Dynamically important magnetic fields near the event horizon of Sgr A*

We study the time-variable linear polarisation of Sgr A* during a bright NIR flare observed with the GRAVITY instrument on July 28, 2018. Motivated by the time evolution of both the observed astrometric and polarimetric signatures, we interpret the data in terms of the polarised emission of a compact region ('hotspot') orbiting a black hole in a fixed, background magnetic field geometry. We calculated a grid of general relativistic ray-tracing models, created mock observations by simulating the instrumental response, and compared predicted polarimetric quantities directly to the measurements. We take into account an improved instrument calibration that now includes the instrument's response as a function of time, and we explore a variety of idealised magnetic field configurations. We find that the linear polarisation angle rotates during the flare, which is consistent with previous results. The hotspot model can explain the observed evolution of the linear polarisation. In order to match the astrometric period of this flare, the near horizon magnetic field is required to have a significant poloidal component, which is associated with strong and dynamically important fields. The observed linear polarisation fraction of $\simeq 30\%$ is smaller than the one predicted by our model ($\simeq 50\%$). The emission is likely beam depolarised, indicating that the flaring emission region resolves the magnetic field structure close to the black hole.


Introduction
There is overwhelming evidence that the Galactic Centre harbours a massive black hole, Sagittarius A* (Sgr A*, Ghez et al. 2008;Genzel et al. 2010) with a mass of M ∼ 4 × 10 6 M as inferred from the orbit of star S2 (Schödel et al. 2002;Ghez et al. 2008;Genzel et al. 2010;Gillessen et al. 2017;Gravity Collaboration et al. 2017, 2018a, 2020bDo et al. 2019a). Due to its close proximity, Sgr A* has the largest angular size of any existing black hole that is observable from Earth, and it provides a unique laboratory to investigate the physical conditions of the matter and the spacetime around the object.
Using precision astrometry with the second generation beam combiner instrument GRAVITY at the Very Large Telescope Interferometer (VLTI) operating in the NIR (Gravity Collaboration et al. 2017), we recently discovered continuous clockwise motion that is associated with three bright flares from Sgr A* (Gravity Collaboration et al. 2018b, 2020c. The scale of the apparent motion 30 − 50 µas is consistent with compact orbiting emission regions ('hotspots', e.g. Broderick & Loeb 2005Hamaus et al. 2009) at 3−5R S , where R S = 2GM/c 2 10 µas, is the Schwarzschild radius. In each flare, we also find evidence for a continuous rotation of the linear polarisation angle. The period of the polarisation angle rotation matches what is inferred from astrometry. An orbiting hotspot sampling a background magnetic field can explain the polarisation angle rotation, as long as the magnetic field configuration contains a significant poloidal component. For a rotating, magnetised fluid, remaining poloidal in the presence of orbital shear implies a dynamically important magnetic field in the flare emission region.
Here, we analyse the GRAVITY flare polarisation data in more detail, accounting for an improved instrument calibration that now includes the VLTI's response as a function of time (Section 2). We find general agreement with our previous results of an intrinsic rotation of the polarisation angle during the flare by using numerical ray tracing simulations (Section 3); we created mock observations by folding hotspot models forward through the observing process. We compare this directly to the data to show that the hotspot model can explain the observed polarisation evolution as well as to constrain the underlying magnetic field geometry and viewer's inclination (Section 4). Matching the observed astrometric period and linear polarisation fraction requires a significant poloidal component of the magnetic field structure on horizon scales around the black hole as well as an emission size that is big enough to resolve it. We discuss the implications of our results and limitations of the simple model in Section 5.

GRAVITY Sgr A* flare polarimetry
GRAVITY observations of Sgr A* have been carried out in splitpolarisation mode, where interferometric visibilities are simultaneously measured in two separate orthogonal linear polarisations. A rotating half-wave plate can be used to alternate be-tween the linear polarisation directions P 00 -P 90 and P −45 -P 45 . As a function of these polarised feeds, the Stokes parameters, as measured by GRAVITY, are I = (P 00 + P 90 )/2, Q = (P 00 − P 90 )/2 and U = (P 45 − P −45 )/2. The circularly polarised component V cannot be recorded with GRAVITY.
We relate on-sky (unprimed) polarised quantities with their GRAVITY measured (primed) counterparts bȳ whereS andS are the on-sky and GRAVITY Stokes vectors, respectively, and M is a matrix that characterises the VLTI's optical beam train response as a function of time, taking into account the rotation of the field of view during the course of the observations and birefringence. The former was calculated from the varying position of the telescopes during the observations and calibrated on sky by observing stars in the Galactic Centre (Gravity Collaboration et al. 2018b). The latter are newly introduced in the analysis here and they were obtained from modelling the effects of reflections on a long optical path through the individual UT telescopes and the VLTI. During 2018, GRAVITY observed several NIR flares from Sgr A* (Gravity Collaboration et al. 2018b). Figure 1 shows the linear polarisation Stokes parameters for four of them as measured by the instrument. On the top left, top right, and bottom left, the flares on May 27, June 27, and July 22 are shown, respectively. Only Stokes Q was measured on those nights. For the July 28 flare (bottom right), both Q and U were measured. All of the flares observed during 2018 exhibit a change in the sign of the Stokes parameters during the flare, which is consistent with a rotation of the polarisation angle with time. The linear polarisation fractions are 10 − 40%, which is in agreement with past measurements (Eckart et al. 2006;Trippe et al. 2007;Eckart et al. 2008a). Polarisation angle swings have also been previously seen in NIR flares with NACO (e.g. Zamaninasab et al. 2010). The smooth polarisation swings in both flares and the July 28 single loop in U versus Q ('QU loop', Marrone et al. 2006, Figure 2) support the astrometric result of orbital motion of a hotspot close to event horizon scales of Sgr A*.
Two assumptions have been made in the calculation of this loop. Since GRAVITY cannot register both linear Stokes parameters simultaneously, one has to interpolate the value of one quantity while the other is measured. In the case of Figure 2, this has been done by linearly interpolating between the median values over each exposure of 5 min. Second, no circular polarisation data are recorded (Stokes V ). This implies that transforming the GRAVITY measured Stokes parameters (primed) to on-sky values (unprimed) not only requires a careful calibration of the instrument systematics (contained in the matrix M, Eq. 1), but an assumption on Stokes V . In Figure 2, the assumption is that V = 0. While in theoretical models Stokes V = 0 is well justified for synchrotron radiation from highly relativistic electrons, birefrigence in the VLTI introduces a non-zero V . It is therefore important to characterize it properly.
In this work, we adopt a forward modelling approach. We take intrinsic Stokes parameters Q and U from numerical calculations of a hotspot orbiting a black hole in a given magnetic field geometry, transform them to the GRAVITY observables Q and U following Eq. (1), and compare them to the data. This not only allows us to fit the July 28 polarisation data directly without having to make assumptions on Stokes V or interpolate between gaps of data due to the lack of simultaneous measurements of the Stokes parameters, but to make predictions for Q when it is the only quantity measured, as is the case for the other 2018 flares.

Polarised synchrotron radiation in orbiting hotspot models
An optically thin hotspot orbiting a black hole produces timevariable polarised emission, depending on the spatial structure of the polarisation map (Connors & Stark 1977). For the case of synchrotron radiation, the polarisation traces the underlying magnetic field geometry (Broderick & Loeb 2005). We first discuss an analytic approximation to demonstrate the polarisation signatures generated by a hotspot in simplified magnetic field configurations, before describing the full numerical calculation of polarisation maps used for comparison to the data.

Analytic approximation
We define the observer's camera centred on the black hole with impact parametersα andβ, which are perpendicular and parallel to the spin axis, with a line of sight directionk (Bardeen 1973). In terms of these directions and assuming flat space, the Cartesian coordinates are expressed bŷ x =α,ŷ = cos iβ − sin ik,ẑ = sin iβ + cos ik, where i is the inclination of the spin axis to the line of sight. Equivalently, α =x,β = cos iŷ + sin iẑ,k = − sin iŷ + cos iẑ.
A&A proofs: manuscript no. GRAVITY_flare_pol Fig. 3. Lab frame diagram of a hotspot orbiting in thexŷ plane with position vectorh = R 0r , wherer is the unit vector in the radial direction. We note thath makes an angle ξ(t) withx. The magnetic fieldB is a function of ξ and consists of a vertical plus radial component. The strength of the latter is given by tan θ, θ, the angle between the vertical, andB. The observer's camera is defined by impact parametersα,β, and a flat space line of sightk. The line of sight makes an angle i with the spin axis of the black hole. The observer's view is shown on the right. Lastly,φ is the unit vector in the azimuthal direction.
The colour gradient denotes the periodic evolution of the hotspot along its orbit over one revolution. The only reason the width of the curves vary is for visualisation purposes. Top: completely vertical magnetic field (θ = 0). We note that Q and U are constants in time and have static values in QU space. Bottom: significantly radial magnetic field with θ = 80; Q and U oscillate and trace two QU loops in time that change in amplitude with inclination. High inclination counteracts the presence of QU loops.
When face-on,k points alongẑ andβ points alongŷ. When edgeon,k points along −ŷ andβ points alongẑ. Let a hotspot be orbiting in thexŷ plane ( Figure 3). In terms ofα,β, andk, the hotspot's position vectorh is given bȳ wherer is the canonical radial vector, R 0 is the orbital radius, and ξ is the angle betweenα andr. Let us consider the magnetic field with vertical and radial components given bȳ where B 0 is the magnitude ofB and θ is the angle betweenẑ and B. The polarisation is given asP =k ×B. In flat space and in the absence of motion (no light bending or aberration), The polarisation angle on the observer's camera is tan ψ = P ·β/P ·α, so that Given that U/Q = 1/2 tan ψ, the Stokes parameters as a function of the polarisation angle are With equations (6), (7), and (8), Stokes Q and U are obtained.
It is important to note that a single choice of i and θ returns Q=Q(ξ) and U=U(ξ). Assuming a constant velocity along the orbit, the angle ξ can be mapped linearly to a time value by setting the duration of the orbital period and an initial position where the ξ = 0.
Additionally, an inclination of i = i 0 < 90 • and i = 180 • − i 0 produces the same polarised curves but they are reversed in ξ with respect to each other. This is expected since, for an observer at i = i 0 and one at i = 180 • − i 0 , the hotspot samples the same magnetic field geometry, but they appear to be moving in opposite directions with respect to each other. This means that the relative order in which the peaks in Q and U appear are reversed between observers at i = i 0 and at i = 180 • − i 0 .
Given that light bending has not been considered in this approximation, in a significantly vertical field (θ 0, top of Fig.  4), the polarisation remains constant in ξ (and time) proportional to − sin i. In QU space, this means a static value as the hotspot goes around the black hole. A particular case of this isP 0 at i 0, sincek andB are parallel. As θ − → π/2, tan θ − → ∞ (bottom of Fig. 4), and the magnetic field becomes radial. In this case and at low inclinations, the polarisation configuration is toroidal (P ∝φ, the azimuthal canonical vector, Eq. B.1). As the hotspot orbits the black hole, Q and U show oscillations of the same amplitude. In one revolution, two superimposed QU loops can be traced. If the viewer's inclination increases, one of the loops decreases more in size than the other and eventually disappears at very high inclinations, leaving only one behind. Increasing inclination, therefore, counteracts the presence of QU loops in an analytical model with a vertical plus radial magnetic field. It is noted that the normalised polarisation configurations of a completely radial magnetic field and a toroidal one are equivalent with just a phase offset of 90 • in ξ (Eq. B.2 in Appendix B).

Ray-tracing calculations
Next, we use numerical calculations to include general relativistic effects. We used the general relativistic ray tracing code grtrans (Dexter & Agol 2009;Dexter 2016) to calculate synchrotron radiation from orbiting hotspots in the Kerr metric.
The hotspot model is taken from Broderick & Loeb (2006), and it consists of a finite emission region orbiting in the equatorial plane at radius R 0 . The orbital speed is constant for the entire emission region, and it matches that of a test particle motion at its centre. The maximum particle density n spot ∼ 2 × 10 7 cm −3 falls off as a three-dimensional Gaussian with a characteristic size of R spot . The magnetic field has a vertical plus radial component 1 . Its strength is taken from an equipartition assumption, where we further assume a virial ion temperature of kT i = (n spot /n tot ) (m p c 2 /R), (n spot /n tot ) = 5, where n tot is the total particle density in the hotspot. For the models considered here, a typical magnetic field strength in the emission region is B 100 G. We calculated synchrotron radiation from a power law distribution of electrons with a minimum Lorentz factor of 1.5 × 10 3 and considered a black hole with a spin of zero. 2 . The model parameters for field strength, density, and minimum Lorentz factor were chosen as typical values for models of Sgr A* which can match the observed NIR flux. Other combinations are possible.
Example snapshots of a hotspot model in a vertical field (θ = 0) and the resulting polarisation configuration are shown in Figure 5. The effects of lensing can be appreciated in the form of secondary images. It can be seen as well that as the hotspot moves along its orbit around the black hole, it samples the magnetic field geometry in time, so that the time-resolved polarisation encodes information about the spatial structure of the magnetic field. Figure 6 shows the numeric calculations of hotspot models with the same magnetic field angles as those in the analytic approximation. Inclination and θ are key parameters in the observed number and shape of QU loops. In contrast to the analytic case, in a significantly vertical field (θ 0, top of Fig. 6), the polarisation is not zero. This is mainly due to light bending, which introduces an effective radial component to the wave-vector in the plane of the observer's camera. This radial component ofk leads to an additional azimuthal contribution toP. The θ = 0 cases show that this effect alone is able to generate QU loops. We see again that increasing inclination leads to a change from two QU loops per hotspot revolution at low inclinations to a single QU loop at high inclinations.
The cases where θ − → 90 • (bottom of Fig. 6) show that increasing this parameter also leads to scenarios with two QU loops per hotspot orbit. The shape of the numerical Q and U curves is similar to the analytic versions. The differences are due to the inclusion of relativistic effects in the ray-tracing calculations. We note that numerical models with a vertical plus toroidal magnetic field show similar features and behaviour to those in the vertical plus radial case (see Appendix C).

Model fitting
We calculated normalised Stokes parameters Q/I and U/I from ray tracing simulations of a grid of hotspot models, folded them through the instrumental response (Eq. 1), and compared them A&A proofs: manuscript no. GRAVITY_flare_pol to GRAVITY's measured Q /I and U /I . The parameters of the numerical model are the orbital radius R 0 , the size of the hotspot R spot , the viewing angle i, and the tilt angle of the magnetic field direction θ. We understand qualitatively how the hotspot size and the orbital radius affect the Q and U curves. 'Smoother' curves, where the amplitude of the oscillations is reduced, are produced either with increasing hotspot sizes at fixed orbital radius or with decreasing R 0 at a fixed hotspot size, due to beam depolarisation (see Appendix F). Since performing full ray tracing simulations is computationally very expensive, and due to the fact that the curves change smoothly and gradually with R 0 and R spot , we chose to fix their values to R 0 = 8R g and R spot = 3R g , R g the gravitational radius. We then scaled them in both period and amplitude to match the data better in the following manner.
Given the duration of a flare ∆t, we could scale a hotspot's period by a factor nT to set the fraction of orbital periods that fit into this time window. The new radius of the orbit is then R ∝ (∆t/nT ) 2/3 . This rescaling introduced small changes in fit quality compared to re-calculating new models, within our parameter range of interest (see Appendix E). We absorbed the effect of beam depolarisation into a factor s that scales the overall amplitude of both Q and U and, therefore, the linear polarisation fraction as well.
Given a hotspot's period, the relative phase reflects the hotspot position relative to an initial position measured at some initial time, where the phase is defined to be zero. We chose the initial position of the hotspot based on the astrometric measurement of the orbital motion of the flare in Gravity Collaboration et al. (2020c). Specifically, we chose the initial phase ξ to match the initial position of the best-fit orbital model to the astrometry.

Application to the July 28 flare
The observed Q /I and U /I were measured from fitting interferometric binary models to GRAVITY data. The binary model measures the separation of Sgr A* and the star S2, which were both in the GRAVITY interferometric field of view ( 50 mas) during 2018. For more details, see Gravity Collaboration et al. (2020a). We measured polarisation fractions assuming that S2's NIR emission is unpolarised. The 70 minute time period analysed is limited by signal-to-noise: binary signatures are largest when Sgr A* is brightest. As a result, we focused on data taken during the flare. We fitted to data binned by 30 seconds since the flux ratio can be rapidly variable. We further adopted error bars on polarisation fractions using the rms of measurements within 300s time intervals since direct binary model fits generally have χ 2 > 1, and as a result underestimate the fit uncertainties.
We computed a grid of models with i, θ, s, and nT as pa- , ∆s = 0.05, and nT such that the allowed range of radii for the fit is R = 8 − 11 R g with ∆R = 0.2. We have included this prior in radii to match the constraint from the combined astrometry of the three bright GRAVITY 2018 flares (Gravity Collaboration et al. 2020c). The best fit parameters and corresponding polarised curves are shown in Figure 7. We find that the curves qualitatively reproduce the data and that the statistically preferred parameter combination for July 28, with a reduced χ 2 ∼ 3.1, favours a radius of 8 R g and moderate i and θ values (left panel of Figure 7). In QU space, these parameters produce two intertwined and embedded QU loops of very different amplitudes in time (right panel of Figure 7). The outer one is fairly circular, centred approximately around zero and with an average radius of 0.18. The inner one has a horizontal oblate shape with a QU axis ratio of approximately 2:1, does not go around zero, and represents a much smaller fraction of the orbit than the larger loop. These moderate values of θ imply that a magnetic field with significant components in both the radial and vertical directions is favoured.
The hotspot is free to trace a clockwise (i > 90 • ) or counterclockwise (i < 90 • ) motion on-sky. At fixed θ, this change in apparent motion results in an inversion of the order in which the maxima of the Q and U curves appear 3 . This effect is due to relativistic motion (Blandford & Königl 1979;Bjornsson 1982). When the magnetic field is purely toroidal (velocity parallel toB), the polarisation angle is independent of velocity. When there is a field component perpendicular to the velocity (poloidal field), relativistic motion induces an additional swing of the polarisation angle in the direction of movement where magnitude depends on the velocity. We ignore this effect in the analytic approximation above, but it is included in our numerical calculations.
The data favour models where the maxima in U /I precede those of Q /I . This behaviour is observed in the case of clockwise motion (i > 90 • ) with θ ∈ [0 • − 90 • ] and in counterclockwise motion (i < 90 • ) with θ ∈ [90 • − 180 • ]. In fact, model curves at a given i > 90 • and θ ∈ [0 • − 90 • ] are identical to those with their 'mirrored' values i = 180 • −i and θ = 180 • −θ. In our analysis, we consider θ ∈ [0 • − 90 • ], which favours a clockwise motion. However, we cannot uniquely determine the apparent direction of motion of the hotspot due to this degeneracy.
Our models overproduce the observed linear polarisation fraction by a factor of ∼ 1.7 (scaling factor s 0.4 < 1). The maximum observed polarisation fraction is 30%, while it is 50% in our models. The degree of depolarisation introduced by the VLTI is not substantial enough to reduce the model linear polarisation fraction to the observed one. Moreover, in the NIR, there are no significant depolarisation contributions from absorption or Faraday effects. As a result, we conclude that the low observed polarisation fraction is likely the result of beam depolarisation. The observed low polarisation fraction implies that the flare emission region is big enough to resolve the underlying magnetic field structure. In the context of our model, this could imply a larger spot size. It could also indicate a degree of disorder in the background magnetic field structure, for example as a result of turbulence.

Application to the July 22 flare
July 28 is the only night with an observed infrared flare in which GRAVITY recorded both Stokes Q and U . Since a single polarisation channel is insufficient to constrain the full parameter space used in our numerical models, we restricted ourselves to the night of July 22, as this observation has the highest precision astrometry 4 , and fixed the viewer inclination and magnetic field geometry to be the same as the best fit model to the July 28 data. We scaled the curves in amplitude with s ∈ [0.05 − 0.35], ∆s = 0.05.
The initial position on sky for both flares is constrained by astrometric data and, therefore, so is the phase offset between both curves. With a fixed phase difference between the curves and free range of radii, we find that the July 22 data favours ex-  8. Fit to the July 22 NIR flare without restricting the phase difference between this night and that of July 28. The colour gradient denotes the evolution of the hotspot as it completes one revolution. The viewer's inclination, magnetic field geometry, and orbital direction have been fixed to the values found for the July 28 flare. The fit favours values of R 0 ∼ 11 R g and there is no initial phase difference between the nights (no difference in starting position on-sky), which is out of the allowed uncertainty range for the astrometry.
tremely large values of R 0 > 20 R g , which are outside of the allowed range obtained from astrometric measurements. In allowing the phase difference to be free and constraining the radii to 8 − 11 R g , with ∆R = 0.2, we find that the data tend to values of R 0 ∼ 11 R g and a phase difference between curves of 0 • (Figure 8). This phase difference value (and position difference associated with it) is outside of the allowed uncertainties in the initial position indicated by the astrometric data. The fact that the magnetic field parameters that describe the July 28 flare fail to adequately fit the data from July 22 may indicate that the background magnetic field geometry changes on a several-day timescale.

Summary and discussion
In this work, we present an extension of the initial analysis of polarisation data performed in Gravity Collaboration et al. (2018b). We forward modelled Q and U Stokes parameters obtained from ray-tracing calculations of a variety of hotspot models in different magnetic field geometries, transformed them into quantities as seen by the instrument, and fitted them directly to the polarised data taken with GRAVITY. This allowed us to not only fit data directly without making assumptions about Stokes V or the interpolation of data in nonsimultaneous Q and U measurements, but also to predict the behaviour in time of the polarised curves and loops for the cases where only one of the parameters was measured.
We have shown that the hotspot model serves to qualitatively reproduce the features seen in the polarisation data measured with GRAVITY. A moderate inclination and moderate mix of both vertical and radial fields provide the best statistical fit to the data. Consistent results are found by fitting the data with a vertical plus toroidal field component (Appendix C). We note that this result does not rely on the assigned strength of the magnetic field, since the model curves are scaled in amplitude, but rather it is only from the geometry of the field. Magnetic fields with a non-zero vertical component fit the data statistically better. This supports the idea that there is some amount of ordered magnetic field in the region near the event horizon with a significant poloidal field component. The presence of this component is associated with magnetic fields that are dynamically important and it confirms the previous finding of strong fields in Gravity Collaboration et al. (2018b). Spatially resolved observations at 1.3mm also found linear polarisation structure consistent with a mix of ordered and disordered magnetic field (Johnson et al. 2015).
Matching the clockwise direction of motion inferred by the astrometric data would require that θ ∈ [0 • − 90 • ]. Under this assumption, the results are also in accordance with the angular momentum direction and orientation of the clockwise stellar disc and gas cloud G2 (Bartko et al. 2009;Gillessen et al. 2019;Pfuhl et al. 2015;Plewa et al. 2017).
We have chosen the bright NIR flare on July 28, 2018 since it is the only one for which both linear Stokes parameters have been measured. Naturally, increasing the number of full data sets in future flares will be useful in constraining the parameter range more.
Our models overproduce the observed NIR linear polarisation fraction of ∼ 30% by a factor of ∼ 1.7, and they must be scaled down to fit the data. In the compact hotspot model context, this implies that an emission region size larger than 3 R g is needed to depolarize the NIR emission through beam depolarisation. Including shear in the models would naturally introduce depolarisation since a larger spread of polarisation vector directions (or equivalently, the magnetic field structure) would be sampled at any moment (e.g. Gravity Collaboration et al. 2020c; Tiede et al. 2020). However, this might smooth out the fitted curves and would probably change the fits. In any case, the observed low NIR polarisation fraction means that the observed emission region resolves the magnetic field structure around the black hole.
Though simplistic, the hotspot model appears to be viable for explaining the general behaviour of the data. It would be interesting to study the polarisation features of more complex, total emission scenarios explored in other works. Ball et al. (2020) study orbiting plasmoids that result from magnetic reconnection events close to the black hole, where some variability in the polarisation should be caused by the reconnecting field itself. Dexter et al. (2020) find that material ejected due to the build-up of strong magnetic fields close to the event horizon can produce flaring events where the emission region follows a spiral trajectory around the black hole. In their calculations, ordered magnetic fields result in a similar polarisation angle evolution as we have studied here. Disorder caused by turbulence reduces the linear polarisation fraction to be consistent with what is observed.
Spatially resolved polarisation data are broadly consistent with the predicted evolution in a hotspot model. This first effort comparing these types of models directly to GRAVITY data shows the promise of using the observations to study magnetic field structure and strength on event horizon scales around black holes.

Appendix A: Vertical plus radial field in Boyer-Lindquist coordinates
In the Boyer-Lindquist coordinate frame, a magnetic field with a vertical plus radial components can be written as: where B µ are the contravariant components of B and δ c ≡ B r /B θ . The magnetic field must satisfy the following conditions: where u µ are the contravariant components of the four-velocity, B is the magnitude of B, and g µν are the covariant components of the Kerr metric. In Boyer-Lindquist coordinates with G = c = M = 1, the non-zero components of the metric are: where a is the dimensionless angular momentum of the black hole. Using Eq. (A.1), (A.2), and (A.3), it follows that the Boyer-Lindquist coordinate frame contravariant components of the magnetic field are δ c g rr u r + g θθ u θ g tt u t + g tφ u φ () (A.4) and where δ LNRF is the ratio of the radial and poloidal magnetic field components in the locally non-rotating frame (LNRF, Bardeen 1973) and B (µ) are the contravariant components of B in the LNRF: where the expression to the far right is obtained by assuming r a (as it is in the hotspot case). The variable δ used in the main text (Eq. (5)) corresponds to δ LNRF defined here as being calculated using the r a approximation.

Appendix B: Analytic approximation with a vertical plus toroidal magnetic field
In the case of a vertical plus toroidal magnetic field, the magnetic field can be written asB ∝ẑ+λφ, where λ ∝ tan θ T is the strength of the toroidal component, θ T is the angle measured from the toroidal component to the vertical component (θ T = 0 denotes a completely toroidal field), and is the canonical vector in the azimuthal direction ( Figure 3). We note thatr ·φ = 0. The polarisation vector in a flat space given byk ×B is then P ∝ −(sin i + λ cos i cos ξ)α − λ sin ξβ (B.2) and the polarisation angle is given by It can be seen from expression (B.2) that at low inclinations or when λ >> 1 (complete toroidal magnetic field), the polarisation has a radial configuration (P ∝r, Eq. 4). This is geometrically equivalent to the polarisation having a toroidal configuration (similar to the one generated by a completely radial magnetic field, see Section 3) with a phase offset of π/2 in Q and U. In this case, we would expect to have two superimposed QU loops in one revolution of the hotspot. Figure B.1 shows a comparison between the analytic (top) and numeric (bottom) calculations for a vertical plus toroidal magnetic field (Appendix C). As expected, in the analytic case, there are always two superimposed loops in QU space in the case of a completely toroidal field. In the numeric calculations, this is also the case given that light bending favours the presence of loops. As a vertical component in the field is introduced, the loops no longer overlay on each other. This effect increases with viewer inclination. It can also be seen that the completely toroidal and radial cases produce the same Q and U curves at low inclinations, save for a phase offset and scaling factor. It can also be seen that toroidal and completely radial configurations produce the same curves, save for a a scaling factor and a phase offset.

Appendix C: Vertical plus toroidal field in Boyer-Lindquist coordinates
In the Boyer-Lindquist coordinate frame, a magnetic field with a vertical plus toroidal components can be written as: where B µ are the contravariant components of B and η c ≡ B θ /B φ . Just as in the vertical plus radial case, the magnetic field must satisfy Eqs. (A.2). Using Eqs. (C.1), (A.2), and (A.3), it follows that the Boyer-Lindquist coordinate frame contravariant components of the magnetic field are η LNRF = B (θ) /B (φ) = tan θ T the ratio of the poloidal and toroidal magnetic field components in the LNRF (Eq. (A.5)), and θ T is the angle measured from the toroidal component to the vertical (θ T = 0 implies a completely toroidal field, Appendix B). We fitted the July 28 data considering this magnetic geometry. Just as in the vertical plus radial case, we computed a grid of models with i, θ, s, and nT as parameters: i ∈ [0 − 180] in increments of ∆i = 4 • ; θ T ∈ [0 − 90], ∆θ T = 5 • ; s ∈ [0.4 − 0.8], ∆s = 0.05, and nT such that the allowed range of radii for the fit is R = 8 − 11 R g with ∆R = 0.2. The best fit is shown in Figure C.1. Though a better reduced χ 2 is found at a somewhat higher inclination than the best fit with a vertical plus radial magnetic field (Fig. 7), the presence of a poloidal component in the magnetic field is still needed. Considering θ T ∈ [0 • − 90 • ], a clockwise motion is preferred (i > 90 • ). Identical curves can be obtained when the direction of motion is counterclockwise (i < 90 • ) and the magnetic field angle is θ T = 180 • − θ T . Figure  C.2 presents a model of a vertical plus toroidal magnetic field with similar parameters to those of the vertical plus radial field best fit.

Appendix D: Spin effects
We present the effects of spin in our calculations. Figure D.1 shows three models with the best fit parameters found for the July 28 flare, at three different dimensionless spin values a=0.0, 0.9, −0.9. The corresponding reduced χ 2 values are reported in Table D

Appendix E: Scaling period effects
We explore the effects of scaling the period of model curves. Figure E.1 shows the best fit model found for the July 28 flare and one calculated at R = 11 R g scaled down to match the period at 8 R g , with the rest of the parameters fixed to those of the best fit. The corresponding reduced χ 2 values are reported in Table E.1. It can be seen that the curves show similar behaviours. Scaled models might have a better reduced χ 2 than their non-scaled versions, but they are still not better than the best fit.

Appendix F: Qualitative beam depolarisation
In the absence of other mechanisms, such as self-absorption or Faraday rotation and conversion, infrared emission from an orbiting hotspot is depolarised by beam depolarisation. . Models calculated at R = 8 R g and at R = 11 R g , the latter was scaled down to match the orbital period at 8 R g . The rest of the parameters are those found for the best fit for the July 28 flare. The reduced χ 2 are reported in Table E.1. For better clarity, the R = 11 R g non-scaled model fit is not shown, but the χ 2 is reported. Comparison of three numerical calculations with all identical parameters, except for R spot : 1, 3, and 5 R g . As the hotspot size increases, the curve features are smoothed from beam depolarisation by sampling larger magnetic field regions and averaging out the different polarisation directions in time.
Table E.1. Reduced χ 2 of models calculated at R = 8 R g and at R = 11 R g , the latter was scaled down to match the orbital period at 8 R g .
More beam depolarisation occurs, the larger the emitting region that samples the underlying magnetic field is, or the more disordered the field itself is. Given the simple magnetic field geometries considered in this work, disorder at small scales is nonexistent. We discuss qualitatively the impact of emission size in the following.
As the hotspot goes around the black hole, it samples a wedge of angles in the azimuthal direction with an arc length of R spot /R 0 . Larger beam depolarisation occurs with the increase of this factor. Figure F.1 shows example curves of numerical calculations at a moderate inclination and magnetic field tilt, where only the hotspot size has been changed. As expected, with increasing R spot at a fixed orbital radius, not only does the amplitude of the polarised curves and QU loops diminish (and with it, the linear polarisation fraction), but the features in them are smoothed out as well. Within the hotspot model, beam depolari-sation can therefore be used to constrain the size of the emitting region as a function of the observed linear polarisation fraction.