Polarimetric Geometric Modeling for mm-VLBI Observations of Black Holes

The Event Horizon Telescope (EHT) is a millimeter very long baseline interferometry (VLBI) array that has imaged the apparent shadows of the supermassive black holes M87* and Sagittarius A*. Polarimetric data from these observations contain a wealth of information on the black hole and accretion flow properties. In this work, we develop polarimetric geometric modeling methods for mm-VLBI data, focusing on approaches that fit data products with differing degrees of invariance to broad classes of calibration errors. We establish a fitting procedure using a polarimetric “m-ring” model to approximate the image structure near a black hole. By fitting this model to synthetic EHT data from general relativistic magnetohydrodynamic models, we show that the linear and circular polarization structure can be successfully approximated with relatively few model parameters. We then fit this model to EHT observations of M87* taken in 2017. In total intensity and linear polarization, the m-ring fits are consistent with previous results from imaging methods. In circular polarization, the m-ring fits indicate the presence of event-horizon-scale circular polarization structure, with a persistent dipolar asymmetry and orientation across several days. The same structure was recovered independently of observing band, used data products, and model assumptions. Despite this broad agreement, imaging methods do not produce similarly consistent results. Our circular polarization results, which imposed additional assumptions on the source structure, should thus be interpreted with some caution. Polarimetric geometric modeling provides a useful and powerful method to constrain the properties of horizon-scale polarized emission, particularly for sparse arrays like the EHT.


INTRODUCTION
The Event Horizon Telescope has imaged M87 * , the 6.5 × 10 9 M ⊙ supermassive black hole in the M87 galaxy, in both total intensity (Event Horizon Telescope Collaboration et al. 2019a,b,c,d,e,f) and linear polarization (Event Horizon Telescope Collaboration et al. 2021a,b) using data from its 2017 campaign.More recently, images have been produced in circular polarization; however, these show inconsistent structure among different imaging and calibration methods due to the weakness of the circular polarization signal, although some secure inferences on the structure could be made (Event Horizon Telescope Collaboration et al. 2023, hereafter Paper IX).The presence of resolved circular polarization structure on event horizon scales was established unambiguously, and a ∼ 4% upper limit on the imageaveraged resolved circular polarization fraction was obtained.While imaging methods showed different circular polarization structure between, e.g., observing epochs and frequency bands, circular polarization geometric model fitting indicated a consistent dipolar asymmetry across the multi-day observing window.
In this paper, we describe and test the polarimetric modeling methods used to obtain this circular polarization modeling result.Since our modeling methods solve for the total intensity and linear polarization structure as well, we also compare our fits to previously obtained results from EHT imaging and other geometric modeling methods.Before introducing the modeling methods, we will next briefly summarize the origin, utility, and previous measurements of linear polarization and circular polarization and the challenges of studying these quantities using very-long-baseline interferometry (VLBI).

Linear Polarization on Event Horizon Scales
In radiatively inefficient accretion flows onto supermassive black holes, (sub)millimeter emission is produced near the event horizon as synchrotron radiation, which is intrinsically linearly polarized at a level of ∼ 70% (see, e.g., Yuan & Narayan 2014).The electric vector polarization angle (EVPA) is orthogonal to the orientation of the magnetic field.Polarimetric imaging and modeling of this emission thus probes the magnetic field structure.
However, on the photon trajectory towards the observer, the linear polarization fraction and EVPA may be altered by two effects.First, photon propagation along curved geodesics near the event horizon affects the EVPA and may lead to depolarization in the observed image (e.g., Connors & Stark 1977;Mościbrodzka et al. 2017;Narayan et al. 2021;Palumbo & Wong 2022;Ricarte et al. 2022a).Second, propagation through the magnetized accretion flow plasma results in Faraday rotation of the EVPA, with depolarizing effects as well.(e.g., Mościbrodzka et al. 2017;Jiménez-Rosales & Dexter 2018;Ricarte et al. 2020).A polarized image of the accretion flow hence probes the spacetime as well as the plasma properties.These effects may be difficult to untangle, although Palumbo et al. (2020) found that the black hole spin can be constrained from the twistiness of the polarization pattern.
Ray-traced general relativistic magnetohydrodynamic (GRMHD) simulations show that the appearance of the horizon-scale accretion flow may indeed depend strongly on black hole and accretion parameters.In particular, magnetically arrested disk (MAD, Narayan et al. 2003) models, with large magnetic flux permeating the event horizon, produce significantly more ordered EVPA patterns than standard and normal evolution (SANE) models, due to the much larger Faraday depth and hence EVPA rotation and Faraday depolarization of the latter (e.g., Event Horizon Telescope Collaboration et al. 2021b).Also, a lower electron temperature results in a more disordered EVPA pattern in these simulations for both SANE and MAD, again due to a larger Faraday depth.Palumbo et al. (2020) found that, despite the plasma effects, the twistiness of the EVPA pattern (the β P,2 -mode, see also Section 2.3) is a proxy for black hole spin in GRMHD simulations, with larger black hole spins often resulting in more radial EVPA patterns.Ricarte et al. (2022b) showed that the handedness of the EVPA pattern switches sign as a function of radius for retrograde accretion flows, which have a black hole spin direction opposite from the large-scale plasma rotation direction.
While the total intensity EHT data of M87 * from 2017 ruled out a few models from the EHT GRMHD simulation library, the linear polarization data of M87 * provided stronger constraints, ruling out a significant fraction of the models in the library.These constraints favored MAD models over SANE models (Event Horizon Telescope Collaboration et al. 2021a,b).In particular, the constraint on β P,2 was a strong discriminator for these results.In general, our theoretical models are increasingly challenged by EHT observations, even more so for Sgr A * (Event Horizon Telescope Collab-oration et al. 2022e), where no single GRMHD library model was able to fit all constraints on the (Stokes I) source structure and light curve variability (Event Horizon Telescope Collaboration et al. 2022e;Wielgus et al. 2022).Additional physics or parameter space may need to be explored.

Circular Polarization on Event Horizon Scales
Circular polarization in millimeter emission from black hole accretion flows may arise from two distinct physical processes (e.g., Wardle & Homan 2003;Mościbrodzka et al. 2021;Ricarte et al. 2021).First, synchrotron radiation is intrinsically polarized depending on the observing frequency and magnetic field strength and configuration.The maximum intrinsic circular polarization from synchrotron radiation is of order 1/γ, where γ is the Lorentz factor of the (relativistic) synchrotron emitting electrons (e.g.Wardle et al. 1998;Wardle & Homan 2003).In numerical simulations of supermassive black hole accretion flows, the intrinsic circular polarization fraction comes down to ∼ 1%, which is substantially lower than the intrinsic linear polarization fraction of ∼ 70% (Ricarte et al. 2021).Circular polarization is therefore more difficult to detect than linear polarization.The sign of intrinsic circular polarization (before any propagation effects occur) directly maps to the magnetic field orientation with respect to the emission direction, with a positive sign of the observed circular polarization corresponding to a magnetic field orientation pointing toward the observer.
Second, circular polarization may be produced from linear polarization through Faraday conversion.In the local plasma frame, Faraday conversion only operates on Stokes U, while linear polarization in synchrotron emission is intrinsically produced only in Stokes Q (perpendicular to both the magnetic field and the photon propagation direction).Along the photon propagation path, part of the Stokes Q thus has to be recast into Stokes U for Faraday conversion to occur.Such an exchange between Stokes Q and U occurs in the case of Faraday rotation, where the linear polarization direction rotates depending on electron density and magnetic field component parallel to the photon propagation direction.Another pathway for Faraday conversion is a rotation of the magnetic field component perpendicular to the photon propagation direction due to a twist in the magnetic field along the photon propagation direction.Faraday conversion through Faraday rotation produces circular polarization in the same direction as the intrinsic emission (assuming a constant magnetic field).Faraday conversion through a positive twist of the magnetic field (counter-clockwise with respect to the photon propaga-tion direction) produces negative circular polarization (see, e.g., Ricarte et al. 2021, and references therein).
In general, the fraction and direction of circular polarization depend on the magnetic field structure, the electron temperature and density, and plasma composition (Jones & O'Dell 1977;Kennett & Melrose 1998;Wardle et al. 1998).These dependencies make circular polarization an excellent probe for constraining the plasma properties of black hole accretion flows.
GRMHD simulations with resolved circular polarization on sub-event-horizon scales show that the Stokes V image structure strongly depends on the inclination of the viewing angle with respect to the black hole spin axis (Ricarte et al. 2021).For high inclinations (edge-on view with respect to the spin axis), the circular polarization image has a clear quadrupolar structure-especially for MAD models-set by the helical magnetic field structure of the jet in combination with the viewing angle.
For low inclinations (face-on view with respect to the spin axis), the relation between the magnetic field geometry and the Stokes V image is less straightforward.The image contains contributions from features above and below the mid-plane, which have opposite sign.Stokes V images from MAD models are visually dominated by the (n = 1) photon ring (even though its contribution to the total Stokes V emission is marginal), which has an opposite sign from the direct emission due to lensing effects (photons making a half orbit around the black hole) in combination with the magnetic field geometry (Mościbrodzka et al. 2021).Stokes V images from SANE models are less clearly structured, with turbulent features depending on the details of the more turbulent magnetic field and Faraday effects (Ricarte et al. 2021).
In GRMHD and semi-analytic models, the circular polarization fraction generally increases at submillimeter wavelengths as a function of positron fraction due to an increase in Faraday conversion (Emami et al. 2021;Anantua et al. 2020).However, degeneracies with other simulation parameters exist (Emami et al. 2023).

Polarimetric VLBI data
Polarization poses particular challenges for the calibration of VLBI data.The polarization signal is often weak (≲ 1 − 10%), especially for circular polarization.Most VLBI experiments, including the EHT, use orthogonal circular feeds, which are ideal for measuring linear polarization but require precise calibration of the right/left (R/L) gain ratios to measure circular polarization (e.g.Homan & Wardle 1999;Homan et al. 2001;Homan & Wardle 2004;Homan & Lister 2006;Gabuzda et al. 2008, see also Section 6.2.2).Besides systematics that also affect total intensity measurements, such as rapid atmospheric phase fluctuations, atmospheric opacity, and antenna pointing offsets, polarimetric VLBI data is particularly sensitive to "leakage" effects between orthogonal feeds (see Section 3.1), which often give spurious signals that exceed the true polarized signal, especially in linear polarization.The combination of weak signals and the additional systematics has often limited polarimetric measurements, especially for circular polarization.In this work, we therefore focus on constructing data products that are invariant to most calibration errors.In this process, we can directly explore the effects of calibration errors in both the bias and uncertainty of the measurements of polarized image structure.

Outline
In this work, we provide a framework for fitting geometric models to polarimetric VLBI data, and we apply these methods to synthetic data generated from GRMHD models and to EHT observations of M87 * .We introduce our polarimetric "m-ring" model in Section 2, and outline our procedure to fit this or other geometric models to VLBI data in Section 3. We test our model fitting framework on synthetic data from geometric models in Section 4, and apply it to synthetic data from GRMHD models in Section 5.In Section 6, we apply our methods to EHT observations of M87 * in 2017, estimating the source properties in both linear and circular polarization.We conclude and provide an outlook for future work in Section 7.

Geometric Modeling of VLBI Data
VLBI measurements, such as those made by the EHT, sample interferometric "visibilities" on each baseline connecting a pair of telescopes with mutual visibility of a target source.These visibilities are given by the correlation of the narrow-band complex electric fields sampled at the telescopes: Because each telescope can record two orthogonal polarization products (typically right and left circular, or orthogonal linear polarizations), each baseline can measure four correlations that can be easily mapped to the four Stokes parameters (see Section 3.1).By the van Cittert-Zernike theorem, these visibilities correspond to samples of the Fourier transform of the sky image, with the wavenumber given by the dimensionless baseline length (measured in wavelengths).For instance, the visibilities in total intensity, Ĩ are given by (see, e.g., Thompson et al. 2017) (1) Here, ⃗ u is the dimensionless baseline vector projected orthogonally to the line of sight, ⃗ x is the angular sky coordinate in radians, and I is the sky brightness distribution.The brightness distribution I(⃗ x) is real, so the corresponding visibilities have a conjugation symmetry: Because an interferometer only sparsely samples the Fourier domain, geometric modeling of interferometric data provides a powerful alternative to imaging (e.g., Pearson 1999;Event Horizon Telescope Collaboration et al. 2019f, 2022d).In particular, for sources with relatively simple morphology, geometric models may be parameterized using far fewer parameters than imaging.Geometric models are also flexible; multiple geometric models can be easily added to describe sources with complex morphologies because the Fourier transform is linear.Geometric modeling is an especially effective analysis strategy for VLBI arrays that have a small number of baselines (such as the EHT) or in cases where the signalto-noise is low (as is frequently the case for polarimetric visibilities).The danger of using geometric models is that the choice of model may significantly affect the inferences, and model misspecification (i.e. using a model that does not fully describe the underlying structure) can result in parameter biases.
Because EHT images of M87 * and Sgr A * are dominated by a prominent ring with an azimuthal brightness asymmetry (Event Horizon Telescope Collaboration et al. 2019d, 2022c), both "crescent" (Kamruddin & Dexter 2013;Benkevitch et al. 2016;Event Horizon Telescope Collaboration et al. 2019f;Wielgus et al. 2020;Lockhart & Gralla 2022) and geometric ring models (Event Horizon Telescope Collaboration et al. 2019f;Johnson et al. 2020;Event Horizon Telescope Collaboration et al. 2022c) provide good fits to EHT data from these sources.

m-ring Model
In this paper, we will focus on using extensions of the "m-ring" model from Johnson et al. (2020) to fit polarimetric EHT data.For our purposes, the principal benefits of this model are: • Efficiency: the model has a simple analytic form in both the image and visibility domains, with analytic gradients.
• Flexibility: the model can describe arbitrary azimuthal variations in brightness, ring shape asymmetry, and radial structure controlled by a Gaussian blurring kernel.
• Polarimetry: the model naturally includes complex polarized structure in both linear and circular polarization.
• Interpretability: the m-ring model naturally describes key image features that are useful for physical interpretation, such as the ring's diameter, brightness asymmetry, shape asymmetry, and rotationally-invariant polarization.
Specifically, the m-ring model is constructed from a thin ring with non-uniform brightness in azimuth expressed as a Fourier series.Written in polar image coordinates (ρ, φ), it takes the form where δ is the Dirac delta distribution, β −k ≡ β * k since the image is real, and β 0 ≡ 1 so that F > 0 gives the total flux density of the ring.By increasing the m-ring order m, increasingly complex azimuthal structures can be modeled.The corresponding visibility function in polar coordinates (u, ϕ) is given by where J k denotes the k th Bessel function of the first kind.Notably, the azimuthal Fourier coefficients in the image and visibility domains are identical up to a constant rescaling.
Two natural extensions of this model are to include shape asymmetry and to introduce finite ring width.For the former, the m-ring can be stretched in any direction using the similarity property of the Fourier transform to compute the associated visibility function: if I(x, y) → I ′ (x, y) = I(ax, by), where the arrow indicates the stretch transformation and For the latter, the m-ring can be easily convolved with Gaussian of full width at half maximum α.This blurred mring has visibility and image functions given by: where I k denotes the k th modified Bessel function of the first kind.

Polarimetric m-ring Model
The m-ring model can be easily generalized to include linear and circular polarization.Each polarization product has an image determined by an associated set of Fourier coefficients; we use {β I,k } for total intensity, {β P,k } for the linear polarization,1 and {β V,k } for circular polarization.Because the linear polarization is complex, there is no conjugation symmetry in the associated β P,k .The circular polarization image is real, so β V,−k = β * V,k .The image-integrated linear and circular polarization fractions are given by m net ≡ β P,0 ∈ C (a complex number) and V net ≡ β V,0 ∈ R (a real number), respectively.
The only difficulty in specifying parameter ranges for the m-ring model is that it is non-trivial to enforce image positivity (I(⃗ x) > 0) and a physical polarization limit ( To approximate these conditions, we typically require |β I,k | < 0.5 (for k ̸ = 0), |β P,k | < 1, and |β V,k | < 1. Specifying the precise physical parameter domain is not problematic in practice because the total polarization is typically much smaller than the intensity: Figure 1 shows examples of polarized m-rings with different parameters.

Data products
In VLBI observations, each antenna feed records the complex electric field E. For EHT antennas, these feeds are left and right circularly polarized2 , so an antenna j records the electric fields E Lj and E Rj , respectively.For each pair of antennas in the array, these signals are then cross-correlated to form the cross-correlation matrix ρ jk , which in the absence of any other observational effects can be written in terms of the Stokes visibility components Ĩ, Q, Ũ, and Ṽ as However, this relation does not hold for imperfect instruments.Observational effects affecting this relation can be categorized in the gain matrix G, the leakage matrix D, and the field rotation matrix Φ.For each antenna j, they are combined in the Jones matrix (5) The measured correlation matrix is then given by the Radio Interferometer Measurement Equation (RIME; Hamaker et al. 1996;Smirnov 2011) The Stokes I and V information is primarily contained in the parallel-hand visibilities (Roberts et al. 1994): From these parallel-hand visibilities, we can construct two data products that are especially suitable for fitting the Stokes V structure.First, we can fit to the parallelhand visibility ratios, R j R * k /L j L * k .Assuming the leak-age terms have been well corrected and the fractional linear polarization is small, dropping the terms proportional to D 2 and D P, R j R * k /L j L * k (hereafter referred to as RR/LL) to first order depends on the fractional circular polarization in the visibility domain: This data product has the advantage of cancelling rapid atmospheric phase variations, since the atmosphere is not significantly birefringent at millimeter wavelengths (i.e., its refractive index is independent of polarization).However, the visibility ratios depend on the R/L gain ratios.These are often stable over many hours and can be corrected, although some EHT sites have shown rapidly variable R/L gain ratios (Paper IX, see also Section 6.2.2).Since the gain ratios are antenna based and multiplicative while the circular polarization signal is baseline-based and additive, complex circular polarization structure may be extracted from the RR/LL data product even if the R/L gain ratio calibration is imperfect (see also Homan & Wardle 1999).
Alternatively, we can fit to the parallel-hand closure phases and closure amplitudes.Closure phase is the sum of visibility phases on a triangle of baselines, and closure amplitudes are ratios of visibility products on a quadrangle of baselines (e.g., Thompson et al. 2017).These data products are independent of multiplicative stationbased calibration errors, including the gains as they cancel in the sums and products, respectively.The closure products are not independent of other station-based calibration errors, such as polarimetric leakage and bandpass errors.For EHT data, estimated residual leakages are only ∼1% (Event Horizon Telescope Collaboration et al. 2021a), and thus have a negligible effect unless polarization fractions are very high.Noting that , it becomes apparent that a non-constant fractional circular polarization leads to phase and amplitude differences between the parallel hands, which can be robustly detected by investigating the gain-invariant closure products.However, closure quantities contain less information than the baseline-based visibility ratios when prior knowledge on the gains is available (e.g.Blackburn et al. 2020).In total intensity, closure phases deviating from 0 or 180 degrees indicate the presence of non-pointsymmetric structure (e.g.Monnier 2007).In circular polarization, non-zero differences between LL * and RR * closure phases on a given triangle and time indicate the presence of non-constant fractional circular polarization structure (see Figure 2 for examples).
Thus, comparing the results of fits to these two types of data products brackets the range of uncertainty in the Stokes V structure, with smaller uncertainties expected for the visibility ratios (tied to our confidence in the apriori R/L gain calibration) and larger uncertainties for the closure products (with fewer calibration assumptions required).
In addition to this information in the parallel-hand visibilities, the cross-hand visibilities contain information about the linear polarization structure of the source (see, e.g., Roberts et al. 1994): Since the leakage terms here enter in products D Ĩ rather than D P (Equation 7), it is of greater importance to calibrate them for a faithful linear polarization source reconstruction.Neglecting D 2 , D P, and V terms, we can use Equations 7 and 9 to construct the cross-hand to parallel-hand visibility ratios (Roberts et al. 1994;Johnson et al. 2015) Here, mjk = Pjk / Ĩjk is the polarimetric ratio in the visibility domain.This quantity, while not the Fourier transform of its image-domain analog m = P/I, has proven useful in capturing linear polarization source structure (Johnson et al. 2015;Gold et al. 2017).Phase fluctuations cancel in the correlation ratios, and the correlation ratios only depend on left-right gain ratios rather than left and right gains separately.Instrumental polarization leakage remains a source of systematic error, and it can be calibrated by observing the source and a set of calibrators across a wide range of field rotation angle.
Another calibration-invariant data product containing polarization information is the closure trace, which is formed on a quadrangle of baselines (Broderick & Pesce 2020).Closure traces are a natural extension of closure amplitudes and closure phases into full Stokes.They are insensitive to both station gains and polarimetric leakage, and hence allow for maximum calibration freedom.They may also provide an unambiguous detection of non-trivial polarization image structure, although linear and circular polarization cannot be distinguished in such a test.Since we have a decent handle on the EHT 2017 D-terms and gains, we utilize that information in our fits by fitting to m rather than closure traces.We use closure traces for consistency checks in Paper IX.

Fitting implementation
Our polarimetric model fitting methods have been implemented in eht-imaging (Chael et al. 2016(Chael et al. , 2018(Chael et al. , 2019)).This Python library contains many utilities for the analysis of VLBI data, including functions for common operations (e.g., data flagging, averaging, network calibration) on VLBI datasets, plotting utilities, static and dynamic imaging tools, synthetic data generation tools, an interstellar scattering module, and most recently geometric modeling (e.g., Event Horizon Telescope Collaboration et al. 2022d).Within the modeling module, several geometric models have been implemented apart from the m-ring model, including point sources, Gaussians, disks, crescents, and rings.These models can be flexibly combined to form a multicomponent source model.For our m-ring fits, we set flat priors (uniform between minimum and maximum values) for all source parameters.For posterior exploration, we use the dynesty sampler (Speagle 2020).

Data pre-processing and treatment of zero-baseline parameters
Before performing any fitting, we pre-process our datasets in several ways.As typically done for analysis of EHT datasets (e.g.Event Horizon Telescope Collaboration et al. 2019dCollaboration et al. , 2022b,c,d),c,d), we add a small fraction (estimated at around 2%; Event Horizon Telescope Collaboration et al. 2019c) of the visibility amplitudes to the thermal error budget in quadrature, effectively acting as a regularization and imposing a maximal signal-to-noise ratio in order to represent residual systematic errors.These systematic uncertainties are added to each data point, thus they impact both visibilities and interferometric closure products.For our fits to RR/LL visibility ratios, this fractional noise absorbs systematic uncertainties in the R/L gain ratios.For our fits to closure data products, the fractional noise covers non-closing errors.Apart from polarimetric leakage, the main source of these additional non-closing uncertainties is related to gain-calibration of wide frequency bands (e.g., Natarajan et al. 2022).We add the 2% noise to GRMHD synthetic datasets generated without non-closing errors as well in order to maintain consistency in the data pre-processing, but the effect on the closure product noise budget is generally small.Apart from adding fractional noise, we also pre-process our data by performing scan-averages.This operation increases signal-to-noise ratios and reduces the number of data points that need to be fit, making the process more efficient.Finally, we may rescale zero-baseline fluxes depending on the dataset.During the generation of the synthetic GRMHD datasets (Paper IX, Section 5), a large-scale component was added to the visibilities to mimic large-scale structure seen in the EHT M87 * data.For our fits to these datasets, we added a large-scale circular Gaussian (FWHM 2 mas) model component with the same total flux and polarization parameters as this added component, and kept these parameters fixed while fitting the compact structure with the polarized m-ring model.For the real EHT M87 * data, we used the datasets where the large-scale structure was taken out by rescaling the zero baselines (Paper IX).
Circular polarization fitting to the parallel-hand closure products does not constrain the integrated fractional circular polarization V net /I tot .The zero-baseline LL visibilities measure I tot + V net and the RR visibilities are sensitive to I tot − V net (Equation 7), but the closure products containing zero-baselines cannot distinguish between these.During the fitting, we therefore fix V net to the ground-truth value for synthetic datasets, and to the measured V net from zero-baseline observations (ALMA-only, Goddi et al. 2021) for real M87 * data.For consistency, we also fix V net for our fits to right-left visibility ratios.We investigate the effect of varying V net on our M87 * fits in Section 6.2.

TESTS ON SYNTHETIC DATA FROM GEOMETRIC MODELS
To further outline, motivate, and test our polarimetric fitting procedure, we start by fitting polarized m-rings to synthetic EHT data generated from the same model.Thus, the model specification is perfect for these tests.In particular, we use these tests on geometric models Figure 2. LL * (blue) and RR * (orange) closure phases and their differences (green) on the ALMA-SMT-LMT triangle for m-ring models with different combinations of point-symmetric and non-point-symmetric total intensity and circular polarization structures, simulated with 2017 April 11 EHT coverage and thermal noise.The βI parameters are the same as those in Figure 1, except for the rightmost panels, where βI,1 = 0.In models with asymmetric Stokes V structure, the βV parameters are the same as those in the bottom center panel of Figure 1, and for the models with symmetric Stokes V structure, βV,0 = 0.1 and βV,1 = 0, with βV,2 = 0.2 in the top right panel.The bottom center model is identical to the bottom center model in Figure 1.A constant fractional circular polarization structure results in identical closure phases in both parallel hands (top left and middle panels).If the fractional circular polarization structure is not constant, the parallel-hand closure phase differences are non-zero (bottom and top right panels).In the top right panel, the fractional circular polarization is point-symmetric but not constant, and the closure phase differences are 0 or 180 degrees.
to establish a preferred fitting procedure that is free of biases, at least in these idealized cases.As shown below, biases may be introduced by not fitting the Stokes I and V structure simultaneously, by fitting the Stokes V structure without taking the linear polarization structure and leakage effects into account, and by fitting to the RR/LL data product in the presence of non-unity R/L gain ratios.Once the fitting procedures have been established and tested, we move on to the more realistic case of GRMHD models in Section 5.

Model description and synthetic data generation
For the geometric model tests, we used a circularly polarized m-ring model with F = 0.5 Jy, d = 40 µas, α = 10 µas, β I,1 = 0.2 − 0.1i, β V,0 = 0.05, and β V,1 = 0.1 − 0.1i.This model is shown in the middle panel of Figure 1.The net circular polarization fraction of 5% is substantially higher than observed for most VLBI sources, as is the resolved circular polarization fraction of up to ∼40% (Section 5 shows more realistic cases).In order to test the effect of linear polarization structure and leakage, we also generated a model with added linear polarization by setting β P,0 = 0.1, β P,−1 = 0.1 + 0.2i, and β P,1 = −0.1 + 0.1i.
Synthetic data were generated with eht-imaging (Chael et al. 2016(Chael et al. , 2018)), using the uv-coverage and thermal noise from the synthetic datasets generated for the imaging and modeling method tests in Paper IX (see also Section 5.1), corresponding to the low-band EHT M87 * dataset from 11 April 2017.Since these tests focus on the effect of fitting procedure choices and the presence of linear polarization and leakage, we did not introduce any systematic gain offsets for these tests.Such effects are introduced in our GRMHD model fits (Section 5).For the tests with linear polarization, we set the left leakage terms D jL = 0.04 + 0.04i, and the right leakage terms D jR = 0.03 + 0.03i for all stations.

Fitting procedures and results
Figure 3 shows β V,1 posteriors from fits to our geometric models using different fitting procedures and data products.In general, fits to closure quantities result in wider posteriors than fits to RR/LL visibility ratios, which is expected given that there is more information in the latter.
The top row in Figure 3 shows the importance of fitting for the Stokes I and V structure simultaneously.Polarimetric reconstructions are often made by first reconstructing the Stokes I structure, and keeping that frozen while reconstructing the polarimetric structure (e.g., Event Horizon Telescope Collaboration et al. 2021a).Taking a similar approach here results in the blue curves in the top row panels of Figure 3. Here, the Stokes I m-ring parameters were fixed to the posterior maximum (MAP) of the Stokes I fit before fitting the Stokes V structure.This strategy results in a small (a few percent) but statistically significant bias.Small errors in the Stokes I parameters propagate to a biased estimate of the Stokes V parameters, as the parallelhand visibilities, which are the data products used for the fits, contain contributions from both (Equation 7).Fitting for the Stokes I and V structure simultaneously (orange curves) is computationally more expensive, but removes the biases.
The middle row in Figure 3 shows another potential source of biases in the Stokes V posteriors.For these fits, the ground truth source model included linear polarization structure, and polarization leakage was introduced in the synthetic data generation (Section 4).The blue curves show posteriors resulting from fitting the Stokes I and V structure simultaneously, but ignoring the linear polarization structure and leakage.While the fit to closure quantities is acceptable, a significant bias is present for the fit to the RR/LL visibility ratios.Even though the linear polarization structure and leakage enters the parallel-hand visibilities only as a second-order effect (Equation 7), they may still cause the circular polarization fits to be biased and therefore should be included in the Stokes V fitting process.The orange curves were obtained by first fitting the Stokes I structure, then fitting the linear polarization structure and leakage parameters, and subsequently fixing the linear polarization structure and leakage parameters to the MAP while fitting for the Stokes I and V structure simultaneously.This strategy removes the biases introduced when ignoring the linear polarization structure and leakage effects.
Finally, the bottom row in Figure 3 shows that when introducing R/L gain amplitude offsets (here set to be constant in time), the calibration-invariant closure-only posteriors are not affected, while the R/L gain-sensitive RR/LL posteriors (Equation 7) show increasing biases with increasing R/L gain offsets.In practice, it is thus important that the R/L gain ratios are calibrated as well as possible when fitting to RR/LL visibility ratios from real datasets.Since closure products are not affected by these gain offsets, checking for consistency with closureonly fits is recommended (see also Section 6).

APPLICATION TO SYNTHETIC DATA FROM GRMHD MODELS
In this section, we apply our polarimetric m-ring fitting procedures to synthetic EHT data from a set of three GRMHD models, investigating in particular how well the basic (asymmetric) linear and circular structure can be recovered for different GRMHD parameters, and how the geometric fits behave as a function of the Stokes V m-ring order m V .The synthetic datasets were generated for the circular polarization imaging and modeling tests described in Paper IX. Here, we discuss the m-ring modeling results in greater depth.
Synthetic data were generated using eht-imaging for low-band EHT M87 * uv-coverage on 11 April 2017.The bottom row shows comparisons between applying different R/L gain ratio offsets to the synthetic data before fitting.The contours show 1σ, 2σ, and 3σ levels.In general, the fits using only closure quantities have slightly weaker constraints, and both data products show biases when fitting Stokes V separate from I; RR/LL visibility ratios show biases when ignoring the presence of linear polarization and leakage effects, and when R/L gain offsets are present in the data.

RR/LL visibilities
Thermal noise and systematic gain and leakage terms were added, and non-unity left-right gain ratios G R /G L were introduced for all sites except ALMA.The G R /G L amplitudes were sampled from a Gaussian distribution with unity mean and a 20% standard deviation, with a 2-hour correlation timescale.For the G R /G L phases, a standard deviation of 10 • (40 • for the SMA station) and a 24-hour correlation timescale were used.These numbers were motivated by a priori limits estimated for the 2017 EHT data (Paper IX).More details on the GRMHD models and data generation are reported in Paper IX.
To quantitatively compare our fits to the ground-truth model, we compute the ground-truth β P,k (and analogously the β V,k ) in the image domain (see, e.g., Palumbo et al. 2020) as where we set the inner radius ρ min to zero and the outer radius ρ max to a large value (outside the field of view) in order to capture the full model image.

Linear polarization results
Figure 4 shows total intensity and linear polarization m-ring fits (m I = 3, m P = 3) to synthetic data from the three GRMHD models.While a comparison by eye shows many low-order features being recovered by the modeling, some systematic offsets clearly remain, which we attribute to model misspecification.The total polarization fraction is recovered least accurately for model 3, which was challenging to fit with the m-ring model due to the extended emission outside the photon ring and the high degree of asymmetry, concentrating most total intensity and polarized emission in the South.The net EVPA is recovered within a few degrees for models 2 and 3 but not for model 1, which has small net polarization fraction and the most complex EVPA structure.∠β P,2 is recovered to within 12 − 34 degrees, and |β P,2 | to within 0.01.
Overall, model 2 is fit best, which can be attributed to a simple twisty polarization pattern centered on the photon ring, which lends itself especially well to m-ring modeling.Such a polarization pattern is seen in many MAD GRMHD models.Models 1 and 3 both show most emission outside the photon ring, due to model 1 being a model with retrograde spin and model 3 being a SANE model with zero spin (SANE and zero-spin models have been ruled out for M87 * , Event Horizon Telescope Collaboration et al. 2019g, 2021b.As is made clear from these results, the ability of m-ring modeling to constrain the polarization structure depends on the similarity of the model to the ground truth, which is indeed the most important caveat for any geometric modeling result.As shown in Section 6, it appears that M87 * as observed by the EHT is well-suited for linear polarization m-ring modeling, given the excellent agreement between imaging and modeling results.

Circular polarization results
Figure 5 shows the posterior maxima of the Stokes V m-ring fits to synthetic data from the three GRMHD models.These were produced as outlined in Section 4.2, with the assumption of a perfect leakage calibration.The m-ring model is suitable for recovering the basic, low-order properties of the complex Stokes V structure in the GRMHD models, although like for linear polarization the performance varies depending on the groundtruth model.For model 1, the first-order orientation of the Stokes V structure, ∠β V,1 , is reproduced to within a few degrees for m V = 1 (see also Figure 6, which shows the complex β V,1 posteriors compared to the groundtruth values), and to within 30 − 50 degrees for m V = 2.The ∠β V,1 of the m V = 3 fit to the visibility ratios deviates most significantly, which is likely due to the multi-lobe structure of the m V = 3 model in combination with a low total circular polarization fraction, so that a small deviation in one of the lobes can result in a large deviation of the first-order orientation ∠β V,1 .
The Stokes V structure of model 2 is more complex, but ∠β V,1 is nevertheless recovered to within less than 40 degrees.The higher m-order fits are more informative here, recovering the alternating regions of positive and negative Stokes V along the ring.
The Stokes V structure of model 3 is less ring-like than for the other models, with most of the Stokes V emission concentrated in the South.The β V,1 posteriors (Figure 6) are further away from the ground truth than for the other models, despite the fact that the imageaveraged circular polarization fraction ⟨|V/I|⟩ is highest for this model.The low m-order fits nevertheless limit the first-order orientation offset to within 30 − 40 degrees.Interestingly, the higher m-order fits to the visibility ratios favor a much smaller ring size in order to capture the compact Stokes V emission in the South.
To summarize, the m-ring model is able to recover low-order Stokes V structures for these three quite distinct GRMHD models with varying accuracy.Since the Stokes V structure is often complex, there are systematics due to model misspecification (the low-order m-ring model does not fully describe the physical characteristics of the Stokes V emission), but the first-order asymmetry orientation is recovered to within a few tens of degrees for m V = 1, 2. Higher-order m-ring fits per-  (2019a,d, 2021a) in Table 1.
form well if the ground-truth Stokes V structure is well described by alternating lobes of positive and negative Stokes V (model 2), but these fits are often less consistent in other cases and hence should be applied with caution in practice.We do not see a clear trend of increasing Stokes V reproducibility with increasing ⟨|V/I|⟩ for these datasets.As discussed in Paper IX, imaging methods showed similar and often greater difficulty in recovering the circular polarization structure from these datasets, and m-ring modeling often outperforms imaging in the recovery of quantities like the image-averaged circular polarization fraction or the first-order orientation, especially when the net circular polarization fraction is low.m-ring modeling is therefore a useful tool for studying low-order circular polarization structures in mm-VLBI observations of black holes.

APPLICATION TO EHT DATA OF M87 *
In this Section, we apply our polarimetric m-ring modeling framework to EHT 2017 observations of M87 * .We use data calibrated through the EHT-HOPS pipeline (Blackburn et al. 2019;Event Horizon Telescope Collaboration et al. 2019c, 2023), unless otherwise specified (Section 6.2).We use both low-band (LO, with a central frequency of 227.1 GHz) and high-band (HI, with a central frequency of 229.1 GHz) data for our analyses.Both bands have a bandwidth of 2 GHz.All datasets were leakage-calibrated using the estimated Dterms from Event Horizon Telescope Collaboration et al. (2021a); Issaoun et al. (2022), which have estimated residual leakages of only ∼1%.We show our recovery of the previously imaged linear polarization structure (Event Horizon Telescope Collaboration et al. 2021a) in Section 6.1.In Section 6.2, we then show our circular polarization fits to M87 * data, providing an extension and more detailed exploration of the fits presented in Paper IX.

Linear polarization
Figure 7 shows our total intensity and linear polarization fits (m I = 3, m P = 3) to EHT 2017 M87 * data on four days (April 5,6,10,and 11), with and without blurring the images with a Gaussian kernel with a FWHM of 20 µas.This blurring kernel is representative for the blurring kernels used for the images in Event Horizon Telescope Collaboration et al. (2019dCollaboration et al. ( , 2021a)).Table 1 compares our fitted parameters to previous EHT results (Event Horizon Telescope Collaboration et al. 2019a,d,f, 2021a).The structure recovered with our m-ring fits is remarkably consistent across days and in excellent agreement with previous EHT imaging and modeling results, especially when our fits are blurred.
For consistency with Event Horizon Telescope Collaboration et al. (2019a), we report the shifted diameter in Table 1.This shifted diameter accounts for the change in peak brightness radius for a thick ring with FWHM α as compared to an infinitesimally thin ring (Event Horizon Telescope Collaboration et al. 2019d).
The diameter change is of order 2 µas for the values reported in Table 1.
For a blurring kernel with FWHM W , the change in ring thickness can be approximated as (Event Horizon Telescope Collaboration et al. 2019d) For the blurred m-ring posterior ranges reported in Table 1 (5th column), our α-posteriors were transformed following this approximation, with W = 20 µas.These values were then used to compute the blurred d ′ posterior ranges following Equation 13.The β-values for the blurred fits were computed by generating 1000 image samples from the posteriors, blurring them with a 20 µas beam, and computing the values from the blurred image samples.Most fitted quantities are insensitive or only weakly sensitive to image blurring.An exception is |β P,2 |, which reduces significantly after blurring.
As found in the previous EHT analyses, the peak brightness in total intensity moves from the southeast towards the southwest for the later days.The total linear polarization between about 1.5 and 4.3% is consistent, within error bars, with that recovered by the polarimetric imaging methods in Event Horizon Telescope Collaboration et al. (2021a), which reported values between 1.0 and 3.7%.
Our fitted ∠β P,2 between about -136 and -121 • is also in agreement with the EHT imaging results, which reported values between -163 • and -127 • (except for a slight offset for the fit to high-band data on April 6).There is a significant discrepancy for |β P,2 |, but as shown in Figure 7 and Table 1 this quantity is sensitive to the applied blurring kernel, and the values are in full agreement when blurring our models with a 20 µas Gaussian kernel.Our posterior widths (Table 1) are generally much smaller than the ranges spanned by the EHT imaging methods, indicating that systematic offsets (model misspecifications) are likely dominant over the statistical uncertainties from fitting a specific model to the data.For all quantities, the statistical uncertainty is largest for April 10 data, which indeed has the smallest number of M87 * scans (amounting to less than 30 minutes on-source).⟨|V/I|⟩ (%) Apr 5 6.2.Circular polarization

Fit results
Figure 8 shows our circular polarization m-ring fits of EHT M87 * data on April 5,6,10,and 11. Figure 9 shows complete polarization ellipse plots for the same fits, which include total intensity, linear, and circular polarization information.Corresponding posterior ranges are reported in Table 1.These figures and table represent fits with an m V = 1 m-ring to closure products; fits with varying Stokes V m-ring order and fits to RR/LL visibility ratios are explored in Figures 10 and  11.Our fits identify a first-order (dipolar) circular polarization asymmetry that is broadly consistent across the four observing epochs spanning a six-day window, with more negative Stokes V in the South and more positive Stokes V in the North.The strength of the dipolar asymmetry slightly increases towards the later epochs, as indicated by the increase in |β V,1 | and ⟨|V/I|⟩.The closure-only fit to the April 10 high-band data is an outlier; the β V,1 posterior is too broad for significant dipole structure to be detected (Table 1).
As shown in Figure 10, the overall Stokes V morphology of the fits is remarkably consistent between fitting to different data products and fitting an m V = 1 or m V = 2 model, on all days.This consistency starts to break down for m V = 3, although it persists for fits using data from April 10 and 11, where the asymmetry for the low m-order fits is most prominent.
The β V,1 posteriors in Figure 11 indicate that a significant dipolar signal is found by nearly all our fits, since β V,1 is nonzero at a > 3σ level, especially for the m V = 1 and m V = 2 fits.The posteriors are generally broader for the closure only fits.The posteriors also indicate a preferred β V,1 orientation close to the positive real axis, indeed corresponding to positive Stokes V in the North.
Considering the Bayesian evidence ln Z and reduced χ 2 as shown in Figure 12, the preferred Stokes V mring order depends on the data product, day, and band that is fit.For the closure only fits, the Bayesian evidence mostly decreases and the goodness of fit remains approximately equal towards higher m V , indicating a preference for a low-order m-ring.Interestingly, ln Z is larger for m V = 1 than for m V = 0 for the April 11 low-band fit (m V = 0 corresponds to constant V along the ring), while the m V = 1 values are slightly lower for the other data sets.Additionally, the β V,1 posteriors (Figure 11) are furthest from zero compared to the other days.These trends indicate that the evidence for the presence of dipolar circular polarization structure is largest for April 11 (especially low band).
For the fits to visibility ratios, the Bayesian evidence mostly increases and the χ 2 decreases towards higher m V , indicating a preference for higher-order m-rings.The difference in Bayesian evidence is especially large between m V = 0 and m V = 1 on April 6 and 11, which is a strong indicator of the presence of horizon-scale Stokes V structure.The Bayesian evidence peaks at m V = 1 for April 6 and 11 (low band), indicating a preference for a dipole structure.Considering the increased inconsistency among the fit Stokes V structure for m V = 3 (Figure 10), the same trends occurring in our GRMHD fits, and the fact that the visibility ratio fits could be affected by R/L gain calibration uncertainties, we have presented the closure-only m V = 1 fits as the main modeling result in Paper IX.We do not deem any further fitted substructure trustworthy.The m V = 3 fits to visibility ratios may be picking up on smaller-scale structure that the lower-order m-rings cannot account for, but in some of our GRMHD tests we have seen that these fits may present images that do not correctly reproduce this smaller-scale structure (Figure 5).Future EHT datasets with better uv-coverage and sensitivity will allow us to detect circular polarization structure more strongly and in more detail than these first-order structure results.
Figure 13 shows that the basic structure of the posterior maxima is independent of the assumed V net , within the range reported from ALMA-only measurements by Goddi et al. (2021).

Sensitivity to R/L gain calibration strategy
As noted in Section 3.1, the RR/LL visibility ratio data product is affected by non-unity R/L gain ratios, while the parallel-hand closure data products are not.In this section, we explore the sensitivity of our M87 * fit results to the gain calibration strategy.The R/L gain ratios in the HOPS data (used in Section 6.1 and Section 6.2.1) were calibrated by fitting a polynomial to the RR and LL visibility offsets (amplitude ratios and phase differences) as a function of time for ten sources observed during the five days of the 2017 EHT campaign.By using data from multiple sources and days, circular polarization signatures of individual sources (assumed to be independent and thus averaging out) could be separated from instrumental R/L gain offsets (shared between the sources).The visibility differences could be fit with a constant complex gain ratio for all stations, except for APEX and SMA, which showed stronger timedependence of the RR and LL visibility phase differences and hence required a time-dependent polynomial fit for the R/L gain phases.Some more details are given in Paper IX, with an illustration of a R/L phase fit shown in Fig. 14 therein, indicating a well-behaving set of sources, without any strongly polarized outlier.The general similarity of our RR/LL fits to our closure-only fits (Figure 10) can be taken as an indication that the effect of any residual R/L gain ratios on our fits is limited.
The robustness of our RR/LL fits in particular can also be tested in an exercise where we attempt to remove the circular polarization signal by self-calibrating the R and L gains separately to our total intensity model assuming V = 0, before fitting the Stokes V structure (e.g.Homan & Wardle 2004).Figure 14 shows the result of performing such an exercise on the EHT HOPS data.Here, we have set a solution interval of two hours for both the gain amplitudes and gain phases.As expected, the closure-only fits are unaffected by this operation, since the closure products are robust against station gains.The RR/LL fits are clearly affected, showing substantially weaker circular polarization structure (note the difference in scale between the two rows), and a different orientation on most days.The exception is April 11, where the zero-V self-calibration failed to remove the dipole structure with approximately North-South orientation, although it is substantially weaker.
To test the sensitivity of our model fitting procedure to upstream calibration choices, we also fit our polarimetric m-ring model to a new version of the 2017 EHT data, which has been calibrated in a slightly different way.We utilize the CASA-VLBI-based (van Bemmel et al. 2022) rPICARD (Janssen et al. 2019) pipeline to solve for instrumental offsets (Event Horizon Telescope Collaboration et al. 2019c) and then combine the two LO plus HI frequency bands and all polarization correlation products to solve for fringes and atmospheric phases (Janssen et al. in prep.).In contrast to the multi-source polynomial fit R/L gain calibration described above, either no R/L amplitude gain ratios are applied to the CASA data used here, or an R/L gain amplitude ratio calibration was performed assuming zero circular polarization.
Figure 15 shows the polarimetric modeling results of the new CASA data.As expected, the R/L gaininsensitive closure-based fitting is in excellent agreement with the HOPS data results.Even without any R/L gain calibration or assuming V = 0, consistent circular polarization signals are retrieved here.However, for the RR/LL visibility ratio fits, the differences in Stokes V structures with the HOPS results and the inconsistency between observing epochs demonstrate a significant sensitivity to the R/L gain calibration strategy (since these are posterior maxima, the structure itself does not look more noisy than for the HOPS data).For April 11 data, on which the HOPS fits indicated the strongest evidence for the presence of dipolar Stokes V structure, the CASA RR/LL fits are in agreement with the closure and HOPS data fits regardless of the R/L gain calibration strategy, which may indicate a reduced sensitivity to R/L gains due to the stronger Stokes V signal on this day.
The consistency of the results with the previously used, well vetted, 2017 calibrated EHT data demonstrates the robustness of our method relative to different calibration assumptions and serves as a first validation of the updated CASA/rPICARD data reduction pathway.Our fits across different data sets, data products, and modeling assumptions support the presence of a persistent dipolar asymmetry in the circular polarization of M87 * especially on April 11, where this asymmetry persists in the RR/LL fits regardless of calibration strategy.

SUMMARY AND OUTLOOK
In this work, we have developed a novel method to reconstruct polarimetric image structure from VLBI observations, making use of data products with different levels of calibration-invariance and simple geometric models.Specifically, polarimetric m-ring fitting is a useful method to obtain information on the polarimetric structure of horizon-scale observations of supermassive black holes.We have shown that ground-truth polarization parameters can be recovered from synthetic EHT data from geometric m-ring models and GRMHD models, with accuracies depending on the level of model misspecification.Even with total and resolved circular polarization fractions as low as 0.5%, the first-order circular polarization asymmetry can be recovered from GRMHD models to within a few to ∼30−40 degrees, depending on the model.Polarized structure is recovered most faithfully in image regions with high total intensity.
Application to EHT M87 * data has shown that the linear polarization structure imaged by Event Horizon Telescope Collaboration et al. ( 2021a) is recovered well with our m-ring modeling framework.Our fits also indicate the presence of a persistent horizon-scale circular polarization asymmetry, with increased negative circular polarization in the South, near the total intensity maximum.This asymmetry persists across fits to different observing epochs, bands, and data products, across fits with circular polarization m-ring orders m V of 1 and 2, for fits assuming different image-integrated circular polarization fractions, and for fits to data calibrated with different calibration pipelines and strategies.Support for the presence of the asymmetry is largest for April 11 data.For this day, a dipolar structure is favored by the Bayesian evidence, the structure persists in fits up to m V = 3, and in R/L gain-sensitive RR/LL visibility ratio fits regardless of the R/L gain calibration strategy used.Some imaging methods reconstruct similar structure on this day (Paper IX).However, given the overall weakness of the circular polarization signal, the sensitivity of our RR/LL fit results to the R/L gain calibration strategy on three days, and the difficulty for imaging methods to reconstruct similarly consistent structure, caution should be exercised in interpreting this result and we are not reporting an unambiguous detection of dipolar structure with a specific orientation.While m-ring fitting reliably reconstructed the first-order circular polarization asymmetry in our GRMHD synthetic data tests (with an orientation offset of a few degrees up to ∼40 degrees), there is no guarantee that the underlying circular polarization structure of M87* has a strong dipolar component.Even for a strong asymmetry detection, GRMHD models show degeneracies in black hole and plasma parameters that may produce such asymmetry, and it may be short-lived due to plasma variability.
As more stations are added to the EHT and antenna sensitivity improves, the circular polarization structure of M87 * will become easier to detect and reconstruct.While the EHT 2017 data only provide upper limits on the resolved circular polarization fraction and tentative circular polarization images with large uncertainties, the future prospects for imaging and modeling M87 * in circular polarization are excellent.
Finally, our modeling methods have other applications as well.Apart from fitting horizon-scale structure with m-rings, polarimetric geometric modeling with different model prescriptions (e.g., a set of polarized Gaussians) may also be used to reconstruct the polarized structure of non-horizon active galactic nuclei.For Sgr A * , snapshot geometric modeling (i.e., fitting to short time snippets of data and then combining the posteriors; Event Horizon Telescope Collaboration et al. 2022d) can utilize our polarimetric m-ring model to mitigate the rapid source variability and constrain the polarimetric structure with only a small number of baselines.With future arrays with many more stations, such as the Next-Generation EHT (ngEHT; Doeleman et al. 2019;Johnson et al. 2023;Doeleman et al. 2023), snapshot modeling may even be used to reconstruct real-time polarimetric black hole movies of Sgr A * .

Figure 1 .
Figure 1.Examples of three m-ring models in Stokes I and P (upper panels), and Stokes I and V (lower panels).Throughout the panels, the Stokes I structure (heat map) is kept constant with F = 0.5 Jy, d = 40 µas, α = 10 µas, and βI,1 = 0.2 − 0.1i.The top center panel shows a linear polarization structure with mnet ≡ βP,0 = 0.1, βP,−1 = 0.1 + 0.2i, and βP,1 = −0.1 + 0.1i.In the top left and right panel, nonzero βP,2 components have been added with opposite sign.The bottom left panel shows a dipolar circular polarization structure (contours) oriented towards the North (βV,1 = 0.14).The net circular polarization is zero, so that the North and South half of the ring are identical with opposite sign in Stokes βV .In the bottom center panel, we have rotated the circular polarization structure by -45 • and introduced a nonzero net circular polarization (Vnet ≡ βV,0 = 0.05), so that the symmetry is broken.Finally, in the bottom right panel we have added a nonzero βV,2 component, increasing the complexity of the azimuthal structure in Stokes V.The model shown in the center panels is used for our geometric tests (Section 4, Figure 3).

Figure 3 .
Figure3.βV,1 posteriors from fitting a circularly polarized m-ring to synthetic EHT 2017 data generated from two different m-ring models.Ground truth values are indicated with red vertical and horizontal lines and correspond to the middle panel of Figure1.Fits using closure quantities are shown on the left, and fits with RR/LL visibility ratios are shown on the right.The posteriors on the top row were computed from fits to data generated from a model with zero linear polarization and without any leakage corruptions added to the data.Each top row panel compares separate (consecutive) Stokes I and V fits (blue) versus simultaneous Stokes I and V fits (orange).The posteriors in the middle row were computed from simultaneous Stokes I and V fits to data generated from a model with nonzero linear polarization and with leakage corruptions added to the data.Each middle row panel compares fits ignoring linear polarization (blue) to fits including linear polarization and leakage fits (orange).The bottom row shows comparisons between applying different R/L gain ratio offsets to the synthetic data before fitting.The contours show 1σ, 2σ, and 3σ levels.In general, the fits using only closure quantities have slightly weaker constraints, and both data products show biases when fitting Stokes V separate from I; RR/LL visibility ratios show biases when ignoring the presence of linear polarization and leakage effects, and when R/L gain offsets are present in the data.

Figure 4 .
Figure 4. Total intensity and linear polarization reconstructions of our three GRMHD models (left to right), from synthetic data with EHT 2017 coverage.The unblurred and blurred (FWHM 10 µas) ground truth images are shown in the top and middle rows, respectively.The bottom row shows the posterior maxima of m-ring fits with mI = 3 and mP = 3.In each panel, the Stokes I structure is indicated by the heat map, and the scale is normalized to the brightest pixel in each panel.The tick length indicates the polarized intensity, the tick orientation indicates the EVPA, and the tick color indicates the fractional linear polarization.The m-ring fit posteriors for several key parameters are compared to the ranges found by Event Horizon Telescope Collaboration et al.(2019a,d, 2021a)  in Table1.

Figure 5 .Figure 6 .
Figure 5. GRMHD ground truth Stokes V images and m-ring fits from EHT April 11 2017 coverage (low band).Each model covers a set of two rows.The left column shows the ground truth without (upper panels) and with (lower panels) Gaussian blurring (FWHM 20 µas).The right three columns show the Stokes V m-ring fits for mV = 1, 2, and 3, fitting to closure quantities (upper panels for each model) and RR/LL visibility ratios (lower panels for each model).

Figure 7 .
Figure 7. EHT 2017 method-averaged total intensity and linear polarization images of M87 * (top row; Event Horizon Telescope Collaboration et al. 2021a) on April 5, 6, 10, and 11 (left to right), compared to our total intensity and linear polarization mring fits (posterior maxima; mI = 3, mP = 3) to the same data, without (middle row) and with (bottom row) blurring with a Gaussian kernel with a FWHM of 20 µas.

Figure 8 .Figure 9 .
Figure 8. Circular polarization m-ring fits (mV = 1, mI = 3; posterior maxima) to closure products of low-band EHT 2017 M87 * data on April 5, 6, 10, and 11 (left to right).The heat map indicates the Stokes I structure, and the contours indicate the Stokes V structure.These fits are also presented in Paper IX.

Table 1 .
Parameters describing the total intensity and polarization structure of M87 * as observed by the EHT in 2017 extracted through imaging and crescent fitting(Event Horizon Telescope Collaboration et al. 2019a,d, 2021a) and m-ring fitting (this work).The fitted model has mI = 3, mP = 3, and mV = 1, with circular polarization fits to closure quantities.The m-ring fit values indicate the posterior means with 2σ ranges.The ⟨|V/I|⟩ values were computed from posterior image samples.The blurred m-ring parameters were obtained by propagation of the fit posteriors for d ′ , α, and |m|net, and from blurred image samples for the β parameters and ⟨|V/I|⟩ (see text for further details).