Polarimetry and astrometry of NIR flares as event horizon scale, dynamical probes for the mass of Sgr A*

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Introduction
The presence of a massive black hole (MBH) in the Galactic center (GC) has been established, beyond any reasonable doubt, in a four-decade-long research by several teams (e.g., Genzel 2021).Not only is Sagittarius A* (Sgr A*) the best case for the existence of black holes, but unlike any other known black hole, it can be resolved both spatially and temporally, allowing dynamical tests long considered unfeasible.The combination of precise stellar astrometry and spectroscopy has revealed the gravitational redshift signature in the orbit of the star S2 (GRAVITY Collaboration 2018b; Do et al. 2019a).Furthermore, the prograde relativistic precession of the orbit yielding an astrometric signature of around 0.5 mas per revolution was detected employing the ultra-high resolution of the GRAVITY interferometer (GRAVITY Collaboration 2020a, 2022b).Emission from a distance of just a few R g (R g = GM/c 2 corresponds to 5.12 µas at our assumed R 0 = 8.277 kpc and for M = 4.297 × 10 6 M from GRAVITY Collaboration 2022b) was detected in the X-ray (Baganoff et al. 2001;Neilsen et al. 2013), near-infrared (NIR, Genzel et al. 2003a;Eisenhauer et al. 2005;Dodds-Eden et al. 2011;Do et al. 2019b;Witzel et al. 2021), and radio bands (Balick & Brown 1974;Lo et al. 1998;Krichbaum et al. 1998;Shen et al. 2005;Yusef-Zadeh et al. 2006;Doeleman & Weintroub 2008;Johnson et al. 2015;Event Horizon Telescope Collaboration 2022).The small spatial scales imply that the light arises deep in the strong gravitational field of the MBH, and that its propagation is subject to strong relativistic effects.
The NIR emission of Sgr A* is highly variable, with flares of ten times the quiescent flux occurring once or twice per day (Dodds-Eden et al. 2011;Witzel et al. 2018;GRAVITY Collaboration 2020c).The flare emission most likely comes from locally heated electrons emitting synchrotron emission from the NIR, possibly up to the X-ray band (Yuan et al. 2003;Dodds-Eden et al. 2009;Ponti et al. 2017;GRAVITY Collaboration 2021a).The synchrotron model with emission coming from a compact source also explains the observed variable NIR polarization of a few tens of percent linear polarization and a negligible amount of circular polarization (Eckart et al. 2006(Eckart et al. , 2008;;Trippe et al. 2007;Zamaninasab et al. 2010;Witzel et al. 2010;Shahzamanian et al. 2015; GRAVITY Collaboration 2020a), as well as the co-evolution of the light curves in the two bands.The estimated source sizes of around 1 R g and the required magnetic field strengths of around 30-80 G (Dodds-Eden et al. 2010;Eatough et al. 2013;Ripperda et al. 2020) are in agreement with the "hot spot model" with orbital radii close to the innermost circular orbit (Broderick & Loeb 2005, 2006;Meyer et al. 2006;Hamaus et al. 2009;Ripperda et al. 2020;GRAVITY Collaboration 2020d).This model has observable signatures in the astrometry and polarimetry of the NIR emission, where the astrometric amplitudes of a few 10 µas were one key driver for the development of GRAVITY (Paumard et al. 2006;GRAVITY Collaboration 2017).Moreover, Dexter et al. (2020) show by means of general relativistic magnetohydrodynamic simulations that for magnetically arrested disks (MADs), flares with properties similar to the observed ones might arise from magnetic reconnection events.
GRAVITY combines the light from either the four Auxiliary Telescopes (1.8 m diameter) or the four Unit Telescopes (8 m diameter) at the Paranal observatory.The spatial resolution in K-band is around 3 mas, and the accuracy by which sources can be located is of a few 10 µas -good enough to follow the motions of orbiting hot spots.In 2018 we observed in three nights flares from Sgr A*, all of which showed astrometrically clockwise motions at radii compatible with the hot spot model.For one flare we also obtained polarimetry in 2018, showing a full rotation of the polarization vector (GRAVITY Collaboration 2018a).
Clearly, the number of such high-precision flares was still small.Hence, we embedded in our nightly observing strategy regular visits to Sgr A* and analyzed in real time the incoming data to monitor for flux increases.If a flare occurred, we continued to observe Sgr A*.
Here we report on one new flare, for which the astrometric and polarimetric data are good enough to resolve orbital motion, and on four additional flares, for which only the polarimetry is of sufficient quality.

Observations
We observe the GC typically six months per year, with a total allocation of around 90 h.The data serve for the astrometric monitoring of the stellar orbits (GRAVITY Collaboration 2018b, 2022b) and for observing the emission from Sgr A* itself (GRAVITY Collaboration 2018a).We report on as-yetunpublished data obtained from observations in 2019, 2021, and 2022 during which Sgr A* exhibited flares.We analyzed them together with previous flare observations.Specifically, we used the observations listed in Table 1.

Data
Our GRAVITY data deliver position and polarization measurements simultaneously.The former are obtained from the phase measurements, the latter by comparing (interferometric) fluxes behind a Wollaston prism.

Astrometry
For the GC, GRAVITY uses optical fibers to simultaneously observe two objects and record the interference of the two targets on the six baselines formed by the four 8 m telescopes.One target is the star IRS 16C (m K = 9.7), used as a phase reference, and the other is the science target.Each GRAVITY fiber has a field of view of approximately 70 mas, comparable to the diffraction limit of the individual telescopes.
Since the beginning of the GRAVITY observations, stars were always close enough to Sgr A* to be observed in the same field of view, yielding binary, triple, or multiple signatures in the interference pattern.In 2018 the star S2 (m K = 14.0) was observed together with Sgr A* in this way.In this single-beam measurement, information such as the position vector and flux ratio between the two sources, is directly inferred from the measured closure phases and amplitudes.Since 2019, S2 has moved so far out of the interferometric field of view that other fainter stars have been detected close to Sgr A* (GRAVITY Collaboration 2021b, 2022b).However, during a flare, Sgr A* outshines these fainter stars and effectively appears as a single source.In this case the astrometry of Sgr A* has to be derived from the visibility phases.These are measured relative to the reference target via GRAVITY's metrology system, which traces the optical path differences between the telescopes and the instrument (GRAVITY Collaboration 2020a, Appendix A therein).
Hence, for the new, dual-beam data sets, the astrometry is obtained as a sum of three measured phases: of Sgr A*, of IRS 16C, and the metrology linkage.The source models are simply unary fits in this case.In practice, we measure the Sgr A* phases and, every few exposures, that of a local phase reference for which we chose S2.Since we reference the Sgr A* positions to S2.The proper motion of S2 during a flare is small enough not to affect our measurements.A significant systematic error in our dual-beam flare astrometry is due to high-order imperfections in the GRAVITY optics.As the metrology sensors cover only four points of the pupil, the high-order aberrations translate into phase errors, and thus astrometric uncertainty.These errors average out effectively by the pupil rotation over the course of the night so that stellar orbits are routinely measured with an accuracy down to 20−50 µas (e.g., GRAVITY Collaboration 2022b).For the individual short exposures of flare observations, however, these metrology footprint errors dominate over the statistical errors.
Since 2022, we circumvent this source of error by simultaneously modulating the telescope pupils relative to the metrology receivers such that the high-order aberrations average out.This restores the astrometry to a level similar to what we achieved in the single-beam case.This new observing mode was not yet in place in 2019 and 2021, and the accuracy of the dual-beam astrometry was not at the level needed for flare orbits during this period.

Polarimetry
In its polarimetric mode, GRAVITY splits the light post-beam combination by means of a Wollaston prism into two orthogonal, linear polarization states.Employing a half-wave plate, we rotate the polarization by 45 • between exposures, such that for each pair of exposures, we measure the polarized flux in the 0 where f θ represents the correlated flux along direction θ and the primes indicate that the values are measured in the detector coordinate system.To get the on-sky polarization state Q and U, it is necessary to correct for the geometric beam propagation and the instrumental polarization.This is done with the model developed by GRAVITY Collaboration (2023).As GRAVITY cannot measure circular polarization, we correct for birefringence with the model presented by Witzel et al. (2010), using the fact that the polarized NIR emission of Sgr A* is predominantly linearly polarized.
Our Q and U values are fractional values normalized to the observed intensity.Unlike at longer wavelengths, the variations of the NIR polarized fluxes are completely dominated by the overall brightness variations -a result of the much shorter heating and cooling timescales in the NIR (von Fellenberg et al. 2023).Thus, the fractional values carry the information on the magnetic field and geometry of the Sgr A* system.
Following GRAVITY Collaboration (2020c), there are two different ways to measure the required fluxes.In the first, if Sgr A* is the dominant (or only) source in the (interferometric) field of view, the coherent flux directly measures the flux of Sgr A*.In the second, if there are multiple sources in the field of view, a multiple source fit to these sources includes the flux ratio of Sgr A* to each star, which, when multiplied with the coherent flux, yields the desired Sgr A* flux.Flux measurements for observations up to 2020 are reported in GRAVITY Collaboration (2020c).For 2021 and 2022 we used binary fits with the stars S29 (2021) and S38 (2022) as the second object beyond Sgr A* (see, e.g., GRAVITY Collaboration 2022a for images of the central region where the stars S29 and S38 are shown).

Analysis
Our sample of flares with polarimetric coverage has grown from one (GRAVITY Collaboration 2018a, 2020b) to six, all of which show a single clockwise loop, or part thereof, in the Q−U plane (Fig. 1).The sample of astrometric measurements has also grown and consists now of four events: the three flares from GRAVITY Collaboration (2020c) and the addition from 2022 May 19 (Fig. 2).
In contrast to the flare astrometry, where we need a precision of ≈10−30 µas per five-minute exposure, polarimetry is less sensitive to observing conditions and is not penalized by the optical imperfections.As a result, we obtained valid polarimetry for flares even if the conditions were not good enough for astrometry.This is the case for the flare from 2019, the two from 2021, and the one from August 2022.
The similarity between the flares motivated us to look at them in an averaged way.We combined the data of the six polarimetric and four astrometric flares by allowing only a relative time shift between the individual events and finding the solution that best represents a sinusoidal in both sky coordinates for the combined flare data.This yields the data shown in Fig. 3.Both averaged data sets are very well described by elliptical figures with a common period of around 60 min and constant angular velocity of 6 • /min.
The mean astrometric loop size in Fig. 3 of around r = 61 ± 9 µas (corresponding to 0.50 ± 0.07 AU for our assumed R 0 ) and the polarization period of P = 60 ± 3 min allows us to estimate the enclosed mass M enc .Assuming Keplerian orbital motion, this yields M enc,estimate = (5.1 ± 1.6) × 10 6 M (Appendix A).Within the error, this value agrees with that known from stellar orbits,  M = (4.297± 0.012)×10 6 M (GRAVITY Collaboration 2022b) and shows that (within the uncertainties) the mass of Sgr A* is enclosed within the flare orbit.
In the following we refine the value of M encl by modeling the astrometry and polarimetry of the flares.We model the astrometric motion of the combined data set as in GRAVITY Collaboration (2020d) by an orbiting relativistic hot spot, including the effects of lensing via ray tracing.However, here we enhance the pure astrometric fit by including more information, mainly from the polarimetry.First, we identify the period of the polarization loop with that of the astrometric loop.Technically this sets a prior on the orbital period.We use P pol = 60 ± 3 min.Second, we show that observing a single polarization loop per astrometric loop yields a strong prior on the allowed range of inclinations (see Sect. 4.1).The angular velocity of 6 • /min constrains the inclination to i = 157 • ± 5 • .Third, the absence of strong spikes from Doppler beaming in the light curves also limits the inclination to be close to face-on.Following Hamaus et al. (2009) and GRAVITY Collaboration (2018b; Appendix C therein), we place a conservative prior of i = 180 • ± 40 • .Finally, the Event Horizon Telescope (EHT) image of Sgr A* constrains the inclination to i = 180 • ± 50 • (Event Horizon Telescope Collaboration 2022).
We took into account all these constraints by including them in the likelihood function and determined a globally best-fitting orbiting hot spot model.The parameters of the fit are the hot spot's orbital radius R, the inclination i, the position angle Ω, and the enclosed mass M enc .

Modeling the polarization loop
The polarization signal of an orbiting hot spot probes the magnetic field around Sgr A* (Broderick & Loeb 2005;GRAVITY Collaboration 2018a, 2020b,d;Wielgus et al. 2022;Michail et al. 2023).Simulations and analytical methods have been used by GRAVITY Collaboration (2020b) and Narayan et al. (2021) to investigate the polarization changes associated with such hotspots.A dominantly toroidal or radial field lead in general to two full loops in the Q−U plane, while a dominantly poloidal field produces either no loops for edgeon geometries (i ∼ 90), one loop for moderate inclinations (i ∼ 30 • /150 • ), or two loops for face-on inclinations (i ∼ 0 • /180 • ).Hence, observing one loop per astrometric revolution severely constrains the configuration space and argues for a vertical magnetic field and moderate inclination.The repeated observations of single loops in the Q−U plane also indicate that the global magnetic field configuration has remained stable over recent years, consistent with the small changes in the circularly polarized emission at sub-millimeter wavelengths (Muñoz et al. 2012).
To derive the inclination constraint, we used the model introduced by Gelles et al. (2021).This semi-analytic model implements an equatorial point-like hot spot orbiting on a Keplerian orbit around a Kerr black hole with a local magnetic field.The model uses the semi-analytic solution for light rays (Gralla & Lupsasca 2020) to obtain a polarized image of the synchrotron emitting source.
Since the model describes purely linear polarized emission, we extend it to take into account the observed polarization fraction in the data.Although coming from a point-source emitter, the model captures the fundamental features of extended emission regions (GRAVITY Collaboration 2020b; Vos et al. 2022).A more detailed description of the model can be found in Appendix B.
The local magnetic field is given in cylindrical coordinates and allows a clear distinction between radial, toroidal, and poloidal magnetic field configurations.Other free parameters are the inclination of the orbit, the orbital radius, the local boost, and the spin of the black hole.The last only has a minor impact on the polarization signal (Gelles et al. 2021), and hence we choose to work with zero-spin models.Further, we fix the radius to 9R g as this matches the observed astrometric and polarimetric periods.Using radii in the 8−10 R g range does not alter our results (see Fig. C.1). Fixing the radius also fixes the Lorentz boosting factor to β = 0.37.The model does not take radiative transfer effects into account.Further, time delays and luminosity changes due to the combination of a non-flat spectral energy distribution and Doppler boosting are neglected.
As expected from the discussion above, the rotation speed distinguishes very well between the poloidal and toroidal/radial fields.Observing one polarimetric loop per astrometric loop only occurs for poloidal fields.The inclination can be constrained from the observed angular velocity with which the Q−U loop is traversed (Fig. 4).The observed angular velocity of 6 • /min for the astrometry is only compatible with poloidal field configurations with an inclination of i = 157 • ± 5 • .
Figure 5 shows that the model for i = 157 • is an excellent description of the observed polarization changes.This inclination matches the one measured by GRAVITY Collaboration (2018a) and the preferred inclination of the EHT ring (Event Horizon Telescope Collaboration 2022).We allowed the position angle on the sky and the arbitrary zero point in time to be adjusted, and picked the combination that yields the In summary, the analysis of the polarization loops gives robust constraints and a result compatible with the astrometric signature.In the following, we combine this information in a combined fit.

Combined fit
Our explicit combined fit (Figs. 6 and D.1) uses the astrometric data and the constraints derived from the polarimetry.We obtained the set of parameters presented in Table 2. Within the uncertainties, the flare radius and the orientation agree with what GRAVITY Collaboration (2018bCollaboration ( , 2020b,d) ,d) found from the previous smaller data set.The novel result is L10, page 5 of 10   the mass constraint, that stems from the new data, in particular from the strong independent period constraint from the polarimetry.The value for the mass inside 9 R g matches the simple estimate, and again also the mass derived using stellar orbits.

Results
The most important result we find is that the mass of Sgr A* measured from stellar orbits, M = (4.297± 0.012) × 10 6 M , is enclosed within 9 R g , corresponding to 0.38 AU, or roughly the orbital radius of planet Mercury.A similarly strong constraint is obtained from the radio-VLBI image of Sgr A* (Event Horizon Telescope Collaboration 2022).
The stellar orbits with the sub-percent precise mass measurement, on the other hand, limit the size of Sgr A* to be smaller than the smallest observed pericenter passages, which are those of S29 with 100 AU in 2021 (directly observed by GRAVITY Collaboration 2022b) and S14 with 26 AU in 2000 (well-determined orbit, but poorer coverage; Gillessen et al. 2017 and newer data).We show an updated version of the enclosed mass vs. radius in Fig. 7.
Other than the mass determination from the stellar orbits, both inner constraints are not assumption-free.The NIR flare constraint assumes that the motion of the emission is near Keplerian (i.e., governed by the forces of gravity), which is well motivated by accretion flow models (Yuan et al. 2003;Narayan & Quataert 2023).The EHT image needs to assume a certain (again well-motivated) structure of the accretion flow, as other gas configurations could mimic the ring-like appearance.Taken together, these two independent estimates make the case for a black hole ever more convincing.Excluded are, for example, self-gravitating fermionic dark matter models with typ-ical core radii of ≈10 −3 pc ≈ 200 AU (Argüelles & Krut 2019;Becerra-Vergara et al. 2020).
The fact that the azimuthal speed of the hot spots is near Keplerian is, however, non-trivial.In low-density advectiondominated accretion flows (ADAFs), the gas motions generally are assumed to be sub-Keplerian (e.g., Yuan & Narayan 2014).The same is not true for magnetically dominated flows, where the speed can depend on the spin of the black hole, and both sub-and super-Keplerian motions occur.We note that a number of non-Keplerian models have also been proposed.In addition to the enclosed mass, we also measured the orientation of the flare orbits (see Fig. 8).The measured orientation, which is the direction of orbital angular momentum, is close to that of the clockwise stellar disk (Levin & Beloborodov 2003;Genzel et al. 2003b;Paumard et al. 2006;Lu et al. 2008;von Fellenberg et al. 2022, Fig. 9), suggesting a physical connection.
The stars that make up the disk are predominantly O-type and Wolf-Rayet-type stars that have strong winds.Simulations by Ressler et al. (2020) indeed show that the almost spherical accretion flow, fueled by the stellar winds from disk stars, carries the initial angular momentum down to event horizon scales, where the gas settles into a disk-like structure.This disk can tilt by a moderate angle with respect to the initial angular momentum, perhaps responsible for the misalignment between the stellar disk and flare angular momentum directions in Fig. 8.The simulations also predict that the infalling gas settles into a magnetically arrested disk (MAD), which naturally carries a poloidal field geometry.Only for strong fields can the geometry withstand the dragging with the fluid that would lead to a toroidal field geometry.Our data, favoring a poloidal field, directly support the MAD state.The MAD scenario is further supported by the estimated magnetic field strength (30-80 G;Ripperda et al. 2020) and measured density profile (see Fig. 6 in Gillessen et al. 2019 and references therein).McKinney et al. (2013) and Ressler et al. (2023) find that for MAD situations, the innermost disk-like structure tends to align with the spin axis of the black hole.The dynamics of NIR flares may, in that case, actually carry information on the black hole spin rather than just on the initial conditions of the inflow.
L10, page 6 of 10 Fig. 7. Central mass distribution in the GC.Right: enclosed mass in the GC as a function of radius.The flare motions constrain the black hole mass of M = (4.2± 2.0) × 10 6 M to be enclosed within less than 9 R g .Out to 10 6 R g the potential is dominated by the MBH, as the comparison with theoretical estimates shows (solid and dashed lines).Left: astrometric flare data overlayed on the EHT observation of Sgr A* (Event Horizon Telescope Collaboration 2022) that provides a constraint at a similar radius.Paumard et al. 2006;Lu et al. 2008), of the Sgr A* flares (this work), and of the gas cloud G2 (Gillessen et al. 2011) are very close to each other.Other features shown are the rotation of the Galaxy, as well as the Circumnuclear Disk (CND) in the GC and the Eastern and Northern Arm gas features.The orientation of the flare angular momentum vector is consistent with the clockwise disk features.

Fig. 1 .
Fig. 1.Flares with polarimetric signature.Top row: flux evolution for each flare.Bottom row: polarimetric signal in Q/I and U/I, color-coded according to time.The flare from 28 July 2018 was published and analyzed in GRAVITY Collaboration (2018a, 2020b).The other five are newly reported here.

Fig. 4 .
Fig. 4. Comparison of the average angular velocity of the Q−U loop for different magnetic field configurations as a function of inclination.Our measured values are consistent with a poloidal magnetic field and an inclination of i = 157 • ± 5 • .

Fig. 6 .
Fig. 6.Combined fit of the astrometric flare data, taking into account the polarimetry constraints.Left: on-sky motion.The gray disk corresponds to the shadow size of a Schwarzschild black hole 3 √ 3R g .Middle and right panels: individual coordinates as a function of time.The gray data are the full data set and the colored points are bins of five minutes.
Aimar et al. (2023) consider the outward motion of a hot spot along a conical trajectory, and Lin et al. (2023) propose that, similar to coronal mass ejections on the Sun, Sgr A* flux ropes are ejected out and filled with energetic electrons.Matsumoto et al. (2020) discuss whether the observed motions are actually pattern motions.

Fig. 8 .
Fig. 8. Compilation of orientations of angular momentum vectors of different dynamical structures in the GC.The colored tori illustrate the definitions of the angles.Inclinations between 0 • and 90 • correspond to counterclockwise motion on the sky (bottom half) and 90 • −180 • to clockwise motion (upper half).The vectors of the clockwise stellar disk (shown are the features called CW1 and CW2 in von Fellenberg et al. 2022; see alsoPaumard et al. 2006;Lu et al. 2008), of the Sgr A* flares (this work), and of the gas cloud G2(Gillessen et al. 2011) are very close to each other.Other features shown are the rotation of the Galaxy, as well as the Circumnuclear Disk (CND) in the GC and the Eastern and Northern Arm gas features.The orientation of the flare angular momentum vector is consistent with the clockwise disk features.

Table 2 .
Posterior mean values of the astrometry fit.Notes.The model corresponding to this set of parameters is displayed in Fig.6(for details of the Bayesian analysis see Appendix D).