Dynamically important magnetic fields near the event horizon of Sgr A*
GRAVITY Collaboration, A. Jiménez-Rosales, J. Dexter, F. Widmann, M. Bauböck, R. Abuter, A. Amorim, J.P. Berger, H. Bonnet, W. Brandner, Y. Clénet, P.T. de Zeeuw, A. Eckart, F. Eisenhauer, N.M. Förster Schreiber, P. Garcia, F. Gao, E. Gendron, R. Genzel, S. Gillessen, M. Habibi, X. Haubois, G. Heissel, T. Henning, S. Hippler, M. Horrobin, L. Jochum, L. Jocou, A. Kaufer, P. Kervella, S. Lacour, V. Lapeyrère, J.-B. Le Bouquin, P. Léna, M. Nowak, T. Ott, T. Paumard, K. Perraut, G. Perrin, O. Pfuhl, G. Rodríguez-Coira, J. Shangguan, S. Scheithauer, J. Stadler, O. Straub, C. Straubmeier, E. Sturm, L.J. Tacconi, F. Vincent, S. von Fellenberg, I. Waisberg, E. Wieprecht, E. Wiezorrek, J. Woillez, S. Yazici, G. Zins
AAstronomy & Astrophysics manuscript no. GRAVITY_flare_pol c (cid:13)
ESO 2020September 15, 2020
Dynamically important magnetic fields near the event horizon ofSgr A*
GRAVITY Collaboration (cid:63) : A. Jiménez-Rosales , J. Dexter , , F. Widmann , M. Bauböck , R. Abuter ,A. Amorim , , J.P. Berger , H. Bonnet , W. Brandner , Y. Clénet , P.T. de Zeeuw , , A. Eckart , , F. Eisenhauer ,N.M. Förster Schreiber , P. Garcia , , F. Gao , E. Gendron , R. Genzel , , S. Gillessen , M. Habibi , X. Haubois ,G. Heissel , T. Henning , S. Hippler , M. Horrobin , L. Jochum , L. Jocou , A. Kaufer , P. Kervella , S. Lacour ,V. Lapeyrère , J.-B. Le Bouquin , P. Léna , M. Nowak , , T. Ott , T. Paumard , K. Perraut , G. Perrin , O. Pfuhl , ,G. Rodríguez-Coira , J. Shangguan , S. Scheithauer , J. Stadler , O. Straub , C. Straubmeier , E. Sturm ,L.J. Tacconi , F. Vincent , S. von Fellenberg , I. Waisberg , , E. Wieprecht , E. Wiezorrek , J. Woillez , S. Yazici , ,and G. Zins Max Planck Institute for Extraterrestrial Physics, Giessenbachstraße 1, 85748 Garching, Germany LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195Meudon, France Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany st Institute of Physics, University of Cologne, Zülpicher Straße 77, 50937 Cologne, Germany Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France Universidade de Lisboa - Faculdade de Ciências, Campo Grande, 1749-016 Lisboa, Portugal Faculdade de Engenharia, Universidade do Porto, rua Dr. Roberto Frias, 4200-465 Porto, Portugal European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching, Germany European Southern Observatory, Casilla 19001, Santiago 19, Chile Max Planck Institute for Radio Astronomy, Auf dem Hügel 69, 53121 Bonn, Germany Sterrewacht Leiden, Leiden University, Postbus 9513, 2300 RA Leiden, The Netherlands Departments of Physics and Astronomy, Le Conte Hall, University of California, Berkeley, CA 94720, USA CENTRA - Centro de Astrofísica e Gravitação, IST, Universidade de Lisboa, 1049-001 Lisboa, Portugal Department of Astrophysical & Planetary Sciences, JILA, Duane Physics Bldg., 2000 Colorado Ave, University of Colorado,Boulder, CO 80309, USA Department of Particle Physics & Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK
ABSTRACT
We study the time-variable linear polarisation of Sgr A* during a bright NIR flare observed with the GRAVITY instrument on July28, 2018. Motivated by the time evolution of both the observed astrometric and polarimetric signatures, we interpret the data in termsof the polarised emission of a compact region (‘hotspot’) orbiting a black hole in a fixed, background magnetic field geometry. Wecalculated a grid of general relativistic ray-tracing models, created mock observations by simulating the instrumental response, andcompared predicted polarimetric quantities directly to the measurements. We take into account an improved instrument calibrationthat 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 canexplain the observed evolution of the linear polarisation. In order to match the astrometric period of this flare, the near horizonmagnetic field is required to have a significant poloidal component, which is associated with strong and dynamically important fields.The observed linear polarisation fraction of (cid:39)
30% is smaller than the one predicted by our model ( (cid:39)
Key words.
Galaxy: center — black hole physics — polarization — relativistic processes (cid:63)
GRAVITY is developed in a collaboration between the MaxPlanck Institute for extraterrestrial Physics, LESIA of Observatoire deParis / Université PSL / CNRS / Sorbonne Université / Université de Parisand IPAG of Université Grenoble Alpes / CNRS, the Max Planck Insti-tute for Astronomy, the University of Cologne, the CENTRA - Centrode Astrofisica e Gravitação, and the European Southern Observatory.Corresponding author: A. Jiménez-Rosales ([email protected])
1. Introduction
There is overwhelming evidence that the Galactic Centre har-bours a massive black hole, Sagittarius A* (Sgr A*, Ghez et al.2008; Genzel et al. 2010) with a mass of M ∼ × M (cid:12) as in-ferred from the orbit of star S2 (Schödel et al. 2002; Ghez et al.2008; Genzel et al. 2010; Gillessen et al. 2017; Gravity Collab-oration et al. 2017, 2018a, 2019, 2020b; Do et al. 2019a). Dueto its close proximity, Sgr A* has the largest angular size of anyexisting black hole that is observable from Earth, and it provides Article number, page 1 of 13 a r X i v : . [ a s t r o - ph . H E ] S e p & A proofs: manuscript no. GRAVITY_flare_pol a unique laboratory to investigate the physical conditions of thematter and the spacetime around the object.The observed emission from Sgr A* is variable at all wave-lengths from the radio to X-rays (e.g. Bagano ff et al. 2001;Zhao et al. 2001; Genzel et al. 2003; Ghez et al. 2004; Eisen-hauer et al. 2005; Macquart & Bower 2006; Marrone et al. 2008;Eckart et al. 2008a; Do et al. 2009; Witzel et al. 2018; Do et al.2019b). The simultaneous, large amplitude variations (‘flares’)seen in the near-infrared (NIR) and X-ray (Yusef-Zadeh et al.2006; Eckart et al. 2008b) are the result of transiently heated rel-ativistic electrons near the black hole, which are likely heated inshocks or by magnetic reconnection (Marko ff et al. 2001; Yuanet al. 2003; Barrière et al. 2014; Haggard et al. 2019).The linear polarisation fraction of (cid:39) −
40% (Eckart et al.2006; Trippe et al. 2007; Eckart et al. 2008a; Zamaninasab et al.2010; Shahzamanian et al. 2015) implies that the NIR emissionis the result of synchrotron emission from relativistic electrons.The NIR to X-ray spectral shape favours direct synchrotron ra-diation from electrons up to high energies ( γ ∼ , Dodds-Eden et al. 2009; Li et al. 2015; Ponti et al. 2017), althoughinverse Compton scenarios may remain viable (Porquet et al.2003; Eckart et al. 2010; Yusef-Zadeh et al. 2012).Using precision astrometry with the second generation beamcombiner instrument GRAVITY at the Very Large Telescope In-terferometer (VLTI) operating in the NIR (Gravity Collabora-tion et al. 2017), we recently discovered continuous clockwisemotion that is associated with three bright flares from Sgr A*(Gravity Collaboration et al. 2018b, 2020c). The scale of the ap-parent motion (cid:39) − µ as is consistent with compact orbitingemission regions (‘hotspots’, e.g. Broderick & Loeb 2005, 2006;Hamaus et al. 2009) at (cid:39) − R S , where R S = GM / c (cid:39) µ as,is the Schwarzschild radius. In each flare, we also find evidencefor a continuous rotation of the linear polarisation angle. The pe-riod of the polarisation angle rotation matches what is inferredfrom astrometry. An orbiting hotspot sampling a backgroundmagnetic field can explain the polarisation angle rotation, as longas the magnetic field configuration contains a significant poloidalcomponent. For a rotating, magnetised fluid, remaining poloidalin the presence of orbital shear implies a dynamically importantmagnetic field in the flare emission region.Here, we analyse the GRAVITY flare polarisation data inmore detail, accounting for an improved instrument calibrationthat now includes the VLTI’s response as a function of time (Sec-tion 2). We find general agreement with our previous results ofan intrinsic rotation of the polarisation angle during the flare byusing numerical ray tracing simulations (Section 3); we createdmock observations by folding hotspot models forward throughthe observing process. We compare this directly to the data toshow that the hotspot model can explain the observed polarisa-tion evolution as well as to constrain the underlying magneticfield geometry and viewer’s inclination (Section 4). Matchingthe observed astrometric period and linear polarisation fractionrequires a significant poloidal component of the magnetic fieldstructure on horizon scales around the black hole as well as anemission size that is big enough to resolve it. We discuss the im-plications of our results and limitations of the simple model inSection 5.
2. GRAVITY Sgr A* flare polarimetry
GRAVITY observations of Sgr A* have been carried out in split-polarisation mode, where interferometric visibilities are simul-taneously measured in two separate orthogonal linear polarisa-tions. A rotating half-wave plate can be used to alternate be- tween the linear polarisation directions P — P and P − — P . As a function of these polarised feeds, the Stokes pa-rameters, as measured by GRAVITY, are I (cid:48) = ( P + P ) / Q (cid:48) = ( P − P ) / U (cid:48) = ( P − P − ) /
2. The circularlypolarised component V (cid:48) cannot be recorded with GRAVITY.We relate on-sky (unprimed) polarised quantities with theirGRAVITY measured (primed) counterparts by¯ S = M ¯ S (cid:48) , (1)where ¯ S and ¯ S (cid:48) are the on-sky and GRAVITY Stokes vectors,respectively, and M is a matrix that characterises the VLTI’s op-tical beam train response as a function of time, taking into ac-count the rotation of the field of view during the course of theobservations and birefringence. The former was calculated fromthe varying position of the telescopes during the observationsand calibrated on sky by observing stars in the Galactic Centre(Gravity Collaboration et al. 2018b). The latter are newly intro-duced in the analysis here and they were obtained from mod-elling the e ff ects of reflections on a long optical path through theindividual UT telescopes and the VLTI.During 2018, GRAVITY observed several NIR flares fromSgr A* (Gravity Collaboration et al. 2018b). Figure 1 shows thelinear polarisation Stokes parameters for four of them as mea-sured by the instrument. On the top left, top right, and bottomleft, the flares on May 27, June 27 , and July 22 are shown, re-spectively. Only Stokes Q (cid:48) was measured on those nights. Forthe July 28 flare (bottom right), both Q (cid:48) and U (cid:48) were measured.All of the flares observed during 2018 exhibit a change in thesign of the Stokes parameters during the flare, which is con-sistent with a rotation of the polarisation angle with time. Thelinear polarisation fractions are (cid:38) − U versus Q (‘ QU loop’, Marroneet al. 2006, Figure 2) support the astrometric result of orbitalmotion of a hotspot close to event horizon scales of Sgr A*.Two assumptions have been made in the calculation of thisloop. Since GRAVITY cannot register both linear Stokes pa-rameters simultaneously, one has to interpolate the value of onequantity while the other is measured. In the case of Figure 2,this has been done by linearly interpolating between the medianvalues over each exposure of (cid:39) V (cid:48) ). This implies that trans-forming the GRAVITY measured Stokes parameters (primed) toon-sky values (unprimed) not only requires a careful calibrationof the instrument systematics (contained in the matrix M , Eq. 1),but an assumption on Stokes V (cid:48) . In Figure 2, the assumption isthat V (cid:48) =
0. While in theoretical models Stokes V = V (cid:48) . It istherefore important to characterize it properly.In this work, we adopt a forward modelling approach. Wetake intrinsic Stokes parameters Q and U from numerical cal-culations of a hotspot orbiting a black hole in a given magneticfield geometry, transform them to the GRAVITY observables Q (cid:48) and U (cid:48) following Eq. (1), and compare them to the data. This notonly allows us to fit the July 28 polarisation data directly withouthaving to make assumptions on Stokes V (cid:48) or interpolate betweengaps of data due to the lack of simultaneous measurements of theStokes parameters, but to make predictions for Q (cid:48) when it is theonly quantity measured, as is the case for the other 2018 flares. Article number, page 2 of 13RAVITY Collaboration: A. Jiménez-Rosales et al.: Strong magnetic fields near Sgr A* Q ' / I ' May 27 th Q ' / I ' June 27 th Q ' / I ' July 22 nd th Fig. 1.
Linear polarisation Stokes parameters of four Sgr A* NIR flares observed by GRAVITY during 2018. The prime notation denotes thequantities as recorded by the instrument (including the e ff ects of field rotation and systematics). Top left: Stokes Q (cid:48) on May 27. Top right: Stokes Q (cid:48) on June 27. Bottom left: Stokes Q (cid:48) on July 22. Bottom right: Stokes Q (cid:48) and U (cid:48) on July 28. All of the flares show (cid:38) −
40% linear polarisation.A common, continuous evolution is seen on all nights. In three cases, Q (cid:48) shows a change in sign, consistent with rotation of the polarisation angle.The implied period of the polarisation evolution matches what is seen in astrometry. Q/I U / I Field rotation
Q/I
Full calibration T i m e [ m i n ] Fig. 2.
Reconstructed evolution of the on-sky linear Stokes parameters in QU space for the July 28 flare, linearly interpolating to fill in U (cid:48) and Q (cid:48) where the other is measured. Colour indicates time in minutes. Left: previous calibration where the quantities have only been subjected to a fieldrotation correction (Gravity Collaboration et al. 2018b). Right: full new calibration including VLTI systematics and Stokes V (cid:48) reconstruction. Inboth cases, the flare traces 1.5 loops during its 60 −
70 minute evolution.
3. Polarised synchrotron radiation in orbitinghotspot models
An optically thin hotspot orbiting a black hole produces time-variable polarised emission, depending on the spatial structureof the polarisation map (Connors & Stark 1977). For the caseof synchrotron radiation, the polarisation traces the underlyingmagnetic field geometry (Broderick & Loeb 2005). We first dis-cuss an analytic approximation to demonstrate the polarisationsignatures generated by a hotspot in simplified magnetic fieldconfigurations, before describing the full numerical calculationof polarisation maps used for comparison to the data.
We define the observer’s camera centred on the black hole withimpact parameters ˆ α and ˆ β , which are perpendicular and parallelto the spin axis, with a line of sight direction ˆ k (Bardeen 1973). Interms of these directions and assuming flat space, the Cartesiancoordinates are expressed byˆ x = ˆ α, ˆ y = cos i ˆ β − sin i ˆ k , ˆ z = sin i ˆ β + cos i ˆ k , (2)where i is the inclination of the spin axis to the line of sight.Equivalently,ˆ α = ˆ x , ˆ β = cos i ˆ y + sin i ˆ z , ˆ k = − sin i ˆ y + cos i ˆ z . (3) Article number, page 3 of 13 & A proofs: manuscript no. GRAVITY_flare_pol
OBSERVER’S CAMERA x , 𝛂 k h r 𝜃 B z , 𝜷 R 𝜉 𝟇 𝜷 𝜉 x , 𝛂 i k -k y-y z h 𝟇 r 𝜃 B OBSERVER’S
CAMERAR Fig. 3.
Lab frame diagram of a hotspot orbiting in the ˆ x ˆ y plane with position vector ¯ h = R ˆ r , where ˆ r is the unit vector in the radial direction. Wenote that ¯ h makes an angle ξ ( t ) with ˆ x . The magnetic field ¯ B is a function of ξ and consists of a vertical plus radial component. The strength of thelatter is given by tan θ , θ, the angle between the vertical, and ¯ B . The observer’s camera is defined by impact parameters ˆ α , ˆ β , and a flat space lineof sight ˆ k . 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, ˆ φ is the unitvector in the azimuthal direction. Q / I U / I U / I Analytic - Completely Vertical B field = 0.00 Q / I U / I U / I Analytic - Vertical plus Radial B field = 80.00
Fig. 4.
Analytic non-relativistic calculations of the linear Stokes parameters Q and U in a vertical plus radial magnetic field at three di ff erentviewer inclinations: i = ◦ , ◦ , ◦ . The colour gradient denotes the periodic evolution of the hotspot along its orbit over one revolution. Theonly reason the width of the curves vary is for visualisation purposes. Top: completely vertical magnetic field ( θ = Q and U areconstants in time and have static values in QU space. Bottom: significantly radial magnetic field with θ = 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.Article number, page 4 of 13RAVITY Collaboration: A. Jiménez-Rosales et al.: Strong magnetic fields near Sgr A* When face-on, ˆ k points along ˆ z and ˆ β points along ˆ y . When edge-on, ˆ k points along − ˆ y and ˆ β points along ˆ z .Let a hotspot be orbiting in the ˆ x ˆ y plane (Figure 3). In termsof ˆ α , ˆ β , and ˆ k , the hotspot’s position vector ¯ h is given by¯ h = R ˆ r = R (cos ξ ˆ α + cos i sin ξ ˆ β − sin i sin ξ ˆ k ) , (4)where ˆ r is the canonical radial vector, R is the orbital radius,and ξ is the angle between ˆ α and ˆ r .Let us consider the magnetic field with vertical and radialcomponents given by¯ B = B √ + δ (ˆ z + δ ˆ r ) ; δ ≡ tan θ, (5)where B is the magnitude of ¯ B and θ is the angle between ˆ z and¯ B . The polarisation is given as ¯ P = ˆ k × ¯ B . In flat space and in theabsence of motion (no light bending or aberration),¯ P ∝ ˆ k × (ˆ z + δ ˆ r ) ∝ − (sin i + tan θ cos i sin ξ ) ˆ α + tan θ cos ξ ˆ β . (6)The polarisation angle on the observer’s camera is tan ψ = ¯ P · ˆ β/ ¯ P · ˆ α , so that ψ = tan − (cid:32) − tan θ cos ξ sin i + tan θ cos i sin ξ (cid:33) . (7)Given that U / Q = / ψ , the Stokes parameters as a func-tion of the polarisation angle are Q = | ¯ P | cos 2 ψ, U = | ¯ P | sin 2 ψ. (8)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 theorbit, the angle ξ can be mapped linearly to a time value bysetting the duration of the orbital period and an initial positionwhere the ξ = i = i < ◦ and i = ◦ − i produces the same polarised curves but they are reversed in ξ with respect to each other. This is expected since, for an observerat i = i and one at i = ◦ − i , the hotspot samples the samemagnetic field geometry, but they appear to be moving in op-posite directions with respect to each other. This means that therelative order in which the peaks in Q and U appear are reversedbetween observers at i = i and at i = ◦ − i .Given that light bending has not been considered in this ap-proximation, in a significantly vertical field ( θ (cid:39)
0, top of Fig.4), the polarisation remains constant in ξ (and time) proportionalto − sin i . In QU space, this means a static value as the hotspotgoes around the black hole. A particular case of this is ¯ P (cid:39) i (cid:39)
0, since ˆ k and ¯ B are parallel. As θ −→ π/
2, tan θ −→ ∞ (bottomof Fig. 4), and the magnetic field becomes radial. In this caseand at low inclinations, the polarisation configuration is toroidal( ¯ P ∝ ˆ φ , the azimuthal canonical vector, Eq. B.1). As the hotspotorbits the black hole, Q and U show oscillations of the same am-plitude. In one revolution, two superimposed QU loops can betraced. If the viewer’s inclination increases, one of the loops de-creases more in size than the other and eventually disappears atvery high inclinations, leaving only one behind. Increasing in-clination, therefore, counteracts the presence of QU loops in ananalytical model with a vertical plus radial magnetic field. It isnoted that the normalised polarisation configurations of a com-pletely radial magnetic field and a toroidal one are equivalentwith just a phase o ff set of 90 ◦ in ξ (Eq. B.2 in Appendix B). Next, we use numerical calculations to include general rela-tivistic e ff ects. We used the general relativistic ray tracing code grtrans (Dexter & Agol 2009; Dexter 2016) to calculate syn-chrotron 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 equato-rial plane at radius R . The orbital speed is constant for the entireemission region, and it matches that of a test particle motion atits centre. The maximum particle density n spot ∼ × cm − falls o ff as a three-dimensional Gaussian with a characteris-tic size of R spot . The magnetic field has a vertical plus radialcomponent . Its strength is taken from an equipartition assump-tion, where we further assume a virial ion temperature of kT i = ( n spot / n tot ) ( m p c / R ) , ( n spot / n tot ) =
5, where n tot is the total parti-cle density in the hotspot. For the models considered here, a typ-ical magnetic field strength in the emission region is B (cid:39)
100 G.We calculated synchrotron radiation from a power law distribu-tion of electrons with a minimum Lorentz factor of 1 . × andconsidered a black hole with a spin of zero. . The model parame-ters for field strength, density, and minimum Lorentz factor werechosen as typical values for models of Sgr A* which can matchthe 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 shownin Figure 5. The e ff ects of lensing can be appreciated in the formof secondary images. It can be seen as well that as the hotspotmoves along its orbit around the black hole, it samples the mag-netic field geometry in time, so that the time-resolved polarisa-tion encodes information about the spatial structure of the mag-netic field.Figure 6 shows the numeric calculations of hotspot mod-els with the same magnetic field angles as those in the analyticapproximation. Inclination and θ are key parameters in the ob-served number and shape of QU loops. In contrast to the analyticcase, in a significantly vertical field ( θ (cid:39)
0, top of Fig. 6), the po-larisation is not zero. This is mainly due to light bending, whichintroduces an e ff ective radial component to the wave-vector inthe plane of the observer’s camera. This radial component of ˆ k leads to an additional azimuthal contribution to ¯ P . The θ = ff ect alone is able to generate QU loops.We see again that increasing inclination leads to a change fromtwo QU loops per hotspot revolution at low inclinations to a sin-gle QU loop at high inclinations.The cases where θ −→ ◦ (bottom of Fig. 6) show that in-creasing 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 di ff erences are dueto the inclusion of relativistic e ff ects in the ray-tracing calcula-tions. We note that numerical models with a vertical plus toroidalmagnetic field show similar features and behaviour to those inthe vertical plus radial case (see Appendix C).
4. Model fitting
We calculated normalised Stokes parameters Q / I and U / I fromray tracing simulations of a grid of hotspot models, folded themthrough the instrumental response (Eq. 1), and compared them See Appendix A for details. Given the scales at which the hotspot is orbiting, a change in the spinof the black hole does not alter the results significantly. See AppendixD for more details Article number, page 5 of 13 & A proofs: manuscript no. GRAVITY_flare_pol
50 0 -50 𝜇 as 50 0 -50 𝜇 as 50 0 -50 𝜇 as50 0 -50 𝜇 as50 0 -50 𝜇 as50 0 -50 𝜇 as - 𝜇 a s - 𝜇 a s TIME T I M E TIME
Fig. 5.
Snapshots of the hotspot as it orbits the black hole clockwise on sky in a vertical magnetic field. The orbital radius is eight gravitationalradii. Total intensity is shown as false colour in the background. Polarisation direction is shown as white ticks in the foreground. Their length isproportional to the linear polarisation fraction in that pixel. The hotspot samples the magnetic field geometry in time as it moves along the orbit,so that the time-resolved polarisation encodes information about the spatial structure of the magnetic field. Q / I U / I U / I Numeric - Completely Vertical B field = 0.00 Q / I U / I U / I Numeric - Vertical plus Radial B field = 80.00
Fig. 6.
Ray-tracing calculations of the linear Stokes parameters Q and U in a vertical plus radial magnetic field with the same θ as those in theanalytic model. The coarse QU loops are due to the time sampling in our simulations. Top: magnetic field inclination of θ = θ = Q and U oscillate in time and trace loops in QU space due to light bending.Article number, page 6 of 13RAVITY Collaboration: A. Jiménez-Rosales et al.: Strong magnetic fields near Sgr A* to GRAVITY’s measured Q (cid:48) / I (cid:48) and U (cid:48) / I (cid:48) . The parameters of thenumerical model are the orbital radius R , the size of the hotspot R spot , the viewing angle i , and the tilt angle of the magnetic fielddirection θ .We understand qualitatively how the hotspot size and the or-bital radius a ff ect the Q and U curves. ‘Smoother’ curves, wherethe amplitude of the oscillations is reduced, are produced eitherwith increasing hotspot sizes at fixed orbital radius or with de-creasing R at a fixed hotspot size, due to beam depolarisation(see Appendix F). Since performing full ray tracing simulationsis computationally very expensive, and due to the fact that thecurves change smoothly and gradually with R and R spot , wechose to fix their values to R = R g and R spot = R g , R g thegravitational radius. We then scaled them in both period and am-plitude to match the data better in the following manner.Given the duration of a flare ∆ t , we could scale a hotspot’speriod by a factor nT to set the fraction of orbital periods thatfit into this time window. The new radius of the orbit is then R ∝ ( ∆ t / nT ) / . This rescaling introduced small changes in fitquality compared to re-calculating new models, within our pa-rameter range of interest (see Appendix E). We absorbed the ef-fect of beam depolarisation into a factor s that scales the overallamplitude of both Q and U and, therefore, the linear polarisationfraction as well.Given a hotspot’s period, the relative phase reflects thehotspot position relative to an initial position measured at someinitial time, where the phase is defined to be zero. We chose theinitial position of the hotspot based on the astrometric measure-ment of the orbital motion of the flare in Gravity Collaborationet al. (2020c). Specifically, we chose the initial phase ξ to matchthe initial position of the best-fit orbital model to the astrometry. flare The observed Q (cid:48) / I (cid:48) and U (cid:48) / I (cid:48) were measured from fitting inter-ferometric binary models to GRAVITY data. The binary modelmeasures the separation of Sgr A* and the star S2, which wereboth in the GRAVITY interferometric field of view ( (cid:39)
50 mas)during 2018. For more details, see Gravity Collaboration et al.(2020a). We measured polarisation fractions assuming that S2’sNIR emission is unpolarised. The 70 minute time period anal-ysed is limited by signal-to-noise: binary signatures are largestwhen Sgr A* is brightest. As a result, we focused on data takenduring the flare. We fitted to data binned by 30 seconds since theflux ratio can be rapidly variable. We further adopted error barson polarisation fractions using the rms of measurements within300s time intervals since direct binary model fits generally have χ >
1, and as a result underestimate the fit uncertainties.We computed a grid of models with i , θ , s , and nT as pa-rameters: i ∈ [0 − ∆ i = ◦ ; θ ∈ [0 − ∆ θ = ◦ ; s ∈ [0 . − . ∆ s = .
05, and nT such that the allowedrange of radii for the fit is R = − R g with ∆ R = .
2. Wehave included this prior in radii to match the constraint from thecombined astrometry of the three bright GRAVITY 2018 flares(Gravity Collaboration et al. 2020c). The best fit parameters andcorresponding polarised curves are shown in Figure 7. We findthat the curves qualitatively reproduce the data and that the sta-tistically preferred parameter combination for July 28, with areduced χ ∼ .
1, favours a radius of 8 R g and moderate i and θ values (left panel of Figure 7). In QU space, these parametersproduce two intertwined and embedded QU loops of very di ff er-ent amplitudes in time (right panel of Figure 7). The outer oneis fairly circular, centred approximately around zero and withan average radius of 0 .
18. The inner one has a horizontal oblate shape with a QU axis ratio of approximately 2:1, does not goaround zero, and represents a much smaller fraction of the or-bit than the larger loop. These moderate values of θ imply thata magnetic field with significant components in both the radialand vertical directions is favoured.The hotspot is free to trace a clockwise ( i > ◦ ) or coun-terclockwise ( i < ◦ ) motion on-sky. At fixed θ , this change inapparent motion results in an inversion of the order in which themaxima of the Q and U curves appear .This e ff ect is due to relativistic motion (Blandford & Königl1979; Bjornsson 1982). When the magnetic field is purelytoroidal (velocity parallel to ¯ B ), the polarisation angle is inde-pendent of velocity. When there is a field component perpendic-ular to the velocity (poloidal field), relativistic motion inducesan additional swing of the polarisation angle in the direction ofmovement where magnitude depends on the velocity. We ignorethis e ff ect in the analytic approximation above, but it is includedin our numerical calculations.The data favour models where the maxima in U (cid:48) / I (cid:48) precedethose of Q (cid:48) / I (cid:48) . This behaviour is observed in the case of clock-wise motion ( i > ◦ ) with θ ∈ [0 ◦ − ◦ ] and in counterclock-wise motion ( i < ◦ ) with θ ∈ [90 ◦ − ◦ ]. In fact, modelcurves at a given i > ◦ and θ ∈ [0 ◦ − ◦ ] are identical to thosewith their ‘mirrored’ values i (cid:48) = ◦ − i and θ (cid:48) = ◦ − θ . In ouranalysis, we consider θ ∈ [0 ◦ − ◦ ], which favours a clockwisemotion. However, we cannot uniquely determine the apparentdirection of motion of the hotspot due to this degeneracy.Our models overproduce the observed linear polarisationfraction by a factor of ∼ . s (cid:39) . < (cid:39) (cid:39)
50% in our models. The degree of depolarisation introducedby the VLTI is not substantial enough to reduce the model lin-ear polarisation fraction to the observed one. Moreover, in theNIR, there are no significant depolarisation contributions fromabsorption or Faraday e ff ects. As a result, we conclude that thelow observed polarisation fraction is likely the result of beam de-polarisation. The observed low polarisation fraction implies thatthe flare emission region is big enough to resolve the underlyingmagnetic field structure. In the context of our model, this couldimply a larger spot size. It could also indicate a degree of disor-der in the background magnetic field structure, for example as aresult of turbulence. flare July 28 is the only night with an observed infrared flare in whichGRAVITY recorded both Stokes Q (cid:48) and U (cid:48) . Since a single po-larisation channel is insu ffi cient to constrain the full parameterspace used in our numerical models, we restricted ourselves tothe night of July 22, as this observation has the highest preci-sion astrometry , and fixed the viewer inclination and magneticfield geometry to be the same as the best fit model to the July 28data. We scaled the curves in amplitude with s ∈ [0 . − . ∆ s = . ff set betweenboth curves. With a fixed phase di ff erence between the curvesand free range of radii, we find that the July 22 data favours ex- This is also equivalent to an inversion of the curve in time and doesnot modify the features of the curve. The astrometry of the May 27 and June 27 flares is not good enoughto pin point their starting location on sky, so it is not at all possible torestrict the phase di ff erence between them and the July 28 flare.Article number, page 7 of 13 & A proofs: manuscript no. GRAVITY_flare_pol Q ' / I ' , U ' / I ' Reduced : 3.104 Q' U'0 10 20 30 40 50 60 70
Time [min] R e s i d u a l s Q/I U / I st orbit2 nd orbitJuly 28 th Vertical plus Radial fieldClockwise Hotspot size = 3 Rgi [deg] = 136 [deg] = 60.0 R [Rg]=8.0 s = 0.45
Fig. 7.
Best fit to the July 28 NIR flare. The colour gradient denotes the periodic evolution of the hotspot along its orbit, moving from darkershades to lighter as the hotpot completes one revolution. The curves qualitatively reproduce the data. The preferred parameter combination favoursa radius of 8 R g and both moderate i and θ values. Q ' / I ' Q'0 5 10 15 20 25 30 35 40Time [min]0.150.000.15 R e s i d u a l s U / I July 22 th Clockwise Hotspot size = 3 Rgi [deg] = 136 [deg] = 60.00 R [Rg] = 11.00 s = 0.10
Fig. 8.
Fit to the July 22 NIR flare without restricting the phase di ff erence between this night and that of July 28. The colour gradient denotes theevolution of the hotspot as it completes one revolution. The viewer’s inclination, magnetic field geometry, and orbital direction have been fixed tothe values found for the July 28 flare. The fit favours values of R ∼ R g and there is no initial phase di ff erence between the nights (no di ff erencein starting position on-sky), which is out of the allowed uncertainty range for the astrometry. tremely large values of R > R g , which are outside of theallowed range obtained from astrometric measurements. In al-lowing the phase di ff erence to be free and constraining the radiito 8 − R g , with ∆ R = .
2, we find that the data tend to val-ues of R ∼ R g and a phase di ff erence between curves of0 ◦ (Figure 8). This phase di ff erence value (and position di ff er-ence associated with it) is outside of the allowed uncertaintiesin the initial position indicated by the astrometric data. The factthat the magnetic field parameters that describe the July 28 flarefail to adequately fit the data from July 22 may indicate that thebackground magnetic field geometry changes on a several-daytimescale.
5. Summary and discussion
In this work, we present an extension of the initial analysis of po-larisation data performed in Gravity Collaboration et al. (2018b).We forward modelled Q and U Stokes parameters obtained fromray-tracing calculations of a variety of hotspot models in di ff er-ent magnetic field geometries, transformed them into quantities as seen by the instrument, and fitted them directly to the po-larised data taken with GRAVITY.This allowed us to not only fit data directly without makingassumptions about Stokes V or the interpolation of data in non-simultaneous Q and U measurements, but also to predict the be-haviour in time of the polarised curves and loops for the caseswhere only one of the parameters was measured.We have shown that the hotspot model serves to qualitativelyreproduce the features seen in the polarisation data measuredwith GRAVITY. A moderate inclination and moderate mix ofboth vertical and radial fields provide the best statistical fit tothe data. Consistent results are found by fitting the data with avertical plus toroidal field component (Appendix C). We notethat this result does not rely on the assigned strength of the mag-netic field, since the model curves are scaled in amplitude, butrather it is only from the geometry of the field. Magnetic fieldswith a non-zero vertical component fit the data statistically bet-ter. This supports the idea that there is some amount of orderedmagnetic field in the region near the event horizon with a signif-icant poloidal field component. The presence of this component Article number, page 8 of 13RAVITY Collaboration: A. Jiménez-Rosales et al.: Strong magnetic fields near Sgr A* is associated with magnetic fields that are dynamically importantand it confirms the previous finding of strong fields in GravityCollaboration et al. (2018b). Spatially resolved observations at1 . θ ∈ [0 ◦ − ◦ ]. Under thisassumption, the results are also in accordance with the angularmomentum direction and orientation of the clockwise stellar discand gas cloud G2 (Bartko et al. 2009; Gillessen et al. 2019; Pfuhlet al. 2015; Plewa et al. 2017).We have chosen the bright NIR flare on July 28, 2018 sinceit is the only one for which both linear Stokes parameters havebeen measured. Naturally, increasing the number of full data setsin future flares will be useful in constraining the parameter rangemore.Our models overproduce the observed NIR linear polarisa-tion fraction of ∼
30% by a factor of ∼ .
7, and they must bescaled down to fit the data. In the compact hotspot model con-text, this implies that an emission region size larger than 3 R g is needed to depolarize the NIR emission through beam depo-larisation. Including shear in the models would naturally intro-duce depolarisation since a larger spread of polarisation vectordirections (or equivalently, the magnetic field structure) wouldbe sampled at any moment (e.g. Gravity Collaboration et al.2020c; Tiede et al. 2020). However, this might smooth out thefitted curves and would probably change the fits. In any case, theobserved low NIR polarisation fraction means that the observedemission region resolves the magnetic field structure around theblack hole.Though simplistic, the hotspot model appears to be viable forexplaining the general behaviour of the data. It would be inter-esting to study the polarisation features of more complex, totalemission scenarios explored in other works. Ball et al. (2020)study orbiting plasmoids that result from magnetic reconnectionevents close to the black hole, where some variability in the po-larisation should be caused by the reconnecting field itself. Dex-ter et al. (2020) find that material ejected due to the build-up ofstrong magnetic fields close to the event horizon can produceflaring events where the emission region follows a spiral trajec-tory around the black hole. In their calculations, ordered mag-netic fields result in a similar polarisation angle evolution as wehave studied here. Disorder caused by turbulence reduces the lin-ear polarisation fraction to be consistent with what is observed.Spatially resolved polarisation data are broadly consistentwith the predicted evolution in a hotspot model. This first ef-fort comparing these types of models directly to GRAVITY datashows the promise of using the observations to study magneticfield structure and strength on event horizon scales around blackholes. Acknowledgements.
JD is pleased to thank D.P. Marrone, J. Moran, M.D. John-son, G.C. Bower, and A.E. Broderick for helpful discussions related to signa-tures of orbital motion around black holes from polarized synchrotron radiation.We thank the anonymous referee for their constructive comments. This workwas supported by a CONACyT / DAAD grant (57265507) and by a Sofja Ko-valevskaja award from the Alexander von Humboldt foundation. A.A. and P.G.were supported by Fundação para a Ciência e a Tecnologia, with grants referenceUIDB / / / BSAB / / References
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Appendix A: Vertical plus radial field inBoyer-Lindquist coordinates
In the Boyer-Lindquist coordinate frame, a magnetic field with avertical plus radial components can be written as: B = ( B t , B r , B θ , B φ ) = ( B t , δ c B θ , B θ , , (A.1)where B µ are the contravariant components of B and δ c ≡ B r / B θ .The magnetic field must satisfy the following conditions: B µ u µ = g µν B ν u µ = B µ B µ = g µν B ν B µ = B , (A.2)where u µ are the contravariant components of the four-velocity, B is the magnitude of B , and g µν are the covariant componentsof the Kerr metric. In Boyer-Lindquist coordinates with G = c = M =
1, the non-zero components of the metric are: g tt = − (cid:32) − r Σ (cid:33) g rr = Σ∆ g θθ = Σ g t φ = g φ t = − r Σ a sin θ g φφ = (cid:34) r + a + ra Σ sin θ (cid:35) sin θ, (A.3)where ∆ ≡ r − r + a Σ ≡ r + a cos θ, where a is the dimensionless angular momentum of the blackhole.Using Eq. (A.1), (A.2), and (A.3), it follows that the Boyer-Lindquist coordinate frame contravariant components of themagnetic field are B t = − CB θ ; B r = δ c B θ = B θ δ LNRF / r ; B θ = B ( g tt C + g rr δ c + g θθ ) − / ; B φ = C ≡ δ c g rr u r + g θθ u θ g tt u t + g t φ u φ () (A.4)and δ c = δ LNRF / r ; δ LNRF = B ( r ) B ( θ ) where δ LNRF is the ratio of the radial and poloidal magnetic fieldcomponents in the locally non-rotating frame (LNRF, Bardeen1973) and B ( µ ) are the contravariant components of B in theLNRF: B ( t ) = ( Σ∆ / A ) / B t ∼ B t ; B ( r ) = ( Σ / ∆ ) / B r ∼ B r ; B ( θ ) = Σ / B θ ∼ rB θ ; B ( φ ) = − ra sin θ ( Σ A ) / B t + ( A / Σ ) / sin θ B φ ∼ r sin θ B φ ; (A.5)with A ≡ ( r + a ) − a ∆ sin θ, where the expression to the far right is obtained by assuming r (cid:29) a (as it is in the hotspot case). The variable δ used in themain text (Eq. (5)) corresponds to δ LNRF defined here as beingcalculated using the r (cid:29) a approximation. Appendix B: Analytic approximation with a verticalplus toroidal magnetic field
In the case of a vertical plus toroidal magnetic field, the magneticfield can be written as ¯ B ∝ ˆ z + λ ˆ φ , where λ ∝ tan θ T is the strengthof the toroidal component, θ T is the angle measured from thetoroidal component to the vertical component ( θ T = φ = − sin ξ ˆ α + cos i cos ξ ˆ β − sin i cos ξ ˆ k (B.1)is the canonical vector in the azimuthal direction (Figure 3). Wenote that ˆ r · ˆ φ = k × ¯ B is then¯ P ∝ − (sin i + λ cos i cos ξ ) ˆ α − λ sin ξ ˆ β (B.2)and the polarisation angle is given by ψ = tan − (cid:32) λ sin ξ sin i + λ cos i cos ξ (cid:33) . (B.3)It can be seen from expression (B.2) that at low inclinationsor when λ >> P ∝ ˆ r , Eq. 4). This is geo-metrically equivalent to the polarisation having a toroidal con-figuration (similar to the one generated by a completely radialmagnetic field, see Section 3) with a phase o ff set of π/ 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 toroidalmagnetic field (Appendix C). As expected, in the analytic case,there are always two superimposed loops in QU space in thecase of a completely toroidal field. In the numeric calculations,this is also the case given that light bending favours the pres-ence of loops. As a vertical component in the field is introduced,the loops no longer overlay on each other. This e ff ect increaseswith viewer inclination. It can also be seen that the completelytoroidal and radial cases produce the same Q and U curves atlow inclinations, save for a phase o ff set and scaling factor. Article number, page 10 of 13RAVITY Collaboration: A. Jiménez-Rosales et al.: Strong magnetic fields near Sgr A* Q / I U / I U / I Analytic - Completely Toroidal B field T = 0.00 Q / I U / I U / I Analytic - Vertical plus Toroidal B field T = 10.00 Q / I U / I U / I Numeric - Completely Toroidal B field T = 0.00 Q / I U / I U / I Numeric - Vertical plus Toroidal B field T = 10.00 Fig. B.1.
Analytic and ray-tracing calculations of Q and U curves in the case of a toroidal magnetic field. Two loops are always observed. In thecase of the analytic case (top), both are superimposed. This is broken by the accounting for light bending in the ray-tracing calculations (bottom).It can also be seen that toroidal and completely radial configurations produce the same curves, save for a a scaling factor and a phase o ff set. Appendix C: Vertical plus toroidal field inBoyer-Lindquist coordinates
In the Boyer-Lindquist coordinate frame, a magnetic field with avertical plus toroidal components can be written as: B = ( B t , B r , B θ , B φ ) = ( B t , , η c B θ , B φ ) (C.1)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 mustsatisfy Eqs. (A.2).Using Eqs. (C.1), (A.2), and (A.3), it follows that the Boyer-Lindquist coordinate frame contravariant components of themagnetic field are B t = B θ / C ; B r = B θ = √ A ρ η LNRF sin θ ( C − ω ) B t ; B φ = C B (cid:113) g tt + g t φ C + g θθ B θ B t + g φφ C ;with (C.2) C ≡ − g tt u t + g t φ u φ g t φ u t + g φφ u φ ; ω = raA ; (C.3) η LNRF = B ( θ ) / B ( φ ) = tan θ T the ratio of the poloidal and toroidalmagnetic field components in the LNRF (Eq. (A.5)), and θ T isthe angle measured from the toroidal component to the vertical( θ T = Table D.1.
Reduced χ of best fit of the July 28 flare data with threedimensionless spins: 0 . , . , − . R [R g ] a χ . . − . . i , θ , s , and nT as parameters: i ∈ [0 − ∆ i = ◦ ; θ T ∈ [0 − ∆ θ T = ◦ ; s ∈ [0 . − . ∆ s = .
05, and nT such that the allowed range of radii for thefit is R = − R g with ∆ R = .
2. The best fit is shown inFigure C.1. Though a better reduced χ is found at a somewhathigher inclination than the best fit with a vertical plus radial mag-netic field (Fig. 7), the presence of a poloidal component in themagnetic field is still needed. Considering θ T ∈ [0 ◦ − ◦ ], aclockwise motion is preferred ( i > ◦ ). Identical curves canbe obtained when the direction of motion is counterclockwise( i < ◦ ) and the magnetic field angle is θ (cid:48) T = ◦ − θ T . FigureC.2 presents a model of a vertical plus toroidal magnetic fieldwith similar parameters to those of the vertical plus radial fieldbest fit. Appendix D: Spin effects
We present the e ff ects of spin in our calculations. Figure D.1shows three models with the best fit parameters found forthe July 28 flare, at three di ff erent dimensionless spin valuesa = . , . , − .
9. The corresponding reduced χ values are re-ported in Table D.1. It can be seen that changes in spin do notalter the curves significantly and they can therefore be ignored. Article number, page 11 of 13 & A proofs: manuscript no. GRAVITY_flare_pol Q ' / I ' , U ' / I ' Reduced : 2.592 Q' U'0 10 20 30 40 50 60 70
Time [min] R e s i d u a l s Q/I U / I st orbit2 nd orbitJuly 28 th Vertical plus Toroidal fieldClockwise Hotspot size = 3 Rgi [deg] = 106 T [deg] = 20.0 R [Rg]=8.0 s = 0.40 Fig. C.1.
Best fit to the July 28 flare with a vertical plus toroidal magnetic field. The colour gradient denotes the periodic evolution of the hotspotalong its orbit, moving from darker shades to lighter as the hotpot completes a revolution. Considering θ T ∈ [0 ◦ − ◦ ], a clockwise motion ispreferred. The fit has a smaller reduced χ at a slightly higher inclination than the best fit with a vertical plus radial field. The presence of a verticalcomponent in the magnetic field is still required to fit the data better. Q ' / I ' , U ' / I ' Reduced : 3.312 Q' U'0 10 20 30 40 50 60 70
Time [min] R e s i d u a l s Q/I U / I st orbit2 nd orbitJuly 28 th Vertical plus Toroidal fieldClockwise Hotspot size = 3 Rgi [deg] = 130 T [deg] = 30.0 R [Rg]=8.0 s = 0.40 Fig. C.2.
Vertical plus toroidal model fit with similar parameters to those of the best fit with a vertical plus radial field.
Time [min] Q ' / I ' , U ' / I ' July 28 th Clockwise Hotspot size = 3 Rgi [deg] = 136 [deg] = 60.0 R [Rg]=8.0 s = 0.45Q'/I' a=0.0U'/I' a=0.0 Q'/I' a=0.9U'/I' a=0.9 Q'/I' a=-0.9U'/I' a=-0.9
Fig. D.1.
Best fit model of the July 28 flare calculated with three di ff erent values of dimensionless spin (a = . , . , − . χ arereported in Table D.1. Changes in spin do not a ff ect the curves significantly. Appendix E: Scaling period effects
We explore the e ff ects of scaling the period of model curves. Fig-ure E.1 shows the best fit model found for the July 28 flare andone calculated at R = R g scaled down to match the period at8 R g , with the rest of the parameters fixed to those of the best fit.The corresponding reduced χ 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 than their non-scaled ver-sions, 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 orFaraday rotation and conversion, infrared emission from an or-biting hotspot is depolarised by beam depolarisation. Beam de-
Article number, page 12 of 13RAVITY Collaboration: A. Jiménez-Rosales et al.: Strong magnetic fields near Sgr A*
Time [min] Q ' / I ' , U ' / I ' Q'/I' r=8.0U'/I' r=8.0 Q'/I' r=11.0 (scaled)U'/I' r=11.0 (scaled)
Fig. E.1.
Models calculated at R = R g and at R = R g , the latter was scaled down to match the orbital period at 8 R g . The rest of the parametersare those found for the best fit for the July 28 flare. The reduced χ are reported in Table E.1. For better clarity, the R = R g non-scaled modelfit is not shown, but the χ is reported. Q / I U / I Q/I U / I R g R g R g i [deg] = 136 = 60.00 R [Rg] = 8.00 s = 1.00 Fig. F.1.
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 di ff erent polarisationdirections in time. Table E.1.
Reduced χ of models calculated at R = R g and at R = R g , the latter was scaled down to match the orbital period at 8 R g . R [R g ] a χ . . . ff erent contributions from po-larisation (or magnetic field) structure and averaging them out.More beam depolarisation occurs, the larger the emitting re-gion that samples the underlying magnetic field is, or the moredisordered the field itself is. Given the simple magnetic field ge-ometries considered in this work, disorder at small scales is non-existent. We discuss qualitatively the impact of emission size inthe following.As the hotspot goes around the black hole, it samples awedge of angles in the azimuthal direction with an arc lengthof R spot / R . Larger beam depolarisation occurs with the increaseof this factor. Figure F.1 shows example curves of numerical cal-culations at a moderate inclination and magnetic field tilt, whereonly the hotspot size has been changed. As expected, with in-creasing R spot at a fixed orbital radius, not only does the ampli-tude of the polarised curves and QU loops diminish (and withit, the linear polarisation fraction), but the features in them aresmoothed out as well. Within the hotspot model, beam depolari- sation can therefore be used to constrain the size of the emittingregion as a function of the observed linear polarisation fraction.loops diminish (and withit, the linear polarisation fraction), but the features in them aresmoothed out as well. Within the hotspot model, beam depolari- sation can therefore be used to constrain the size of the emittingregion as a function of the observed linear polarisation fraction.