First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole

We model the plasma flow around Sgr A* using ideal, nonradiative GRMHD in the Kerr metric, with a_* a free parameter (see, e.g., Koide et al. 1999; Komissarov 2001; Gammie et al. 2003; De Villiers & Hawley 2003; Anninos et al.2005; Del Zanna et al. 2007).

We integrate the GRMHD equations in three spatial dimensions using multiple algorithms: KHARMA (Prather et al. 2021), BHAC (Porth et al. 2017; Olivares et al. 2019), H-AMR (Liska et al. 2019), koral (Sądowski et al. 2013), and Athena++ (White et al. 2016); see Porth et al. (2019) and H. Olivares et al. (2022 in preparation) for comparisons of GRMHD codes. All simulations assume constant adiabatic index Γ_ad.

Unless stated otherwise, the initial conditions for the GRMHD simulations are constant angular momentum hydrodynamic equilibrium tori (Fishbone & Moncrief 1976), with orbital angular momentum that is parallel or antiparallel to the black hole spin. The tori are seeded with a weak, poloidal magnetic field. The simulations use varying torus pressure maximum radius (from ∼ 15 GM/c² to 40 GM/c²), peak temperature, adiabatic index, and initial field configurations. These variations permit us to test the robustness of our results (see Appendix A).

The torus initial conditions are motivated by the notion that the near-horizon flow, where most of the emission is generated (M87* Paper V), relaxes to a statistically steady state that is nearly independent of the flow at larger radius. This notion is challenged in the stellar-wind-fed models of Ressler et al. (2020b), which are included in our study.

All simulations are run in Kerr−Schild-like coordinates, which are regular on the horizon. Unless stated otherwise, boundary conditions are outflow at the inner boundary, which is located inside the event horizon, and outflow at the outer boundary, which is located at ${r}_{\max }\geqslant 1000\,{GM}/{c}^{2}$. Most simulations are evolved to t_final ≥ 30,000 GM/c³.

Once the evolution has started, the magnetorotational instability (MRI; Balbus & Hawley 1992) and possibly other instabilities, such as, for MAD models, magnetic Rayleigh–Taylor instabilities (Marshall et al. 2018), drive the torus to a turbulent, fluctuating state. Defining P_gas ≡ gas pressure and P_mag ≡ B²/(8π) ≡ magnetic pressure, the standard accretion flow models can be divided by latitude into three zones: (i) an equatorial inflow, (ii) a midlatitude disk wind/corona with β ≡ P_gas/P_mag ∼ 1, and (iii) a polar "funnel" with σ ≡ B²/(4π ρ c²) ≫ 1.

The magnetic flux through the horizon, characterized by $\phi \equiv {{\rm{\Phi }}}_{\mathrm{BH}}{(\dot{M}{r}_{{\rm{g}}}^{2}c)}^{-1/2}$ (Φ_BH ≡ magnetic flux interior to the black hole equator, $\dot{M}\equiv$ mass accretion rate), divides the outcome into two states: the MAD state (e.g., Bisnovatyi-Kogan & Ruzmaikin 1974; Igumenshchev et al. 2003; Narayan et al. 2003; Tchekhovskoy et al. 2011) and the SANE state (e.g., De Villiers et al. 2003; Gammie et al. 2003; Narayan et al. 2012). MAD models have ϕ ∼ ϕ_crit ∼ 60. ¹⁵⁵ In MAD models, magnetic flux accretes onto the hole until ϕ ≳ ϕ_crit. Accretion of additional flux leads to flux expulsion events so that the flow maintains ϕ ∼ ϕ_crit. Our SANE models, in contrast, typically have ϕ ∼ 1.

We consider two GRMHD simulations with initial conditions that differ from the fiducial aligned torus: strongly magnetized non-MAD tilted torus simulations (Liska et al. 2018; Chatterjee et al. 2020), and a simulation in which Sgr A* is fed directly by winds from stars in its vicinity (Ressler et al. 2020b). The wind-fed simulations result in a mode of accretion that is similar to MAD but typically has lower mean angular momentum and is less well organized. The wind-fed models have a_* = 0.

The GRMHD simulation library is summarized in Table 1. Figure 1 shows a few examples of GRMHD simulations for an aligned SANE, an aligned MAD, a tilted torus, and a wind-fed simulation. These simulations vary in numerical method and in numerical resolution. We present more information on numerical methods in Appendices A and B.

Zoom In Zoom Out Reset image size

Table 1. EHT GRMHD Simulation Library

Setup	Code	a*	Mode	Γad	tfinal	rout	Resolution
Torus	KHARMA a	0, ± 0.5, ± 0.94	MAD/SANE	$\tfrac{4}{3}/\tfrac{4}{3}$	30,000	1000	288 × 128 × 128
Torus	BHAC b	0, ± 0.5, ± 0.94	MAD/SANE	$\tfrac{4}{3}/\tfrac{4}{3}$	30,000	3,333	512 × 192 × 192
Torus	H-AMR c	0, ± 0.5, ± 0.94	MAD/SANE	$\tfrac{13}{9}/\tfrac{5}{3}$	35,000	1000/200	348/240 × 192 × 192
Torus	koral d	0, ± 0.3, ± 0.5, ± 0.7, ± 0.9	MAD	$\tfrac{13}{9}$	101,000	100,000	288 × 192 × 144
Tilted	H-AMR e	0.94	SANE f	$\tfrac{5}{3}$	105,000	100,000	448 × 144 × 240
Wind-fed	Athena++ g	0	MAD	$\tfrac{5}{3}$	20,000	2,400	356 × 128 × 128

Notes. Summary of the EHT Sgr A* GRMHD simulation library. The last column is N₁ × N₂ × N₃, with coordinate x₁ monotonic in radius, x₂ monotonic in colatitude θ, and x₃ proportional to longitude ϕ. The first four entries use aligned torus initial conditions. The last two entries are tilted accretion models and two realizations of the wind-fed accretion model that differ in stellar wind magnetization. Times are given in units of GM/c³ = 20.4 s and radii in units of GM/c².

^a See Prather et al. (2021); KHARMA is a GPU-enabled version of the iharm3d code. ^b Porth et al. (2017); Olivares et al. (2019); Mizuno et al. (2021); Cruz-Osorio et al. (2021). ^c Liska et al. (2019) ^d Narayan et al. (2022). ^e Chatterjee et al. (2020). ^f ϕ/ϕ_crit ≃ 0.8. ^g White et al. (2016); Ressler et al. (2020b).

Download table as:  ASCIITypeset image

The gas temperature profile is a critical feature of the GRMHD simulations. Figure 2 shows the time- and azimuth-averaged profiles of the midplane dimensionless gas temperature P/(ρ c²) in a set of aligned GRMHD simulations. The temperature profiles exhibit trends with spin and magnetic state (MAD or SANE) that drive many of the trends seen in the radiative models: MAD models are a factor of several hotter than SANE models, and both MAD and SANE become hotter as a_* increases.

Zoom In Zoom Out Reset image size

Synthetic images are generated from the GRMHD simulations in a radiative transfer step. The transfer step requires (i) a model for the eDF, (ii) assignment of a density scale to the GRMHD simulation, (iii) the inclination i (angle between the torus angular momentum and the line of sight), and (iv) a numerical integration performed as a post-processing step that assumes that the plasma evolution is unaffected by radiation.

Electron Distribution Function.—Thermal models have electron energies distributed according to the Maxwell–Jüttner distribution function:

$$\begin{eqnarray}&&\frac{1}{{n}_{e}}\frac{{{dn}}_{e}}{d\gamma }=\frac{{\gamma }^{2}\sqrt{1-1/{\gamma }^{2}}}{{{\rm{\Theta }}}_{e}{K}_{2}(1/{{\rm{\Theta }}}_{e})}\exp \left(-\frac{\gamma }{{{\rm{\Theta }}}_{e}}\right),\end{eqnarray} \tag{ 11 }$$

where K₂ is a modified Bessel function of the second kind and γ is the electron Lorentz factor. Recall Θ_e = k_B T_e /(m_e c²), which is determined by the ion−electron temperature ratio R ≡ T_i /T_e :

$$\begin{eqnarray}&&{T}_{e}=\displaystyle \frac{2{m}_{p}u}{3{k}_{{\rm{B}}}\rho (2+R)}.\end{eqnarray} \tag{ 12 }$$

Here u and ρ are the internal energy density and rest-mass density from the GRMHD simulation, respectively, and we have assumed that the ions are nonrelativistic with adiabatic index 5/3 and the electrons are relativistic with adiabatic index 4/3. Thermal models are motivated by the idea that wave-particle scattering drives partial relaxation of the eDF, even though Coulomb scattering is ineffective.

The temperature ratio depends on a balance between microphysical dissipation, radiative cooling, and fluid transport. Models for collisionless dissipation vary widely in their predictions for the ratio of heat deposited in ions and electrons but depend most strongly on the local magnetic field strength. This motivates a prescription in which the temperature ratio depends solely on the plasma β ≡ P_gas/P_mag (Chan et al. 2015b). We adopt the same model as M87* Paper V and Event Horizon Telescope Collaboration et al. 2021a, hereafter M87* Paper VIII, where

$$\begin{eqnarray}&&R=\displaystyle \frac{{T}_{i}}{{T}_{e}}={R}_{\mathrm{high}}\displaystyle \frac{{b}^{2}}{{b}^{2}+1}+{R}_{\mathrm{low}}\displaystyle \frac{1}{{b}^{2}+1}\end{eqnarray} \tag{ 13 }$$

(Mościbrodzka et al. 2016) and b ≡ β/β_crit. This model has three free parameters: β_crit, R_low, and R_high. We fix R_low = 1 (consistent with the long cooling time in Sgr A*; see discussion in Event Horizon Telescope Collaboration et al. 2021a) and β_crit = 1, but we allow R_high to vary from 1 to 160. When R_high ≫ 1, emission is shifted away from the midplane and toward the poles.

In nonthermal models, the eDF has a power-law tail extending to high energy. We explore two implementations: (i) a power-law distribution function

$$\begin{eqnarray}&&\displaystyle \frac{1}{{n}_{e}}\displaystyle \frac{{{dn}}_{e}}{d\gamma }=\displaystyle \frac{p-1}{{\gamma }_{\min }^{1-p}-{\gamma }_{\max }^{1-p}}{\gamma }^{-p},\end{eqnarray} \tag{ 14 }$$

with power-law index p and upper and lower limits ${\gamma }_{\min }$ and ${\gamma }_{\max };$ and (ii) a so-called κ distribution function, inspired by observations of the solar wind and by results of collisionless plasma simulations (e.g., Kunz et al. 2015, and references therein)

$$\begin{eqnarray}&&\displaystyle \frac{1}{{n}_{e}}\displaystyle \frac{{{dn}}_{e}}{d\gamma }=\gamma \sqrt{{\gamma }^{2}-1}{\left(1+\displaystyle \frac{\gamma -1}{\kappa w}\right)}^{-(\kappa +1)},\end{eqnarray} \tag{ 15 }$$

which has width parameter w and power-law index parameter κ.

Evidently, any eDF assignment scheme is an approximation since the eDF depends in general on both local conditions and particle histories. Notice that we also assume that the eDF is isotropic and neglect electron−positron pairs.

Once the eDF is specified, the radiative transfer coefficients (emissivities, absorptivities, and rotativities) can be readily calculated; see Marszewski et al. (2021) for a recent summary.

Model Scaling.—With the exception of the stellar-wind-fed simulations, the GRMHD simulations considered in this work contain a characteristic speed, c, but are otherwise scale-free; they set GM = c = 1. Physical scales are assigned during the radiative transfer step. The black hole mass M fixes the length unit GM/c² and time unit GM/c³. Because the GRMHD simulations are non-self-gravitating, one is free to set a density scale, or equivalently the accretion rate $\dot{M}$ or plasma mass scale ${ \mathcal M }$.

The plasma mass scale parameter ${ \mathcal M }$ controls the plasma emissivity and the plasma optical depth and thus the source brightness. We adjust ${ \mathcal M }$ iteratively until the time-averaged 230 GHz flux densities of the models are within a few percent of the 2.4 Jy mean observed during the 2017 campaign. Notice that, in this work, model parameters are always varied at constant time-averaged millimeter flux density.

Radiative Transfer Calculation.—Given an eDF, density scale ${ \mathcal M }$, inclination i, and radiative transfer coefficients based on local properties of the plasma, the emergent radiation is obtained by integrating the radiative transfer equation. We use two classes of numerical methods: observer-to-emitter ray-tracing to generate synthetic images (ipole, Mościbrodzka & Gammie 2018; BHOSS, Younsi et al. 2012), and emitter-to-observer Monte Carlo to generate spectral energy distributions (SEDs, using grmonty; Dolence et al. 2009).

All radiative transport calculations are carried out using the fast-light approximation, in which plasma variables are read from a GRMHD output file at constant Kerr−Schild time and are assumed not to change during ray-tracing. Including light-travel time effects in the model introduces minor changes to light curves and images (Dexter et al. 2010; Mościbrodzka et al. 2021). Further detail on numerical methods is given in Appendix A.1. Comparisons of radiative transfer codes (Gold et al. 2020; B. Prather et al. 2022, in preparation) show that differences between codes do not contribute substantially to the error budget.

Images are produced at 86 GHz, 230 GHz, and 2.2 μm. Direct imaging includes synchrotron and bremsstrahlung (both ion−electron and electron−electron; see Yarza et al. 2020, for a recent review). Unless stated otherwise, the image library has a field of view (full width), resolution (pixel count), and half-width angular size of 800 μas, 200 × 200, 80ϑ _g at 86 GHz; 200 μas, 400 × 400, 20ϑ _g at 230 GHz; and 100 μas, 200 × 200, 10ϑ _g at 2.2 μm.

SEDs are produced for narrow bins in inclination angle. At each inclination, the SED is averaged over azimuth. The SED includes synchrotron, bremsstrahlung, and Compton scattering.

We find that 2.2 μm emission is usually dominated by synchrotron, but occasionally 2.2 μm synchrotron is so weak that Compton scattering dominates. We also find that the X-ray can be dominated by either Compton scattering or bremsstrahlung, with the latter dominating in models with a large population of cold electrons at large radius. Figures 3 and 4 show examples of model images and multiwavelength SEDs from our library.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The GRMHD simulation-derived temperatures are unreliable in regions where σ ≡ B²/(8π ρ c²) is large because truncation error in integration of the total energy equation produces large fractional errors in temperature. All radiative transfer models therefore set the emissivity, absorptivity, and inverse Compton scattering cross sections to 0 for the regions with σ > σ_cut = 1.

The source size can be characterized using the second moments of the source image on the sky. The second moments in the image domain map to second derivatives of the visibilities near zero baseline in the (u, v) domain, so short-baseline VAs can be used to directly estimate the source size.

This procedure is used in Paper II to set an upper limit of 95 μas FWHM and lower limit of 38 μas FWHM for the second moment along a direction through the source corresponding to the orientation of the short baselines (SMT-LMT and ALMA-LMT). This is done without any assumption about the structure of the source and is therefore quite permissive.

These limits do not include scattering. The scattering kernel is estimated to have 16.2 μas FWHM along the relevant EHT baselines. To descatter the sky image size, we subtract this value in quadrature, which produces a scattering-corrected 93.6 μas FWHM upper limit and 34.4 μas FWHM lower limit.

To score a model, we evaluate the second-moment tensor for each simulated 230 GHz image and find its eigenvalues ${\lambda }_{\mathrm{maj}}^{2}/(8\mathrm{ln}2)$ and ${\lambda }_{\min }^{2}/(8\mathrm{ln}2)$, where λ_maj and ${\lambda }_{\min }$ are the major- and minor-axis FWHM, respectively. The image is deemed compliant if there exists any position angle (PA) for which the second moment would satisfy the size constraints, i.e., it is compliant if for any λ such that ${\lambda }_{\min }\leqslant \lambda \leqslant {\lambda }_{\mathrm{maj}}$, λ lies between the scattering-corrected upper and lower limits. We reject models with compliance fraction <0.01.

The second constraint provides a morphological check on the VAs. We ask two questions of each model snapshot: (i) is the first minimum in the visibilities—"the null"—at about the right place, and (ii) are the long-baseline VAs comparable to the data? The null locations and long-baseline amplitudes are sensitive to the source structure. For example, if the source is a simple, circularly symmetric ring of finite width, then the location of the first minimum depends only on the ring diameter, while the VAs on long baselines depend mainly on ring width. GRMHD models are more complicated, with fluctuations in the null locations and long-baseline amplitudes (e.g., Medeiros et al. 2018; M87* Paper V).

We compare with data from April 7, which have the best (u, v) coverage near the minima in the VAs. The first visibility minimum in both the N–S and E–W directions in the data always occurs between 2.5 and 3.5 Gλ (see Paper II, for details). For the long-baseline interval between 6 and 8 Gλ in the data, the VAs have ≲4% of the zero-baseline flux. One complication when comparing models to data on long baselines is the effect of interstellar scattering. Diffractive scattering effectively convolves the image with a smooth kernel and can reduce the amplitudes to ∼50% of their descattered values in the 6–8 Gλ range; refractive scattering, on the other hand, introduces noise at all baselines of order 0.5%–3%, depending on the characteristics of the scattering screen (Johnson et al. 2018; Psaltis et al. 2018).

To apply this constraint, we compute the VA of each model snapshot along PAs of 0°, 45°, 90°, and 135° (because of Hermitian symmetry we need only consider PAs in the 0°–180° range). We find the first minimum numerically and compute the median VAs between 6 and 8 Gλ. We classify a snapshot as compliant if (i) for at least one PA the first minimum falls between 2.5 and 3.5 Gλ and (ii) at no PA do the median VAs exceed 4%/50% = 0.08 of the zero-baseline flux. We reject models with compliance fraction <1%.

Following Paper IV, we fit an m-ring image-plane model to snapshots from EHT data and from simulated data and then compare the distributions of fit parameters.

The m-ring is a δ function in radius with diameter d multiplied by a truncated (up to m = 3; notice that Paper IV truncates at m = 4) Fourier series, convolved with a Gaussian of width w. The model also contains a centered Gaussian component, with amplitude and width as free parameters, to absorb large-scale emission and emission interior to the ring. ¹⁵⁷ The m-ring model has 10 parameters. ¹⁵⁸ We use three of the parameters that are well constrained and physically interpretable: the m-ring diameter d, the m-ring width w (FWHM of the convolving Gaussian), and the m = 1 relative amplitude β₁ (the "asymmetry"). For more details about the m-ring model see Section 4.3 of Paper IV.

We fit the m-ring independently to snapshots consisting of 2-minute intervals of EHT data (this averaging interval is consistent with that used in Paper IV). Over these short intervals, we approximate the source as static. Uncertainties in the fitted m-ring parameters are dominated by the limited baseline coverage during these snapshots rather than by calibration uncertainties or thermal noise. Because snapshots that are close in time sample nearly identical baselines, they do not provide additional model constraints.

To compare fitted m-ring parameters from EHT data to those from synthetic data, we select a subset of ten 120 s scans that have detections on more than ten baselines and integration times at all stations > 40 s. The selected scans are as widely separated in time as possible so that they sample distinct baseline coverage, with an average separation of ≃ 1240 s ≃ 60 GM/c³, which is small compared to the VA correlation time in the models (Georgiev et al. 2022). Note that the selected scans overlap with those found in Farah et al. (2022). Only small changes in model selection were observed if any one scan was removed from the comparison. The data were descattered before fitting, that is, the VAs were divided by the scattering kernel.

The maximum likelihood m-ring parameters for the 10 selected EHT scans are listed in Table 3. Evidently the fit parameters are noisy. The fits for d range from 39 to 84 μas, for w from 9 to 21 μas, and for β₁ from 0.04 to 0.48. The variation in fit parameters could be caused by source variability, thermal noise, or gain variations. In the models the main driver of fit variations is source variability.

For the models, we read in a series of model images, generate synthetic data for each image for each scan at four PAs (0°, 45°, 90°, 135°), and fit m-rings to the synthetic data. This produces a distribution of m-ring parameters for each model.

The synthetic data are generated as follows. A model image I(x, y) is Fourier transformed to complex visibilities V(u, v) with an assumed PA and then sampled on baselines i drawn from the comparison scan, V_i ≡ V(u_i , v_i ). Normally distributed thermal noise δ V_th,i with amplitude based on telescope performance during the scan is added, and multiplicative, normally distributed noise with unit variance N is added to crudely model gain corrections: ${\tilde{V}}_{i}={V}_{i}(1+\epsilon N)+\delta {V}_{\mathrm{th},i}$. We set = 0.05, but no substantial changes in fit parameters were observed for = 0.02. We then fit to the VAs $| {\tilde{V}}_{i}|$ and closure phases. ¹⁵⁹

We sample the model images once per 500 GM/c³, which is comparable to a correlation time. A model with a 15,000 GM/c³ imaging window thus produces 30 fits per scan per PA.

In comparing the models to the data, we (i) generate the distribution of fit parameters at each PA; (ii) use a Kolmogorov–Smirnov (K-S) test to compare the distribution of ∼300 synthetic data fits with the distribution of 10 observational fits, and obtain a p-value (what is the probability they are drawn from the same underlying distribution?); (iii) average the p-values over the four sampled PAs (i.e., marginalize over PA; the models do not show a significant PA preference); and (iv) reject the model if p < 0.01.

The Global Millimeter VLBI Array (GMVA) observed Sgr A* on 2017 April 3, just 3 days ( ≃ 13,000 GM/c³) before the EHT campaign. Issaoun et al. (2019) estimate that the compact flux during this observation was F₈₆ = 2.0 ± 0.2 Jy (2σ errors).

To test the models, we compute a library of 86 GHz images for all GRMHD snapshots for all models and integrate over them to obtain F₈₆. We assume normally distributed measurement errors with σ = 0.1 Jy and convolve the F₈₆ distribution for each model with the resulting Gaussian. We reject models where the value of the error-broadened cumulative distribution function (CDF) at 2.0 Jy is <1% or >99%.

The GMVA observations from 2017 April 3 constrain the FWHM of the source major axis. Notice that two different values for the major-axis FWHM have been published in the literature: 120 ± 34 μas (Issaoun et al. 2019) and ${\mathrm{FWHM}}_{\mathrm{maj}}={146}_{-12}^{+11}\,\mu \mathrm{as}$ (95% confidence; Issaoun et al. 2021). We adopt the latter analysis.

We compute the major-axis FWHM for each image in the 86 GHz image library. We assume normally distributed errors with σ = 6 μas and convolve the model major-axis distribution with the normal distribution. We reject models with error-broadened CDF < 1% or >99% at 146 μas.

Our synthetic 86 GHz images have a 800 μas field of view. A 200 μas field of view cuts off enough emission that the major axis is biased downward in many models by ∼ 20%. Increasing the field of view beyond 800 μas has negligible effect.

Sgr A* has a quiescent and a flaring component in the near-infrared (NIR), with flares occurring a few times per day (1 day ≃ 4, 200 GM/c³; Witzel et al. 2018). Since there is as yet no generally accepted model for NIR flares, we accept models that do not produce flares (indeed, none of our models reliably produce flares, even those with nonthermal eDFs). Our working hypothesis is that the models can be made to produce flares by introducing a process that accelerates a small fraction of electrons into an NIR-bright tail of the eDF. If the model overproduces quiescent 2.2 μm emission, however, then we reject it.

Sgr A* had a median 2.2 μm flux = 0.8 ± 0.3 mJy in 2017 (Gravity Collaboration et al. 2020a; see Table 1). The median flux density likely overestimates the median quiescent flux density since it includes flares.

We compute the model median 2.2 μm flux density using one of two procedures. If a full SED—which includes Compton scattering—is available, then we use it. The SEDs are generated by the grmonty Monte Carlo code (Dolence et al. 2009; Wong et al. 2022). If a full SED is not available (see Table 2), then we compute a 2.2 μm image that includes only synchrotron emission (synchrotron absorption is negligible at 2.2 μm for Sgr A*).

A rigorous model evaluation procedure would correct for the upward bias in median quiescent flux density from flares and allow for errors in the model and observed median flux density, but these refinements are sufficiently uncertain that, instead, we set a conservative threshold of 1.0 mJy and reject the model if its median 2.2 μm flux density exceeds the threshold.

Sgr A* flares in the X-ray less than about once per day (see Yuan et al. 2018, and references therein). Chandra observations during the 2017 campaign suggest a conservative upper limit on the median (quiescent) ν L_ν at 6 keV of 10³³ erg s⁻¹ (Paper II).

As for the model 2.2 μm flux density, we estimate ${\left.\nu {L}_{\nu }\right|}_{h\nu =6\,\mathrm{keV}}$ in two ways. The SED, which incorporates Compton scattering and bremsstrahlung, is used if it is available. If the SED is not available, then we compute an X-ray image that includes only bremsstrahlung (which dominates the X-ray emission in thermal SANE models with R_high = 40,160) enabling us to eliminate a few additional models.

We reject the model if its median ${\left.\nu {L}_{\nu }\right|}_{h\nu =6\,\mathrm{keV}}\,\gt {10}^{33}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}$.

We compare variability in the models to light-curve observations of Sgr A* from 2005 to 2017 using the 3 hr modulation index M₃, where M_ΔT ≡ σ_ΔT /μ_ΔT , σ_ΔT is the standard deviation measured over an interval ΔT (in hours), and μ_ΔT is the mean measured over the same interval.

Following Chan et al. (2015a), we use M_ΔT because it is easy to describe, easy to compute, commonly used in the literature (in the X-ray astronomy literature it is "rms %"), and closely related to the structure function, since the expectation value for ${\sigma }_{T}^{2}$ is given by an integral over the structure function (see D. Lee et al. 2022, in preparation).

We use ΔT = 3 hr ( ∼ 530 GM/c³) because it is long enough to be comparable to the characteristic timescale measured in damped random walk fits to the ALMA light curve (see Table 10 of Wielgus et al. 2022) but short enough that the model light curves provide a sample that is large enough to be constraining. In extracting a sample of M₃ from the light curves, we use as many 3 hr segments as possible, equally spaced away from the light-curve endpoints and each other, and calculate M₃ on each segment. We treat consecutive measurements of M₃ as independent, consistent with the minimal correlation expected for a damped random walk (D. Lee et al. 2022, in preparation).

We must select an observed distribution of M₃. The April 7 data alone provide only a weak constraint because there are only three samples. The M₃ measured from EHT 2017 observations on April 5–11 provide seven samples, while the M₃ measured from all available light curves longer than 3 hr, including earlier SMA and CARMA data (the "historical distribution"; see Wielgus et al. 2022), yield 42 samples. The 2017 distribution is consistent with being drawn from the historical distribution, although April 6 has one of the quietest segments on record, and April 11 one of the most variable. We selected the historical distribution and note that the 2017 distribution rejects slightly more models but leads to identical conclusions.

For each model we use a two-sample K-S test to estimate the probability p that the model and observed M₃ values are drawn from the same underlying distribution. We reject the model if p < 0.01.

Through the K-S test, the strength of the M₃ constraint depends on the number of data and model samples. The fiducial models have duration 10⁴ or 1.5 × 10⁴ GM/c³ (18 or 28 samples), whereas most exploratory models have duration 5 × 10³ GM/c³ (nine samples). The M₃ constraint is therefore weaker for the exploratory models: an exploratory model that passes the constraint may be more variable than a fiducial model that fails.

Fluctuations in the spatial structure of the source produce fluctuations in the VAs. Here we compare the power spectrum of structural variability from EHT observations with predictions from GRMHD models.

A nonparametric technique to measure the variance of the spatially detrended VAs at a location in the (u, v)-plane is described in Broderick et al. (2022) and briefly summarized here. We use EHT observations of Sgr A* from April 5, 6, 7, and 10 (April 11 was excluded). To remove correlations associated with variations in the total flux, we normalize the VA data with the contemporaneous intrasite light curve (Georgiev et al. 2022). The light-curve-normalized VAs are then linearly detrended, and variances are computed and azimuthally averaged (Broderick et al. 2022). The resulting ${\sigma }_{\mathrm{var}}^{2}(| u| )$ is a measure of the fractional structural variability as a function of baseline length ∣u∣. The ${\sigma }_{\mathrm{var}}^{2}(| u| )$ is included in an inflated error budget when making images of and fitting models to the 2017 EHT observations of Sgr A* (Paper III).

We measured this quantity from the GRMHD simulations (see Georgiev et al. 2022, for details). For all simulations reported here, ${\sigma }_{\mathrm{var}}^{2}$ is well approximated by a broken power law with parameters that are nearly universal among simulations. The ${\sigma }_{\mathrm{var}}^{2}$ is measured over a 4-day period, which is longer than the typical model duration. We therefore expect that model values will be biased downward compared to the data. Furthermore, each GRMHD simulation can only give one draw from a distribution that is broader than if the simulation spanned 4 days. This secondary effect negates the downward bias, which is further unimportant, as we do not exclude models for being not variable enough. To measure the larger broadness of the distribution, we use multiple simulations with the same parameters and subdivide the analysis of long simulations into windows. The uncertainties in the measurement from the GRMHD simulations due to simulation resolution, the fast-light approximation, and code differences are small compared to the uncertainty due to the variability of ${\sigma }_{\mathrm{var}}^{2}$ due to short simulations (Georgiev et al. 2022).

The measured ${\sigma }_{\mathrm{var}}^{2}$ is well characterized by a power law for 2 Gλ < ∣u∣ < 6 Gλ (Georgiev et al. 2022). For comparison with the models presented here, we distill the ${\sigma }_{\mathrm{var}}^{2}$ to two numbers: the amplitude a₄ ² at 4 Gλ and a power-law index b. Because the normalization is done in the center of the fit range, the estimated ${\mathrm{log}}_{10}({a}_{4}^{2})=-3.4\pm 0.1$ and b = 2.4 ± 0.8 are essentially uncorrelated.

Model predictions for a₄ ² and b are computed using the power spectral densities from Georgiev et al. (2022). ¹⁶⁰ The anisotropic diffractive scattering kernel from Johnson et al. (2018) is applied to ${\sigma }_{\mathrm{var}}^{2}(| u| )$ and averaged over relative orientations of the major axis of the scattering kernel and the black hole spin. These estimates are then azimuthally averaged, and the parameters a₄ ² and b are determined from a least-squares linear fit to ${\sigma }_{\mathrm{var}}^{2}(| u| )$ in 2 Gλ < ∣u∣ < 6 Gλ.

For each model the fits for a₄ ² and b are done separately on each window of length 5 × 10³ GM/c³, giving at most three measurements for most models. This makes a direct comparison with the measured value difficult, as the model distribution is poorly constrained.

Georgiev et al. (2022) estimates that the typical width of a model distribution is ${\mathrm{log}}_{10}({a}_{4}^{2})\pm 0.1$. We can obtain a rough estimate for how the models fare compared to the measurement by taking the mean across windows, assuming that the width of the distribution is σ = 0.1, and comparing this with the observed distribution under the assumption that both are distributed normally. We reject models with error-broadened CDF <1% or >99% at ${\mathrm{log}}_{10}({a}_{4}^{2})=-3.4$.

Second Moment.—Without assuming a ring, the EHT data allow a wide range of second moments. The second moment constraint passes 98% of all models. Here and in what follows, the quoted passing fraction for the model describes the fraction of points in parameter space for which the existing model sets (KHARMA, BHAC, and when present H-AMR) agree that the model passes the constraint. In short, nearly all fiducial models are about the right size once we use the 230 GHz to fix the mass unit ${ \mathcal M }$. The few rejected models are a_* ≤ 0, face-on, SANE models with R_high = 1. These models have extended emission on scales large compared to the critical impact parameter ${b}_{c}=\sqrt{27}\,{GM}/{c}^{2}$. The right panel of Figure 5 shows an example of one of these failed models. The left panel shows an example of a passing model.

Zoom In Zoom Out Reset image size

Visibility Amplitude Morphology.—The VA morphology constraint tests the null location and long-baseline VAs. Figure 6 shows an example of a passing and failing model. The constraint disfavors edge-on models at positive spin and a few large-R_high SANE models. This is mainly because the edge-on models contain bright spots, corresponding to the approaching side of the rotating accretion flow, and faint rings, so the first nulls get washed out by the bright features. The VA morphology constraint passes 79% of all models.

Zoom In Zoom Out Reset image size

M-ring Fits.—The m-ring asymmetry, diameter, and width are treated as separate constraints. Recall that we compare the distribution from the data to that from the model using a two-sample K-S test.

The asymmetry parameter is typically not well constrained. Many rejected models are at high inclination and have a_* = 0.94. These models have asymmetries that are large and detectable because Doppler boosting concentrates emission in an equatorial spot on the approaching side of the disk. The asymmetry parameter constraint passes 91% of all models.

The m-ring diameter, which depends on the diameter of the shadow and the ring width, is better constrained than the asymmetry parameter and varies systematically from model to model. The ring diameter constraint passes 54% of all models.

Most of the models that fail are low-inclination models with ring diameters that are too large. Only two BHAC models fail because the ring diameter is too small. Most of the rejected models are low-inclination models at a_* < 0.

The m-ring width w is the most tightly constrained of the three m-ring parameters. Although the closure phases constrain w as well, it is easiest to see how w affects VAs at long baselines. For example, for a circularly symmetric ring the VAs are a Bessel function multiplied by a Gaussian with width ∼ 1/w. Increasing w therefore decreases the amplitude of the long baselines. Figure 7 shows examples of models that pass and fail the m-ring width constraint.

Zoom In Zoom Out Reset image size

Figure 8 summarizes the pass/fail status of the fiducial models for the m-ring width. All rejected models have median w that is below the median of the data, ≃ 17.5 μas. The rejected models include all MAD models at a_* ≤ 0 and all edge-on (i = 90°) models in the KHARMA, BHAC, and H-AMR fiducial models. MAD models exhibit a strong trend toward smaller w as i increases. SANE models exhibit a similar but weaker trend. The SANE model images have higher optical depth, broader rings, and more substructure than the MAD models. Their w distributions are typically broad, with mode well below 17.5 μas. Only for a_* = 0.94, where the optical depth is lower owing to higher temperatures in the emitting region, do most of the models exhibit a sharply peaked w distribution centered at 17.5 μas.

Zoom In Zoom Out Reset image size

EHT Constraint Summary.—We can combine all EHT constraint cuts with a logical and operation. The results are summarized in Figure 9. Evidently EHT data alone are capable of discriminating between models. The edge-on (i = 90°) models all fail, with some failing m-ring width, diameter, asymmetry, and the VA morphology constraint. The cuts clearly favor a_* > 0 models, with a few exceptions. There are two clusters of models that do not fail any constraints in any models: positive-spin MAD models at low inclination, and positive-spin SANE models, also at low inclination.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

86 GHz Flux Density.—In a simplified picture Sgr A*'s millimeter flux is produced in a photosphere that decreases in size as frequency increases. Because optical depth is not large at 230 GHz (∼0.4 in the one-zone model) and the source structure is complicated (the optical depth varies across the image), the simplified picture is imprecise. Nevertheless, 86 GHz emission is on average produced at larger radius than 230 GHz emission, and the 86 GHz source size is larger than the 230 GHz source size. The ratio of 86 GHz to 230 GHz flux density is therefore sensitive to the radial structure of the source plasma.

Figure 28 records the results of applying this constraint. Most R_high = 1 models, both MAD and SANE, fail the 86 GHz flux density test. The 86 GHz flux density is quite sensitive to R_high. For example, SANE a_* = 0.5, i = 70°, 90° models are too bright at R_high = 1 and too dim at R_high = 10. This suggests that there are passing models in between, and that the parameter space is not sampled densely enough. Finally, the 86 GHz flux constraint strongly favors MAD models over SANE models in all three fiducial model sets.

86 GHz Major Axis.—As for the 86 GHz flux, the 86 GHz size is sensitive to optical depth as a function of radius in the source plasma. Figure 29 in Appendix C shows the full results of applying this constraint.

The 86 GHz size is sensitive to inclination. For example, the SANE, a_* = 0, R_high = 40 models are too small at low inclination and too large when seen edge-on, because the edge-on models have prominent limb-brightened jet walls that are visible to 100 μas. The 86 GHz size constraint passes only 58% of models and is therefore one of the tightest constraints.

The physical picture for 86 GHz source size is complicated, as is the extraction of the constraint itself from observations. Notice that (i) two different values for the 86 GHz intrinsic source size have been reported in the literature (see Section 3.2.2), (ii) scattering is 7 times stronger at 86 GHz than at 230 GHz, (iii) scattering must be subtracted accurately to obtain the intrinsic source size, and (iv) the error bars for the 86 GHz source size are narrow and this plays a key role in determining the strength of the constraint.

2.2 μm Median Flux Density.—2.2 μm photons are produced by the synchrotron process from electrons on the high-energy end of the eDF. For the one-zone model with B = 30 G and Θ_e = 10, the mean Lorentz factor is γ = 30 and the synchrotron critical frequency ν_crit = γ² eB/(2π m_e c) ≃ 80 GHz. Emission at 2.2 μm is produced by electrons with Lorentz factor γ ≃ 10³, so 2.2 μm flux density is sensitive to Θ_e and B. Both increase toward the horizon, and field strength is nearly independent of latitude, so 2.2 μm photons are produced at small radius in regions where Θ_e is highest.

The sensitivity to Θ_e implies that 2.2 μm flux density will be highest for models with higher temperatures. For SANEs the midplane gas temperature, and therefore electron temperature in the R_high prescription, increases with a_*, so the highest 2.2 μm flux density is at positive a_*.

The sensitivity to B implies that 2.2 μm flux density will be highest for parameters with stronger fields. B depends on the GRMHD flow configuration and also on the accretion rate, which is fixed by the observed F₂₃₀, so when all else is equal the 2.2 μm flux density is highest when the accretion rate is largest. The dependence of accretion rate on model parameters is discussed in Section 5.5. In brief, for SANE models the accretion rate declines as a_* increases and R_high decreases. For MAD models the accretion rate dependence on a_* and R_high is relatively weak.

Finally, the 2.2 μm flux density is also sensitive to inclination. A combination of Doppler boosting and the rapid falloff in emissivity in the NIR means that at large inclination lower-frequency emission from the approaching side of the accretion flow is boosted into the NIR, and thus 2.2 μm flux is higher at high inclination.

Figure 10 shows sample SEDs from our model library, where the left panel is a model that passes the 2.2 μm flux limit and the right panel is a model that fails. Models that pass the 2.2 μm flux limit are shown in Appendix C in Figure 30. The rejected SANE models (7% rejected by all of KHARMA, BHAC, and H-AMR) tend to be at high inclination: their images are dominated by a bright spot on the approaching side of the disk. The rejected MAD models (53%) include nearly all models at R_high = 1 and R_high = 10, where Θ_e tends to be larger, and the majority of high-inclination models, where the effect of Doppler boosting is largest.

We find that some models are Compton dominated at 2.2 μm. For example, a_* = − 0.94 SANE models become optically thin at relatively low frequency as R_high goes to 1, and thus synchrotron emission drops off rapidly as frequency increases. When the synchrotron is weak enough, the underlying bump of Comptonized millimeter photons dominates.

X-ray Luminosity.—X-ray production in fiducial models is typically dominated by Compton upscattering of thermal synchrotron photons. In the first Compton bump ν L_ν is thus proportional to the y-parameter $y\sim 16{{\rm{\Theta }}}_{e}^{2}{\tau }_{e}$, where τ_e is a characteristic electron-scattering optical depth and Θ_e is a typical dimensionless electron temperature. At R_high = 1 the X-ray band lies in the first Compton bump, while at larger R_high the bumps move to lower energy because the bulk of the Thomson depth is in the midplane where Θ_e ∝ 1/R_high.

We find that in a few large-R_high SANE models, however, X-ray emission is dominated by bremsstrahlung (synchrotron never dominates the X-ray in thermal models). Bremsstrahlung emissivity j_ν,b ∝ n², so at fixed temperature bremsstrahlung increases rapidly with density. Notice that ${j}_{\nu ,b}\propto {{\rm{\Theta }}}_{e}^{1/2}$ for Θ_e > 1 and ${{\rm{\Theta }}}_{e}^{-1/2}$ for Θ_e < 1, so cool disks enhance bremsstrahlung. Bremsstrahlung therefore dominates Compton in models with high density and low temperature, i.e., some models with large R_high (see Section 5).

In models with bremsstrahlung-dominated X-ray emission the median radius of emission is ≃20GM/c². Although the models are equilibrated at this radius, the X-ray luminosity may be partially contaminated by emission from unequilibrated plasma at larger radii. Because the fiducial models start with a torus of finite radial extent, however, they are also missing bremsstrahlung emission from outside the initial torus. A full assessment of the associated uncertainty requires large, long runs. Notice that because bremsstrahlung arises at large radii, it varies more slowly than the synchrotron and Compton-upscattered X-ray emission and is therefore potentially distinguishable (Neilsen et al. 2013).

The left panel of Figure 10 shows a model that passes the X-ray flux limit, while the right panel shows a model that fails. The X-ray cuts are shown in Appendix C, Figure 31. Some large-R_high SANE models fail owing to excess bremsstrahlung, although there is notable disagreement between BHAC and KHARMA for SANE X-ray fluxes. MAD models that fail have small R_high and are Compton dominated in the X-ray. Nearly all R_high = 1 MAD models fail the X-ray constraint, as do many at R_high = 10. This is because the midplane Θ_e increases as R_high goes to 1. Since the midplane contributes most of the electron-scattering optical depth, small-R_high models have the largest y-parameter and are at greatest risk of overproducing X-rays.

Summary of Non-EHT Constraints.—Applying only non-EHT constraints leaves 6% of models as shown in Figure 11. The surviving models are the result of applying a heterogeneous and noisy set of constraints using a hard cutoff, which somewhat obscures the underlying physical picture. Nevertheless, the surviving 13 models are all MAD and all have R_high > 10. All but two have i < 70°. This leaves a cluster of surviving MAD models at large R_high and low to moderate inclination.

Zoom In Zoom Out Reset image size

Variability is central to the interpretation of EHT observations of Sgr A*: an 8 hr observation of Sgr A* lasts 1400 GM/c³, a timescale over which most models vary substantially. In contrast, an 8 hr observation of M87* is ∼ GM/c³, and on this timescale M87* hardly varies at all.

Recall that we consider two variability constraints, one on the 230 GHz light curve and the other on 230 GHz VAs. We find that SANE models are less variable than MAD models. Only 3.5% of models, all SANE, pass both variability constraints. A possible interpretation of this result is that the models are missing a physical ingredient that would reduce variability, and this is discussed in Section 5.

Modulation Index.—The distributions of 3 hr modulation index (M₃) across all fiducial SANE models, across all fiducial MAD models, and across the historical data set are shown in Figure 12. The plot also shows distributions for individual models with the lowest and highest median M₃.

Zoom In Zoom Out Reset image size

The M₃ cuts are summarized in Appendix C, Figure 34. We find that (i) as a group, the fiducial models are more variable than the data; (ii) the MAD models are more variable than SANE models; (iii) 11 individual models pass the constraint for all fiducial model sets, and these are exclusively SANE models; (iv) there are some differences between variability in the fiducial model sets, with H-AMR models notably more variable than KHARMA and BHAC models; and (v) the pass fractions for the fiducial model sets are 20% for KHARMA, 27% for BHAC, and 7% for H-AMR. The modulation index is the tightest single constraint on the models.

4 Gλ Visibility Amplitude Variability.—The power-law index b of the variance ${\sigma }_{\mathrm{var}}^{2}(| u| )$ at 2–6 Gλ of the models is generally in good agreement with the value measured from the 2017 EHT campaign (excluding April 11). The amplitude a₄ ², however, varies depending on the model.

Figure 13 shows the distribution of $({a}_{4}^{2},b)$ from the EHT observation, along with the distributions across all fiducial models. For a single model, the number of measurements of $({a}_{4}^{2},b)$ is equal to the number of windows for that model (three in most cases). The koral models appear more variable because they include only R_high = 20 MAD models at various spins.

Zoom In Zoom Out Reset image size

The models tend to be more variable than the observations, with face-on models performing better than edge-on models. For SANE models, R_high = 10 tends to be more variable than others. For MAD models, there is a slight preference for lower R_high.

Long-duration koral Models.—We have imaged a set of MAD models run with the koral code out to ∼ 100, 000 GM/c³. These long-duration models have R_high = 20, which lies off our fiducial model parameter grid. They enable us to assess the importance of integration time for application of the constraints and provide a more accurate distribution for, e.g., M₃.

The koral models are discussed in Appendix B. In brief, we find no evidence for significantly different variability when comparing the first and second half of the koral runs, consistent with no long-term evolution of the variability. We also find no significant differences when comparing the koral runs to nearby models on the fiducial model parameter grid. Notice that in Figure 13 the koral models are more variable than the other model sets only because the other model sets contain lower-variability SANE models.

None of the fiducial models survive the full gauntlet of constraints. The pass fractions for individual constraints for the BHAC, KHARMA, and H-AMR fiducial models are listed in Table 4. M₃ is the most severe constraint, followed by the m-ring width constraint. Together the variability constraints pass only 4% of fiducial models and prefer SANEs, which are less variable than MADs, while the remaining constraints prefer MAD models.

It is likely that the models are physically incomplete. It is also possible, however, that one of the constraints is measured incorrectly, that one of the constraints is applied incorrectly, or that one of the constraints is poorly predicted for numerical reasons. To investigate this, we identify all models that fail only one constraint in Table 5. We find that the critical constraints are 86 GHz size, m-ring diameter, and M₃. Notice that there is overlap between KHARMA and BHAC in MAD models that fail the M₃ constraint. The H-AMR models fare significantly worse than the KHARMA and BHAC models in the M₃ constraint: only 7% of models pass. The remaining models all fail at least one additional constraint, leading to their exclusion from Table 5.

The R_high prescription provides a convenient, one-parameter model for assigning electron temperatures, but here is a vast function space of possible alternative parameterizations. One well-motivated choice is the critical beta model (Anantua et al. 2020b), which sets T_e = T_e (R) and $R=f\exp (-\beta /{\beta }_{c})$ (see Equation (11)). This "critical beta" model has two parameters, f and β_c . We consider a single point in the parameter space: f = 0.5, β_c = 1. Compared to the R_high temperature prescription, the main new characteristic of the critical beta models is that the electron-to-ion temperature ratio approaches 0 at high β instead of 1/R_high.

We have run all tests except X-ray for the critical beta models. The 2.2 μm flux is calculated by imaging only and therefore does not include Compton scattering.

All critical beta models fail the non-EHT constraints, with the 86 GHz size constraint rejecting most models as too small. The variability constraints pass 23% of the models. No models survive the combined EHT and non-EHT constraints even if variability constraints are excluded. Notice that this does not imply that critical beta models are ruled out, since we have only tested a single point in the f, β_crit parameter space.

So far we have assumed a thermal eDF (Equation (11)). Fully kinetic simulations and resistive MHD predict that reconnection in current sheets within the accretion flow and in the jet sheath leads to the acceleration of particles to higher energies, resulting in the emergence of a power-law tail (e.g., Sironi et al. 2021, and references therein). Such acceleration events are thought to be the origin of NIR and X-ray flares detected in Sgr A*. Here we do not address flare mechanisms but seek to constrain the contribution of nonthermal electrons to the quiescent emission of Sgr A*.

Below we assume different forms of the eDF assuming that a fraction of the electron population is accelerated into a nonthermal tail. There are multiple ways of doing this, but we will continue to assume that the eDF depends instantaneously on local conditions and set the accretion rate so that the 230 GHz time-averaged compact flux is 2.4 Jy.

First, we consider a hybrid thermal/power-law distribution using H-AMR/BHOSS. Since we are modeling quiescent emission, we assume a steep power-law index of p = 4 with a constant nonthermal acceleration efficiency = n_{e, power-law}/n_{e, thermal} = 0.1, typical of PIC simulations (e.g., Sironi et al. 2015; Crumley et al. 2019). Following Chatterjee et al. (2021), the power-law tail is stitched to the thermal core by choosing the minimum Lorentz factor limit of the power law, ${\gamma }_{\min }$, to be at the peak of the Maxwellian component. The upper end of the power law is set to ${10}^{5}{\gamma }_{\min }$ (see Equation (14)). The temperature of the thermal component is set by the R_high prescription (Equation (13)). We find that the accretion rate is slightly smaller than for corresponding thermal models, consistent with a small contribution from the power-law component to the 230 GHz total intensity.

230 GHz VLBI Pre-image Size.—Hybrid thermal/power-law models have larger 230 GHz VLBI pre-image sizes compared to their purely thermal counterparts. This is because the power-law component of the eDF allows high-energy electrons in weak magnetic fields at distances more than a few gravitational radii (i.e., larger than the typical emission radius of the 230 GHz images) to contribute to the total image. However, the extension in the images is much smaller for MAD models, with most MAD images displaying an increase in size of <10%.

86 GHz Flux and Image Size.—In general, the R_high = 1 models produce too much 86 GHz flux. Since the lower limit of the power law ${\gamma }_{\min }$ is directly affected by the local electron temperature, the highest-energy electrons are located in the jet sheath where T_i ≈ T_e . Indeed, this is why SANE models produce more 86 GHz flux when nonthermal electrons are introduced, especially at larger R_high values. On the other hand, MAD thermal and mixed thermal/nonthermal models behave similarly, as the bulk of the emission is produced in the inner disk.

The 86 GHz image sizes for the hybrid H-AMR models are, on average, larger than their thermal-only counterparts, similar to the 230 GHz image sizes. The higher-energy electrons of a hybrid thermal/power-law population emit at higher frequencies than their thermal core, thereby extending the image size. This effect increases the image size of MAD models by only a few percent.

2.2 μm Constraint.—The addition of the power-law tail increases the flux at 2.2 μm, and thus the GRAVITY-based 2.2 μm median flux density threshold of 1.0 mJy provides a strong constraint on the power-law index and the acceleration efficiency. In brief, 59% of the power-law models, especially R_high = 1 and 40 MAD models, are ruled out by the 2.2 μm constraint.

Summary.—Overall, H-AMR hybrid thermal/power-law models behave quite differently from their thermal counterparts. For the thermal models, both EHT and non-EHT constraints are equally successful in ruling out models, with 22% passing for each constraint set. For the power-law model set non-EHT constraints pass 39% of models while EHT constraints pass 10% of models. This disparity occurs for two reasons: (i) introducing nonthermal electrons pushes the 86 GHz image size to the acceptable range, as thermal models typically exhibit small image sizes; and (ii) the m-ring width is found to be smaller for the hybrid models. This could be due to a change in the gas density scaling that is required to match the 230 GHz flux. Nonthermal models require a smaller normalization value, meaning a smaller electron number density as compared to the corresponding thermal models. A decrease in the number density lowers the optical depth, leading to a thinner photon ring. For the initial 5000 GM/c³ survey, two mid-inclination power-law models survive: a SANE a_* = 0.94 model and a MAD a_* = 0.5, R_high = 1 model (see Table 6), although ultimately both models are ruled out when extended to 15,000 GM/c³.