Circular Polarization of Simulated Images of Black Holes

We investigate the origin of circular polarization (CP) using first-principles models of synchrotron-emitting systems, and study the distribution of the expected CP across a set of models at varying spin, magnetization, and electron distribution functions (eDFs).

The 2017 Event Horizon Telescope (EHT) campaign produced total intensity images of the supermassive black hole at the center of M87 (hereafter, M87*) and the Milky Way (hereafter, Sgr A*) at a resolution of ∼25 μas (Event Horizon Telescope Collaboration et al. 2019a, 2019b, 2019c, 2019d, 2019e, 2019f—hereafter, EHTC M87 I–VI; and Event Horizon Telescope Collaboration et al. 2022a, 2022b, 2022c, 2022d, 2022e, 2022f—hereafter, EHTC SgrA I–VI). Both reconstructed images show a ring surrounding a central flux depression. The ring is produced by synchrotron emission from hot gas surrounding the black hole, and the central depression corresponds to lines of sight that cross the event horizon (the black hole "shadow").

EHT images have been interpreted by comparison to a library of numerical models (EHTC M87 V; EHTC SgrA V), in which the spin, flow magnetization, source inclination, and eDF are varied, but the time-averaged 1.3 mm flux density is held fixed and consistent with the 2017 April observations (EHTC M87 IV; Wielgus et al. 2022a). In particular, plasma flow models were generated using general relativistic magnetohydrodynamics (GRMHD) simulations and then ray-traced using a general relativistic radiative transfer (GRRT) scheme; the modeling procedure is described in detail in Wong et al. (2022). The models predict time-dependent image structure in all four Stokes parameters at frequencies where scattering is unimportant, time-dependent unpolarized flux density across the electromagnetic spectrum, and jet power.

For M87*, the model comparison exercise found a subset of library models that satisfied all available observational constraints. The most discriminating observational constraint was a lower limit on a jet power of ${10}^{42}\,{\rm{erg}}\,{{\rm{s}}}^{-1}$. The favored models were highly magnetized—so-called magnetically arrested disk (MAD) models (Igumenshchev et al. 2003; Narayan et al. 2003), in which the magnetic flux through the horizon is large enough to episodically push aside the accreting plasma—and contained a population of relatively cool electrons (EHTC M87 V).

For Sgr A*, 11 observational constraints were used in the model comparison exercise. No models satisfied all constraints (EHTC SgrA V). The most discriminating observational constraint was a measure of fractional variability in the 1.3 mm Atacama Large Millimeter/submillimeter Array (ALMA) light curve; almost all models that failed this test were found to be too variable—most models failed with p < 0.01 for two-sample Kolmogorov–Smirnov (K-S) tests comparing distributions of the ratio of the standard deviation to mean flux averaged over 3 hr timescales. An incomplete list of possible explanations for this variability "crisis" is provided in EHTC SgrA V, including the possible presence of slowly varying, resolved-out features that would make the fractional variability in the ALMA light curve a lower limit on the variability of the compact source. Setting aside the variability constraint, the Sgr A* model comparison identified a set of models that passed all remaining constraints. The favored models were MAD models that contain a population of relatively cool electrons.

EHT also recorded linear polarization (LP) and CP data in the 2017 campaign. LP and CP model comparison studies of M87* (Event Horizon Telescope Collaboration et al. 2021a, 2021b; hereafter, EHTC M87 VII–VIII) included constraints applied to the models based on LP maps and limits to the total CP fraction. Model LP maps are sensitive to the magnetic field configuration of M87* and are highly constraining. Unresolved CP measurements, by contrast, exclude only a few models.

Event Horizon Telescope Collaboration et al. (2023; hereafter, EHTC M87 IX) analyzed CP in the 2017 EHT campaign data and found evidence for nonzero CP. The Stokes V resolved structure could not be constrained, however, in contrast to image reconstructions in Stokes I, Q, and U. EHTC M87 IX placed an upper limit on the magnitude of the resolved CP fraction of 3.7%. Consistent with the results from LP model comparisons (EHTC M87 VIII), the CP constraints favor highly magnetized simulations with relatively cooler electrons in the disk and jet.

ALMA has recorded polarization data in the 1mm and 3mm bands for Sgr A*, M87*, and other active galactic nuclei (AGNs: low-luminosity AGNs, radio-loud AGNs, and blazars for which horizon-scale images are not possible with the current EHT resolution). Bower et al. (2018; 2016 observations), Goddi et al. (2021), and Wielgus et al. (2022b, 2024; 2017 observations) present this ALMA data with detections or limits on the unresolved LP and CP of Sgr A*. Other unresolved CP observations of Sgr A* from the Submillimeter Array at 3 mm and other wavelengths are given in Muñoz et al. (2012) and references therein.

It is expected that at the observing wavelength of 1.3 mm, emission will be produced by the synchrotron process, which is expected to be strongly linearly polarized. The polarization state is modified, however, by propagation through a warm, magnetized plasma, through Faraday rotation and conversion. The combination of emission and propagation effects is complicated, so numerical radiative transfer methods are essential in understanding the polarization of M87* and Sgr A*.

CP structure can be used to understand the magnetic field structure of the accretion disk and field geometry of GRMHD models under certain conditions of observing angles and optical and Faraday thicknesses (Mościbrodzka et al. 2021; Ricarte et al. 2021; Tsunetoe et al. 2021). CP may also be a useful probe of plasma composition, since CP emission and Faraday rotation is sensitive to the electron–positron pair content (Wilson & Weiler 1997; Wardle et al. 1998; Homan et al. 2009; Anantua et al. 2020; Emami et al. 2021).

Unresolved CP measurements, particularly the handedness (or sign) of CP, have been hypothesized to indicate the sense of rotation of the disk (Enßlin 2003) or the magnetic field configuration: the structure and dipole moment of the magnetic field (Beckert & Falcke 2002). Constant handedness across long timescales indicates a constancy in either of these two properties. This is especially interesting in the case of Sgr A*, where the sign of CP has been observed to be constantly negative across decades for 1.3 mm (and larger-wavelength) observations. In this paper, we investigate these hypotheses by comparing observational data at 1.3 mm to a library of simulated images. As we still do not have robust horizon-scale CP images of M87* and Sgr A*, it is useful to compare the integrated fractional CP values obtained from GRRT images to the ALMA data, to see which models are consistent.

The net CP is

$$\begin{eqnarray}&&{v}_{\mathrm{net}}\equiv \frac{\int {d}^{2}x\,V(x,y)}{\int {d}^{2}x\,I(x,y)},\end{eqnarray} \tag{ 1 }$$

where x, y are coordinates on the sky. Here I, V are the Stokes I and V images convolved with the beam. The net CP fraction is accessible from ALMA observations, with an effective beam size of 1'' ∼ 10⁵ GM/c². The GRMHD model library reliably reproduces emission out to 200 μas ∼ 40GM/c² diameter of the source, and to compare our simulations to observations, we assume that there is no significant emission between the two scales. In EHTC SgrA II, comparisons of horizon-scale baselines and short baselines such as ALMA-APEX (100 mas ∼ 2 × 10⁴ GM/c²) suggest that at least ∼90% of the flux density (up to 100 mas) arises from the horizon-scale emission.

We will also occasionally refer to the average absolute CP fraction:

$$\begin{eqnarray}&&\langle | v| \rangle \equiv \displaystyle \frac{\int {d}^{2}x\,| V| }{\int {d}^{2}x\,I}.\end{eqnarray} \tag{ 2 }$$

Evidently, measurement of 〈∣v∣〉 requires a resolved image of the source and depends on the beam size.

The paper is organized as follows. In Section 2, we review the origin of CP in first-principles models in which the dominant emission mechanism is synchrotron emission from a relativistic thermal distribution of electrons. In Section 3, we describe the numerical models used in the analysis, along with the parameters that characterize the resulting model library. In Section 4, we investigate the CP properties of a single GRMHD model, highlighting properties that are generalizable across most of the library. In Section 5, we present the full library of v_net distributions, along with fits for their dependence on model parameters. Using unresolved very long baseline interferometry (VLBI) data of Sgr A* (Bower et al. 2018; Wielgus et al. 2022b), we attempt to constrain our models of Sgr A* in Section 6. In Section 7, we present a discussion on the models, including caveats, followed by a conclusion.

2.1. Polarized Radiative Transfer Equation

The time-independent radiative transfer equation, in flat space, along a ray labeled by s, in the Stokes basis (I, Q, U, V), is:

$$\begin{eqnarray}&&\displaystyle \frac{d}{{ds}}\left(\begin{array}{l}I\\ Q\\ U\\ V\end{array}\right)=\left(\begin{array}{c}{j}_{I}\\ {j}_{Q}\\ {j}_{U}\\ {j}_{V}\end{array}\right)-\left(\begin{array}{cccc}{\alpha }_{I} & {\alpha }_{Q} & {\alpha }_{U} & {\alpha }_{V}\\ {\alpha }_{Q} & {\alpha }_{I} & {\rho }_{V} & -{\rho }_{U}\\ {\alpha }_{U} & -{\rho }_{V} & {\alpha }_{I} & {\rho }_{Q}\\ {\alpha }_{V} & {\rho }_{U} & -{\rho }_{Q} & {\alpha }_{I}\end{array}\right)\left(\begin{array}{l}I\\ Q\\ U\\ V\end{array}\right).\end{eqnarray} \tag{ 3 }$$

Here, j_S , α_S , and ρ_S are the emission, absorption, and Faraday rotation and conversion coefficients ("rotativities") of component S of the Stokes vector. The coefficients depend on the field strength and direction, the energy distribution function of particles (electrons for synchrotron radiation from an electron–ion plasma), and the frame in which they are measured. Scattering processes such as Compton scattering are negligible at millimeter wavelength for M87* and Sgr A* and have been neglected. CP is described by the V component of the Stokes vector. We follow the IEEE convention for definitions of the sign of CP, where V > 0 is right-handed CP, such that the electric field vector rotates in a right-hand direction, at a fixed point in space, with the thumb pointing along the direction of propagation (Hamaker & Bregman 1996).

Separating out the Stokes V equation,

$$\begin{eqnarray}&&\displaystyle \frac{d}{{ds}}V={j}_{V}-{\alpha }_{V}I-{\rho }_{U}Q+{\rho }_{Q}U-{\alpha }_{I}V.\end{eqnarray} \tag{ 4 }$$

Evidently, Stokes V can be altered by direct, circularly polarized emission; polarization-specific absorption (i.e., the plasma acts as a circular polarizing filter); Faraday conversion from LP to circular; and polarization-nonspecific absorption. Polarization-nonspecific absorption α_I does not change the fractional CP V/I.

2.2. Origin of CP

2.3. Transfer Coefficients for Thermal Distribution

We adopt a thermal (Maxwell–Jüttner) electron energy distribution function. This is motivated by the notion that the eDF is likely to have an approximately thermal core extending up to Lorentz factor ∼30 that produces millimeter emission (a hollow distribution, one in which $f({\boldsymbol{p}})$ has a minimum at ${\boldsymbol{p}}=0$, would be kinetically unstable; see Penrose 1960), and the idea that any superthermal tail on the distribution must not overproduce near-IR emission (Section 4.2.3 of EHTC SgrA V); i.e, the tail is constrained by the IR-to-millimeter color.

The emission coefficients are summarized in Dexter (2016), Pandya et al. (2016), and Marszewski et al. (2021); the absorption coefficients follow from Kirchoff's law; and the Faraday coefficients ρ_S are given in Pandya et al. (2018).

It is helpful in understanding the symmetries of the transfer equations to write out the transfer coefficients explicitly in the frame of the plasma. As is conventional in this field, we work in a Stokes basis, where Stokes U corresponds to LP at ±45° to the projection of the magnetic field on the plane perpendicular to the wavevector. Then the emissivity fits are (following Dexter 2016):

$$\begin{eqnarray}&&{j}_{I}\left(\nu ,\theta \right)=\displaystyle \frac{{{ne}}^{2}\nu }{2\sqrt{3}c{{\rm{\Theta }}}_{e}^{2}}{I}_{I}(x),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{j}_{Q}\left(\nu ,\theta \right)=\displaystyle \frac{{{ne}}^{2}\nu }{2\sqrt{3}c{{\rm{\Theta }}}_{e}^{2}}{I}_{Q}(x),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{j}_{U}=0,\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&{j}_{V}\left(\nu ,\theta \right)=\displaystyle \frac{2{{ne}}^{2}\nu \cot (\theta )}{3\sqrt{3}c{{\rm{\Theta }}}_{e}^{2}}{I}_{V}(x).\end{eqnarray} \tag{ 8 }$$

Here, n is the electron number density, −e is the electron charge, ν is the photon frequency, θ is the angle between the wavevector and magnetic field (sometimes called the observer angle), c is the speed of light, Θ_e ≡ k_B T_e /(mc²) is the dimensionless electron temperature, x = ν/ν_c with ${\nu }_{c}=(3/2){\nu }_{B}\sin (\theta ){{\rm{\Theta }}}_{e}^{2}$, ν_B = e∣B∣/(2π mc) (the cyclotron frequency), and m is the electron mass. The I_S do not change sign under field reversal B → − B .

The absorptivities can be found from Kirchoff's law (B_ν is the blackbody function):

$$\begin{eqnarray}&&{j}_{S}={\alpha }_{S}{B}_{\nu }.\end{eqnarray} \tag{ 9 }$$

The Faraday coefficients (fits) are:

$$\begin{eqnarray}&&{\rho }_{Q}=\displaystyle \frac{{{ne}}^{2}{\nu }_{B}^{2}{\sin }^{2}\theta }{{mc}{\nu }^{3}}f(X)\left[\displaystyle \frac{{K}_{1}\left({{\rm{\Theta }}}_{e}^{-1}\right)}{{K}_{2}\left({{\rm{\Theta }}}_{e}^{-1}\right)}+6{{\rm{\Theta }}}_{e}\right],\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\rho }_{U}=0,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\rho }_{V}=\displaystyle \frac{2{{ne}}^{2}{\nu }_{B}}{{mc}{\nu }^{2}}\displaystyle \frac{{K}_{0}\left({{\rm{\Theta }}}_{e}^{-1}\right)}{{K}_{2}\left({{\rm{\Theta }}}_{e}^{-1}\right)}\cos \theta g(X),\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&X={\left(\displaystyle \frac{3}{2\sqrt{2}}{10}^{-3}\displaystyle \frac{\nu }{{\nu }_{c}}\right)}^{-1/2},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&g(X)=1-0.11\,{\rm{ln}}(1+0.035X),\end{eqnarray} \tag{ 14 }$$

and K₀, K₂ are modified Bessel functions of the second kind at orders 0 and 2, respectively. A field reversal transforms θ → π − θ, which thus reverses the sign of j_V , α_V , and ρ_V .

In a field-aligned Stokes basis, j_U = α_U = ρ_U = 0, and thus the Faraday conversion term in the transfer equation reduces to +ρ_Q U. Stokes U in the field-aligned Stokes basis is therefore required to produce Stokes V by Faraday conversion. The Stokes U transfer equation reduces in the field-aligned basis, for a uniform plasma, to dU/ds = ρ_V Q − α_I U − ρ_Q V, so Stokes V can be produced by Faraday rotation of Stokes Q followed by Faraday conversion to Stokes V. We must add a term to the transfer equation, however, if we force the Stokes basis to be field aligned at each point on the ray. This term captures the effect of the rotation of the field through an angle ψ in the plane perpendicular to the line of sight, which interconverts Stokes Q and U, with dU/ds = ... + 2d ψ/dsQ, dQ/ds = ... −2d ψ/dsU. Restated, emission of linearly polarized light elsewhere along the line of sight produces Stokes U locally that can be Faraday-converted to Stokes V. In the end, Faraday conversion acts on linearly polarized light produced by some combination of Faraday rotation and field-line rotation (or "twist"). Ricarte et al. (2021) explore this effect in detail, along with the resulting properties of the magnetic field, which are apparent in Faraday-conversion-produced CP maps of GRMHD models.

2.4. Symmetries of the Coefficients and RTE

2.5. One-zone Model

We can crudely estimate the degree of CP expected in Sgr A* and M87* using a one-zone model. The model consists of a uniform sphere of hot plasma with radius r = 5GM/c². The uniform sphere has constant radiative transfer coefficients, the background radiation vanishes, and the spacetime is Minkowski.

The magnetic field strength and number density of the plasma can be estimated using the one-zone model, following EHTC M87 V and EHTC SgrA V for M87* and Sgr A*. As in those papers, we assume the electrons have a thermal (Maxwell–Jüttner) distribution function. The electron temperature and ion temperature need not be equal to each other, as the plasma is collisionless and there may be preferential heating of ions by turbulent dissipation (Quataert & Gruzinov 1999; Yuan & Narayan 2014; Ressler et al. 2015; Mościbrodzka et al. 2016; Zhdankin et al. 2021). We set the dimensionless electron temperature Θ_e ≡ kT_e /(m_e c²) = 10. The magnetic pressure B²/(8π) is set equal to the gas pressure, assuming the ion-to-electron temperature ratio is 3 and that the gas is pure hydrogen. The angle between the magnetic field and line of sight is set to 60°. The 1.3 mm flux density is set equal to the observed 0.7 Jy for M87* and 2.4 Jy for Sgr A*. Then the number density and magnetic field strength are

$$\begin{eqnarray*}&&{n}_{e,{\rm{M}}87* }=3.3\times {10}^{4}\,{\mathrm{cm}}^{-3};{B}_{{\rm{M}}87* }=4.8\,{\rm{G}},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{n}_{e,\mathrm{Sgr}\,{\rm{A}}* }={10}^{6}\,{\mathrm{cm}}^{-3};{B}_{\mathrm{Sgr}\,{\rm{A}}* }=29\,{\rm{G}},\end{eqnarray*}$$

where we have iterated numerically over n_e to find a solution.

Given the density and magnetic field strength, we can compute the CP fraction using the exact solution to the polarized radiative transfer equation (Landi Degl'Innocenti & Landi Degl'Innocenti 1985; Mościbrodzka & Gammie 2018). We find

$$\begin{eqnarray}&&{\mathrm{CP}}_{{\rm{M}}{87}^{* }}=16.7 \% \end{eqnarray} \tag{ 15 }$$

and

$$\begin{eqnarray}&&{\mathrm{CP}}_{{\mathrm{SgrA}}^{* }}=5.1 \% .\end{eqnarray} \tag{ 16 }$$

These CP values give an estimate of the resolved CP fractions (〈∣v∣〉), which are consistent with the per-pixel CP fractions in simulations (see Figure 1, for example). In comparison, v_net observations (Table 1) and v_net in simulations are much lower (Figures 5 and 16), which suggests cancellations across different parts of the image.

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Table 1. Measurements of CP Fraction at 1.3 mm for EHT Targets

Reference	Source	Mean vnet	Epoch vnet
		(%)	(%)
			−1.1
Muñoz et al. (2012)	Sgr A*	−1.2 ± 0.3	−1.2
			−1.1
			−1.3
Bower et al. (2018)	Sgr A*	−1.1 ± 0.2	−0.9
			−1.3
Goddi et al. (2021)	Sgr A*	[−1.0, −1.5] ± 0.6	N/A
Goddi et al. (2021)	M87*	≲ ± 0.8	N/A
			−1.5
			−1.4
Wielgus et al. (2022b)	Sgr A*	−1.23 ± 0.4	−0.9
			−1.0
			−1.0

Note. Note that for Wielgus et al. (2022b), the epoch values are averages of 3 hr windows. While we report epoch values for Muñoz et al. (2012), we only use the ALMA epoch v_net for comparing our simulations with Sgr A*.

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In the vicinity of the one-zone model parameters, we can probe the effects of intrinsic emission and Faraday conversion on Stokes V. The optical and Faraday depths (α_I r, ρ_Q r, and ρ_V r) are approximately 0.4—in between optically/Faraday thin and thick. In the optically/Faraday-thin regime, Stokes V is well approximated by j_V r. In the moderately optically thick regime, Faraday effects are dominant; the solution to Stokes V with j_V = α_V = 0 is similar to the full solution. At longer wavelengths, absorption effects play an important role. Further analysis of CP versus frequency for the one-zone model is presented in Appendix A.

3.1. GRMHD Models

3.2. Radiative Transfer Numerical Model

3.3. Image Library Parameters

First, consider a single, fiducial model: a MAD, spin +0.5, R_high 160, and inclination 30° (and 150°) model for Sgr A*. This is one of the best-bet models, based on EHT and multiwavelength constraints (EHTC SgrA V).

4.1. Sample Image

4.2. CP Distribution

The CP fluctuates in time. To test a model, we compare the model's distribution of CP to the observed distribution of CP. The top panel in Figure 2 shows the distribution of CP broken down into the aligned- and reversed-field models, as well as the distribution seen from above (inclination 30°) and below (inclination 150°). The time evolution is shown in the bottom two panels for each subset of the model. Evidently, reversing the field, or imaging from a complementary inclination, does not flip the distribution about 0, consistent with the discussion above; however, there appears to be some anticorrelation with reversing the field, which suggests intrinsic CP emission or Faraday rotation being important for v_net in these models. Notice that CP changes sign as a result of the fluctuations in the small patches of polarization seen in Figure 1.

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4.3. Average Images

Time-averaged images provide information about which CP generation mechanism dominates. The mean of the distribution of the integrated CP fraction is not precisely equivalent to the integrated CP fraction of a time-averaged image, but if the total flux of each image is close to the mean value of 2.4 Jy, then the two quantities are comparable. Figure 3 shows the average images for the fiducial model.

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For this particular model, after averaging, the region that dominates the CP map is emission near the disk, close to the black hole (EHTC M87 V). The bright positive feature is the region of the accretion disk where the fluid velocity is aligned with the line of sight, and thus it appears prominently due to Doppler boosting. A clear ring-like structure is seen. Its opposite sign is a consequence of the relatively low Faraday rotation thickness of the image and the imprint of the magnetic fields on Stokes V through Faraday conversion, as observed in Mościbrodzka et al. (2021) and Ricarte et al. (2021). The latter paper, in particular, shows that the sign of CP in the lensed photon ring always has the opposite sign of CP arising from Faraday conversion in Faraday-thin images.

4.4. Contribution of Transfer Coefficients

Here, we probe the relative importance of processes contributing to the net CP by turning off individual radiative transfer coefficients one by one. We re-image the fiducial model with 30 snapshots across 15,000M, turning off j_V (intrinsic CP emission), α_V (CP absorption), ρ_Q (Faraday conversion), and ρ_V (Faraday rotation). Figure 4 shows time series of CP in each case.

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Evidently, CP-selective absorption of unpolarized radiation (α_V ) plays a negligible role. This is because α_V is calculated from the Planck function, j_V is about 2 orders of magnitude smaller than j_I , and the models are optically thin. The remaining three coefficients all affect 〈∣v∣〉, with ρ_Q being the dominant mechanism of CP production, as 〈∣v∣〉 is highly suppressed when excluding Faraday conversion. Although j_V is subdominant, it is non-negligible. This is also observed in MAD model images for M87^* at different R_high values, where we see the inclusion of j_V can increase 〈∣v∣〉 by as much as 50%. Faraday conversion and intrinsic CP emission both contribute to 〈∣v∣〉, but the effect of Faraday rotation varies. The importance of each coefficient also varies in time. We conclude that only α_V is negligible, but all remaining effects need to be accounted for to accurately model CP.

A large number of models were investigated in a similar manner in EHTC M87 IX, with one snapshot from each model that passed all polarimetric constraints, testing the relative effects of j_V , ρ_V , and ρ_Q . While the definitions of 〈∣v∣〉 in this paper, and 〈∣v∣〉 in EHTC M87 IX, differ by a Gaussian blur of 20 μas in the latter, the results are consistent, in that the contribution of j_V is subdominant compared to ρ_Q .

The optical depths, magnetic field strength, and thus CP production mechanisms vary greatly across models. Combined with the nontrivial effect of including the reversed-field distributions, it is not possible to formulate a universally applicable, simple model for CP that is valid across all parameters. By comparing the aligned- and reversed-field distributions, however, we can understand whether each model's CP is produced via ${\boldsymbol{B}}$ field-polarity-invariant pathways (Faraday conversion through field twist) or noninvariant pathways (intrinsic emission or Faraday conversion through Faraday rotation).

Convergence tests of distributions of v_net are given in Appendix B—the v_net distributions appear converged with respect to GRMHD resolution and GRRT image resolution. Figures 5–6 show the v_net distributions (aligned- and reversed-field) of Sgr A* for all models in the library. M87* distributions are given in Appendix C. The first and second moments of each distribution are given in Appendix D.

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Comparing the MAD and SANE models, we see that the SANE models have higher v_net, particularly for lower-R_high values. For the MAD models, almost all models have ∣v_net∣ < 2%, whereas the SANE models have snapshots with ∣v_net∣ > 4%. Given the low detected values of CP for Sgr A* and M87*, some of our SANE models can be ruled out. The MAD models also exhibit cleaner trends across model parameters, whereas the SANE (especially low-R_high) models are more turbulent. Ricarte et al. (2021; see Figures 8, 9, and 13) also find that SANE models have higher Faraday depths than MADs. Higher Faraday depth implies more scrambling of LP, and thus, to the extent that Faraday conversion is important, scrambling of CP. This scrambling hides imprints of the magnetic field in CP measurements of SANEs compared to MADs.

The effect of field reversal is mixed. For some SANE models, v_net is nearly antisymmetric (spin 0, R_high = 10), while for a few models v_net is nearly symmetric (spin 0, R_high = 1). A majority of the SANE models, and all of the MAD models, show imperfect symmetry/antisymmetry under field reversal, indicating that both magnetic field twist and Faraday rotation + intrinsic emission contribute significantly.

In Sgr A*, CP is almost 0 for edge-on models. This can be attributed to the cancellation that occurs across every image due to symmetries in the magnetic field geometry (see Ricarte et al. 2021; Tsunetoe et al. 2021 for detailed descriptions). Edge-on models tend to have higher Faraday depths, which also contribute to the increased cancellation of CP across the image. We find that the Faraday rotation depths for SANE models are 2 orders of magnitude higher than for the corresponding MAD models. Faraday depth is a strong function of R_high (increases), spin (decreases from retrograde to prograde), and inclination (increases until 90°). R_high and spin directly influence the temperature of the electrons, and models with hotter electrons have lower Faraday depths.

5.1. Parameter Dependence of CP Distributions

Here, we focus on parameters that can influence v_net across all models, such as spin, inclination, R_high, and frequency.

5.1.1. Spin Dependence

The black hole spin influences the sign and shape of the v_net distributions. Prograde spin model snapshots contain components with opposite signs of v_net, i.e., more spatial cancellation than in low-spin models. As a result, high-spin prograde model distributions of v_net are broader than low-spin models.

An image in Stokes I (or V) can be divided into the weakly lensed component (n = 0) and a strongly lensed component, where photons wrap around the black hole in n half circles (n = 1, 2, 3...; see Johnson et al. 2020). Each ring n is $\sim \exp (-\pi )$ fainter than the next. In this work, we see effects primarily from the n = 0 and n = 1 images.

For face-on prograde MAD models, the opposite-signed n = 0 and n = 1 portions of the image become important, with the n = 1 ring becoming the dominant source of fractional CP, as seen in Figure 7, which shows average images of CP for a MAD model across spin. The n = 0 mode of the CP image has contributions from intrinsic emission and Faraday conversion, both through Faraday rotation and twist. In the Faraday-thin and optically thin regions with toroidal magnetic fields, the n = 1 photon ring is the opposite sign of the n = 0 component. The opposite sign of the photon ring is a consequence of Faraday conversion through the twist of the magnetic field, as explored in Ricarte et al. (2021). The field structure of retrograde models is less toroidal and thus the n = 1 contribution is reduced. This effect is also reduced as the Faraday thickness increases and thus is less prominent in retrograde and SANE models. While the sign flip in the n = 1 photon ring is prominent at face-on inclinations, this effect of prograde models having opposite-signed v_net distributions is also seen at higher inclinations, although it is uncertain if the same phenomenon is responsible. It is possible that as the spin increases, the contribution of the n = 0 portion of the image decreases, but further investigation would be necessary.

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5.1.2. Inclination Dependence

Images at observer inclination θ ≠ 90° typically contain contributions from the near- and farside regions (z > 0 and z < 0, where z = 0 is the midplane of the system). For both of these regions, ${\boldsymbol{k}}\cdot {\boldsymbol{B}}$ is important, along with the direction of twist of ${\boldsymbol{B}}$ along the geodesic. At low inclinations, photons from the farside region have larger optical and Faraday depths, from gravitational lensing increasing the path length. For higher-R_high models, the cool disk may also increase the Faraday depth for farside photons traveling through it. As a result, the farside contribution can be scrambled or even change sign in certain regions, but it is unlikely to exactly cancel out, or surpass, the nearside component, thus generating a net v_net biased toward the near-field component.

For inclinations θ < 90° and aligned magnetic fields, MAD models on average have positive v_net (ignoring photon ring effects). In the nearside region, ${\boldsymbol{k}}\cdot {\boldsymbol{B}}\gt 0$, so j_V > 0 and ρ_V > 0. For Faraday-thin regions, ρ_V > 0 increases Stokes U, which in turn increases Stokes V via conversion (see Equation (3)). Faraday conversion through the twist in magnetic fields also contributes toward v_net > 0; Ricarte et al. (2021) demonstrate this for a face-on model by considering the twist of ${\boldsymbol{B}}$ along the jet as it (the jet) broadens out (denoted as the vertical twist ξ_V ). The vertical twist ξ_V (and the Stokes basis rotation) is clockwise along the line of sight, which corresponds to a rotation of Q > 0 to U > 0 and thus V > 0 as ρ_Q > 0. Another form of twist explored in Enßlin (2003) and Ricarte et al. (2021), the transverse twist (ξ_T ), occurs for edge-on models. For disks rotating counterclockwise in the sky (θ < 90°), trailing magnetic fields embedded in the disk will be twisted counterclockwise along the line of sight (for the approaching jet), leading to V < 0. As this effect is maximal for edge-on simulations, its imprint on inclination compared to that of the vertical twist is minimal.

While it is difficult to ascertain the sense of twist of ${\boldsymbol{B}}$ for intermediate observer inclinations, one can expect the effect of twist along the jet to reduce as the observer inclination increases. Due to symmetries in the global magnetic field across the midplane, as θ → 90°, any additional effects of twist should cancel out. Also, a complementary inclination angle will reverse ${\boldsymbol{k}}\cdot {\boldsymbol{B}}$ and the twist of ${\boldsymbol{B}}$, thus reversing the sign of CP. Thus, on average, the v_net distributions should follow a cosine function, but fluctuations will prevent individual snapshots from following a neat pattern.

From Figures 5 and 6, we see that this is the case for the aligned- and reversed-field models. The universality of this behavior across all models suggests that the mean across field configurations encodes the sense of twist of the magnetic field, since CP generated from the conversion of LP via a twisted magnetic field is the only mechanism invariant to the sign of ${\boldsymbol{B}}$.

A mean positive v_net thus implies an overall clockwise twist in ${\boldsymbol{B}}$ along ${\boldsymbol{k}}$ for both retrograde and nonspinning models with θ < 90° (where the n = 0 contribution is dominant). It could also imply an overall counterclockwise twist for prograde models with θ > 90°, only in cases where the n = 1 contribution is greater than the n = 0 component, which is most a + 0.94 models and a few a + 0.5 models.

A mean negative v_net implies the same, but with the corresponding opposite sense of twist and observer inclinations. Thus, we find that the sign of the mean v_net is sensitive to not only the global sign of the magnetic field, but also the sense of rotation of the accretion flow.

5.1.3.  R_high Dependence

As mentioned in Section 3.3, a higher R_high implies cooler electrons in the disk. This causes less emission from the midplane of the disk. For higher R_high, a higher mass unit (${ \mathcal M }$) is required to obtain the same output flux as that from a lower-R_high-value simulation, and this implies higher density and therefore an increase in all the radiative transfer coefficients. This causes two important effects. One is an increase in CP for individual pixels, because of increased Faraday conversion and emission. The other is a decrease in the overall CP, arising from an increase in depolarization: scrambled LP from Faraday rotation leads to scrambled CP through Faraday conversion. The presence of a cooler, denser population of electrons in the disk midplane is particularly important for increasing Faraday rotation.

The overall effect is that v_net (unresolved CP) distributions broaden with increasing R_high, and 〈∣v∣〉 (resolved CP) distributions increase in magnitude with R_high. The distributions of v_net and 〈∣v∣〉 for different values of the R_high and M87* parameters are given in Figure 8. Although it is difficult to observe any trend in v_net (likely due to cancellations), a clear increase in 〈∣v∣〉 with R_high is seen. As this effect is seen for most spins, we conclude that resolved observations of Stokes V (〈∣v∣〉) in the future can constrain electron temperature models. This is consistent with the results found in Mościbrodzka et al. (2021).

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Models with R_high = 1 are qualitatively different in v_net, especially in the SANE models, as these models have hot, dense disks that dominate the emission in a relatively concentrated region (in the poloidal direction) in the midplane. Higher-R_high models, in comparison, require a higher scaling factor to obtain the same flux. This causes more contributions from both near and far jet-sheath regions, which combined with a cooler midplane can increase the Faraday depths and emission regions.

5.1.4. Frequency Dependence

While not a parameter explored in the image library, we investigate v_net of a few models versus frequency, shown in Figure 9 for three models: (MAD a+0.5, R_high 160, inclination 30°), (MAD a+0.94, R_high 1, inclination 30°), and (SANE, a0, R_high 40, inclination 130°). We find three different spectral behaviors of Stokes V, which suggest that different mechanisms are dominant in each model. The SANE model, while optically thin, remains Faraday-thick. Thus, even at higher frequencies, when probing inner regions of the accretion flow, the Stokes V signal does not significantly decrease. The MAD models are Faraday-thin near 230 GHz, so Stokes V reduces as frequency increases and the R_high = 160 model decreases faster compared to the R_high = 1 model. From the distributions of v_net, we find that the MAD R_high 160 model does not neatly change sign with ${\boldsymbol{B}}$ reversal, whereas the R_high 40 model does. This suggests that Faraday conversion (through twisted magnetic fields) is important to the former (high-R_high model), but not the latter (lower R_high), where intrinsic emission or Faraday conversion through rotation dominate.

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In Appendix A, we analyze the analytic solution to Stokes V for a single geodesic and find that Stokes V from just Faraday conversion decreases much faster with increasing frequency than Stokes V from intrinsic emission. Qualitatively, comparing these analytic results (Figure 13 in Appendix A) to the numerical models (Figure 9) suggests that a steep ν⁻² scaling in Stokes V, as seen in the MAD a+0.5, R_high 160 model, points to a Faraday-conversion-dominated model and a slow ν⁻¹ scaling, as seen in the MAD a+0.94, R_high 1 model, is intrinsic-emission-dominated. The frequency scaling estimates are heuristic and an extensive frequency analysis across all models is needed for direct comparison with observations.

5.2. Fits to v_net Distributions

The mean and standard deviations of all v_net distributions are provided in Appendix D. Likely because of the mix of physical processes that contribute to CP, it is difficult to provide a simple fit to these moments that covers the entire model space. The Sgr A* MAD models, however, exhibit clear trends. The mean is readily fit by

$$\begin{eqnarray}&&\langle {v}_{\mathrm{net}}\rangle =0.61 \% \,\cos (\theta )\left(1-\frac{0.64}{{R}_{\mathrm{high}}}\right)\left(S-\frac{{\left(1+a\right)}^{2}}{3}+1\right),\end{eqnarray} \tag{ 17 }$$

which we extracted using the PySR symbolic regression code (Cranmer 2020; Cranmer et al. 2020). Here, θ is the observer inclination and S is the sense of the magnetic field (1 for aligned and −1 for reversed cases). The fitting function recovers a cosine dependence on the observer inclination, which we attribute to the overall switch in sign from viewing at opposite poles, and increasing cancellations when observing edge-on, due to the symmetries of the magnetic field.

The mean v_net and the fit (Equation (17)) are shown in Figure 10. The fit accurately measures the mean v_net of roughly 80% of the models to within 0.3% (absolute difference). The fits for the means and standard deviations of the SANE models do not permit such an accurate fit, but the raw data are provided in Appendix D.

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6.1. Sgr A*

Which models produce CP that matches Sgr A* and M87*? The detections and limits of v_net for M87* and Sgr A* are given in Table 1. For Sgr A*, the observations from ALMA given in Bower et al. (2018) and Wielgus et al. (2022b) are used as a constraint on the CP (8 data points, assumed to be uncorrelated, ranging from −1.5% to −0.92%). To test the models, a procedure similar to that in EHTC SgrA V is used. Two-sided K-S test p-values (p) are computed between distributions of the model and observations. The model is sampled every 400M to obtain an approximately uncorrelated sample. The model fails the constraint if both the aligned- and reversed-field distribution give p < 0.01, giving 99% confidence in rejecting the null hypothesis that the model and observations arose from the same distribution. The constraint plots are shown in Figure 11.

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We have 27/180 = 15% of the MAD models pass and 43/180 = 24% of the SANE models pass the CP constraint. The SANE models are less constrained overall than the MAD models. This can be attributed to the higher fractional CP generated in general, increasing the chance that one combination of flow orientation and field configuration will produce sufficiently negative CP to pass the constraint.

All the best-bet models given in EHTC SgrA V fail the CP constraint. These are prograde MAD models with high R_high at a low inclination. This can be attributed to the effect of black hole spin on prograde models, as mentioned in Section 5—cancellation between the n = 1 photon ring and the n = 0 "weakly lensed" emission shifts the distributions toward the center, while the observations are predominantly around the −1% level.

Given that θ > 90° represents a clockwise rotation of the disk in the sky (beyond the ergosphere for retrograde models), there is a clear preference for retrograde MAD models in which the flow orientation is clockwise in the sky. All these models pass when the ${\boldsymbol{B}}$ field is oriented parallel to the disk angular momentum vector (aligned field). This is because the photon ring has a negligible effect and the overall n = 0 emission (in this configuration) is negative. These findings are consistent with the GRAVITY measurements of the orientation of the flow for Sgr A* based on the motion of NIR flares (GRAVITY Collaboration et al. 2018).

For the one passing prograde MAD model, the field is in the reversed configuration, meaning that all the passing MAD models have the dipole moment of the field pointed away from the observer. Given the combination of inclinations and field orientations, we find that the v_net constraint (on MADs) is sensitive to both the sense of rotation of the flow (as proposed and explored in Enßlin 2003 and Mościbrodzka et al. 2021, respectively) and the overall direction or structure of the magnetic field configuration (Beckert & Falcke 2002). If the constraint were insensitive to the sense of rotation of the flow, then models with θ < 90° would pass, and if it were insensitive to the direction of the magnetic field, then both aligned and reversed fields for all passing MAD models would pass.

All the MAD edge-on inclination models fail, and only one of the SANE edge-on inclination models passes the CP constraint. This is because of cancellations that occur across the image domain, due to symmetries in the magnetic field structure for each of the models (Ricarte et al. 2021), yielding a net-zero CP fraction.

Combined with the constraints in EHTC SgrA V, the CP test eliminates all models. The best-bet prograde MAD models do not produce sufficient CP, whereas the retrograde MADs (which pass CP constraints) are mostly eliminated from the m-ring constraints, which compare the ring width, asymmetry, and diameter of the model to the observed image. Most of the SANEs fail the m-ring and non-EHT (multiwavelength) constraints.

6.2. M87*

Here we discuss certain limitations of the theoretical models and provide some expectations for future improvements. As these models are the same underlying GRMHD simulations used in the KHARMA library in EHTC SgrA V, most of the caveats discussed in that paper apply here. Given the diversity of the model parameters, it is difficult to predict the effect of an improvement on an entire library of simulations, and thus each of the suggestions merits a separate analysis that is beyond the scope of the paper.

The primary caveat is that the models are highly variable in Stokes I and V when compared to Sgr A*. While v_net for Sgr A* is consistently observed at the percent level, frequent sign crossings are observed for most models, due to turbulence and rapidly changing optical and Faraday depths. Comparing the mean of distributions could provide a more robust method of comparing simulations to observations (EHTC SgrA V; Wielgus et al. 2022a, 2024). However, for our models, the sign crossings invariably produce mean v_net values lower than the percent level. Improvements to physics in GRMHD models, such as including self-consistent electron heating and cooling mechanisms or the addition of leading-order collisionless corrections, such as viscosity and heat conduction, as given in Chandra et al. (2015), can potentially reduce the fluctuations of Stokes V in the models and thus shift the v_net distributions away from 0.

The initial condition of the current GRMHD library is an equilibrium Fishbone–Moncrief torus solution (Fishbone & Moncrief 1976) seeded with a magnetic field. Most of the emission regions for such simulations occur within 20M, with the time period chosen so as to allow these regions to reach a steady-state solution. However, alternative initial conditions, such as stellar-wind-fed models (Ressler et al. 2020), can yield qualitatively different results in all the Stokes images. Magnetic fields can greatly influence the structure of the CP image. Since the magnetization of the stellar winds is poorly constrained, qualitative features of the polarimetric image are sensitive to the choice of plasma β and further studies are necessary. For higher magnetizations of the stellar wind, the simulations can be expected to settle into a MAD state and show similar properties to the corresponding MAD simulation with similar inclinations and electron temperatures in our library.

Changes to GRMHD fluid parameters can influence the resulting v_net distributions. In a newer set of GRMHD simulations generated using KHARMA (referred to as "v5" as opposed to the current "v3"), the simulations are run with a different adiabatic index (5/3 instead of 4/3) for the fluid, different GRMHD floor prescriptions, a higher resolution (384 × 192 × 192 compared to 288 × 128 × 128 in v3), and are run out until 50 × 10³ GM/c³ in time. A comparison of the same GRRT model parameters between this simulation and the simulation used in the paper is given in Figure 12 (the same model as in Section 4). A two-sample K-S test cannot distinguish the newer simulation distribution from the one used in this paper, and both perform similarly when compared to observational data of Sgr A*, yet this inference cannot be applied to the full library of simulations. The characteristics of the new library, containing densely sampled black hole spins, will be discussed in a later paper.

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Electron–positron plasmas have the capacity to greatly influence the morphology of the Stokes V image without affecting the Stokes I image significantly. With an equal proportion of positrons and electrons, both intrinsic emission and Faraday conversion through Faraday rotation vanish and the resultant CP will encode conversion through the sign of twist (Wardle et al. 1998; Anantua et al. 2020; Emami et al. 2021).

The eDF was chosen to be a relativistic Maxwell–Jüttner distribution. Based on observations of the solar wind and simulations of collisionless plasma simulations via particle-in-cell codes, a power-law tail to the distribution can be modeled—the so-called κ distribution (Kunz et al. 2015 and references therein). Nonthermal eDFs introduce hotter electrons that influence all the radiative transfer coefficients, but a systematic study of the effects on CP is still needed.

The R_high prescription (Mościbrodzka et al. 2016) used to assign electron temperatures in our models is a phenomenological model. The R_high model defines the electron temperatures as a particular function of plasma β. Another parameterization of the accretion flow is the critical beta model (Anantua et al. 2020), which differs in that the electron temperatures approach 0 instead of 1/R_high at high plasma β, in the midplane. Colder electrons in the midplane will enhance Faraday rotation while suppressing intrinsic emission and thus reduce v_net in the images.

As the observed v_net for Sgr A* seem to lie squarely around the −1% value across many decades, the potential effects of an external Faraday screen should be investigated (though recently argued against by Wielgus et al. 2024). Faraday conversion and intrinsic emission is heavily suppressed in cold plasmas compared to Faraday rotation, thus the existence of a screen should not greatly affect the v_net measurements. It is possible, however, for the screen to undergo field reversals along the photon trajectory, in which case Faraday conversion can dominate in a small region where the field is perpendicular to the photon trajectory. Gruzinov & Levin (2019) investigate the effect of such field reversals in cold plasma and find that the resulting CP oscillates quasiperiodically as a function of λ². Since the observations of Sgr A* do not seem to show this oscillation, we may assume that such effects are subdominant in observations of Sgr A*. For M87*, CP(λ) is not yet observed.

While ∼90% of the emission up to 100 mas arises from horizon scales (EHTC SgrA II), emission from large-scale structures between 100 mas and 1 as is not as well constrained. The lack of a conclusively observed jet from Sgr A* at lower frequencies (EHTC SgrA II and references therein) also suggests that any extended emission is of low intensity. However, it is possible that a highly polarized, low-intensity source in this region could influence Stokes V measurements and offset a signal arising from the source. Future improvements to the global VLBI array can improve the limits on extended structure (Raymond et al. 2021).

In this paper, we investigate the CP of simulated images of Sgr A* and M87*, focusing on the image-integrated CP (v_net). We explore a library of GRMHD models spanning different black hole spins and accretion disk magnetic states (MAD and SANE). Ray-traced images of the GRMHD models span different electron temperatures, observer inclinations, and both aligned and reversed polarities of the global magnetic field. To understand the properties of v_net, we first focus on one MAD a+0.5, R_high 160, inclination 30° model. We then plot v_net distributions across an entire model library and find trends with respect to the library parameters along with fitting functions for the Sgr A* MAD models. Models of Sgr A* are constrained by performing a K-S test between simulations and unresolved ALMA observations of Sgr A*. We find the following results:

• 1.  
  Field reversal does not flip v_net distributions, as there exist both symmetric and antisymmetric terms in the radiative transfer equation (Equation (4)). The relative contributions of these terms can vary greatly. Models with symmetric (antisymmetric) terms dominating the equation can have nearly symmetric (antisymmetric) distributions of v_net. In practice, however, most models contain contributions from both terms.
• 2.  
  Large cancellations occur both spatially and temporally in CP due to turbulent fluctuations and symmetries in the magnetic field. Average images and means of v_net distributions can smooth over fluctuations and probe the structure of magnetic fields. The sense of twist of the magnetic field is encoded in the mean of the v_net distributions (when averaged over the field configuration). Inclinations <90° encode an overall clockwise twist of field (positive v_net).
• 3.  
  SANE models produce more CP on average than MADs. For Sgr A*, nearly all the MAD models lie within ∣v_net∣ < 2%, whereas for SANEs this is ∣v_net∣ ≲ 5%.
• 4.  
  When compared to v_net ALMA measurements of Sgr A* via a K-S test, the MAD models that pass contain ${\boldsymbol{B}}$ pointing away from the observer. All but one of the passing models are clockwise in the sky, in agreement with the direction of the putative orbital motion reported by GRAVITY Collaboration et al. (2018) and Wielgus et al. (2022b). SANE models being more variable with larger v_net tend to pass without a clear trend. None of the best-bet models survive the v_net constraint, as most of the MAD models exhibit sign changes in v_net, unlike observations, which lie closely around the −1% region.
• 5.  
  Edge-on models produce v_net ≈ 0, due to symmetries in the magnetic field structure, and are thus disfavored for Sgr A*.
• 6.  
  The black hole spin influences the v_net distribution of a model. High-spin prograde models appear to contain imprints of the photon ring with the opposite sign of CP compared to the weakly lensed component, causing v_net in prograde models to be centered closer to 0 or even have the opposite sign, compared to otherwise similar retrograde models.
• 7.  
  Electron temperature assignment can be constrained by future observations of the resolved CP measurement, 〈∣v∣〉. Higher-R_high models or colder electrons in the disk will have higher 〈∣v∣〉 values with v_net weakly affected.

Overall, we find that while CP in radiatively inefficient accretion flows can be complicated, there are interesting trends and properties with respect to model parameters. The GRMHD models appear to be highly variable in CP, with frequent sign crossings in v_net. The current constraints of v_net for Sgr A* seem to highlight the global direction of the magnetic field and the sense of rotation of the flow. Since v_net observations are possible for point sources, observational data from targets besides Sgr A* and M87* can also be used to infer model properties.

The authors thank Monika Mościbrodzka, Angelo Ricarte, and the anonymous EHT internal referee for valuable comments that improved the manuscript.

This work was supported by NSF grants AST 17-16327 (Horizon), OISE 17-43747, and AST 20-34306. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. This research was done using services provided by the OSG Consortium, which is supported by the National Science Foundation awards #2030508 and #1836650. This research is part of the Delta research computing project, which is supported by the National Science Foundation (award OCI 2005572) and the State of Illinois. Delta is a joint effort of the University of Illinois at Urbana–Champaign and its National Center for Supercomputing Applications. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.00764.S, ADS/JAO.ALMA#2016.1.01154.V, and ADS/JAO.ALMA#2016.1.01404.V. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. C.F.G. was supported by the IBM Einstein Fellow Fund at the Institute for Advanced Study. M.W. acknowledges the support from the European Research Council advanced grant "M2FINDERS—Mapping Magnetic Fields with INterferometry Down to Event hoRizon Scales" (grant No. 101018682).

In the optically/Faraday-thin limit, we can approximate the contributing terms (intrinsic emission and Faraday conversion) to Stokes V from the full solution of the radiative transfer equation (Equation (3)) as follows:

$$\begin{eqnarray}&&{V}_{\mathrm{emission}}\approx {j}_{V}L,\end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray}&&{V}_{\mathrm{conversion}}\approx {\rho }_{Q}{\rho }_{V}{j}_{Q}{L}^{3},\end{eqnarray} \tag{ A2 }$$

where Stokes V from Faraday conversion in a uniform field geometry only arises from Faraday conversion of linearly polarized light rotated from Q to U. L is a characteristic length scale, which we set to the radius of the one-zone sphere L = 5GM/c². Thus, Equations (A1) and (A2) can be used as proxies to probe the general spectral behavior of the components to the full solution of Stokes V.

Figure 13 shows the spectral behavior of the two CP mechanisms along with their approximations in the Faraday-thin limit using one-zone models of Sgr A*. The Stokes V_emission (V_conversion) solution is obtained by setting ρ_Q = 0 (j_V = α_V = 0), respectively. The approximations have the same scaling behavior as the full solutions, with the CP from Faraday conversion decreasing much faster with frequency than intrinsic emission.

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When comparing with numerical models, we see the same qualitative behavior: models in which intrinsic emission is expected to dominate show a slower decrease in frequency compared to conversion-dominated models. The spectral slopes of Stokes V are completely different, but this is not unexpected, given the nontrivial field configuration of the simulations and the integration across many different geodesics (compared to a single geodesic in the analytic solution).

Before making estimates and predictions from the surveyed data, it should be tested that the distributions computed from the EHT imaging library have converged to the true distribution of the process. Here, we investigate the convergence properties of the models with respect to resolution in time, GRMHD, and GRRT modeling.

Have the distributions converged, i.e., has the library been imaged over a long enough interval? Figure 14 displays the effect of increasing the simulation time for our sample MAD, spin +0.5, R_high 160, inclination 30° model. While the mean of the distribution is not fully settled, the changes are within 10%: from 5 to 15 kM time intervals, in increments of 3 kM, the mean v_net changes as 0.51%, 0.78%, 0.67%, 0.79%, and 0.82%, respectively. The distributions also become more unimodal as the number of independent samples increases. The correlation time of this model is about 400M, implying about 38 independent samples and a standard error of $\sigma /\sqrt{n}\approx 0.13 \%$.

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The distributions might have encoded features dependent on the resolution of the GRMHD simulations. Figure 15 shows that this is not the case for a MAD spin +0.94 model, as the resolution does not drastically affect the distribution. The GRMHD resolution used for the EHT imaging in EHTC SgrA V and this paper was 288 × 128 × 128. The fluctuations in the distributions are likely due to limited time sampling of the model (both in cadence and length).

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v_net is a metric that is independent of the image (GRRT) resolution past a certain threshold that represents the scale of critical structures in the image. We generate a snapshot for a MAD spin +0.94 model at resolutions 0.25, 0.5, 1, 2, 4, 8, and 16 μas pixel⁻¹ and find the v_net values to be 0.361%, 0.358%, 0.340%, 0.384%, 0.476%, 0.447%, and 0.335%, respectively. Downsampling the model library (0.5 μas pixel⁻¹) across all parameters shows similar trends, suggesting that the v_net measurements of GRRT models usually vary within 0.1% across image resolution until 8 μas pixel⁻¹, making the existing image resolution of the library sufficient.

Figures 16 and 17 show v_net distributions for the M87* library parameters. As the observer inclination for the M87* library is fixed to the inclination angle of the forward jet, these M87* distributions are mostly a subset of the parameters used for Sgr A*, aside from a denser sampling of R_high. Thus, inferences on the parameter trends based on the Sgr A* distributions also apply to M87*: SANEs produce more v_net than the MAD models, the shift of the v_net distributions goes toward 0 as the spin goes from negative to positive, and an increase in R_high increases and broadens the v_net distributions.

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Here, we provide tables of the first two moments of the distributions for v_net (%) for both Sgr A* and M87* in Table 2.

Table 2. Mean and Standard Deviation for v_net (Fractional CP, %) for all GRMHD Models for SgrA^* and M87^*

Source	Flux	a	Rhigh	θ	Mean vnet	Std vnet	Mean vnet (Rev ${\boldsymbol{B}}$)	Std. vnet (Rev ${\boldsymbol{B}}$)
SgrA*	SANE	−0.94	1	10	0.62	2.80	−0.15	2.85
SgrA*	SANE	−0.94	1	30	−0.71	1.32	−1.00	1.36
SgrA*	SANE	−0.94	1	50	−1.61	1.96	−1.96	2.01
SgrA*	SANE	−0.94	1	70	−1.66	2.38	−2.16	2.49
...		...			...	...	...	...
M87*	MAD	+0.94	40	163	0.09	0.36	0.45	0.23
M87*	MAD	+0.94	80	163	0.08	0.44	0.45	0.26
M87*	MAD	+0.94	160	163	0.13	0.48	0.37	0.33

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  Machine-readable (MRT)Typeset image