First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring

All GRMHD simulations are initialized with a weakly magnetized torus of plasma orbiting in the equatorial plane of the black hole (e.g., De Villiers et al. 2003; Gammie et al. 2003; McKinney & Blandford 2009; Porth et al. 2017). We do not consider tilted models, in which the accretion flow angular momentum is misaligned with the black hole spin. The limitations of this approach are discussed in Section 7.

The initial torus is driven to a turbulent state by instabilities, including the magnetorotational instability (see e.g., Balbus & Hawley 1991). In all cases the outcome contains a moderately magnetized midplane with orbital frequency comparable to the Keplerian orbital frequency, a corona with gas-to-magnetic-pressure ratio ${\beta }_{{\rm{p}}}\equiv {p}_{\mathrm{gas}}/{p}_{\mathrm{mag}}\sim 1$, and a strongly magnetized region over both poles of the black hole with ${B}^{2}/\rho {c}^{2}\gg 1$. We refer to the strongly magnetized region as the funnel, and the boundary between the funnel and the corona as the funnel wall (De Villiers et al. 2005; Hawley & Krolik 2006). All models in the library are evolved from t = 0 to $t={10}^{4}\,{r}_{{\rm{g}}}{c}^{-1}$.

The simulation outcome depends on the initial magnetic field strength and geometry insofar as these affect the magnetic flux through the disk, as discussed below. Once the simulation is initiated the disk transitions to a turbulent state and loses memory of most of the details of the initial conditions. This relaxed turbulent state is found inside a characteristic radius that grows over the course of the simulation. To be confident that we are imaging only those regions that have relaxed, we draw snapshots for comparison with the data from $5\times {10}^{3}\le t/{r}_{{\rm{g}}}{c}^{-1}\le {10}^{4}$.

GRMHD models have two key physical parameters. The first is the black hole spin ${a}_{* }$, $-1\lt {a}_{* }\lt 1$. The second parameter is the absolute magnetic flux ${{\rm{\Phi }}}_{\mathrm{BH}}$ crossing one hemisphere of the event horizon (see Tchekhovskoy et al. 2011; O. Porth et al. 2019, in preparation for a definition). It is convenient to recast ${{\rm{\Phi }}}_{\mathrm{BH}}$ in dimensionless form $\phi \equiv {{\rm{\Phi }}}_{\mathrm{BH}}{\left(\dot{M}{r}_{{\rm{g}}}^{2}c\right)}^{-1/2}$.¹¹⁰

The magnetic flux ϕ is nonzero because magnetic field is advected into the event horizon by the accretion flow and sustained by currents in the surrounding plasma. At $\phi \gt {\phi }_{\max }\sim 15$,¹¹¹ numerical simulations show that the accumulated magnetic flux erupts, pushes aside the accretion flow, and escapes (Tchekhovskoy et al. 2011; McKinney et al. 2012). Models with $\phi \sim 1$ are conventionally referred to as Standard and Normal Evolution (SANE; Narayan et al. 2012; Sa̧dowski et al (2013a)) models; models with $\phi \sim {\phi }_{\max }$ are conventionally referred to as Magnetically Arrested Disk (MAD; Igumenshchev et al. 2003; Narayan et al. 2003) models.

The Simulation Library contains SANE models with ${a}_{* }=-0.94$, −0.5, 0, 0.5, 0.75, 0.88, 0.94, 0.97, and 0.98, and MAD models with ${a}_{* }=-0.94$, −0.5, 0, 0.5, 0.75, and 0.94. The Simulation Library occupies 23 TB of disk space and contains a total of 43 GRMHD simulations, with some repeated at multiple resolutions with multiple codes, with consistent results (O. Porth et al. 2019, in preparation).

To produce model images from the simulations for comparison with EHT observations we use GRRT to generate a large number of synthetic images and derived VLBI data products. To make the synthetic images we need to specify the following: (1) the magnetic field, velocity field, and density as a function of position and time; (2) the emission and absorption coefficients as a function of position and time; and (3) the inclination angle between the accretion flow angular momentum vector and the line of sight i, the position angle $\mathrm{PA}$, the black hole mass $M$, and the distance D to the observer. In the following we discuss each input in turn. The reader who is only interested in a high-level description of the Image Library may skip ahead to Section 3.3.

(1) GRMHD models provide the absolute velocity field of the plasma flow. Nonradiative GRMHD evolutions are invariant, however, under a rescaling of the density by a factor ${\mathscr{M}}$. In particular, they are invariant under $\rho \to {\mathscr{M}}\rho$, field strength $B\to {{\mathscr{M}}}^{1/2}B$, and internal energy $u\to {\mathscr{M}}u$ (the Alfvén speed $B/{\rho }^{1/2}$ and sound speed $\propto \sqrt{u/\rho }$ are invariant). That is, there is no intrinsic mass scale in a nonradiative model as long as the mass of the accretion flow is negligible in comparison to $M$.¹¹² We use this freedom to adjust ${\mathscr{M}}$ so that the average image from a GRMHD model has a 1.3 mm flux density ≈0.5 Jy (see Paper IV). Once ${\mathscr{M}}$ is set, the density, internal energy, and magnetic field are fully specified.

The mass unit ${\mathscr{M}}$ determines $\dot{M}$. In our ensemble of models $\dot{M}$ ranges from $2\times {10}^{-7}{\dot{M}}_{\mathrm{Edd}}$ to $4\times {10}^{-4}{\dot{M}}_{\mathrm{Edd}}$. Accretion rates vary by model category. The mean accretion rate for MAD models is $\sim {10}^{-6}{\dot{M}}_{\mathrm{Edd}}$. For SANE models with ${a}_{* }\gt 0$ it is $\sim 5\times {10}^{-5}{\dot{M}}_{\mathrm{Edd}};$ and for ${a}_{* }\lt 0$ it is $\sim 2\times {10}^{-4}{\dot{M}}_{\mathrm{Edd}}$.

(2) The observed radio spectral energy distributions (SEDs) and the polarization characteristics of the source make clear that the 1.3 mm emission is synchrotron radiation, as is typical for active galactic nuclei (AGNs). Synchrotron absorption and emission coefficients depend on the eDF. In what follows, we adopt a relativistic, thermal model for the eDF (a Maxwell-Jüttner distribution; Jüttner 1911; Rezzolla & Zanotti 2013). We discuss the limitations of this approach in Section 7.

All of our models of M87 are in a sufficiently low-density, high-temperature regime that the plasma is collisionless (see Ryan et al. 2018, for a discussion of Coulomb coupling in M87). Therefore, T_e likely does not equal the ion temperature T_i, which is provided by the simulations. We set T_e using the GRMHD density ρ, internal energy density u, and plasma ${\beta }_{{\rm{p}}}$ using a simple model:

$$\begin{eqnarray}&&{T}_{e}=\displaystyle \frac{2{m}_{p}u}{3k\rho (2+R)},\end{eqnarray} \tag{ 7 }$$

where we have assumed that the plasma is composed of hydrogen, the ions are nonrelativistic, and the electrons are relativistic. Here $R\equiv {T}_{i}/{T}_{e}$ and

$$\begin{eqnarray}&&R={R}_{\mathrm{high}}\displaystyle \frac{{\beta }_{{\rm{p}}}^{2}}{1+{\beta }_{{\rm{p}}}^{2}}+\displaystyle \frac{1}{1+{\beta }_{{\rm{p}}}^{2}}.\end{eqnarray} \tag{ 8 }$$

This prescription has one parameter, ${R}_{\mathrm{high}}$, and sets ${T}_{e}\simeq {T}_{i}$ in low ${\beta }_{{\rm{p}}}$ regions and ${T}_{e}\simeq {T}_{i}/{R}_{\mathrm{high}}$ in the midplane of the disk. It is adapted from Mościbrodzka et al. (2016) and motivated by models for electron heating in a turbulent, collisionless plasma that preferentially heats the ions for ${\beta }_{{\rm{p}}}\gtrsim 1$ (e.g., Howes 2010; Kawazura et al. 2018).

(3) We must specify the observer inclination i, the orientation of the observer through the position angle $\mathrm{PA}$, the black hole mass $M$, and the distance D to the source. Non-EHT constraints on i, $\mathrm{PA}$, and $M$ are considered below; we have generated images at $i=12^\circ ,17^\circ ,22^\circ ,158^\circ ,163^\circ$, and 168° and a few at i = 148°. The position angle (PA) can be changed by simply rotating the image. All features of the models that we have examined, including $\dot{M}$, are insensitive to small changes in i. The image morphology does depend on whether i is greater than or less than 90°, as we will show below.

The model images are generated with a $160\times 160\,\mu \mathrm{as}$ field of view and $1\mu \mathrm{as}$ pixels, which are small compared to the $\sim 20\,\mu \mathrm{as}$ nominal resolution of EHT2017. Our analysis is insensitive to changes in the field of view and the pixel scale.

For $M$ we use the most likely value from the stellar absorption-line work, $6.2\times {10}^{9}{M}_{\odot }$ (Gebhardt et al. 2011). For the distance D we use $16.9\,\,\mathrm{Mpc}$, which is very close to that employed in Paper VI. The ratio ${GM}/({c}^{2}D)=3.62\,\mu \mathrm{as}$ (hereafter M/D) determines the angular scale of the images. For some models we have also generated images with $M=3.5\times {10}^{9}\,{M}_{\odot }$ to check that the analysis results are not predetermined by the input black hole mass.

The Image Library contains of order 60,000 images. We generate images from 100 to 500 distinct output files from each of the GRMHD models at each of ${R}_{\mathrm{high}}=1,10,20,40,80$, and 160. In comparing to the data we adjust the $\mathrm{PA}$ by rotation and the total flux and angular scale of the image by simply rescaling images from the standard parameters in the Image Library (see Figure 29 in Paper VI). Tests indicate that comparisons with the data are insensitive to the rescaling procedure unless the angular scaling factor or flux scaling factor is large.¹¹³

The comparisons with the data are also insensitive to image resolution.¹¹⁴

A representative set of time-averaged images from the Image Library are shown in Figures 2 and 3. From these figures it is clear that varying the parameters ${a}_{* }$, ϕ, and ${R}_{\mathrm{high}}$ can change the width and asymmetry of the photon ring and introduce additional structures exterior and interior to the photon ring.

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The location of the emitting plasma is shown in Figure 4, which shows a map of time- and azimuth-averaged emission regions for four representative ${a}_{* }\gt 0$ models. For SANE models, if ${R}_{\mathrm{high}}$ is low (high), emission is concentrated more in the disk (funnel wall), and the bright section of the ring is dominated by the disk (funnel wall).¹¹⁵ Appendix B shows images generated by considering emission only from particular regions of the flow, and the results are consistent with Figure 4.

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Figures 2 and 3 show that for both MAD and SANE models the bright section of the ring, which is generated by Doppler beaming, shifts from the top for negative spin, to a nearly symmetric ring at ${a}_{* }=0$, to the bottom for ${a}_{* }\gt 0$ (except the SANE ${R}_{\mathrm{high}}=1$ case, where the bright section is always at the bottom when i > 90°). That is, the location of the peak flux in the ring is controlled by the black hole spin: it always lies roughly 90 degrees counterclockwise from the projection of the spin vector on the sky. Some of the ring emission originates in the funnel wall at $r\lesssim 8\,{r}_{{\rm{g}}}$. The rotation of plasma in the funnel wall is in the same sense as plasma in the funnel, which is controlled by the dragging of magnetic field lines by the black hole. The funnel wall thus rotates opposite to the accretion flow if ${a}_{* }\lt 0$. This effect will be studied further in a later publication (Wong et al. 2019). The resulting relationships between disk angular momentum, black hole angular momentum, and observed ring asymmetry are illustrated in Figure 5.

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The time-averaged MAD images are almost independent of ${R}_{\mathrm{high}}$ and depend mainly on ${a}_{* }$. In MAD models much of the emission arises in regions with ${\beta }_{{\rm{p}}}\sim 1$, where ${R}_{\mathrm{high}}$ has little influence over the electron temperature, so the insensitivity to ${R}_{\mathrm{high}}$ is natural (see Figure 4). In SANE models emission arises at ${\beta }_{{\rm{p}}}\sim 10$, so the time-averaged SANE images, by contrast, depend strongly on ${R}_{\mathrm{high}}$. In low ${R}_{\mathrm{high}}$ SANE models, extended emission outside the photon ring, arising near the equatorial plane, is evident at ${R}_{\mathrm{high}}=1$. In large ${R}_{\mathrm{high}}$ SANE models the inner ring emission arises from the funnel wall, and once again the image looks like a thin ring (see Figure 4).

Figure 6 and the accompanying animation show the evolution of the images, visibility amplitudes, and closure phases over a $5000\,{r}_{{\rm{g}}}{c}^{-1}\approx 5\,\mathrm{yr}$ interval in a single simulation for M87. It is evident from the animation that turbulence in the simulations produces large fluctuations in the images, which imply changes in visibility amplitudes and closure phases that are large compared to measurement errors. The fluctuations are central to our procedure for comparing models with the data, described briefly below and in detail in Paper VI.

The timescale between frames in the animation is $50\,{r}_{{\rm{g}}}{c}^{-1}\simeq 18$ days, which is long compared to EHT2017 observing campaign. The images are highly correlated on timescales less than the innermost stable circular orbit (ISCO) orbital period, which for ${a}_{* }=0$ is $\simeq 15\ {r}_{{\rm{g}}}{c}^{-1}\simeq 5$ days, i.e., comparable to the duration of the EHT2017 campaign. If drawn from one of our models, we would expect the EHT2017 data to look like a single snapshot (Figures 6) rather than their time averages (Figures 2 and 3).

The model must be close to radiative equilibrium. The GRMHD models in the Simulation Library do not include radiative cooling, nor do they include a detailed prescription for particle energization. In nature the accretion flow and jet are expected to be cooled and heated by a combination of synchrotron and Compton cooling, turbulent dissipation, and Coulomb heating, which transfers energy from the hot ions to the cooler electrons. In our suite of simulations the parameter ${R}_{\mathrm{high}}$ can be thought of as a proxy for the sum of these processes. In a fully self-consistent treatment, some models would rapidly cool and settle to a lower electron temperature (see Mościbrodzka et al. 2011; Ryan et al. 2018; Chael et al. 2019). We crudely test for this by calculating the radiative efficiency $\epsilon \equiv {L}_{\mathrm{bol}}/(\dot{M}{c}^{2})$, where ${L}_{\mathrm{bol}}$ is the bolometric luminosity. If it is larger than the radiative efficiency of a thin, radiatively efficient disk,¹¹⁷ which depends only on ${a}_{* }$ (Novikov & Thorne 1973), then we reject the model as physically inconsistent.

We calculate ${L}_{\mathrm{bol}}$ with the Monte Carlo code grmonty  (Dolence et al. 2009), which incorporates synchrotron emission, absorption, Compton scattering at all orders, and bremsstrahlung. It assumes the same thermal eDF used in generating the Image Library. We calculate ${L}_{\mathrm{bol}}$ for 20% of the snapshots to minimize computational cost. We then average over snapshots to find $\langle {L}_{\mathrm{bol}}\rangle$. The mass accretion rate $\dot{M}$ is likewise computed for each snapshot and averaged over time. We reject models with  that is larger than the classical thin disk model. (Table 3 in Appendix A lists  for a large set of models.) All but two of the radiatively inconsistent models are MADs with ${a}_{* }\geqslant 0$ and ${R}_{\mathrm{high}}=1$. Eliminating all MAD models with ${a}_{* }\geqslant 0$ and ${R}_{\mathrm{high}}=1$ does not change any of our earlier conclusions.

As part of the EHT2017 campaign, we simultaneously observed M87 with the Chandra X-ray observatory and the Nuclear Spectroscopic Telescope Array (NuSTAR). The best fit to simultaneous Chandra and NuSTAR observations on 2017 April 12 and 14 implies a $2\mbox{--}10\,\mathrm{keV}$ luminosity of ${L}_{{{\rm{X}}}_{\mathrm{obs}}}\,=4.4\pm 0.1\times {10}^{40}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. We used the SEDs generated from the simulations while calculating ${L}_{\mathrm{bol}}$ to reject models that consistently overproduce X-rays; specifically, we reject models with $\mathrm{log}{L}_{{{\rm{X}}}_{\mathrm{obs}}}\lt \mathrm{log}\langle {L}_{{\rm{X}}}\rangle -2\sigma (\mathrm{log}{L}_{{\rm{X}}})$. We do not reject underluminous models because the X-rays could in principle be produced by direct synchrotron emission from nonthermal electrons or by other unresolved sources. Notice that ${L}_{{\rm{X}}}$ is highly variable in all models so that the X-ray observations currently reject only a few models. Table 3 in Appendix A shows $\langle {L}_{{\rm{X}}}\rangle$ as well as upper and lower limits for a set of models that is distributed uniformly across the parameter space.

In our models the X-ray flux is produced by inverse Compton scattering of synchrotron photons. The X-ray flux is an increasing function of ${\tau }_{T}{T}_{e}^{2}$ where τ_T is a characteristic Thomson optical depth (${\tau }_{T}\sim {10}^{-5}$), and the characteristic amplification factor for photon energies is $\propto {T}_{e}^{2}$ because the X-ray band is dominated by singly scattered photons interacting with relativistic electrons (we include all scattering orders in the Monte Carlo calculation). Increasing ${R}_{\mathrm{high}}$ at fixed ${F}_{\nu }(230\,\ \mathrm{GHz})$ tends to increase $\dot{M}$ and therefore τ_T and decrease T_e. The increase in T_e dominates in our ensemble of models, and so models with small ${R}_{\mathrm{high}}$ have larger ${L}_{{\rm{X}}}$, while models with large ${R}_{\mathrm{high}}$ have smaller ${L}_{{\rm{X}}}$. The effect is not strictly monotonic, however, because of noise in our sampling process and the highly variable nature of the X-ray emission.

The overluminous models are mostly SANE models with ${R}_{\mathrm{high}}\leqslant 20$. The model with the highest $\langle {L}_{{\rm{X}}}\rangle =4.2\,\times {10}^{42}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ is a SANE, ${a}_{* }=0$, ${R}_{\mathrm{high}}=10$ model. The corresponding model with ${R}_{\mathrm{high}}=1$ has $\langle {L}_{{\rm{X}}}\rangle =2.1\,\times {10}^{41}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, and the difference between these two indicates the level of variability and the sensitivity of the average to the brightest snapshot. The upshot of application of the ${L}_{{\rm{X}}}$ constraints is that ${L}_{{\rm{X}}}$ is sensitive to ${R}_{\mathrm{high}}$. Very low values of ${R}_{\mathrm{high}}$ are disfavored. ${L}_{{\rm{X}}}$ thus most directly constrains the electron temperature model.

Estimates of M87's jet power (${P}_{\mathrm{jet}}$) have been reviewed in Reynolds et al. (1996), Li et al. (2009), de Gasperin et al. (2012), Broderick et al. (2015), and Prieto et al. (2016). The estimates range from 10⁴² to ${10}^{45}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. This wide range is a consequence of both physical uncertainties in the models used to estimate ${P}_{\mathrm{jet}}$ and the wide range in length and timescales probed by the observations. Some estimates may sample a different epoch and thus provide little information on the state of the central engine during EHT2017. Nevertheless, observations of HST-1 yield ${P}_{\mathrm{jet}}\sim {10}^{44}\ \,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (e.g., Stawarz et al. 2006). HST-1 is within $\sim 70\,\mathrm{pc}$ of the central engine and, taking account of relativistic time foreshortening, may be sampling the central engine ${P}_{\mathrm{jet}}$ over the last few decades. Furthermore, the 1.3 mm light curve of M87 as observed by SMA shows $\lesssim 50 \%$ variability over decade timescales (Bower et al. 2015). Based on these considerations it seems reasonable to adopt a very conservative lower limit on jet power $\equiv {P}_{\mathrm{jet},\min }={10}^{42}\ \,\mathrm{erg}\,{{\rm{s}}}^{-1}$.

To apply this constraint we must define and measure ${P}_{\mathrm{jet}}$ in our models. Our procedure is discussed in detail in Appendix A. In brief, we measure the total energy flux in outflowing regions over the polar caps of the black hole in which the energy per unit rest mass exceeds $2.2\,{c}^{2}$, which corresponds to βγ = 1, where $\beta \equiv v/c$ and γ is Lorentz factor. The effect of changing this cutoff is also discussed in Appendix A. Because the cutoff is somewhat arbitrary, we also calculate ${P}_{\mathrm{out}}$ by including the energy flux in all outflowing regions over the polar caps of the black hole; that is, it includes the energy flux in any wide-angle, low-velocity wind. ${P}_{\mathrm{out}}$ represents a maximal definition of jet power. Table 3 in Appendix A shows ${P}_{\mathrm{jet}}$ as well as a total outflow power ${P}_{\mathrm{out}}$.

The constraint ${P}_{\mathrm{jet}}\gt {P}_{\mathrm{jet},\min }={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ rejects all ${a}_{* }=0$ models. This conclusion is not sensitive to the definition of ${P}_{\mathrm{jet}}$: all ${a}_{* }=0$ models also have total outflow power ${P}_{\mathrm{out}}\,\lt {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. The most powerful ${a}_{* }=0$ model is a MAD model with ${R}_{\mathrm{high}}=160$, which has ${P}_{\mathrm{out}}=3.7\times {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and ${P}_{\mathrm{jet}}$ consistent with 0. We conclude that our ${a}_{* }=0$ models are ruled out.

Can the ${a}_{* }=0$ models be saved by changing the eDF? Probably not. There is no evidence from the GRMHD simulations that these models are capable of producing a relativistic outflow with $\beta \gamma \gt 1$. Suppose, however, that we are willing to identify the nonrelativistic outflow, whose power is measured by ${P}_{\mathrm{out}}$, with the jet. Can ${P}_{\mathrm{out}}$ be raised to meet our conservative threshold on jet power? Here the answer is yes, in principle, and this can be done by changing the eDF. The eDF and ${P}_{\mathrm{out}}$ are coupled because ${P}_{\mathrm{out}}$ is determined by $\dot{M}$, and $\dot{M}$ is adjusted to produce the observed compact mm flux. The relationship between $\dot{M}$ and mm flux depends upon the eDF. If the eDF is altered to produce mm photons less efficiently (for example, by lowering T_e in a thermal model), then $\dot{M}$ and therefore ${P}_{\mathrm{out}}$ increase. A typical nonthermal eDF, by contrast, is likely to produce mm photons with greater efficiency by shifting electrons out of the thermal core and into a nonthermal tail. It will therefore lower $\dot{M}$ and thus ${P}_{\mathrm{out}}$. A thermal eDF with lower T_e could have higher ${P}_{\mathrm{out}}$, as is evident in the large ${R}_{\mathrm{high}}$ SANE models in Table 3. There are observational and theoretical lower limits on T_e, however, including a lower limit provided by the observed brightness temeprature. As T_e declines, n_e and B increase and that has implications for source linear polarization (Mościbrodzka et al. 2017; Jiménez-Rosales & Dexter 2018), which will be explored in future work. As T_e declines and n_e and n_i increase there is also an increase in energy transfer from ions to electrons by Coulomb coupling, and this sets a floor on T_e.

The requirement that ${P}_{\mathrm{jet}}\gt {P}_{\mathrm{jet},\min }$ eliminates many models other than the ${a}_{* }=0$ models. All SANE models with $| {a}_{* }| =0.5$ fail to produce jets with the required minimum power. Indeed, they also fail the less restrictive condition ${P}_{\mathrm{out}}\gt {P}_{\mathrm{jet},\min }$, so this conclusion is insensitive to the definition of the jet. We conclude that among the SANE models, only high-spin models survive.

At this point it is worth revisiting the SANE, ${R}_{\mathrm{high}}=1$, ${a}_{* }=-0.94$ model that favored a low black hole mass in Section 5. These models are not rejected by a naive application of the ${P}_{\mathrm{jet}}\gt {P}_{\mathrm{jet},\min }$ criterion, but they are marginal. Notice, however, that we needed to assume a mass in applying the this criterion. We have consistently assumed $M=6.2\times {10}^{9}\,{M}_{\odot }$. If we use the $M\sim 3\times {10}^{9}\,{M}_{\odot }$ implied by the best-fit M/D, then $\dot{M}$ drops by a factor of two, ${P}_{\mathrm{jet}}$ drops below the threshold and the model is rejected.

The lower limit on jet power ${P}_{\mathrm{jet},\min }={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ is conservative and the true jet power is likely higher. If we increased ${P}_{\mathrm{jet},\min }$ to $3\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, the only surviving models would have $| {a}_{* }| =0.94$ and ${R}_{\mathrm{high}}\geqslant 10$. This conclusion is also not sensitive to the definition of the jet power: applying the same cut to ${P}_{\mathrm{out}}$ adds only a single model with $| {a}_{* }| \lt 0.94$, the ${R}_{\mathrm{high}}=160$, ${a}_{* }=0.5$ MAD model. The remainder have ${a}_{* }=0.94$. Interestingly, the most powerful jets in our ensemble of models are produced by SANE, ${a}_{* }=-0.94$, ${R}_{\mathrm{high}}=160$ models, with ${P}_{\mathrm{jet}}\simeq {10}^{43}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}$.

Estimates for ${P}_{\mathrm{jet}}$ extend to ${10}^{45}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, but in our ensemble of models the maximum ${P}_{\mathrm{jet}}\sim {10}^{43}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. Possible explanations include: (1) ${P}_{\mathrm{jet}}$ is variable and the estimates probe the central engine power at earlier epochs (discussed above); (2) the ${P}_{\mathrm{jet}}$ estimates are too large; or (3) the models are in error. How might our models be modified to produce a larger ${P}_{\mathrm{jet}}$? For a given magnetic field configuration the jet power scales with $\dot{M}{c}^{2}$. To increase ${P}_{\mathrm{jet}}$, then, one must reduce the mm flux per accreted nucleon so that at fixed mm flux density $\dot{M}$ increases.¹¹⁸ Lowering T_e in a thermal model is unlikely to work because lower T_e implies higher synchrotron optical depth, which increases the ring width. We have done a limited series of experiments that suggest that even a modest decrease in T_e would produce a broad ring that is inconsistent with EHT2017 (Paper VI). What is required, then, is a nonthermal (or multitemperature) model with a large population of cold electrons that are invisible at mm wavelength (for a thermal subpopulation, ${{\rm{\Theta }}}_{e,\mathrm{cold}}\lt 1$), and a population of higher-energy electrons that produces the observed mm flux (see Falcke & Biermann 1995). We have not considered such models here, but we note that they are in tension with current ideas about dissipation of turbulence because they require efficient suppression of electron heating.

The ${P}_{\mathrm{jet}}$ in our models is dominated by Poynting flux in the force-free region around the axis (the "funnel"), as in the Blandford & Znajek (1977) force-free magnetosphere model. The energy flux is concentrated along the walls of the funnel.¹¹⁹ Tchekhovskoy et al. (2011) provided an expression for the energy flux in the funnel, the so-called Blandford–Znajek power ${P}_{\mathrm{BZ}}$, which becomes, in our units,

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{BZ}} & = & 2.8\,f({a}_{* }){\left(\displaystyle \frac{\phi }{15}\right)}^{2}\dot{M}{c}^{2}\\ & = & 2.2\times {10}^{43}\,f({a}_{* }){\left(\displaystyle \frac{\phi }{15}\right)}^{2}\\ & & \times \,\left(\displaystyle \frac{\dot{M}}{{10}^{-6}{\dot{M}}_{\mathrm{Edd}}}\right)\left(\displaystyle \frac{M}{6.2\times {10}^{9}{M}_{\odot }}\right)\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}\end{array}\end{eqnarray} \tag{ 9 }$$

where $f({a}_{* })\approx {a}_{* }^{2}{\left(1+\sqrt{1-{a}_{* }^{2}}\right)}^{-2}$ (a good approximation for ${a}_{* }\lt 0.95$) and ${\dot{M}}_{\mathrm{Edd}}=137\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ for $M=6.2\times {10}^{9}\,{M}_{\odot }$. This expression was developed for models with a thin disk in the equatorial plane. ${P}_{\mathrm{BZ}}$ is lower for models where the force-free region is excluded by a thicker disk around the equatorial plane. Clearly ${P}_{\mathrm{BZ}}$ is comparable to observational estimates of ${P}_{\mathrm{jet}}$.

In our models (see Table 3) ${P}_{\mathrm{jet}}$ follows the above scaling relation but with a smaller coefficient. The ratio of coefficients is model dependent and varies from 0.15 to 0.83. This is likely because the force-free region is restricted to a cone around the poles of the black hole, and the width of the cone varies by model. Indeed, the coefficient is larger for MAD than for SANE models, which is consistent with this idea because MAD models have a wide funnel and SANE models have a narrow funnel. This also suggests that future comparison of synthetic 43 and 86 GHz images from our models with lower-frequency VLBI data may further constrain the magnetic flux on the black hole.

The connection between the Poynting flux in the funnel and black hole spin has been discussed for some time in the simulation literature, beginning with McKinney & Gammie (2004; see also McKinney 2006; McKinney & Narayan 2007). The structure of the funnel magnetic field can be time-averaged and shown to match the analytic solution of Blandford & Znajek (1977). Furthermore, the energy flux density can be time-averaged and traced back to the event horizon. Is the energy contained in black hole spin sufficient to drive the observed jet over the jet lifetime? The spindown timescale is $\tau =(M-{M}_{\mathrm{irr}}){c}^{2}/{P}_{\mathrm{jet}}$, where ${M}_{\mathrm{irr}}\equiv M{\left(\left(1+\sqrt{1-{a}_{* }^{2}}\right)/2\right)}^{1/2}$ is the irreducible mass of the black hole. For the ${a}_{* }=0.94$ MAD model with ${R}_{\mathrm{high}}=160$, $\tau =7.3\times {10}^{12}\,\mathrm{yr}$, which is long compared to a Hubble time ($\sim {10}^{10}$ yr). Indeed, the spindown time for all models is long compared to the Hubble time.

We conclude that for models that have sufficiently powerful jets and are consistent with EHT2017, ${P}_{\mathrm{jet}}$ is driven by extraction of black hole spin energy through the Blandford–Znajek process.

We have applied constraints from AIS, a radiative self-consistency constraint, a constraint on maximum X-ray luminosity, and a constraint on minimum jet power. Which models survive? Here we consider only models for which we have calculated ${L}_{{\rm{X}}}$ and ${L}_{\mathrm{bol}}$. Table 2 summarizes the results. Here we consider only i = 163° (for ${a}_{* }\geqslant 0$) and i = 17° (for ${a}_{* }\lt 0$). The first three columns give the model parameters. The next four columns show the result of application of each constraint: Themis-AIS (here broken out by individual model rather than groups of models), radiative efficiency ($\epsilon \lt {\epsilon }_{\mathrm{thin}\mathrm{disk}}$), ${L}_{{\rm{X}}}$, and ${P}_{\mathrm{jet}}$.

Table 2.  Rejection Table

Fluxa	${a}_{* }$ b	${R}_{\mathrm{high}}$ c	AISd	e	${L}_{{\rm{X}}}$ f	${P}_{\mathrm{jet}}$ g
SANE	−0.94	1	Fail	Pass	Pass	Pass	Fail
SANE	−0.94	10	Pass	Pass	Pass	Pass	Pass
SANE	−0.94	20	Pass	Pass	Pass	Pass	Pass
SANE	−0.94	40	Pass	Pass	Pass	Pass	Pass
SANE	−0.94	80	Pass	Pass	Pass	Pass	Pass
SANE	−0.94	160	Fail	Pass	Pass	Pass	Fail
SANE	−0.5	1	Pass	Pass	Fail	Fail	Fail
SANE	−0.5	10	Pass	Pass	Fail	Fail	Fail
SANE	−0.5	20	Pass	Pass	Pass	Fail	Fail
SANE	−0.5	40	Pass	Pass	Pass	Fail	Fail
SANE	−0.5	80	Fail	Pass	Pass	Fail	Fail
SANE	−0.5	160	Pass	Pass	Pass	Fail	Fail
SANE	0	1	Pass	Pass	Pass	Fail	Fail
SANE	0	10	Pass	Pass	Pass	Fail	Fail
SANE	0	20	Pass	Pass	Fail	Fail	Fail
SANE	0	40	Pass	Pass	Pass	Fail	Fail
SANE	0	80	Pass	Pass	Pass	Fail	Fail
SANE	0	160	Pass	Pass	Pass	Fail	Fail
SANE	+0.5	1	Pass	Pass	Pass	Fail	Fail
SANE	+0.5	10	Pass	Pass	Pass	Fail	Fail
SANE	+0.5	20	Pass	Pass	Pass	Fail	Fail
SANE	+0.5	40	Pass	Pass	Pass	Fail	Fail
SANE	+0.5	80	Pass	Pass	Pass	Fail	Fail
SANE	+0.5	160	Pass	Pass	Pass	Fail	Fail
SANE	+0.94	1	Pass	Fail	Pass	Fail	Fail
SANE	+0.94	10	Pass	Fail	Pass	Fail	Fail
SANE	+0.94	20	Pass	Pass	Pass	Fail	Fail
SANE	+0.94	40	Pass	Pass	Pass	Fail	Fail
SANE	+0.94	80	Pass	Pass	Pass	Pass	Pass
SANE	+0.94	160	Pass	Pass	Pass	Pass	Pass
MAD	−0.94	1	Fail	Fail	Pass	Pass	Fail
MAD	−0.94	10	Fail	Pass	Pass	Pass	Fail
MAD	−0.94	20	Fail	Pass	Pass	Pass	Fail
MAD	−0.94	40	Fail	Pass	Pass	Pass	Fail
MAD	−0.94	80	Fail	Pass	Pass	Pass	Fail
MAD	−0.94	160	Fail	Pass	Pass	Pass	Fail
MAD	−0.5	1	Pass	Fail	Pass	Fail	Fail
MAD	−0.5	10	Pass	Pass	Pass	Fail	Fail
MAD	−0.5	20	Pass	Pass	Pass	Pass	Pass
MAD	−0.5	40	Pass	Pass	Pass	Pass	Pass
MAD	−0.5	80	Pass	Pass	Pass	Pass	Pass
MAD	−0.5	160	Pass	Pass	Pass	Pass	Pass
MAD	0	1	Pass	Fail	Pass	Fail	Fail
MAD	0	10	Pass	Pass	Pass	Fail	Fail
MAD	0	20	Pass	Pass	Pass	Fail	Fail
MAD	0	40	Pass	Pass	Pass	Fail	Fail
MAD	0	80	Pass	Pass	Pass	Fail	Fail
MAD	0	160	Pass	Pass	Pass	Fail	Fail
MAD	+0.5	1	Pass	Fail	Pass	Fail	Fail
MAD	+0.5	10	Pass	Pass	Pass	Pass	Pass
MAD	+0.5	20	Pass	Pass	Pass	Pass	Pass
MAD	+0.5	40	Pass	Pass	Pass	Pass	Pass
MAD	+0.5	80	Pass	Pass	Pass	Pass	Pass
MAD	+0.5	160	Pass	Pass	Pass	Pass	Pass
MAD	+0.94	1	Pass	Fail	Fail	Pass	Fail
MAD	+0.94	10	Pass	Fail	Pass	Pass	Fail
MAD	+0.94	20	Pass	Pass	Pass	Pass	Pass
MAD	+0.94	40	Pass	Pass	Pass	Pass	Pass
MAD	+0.94	80	Pass	Pass	Pass	Pass	Pass
MAD	+0.94	160	Pass	Pass	Pass	Pass	Pass

Notes.

^aflux: net magnetic flux on the black hole (MAD, SANE). ^b ${a}_{* }$: dimensionless black hole spin. ^c ${R}_{\mathrm{high}}$: electron temperature parameter. See Equation (8). ^dAverage Image Scoring (Themis-AIS), models are rejected if $\langle p\rangle \leqslant 0.01$. See Section 4 and Table 1. ^e: radiative efficiency, models are rejected if is larger than the corresponding thin disk efficiency. See Section 6.1. ^f ${L}_{{\rm{X}}}$: X-ray luminosity; models are rejected if $\langle {L}_{{\rm{X}}}\rangle {10}^{-2\sigma }\gt 4.4\times {10}^{40}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. See Section 6.2. ^g ${P}_{\mathrm{jet}}$: jet power, models are rejected if ${P}_{\mathrm{jet}}\leqslant {10}^{42}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. See Section 6.3.

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The final column gives the logical AND of the previous four columns, and allows a model to pass only if it passes all tests. Evidently most of the SANE models fail, with the exception of some ${a}_{* }=-0.94$ models and a few ${a}_{* }=0.94$ models with large ${R}_{\mathrm{high}}$. A much larger fraction of the MAD models pass, although ${a}_{* }=0$ models all fail because of inadequate jet power. MAD models with small ${R}_{\mathrm{high}}$ also fail. It is the jet power constraint that rejects the largest number of models.

Post-processed GRMHD simulations that are consistent with EHT data and the flux density of 1.3 mm emission in M87 can yield unphysically large radiative efficiencies (see Section 6). This implies that the radiative cooling timescale is comparable to or less than the advection timescale. As a consequence, including radiative cooling in simulations may be necessary to recover self-consistent models (see Mościbrodzka et al. 2011; Dibi et al. 2012). In our models we use a single parameter, ${R}_{\mathrm{high}}$, to adjust T_e and account for all effects that might influence the electron energy density. How good is this approximation?

The importance of radiative cooling can be assessed using newly developed, state-of-the-art general relativistic radiation GRMHD ("radiation GRMHD") codes. Sa̧dowski et al. (2013b; see also Sa̧dowski et al. 2014, 2017; McKinney et al. 2014) applied the M1 closure (Levermore 1984), which treats the radiation as a relativistic fluid. Ryan et al. (2015) introduced a Monte Carlo radiation GRMHD method, allowing for full frequency-dependent radiation transport. Models for turbulent dissipation into the electrons and ions, as well as heating and cooling physics that sets the temperature ratio T_i/T_e, have been added to GRMHD and radiative GRMHD codes and used in simulations of Sgr A* (Ressler et al. 2015, 2017; Chael et al. 2018) and M87 (Ryan et al. 2018; Chael et al. 2019). While the radiative cooling and Coulomb coupling physics in these simulations is well understood, the particle heating process, especially the relative heating rates of ions and electrons, remains uncertain.

Radiation GRMHD models are computationally expensive per run and do not have the same scaling freedom as the GRMHD models, so they need to be repeatedly re-run with different initial conditions until they produce the correct 1.3 mm flux density. It is therefore impractical to survey the parameter space using radiation GRMHD. It is possible, however, to check individual GRMHD models against existing radiation GRMHD models of M87 (Ryan et al. 2018; Chael et al. 2019).

The SANE radiation GRMHD models of Ryan et al. (2018) with ${a}_{* }=0.94$ and $M=6\times {10}^{9}\,{M}_{\odot }$ can be compared to GRMHD SANE ${a}_{* }=0.94$ models at various values of ${R}_{\mathrm{high}}$. The radiative models have $\dot{M}/{\dot{M}}_{\mathrm{Edd}}=5.2\times {10}^{-6}$ and ${P}_{\mathrm{jet}}\,=5.1\times {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. The GRMHD models in this work have, for $1\leqslant {R}_{\mathrm{high}}\leqslant 160$, $0.36\times {10}^{-6}\leqslant \dot{M}/{\dot{M}}_{\mathrm{Edd}}\leqslant 20\times {10}^{-6}$, and $0.22\leqslant {P}_{\mathrm{jet}}/({10}^{41}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1})\leqslant 12$ (Table 3). Evidently the mass accretion rates and jet powers in the GRMHD models span a wide range that depends on ${R}_{\mathrm{high}}$, but when we choose ${R}_{\mathrm{high}}$ = 10 − 20 they are similar to what is found in the radiative GRMHD model when using the turbulent electron heating model (Howes 2010).

Table 3.  Model Table

Flux	ϕ	Spin	${R}_{\mathrm{high}}$	${L}_{\mathrm{bol}}/\left(\dot{M}\,{c}^{2}\right)$	${L}_{{\rm{X}}}$ (cgs)	${P}_{\mathrm{jet}}$ (cgs)	${P}_{\mathrm{jet}}/\left(\dot{M}\,{c}^{2}\right)$	${P}_{\mathrm{out}}$ (cgs)	${P}_{\mathrm{out}}/\left(\dot{M}\,{c}^{2}\right)$	${P}_{\mathrm{jet},\mathrm{em}}/{P}_{\mathrm{jet}}$	$\dot{M}/{\dot{M}}_{\mathrm{Edd}}$
SANE	1.02	−0.94	1	1.27 × 10−2	${3.18}_{\gt 0.20}^{\lt 49.55}\times {10}^{41}$	1.16 × 1042	5.34 × 10−3	1.19 × 1042	5.48 × 10−3	0.84	2.77 × 10−5
SANE	1.02	−0.94	10	1.6 × 10−3	${9.62}_{\gt 1.44}^{\lt 64.42}\times {10}^{40}$	4.94 × 1042	5.34 × 10−3	5.07 × 1042	5.48 × 10−3	0.84	1.19 × 10−4
SANE	1.02	−0.94	20	6.09 × 10−4	${3.26}_{\gt 0.90}^{\lt 11.86}\times {10}^{40}$	5.8 × 1042	5.34 × 10−3	5.96 × 1042	5.48 × 10−3	0.84	1.39 × 10−4
SANE	1.02	−0.94	40	2.45 × 10−4	${8.89}_{\gt 1.56}^{\lt 50.53}\times {10}^{39}$	7.02 × 1042	5.34 × 10−3	7.21 × 1042	5.48 × 10−3	0.84	1.69 × 10−4
SANE	1.02	−0.94	80	1.33 × 10−4	${2.65}_{\gt 0.39}^{\lt 18.26}\times {10}^{39}$	8.89 × 1042	5.34 × 10−3	9.13 × 1042	5.48 × 10−3	0.84	2.13 × 10−4
SANE	1.02	−0.94	160	7.12 × 10−5	${6.36}_{\gt 0.73}^{\lt 55.27}\times {10}^{38}$	1.2 × 1043	5.34 × 10−3	1.23 × 1043	5.48 × 10−3	0.84	2.87 × 10−4
SANE	1.11	−0.5	1	1.62 × 10−2	${1.97}_{\gt 0.98}^{\lt 3.94}\times {10}^{41}$	2.62 × 1040	1.86 × 10−4	3.84 × 1040	2.72 × 10−4	0.88	1.81 × 10−5
SANE	1.11	−0.5	10	2.17 × 10−3	${1.94}_{\gt 0.69}^{\lt 5.40}\times {10}^{41}$	1.95 × 1041	1.86 × 10−4	2.85 × 1041	2.72 × 10−4	0.88	1.34 × 10−4
SANE	1.11	−0.5	20	6.69 × 10−4	${3.72}_{\gt 1.80}^{\lt 7.72}\times {10}^{40}$	2.26 × 1041	1.86 × 10−4	3.31 × 1041	2.72 × 10−4	0.88	1.56 × 10−4
SANE	1.11	−0.5	40	2.47 × 10−4	${9.44}_{\gt 6.67}^{\lt 13.37}\times {10}^{39}$	2.62 × 1041	1.86 × 10−4	3.83 × 1041	2.72 × 10−4	0.88	1.81 × 10−4
SANE	1.11	−0.5	80	1.26 × 10−4	${1.23}_{\gt 0.33}^{\lt 4.58}\times {10}^{39}$	3.2 × 1041	1.86 × 10−4	4.68 × 1041	2.72 × 10−4	0.88	2.21 × 10−4
SANE	1.11	−0.5	160	7.86 × 10−5	${3.72}_{\gt 0.83}^{\lt 16.68}\times {10}^{38}$	4.21 × 1041	1.86 × 10−4	6.16 × 1041	2.72 × 10−4	0.88	2.9 × 10−4
SANE	0.99	0	1	3.17 × 10−2	${2.08}_{\gt 0.02}^{\lt 194.22}\times {10}^{41}$	2.24 × 1036	4.4 × 10−8	5.22 × 1039	1.03 × 10−4	1.01	6.5 × 10−6
SANE	0.99	0	10	1.88 × 10−2	${4.2}_{\gt 0.04}^{\lt 425.40}\times {10}^{42}$	4.38 × 1037	4.4 × 10−8	1.02 × 1041	1.03 × 10−4	1.01	1.27 × 10−4
SANE	0.99	0	20	5.83 × 10−3	${1.57}_{\gt 0.06}^{\lt 39.69}\times {10}^{42}$	8.02 × 1037	4.4 × 10−8	1.87 × 1041	1.03 × 10−4	1.01	2.33 × 10−4
SANE	0.99	0	40	7.8 × 10−4	${8.92}_{\gt 1.92}^{\lt 41.45}\times {10}^{40}$	9.16 × 1037	4.4 × 10−8	2.14 × 1041	1.03 × 10−4	1.01	2.66 × 10−4
SANE	0.99	0	80	1.69 × 10−4	${2.5}_{\gt 0.33}^{\lt 19.17}\times {10}^{39}$	1.03 × 1038	4.4 × 10−8	2.41 × 1041	1.03 × 10−4	1.01	3 × 10−4
SANE	0.99	0	160	1.08 × 10−4	${3.44}_{\gt 0.89}^{\lt 13.32}\times {10}^{38}$	1.23 × 1038	4.4 × 10−8	2.87 × 1041	1.03 × 10−4	1.01	3.57 × 10−4
SANE	1.10	0.5	1	4.97 × 10−2	${5.5}_{\gt 0.88}^{\lt 34.41}\times {10}^{40}$	2.57 × 1039	1.63 × 10−4	9.19 × 1039	5.86 × 10−4	0.88	2.01 × 10−6
SANE	1.10	0.5	10	5.98 × 10−3	${4.73}_{\gt 0.25}^{\lt 88.59}\times {10}^{40}$	1.91 × 1040	1.64 × 10−4	6.84 × 1040	5.86 × 10−4	0.88	1.5 × 10−5
SANE	1.10	0.5	20	3.33 × 10−3	${3.83}_{\gt 0.30}^{\lt 49.18}\times {10}^{40}$	4.09 × 1040	1.64 × 10−4	1.47 × 1041	5.86 × 10−4	0.88	3.2 × 10−5
SANE	1.10	0.5	40	1.74 × 10−3	${2.52}_{\gt 0.28}^{\lt 22.73}\times {10}^{40}$	8.02 × 1040	1.64 × 10−4	2.87 × 1041	5.86 × 10−4	0.88	6.28 × 10−5
SANE	1.10	0.5	80	6.95 × 10−4	${7.84}_{\gt 0.67}^{\lt 91.92}\times {10}^{39}$	1.27 × 1041	1.64 × 10−4	4.55 × 1041	5.86 × 10−4	0.88	9.95 × 10−5
SANE	1.10	0.5	160	2.78 × 10−4	${1.37}_{\gt 0.08}^{\lt 22.85}\times {10}^{39}$	1.69 × 1041	1.63 × 10−4	6.06 × 1041	5.86 × 10−4	0.88	1.33 × 10−4
SANE	1.64	0.94	1	1.4	${2.38}_{\gt 0.02}^{\lt 359.03}\times {10}^{41}$	2.2 × 1040	7.76 × 10−3	3.38 × 1040	1.19 × 10−2	0.82	3.63 × 10−7
SANE	1.64	0.94	10	2.7 × 10−1	${2.79}_{\gt 0.02}^{\lt 508.99}\times {10}^{41}$	1.4 × 1041	7.76 × 10−3	2.15 × 1041	1.19 × 10−2	0.82	2.31 × 10−6
SANE	1.64	0.94	20	1.74 × 10−1	${5.75}_{\gt 0.02}^{\lt 1685.98}\times {10}^{41}$	3.22 × 1041	7.76 × 10−3	4.94 × 1041	1.19 × 10−2	0.82	5.31 × 10−6
SANE	1.64	0.94	40	7.2 × 10−2	${4.71}_{\gt 0.01}^{\lt 2490.36}\times {10}^{41}$	5.97 × 1041	7.76 × 10−3	9.17 × 1041	1.19 × 10−2	0.82	9.84 × 10−6
SANE	1.64	0.94	80	2.38 × 10−2	${1.42}_{\gt 0.00}^{\lt 860.83}\times {10}^{41}$	8.87 × 1041	7.76 × 10−3	1.36 × 1042	1.19 × 10−2	0.82	1.46 × 10−5
SANE	1.64	0.94	160	8.45 × 10−3	${3.22}_{\gt 0.01}^{\lt 1687.88}\times {10}^{40}$	1.23 × 1042	7.76 × 10−3	1.89 × 1042	1.19 × 10−2	0.82	2.03 × 10−5
MAD	8.04	−0.94	1	7.61 × 10−1	${2.12}_{\gt 0.25}^{\lt 17.74}\times {10}^{41}$	1.36 × 1042	2.09 × 10−1	1.6 × 1042	2.46 × 10−1	0.75	8.32 × 10−7
MAD	8.04	−0.94	10	7.54 × 10−2	${5.76}_{\gt 0.49}^{\lt 68.06}\times {10}^{40}$	1.97 × 1042	2.09 × 10−1	2.32 × 1042	2.46 × 10−1	0.75	1.21 × 10−6
MAD	8.04	−0.94	20	3.76 × 10−2	${2.27}_{\gt 0.18}^{\lt 29.09}\times {10}^{40}$	2.38 × 1042	2.09 × 10−1	2.8 × 1042	2.46 × 10−1	0.75	1.46 × 10−6
MAD	8.04	−0.94	40	2.07 × 10−2	${6.18}_{\gt 0.49}^{\lt 77.36}\times {10}^{39}$	3 × 1042	2.09 × 10−1	3.54 × 1042	2.46 × 10−1	0.75	1.84 × 10−6
MAD	8.04	−0.94	80	1.17 × 10−2	${1.32}_{\gt 0.07}^{\lt 26.36}\times {10}^{39}$	3.99 × 1042	2.09 × 10−1	4.71 × 1042	2.46 × 10−1	0.75	2.45 × 10−6
MAD	8.04	−0.94	160	6.52 × 10−3	${2.57}_{\gt 0.14}^{\lt 46.76}\times {10}^{38}$	5.7 × 1042	2.09 × 10−1	6.73 × 1042	2.46 × 10−1	0.75	3.5 × 10−6
MAD	12.25	−0.5	1	2.96 × 10−1	${1.39}_{\gt 0.17}^{\lt 11.56}\times {10}^{41}$	3.43 × 1041	4.91 × 10−2	6.04 × 1041	8.64 × 10−2	0.82	8.95 × 10−7
MAD	12.25	−0.5	10	4.53 × 10−2	${2.43}_{\gt 0.30}^{\lt 19.86}\times {10}^{40}$	5.31 × 1041	4.92 × 10−2	9.33 × 1041	8.64 × 10−2	0.82	1.38 × 10−6
MAD	12.25	−0.5	20	2.67 × 10−2	${8.18}_{\gt 0.86}^{\lt 77.51}\times {10}^{39}$	6.45 × 1041	4.92 × 10−2	1.13 × 1042	8.64 × 10−2	0.82	1.68 × 10−6
MAD	12.25	−0.5	40	1.69 × 10−2	${2.17}_{\gt 0.21}^{\lt 22.33}\times {10}^{39}$	8.07 × 1041	4.92 × 10−2	1.42 × 1042	8.64 × 10−2	0.82	2.1 × 10−6
MAD	12.25	−0.5	80	1.07 × 10−2	${4.87}_{\gt 0.47}^{\lt 50.76}\times {10}^{38}$	1.05 × 1042	4.92 × 10−2	1.85 × 1042	8.64 × 10−2	0.82	2.74 × 10−6
MAD	12.25	−0.5	160	6.43 × 10−3	${1.09}_{\gt 0.17}^{\lt 7.06}\times {10}^{38}$	1.46 × 1042	4.92 × 10−2	2.57 × 1042	8.64 × 10−2	0.82	3.81 × 10−6
MAD	15.44	0	1	2.67 × 10−1	${1.22}_{\gt 0.10}^{\lt 14.60}\times {10}^{41}$	0.0	0.0	8.39 × 1040	1.51 × 10−2	0.00	7.12 × 10−7
MAD	15.44	0	10	4.53 × 10−2	${1.86}_{\gt 0.11}^{\lt 31.55}\times {10}^{40}$	0.0	0.0	1.39 × 1041	1.51 × 10−2	0.00	1.18 × 10−6
MAD	15.44	0	20	2.81 × 10−2	${5.98}_{\gt 0.35}^{\lt 101.81}\times {10}^{39}$	0.0	0.0	1.71 × 1041	1.51 × 10−2	0.00	1.46 × 10−6
MAD	15.44	0	40	1.85 × 10−2	${1.63}_{\gt 0.10}^{\lt 27.75}\times {10}^{39}$	0.0	0.0	2.15 × 1041	1.51 × 10−2	0.00	1.82 × 10−6
MAD	15.44	0	80	1.21 × 10−2	${3.51}_{\gt 0.20}^{\lt 61.34}\times {10}^{38}$	0.0	0.0	2.77 × 1041	1.51 × 10−2	0.00	2.35 × 10−6
MAD	15.44	0	160	7.63 × 10−3	${8.06}_{\gt 0.81}^{\lt 80.62}\times {10}^{37}$	0.0	0.0	3.73 × 1041	1.51 × 10−2	0.00	3.17 × 10−6
MAD	15.95	0.5	1	5.45 × 10−1	${1.57}_{\gt 0.21}^{\lt 11.98}\times {10}^{41}$	4.64 × 1041	1.16 × 10−1	6.74 × 1041	1.69 × 10−1	0.85	5.11 × 10−7
MAD	15.95	0.5	10	9.45 × 10−2	${2.71}_{\gt 0.20}^{\lt 36.30}\times {10}^{40}$	8.07 × 1041	1.16 × 10−1	1.17 × 1042	1.69 × 10−1	0.85	8.89 × 10−7
MAD	15.95	0.5	20	5.54 × 10−2	${9.67}_{\gt 0.74}^{\lt 126.69}\times {10}^{39}$	1.02 × 1042	1.16 × 10−1	1.49 × 1042	1.69 × 10−1	0.85	1.13 × 10−6
MAD	15.95	0.5	40	3.5 × 10−2	${3.3}_{\gt 0.28}^{\lt 39.01}\times {10}^{39}$	1.32 × 1042	1.16 × 10−1	1.92 × 1042	1.69 × 10−1	0.85	1.45 × 10−6
MAD	15.95	0.5	80	2.22 × 10−2	${8}_{\gt 0.70}^{\lt 91.84}\times {10}^{38}$	1.74 × 1042	1.16 × 10−1	2.52 × 1042	1.69 × 10−1	0.85	1.92 × 10−6
MAD	15.95	0.5	160	1.35 × 10−2	${1.79}_{\gt 0.38}^{\lt 8.44}\times {10}^{38}$	2.38 × 1042	1.16 × 10−1	3.46 × 1042	1.69 × 10−1	0.85	2.62 × 10−6
MAD	12.78	0.94	1	3.65	${5.19}_{\gt 0.62}^{\lt 43.60}\times {10}^{41}$	1.97 × 1042	8.23 × 10−1	2.29 × 1042	9.55 × 10−1	0.80	3.07 × 10−7
MAD	12.78	0.94	10	3.68 × 10−1	${1.3}_{\gt 0.13}^{\lt 13.22}\times {10}^{41}$	3.04 × 1042	8.23 × 10−1	3.52 × 1042	9.55 × 10−1	0.80	4.73 × 10−7
MAD	12.78	0.94	20	1.79 × 10−1	${5}_{\gt 0.44}^{\lt 56.22}\times {10}^{40}$	3.73 × 1042	8.23 × 10−1	4.33 × 1042	9.55 × 10−1	0.80	5.81 × 10−7
MAD	12.78	0.94	40	9.43 × 10−2	${1.54}_{\gt 0.11}^{\lt 22.13}\times {10}^{40}$	4.74 × 1042	8.23 × 10−1	5.5 × 1042	9.55 × 10−1	0.80	7.38 × 10−7
MAD	12.78	0.94	80	5.19 × 10−2	${3.74}_{\gt 0.17}^{\lt 80.85}\times {10}^{39}$	6.26 × 1042	8.23 × 10−1	7.27 × 1042	9.55 × 10−1	0.80	9.75 × 10−7
MAD	12.78	0.94	160	2.82 × 10−2	${6.97}_{\gt 0.26}^{\lt 186.48}\times {10}^{38}$	8.75 × 1042	8.23 × 10−1	1.02 × 1043	9.55 × 10−1	0.80	1.36 × 10−6

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