, give the GC distance based on a combination of various methods. measurements into agreement as shown in Fig. 11 and Tab. 3.We further note the following: – In contrast to Gravity Collaboration et al. (2020), we alsoapply a correction for the 2018 data, where S2 and Sgr A*were close to each other and close to the field center. Yet,the small aberration corrections lead to a small upward cor-rection of R of around 30 pc, comparable to the systematicerror. – The orbit is particularly sensitive to the pericenter data. Thisleads to the e ff ect that the statistical uncertainty decreases strongly with time, while the systematic uncertainty even in-creases slightly during this time frame, since varying the as-sumptions then leads to stronger variations in the fit result. We estimate that the accuracy of our VLT-based result is at the40 pc level. However, it deviates significantly from the Keck-based value reported in Do et al. (2019), with the di ff erencebeing at the 300 pc level. Since both works use the orbit of S2 Article number, page 11 of 17 & A proofs: manuscript no. gravity-phasemapsparameter value a r mas s . ˘ . e . ˘ . i r deg s . ˘ . ω r deg s . ˘ . Ω r deg s . ˘ . P r yr s . ˘ . t peri r yr s . ˘ . x r mas s ´ . ˘ . y r mas s . ˘ . x r mas { yr s . ˘ . y r mas { yr s . ˘ . z r mas { yr s ´ . ˘ . M ‚ “ M d ‰ . ˘ . R r pc s . ˘ . Table 2.
Orbital parameters of S2 with their statistical uncertainties.
Phasemaps None 2017 only 2017 and 2018GRAV. coll 2018 ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ Table 3.
Published values of R (bold) and the corresponding values ifthe aberrations are taken into account (right column). All values in pc. around Sgr A* for the determination of R , it is important to in-vestigate where the discrepancy is arising, and we address thisin App. C. Overall, we conclude that the combination of – a di ff erence in the radial velocity data and – a modest o ff set of the Keck coordinate system in the decli-nation directionmight explain the discrepancy. Both e ff ects contribute roughly50%.About 20% of the radial velocity di ff erence can be attributedto the Doppler formula in StarKit used implicitly by Do et al.(2019). The remaining 80% are unexplained and could be in ei-ther the Keck or the VLT data.The origin of the coordinate system o ff set is unclear as well.Trying to explain the o ff set with a shift of the VLT coordinatesystems is much harder than imposing a shift of the Keck onedue to the high precision of the GRAVITY data.
6. Conclusions
GRAVITY delivers high-resolution astrometry which, in combi-nation with spectroscopic data, allows for a very precise determi-nation of the Galactic Center distance. The values inferred fromdi ff erent epochs (Gravity Collaboration et al. 2018, 2019, 2020)show a small discrepancy at the 1% level, which nevertheless issignificant due to the high precision of the measurement.We were able to relate this shift to optical aberrations intro-duced in the instrument, which lead to a field-dependent distor-tion of the visibility phase. Their e ff ect is the stronger, the fur-ther o ff -axis an object lies within the FOV. In particular Galac-tic Center observations close to the S2 pericenter passage area ff ected, where S2 and Sgr A* are detected simultaneously in R A [ a s ] Not correctedCorrectedDual BeamS2 Orbit2017.25 2017.50Date [yrs]0.0200.0250.0300.035 D e c [ a s ] Fig. 12.
Detailed view of the S2 orbit in 2017. Dual-beam points donot su ff er from aberration-related systematic errors and agree very wellwith our corrected data points. a single fiber pointing but at a separation comparable to theFOV. In earlier and later epochs, in contrast, we employed theso-called dual-beam method and targeted each source individ-ually. In this case, as for most other GRAVITY science observ-able, each source can be well centered and aberration correctionsbecome irrelevant. The dual-beam observation mode was alsoassumed to derive the astrometric error budget in Lacour et al.(2014), which did not include the e ff ect of phase maps for thisprecise reason.The full analytical description which we developed here al-lows us to propagate the e ff ect of optical aberrations at fiber in-jection to the measured visibilities. Fitting this model to dedi-cated calibration measurements confirms its validity and enablesus to account for the e ff ect in the data analysis. We further verifythe approach with dedicated test-case observations.The formalism which we developed is applicable beyondGRAVITY to any optical / near-IR interferometer where aber-rations are introduced in the pupil or the focal plane. Therehave been several cases in the literature with more than one ob-ject lying in the interferometer’s FOV, for example some Keck(Colavita et al. 2013), CHARA (ten Brummelaar et al. 2005)or NPOI (Armstrong et al. 1998) results on binary stars. Howseverely aberrations a ff ect an observation, however, depends notonly on their strength for a particular instrument but also onthe o ff -axis distance considered and on the statistical noise inthe measurement. In the example of GRAVITY on the UTs, themean phase error introduced at 20 mas separation is 4 ´ ´
20 degrees at 50 mas. Whilea binary test case as presented in Sec. 4.1 can serve as a gen-eral strategy to diagnose whether aberration-induced systematicsare an issue, dedicated calibration measurement are required fortheir correction in the analysis for each individual instrument.With the results from the GRAVITY Calibration Unit mea-surements and our refined analysis scheme, we are able to fur-ther improve the separation between S2 and Sgr A* in 2017and 2018, introducing shifts up to 0 . ff er from phase aberrations. Indeed, theimproved data agrees very well with these positions. Article number, page 12 of 17RAVITY Collaboration: R. Abuter et al.: Improved GRAVITY astrometric accuracy from modeling of optical aberrations
Of all orbital parameters, the distance to the Galactic Center R is most strongly a ff ected by the change in the S2 position.This can be easily understood if one views R as the scalingfactor between angular and proper velocity. As such, the field-dependent phase errors discussed in this work fully explain theshift between earlier R measurements with GRAVITY data. Ap-plying the analysis scheme developed here lifts any such discrep-ancies (see Sec. 5.2). In particular Fig. 11 demonstrates that be-latedly corrected data sets of earlier publications give fully con-sistent results whose accuracy increases with time. References
Armstrong, J. T., Mozurkewich, D., Rickard, L. J., et al. 1998, ApJ, 496, 550Bhatnagar, S., Cornwell, T. J., Golap, K., & Uson, J. M. 2008, A&A, 487, 419Bland-Hawthorn, J. & Gerhard, O. 2016, ARA&A, 54, 529Blind, N., Eisenhauer, F., Haug, M., et al. 2014, in Society of Photo-OpticalInstrumentation Engineers (SPIE) Conference Series, Vol. 9146, Proc. SPIE,91461UBoehle, A., Ghez, A. M., Schödel, R., et al. 2016, ApJ, 830, 17Chatzopoulos, S., Fritz, T. K., Gerhard, O., et al. 2015, MNRAS, 447, 948Chu, D. S., Do, T., Hees, A., et al. 2018, ApJ, 854, 12Colavita, M. M., Wizinowich, P. L., Akeson, R. L., et al. 2013, PASP, 125, 1226Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, VizieR Online DataCatalog, II / Max Planck Institute for extraterrestrial Physics, Giessenbach-straße 1, 85748 Garching, Germany LESIA, Observatoire de Paris, Université PSL, CNRS, SorbonneUniversité, Université de Paris, 5 place Jules Janssen, 92195Meudon, France Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidel-berg, Germany st Institute of Physics, University of Cologne, Zülpicher Straße 77,50937 Cologne, Germany Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France Universidade de Lisboa - Faculdade de Ciências, Campo Grande,1749-016 Lisboa, Portugal Faculdade de Engenharia, Universidade do Porto, rua Dr. RobertoFrias, 4200-465 Porto, Portugal European Southern Observatory, Karl-Schwarzschild-Straße 2,85748 Garching, Germany European Southern Observatory, Casilla 19001, Santiago 19, Chile Sterrewacht Leiden, Leiden University, Postbus 9513, 2300 RA Lei-den, The Netherlands Departments of Physics and Astronomy, Le Conte Hall, Universityof California, Berkeley, CA 94720, USA CENTRA - Centro de Astrofísica e Gravitação, IST, Universidadede Lisboa, 1049-001 Lisboa, Portugal Department of Astrophysical & Planetary Sciences, JILA, DuanePhysics Bldg., 2000 Colorado Ave, University of Colorado, Boulder,CO 80309, USA Department of Particle Physics & Astrophysics, Weizmann Instituteof Science, Rehovot 76100, Israel Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK Department of Physics, Technical University Munich, James-Franck-Straße 1, 85748 Garching, Germany Max Planck Institute for Radio Astronomy, Auf dem Hügel 69,53121 Bonn, Germany Article number, page 13 of 17 & A proofs: manuscript no. gravity-phasemaps
GV1 GV2 GV3 GV4 A ´ . ´ . ´ .
019 -0.014 A ´ .
000 0 .
008 0 .
062 -0.014 A . ´ .
030 0 .
053 0.022 A ´ . ´ .
009 0 .
028 0.010 A ´ . ´ .
012 0 .
015 -0.035 A ´ . ´ . ´ .
016 -0.002 A ´ . ´ .
042 0 .
028 0.065 A .
032 0 .
071 0 .
081 0.013 A ´ ´ .
056 0 .
011 0 .
032 0.021 A . ´ . ´ .
026 0.054 A ´ ´ . ´ . ´ .
036 -0.016 A ´ . ´ . ´ .
046 -0.034 A ´ . ´ .
005 0 .
049 0.002 A ´ . ´ . ´ .
029 -0.012 A ´ . ´ . ´ .
023 -0.019 A ´ .
011 0 .
030 0 .
013 0.014 A ´ . ´ .
026 0 .
032 -0.032 A ´ . ´ . ´ .
015 -0.013 A ´ .
020 0 . ´ .
026 -0.030 A ´ . ´ .
018 0 .
008 -0.027 A . ´ .
003 0 .
047 -0.002 A ´ ´ .
003 0 .
009 0 .
018 -0.001 A ´ .
009 0 . ´ .
019 0.015 A ´ ´ .
002 0 . ´ .
017 0.002 A .
021 0 .
000 0 .
018 0.018 A ´ . ´ .
001 0 .
002 -0.000 A .
003 0 .
002 0 .
003 0.002 A .
024 0 .
001 0 .
024 0.007 B ´ .
010 0 .
113 0 .
065 0.033 B . ´ .
043 0 .
062 0.042 B ´ . ´ .
011 0 .
005 0.007 B ´ ´ .
045 0 . ´ .
086 0.024 B .
011 0 . ´ .
004 0.031
Table A.1.
Zernike coe ffi cients for science channel aberrations fitted tothe calibration measurement on 03 / /
19. All coe ffi cient are given inunits of µ m. Appendix A: List of Zernike coefficients
The Zernike coe ffi cients obtained by fitting the Calibration Unitmeasurements from late-2019 and early-2020 are summarized inTabs. A.1 and A.2, respectively. We provide the science channelresults for all for GRAVITY beams (GV1 to GV4) in units of µ maccording to the definitions in Eqs. (5) and (33), where A mn labelspupil-plane aberrations and B mn those in the focal plane. Appendix B: Data
We use the data set presented in Gravity Collaboration et al.(2020) with the following changes: – Each single-beam astrometric position is corrected accordingto Eq. (37), and we add the statistical error of this correctionin quadrature, which increases the individual uncertaintiesby around 15 µ as. GV1 GV2 GV3 GV4 A ´ . ´ . ´ .
019 -0.027 A ´ ´ .
018 0 .
034 0 .
066 -0.003 A .
008 0 .
016 0 .
045 0.043 A ´ . ´ .
005 0 .
047 0.006 A ´ . ´ .
010 0 .
019 -0.023 A ´ . ´ . ´ .
024 0.012 A ´ . ´ .
039 0 .
038 0.075 A .
042 0 .
079 0 .
063 0.026 A ´ ´ .
031 0 .
009 0 .
029 0.023 A ´ . ´ .
006 0 .
022 0.032 A ´ ´ . ´ . ´ .
042 -0.014 A ´ . ´ . ´ .
019 -0.017 A ´ . ´ .
014 0 .
023 -0.014 A ´ . ´ . ´ .
016 -0.016 A ´ . ´ .
027 0 .
001 -0.023 A ´ .
022 0 . ´ .
000 -0.000 A ´ . ´ .
031 0 .
034 -0.041 A ´ . ´ . ´ .
011 -0.017 A ´ . ´ . ´ .
025 -0.017 A ´ . ´ .
015 0 .
008 -0.007 A ´ .
008 0 .
001 0 .
058 0.004 A ´ ´ .
006 0 .
018 0 .
040 0.014 A .
001 0 . ´ .
002 0.008 A ´ .
013 0 . ´ .
005 0.001 A .
012 0 .
021 0 .
014 0.015 A ´ ´ .
001 0 .
002 0 .
003 0.001 A ´ . ´ .
001 0 .
006 0.004 A .
030 0 .
007 0 .
016 0.009 B ´ .
002 0 .
115 0 .
036 0.023 B . ´ .
032 0 .
086 0.035 B ´ . ´ .
000 0 .
004 0.015 B ´ ´ .
027 0 . ´ .
076 0.012 B .
043 0 . ´ .
040 0.008
Table A.2.
Zernike coe ffi cients for science channel aberrations fitted tothe calibration measurement on 03 / /
20. All coe ffi cient are given inunits of µ m. – We corrected the radial velocity of the epoch 2018.1277,which was 13 km / s too high in the previous data set. – Further, we are able to add one interferometric position mea-surement of S2 from early March 2020. Like in 2019, theseparation between S2 and Sgr A* exceeds the fiber field ofview, and hence a dual-beam measurement needed to be em-ployed.Our data set consists of 128 AO-based astrometric points, 58GRAVITY-based astrometric points and 97 radial velocities, ofwhich the first three before 2003 are from Do et al. (2019).
Appendix B.1: Dual-beam measurement in 2020
Due to the limited observability of the GC in early March andexpecting observations in the following months, we did not at-tempt to observe Sgr A* in March 2020, but only pointed to S2
Article number, page 14 of 17RAVITY Collaboration: R. Abuter et al.: Improved GRAVITY astrometric accuracy from modeling of optical aberrations and to our usual calibrator star R2, with the aim of testing thestability of the GRAVITY astrometry. Pointings to Sgr A* wereplanned for later in the year. They had to be canceled due to thepandemic-related closure of the VLT(I) from mid-March on. Tostill determine the S2 – Sgr A* separation vector from this ob-servation, we need to proceed in two steps and first measure theS2 – R2 distance, then we reference R2 to Sgr A*.The distance between S2 and R2 is measured with the dual-beam method (Sec. 4.2), where we calibrate the S2 files withR2. In addition to the 2020 measurement, this separation is alsoavailable for 56 epochs in the years 2017, 2018 and 2019. It canbe measured very precisely due to the brightness of the two stars.Since the S2 – Sgr A* vectors have already been determinedin Gravity Collaboration et al. (2020), we can also refer R2 toSgr A* in those earlier epochs. We then fit a simple quadraticfunction for the time evolution of the R2 coordinates relative toSgr A* and extrapolate it to March 2020. Given the large num-ber of data and the small time range to extrapolate for, the extrauncertainty introduced is well below the 100 µ as level.We derive the S2 position in 2020 from the four scientificallyusable exposures as their mean. We assign an error of 150 µ as toeach coordinate for this data point, reflecting both the smallernumber of files compared to what we typically had available in2019 and the extra uncertainty due to the additional step of ref-erencing via R2. The new data point falls well onto the expectedorbit, but its error bar is too large to have a significant impact onthe fitted parameters. Appendix C: Analysis of the difference between R determinations from Keck and VLT data sets While we believe our determination of R is accurate to the 40 pclevel, we note that the value published in Do et al. (2019) isdiscrepant at the 300 pc level. Both teams use the orbit of thestar S2 around Sgr A* for the R determination, and hence it isnatural to ask where the di ff erences are. Appendix C.1: Data
Beyond our ("VLT") data set (App. B), we use the Keck dataset published in Do et al. (2019). We apply the NIRC2 radialvelocity o ff set of `
80 km / s as determined in Do et al. (2019) tothe NIRC2 data, i.e. we add 80 km / s to these radial velocities.Unlike Do et al. (2019), we then don’t fit for this o ff set. Further,we drop the last astrometric data point (epoch 2018.67148268),as suggested by the authors in a private communication. The dataset consists of 45 astrometric points and 116 radial velocities, ofwhich 41 are actually from the VLT data set between 2003 and2016. The published table also includes one radial velocity fromthe epoch 2019.3567, which possibly was not part of the data setactually used in Do et al. (2019). Appendix C.2: The difference in R We fit the orbit with a simple, 13-parameter model: The six or-bital elements of the star (corresponding to the initial conditionsof the star in phase space), six parameters for the position and ve-locity of the MBH, and the mass of the MBH. The fits are doneusing the relativistic corrections as in Gravity Collaboration et al.(2020), i.e. we fix f RS “ f SP “
1. For this non-Keplerian motion,the meaning of the orbital elements is that they are osculating ata reference epoch, for which we choose T = ff set,on which we set priors following the work from Plewa et al.(2015), and we include the NACO flare positions as an additionalconstraint for locating the mass. This fit yields R “ . ˘ . a “ . ˘ .
034 mas i “ . ˘ . ˝ Ω “ . ˘ . ˝ , (C.1)where a is the semi-major axis, i the inclination and Ω the posi-tion angle of ascending node of the S2 orbit, and the errors arethe statistical fit uncertainties. The VLT astrometry is dominatedby the GRAVITY points, as illustrated by dropping all AO datapoints, which results in R “ ˘
10 pc.Fitting the Keck data set with the same 13-parameter modelas used for Eq. C.1 yields R “ ˘
44 pc a “ . ˘ .
27 mas i “ . ˘ . ˝ Ω “ . ˘ . ˝ . (C.2)This is not the exact same number as in Do et al. (2019), where R “ ˘
59 pc is reported. The small (and statisticallyinsignificant) di ff erence is most likely due to the noise modelwhich Do et al. (2019) include in their analysis, which we donot have readily available. Applying the noise model at hand(Plewa & Sari 2018; Gravity Collaboration et al. 2019) yields R “ ˘
56 pc. Hence, the value reported by Do et al. (2019)lies between the two numbers we get by re-fitting their data. Inthe following, we will use for simplicity, and for equal treatmentof the data, the value and approach as in Eq. C.2. We have thusa di ff erence of ∆ R “ ˘
45 pc.
Appendix C.3: Comparing, combining & adjusting theastrometry
Already, Gillessen et al. (2009a) noticed that a simple attemptto compare the astrometric data sets by plotting them on topof each other fails. One needs to allow for an o ff set and adrift between the two coordinate systems (i.e. four parameters ∆ x , ∆ y, ∆ v x , ∆ v y ). This yields thus a 17-parameter fit. Compar-ing the best-fitting parameters in Eq. C.1 and Eq. C.2 shows thatthey di ff er in Ω significantly. This parameter is fully degeneratewith the angular orientation (called β here) of the coordinate sys-tem. Hence, the di ff erence in Ω suggests that the two astrometricdata sets are rotated with respect to each other.Therefore we extend the combination scheme by an addi-tional, fifth parameter, ∆ β , resulting in a 18-parameter fit. Withthis we fitted both data sets simultaneously, omitting the 41 VLTradial velocities from the Keck data set, whist dropping also thethree Keck ones in the VLT data set. This fit matches the two Article number, page 15 of 17 & A proofs: manuscript no. gravity-phasemaps coordinate systems ideally onto each other and results in R “ ˘ a “ . ˘ .
03 mas i “ . ˘ . ˝ Ω “ . ˘ . ˝ ∆ β “ . ˘ . ˝ , (C.3)Note that the value of ∆ β matches the di ff erence ∆Ω . We con-clude that indeed the Keck and VLT data are rotated with respectto each other. The other parameters are very similar to Eq. C.1,which is due to the considerably smaller astrometric uncertain-ties of the GRAVITY data compared to the adaptive optics data.With the best-fit coordinate system di ff erence in hand, wecan transform the Keck astrometric data into the VLT coordi-nate system and vice versa. We choose to do the former, sincethe VLT data set is more directly calibrated by the interferomet-ric data. After applying the coordinate system di ff erence to theKeck data, we can fit them again with a 13-parameter model.This yields the exact same best-fit parameters as in Eq. C.2 (withthe exception of Ω , of course). Hence, transforming the astrom-etry does not change the more fundamental di ff erences betweenthe two orbits, while a direct comparison is now feasible. Thevalue of Ω can be omitted in the following. Appendix C.4: Discrepancy in the radial velocity data
Chu et al. (2018) have investigated the consistency of the radialvelocity data between the Keck and VLT data sets for the years2000 to 2016, and they concluded that the data are in agreementwith each other. We have repeated the exercise, now also extend-ing into the time of the pericenter passage in 2018 (Fig. C.1).To our surprise, the radial velocities di ff er systematically from « ff erence gets larger as the radial velocityincreases ever more. The di ff erence reaches «
50 km / s in 2018,just before the star swung through pericenter .Hence, it is an obvious question to ask what influence theradial velocities have on R ? For this, we swapped the radialvelocities between the two data sets. Using the VLT-set togetherwith the Keck astrometry yields R “ ˘
32 pc a “ . ˘ .
21 mas i “ . ˘ . ˝ (C.4)Vice versa, using the Keck radial velocities together with theVLT astrometry yields R “ ˘
14 pc. Given that the Keckradial velocity set contains 35% VLT radial velocities, the fit inEq. C.4 is the cleaner test. We thus explain roughly half of thedi ff erence in R with the radial velocity data, i.e. 159 pc.Why do the radial velocities di ff er? So far, we can only of-fer an explanation for «
20% of the radial velocity di ff erence:We applied the stellar atmosphere model-based fitting with theStarKit package used in Do et al. (2019) also to the VLT spec-troscopy. We found a significant di ff erence for large radial veloc-ities, which we were able to trace down to the Doppler formulaused by the StarKit package. While both Do et al. (2019) andGravity Collaboration et al. (2020) state that the spectroscopicobservable is v r “ z c , i.e. the redshift of a given spectrum, the Also, there is one obvious outlier in the Keck data, the earliest2018 point. We have checked that dropping this measurement does notchange the Keck-fit result in any significant way.
StarKit package actually applies a Doppler formula which in-cludes the longitudinal, relativistic correction: λ “ λ b ` v r { c ´ v r { c .In this form, the Doppler formula ignores the (significant) tan-gential motion v t of S2. In order to apply a relativistic correctionone needs to use the full Doppler formula 1 ` z “ ` v r { c ? ´p v r ` v t q{ c (Lindegren & Dravins 2003). For this correction, however, thespectroscopic information is not su ffi cient. One cannot, in gen-eral, Doppler-correct a spectrum in a relativistic way withoutknowing the other motion component. Further, even if one wouldapply the full correction, one would in the following of coursenot be able to fit for the relativistic redshift anymore.The di ff erence between the two formulae is small at veloc-ities much smaller than the speed of light, but becomes impor-tant close to peri-center, when S2 reaches a velocity of nearly8000 km { s. Still, it amounts to «
25 km { s at most and thus issmaller than the observed di ff erence in Fig. C.1. This di ff erenceis also visible in Fig. 1 of Do et al. (2019): The plotted modelspectra are slightly more redshifted than what the underlyingdata suggest. Changing the Keck radial velocities accordinglyyields a fit with R “ ˘
44 pc, i.e. accounting for 37 pc ofthe 159 pc.Further checks did not yield any clues why there remains asignificant di ff erence in the radial velocities. We note: – We checked whether the time stamps are assigned consis-tently between the two data sets, and did not find a di ff er-ence. – Fig. C.1 bottom right shows that both data sets clearly showthe redshift peak around pericenter.
Appendix C.5: Discrepancy in the astrometry
Comparing the fits in Eq. C.1 and Eq. C.2 shows that they notonly di ff er in R , but also in the size of the semi-major axis a .We find ∆ a { a “ . ˘ .
22 %. The same is not true for thesemi-minor axis though, ∆ b { b is consistent with 0. Interestingly,the projected ellipses as given by the astrometric data in the planeof sky agree in both semi-major and semi-minor axes to within0 . i need to di ff er, which Eq. C.1and Eq. C.2 confirm. We find in accordance with the above 1 ´ sin p i VLT q{ sin p i Keck q « . a but not in b hints towards an o ff setof the center of mass in the declination direction. Indeed, we canshow that introducing an o ff set to either y or v y (the mass positionand velocity in declination) can explain the remaining discrep-ancy. Starting from the fit of the transformed Keck data set, wefix v y to its best fit value of ´ .
15 mas / yr. All other parametersare left free again for a subsequent fit. Additionally using theVLTI velocities in this fit instead of the Keck ones yields: R “ ˘
28 pc a “ . ˘ .
16 mas i “ . ˘ . ˝ . (C.5)This fit yields thus from the Keck astrometry the same value for R as the VLT fit. Also note, that indeed semi-major axis a andinclination i have moved to the VLT values by forcing v y to havean o ff set. Since the mass position is parametrized with a timeorigin at T = y also changes, from ´ .
972 masto 1 .
234 mas. The systematic uncertainty on y and v y estimatedby Do et al. (2019) are 1 .
16 mas and 0 .
066 mas / yr respectively. Article number, page 16 of 17RAVITY Collaboration: R. Abuter et al.: Improved GRAVITY astrometric accuracy from modeling of optical aberrations ���� ���� ���� ���� ���� ���� - ���� - ��������������������� � [ �� ] � � � � [ � � / � ] - - v L S R r e s i dua l [ k m / s ] �� - v L S R r e s i dua l [ k m / s ] - v L S R r e s i dua l [ k m / s ] Fig. C.1.
Comparison of the radial velocity data sets. Blue points are data from the VLT data set, red from the Keck data set. Top left: Radialvelocity as a function of time for the VLT fit (Eq. C.1). Top right: Yearly averages of the residua of the two data sets to the fit from Eq. C.1. Byconstruction the VLT data thus scatter around 0. The Keck data deviate systematically from 2011 on, and the discrepancy increases in the lateryears. Bottom left: The same as the left panel, but zooming in to the period 2015 - 2020, and showing all individual data points. The best fitKeck orbit corresponding to Eq. C.2 is the red line. Apparently, the di ff erence is largest, when the radial velocity gets largest (in the year 2018 atpericenter passage). Bottom Right: Both data sets show a clear peak in radial velocity in 2018 when comparing with the Keplerian part of the VLTfit (Eq. C.1), i.e. both data sets clearly detect the redshift term. - - - r e s i dua l de c li na t i on [ '' ] - - - r e s i dua l de c li na t i on [ '' ] Fig. C.2.
Comparison of the astrometric residual after forcing an o ff set in declination such that the fit to Keck data set matches the VLT one (left)and such that the fit to VLT data set matches the Keck one (right). Lighter blue corresponds to AO data from the VLT data set, darker blue to theGRAVITY data. Hence, the di ff erence one needs to enforce is within « σ of thesystematic uncertainty, and the residuals in fig. C.2 (left) appearto be acceptable. Essentially the same can be achieved by forcingan o ff set to y and leaving v y free instead.Can one can turn the argument around and apply a similaro ff set to the VLT data in order to lower the VLT-based valueof R ? In a first attempt we applied the same o ff set to the VLTAO data. However, even an o ff set 10 x larger (i.e. 1 . / yr),changes R only by «
30 pc. This is not surprising, since theVLT astrometry is completely dominated by the GRAVITY data.Thus, we instead tried varying v y and y for the GRAVITY data,giving up the assumption that the GRAVITY source directlyis the mass center. Also, we exchanged the VLT radial veloc-ities for the Keck ones. We find that we need to change v y by ´ . / yr in order to get a distance similar to the Keck value: R “ ˘
16 pc a “ . ˘ .
05 mas i “ . ˘ . ˝ . (C.6)The fit achieves the lower R by tilting the orbit similar to the fitfrom Eq. C.2. The enforced change of v y is unrealistically large(12 ˆ larger than what was needed for the Keck data), Also, theGRAVITY data show very strong and systematic residuals of upto 0 . χ of the fit increasedfrom 1 .
50 to 2 .63.