Detection of the Gravitational Redshift in the Orbit of the Star S2 Near the Galactic Centre Massive Black Hole? GRAVITY Collaboration??: R

A&A 615, L15 (2018) https://doi.org/10.1051/0004-6361/201833718 Astronomy c ESO 2018 & Astrophysics

LETTER TO THE EDITOR Detection of the gravitational redshift in the orbit of the star S2 near the Galactic centre massive black hole? GRAVITY Collaboration??: R. Abuter8, A. Amorim6,14, N. Anugu7, M. Bauböck1, M. Benisty5, J. P. Berger5,8, N. Blind10, H. Bonnet8, W. Brandner3, A. Buron1, C. Collin2, F. Chapron2, Y. Clénet2, V. Coudé du Foresto2, P. T. de Zeeuw12,1, C. Deen1, F. Delplancke-Ströbele8, R. Dembet8,2, J. Dexter1, G. Duvert5, A. Eckart4,11, F. Eisenhauer1,???, G. Finger8, N. M. Förster Schreiber1, P. Fédou2, P. Garcia7,14, R. Garcia Lopez15,3, F. Gao1, E. Gendron2, R. Genzel1,13, S. Gillessen1, P. Gordo6,14, M. Habibi1, X. Haubois9, M. Haug8, F. Haußmann1, Th. Henning3, S. Hippler3, M. Horrobin4, Z. Hubert2,3, N. Hubin8, A. Jimenez Rosales1, L. Jochum8, L. Jocou5, A. Kaufer9, S. Kellner11, S. Kendrew16,3, P. Kervella2, Y. Kok1, M. Kulas3, S. Lacour2, V. Lapeyrère2, B. Lazareff5, J.-B. Le Bouquin5, P. Léna2, M. Lippa1, R. Lenzen3, A. Mérand8, E. Müler8,3, U. Neumann3, T. Ott1, L. Palanca9, T. Paumard2, L. Pasquini8, K. Perraut5, G. Perrin2, O. Pfuhl1, P. M. Plewa1, S. Rabien1, A. Ramírez9, J. Ramos3, C. Rau1, G. Rodríguez-Coira2, R.-R. Rohloff3, G. Rousset2, J. Sanchez-Bermudez9,3, S. Scheithauer3, M. Schöller8, N. Schuler9, J. Spyromilio8, O. Straub2, C. Straubmeier4, E. Sturm1, L. J. Tacconi1, K. R. W. Tristram9, F. Vincent2, S. von Fellenberg1, I. Wank4, I. Waisberg1, F. Widmann1, E. Wieprecht1, M. Wiest4, E. Wiezorrek1, J. Woillez8, S. Yazici1,4, D. Ziegler2, and G. Zins9 (Affiliations can be found after the references)

Received 26 June 2018 / Accepted 29 June 2018

ABSTRACT

The highly elliptical, 16-year-period orbit of the star S2 around the massive black hole candidate Sgr A* is a sensitive probe of the gravitational 1 field in the Galactic centre. Near pericentre at 120 AU 1400 Schwarzschild radii, the star has an orbital speed of 7650 km s− , such that the first-order effects of Special and General Relativity have≈ now become detectable with current capabilities. Over≈ the past 26 years, we have monitored the radial velocity and motion on the sky of S2, mainly with the SINFONI and NACO adaptive optics instruments on the ESO Very Large Telescope, and since 2016 and leading up to the pericentre approach in May 2018, with the four-telescope interferometric beam-combiner instrument GRAVITY. From data up to and including pericentre, we robustly detect the combined gravitational redshift and relativistic transverse 1 Doppler effect for S2 of z = ∆λ/λ 200 km s− /c with different statistical analysis methods. When parameterising the post-Newtonian contribution from these effects by a factor f ,≈ with f = 0 and f = 1 corresponding to the Newtonian and general relativistic limits, respectively, we find from posterior fitting with different weighting schemes f = 0.90 0.09 0.15 . The S2 data are inconsistent with pure Newtonian dynamics. ± |stat ± |sys Key words. Galaxy: center – gravitation – black hole physics

1. Introduction laboratories probe planetary masses that are about a factor 106 lower than the stellar mass scale. For massive black hole (MBH) 6 10 General Relativity (GR) so far has passed all experimental tests candidates with masses of 10 − M , only indirect evidence with flying colours (Einstein 1916; Will 2014). The most strin- for GR effects has been reported, such as relativistically broad- gent are tests that employ solar mass pulsars in binary systems ened, redshifted iron Kα line emission in nearby active galax- (Kramer et al. 2006), and gravitational waves from 10 to 30 M ies (Tanaka et al. 1995; Fabian et al. 2000). The closest MBH is 6 black hole in-spiral events (Abbott et al. 2016a,b,c). These tests at the centre of the Milky Way (R0 8 kpc, M 4 10 M ), cover a wide range of field strengths and include the strong and its Schwarzschild radius subtends≈ the largest• ≈ angle× on the curvature limit (Fig. A.2). At much lower field strength, Earth sky of all known MBHs (RS 10 µas 0.08 AU). It is coincident with a very compact, variable≈ X-ray,≈ infrared, and radio source, ? This paper is dedicated to Tal Alexander, who passed away about a Sgr A*, which in turn is surrounded by a very dense cluster of week before the pericentre approach of S2. ?? orbiting young and old stars. Radio and infrared observations GRAVITY is developed in a collaboration by the Max Planck have provided detailed information on the distribution, kinemat- Institute for extraterrestrial Physics, LESIA of Paris Observa- ics, and physical properties of this nuclear star cluster and of tory/CNRS/Sorbonne Université/Univ. Paris Diderot and IPAG of Uni- versité Grenoble Alpes/CNRS, the Max Planck Institute for Astronomy, the hot, warm, and cold interstellar gas interspersed in it (cf. the University of Cologne, the CENTRA – Centro de Astrofisica e Grav- Genzel et al. 2010; Morris et al. 2012; Falcke & Markoff 2013). itação, and the European Southern Observatory. High-resolution near-infrared (NIR) speckle and adaptive optics ??? Corresponding author: F. Eisenhauer (AO) assisted imaging and spectroscopy of the nuclear star clus- e-mail: [email protected] ter over the past 26 years, mainly by two groups in Europe (the

Article published by EDP Sciences L15, page 1 of 10 A&A 615, L15 (2018) A&A proofs: manuscript no. s2_20180702_revised_language_edited2 Max Planck Institute for Extraterrestrial Physics, MPE, and the USA (the University of California at Los Angeles, UCLA, at the University of Cologne at the ESO New Technology Telecsope, Keck telescopes) have delivered more than 104 stellar motions NACO NTT, and the Very Large Telescope, VLT) and one group in the GRAVITY and orbit determinations for 45 individual stars (Schödel et al. USA (the University of California at Los Angeles, UCLA, at the IRS16C 2002; Ghez et al. 2003, 2008;≈ Eisenhauer et al. 2005; Gillessen Keck telescopes) have delivered more than 104 stellar motions et al. 2009b, 2017; Schödel et al. 2009; Meyer et al. 2012; Boehle Sgr A* and orbit determinations for 45 individual stars (Schödel et al. et al. 2016; Fritz et al. 2016).≈ These orbits, in particular, the S2 2002, 2009; Ghez et al. 2003, 2008; Eisenhauer et al. 2005; highly eccentric orbit of the main-sequence B-star S2 (or S02 Gillessen et al. 2009b, 2017; Meyer et al. 2012; Boehle et al. 22 mas in the UCLA nomenclature), have demonstrated that the gravita- 2016; Fritz et al. 2016). These orbits, in particular, the highly S2 tional potential is dominated by a compact object of 4 106 M eccentric orbit of the main-sequence B-star S2 (or S02 in the that is concentrated within a pericentre distance from≈ × S2 of 17 UCLA nomenclature), have demonstrated that the gravitational light hours 14 mas or 120 AU from Sgr A*. S2 appears to be potential is≈ dominated by a compact object of 4 106 M March 26 2018 March 2018 a single star (Martins et al. 2008; Habibi et al. 2017; GRAVITY that is concentrated within a pericentre distance≈ from× S2 of Collaboration et al. 2017; Chu et al. 2018), making it an ideal HeI HeI HI 7-4 17 light hours 14 mas or 120 AU from Sgr A*. S2 appears probe for testing GR by diffraction-limited imaging and spec- to be a single≈ star (Martins et al. 2008; Habibi et al. 2017; troscopy (Alexander 2005; Zucker et al. 2006; Parsa et al. 2017), GRAVITY Collaboration 2017; Chu et al. 2018), making it an and interferometry (Grould et al. 2017) through the deviation of S2-SINFONI ideal probe for testing GR by diffraction-limited imaging and its apparent motion from a Keplerian orbit. spectroscopy (Alexander 2005; Zucker et al. 2006; Parsa et al. The radio source Sgr A* is coincident with the mass cen- 14 mas 2017), and interferometry (Grould et al. 2017) through the devi- 100 AU troid to < 1 mas (Plewa et al. 2015), and is itself very com- 1300 R ation of its apparent motion from a Keplerian orbit. 55 mas S2 S pact, with R (1.3 mm) < 18 µas 1.8 RS , based on millimetre The radio source Sgr A* is≈ coincident with the mass cen- S2 troidvery tolong<1 baseline mas (Plewa interferometry et al. 2015), (Falcke and is itself et al. very 2000; compact, Doele- man et al. 2008; Johnson et al. 2017). In addition, Sgr A* shows dirty beam with R (1.3 mm) < 18 µas 1.8 RS, based on millimetre very 2.2 x 4.7 mas July 2017 May / June 2018 longno detectable baseline interferometry intrinsic motion,≈ (Falcke which et supports al. 2000; the Doeleman interpretation et al. that the compact radio source is coincident with the mass (Reid 2008; Johnson et al. 2017). In addition, Sgr A* shows no de- Fig. 1. Monitoring the S2 orbit around Sgr A* with the three VLT(I) tectable& Brunthaler intrinsic 2004; motion, Reid 2009). which The supports most conservative the interpretation explanation for Sgr A* is that it is an MBH, assuming that GR is appli- instrumentsinstruments NACO NACO (AO-assisted, (AO-assisted, single single UT UT imaging), imaging), GRAVITY GRAVITY (in- (in- that the compact radio source is coincident with the mass terferometricterferometric astrometry-imaging astrometry-imaging with with all all four four UTs UTs of of the the VLT) VLT) and and cable (Genzel et al. 2010; Falcke & Markoff 2013; Vincent et al. (Reid & Brunthaler 2004; Reid 2009). The most conservative SINFONI (AO-assisted integral integral field field spectroscopy). spectroscopy). UpperUpper left: left Decon-: De- explanation2016). So far, for Newtonian Sgr A* is orbits that in it a is single an MBH, central assumingforce potential that convolvedvolved NACO NACO K-bandK-band image image of the of the Galactic Galactic centre centre a few a few weeks weeks be- GRcan describeis applicable the motions (Genzel of et all al.stars. 2010 Any; Falcke extended & Marko massff within2013; beforefore the the 2018 2018 pericentre pericentre passage. passage. The The source source S2 S2 appears slightly elon- elon- Vincentthe S2 orbit et al.is 2016 lower). So than far, about Newtonian 1 % of theorbits central in a mass single (Hees cen- gatedgated because because of of confusion confusion with with Sgr Sgr A*. A*.Upper Upper right right:: Nearly simulta- simulta- et al. 2017; Gillessen et al. 2017). neousneous GRAVITY GRAVITY image image of of S2 S2 and and Sgr Sgr A*. A*. The The image image shows shows the the central central tral force potential can describe the motions of all stars. Any 44 150150 mas (0.0059 pc 1.4 1010 RRS)) after after self-calibration self-calibration and and CLEAN- CLEAN- ≈ × S extended mass within the S2 orbit is lower than about 1% of the inging with with AIPS. AIPS. The The image≈ image× is reconstructedis reconstructed from from 34 integrations 34 integrations of 5 ofmin 5 central mass (Hees et al. 2017; Gillessen et al. 2017). 2. Observations eachminutes from each several from nights several at the nights end ofat Marchthe end 2018. of March Sgr A* 2018. was (on Sgr av- A* erage)was (onK average)= 16.6 magK = and16 the.6 mag rms and noise the level rms of noise the level background of the back- after We present1 an analysis of the positions and K-band spectra of cleaningground after is 20 cleaning mag. Both is 20 Sgr mag. A* andBoth S2 Sgr are A* unresolved. and S2 are Here unresolved. and in 2. Observations ≈ ≈ the star S2 from 1992 to 2018 (Figs. 1 and 2). theHere other and GRAVITY in the other images, GRAVITY the elongation images,the is due elongation to the shape is due of the to thein- 1 WeWe present obtainedan analysis sky-projected of the positions andof theK-band star S2 spectra with the of terferometricshape of the interferometric clean beam. Bottom clean: S2 beam. – Sgr Bottom: A* GRAVITY S2 - Sgr images A* GRAV- (co- thespeckle star S2 camera from SHARP 1992 to at2018 the (Figs. NTT1 (1992-2002: and2). Hofmann et al. additionITY images of several (co-addition days) offrom several July days) 2017 from(bottom July left 2017) and (bottom May / June left) 2018,and May a few/ June days 2018, after a pericentre few days ( afterbottom pericentre right). The (bottom inset right).in the bot- The 1993),We butobtained most sky-projected of our imaging positions comes of from the starthe AO-assistedS2 with the speckle camera SHARP at the NTT (1992–2002: Hofmann et al. tominset left in thepanel bottom shows left the panel instantaneous shows the interferometric instantaneous beam, interferometric which is NIR imager NACO at the VLT (2002-2018: Lenzen et al. 2beam,.2 mas which4.7 mas is 2.2 without mas 4 Earth.7 mas rotation. without The Earth inset rotation. in the The middle inset left in × 19931998;), Rousset but most et of al. our 1998) imaging and the comes interferometric from the AO-assisted astrometry- showsthe middle a× co-added left shows SINFONI a co-addedK-band SINFONI spectrum K-band of the spectrum star S2, taken of the from star NIRimager imager GRAVITY NACO withat the all VLT four (2002–2018: Unit Telescopes Lenzen (UTs) et al. of1998 the; HabibiS2, taken et al. from(2017 Habibi). et al. (2017). RoussetVLT interferometer et al. 1998) (GRAVITY and the interferometric Collaboration astrometry-imager et al. 2017). The GRAVITYSHARP/NACO with all data four deliver Unit Telescopes relative positions (UTs) of between the VLT stars inter- in ferometerthe nuclear (GRAVITY star cluster. Collaboration These are then 2017 registered). The SHARP with /NACO1 mas fibresolved (see sources GRAVITY in > 90 Collaboration % of our individual 2017). integrations S2 and Sgr(5 A* min- are ≤ dataprecision deliver in relative the radio positions frame of between the Galactic stars in centre the nuclear (Reid et star al. simultaneouslyutes each), such detected that the S2 as -two Sgr unresolved A* vector is sources directly in obtained> 90% cluster. These are then registered with 1 mas precision in the of our individual integrations (5 min each), such that the S2 – 2007) using multi-epoch observations of≤ nine SiO maser stars in each of these measurements. radiocommon frame between of the Galactic our infrared centre data (Reid and et the al. 2007 radio) interferome-using multi- Sgr A*Our vector 2003-2018 is directly measurements obtained in of each the Brackett-of these measurements.γ line velocity epochtry, after observations correcting ofNACO nine image SiO maser distortions stars common with observations between wereOur taken 2003–2018 with the AO-assistedmeasurements integral of the field Brackett- spectrometerγ line veloc- SIN- ourof a infrared globular data cluster and calibrated the radio on interferometry, data from the after Hubble correcting Space ityFONI were at the taken VLT with (Eisenhauer the AO-assisted et al. 2003a; integral Bonnet field et spectrome- al. 2004), NACOTelescope image (Plewa distortions et al. 2015). with observations In the GRAVITY of a globular interferomet- cluster terwith SINFONI five additional at the 2000-2003VLT (Eisenhauer slit-spectra et al. from2003a the; Bonnet AO imagers et al. calibratedric observations, on data we from detected the Hubble and Space stabilised Telescope the interferometric (Plewa et al. 2004and spectrometers), with five additional NIRC2 2000–2003at Keck (see slit-spectra Ghez et al. from 2003; the Chu AO 2015). In the GRAVITY interferometric observations, we de- fringes on the stars IRS16C or IRS16NW located 100 from imagerset al. 2018) and and spectrometers NACO (Eisenhauer NIRC2 at et Keck al. 2003b). (see Ghez et al. 2003; tected and stabilised the interferometric fringes on≈ the stars Chu et al. 2018) and NACO (Eisenhauer et al. 2003b). Sgr A*, and observed the “binary” S2 - Sgr A* within the sec- The 1992-2016 speckle and AO-imaging and spectroscopic IRS16C or IRS16NW located 100 from Sgr A*, and observed The 1992–2016 speckle and AO-imaging and spectroscopic ond phase-referenced fibre (see≈ GRAVITY Collaboration et al. data used below have been presented in Gillessen et al. (2017). In the “binary” S2 – Sgr A* within the second phase-referenced data used below have been presented in Gillessen et al.(2017). 2017). S2 and Sgr A* are simultaneously detected as two unre- 2017 and 2018 we increased the cadence of the observations in In 2017 and 2018 we increased the cadence of the observa- 1 Based on observations made with ESO Telescopes at the La Silla preparation for the pericentre approach in May 2018. We added 1 Based on observations made with ESO Telescopes at the La Silla tions in preparation for the pericentre approach in May 2018. Paranal Observatory under programme IDs 075.B-0547, 076.B-0259, 21 epochs of NACO K- and H-band imaging in the 13 mas/pix Paranal Observatory under programme IDs 075.B-0547, 076.B-0259, 077.B-0014, 078.B-0136, 179.B-0261, 183.B-0100, 087.B-0117, We added 21 epochs of NACO K- and H-band imaging in the 077.B-0014, 078.B-0136, 179.B-0261, 183.B-0100, 087.B-0117, scale, and 21 epochs of NACO K-band imaging in the 27 mas/pix 088.B-0308, 288.B-5040, 089.B-0162, 091.B-0081, 091.B-0086, 13 mas pix− scale, and 2 epochs of NACO K-band imaging 088.B-0308, 288.B-5040, 089.B-0162, 091.B-0081, 091.B-0086, scale to measure the1 SiO maser positions (Reid et al. 2007) that 091.B-0088, 092.B-0238, 092.B-0398, 093.B-0217, 093.B-0218, in the 27 mas pix− scale to measure the SiO maser positions 091.B-0088, 092.B-0238, 092.B-0398, 093.B-0217, 093.B-0218, define our coordinate system (Plewa et al. 2015). We obtained 30 594.B-0498, 097.B-0050, 598.B-0043, 299.B-5014, 299.B-5056, (Reid et al. 2007) that define our coordinate system (Plewa et al. 594.B-0498, 097.B-0050, 598.B-0043, 299.B-5014, 299.B-5056, data sets of GRAVITY interferometry, and 26 additional spec- 099.B-0162, 0100.B-0731, 0101.B-0195, and 0101.B-0576. 2015). We obtained 30 data sets of GRAVITY interferometry, 099.B-0162, 0100.B-0731, 0101.B-0195, and 0101.B-0576. troscopy epochs with SINFONI using the 25 mas/pix scale and L15, page 2 of 10 Article number, page 2 of 10 GRAVITYGRAVITY Collaboration: Collaboration: R. Abuter Detection et al.: Detection of gravitational of gravitational redshift redshift

SHARP (corr.) 4000 0.20 NACO (corr.) GRAVITY NIRC2 SINFONI 2000

0.15 0 Year Radial velocity (km/s)

2015

−2000 2010 2000 2005 2010 2015 2020 0.10 5000 R = 400 AU S Year 2005

S2 − GRAVITY 2000 2018 1995 0.05 Jun 23 SgrA* Dec. offset from Sgr A* (arcsec) 0.00 1990

Jun 3 Jun 1 May 30 May 27 Peri−centre SgrA* May 24 0.00 −0.01 March 28 May 4 31 May Apr2 Apr 28 26 500 R = 40 AU S Dec. offset from Sgr A* (arcsec)

0.05 0.00 −0.05 0.02 0.01 0.00 −0.01 −0.02 R.A. offset from Sgr A* (arcsec) R.A. offset from Sgr A* (arcsec)

Fig.Fig. 2. 2.SummarySummary of of the the observational observational results results of of monitoring monitoring the the S2 S2 - –Sgr Sgr A* A* orbit orbit from from 1992 1992 to to 2018. 2018. Left:Left Projected: projected orbit orbit of of the the star star S2 S2 on on thethe sky sky (J2000) (J2000) relative relative to to the the position position of of the the compact compact radio radio source source Sgr Sgr A* A* (brown (brown crossed crossed square square at at the the origin). origin). Triangles Triangles and and circles circles (and (and 1 1σσ uncertainties)uncertainties) denote denote the the position position measurements measurements with with SHARP SHARP at at the the NTT NTT and and NACO NACO at at the the VLT, VLT, colour-coded colour-coded for for time time (colour (colour bar bar on on the the right right , , side).side). All All data data points points are are corrected corrected for for the the best-fit best-fit zero-point zero-point (x (0x,0y0y)0 and) and drifts drifts (x ˙ (0x ˙,0y˙0y˙)0 of) of the the coordinate coordinate system system relative relative to to Sgr Sgr A* A* (see (see Plewa Plewa et et al. al. 2015).2015). Green Green squares squares mark mark the the GRAVITY GRAVITY measurements. measurements. The The bottombottom right right panel panel showsshows a zooma zoom around around pericentre pericentre in in 2018. 2018. TopTop right: right Radial: radial velocityvelocity of of S2 S2 as as a afunction function of of time time (squares: (squares: SINFONI SINFONI/NACO/NACO at at the the VLT; VLT; triangles: triangles: NIRC2 NIRC2 at at Keck). Keck). S2 S2 reached reached pericentre pericentre of of its its orbit orbit at at the the end end ofof April April 2002, 2002, and and then then again again on on May May 19 19th,th, 2018 2018 (MJD (MJD 58257.67). 58257.67). The The data data before before 2017 2017 are are taken taken from from Ghez Ghez et et al. al. (2008),(2008), Boehle Boehle et et al. al. (2016),(2016), ChuChu et et al. al. (2018),(2018), and and GillessenGillessen et et al. al.( (2009b,2017, 2009b 2017).). The 2017/2018/2018 NACO NACO/SINFONI/SINFONI and and GRAVITY GRAVITY data data are are presented presented here here for for the the first first time. time. TheThe cyan cyan curve curve shows shows the the best-fitting best-fitting S2 S2 orbit orbit to to all all these these data, data, including including the the eff eectsffects of of General General and and Special Special Relativity. Relativity. theand combined 26 additional H+K-band spectroscopy grating epochs with a with spectral SINFONI resolution using of the displaysmotion theof the radial star velocityfrom day measurements to day. The upper with right SINFONI panel of at Fig. the2 1 R251500. mas pix− scale and the combined H + K-band grating with a VLTdisplays and NIRC2 the radial at Keck velocity in the measurements 1992-2018 period. with SINFONI at the spectral≈ resolution of R 1500. VLT and NIRC2 at Keck in the 1992–2018 period. For more details on the≈ data analysis of all three instruments, At pericentre Rperi, S2 moves with a total space velocity For more details on the data analysis of all three instruments, At pericentre Rperi, S2 moves with a2 total space veloc- we refer to Appendix A. of 7650 km/s, or β = v/c = 2.55 10− . This means that we refer to AppendixA. ity≈ of 7650 km s 1, or β = v/c = 2.55× 10 2. This means that the first-order≈ parameterised− post-Newtonian× − correction terms (PPN(1)),the first-order due to parameterised Special and General post-Newtonian Relativity, correction beyond the terms or- 3. Results bital(PPN(1)), Doppler due and to Rømer Special eff andects, General are within Relativity, reach of beyond current the mea- or- 3. Results bital Doppler and Rømer effects,2 are within reach of current4 surement precision, PPN(1) β (RS /Rperi) 6.5 10− . 3.1. Relativistic corrections measurement precision, PPN(1)∼ ∼β2 (R /R ∼) 6.5× 10 4. 3.1. Relativistic corrections These terms can be parameterised spectroscopically∼ ∼ S peri as∼ (e.g.× Mis-− The left panel of Fig. 2 shows the combined single-telescope nerThese et al. terms 1973; canAlexander be parameterised 2005; Zucker spectroscopically et al. 2006). as (e.g. andThe interferometric left panel of Fig. astrometry2 shows of the the combined 1992-2018 single-telescope sky-projected Misner et al. 1973; Alexander 2005; Zucker et al. 2006). and interferometric astrometry of the 1992–2018 sky-projected ∆λ orbital motion of S2, where the zero point is the position of the ∆λ 2 2 3 3 orbital motion of S2, where the zero point is the position of the z z== ==BB0 0++BB0.50.β5β++BB1β1β++ (β(β),), (1)(1) central mass and of Sgr A*. All NACO points were corrected λλ OO forcentral a zero-point mass and offset of and Sgr driftA*. Allin R.A. NACO/Dec., points which were are obtained corrected for a zero-point offset and drift in RA/Dec, which are obtained where the PPN(1)z termB =B1 =+B1B,tD + B1,gr,B with= B1,tD== B. 1,gr from the orbit fit. The bottom right panel zooms into the 2018 where the PPN(1)2 z term 1 1,tD 1,gr, with 1,tD 4 1,gr 0 5, from the orbit fit. The bottom right panel zooms into the 2018 = 0.5,2 and β = [RS(1 + e)]/[2a(1 e)] = 64.51 10− for S2. Here section of the orbit around pericentre measured with GRAVITY. and β =[Rs(1+e)]/[2a(1 e)]=6.−51 10− for× S2. Here a is the section of the orbit around pericentre measured with GRAVITY. a is the semi-major axis− and e is the× eccentricity of the S2 orbit. The zoom demonstrates the hundred-fold improvement of as- semi-major axis and e is the eccentricity of the S2 orbit. B0.5β is B β is the Newtonian Doppler shift. trometryThe zoom between demonstrates SHARPthe in the hundred-fold 1990s ( 4 masimprovement precision) of and as- the0 Newtonian.5 Doppler shift. trometry between SHARP in the 1990s≈ ( 4 mas precision) and Equation (1) indicates that PPN(1)z consists in equal terms NACO in the 2000s ( 0.5 mas) to GRAVITY≈ in 2018 (as small Eq. (1) indicates that PPN(1)z consists in equal terms of the NACO in the 2000s≈ ( 0.5 mas) to GRAVITY in 2018 (as small of the special relativistic transverse Doppler effect (B1,tD) and as 30 µas). While the≈ motion on the sky of S2 could be detected special relativistic transverse Doppler effect (B1,tD) and the gen- as≈ 30 µas). While the motion on the sky of S2 could be detected the general relativistic gravitational redshift (B1,gr), totalling with NACO over a month, the GRAVITY observations detect the eral relativistic gravitational redshift (B1,gr), totalling 200 km/s with≈ NACO over a month, the GRAVITY observations detect the 200 km s 1 redshift at pericentre, while at apocentre,≈ it amounts motion of the star from day to day. The upper right panel of Fig. 2 redshift≈ at pericentre,− while at apocentre, it amounts to only Article number,L15, page page 3 3 of of 10 10 A&A proofs: manuscriptA&A no. 615, s2_20180702_revised_language_edited2 L15 (2018)

3.3.3.3. Posterior Posterior analysis Analysis

Because of the uncertainties in the parameters of Kprior, in par- Because of the uncertainties in the parameters of Kprior, in partic- ticular,ular, in in the the strongly strongly correlated correlated mass mass and distance, and distance, a more a conser- more conservativevative approach approach is to determine is to determine the best-fit the best-fit value of value the parame- of the 200 parameterter f a posteriorif a posteriori,, including including all data all data and fittingand fitting for the for optimum the op- timumvalues values of allparameters. of all parameters. In carrying In carrying out the out fitting, the itfitting, is essential it is essentialto realise to that realise the inferredthat the inferredmeasurement measurement uncertainties uncertainties are domi- arenated dominated by systematic by systematic effects, especially effects, especially when evidence when fromevidence three from three or more very different measurement techniques is (km/s) or more very different measurement techniques is combined (see combinedAppendix (see A.6 Appendix for a more A.6 detailed for a more discussion). detailedIn discussion). particular In the

z Kprior 100 particularNACO measurements the NACO measurements are subject are to subject correlated to correlated systematic sys- er- −v

z tematic errors, for example from unrecognised confusion events

v rors, for example from unrecognised confusion events (Plewa & (PlewaSari 2018), & Sari which 2018), typically which typically last for one last year for one and year are compara- and are comparableble in size in to size the statistical to the statistical errors. errors. We therefore We therefore down-sampled down- sampledthe NACO the dataNACO into data 100 into bins 100 with bins equal with path equal lengths path along lengths the along the projected orbit (Fig.4, middle) and gave these data 0 projected orbit (Fig. 4, middle) and gave these data in addition ina addition lower weight a lower of weight 0.5. Depending of 0.5. Depending on exactly on which exactly weighting which weightingor averaging or averaging scheme was scheme chosen, was the chosen, posterior the posterior analysis includ- anal- ysising including all data between all data between 1992 and 1992 2018 and yielded 2018 yieldedf valuesf betweenvalues . . 2015 2017 2019 between0.85 and 0 85 1.09. and With 1 09. a Withweighting a weighting of 0.5 of of the 0.5 NACO of the data,NACO we data, we find f = 0.90 0.09 (Fig.4). GR ( f = 1) is favoured over Year find f = 0.90 0.09± (Fig. 4). GR ( f = 1) is favoured over pure pureNewtonian Newtonian physics± physics ( f = (0)f = at0) the at the10 σ10level.σ level. Fig.Fig. 3. 3.ResidualResidual velocity velocityc∆c∆zz==cc(z(zGRGR zzKK)) for for the the best best fitting fitting prior prior Keple- Ke- ≈ − TheThe error error on onf fisis derived derived from from≈ the the posterior posterior probability probability plerianrian K Kpriorprior( f(=f =0,0, grey) grey) and and the the same same orbit orbit with withf =1f = (red1 (red GRprior GR).prior Kprior). distributions (Fig.4, bottom) of a Markov chain Monte Carlo Kwasprior was constructed constructed from from all all 1992-2018 1992–2018 astrometric astrometric data data with with NACO NACO & distributions (Fig. 4, bottom) of a Markov chain Monte Carlo &GRAVITY GRAVITY and and the the SINFONI SINFONI data data between between 2004 2004 and and 2016 2016 (open (open black (MCMC)(MCMC) analysis. analysis. Fig. Fig. A.1 A.1 shows shows the the full full set set of of correlation correlation plots plots blackcircles). circles). The The 2017 2017/2018/2018 SINFONI SINFONI data data points points (black (black circles circles with with cyan andand probability probability distributions distributions for for the the fit fit parameters. parameters. The The distribu- distribu- cyanshading) shading) can can then then be added be added to test to test if the if thespectroscopic spectroscopic data data around around peri- tionstions are are compact compact and and all all parameters parameters are are well well determined. determined. The The pericentrecentre follow follow K Kpriorpriororor the the GR GRpriorprior predicted from from K Kpriorprior.. The The new new data data best-fit values and uncertainties are given Table A.1. 2 2 best-fit values and uncertainties are given Table A.1. pointspoints near near and and up up to to pericentre, pericentre, where where the theβ βeffeectsffects in in radial radial velocity velocity TheThe superb superb GRAVITY GRAVITY astrometry astrometry demonstrably demonstrably improves improves areare expected expected to to be be important, important, fall fall close close to to the the predicted predicted GR GRpriorpriorcurve,curve, the quality of the fits and is crucial for overcoming the source andand exclude exclude the the Keplerian Keplerian prior prior orbit. orbit. the quality of the fits and is crucial for overcoming the source confusionconfusion between between Sgr Sgr A* A* and and S2 near S2 near pericentre. pericentre. A minimal A minimal detection of PPN(1)z (Eq. (1)) is provided by a combination using detection of PPN(1)z (Eq. (1)) is provided by a combination us- only NACO and SINFONI data ( fNACO + SINFONI = 0.71 0.19, ing only NACO and SINFONI data ( fNACO+SINFONI =0.71± 0.19, 1 3.6 σ), but the inclusion of the GRAVITY data very significantly± to6 onlykm/s. 6 If km the s− total. If orbital the total redshift orbitalz redshiftis separatedztot is into separated a New- 3.6 σ), but the inclusion of the GRAVITY data very significantly tot improves the precision and significance of the fitted parameters: intotonian a Newtonian/Kepler part/Keplerz and part a GRzK correction,and a GR one correction, can write onez = improves the precision and significance of the fitted parameters: K tot the improvement reaches a factor of 2–3. canz + writef (z ztotz =),z whereK + f (fzGRis zerozK for), where purely Newtonianf is zero for physics purely and the improvement reaches a factor of 2 to 3. K GR K − A still more demanding test is to search for any Keplerian Newtonianunity for GR.− physics In the andfollowing unity we for show GR. the In residuals the following∆z=z wez . A still more demanding test is to search for any Keplerian GR− K fit to all data and determine whether its goodness of fit is signifi- showThe Keplerian the residuals part∆ ofz the= zGR orbitz isK. atThe∆z= Keplerian0, and the partPPN(1) of thecor- fit to all data and determine whether its goodness of fit is signifi- − z cantly poorer than the goodness of fit of the best-fitting GR-orbit. orbitrections is at appear∆z = 0, as and an excess. the PPN(1)z corrections appear as an cantly poorer than the goodness of fit of the best-fitting GR-orbit. For linear models the formula presented in Andrae et al.(2010) excess. For linear models the formula presented in Andrae et al. (2010) can be used to estimate the significance. However, the value for can be used to estimate the significance. However, the value for the degrees of freedom (d.o.f.) is not well defined for non-linear 3.2. Analysis with prior Kepler orbit the degrees of freedom (d.o. f.) is not well defined for non-linear 3.2. Analysis with prior Kepler orbit models (Andrae et al. 2010). In our case, we have two models models (Andrae et al. 2010). In our case, we have two models that only differ significantly over a very critical short time-span WeWe define define a aprior prior orbit orbit K Kpriorbyby excluding excluding those those data data for for which which that only differ significantly over a very critical short time-span prior given the uncertainties in the underlying data. We therefore used thethe PPN(1) PPN(1)z corrections matter. matter. For For Kprior K wewe use use the entire the entire 1992- given the uncertainties in the underlying data. We therefore used z prior the number of those data points as d.o.f. for which the two mod- 1992–20182018 SHARP SHARP/NACO/NACO and GRAVITYdata and GRAVITY and data the and SINFONI the SIN- data the number of those data points as d.o. f. for which the two models predict significant differences. The difference in χ2 yields a FONIfrom data 2004 from up to 2004 the up end to of the 2016. end of We 2016. then We obtained then obtained Kprior as els predict significant differences. This is, in effect, the number formal significance of 5 σ or greater in favour of the relativistic Kdescribedas described in Gillessen in Gillessen et al.(2017), et al.(2017 which), which requires requires a simulta- a si- of SINFONI measurements in 2017 and 2018, that is, 26 mea- prior model. multaneousneous fit of fit 13 of parameters. 13 parameters. The TheRømer Rømer delay delay is included is included in the surements. The difference in χ2 amounts to 87 in favour of the For further comments on a Bayesian analysis of our data, see incalculation. the calculation. The resulting The resulting orbit is orbit a modest is a modestupdate of update Gillessen of relativistic model and therefore yields a formal significance of Appendix A.9. Gillessenet al. (2017). et al.(2017 Using). this Using as thisthe prior as the orbit, prior we orbit, then we added then the 8.5 σ. For further comments on a Bayesian analysis of our data, addedradial the velocities radial velocities from 2017 from and 2017 2018 and (Fig. 2018 3). (Fig. The3). 26 The residual 26 see Appendix A.9. residual2017/2018 2017 spectroscopic/2018 spectroscopic data relative data to relative Kprior clearly to Kprior doclearly not fol- dolow not the follow best-fitting the best-fitting Keplerian Keplerian orbit derived orbit from derived all previous from all 51 4. Discussion previousspectroscopic 51 spectroscopic and 196 positions and 196 in positions the past 26in the years past (grey 26 years line in We4. have Discussion reported the first direct detection of the PPN(1) gravita- (greyFig. line 3), but in Fig. instead3), but follow instead the followf = 1 the (i.e.f GR(K= 1 (i.e.prior GR(K)) version)) of prior tionalWe have redshift reported parameter the first around direct the detection MBH in of the the Galactic PPN(1) centre gravi- versionKprior (red ofK line in(red Fig. 3). line This in testFig.3 is). fair: This GR-corrections test is fair: GR- should prior fromtational a data redshift set that parameter extends around up to the and MBH includes in the the Galactic pericen- cen- correctionsonly be detectable should only with be our detectable measurement with our errors measurement within 1 yearer- ff ± tretre approach from a data in May set that 2018. extends Three up di toerent and analysis includes methods the pericen- of rorsofpericentre. within 1 year of pericentre. ourtre data approach suggest in that May this 2018. detection Three favoursdifferent the analysis post-Newtonian methods of This a priori± test demonstrates that the spectroscopic data This a priori test demonstrates that the spectroscopic data modelour data with suggest robust that significance. this detection Further favours improvement the post-Newtonian of our around the pericenter passage are inconsistent with Newtonian results is expected as our monitoring continues post pericen- around the pericenter passage are inconsistent with Newtonian2 model with robust significance. Further improvement of our re- dynamicsdynamics and and consistent consistent with with GR. GR. However, However, both both Kprior K (χ(rχ=2 =21)21) tre. Still, there are reasons to be cautious about the signifi- 2 prior r sults is expected as our monitoring continues post pericentre. and GR(Kprior)(χr =2 8) are poor fits to the data. cance of these early results, mainly because of the systematic and GR(Kprior)(χr =8 ) are poor fits to the data. Still, there are reasons to be cautious about the significance of

L15,Article page number, 4 of 10 page 4 of 10 GRAVITY Collaboration: Detection of gravitational redshift

Fig. 4. Posterior analysis of all data by fitting for f simultaneously with all other parameters. We plot the residuals in spectroscopy (top, NIRC2, NACO, and SINFONI), Dec and RA (middle two panels, filled grey: NACO; open grey: SHARP; green filled blue: GRAVITY) between the best f = 1 fit and the f = 0 (Newtonian) part of that fit for the model (red line) and all data. The black curve includes the Schwarzschild precession. Here, we down-sampled the NACO data into 100 equal bins along the orbit to obtain a constant weighting in spatial coverage. With a weight of 0.5 for the NACO data (in order to account for the systematic errors), this yields a 10 σ result in favour of GR ( f = 0.90 0.09), and χ2 = 0.86. The ± r bottom panel shows the posterior probability distributions for f and its correlation with the mass M and distance R0 of the massive black hole, and the argument of periapsis ω. The distributions are compact and all parameters are well determined.• effects and the validity of our basic assumptions and model. The tion) could still be commensurable with our full new data most important concern probably is that our basic input model set. We find that such an extended mass is lower than 0.35– (a binary consisting of an MBH and a star with much lower 1% of the central mass, depending on the assumed Plummer mass) is incomplete. Additional “luminous” and massive ob- radius. jects around S2 and between S2 and Sgr A* are unlikely given The next relativistic correction term we hope to detect is the the spectroscopic and imaging data. Based on the radial ve- Schwarzschild precession, which per orbital revolution is locities of S2, Chu et al.(2018) excluded any companion with 3πRS Msin(i) < 1.6 M for periods up to 150 days, the longest period ∆Φper orbit = radians 120 for S2. (2) a(1 e2) ≈ for which the binary is not subject to tidal break-up. The GRAV- − ITY imaging data (Fig.1) so far do not show any object near Since the precession is strongly dependent on distance from the Sgr A* and S2 brighter than K 18.5 mag, corresponding to a black hole and S2 is on a highly elliptical orbit, the term man- 2 M main-sequence star. However,≈ massive, non-luminous ob- ifests itself as a kink between the incoming near-Keplerian and jects, such as stellar black holes, might be present and could the outgoing near-Keplerian orbit. In addition, it leads to a west- affect the orbital dynamics of S2 (Gualandris & Merritt 2009; ward drift of all data points around apocentre. The posterior Merritt et al. 2010; Gualandris et al. 2010). We repeated the ex- fit of the current data including the Schwarzschild precession ercise by Gillessen et al.(2017) of testing how much of an yields an f -value still closer to GR than without the preces- extended mass distribution (in form of a Plummer distribu- sion term ( f = 0.94 0.09). The chances for robustly detecting ±

L15, page 5 of 10 A&A 615, L15 (2018) the Schwarzschild precession with further observations are very Mackay, D. J. C. 2003, Information Theory, Inference and Learning Algorithms, high. GRAVITY will continue to be critical for this second phase 640 of the experiment. Our forecast suggests that we will obtain a 5 σ Martins, F., Gillessen, S., Eisenhauer, F., et al. 2008, ApJ, 672, L119 Merritt, D., Alexander, T., Mikkola, S., & Will, C. M. 2010, Phys. Rev. D, 81, posteriori result with GRAVITY by 2020 (Grould et al. 2017). 062002 Meyer, L., Ghez, A. M., Schödel, R., et al. 2012, Science, 338, 84 Acknowledgements. We are very grateful to our funding agencies (MPG, ERC, Misner, C. W., Thorne, K. S., & Wheeler, J. A. 1973, Gravitation CNRS, DFG, BMBF, Paris Observatory, Observatoire des Sciences de l’Univers Morris, M. R., Meyer, L., & Ghez, A. M. 2012, Res. Astron. Astrophys., 12, de Grenoble, and the Fundação para a Ciência e Tecnologia), to ESO and the 995 ESO/Paranal staff, and to the many scientific and technical staff members in Parsa, M., Eckart, A., Shahzamanian, B., et al. 2017, ApJ, 845, 22 our institutions who helped to make NACO, SINFONI, and GRAVITY a reality. Plewa, P. M., Gillessen, S., Eisenhauer, F., et al. 2015, MNRAS, 453, 3234 S.G., P.P., C.D., N.B., and Y.K. acknowledge support from ERC starting grant Plewa, P. M., & Sari, R. 2018, MNRAS, 476, 4372 No. 306311. F.E. and O.P. acknowledge support from ERC synergy grant No. Pound, R. V., & Rebka, G. A. 1959, Phys. Rev. Lett., 3, 439 610058. We also would like to acknowledge the important theoretical contribu- Psaltis, D. 2004, in X-ray Timing 2003: Rossi and Beyond, eds. P. Kaaret, F. K. tions of the late Tal Alexander (Weizmann Institute, Rehovot), whose 2006 paper Lamb, & J. H. Swank, American Institute of Physics Conference Series, 714, with Shay Zucker and members of the MPE group encouraged us to pursue this 29 project. Unfortunately, Tal missed seeing the fruits of this effort by only a few Reid, M. J. 2009, Int. J. Mod. Phys. D, 18, 889 weeks. Reid, M. J., & Brunthaler, A. 2004, ApJ, 616, 872 Reid, M. J., Menten, K. M., Trippe, S., Ott, T., & Genzel, R. 2007, ApJ, 659, 378 References Rousset, G., Lacombe, F., Puget, P., et al. 1998, in Adaptive Optical System Technologies, eds. D. Bonaccini, & R. K. Tyson, Proc. SPIE, 3353, 508 Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016a, ApJ, 818, L22 Sabha, N., Eckart, A., Merritt, D., et al. 2012, A&A, 545, A70 Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016b, Phys. Rev. Lett., 116, Scheithauer, S., Brandner, W., Deen, C., et al. 2016, in Adaptive Optics Systems 241103 V, Proc. SPIE, 9909, 99092L Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016c, Phys. Rev. Lett., 116, Schödel, R., Ott, T., Genzel, R., et al. 2002, Nature, 419, 694 061102 Schödel, R., Merritt, D., & Eckart, A. 2009, A&A, 502, 91 Alexander, T. 2005, Phys. Rep., 419, 65 Schödel, R., Gallego-Cano, E., Dong, H., et al. 2018, A&A, 609, A27 Andrae, R., Schulze-Hartung, T., & Melchior, P. 2010, ArXiv e-prints Tanaka, Y., Nandra, K., Fabian, A. C., et al. 1995, Nature, 375, 659 [arXiv: 1012.3754] Tatulli, E., Millour, F., Chelli, A., et al. 2007, A&A, 464, 29 Anugu, N., Amorim, A., Gordo, P., et al. 2018, MNRAS, 476, 459 Taylor, J. H., & Weisberg, J. M. 1982, ApJ, 253, 908 Bardeen, J. M., & Petterson, J. A. 1975, ApJ, 195, L65 Thiébaut, E. 2008, in Optical and Infrared Interferometry, Proc. SPIE, 7013, Baron, F., Monnier, J. D., & Kloppenborg, B. 2010, in Optical and Infrared 70131I Interferometry II, Proc. SPIE, 7734, 77342I Vincent, F. H., Meliani, Z., Grandclément, P., Gourgoulhon, E., & Straub, O. Barstow, M. A., Bond, H. E., Holberg, J. B., et al. 2005, MNRAS, 362, 2016, Class. Quant. Grav., 33, 105015 1134 Waisberg, I., Dexter, J., Gillessen, S., et al. 2018, MNRAS, 476, 3600 Boehle, A., Ghez, A. M., Schödel, R., et al. 2016, ApJ, 830, 17 Will, C. M. 2008, ApJ, 674, L25 Bonnet, H., Abuter, R., Baker, A., et al. 2004, The Messenger, 117, 17 Will, C. M. 2014, Liv. Rev. Rel., 17, 4 Chu, D. S., Do, T., Hees, A., et al. 2018, ApJ, 854, 12 Zucker, S., Alexander, T., Gillessen, S., Eisenhauer, F., & Genzel, R. 2006, ApJ, Doeleman, S. S., Weintroub, J., Rogers, A. E. E., et al. 2008, Nature, 455, 78 639, L21 Einstein, A. 1916, Ann. Phys., 354, 769 Eisenhauer, F., Abuter, R., Bickert, K., et al. 2003a, in Instrument Design and Performance for Optical/Infrared Ground-based Telescopes, eds. M. Iye, & 1 A. F. M. Moorwood, Proc. SPIE, 4841, 1548 Max Planck Institute for extraterrestrial Physics, Giessenbach- Eisenhauer, F., Schödel, R., Genzel, R., et al. 2003b, ApJ, 597, L121 straße 1, 85748 Garching, Germany Eisenhauer, F., Genzel, R., Alexander, T., et al. 2005, ApJ, 628, 246 2 LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Fabian, A. C., Iwasawa, K., Reynolds, C. S., & Young, A. J. 2000, PASP, 112, Université, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules 1145 Janssen, 92195 Meudon, France Falcke, H., & Markoff, S. B. 2013, Class. Quant. Grav., 30, 244003 3 Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidel- Falcke, H., Melia, F., & Agol, E. 2000, ApJ, 528, L13 berg, Germany Fritz, T. K., Chatzopoulos, S., Gerhard, O., et al. 2016, ApJ, 821, 44 4 1st Institute of Physics, University of Cologne, Zülpicher Straße 77, Gallego-Cano, E., Schödel, R., Dong, H., et al. 2018, A&A, 609, A26 50937 Cologne, Germany Genzel, R., Schödel, R., Ott, T., et al. 2003, ApJ, 594, 812 5 Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, Rev. Mod. Phys., 82, 3121 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 6 Ghez, A. M., Duchêne, G., Matthews, K., et al. 2003, ApJ, 586, L127 Universidade de Lisboa – Faculdade de Ciências, Campo Grande, Ghez, A. M., Salim, S., Weinberg, N. N., et al. 2008, ApJ, 689, 1044 1749-016 Lisboa, Portugal Gillessen, S., Eisenhauer, F., Fritz, T. K., et al. 2009a, ApJ, 707, L114 7 Faculdade de Engenharia, Universidade do Porto, rua Dr. Roberto Gillessen, S., Eisenhauer, F., Trippe, S., et al. 2009b, ApJ, 692, 1075 Frias, 4200-465 Porto, Portugal Gillessen, S., Plewa, P. M., Eisenhauer, F., et al. 2017, ApJ, 837, 30 8 European Southern Observatory, Karl-Schwarzschild-Straße 2, GRAVITY Collaboration (Abuter, R., et al.) 2017, A&A, 602, A94 85748 Garching, Germany Greenstein, J. L., Oke, J. B., & Shipman, H. L. 1971, ApJ, 169, 563 9 European Southern Observatory, Casilla 19001 Santiago 19, Chile Greisen, E. W. 2003, in Information Handling in Astronomy – Historical Vistas, 10 ed. A. Heck, Astrophysics and Space Science Library, 285, 109 Observatoire de Genève, Université de Genève, 51 Ch. des Mail- Grould, M., Vincent, F. H., Paumard, T., & Perrin, G. 2017, A&A, 608, A60 lettes, 1290 Versoix, Switzerland 11 Gualandris, A., Gillessen, S., & Merritt, D. 2010, MNRAS, 409, 1146 Max Planck Institute for Radio Astronomy, Auf dem Hügel 69, Gualandris, A., & Merritt, D. 2009, ApJ, 705, 361 53121 Bonn, Germany Habibi, M., Gillessen, S., Martins, F., et al. 2017, ApJ, 847, 120 12 Sterrewacht Leiden, Leiden University, Postbus 9513, 2300 RA Lei- Hees, A., Do, T., Ghez, A. M., et al. 2017, Phys. Rev. Lett., 118, 211101 den, The Netherlands Hofmann, R., Eckart, A., Genzel, R., & Drapatz, S. 1993, Ap&SS, 205, 1 13 Departments of Physics and Astronomy, Le Conte Hall, University Högbom, J. A. 1974, A&AS, 15, 417 of California, Berkeley, CA 94720, USA Johnson, M. D., Bouman, K. L., Blackburn, L., et al. 2017, ApJ, 850, 172 14 CENTRA – Centro de Astrofísica e Gravitação, IST, Universidade Kramer, M., Stairs, I. H., Manchester, R. N., et al. 2006, Science, 314, 97 de Lisboa, 1049-001 Lisboa, Portugal Lachaume, R., & Berger, J.-P. 2013, MNRAS, 435, 2501 15 Lapeyrere, V., Kervella, P., Lacour, S., et al. 2014, in Optical and Infrared Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Interferometry IV, Proc. SPIE, 9146, 91462D Dublin 2, Ireland Lenzen, R., Hofmann, R., Bizenberger, P., & Tusche, A. 1998, in Infrared 16 European Space Agency, Space Telescope Science Institute, 3700 Astronomical Instrumentation, ed. A. M. Fowler, Proc. SPIE, 3354, 606 San Martin Drive, Baltimore, MD 21218, USA

L15, page 6 of 10 GRAVITY Collaboration: Detection of gravitational redshift

Appendix A: Supplementary material A.2. GRAVITY observations A.1. NACO and SINFONI data analysis The GRAVITY observations were taken at the Very Large Telescope Interferometer in Chile. The instrument coherently The data reduction and analysis tools are almost identical to what combines the light of the four 8m UTs. We chose the we used in Gillessen et al.(2017), such that in the following sec- most sensitive low spectral resolution mode of GRAVITY tion we concentrate on the specific aspects relevant for the excess (GRAVITY Collaboration 2017). In this mode, the science spec- redshift. We do not use the “combined” data set in the sense of trum is dispersed across 14 pixels with a spectral resolving Gillessen et al.(2009a), that is, we do not include the astromet- power of R 20. Nearly all data were taken in split polarisation ric data set presented in Boehle et al.(2016), but use NTT and mode, with≈ a Wollaston prism inserted in the optical train and VLT astrometry only. This facilitates fitting because it removes the two linear polarisations recorded independently. four fit parameters. All four UTs locked their Coudé infrared AO (CIAO, In 2018, the NACO point spread functions (PSF) for S2 Scheithauer et al. 2016) module on the brightest source in the ff and Sgr A* overlap, such that confusion a ects the S2 posi- field, the red supergiant IRS7 (m 6.5 mag, distance from tions at the mas level, similar to the data in 2002 (Ghez et al. K ≈ Sgr A* 5.500). Active field and pupil guiding was enabled 2008; Gillessen et al. 2009b). Fortunately, the GRAVITY as- (Anugu≈ et al. 2018). The interferometric observations started ff trometry is not a ected by this confusion problem and improves with IRS16NW or IRS16NE feeding the fringe-tracker and the 2017/2018 astrometry to a precision of 30 150 µas. In the − IRS16C feeding the science channel. These two bright stars case of NACO, fainter and so-far unknown stars might be present (m 10.0–10.5 mag, separation from Sgr A* 1 ) were used to in the field and could result in 0.5 1 mas positional offsets from K ≈ ≈ 00 − find fringes and to zero the optical delay of the science channel. undetected source confusion throughout the S2 orbit, which typ- The actual observations of S2 and Sgr A* were then made with ically lasts for about one year (Plewa & Sari 2018). IRS16C or IRS16NW as fringe-tracking star. Each science ex- Another critical aspect is the precision of radial velocity posure consists of 30 frames with an individual integration time measurements of S2. It is a K 14 mag star, for which the HI s of 10s each. The typical observing sequence had five such 5-min n = 7 4 recombination line (Brackett-≈ γ, λ = 2.1661 µm) and the exposures on Sgr A*, one exposure on S2, one sky exposure, He-I− (λ = 2.112 µm) line can be detected in absorption at >5 σ and one exposure on R2, a moderately bright (m 12.1 mag, per spectral resolution element in one hour. The wavelength cal- K separation 1.5 ) nearby unresolved giant star, which≈ served as ibration is fine-tuned in each individual exposure by comparing 00 a visibility≈ calibrator. This sequence was repeated several times the positions of the atmospheric OH-emission-lines with their ex- per night. pected positions. In the wavelength regime in which the S2 Brγ line is currently observed, we use approximately a dozen lines, and the scatter of the OH-lines after the fine-tuning around the A.3. GRAVITY data analysis 1 expected positions is below 5 km s− , which is smaller than the scatter in the Brγ data. We hence estimate that our systematic We used the standard GRAVITY pipeline to process the data 1 (Lapeyrere et al. 2014; GRAVITY Collaboration 2017). Each in- uncertainty due to the wavelength calibration is 5 km s− . The dominant error term, however, is the correction from residual dividual exposure was first sky subtracted, flat fielded, and wave- sky features in the data. Given that S2 changes its radial veloc- length calibrated. The data were then reduced based on a pixel- ity quickly and that these residuals vary from one observation to-visibility matrix (P2VM, Tatulli et al. 2007), which represents to the next, we can assume that they essentially act as a ran- the instrument transfer function including throughput, coher- dom error. In practice, typical 1 σ uncertainties of the line centres ence, phase shift, and cross-talk information of each individ- 1 ual pixel. In a second step, the science complex visibilities are are 12 to 20 km s− . At some observation epochs, confusion with± other stars± or extended nebular emission (mainly at low ve- phase-referenced to the fringe-tracker complex visibilities using locities) or atmospheric residuals leads to increased uncertain- the laser metrology and a fiber dispersion model. The observa- ties. The line shape of S2 might be affected by a stellar wind, tory transfer function (i.e. coherence loss due to vibrations, un- although previous analyses suggest that S2 is a main-sequence corrected atmosphere, birefringence, etc.) was calibrated on the dwarf with low rotational velocity, which is not expected to have nearby unresolved calibrator star R2. significant mass loss (Martins et al. 2008; Habibi et al. 2017). The stacked spectrum of S2 with a very high signal-to-noise ratio A.4. Model fitting of 200, newly obtained during the pericentre passage (March– June≈ 2018), does not show a P Cygni profile either, which would The reported astrometric positions are based on a two- be indicative of a wind. If a wind component were to introduce a component binary fit to the (squared) visibilities and clo- constant bias, it would affect the accuracy, but not the precision sure phases. We took into account the flux ratio between S2 by which we measure the redshift of S2. In the fit, this would in and Sgr A*, the colour of Sgr A*, bandwidth smearing (e.g. turn be absorbed into the motion of the reference system in the Lachaume & Berger 2013), and a telescope-dependent injec- direction of the line of sight. If the shape of the spectrum is vari- tion ratio. We developed several independent fitting codes, em- able due to the wind, we would obtain a lower precision on the ploying least-squares minimization, MCMC optimization, and radial velocities. So far, no hints of a variable spectrum of S2 have a combination of these two techniques. A full mathematical been seen, and the classification of S2 as a B2.5 main-sequence derivation of the models is beyond the scope of this paper. Over- star argues against spectral variability (Habibi et al. 2017). More- all, the results agree very well independent of the optimization over, we use a cross-correlation with the observed S2 spectrum technique and the detailed implementation. (Martins et al. 2008; Habibi et al. 2017) in addition to a line fit, which would most likely be affected in a different way than the A.5. Imaging single line. The two ways of determining the radial velocity agree very well, which demonstrates that the line shape of the Brγ line Complementary to the model fitting, we reconstructed im- does not affect our measurement. ages using radio- and optical interferometry imaging tools. By

L15, page 7 of 10 A&A 615, L15 (2018) employing different codes, we checked for consistency and ro- Table A.1. Best-fit orbit parameters with and without Schwarzschild bustness. precession. The radio-interferometry-like imaging was done with the Astronomical Imaging Process System (AIPS, Greisen 2003) Parameter Without Schwarzschild With Schwarzschild Unit developed at NRAO. For each exposure, we reconstructed the precession precession dirty image by a discrete Fourier transform of the complex f 0.901 0.090 0.945 0.090 ± ± visibility data. We then extracted preliminary images of S2 M 4.106 0.034 4.100 0.034 106 M • ± ± using the CLEAN algorithm (e.g. Högbom 1974) with clean R 8127 31 8122 31 pc 0 ± ± boxes only on the brightest features of the dirty image. Af- a 125.38 0.18 125.40 0.18 mas ± ± ter this, we performed phase-based self-calibration of the vis- e 0.88473 0.00018 0.88466 0.00018 ± ± i 133.817 0.093 133.818 0.093 ibility data with the preliminary S2 model to correct for ± ± ◦ ω 66.12 0.12 66.13 0.12 telescope-based errors. Then we re-ran CLEAN on the self- ± ± ◦ Ω 227.82 0.19 227.85 0.19 calibrated data to clean both on S2 and Sgr A* iteratively, re- ± ± ◦ sulting in one image per exposure. In a last step, we com- P 16.0526 16.0518 yr t 2018.37965 0.00015 2018.37974 0.00015 yr bined individual exposures to obtain the final image for each peri ± ± 58257.667 0.054 58257.698 0.054 MJD ± ± night. x 0.88 0.47 1.00 0.47 mas 0 − ± − ± We also made use of MiRA2 (Thiébaut 2008) and Squeeze y0 0.97 0.41 0.99 0.41 mas − ± − ± 1 (Baron et al. 2010), two optical interferometry imaging codes. x˙0 0.070 0.031 0.076 0.031 mas yr− ± ± 1 These codes fit an image to the data using a least-squares min- y˙0 0.178 0.030 0.178 0.030 mas yr− ± ± 1 imization (MiRA2) or MCMC (Squeeze) with some penalty z˙0 2.4 3.0 1.9 3.0 km s− 2 ± ± function to account for the sparsity of the data. The advantage of χred 0.86 0.86 these codes is that they can account for coherence losses due to bandwidth smearing, and that they can also work directly with Notes. For the case of Schwarzschild precession, the orbital parameters the robust closure-phases, thereby avoiding the self-calibration should be interpreted as the osculating orbital parameters. The argument described above. Their weakness, however, is that the result de- of periapsis ω and the time of pericentre passage tperi are given for the pends on the chosen penalty function (priors), and fit conver- epoch of last apocenter in 2010. gence can be an issue. A.7. Rømer effect A.6. Treatment of systematic uncertainties Both gravitational redshift and transverse Doppler effect are of 2 As discussed above, all of the major observational input data the order of β . As Zucker et al.(2006) have pointed out, the in this paper (NACO, SINFONI, GRAVITY) are affected by classical Rømer delay also needs to be taken into account at that strong and different systematic effects. NACO positional mea- level. Including the light travel time requires solving the equation surements are affected by obvious or unrecognized confusion x(tem) events, especially close to Sgr A* (typically lasting one year, tobs = tem + , (A.1) Plewa & Sari 2018) and have to be tied into a long-term ref- c erence frame coupled to the radio interferometric frame. Stars where x denotes the line-of-sight distance, tem the time of emis- common to both frames but spread over tens of arcseconds are sion, and tobs the time of observation. This equation can only be used for this alignment, but they in turn require a careful analysis solved iteratively, which is done in some of our codes. However, of image distortions and mosaicking shifts (Plewa et al. 2015). one can also approximate the correction term by As a result, the S2 positions can at times have systematic un- x(t ) x(t ) v (t ) certainties exceeding 1 mas, although the statistical errors are em obs 1 x obs , (A.2) 0.4 0.6 mas. Depending on the redshift, SINFONI spectro- c ≈ c − c scopic≈ − data can be affected more or less strongly by atmospheric which can be evaluated without iteration, and is thus more suit- sky lines and extended nebular emission in the central cluster. able for a fitting algorithm. For the S2 orbit, the correction term This means that velocity uncertainties can exceed the typical varies by around eight days over the orbit, and the approximation 1 performance in good conditions of 10 15 km s− . GRAVITY as- never differs by more than 10 s from the exact solution. Some of trometry offers by far the best expected− positional information our fitting codes therefore use the approximation to calculate the (30 150 µas). However, the current accuracy is still limited by Rømer delay. Light bending (lensing and Shapiro time delay) − 1 systematics and calibration errors, especially if Sgr A* is par- effects on positions ( 20 µas) and velocities ( 5 km s− ) can be ≈ ≈ ticularly faint, is varying during the integration, or if the S2 – neglected at the current precision. Sgr A* separation is very large and close to the limit of the interferometric and photometric field of view, as was the case in A.8. Degeneracy between special relativistic effects and 2017. gravitational redhift Our approach to account for these effects was to have several team members use different analysis tools for po- In Eq. (1), the special relativistic transverse Doppler effect sitional fitting and extraction, and to compare and aver- and the gravitational redshift are completely degenerate. The age these results, with the analysis scatter providing an es- degeneracy is broken only by the relative motion between timate of the systematic effects. Bootstrapping and removal observer and massive black hole, which is mostly due to the of questionable data sets was performed in all cases. Every solar orbit in the Milky Way. The apparent motion of Sgr A* 1 new observing epoch will add to the understanding of the of about 240 km s− along the Galactic plane towards the south- systematic effects, and we expect to further improve espe- west (Reid & Brunthaler 2004) leads to a correction term that is 1 1 cially the astrometric accuracy of GRAVITY in the coming of the order of +5 km s− at pericentre and 0.2 km s− at apoc- years. entre. Earth’s motion around the Sun will contribute− a term that

L15, page 8 of 10 GRAVITY Collaboration: Detection of gravitational redshift GRAVITY Collaboration: R. Abuter et al.: Detection of gravitational redshift

Fig. A.1. Posterior probability distributions obtained from a Markov chain for the 14-parameter fit including f as a free parameter. All parameters Fig.are wellA.1. Posterior constrained, probability in particular distributions also f . We obtained have omittedfrom a Markov the panels chain corresponding for the 14-parameter to the four fitincluding coordinatef systemas a free parameters parameter. in All this parameters figure for areclarity. well constrained, in particular also f . We have omitted the panels corresponding to the four coordinate system parameters in this figure for clarity. is smaller by an order of magnitude. Overall, the effect of the rel- Keplerian orbit parameters a, e, i, Ω, ω, Tperi, and the parame- twoative posterior motion parameterbetween observer spaces is and almost Sgr A* unity, is too such small that to the break ev- fitter thef bootstrapintroduced sample in section and3 used to distinguishthe standard between deviation Newto- of the ρ ρ χ2/ . idencethe degeneracy. ratio equals We therefore the likelihood use the ratio, standard i.e. local∆ standard2, which of best-fitnian/Keplerian values of dynamicsf as sampling ( f = 0) error. and We the obtain combined∆ f = special0 15. and is 43 in favour of f = 1 (assuming p( f = 0) =≈p( f = 1) a pri- rest≈ (LSR) correction and accept the complete degeneracy. general relativistic effects up to PPN(1)z ( f = 1). Table A.1 lists ori). The differences in the Bayesian information criteria (BIC) the best-fit solution for the entire data set as treated in Fig.4 (left χ2 Appendix A.10: Lense-Thirring effect and the Akaike information criteria (AIC) both equal ∆ in our without, and right with Schwarzschild precession) for a weight case.A.9. MCMCGiven that analysis∆χ2 = 87, we have a “decisive” evidence for Theof 0.5 S2 of experiment the down-sampled delivers a NACO valuable data. confirmation Fig. A.1 gives of GR the in pos- a f =1 when applying Jeffrey’s scale. As discussed in the main text, we carried out posteriori anal- so-farterior unexplored parameter distributions regime at high of masses this fit. (Fig. A.2, adapted from yses of all data by fitting 14 parameters characterising the PsaltisFor 2004). the purpose A further of model goal would comparison be determining in a Bayesian the spin frame- of NACOWe estimated reference the frame sampling relative error to on Sgrf A*by means (x0, y0, ofx˙0 a, y˙ boot-0, z˙0), thework, massive we use black the hole fact that through the posterior the combination probability of frame distribution drag- strapthe centralanalysis, mass duringM which, its we distance randomised from the the astrometric Sun R0, and the gingis well and described quadrupole bymoment, a multivariate the so-called Gaussian Lense-Thirring in both cases ( (LT)f = 0 • spectroscopic data points separately but simultaneously. We re- precession, of PPN(1.5) order (e.g. Misner et al. 1973; Bardeen L15, page 9 of 10 Article number, page 9 of 10 A&A 615, L15 (2018) A&A proofs: manuscript no. s2_20180702_revised_language_edited2

Table A.1. Best-ﬁt orbit parameters with and without Schwarzschild spectroscopic data points separately but simultaneously. We reﬁt

✙ ✏ ✙ ✑

✤ ✠ ✟ ✛✁ ✝

precession. ✵ the bootstrap sample and used the standard deviation of the best-

✓ ✖ ✗ ✕✧

✤ ✛ ✍✝ ☞ ✝ ☞ ✪✌✎ ✚✫

✚ ﬁt values of f as sampling error. We obtain ∆ f = 0.15.

✙

✱ ✲ ✌✄✝

✰✝ -

✮ ✯ ✙ ✑ ✏

✢✝✄ ✎ ✛✁ ☞✌ ✁✄ ✜✝ ✝ ✁ ☛✝ Para- Without Schwarz- With Schwarz- Unit - ✥

✖ ✗ ★ ✧

☞ ✦ ✪✌ ✎ ✆ ✫✬ ✜ ✭ meter schild precession schild precession ✚

A.10. Lense-Thirring effect

✕✧ ★✩ ✏ ✑

☎✝ ✎✝ ✎✌✁✄

) ✿

. . . . ✾ f 0 901 0 090 0 945 0 090 ✶

/ - *

± ± ✽

✖

✁ ☞✞✌✎ ✠ ☞✁✂✄ ☎ ✚ ☞ ✦ 6 ✚✥ The S2 experiment delivers a valuable conﬁrmation of GR in M 4.106 0.034 4.100 0.034 10 M (

• ± ± ✼ a so-far unexplored regime at high masses (Fig. A.2, adapted ✻

R0 8127 31 8122 31 pc ✺

✖ ✑ ✗ ✒ ✑

☞✝ ☎ ✌ ✎ ✚✌☞✌ ✂ ✤ ± ± ☞✠✢✣

a 125.38 0.18 125.40 0.18 mas - ✹ from Psaltis 2004). A further goal would be determining the

✙✑ ✔ ✙ ✙✑

✝ ✜✠ ✁☞ ☛✂ ✠ ☞

e 0.88473± 0.00018 0.88466± 0.00018 ✛✂ - spin of the massive black hole through the combination of frame

✕ ✖ ✗ ✘ ✒✙ ✏

✎ ✝ ✝ ✎✌ ✁✄ ± ± ✌ i 133.817 0.093 133.818 0.093 ◦ dragging and quadrupole moment, the so-called Lense-Thirring

- ✸

✗ ✘ ✙ ✔

✠ ☛✌ ☞✁ ✝ ✠ ± ± ✚

ω 66.12 0.12 66.13 0.12 ◦ (LT) precession, of PPN(1.5) order (e.g. Misner et al. 1973;

✏ ✑✑ ✒ ✓ ✏ ✔

✝ ✌✁✄ ✁ ✝ ☞ ✂☞ ± ± ☞✝

Ω 227.82 0.19 227.85 0.19 ◦ Bardeen & Petterson 1975; Will 2008):

✆✝ ✞✟✠ ✝ ✡☛✝ ☞✌✍✝✄ ✎ . ± . ± ✁✂✄ ☎ -

P 16 0526 16 0518 yr - ✷ ✵

✴ ✵ ✴ ✳ ✶ ✵ ✶✳ ✥✳ 3/2

tperi 2018.37965 0.00015 2018.37974 0.00015 yr

✙ ✖ ✖ ✑✑

✁ ☞ (✍✠ / ) RS 58257.667 ±0.054 58257.698 ±0.054 MJD ∆ω = 2 ξ , (A.3) ± ± a(1 e2) x0 0.88 0.47 1.00 0.47 mas Comparison of tests of General Relativity, inspired by Psaltis − ± − ± Fig. A.2. Comparison of tests of General Relativity, inspired by Psaltis − ! y0 0.97 0.41 0.99 0.41 mas (2004). Shown in black are well-established tests: the Pound & Rebka − ± − ± (2004). Shown in black are well-established tests: the Pound & Rebka where ξ 1 is the dimensionless spin parameter of a Kerr black x˙0 0.070 0.031 0.076 0.031 mas/yr ((1959)1959) experiment, the precession of Mercury ((EinsteinEinstein 1916 1916),), light de- ≤ ± ± hole. For ξ = 0.5, the LT precession of S2 is 900, which is y˙0 0.178 0.030 0.178 0.030 mas/yr flection and the Shapiro delay in the solar system, the Hulse-Taylor ± ± clearly not detectable. It is thus necessary to observe stars yet z˙0 2.4 3.0 1.9 3.0 km/s pulsar (Taylor (Taylor & & Weisberg Weisberg 1982),1982), the the gravitational gravitational redshift redshift of Sirius of 2 ± ± deeper in the potential if the spin of Sgr A* is to be mea- χred 0.86 0.86 SiriusB (Greenstein B (Greenstein et al. 1971; et al. Barstow1971; Barstow et al. 2005), et al. 2005 the LIGO), the LIGOdetections de- tections(Abbott et (Abbott al. 2016a,c,b), et al. 2016a and, theb,c), relativistic and the Fe relativistic Kα line (Tanaka Fe Kα etline al. sured with orbiting stars. Waisberg et al.(2018) have quantita- For the case of Schwarzschild precession, the orbital parameters should (1995;Tanaka Fabian et al. et1995 al. 2000).; Fabian Future et al. tests2000 are). Future shown tests in grey, are shown and this in work, grey, tively analysed the requirements for detecting the LT-precession be interpreted as the osculating orbital parameters. The argument of andwhich this uses work, the which S2 orbit uses around the S2 Sgr orbit A*, around is shown Sgr in A*, blue. is shown in blue. on a star inside the S2 orbit. They find that such a star periapsis ω and the time of pericentre passage tperi are given for the would have to have a combination of semi-major axis a and 2 3/4 epoch of last apocenter in 2010. eccentricity e, a(1 e ) /RS 250, and a significant detec- and f = 1). We can thus use the respective peak (best-fit) pa- tion would require− 10 µas astrometric≤ precision in a campaign rameter values and the covariance matrices to approximate the over several years. Based on the K-band luminosity func- & Petterson 1975; Will 2008): Bayesian evidence integrals (Mackay 2003). The ratio of the so- tion (Genzel et al. 2003; Sabha et al. 2012; Gallego-Cano et al. / called Occam factors describing the ratio of the volumes of the 2018; Schödel et al. 2018) and the eccentricity distribution of R 3 2 ∆ω = 2 ξ S , (A.3) two posterior parameter spaces is almost unity, such that the ev- the S-stars (Gillessen et al. 2017), Waisberg et al.(2018) esti- a(1 e2) ρ ρ ∆χ2/ < − ! idence ratio equals the likelihood ratio, i.e. 2, which mate a probability of 10 % for a K 19 mag star to fulfill is 43 in favour of f = 1 (assuming p( f = 0) = p(≈f = 1) a priori). the above requirement.≈ Still fainter stars would likely be more ξ where 1 is the dimensionless spin parameter of a Kerr The≈ differences in the Bayesian information criteria (BIC) and common. No second star with K < 18.5 mag has so far been ≤ ξ = . , black hole. For 0 5 the LT precession of S2 is 900, which the Akaike information criteria (AIC) both equal ∆χ2 in our case. reliably detected near Sgr A* with GRAVITY, which is con- is clearly not detectable. It is thus necessary to observe stars Given that ∆χ2 = 87, we have a “decisive” evidence for f = 1 sistent with the predictions of Waisberg et al.(2018). We hope yet deeper in the potential if the spin of Sgr A* is to be mea- when applying Jeffrey’s scale. for such a detection in the next years, when S2 has moved sured with orbiting stars. Waisberg et al. (2018) have quantita- We estimated the sampling error on f by means of a boot- away from Sgr A* and cleared the field of view for fainter tively analysed the requirements for detecting the LT-precession strap analysis, during which we randomised the astrometric and objects. on a star inside the S2 orbit. They find that such a star would have to have a combination of semi-major axis a and eccentric- 2 3/4 ity e, a(1 e ) /RS 250, and a significant detection would require 10−µas astrometric≤ precision in a campaign over several years. Based on the K-band luminosity function (Genzel et al. 2003; Sabha et al. 2012; Gallego-Cano et al. 2018; Schödel et al. 2018) and the eccentricity distribution of the S-stars (Gillessen et al. 2017), Waisberg et al. (2018) estimate a probability of 10 % for a K < 19 mag star to fulfill the above requirement. Still≈ fainter stars would likely be more common. No second star with K < 18.5 mag has so far been reliably detected near Sgr A* with GRAVITY, which is consistent with the predictions of Waisberg et al. (2018). We hope for such a detection in the next years, when S2 has moved away from Sgr A* and cleared the field of view for fainter objects.

L15, page 10 of 10 Article number, page 10 of 10