news2019. 4. 11. 05:20



‘사건 지평면 망원경’이 타원형 은하계 M87의 중심부 촘촘한 radio source 지도를 완성했다. 


주파수 1.3 밀리 미터. 선례가 없을 정도 식별 능력이 있는 해상도 (angular resolution).


 2017 ETH 데이터에 나타난 비대칭 고리의 물리학적 의미가 무엇인가? 이를 위해 대규모 라이브러리 모델을 세웠다. 

이 모델의 기초는 일반적 상대적 자기유체역학 (GRMHD :general relativistic magnetohydrodynamic) 시뮬레이션과 일반 상대 광선 추적 방법으로 생산한 종합적 이미지이다. 


우리는 관찰 가시성과 이 라이브러리를 비교하고 다음을 확신하게 되었다. 블랙홀 사건 지평면 근처에 있는 뜨거운 플라즈마 궤도로부터 싱크로트론(synchrotron) 방출 때문에 생긴 강한 중력 효과 예측과 그 비대칭 고리가 서로 일치한다. 


고리 반경과 고리 비대칭은 블랙홀 물질과 회전에 의존한다. 


전반적으로 관찰된 이미지는 일반 상대성에 근거해 예측되었던 ‘회전 커 Kerr’ 블랙홀의 그림자에 대한 예상과 일치한다. 

만약 블랙홀 회전과 대규모 M87 제트가 일직선상에 있으면, 블랙홀 회전 진로는 지구로부터 멀어져 간다.



(발표문 요약 앞부분 번역)



블랙홀 black hole 관찰 사진 발표 논문:






https://iopscience.iop.org/article/10.3847/2041-8213/ab0f43




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




Published 2019 April 10 • © 2019. The American Astronomical Society.




 
Focus on the First Event Horizon Telescope Results





Abstract



The Event Horizon Telescope (EHT) has mapped the central compact radio source of the elliptical galaxy M87 at 1.3 mm with unprecedented angular resolution. 

Here we consider the physical implications of the asymmetric ring seen in the 2017 EHT data.


 To this end, we construct a large library of models based on general relativistic magnetohydrodynamic (GRMHD) simulations and synthetic images produced by general relativistic ray tracing. 

We compare the observed visibilities with this library and confirm that the asymmetric ring is consistent with earlier predictions of strong gravitational lensing of synchrotron emission from a hot plasma orbiting near the black hole event horizon. 


The ring radius and ring asymmetry depend on black hole mass and spin, respectively, and both are therefore expected to be stable when observed in future EHT campaigns. 

Overall, the observed image is consistent with expectations for the shadow of a spinning Kerr black hole as predicted by general relativity.

 If the black hole spin and M87's large scale jet are aligned, then the black hole spin vector is pointed away from Earth.

 Models in our library of non-spinning black holes are inconsistent with the observations as they do not produce sufficiently powerful jets.

 At the same time, in those models that produce a sufficiently powerful jet, the latter is powered by extraction of black hole spin energy through mechanisms akin to the Blandford-Znajek process. 

We briefly consider alternatives to a black hole for the central compact object. Analysis of existing EHT polarization data and data taken simultaneously at other wavelengths will soon enable new tests of the GRMHD models, as will future EHT campaigns at 230 and 345 GHz.

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1. Introduction

In 1918 the galaxy Messier 87 (M87) was observed by Curtis and found to have "a curious straight ray ... apparently connected with the nucleus by a thin line of matter" (Curtis 1918, p. 31). Curtis's ray is now known to be a jet, extending from sub-pc to several kpc scales, and can be observed across the electromagnetic spectrum, from the radio through γ-rays. Very long baseline interferometry (VLBI) observations that zoom in on the nucleus, probing progressively smaller angular scales at progressively higher frequencies up to 86 GHz by the Global mm-VLBI Array (GMVA; e.g., Hada et al. 2016; Boccardi et al. 2017; Kim et al. 2018; Walker et al. 2018), have revealed that the jet emerges from a central core. Models of the stellar velocity distribution imply a mass for the central core $M\approx 6.2\times {10}^{9}\,{M}_{\odot }$ at a distance of $16.9\,\mathrm{Mpc}$ (Gebhardt et al. 2011); models of arcsecond-scale emission lines from ionized gas imply a mass that is lower by about a factor of two (Walsh et al. 2013).

The conventional model for the central object in M87 is a black hole surrounded by a geometrically thick, optically thin, disk accretion flow (e.g., Ichimaru 1977; Rees et al. 1982; Narayan & Yi 19941995; Reynolds et al. 1996). The radiative power of the accretion flow ultimately derives from the gravitational binding energy of the inflowing plasma. There is no consensus model for jet launching, but the two main scenarios are that the jet is a magnetically dominated flow that is ultimately powered by tapping the rotational energy of the black hole (Blandford & Znajek 1977) and that the jet is a magnetically collimated wind from the surrounding accretion disk (Blandford & Payne 1982; Lynden-Bell 2006).

VLBI observations of M87 at frequencies $\gtrsim 230\,\mathrm{GHz}$ with the Event Horizon Telescope (EHT) can resolve angular scales of tens of $\mu \mathrm{as}$, comparable to the scale of the event horizon (Doeleman et al. 2012; Akiyama et al. 2015; EHT Collaboration et al. 2019a2019b2019c, hereafter Paper III, and III). They therefore have the power to probe the nature of the central object and to test models for jet launching. In addition, EHT observations can constrain the key physical parameters of the system, including the black hole mass and spin, accretion rate, and magnetic flux trapped by accreting plasma in the black hole.

In this Letter we adopt the working hypothesis that the central object is a black hole described by the Kerr metric, with mass $M$ and dimensionless spin ${a}_{* }$$-1\lt {a}_{* }\lt 1$. Here ${a}_{* }\equiv {Jc}/{{GM}}^{2}$, where JG, and care, respectively, the black hole angular momentum, gravitational constant, and speed of light. In our convention ${a}_{* }\lt 0$ implies that the angular momentum of the accretion flow and that of the black hole are anti-aligned. Using general relativistic magnetohydrodynamic (GRMHD) models for the accretion flow and synthetic images of these simulations produced by general relativistic radiative transfer calculations, we test whether or not the results of the 2017 EHT observing campaign (hereafter EHT2017) are consistent with the black hole hypothesis.

This Letter is organized as follows. In Section 2 we review salient features of the observations and provide order-of-magnitude estimates for the physical conditions in the source. In Section 3 we describe the numerical models. In Section 4 we outline our procedure for comparing the models to the data in a way that accounts for model variability. In Section 5 we show that many of the models cannot be rejected based on EHT data alone. In Section 6 we combine EHT data with other constraints on the radiative efficiency, X-ray luminosity, and jet power and show that the latter constraint eliminates all ${a}_{* }=0$ models. In Section 7 we discuss limitations of our models and also briefly discuss alternatives to Kerr black hole models. In Section 8 we summarize our results and discuss how further analysis of existing EHT data, future EHT data, and multiwavelength companion observations will sharpen constraints on the models.

2. Review and Estimates

In EHT Collaboration et al. (2019d; hereafter Paper IV) we present images generated from EHT2017 data (for details on the array, 2017 observing campaign, correlation, and calibration, see Paper II and Paper III). A representative image is reproduced in the left panel of Figure 1.




Figure 1. Left panel: an EHT2017 image of M87 from Paper IV of this series (see their Figure 15). Middle panel: a simulated image based on a GRMHD model. Right panel: the model image convolved with a $20\,\mu \mathrm{as}$ FWHM Gaussian beam. Although the most evident features of the model and data are similar, fine features in the model are not resolved by EHT.

 Standard image High-resolution image Export PowerPoint slide

Four features of the image in the left panel of Figure 1 play an important role in our analysis: (1) the ring-like geometry, (2) the peak brightness temperature, (3) the total flux density, and (4) the asymmetry of the ring. We now consider each in turn.

(1) The compact source shows a bright ring with a central dark area without significant extended components. This bears a remarkable similarity to the long-predicted structure for optically thin emission from a hot plasma surrounding a black hole (Falcke et al. 2000). The central hole surrounded by a bright ring arises because of strong gravitational lensing (e.g., Hilbert 1917; von Laue 1921; Bardeen 1973; Luminet 1979). The so-called "photon ring" corresponds to lines of sight that pass close to (unstable) photon orbits (see Teo 2003), linger near the photon orbit, and therefore have a long path length through the emitting plasma. These lines of sight will appear comparatively bright if the emitting plasma is optically thin. The central flux depression is the so-called black hole "shadow" (Falcke et al. 2000), and corresponds to lines of sight that terminate on the event horizon. The shadow could be seen in contrast to surrounding emission from the accretion flow or lensed counter-jet in M87 (Broderick & Loeb 2009).

The photon ring is nearly circular for all black hole spins and all inclinations of the black hole spin axis to the line of sight (e.g., Johannsen & Psaltis 2010). For an ${a}_{* }=0$ black hole of mass $M$ and distance D, the photon ring angular radius on the sky is

Equation (1)

where we have scaled to the most likely mass from Gebhardt et al. (2011) and a distance of $16.9\,\mathrm{Mpc}$(see also EHT Collaboration et al. 2019e, (hereafter Paper VI; Blakeslee et al. 2009; Bird et al. 2010; Cantiello et al. 2018). The photon ring angular radius for other inclinations and values of ${a}_{* }$ differs by at most 13% from Equation (1), and most of this variation occurs at $1-| {a}_{* }| \ll 1$ (e.g., Takahashi 2004; Younsi et al. 2016). Evidently the angular radius of the observed photon ring is approximately $\sim 20\,\mu \mathrm{as}$(Figure 1 and Paper IV), which is close to the prediction of the black hole model given in Equation (1).

(2) The observed peak brightness temperature of the ring in Figure 1 is ${T}_{b,{pk}}\sim 6\times {10}^{9}\,{\rm{K}}$, which is consistent with past EHT mm-VLBI measurements at 230 GHz (Doeleman et al. 2012; Akiyama et al. 2015), and GMVA 3 mm-VLBI measurements of the core region (Kim et al. 2018). Expressed in electron rest-mass (me) units, ${{\rm{\Theta }}}_{b,{pk}}\equiv {k}_{{\rm{B}}}{T}_{b,{pk}}/({m}_{e}{c}^{2})\simeq 1$, where ${k}_{{\rm{B}}}$ is Boltzmann's constant. The true peak brightness temperature of the source is higher if the ring is unresolved by EHT, as is the case for the model image in the center panel of Figure 1.

The 1.3 mm emission from M87 shown in Figure 1 is expected to be generated by the synchrotron process (see Yuan & Narayan 2014, and references therein) and thus depends on the electron distribution function (eDF). If the emitting plasma has a thermal eDF, then it is characterized by an electron temperature ${T}_{e}\geqslant {T}_{b}$, or ${{\rm{\Theta }}}_{e}\equiv {k}_{{\rm{B}}}{T}_{e}/({m}_{e}{c}^{2})\gt 1$, because ${{\rm{\Theta }}}_{e}\gt {{\rm{\Theta }}}_{b,{pk}}$ if the ring is unresolved or optically thin.

Is the observed brightness temperature consistent with what one would expect from phenomenological models of the source? Radiatively inefficient accretion flow models of M87 (Reynolds et al. 1996; Di Matteo et al. 2003) produce mm emission in a geometrically thick donut of plasma around the black hole. The emitting plasma is collisionless: Coulomb scattering is weak at these low densities and high temperatures. Therefore, the electron and ion temperatures need not be the same (e.g., Spitzer 1962). In radiatively inefficient accretion flow models, the ion temperature is slightly less than the ion virial temperature,

Equation (2)

where ${r}_{{\rm{g}}}\equiv {GM}/{c}^{2}$ is the gravitational radius, r is the Boyer–Lindquist or Kerr–Schild radius, and mp is the proton mass. Most models have an electron temperature ${T}_{e}\lt {T}_{i}$ because of electron cooling and preferential heating of the ions by turbulent dissipation (e.g., Yuan & Narayan 2014; Mościbrodzka et al. 2016). If the emission arises at $\sim 5\,{r}_{{\rm{g}}}$, then ${{\rm{\Theta }}}_{e}\simeq 37({T}_{e}/{T}_{i})$, which is then consistent with the observed ${{\rm{\Theta }}}_{b,{pk}}$ if the source is unresolved or optically thin.

(3) The total flux density in the image at $1.3\,\mathrm{mm}$ is $\simeq 0.5$ Jy. With a few assumptions we can use this to estimate the electron number density ne and magnetic field strength B in the source. We adopt a simple, spherical, one-zone model for the source with radius $r\simeq 5\,{r}_{{\rm{g}}}$, pressure ${n}_{i}{{kT}}_{i}+{n}_{e}{{kT}}_{e}={\beta }_{{\rm{p}}}{B}^{2}/(8\pi )$with ${\beta }_{{\rm{p}}}\equiv {p}_{\mathrm{gas}}/{p}_{\mathrm{mag}}\sim 1$, ${T}_{i}\simeq 3{T}_{e}$, and temperature ${\theta }_{e}\simeq 10{\theta }_{b,{pk}}$, which is consistent with the discussion in (2) above. Setting ne = ni (i.e., assuming a fully ionized hydrogen plasma), the values of B and ne required to produce the observed flux density can be found by solving a nonlinear equation (assuming an average angle between the field and line of sight, 60°). The solution can be approximated as a power law:

Equation (3)

Equation (4)

assuming that $M=6.2\times {10}^{9}\,{M}_{\odot }$ and $D=16.9\,\mathrm{Mpc}$, and using the approximate thermal emissivity of Leung et al. (2011). Then the synchrotron optical depth at $1.3\,\mathrm{mm}$ is ~0.2. One can now estimate an accretion rate from (3) using

Equation (5)

assuming spherical symmetry. The Eddington accretion rate is

Equation (6)

where ${L}_{\mathrm{Edd}}\equiv 4\pi {{GMcm}}_{p}/{\sigma }_{T}$ is the Eddington luminosity (${\sigma }_{T}$ is the Thomson cross section). Setting the efficiency $\epsilon =0.1$ and $M=6.2\times {10}^{9}\,{M}_{\odot }$, ${\dot{M}}_{\mathrm{Edd}}=137\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, and therefore $\dot{M}/{\dot{M}}_{\mathrm{Edd}}\,\sim 2.0\times {10}^{-5}$.

This estimate is similar to but slightly larger than the upper limit inferred from the 230 GHz linear polarization properties of M87 (Kuo et al. 2014).

(4) The ring is brighter in the south than the north. This can be explained by a combination of motion in the source and Doppler beaming. As a simple example we consider a luminous, optically thin ring rotating with speed v and an angular momentum vector inclined at a viewing angle i > 0° to the line of sight. Then the approaching side of the ring is Doppler boosted, and the receding side is Doppler dimmed, producing a surface brightness contrast of order unity if v is relativistic. The approaching side of the large-scale jet in M87 is oriented west–northwest (position angle $\mathrm{PA}\approx 288^\circ ;$ in Paper VI this is called ${\mathrm{PA}}_{\mathrm{FJ}}$), or to the right and slightly up in the image. Walker et al. (2018) estimated that the angle between the approaching jet and the line of sight is 17°. If the emission is produced by a rotating ring with an angular momentum vector oriented along the jet axis, then the plasma in the south is approaching Earth and the plasma in the north is receding. This implies a clockwise circulation of the plasma in the source, as projected onto the plane of the sky. This sense of rotation is consistent with the sense of rotation in ionized gas at arcsecond scales (Harms et al. 1994; Walsh et al. 2013). Notice that the asymmetry of the ring is consistent with the asymmetry inferred from 43 GHz observations of the brightness ratio between the north and south sides of the jet and counter-jet (Walker et al. 2018).

All of these estimates present a picture of the source that is remarkably consistent with the expectations of the black hole model and with existing GRMHD models (e.g., Dexter et al. 2012; Mościbrodzka et al. 2016). They even suggest a sense of rotation of gas close to the black hole. A quantitative comparison with GRMHD models can reveal more.

3. Models

Consistent with the discussion in Section 2, we now adopt the working hypothesis that M87 contains a turbulent, magnetized accretion flow surrounding a Kerr black hole. To test this hypothesis quantitatively against the EHT2017 data we have generated a Simulation Library of 3D time-dependent ideal GRMHD models. To generate this computationally expensive library efficiently and with independent checks on the results, we used several different codes that evolved matching initial conditions using the equations of ideal GRMHD. The codes used include BHAC  (Porth et al. 2017), H-AMR  (Liska et al. 2018; K. Chatterjee et al. 2019, in preparation), iharm  (Gammie et al. 2003), and KORAL  (Sa̧dowski et al. 2013b, 2014). A comparison of these and other GRMHD codes can be found in O. Porth et al. 2019 (in preparation), which shows that the differences between integrations of a standard accretion model with different codes is smaller than the fluctuations in individual simulations.

From the Simulation Library we have generated a large Image Library of synthetic images. Snapshots of the GRMHD evolutions were produced using the general relativistic ray-tracing (GRRT) schemes ipole  (Mościbrodzka & Gammie 2018), RAPTOR  (Bronzwaer et al. 2018), or BHOSS (Z. Younsi et al. 2019b, in preparation). A comparison of these and other GRRT codes can be found in Gold et al. (2019), which shows that the differences between codes is small.

In the GRMHD models the bulk of the 1.3 mm emission is produced within $\lesssim 10\,{r}_{{\rm{g}}}$ of the black hole, where the models can reach a statistically steady state. It is therefore possible to compute predictive radiative models for this compact component of the source without accurately representing the accretion flow at all radii.

We note that the current state-of-the-art models for M87 are radiation GRMHD models that include radiative feedback and electron-ion thermodynamics (Ryan et al. 2018; Chael et al. 2019). These models are too computationally expensive for a wide survey of parameter space, so that in this Letter we consider only nonradiative GRMHD models with a parameterized treatment of the electron thermodynamics.

3.1. Simulation Library

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 phgr 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).

3.2. Image Library Generation

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, Te likely does not equal the ion temperature Ti, which is provided by the simulations. We set Te using the GRMHD density ρ, internal energy density u, and plasma ${\beta }_{{\rm{p}}}$ using a simple model:

Equation (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

Equation (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.

3.3. Image Library Summary

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 ${a}_{* }$, phgr, 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.

Figure 2.

Figure 2. Time-averaged 1.3 mm images generated by five SANE GRMHD simulations with varying spin (${a}_{* }=-0.94$ to ${a}_{* }=+0.97$ from left to right) and ${R}_{\mathrm{high}}$ (${R}_{\mathrm{high}}=1$ to ${R}_{\mathrm{high}}=160$ from top to bottom; increasing ${R}_{\mathrm{high}}$ corresponds to decreasing electron temperature). The colormap is linear. All models are imaged at i = 163°. The jet that is approaching Earth is on the right (west) in all the images. The black hole spin vector projected onto the plane of the sky is marked with an arrow and aligned in the east–west direction. When the arrow is pointing left the black hole rotates in a clockwise direction, and when the arrow is pointing right the black hole rotates in a counterclockwise direction. The field of view for each model image is $80\,\mu \mathrm{as}$(half of that used for the image libraries) with resolution equal to $1\mu \mathrm{as}$/pixel (20 times finer than the nominal resolution of EHT2017, and the same employed in the library images).

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Figure 3.

Figure 3. Same as in Figure 2 but for selected MAD models.

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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.

Figure 4.

Figure 4. Binned location of the point of origin for all photons that make up an image, summed over azimuth, and averaged over all snapshots from the simulation. The colormap is linear. The event horizon is indicated by the solid white semicircle and the black hole spin axis is along the figure vertical axis. This set of four images shows MAD and SANE models with ${R}_{\mathrm{high}}=10$ and 160, all with ${a}_{* }=0.94$. The region between the dashed curves is the locus of existence of (unstable) photon orbits (Teo 2003). The green cross marks the location of the innermost stable circular orbit (ISCO) in the equatorial plane. In these images the line of sight (marked by an arrow) is located below the midplane and makes a 163° angle with the disk angular momentum, which coincides with the spin axis of the black hole.

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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.

Figure 5.

Figure 5. Illustration of the effect of black hole and disk angular momentum on ring asymmetry. The asymmetry is produced primarily by Doppler beaming: the bright region corresponds to the approaching side. In GRMHD models that fit the data comparatively well, the asymmetry arises in emission generated in the funnel wall. The sense of rotation of both the jet and funnel wall are controlled by the black hole spin. If the black hole spin axis is aligned with the large-scale jet, which points to the right, then the asymmetry implies that the black hole spin is pointing away from Earth (rotation of the black hole is clockwise as viewed from Earth). The blue ribbon arrow shows the sense of disk rotation, and the black ribbon arrow shows black hole spin. Inclination i is defined as the angle between the disk angular momentum vector and the line of sight.

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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.

Figure 6. Single frame from the accompanying animation. This shows the visibility amplitudes (top), closure phases plotted by Euclidean distance in 6D space (middle), and associated model images at full resolution (lower left) and convolved with the EHT2017 beam (lower right). Data from 2017 April 6 high-band are also shown in the top two plots. The video shows frames 1 through 100 and has a duration of 10 s.

(An animation of this figure is available.)

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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).

4. Procedure for Comparison of Models with Data

As described above, each model in the Simulation Library has two dimensionless parameters: black hole spin ${a}_{* }$ and magnetic flux phgr. Imaging the model from each simulation adds five new parameters: ${R}_{\mathrm{high}}$, i, $\mathrm{PA}$, $M$, and D, which we set to $16.9\,\mathrm{Mpc}$. After fixing these parameters we draw snapshots from the time evolution at a cadence of 10 to $50\,{r}_{{\rm{g}}}{c}^{-1}$. We then compare these snapshots to the data.

The simplest comparison computes the ${\chi }_{\nu }^{2}$ (reduced chi square) distance between the data and a snapshot. In the course of computing ${\chi }_{\nu }^{2}$ we vary the image scale M/D, flux density Fν, position angle $\mathrm{PA}$, and the gain at each VLBI station in order to give each image every opportunity to fit the data. The best-fit parameters $(M/D,{F}_{\nu },\mathrm{PA})$ for each snapshot are found by two pipelines independently: the Themis pipeline using a Markov chain Monte Carlo method (A. E. Broderick et al. 2019a, in preparation), and the GENA  pipeline using an evolutionary algorithm for multidimensional minimization (Fromm et al. 2019a; C. Fromm et al. 2019b, in preparation; see also Section 4 of Paper VI for details). The best-fit parameters contain information about the source and we use the distribution of best-fit parameters to test the model by asking whether or not they are consistent with existing measurements of M/D and estimates of the jet $\mathrm{PA}$ on larger scales.

The ${\chi }_{\nu }^{2}$ comparison alone does not provide a sharp test of the models. Fluctuations in the underlying GRMHD model, combined with the high signal-to-noise ratio for EHT2017 data, imply that individual snapshots are highly unlikely to provide a formally acceptable fit with ${\chi }_{\nu }^{2}\simeq 1$. This is borne out in practice with the minimum ${\chi }_{\nu }^{2}=1.79$ over the entire set of the more than 60,000 individual images in the Image Library. Nevertheless, it is possible to test if the ${\chi }_{\nu }^{2}$ from the fit to the data is consistent with the underlying model, using "Average Image Scoring" with Themis (Themis-AIS), as described in detail in Appendix F of Paper VI). Themis-AIS measures a ${\chi }_{\nu }^{2}$ distance (on the space of visibility amplitudes and closure phases) between a trial image and the data. In practice we use the average of the images from a given model as the trial image (hence Themis-AIS), but other choices are possible. We compute the ${\chi }_{\nu }^{2}$ distance between the trial image and synthetic data produced from each snapshot. The model can then be tested by asking whether the data's ${\chi }_{\nu }^{2}$ is likely to have been drawn from the model's distribution of ${\chi }_{\nu }^{2}$. In particular, we can assign a probability p that the data is drawn from a specific model's distribution.

In this Letter we focus on comparisons with a single data set, the 2017 April 6 high-band data (Paper III). The eight EHT2017 data sets, spanning four days with two bands on each day, are highly correlated. Assessing what correlation is expected in the models is a complicated task that we defer to later publications. The 2017 April 6 data set has the largest number of scans, 284 detections in 25 scans (see Paper III) and is therefore expected to be the most constraining.116

5. Model Constraints: EHT2017 Alone

The resolved ring-like structure obtained from the EHT2017 data provides an estimate of M/D(discussed in detail in Paper VI) and the jet $\mathrm{PA}$ from the immediate environment of the central black hole. As a first test of the models we can ask whether or not these are consistent with what is known from other mass measurements and from the orientation of the large-scale jet.

Figure 7 shows the distributions of best-fit values of M/D for a subset of the models for which spectra and jet power estimates are available (see below). The three lines show the M/D distribution for all snapshots (dotted lines), the best-fit 10% of snapshots (dashed lines), and the best-fit 1% of snapshots (solid lines) within each model. Evidently, as better fits are required, the distribution narrows and peaks close to $M/D\sim 3.6\,\mu \mathrm{as}$ with a width of about $0.5\mu \mathrm{as}$.

Figure 7.

Figure 7. Distribution of M/D obtained by fitting Image Library snapshots to the 2017 April 6 data, in $\mu \mathrm{as}$, measured independently using the (left panel) Themis and (right panel) GENA pipelines with qualitatively similar results. Smooth lines were drawn with a Gaussian kernel density estimator. The three lines show the best-fit 1% within each model (solid); the best-fit 10% within each model (dashed); and all model images (dotted). The vertical lines show $M/D=2.04$ (dashed) and $3.62\,\mu \mathrm{as}$ (solid), corresponding to M = 3.5 and $6.2\times {10}^{9}\,{M}_{\odot }$. The distribution uses a subset of models for which spectra and jet power estimates are available (see Section 6). Only images with ${a}_{* }\gt 0$, i > 90° and ${a}_{* }\lt 0$, i < 90° (see also the left panel of Figure 5) are considered.

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The distribution of M/D for the best-fit $\lt 10 \% $ of snapshots is qualitatively similar if we include only MAD or SANE models, only models produced by individual codes (BHAC, H-AMR, iharm, or KORAL), or only individual spins. As the thrust of this Letter is to test the models, we simply note that Figure 7indicates that the models are broadly consistent with earlier mass estimates (see Paper VI for a detailed discussion). This did not have to be the case: the ring radius could have been significantly larger than $3.6\,\mu \mathrm{as}$.

We can go somewhat further and ask if any of the individual models favor large or small masses. Figure 8 shows the distributions of best-fit values of M/D for each model (different ${a}_{* }$, ${R}_{\mathrm{high}}$, and magnetic flux). Most individual models favor M/D close to $3.6\,\mu \mathrm{as}$. The exceptions are ${a}_{* }\leqslant 0$ SANE models with ${R}_{\mathrm{high}}=1$, which produce the bump in the M/D distribution near $2\mu \mathrm{as}$. In these models, the emission is produced at comparatively large radius in the disk (see Figure 2) because the inner edge of the disk (the ISCO) is at a large radius in a counter-rotating disk around a black hole with $| {a}_{* }| \sim 1$. For these models, the fitting procedure identifies EHT2017's ring with this outer ring, which forces the photon ring, and therefore M/D, to be small. As we will show later, these models can be rejected because they produce weak jets that are inconsistent with existing jet power estimates (see Section 6.3).

Figure 8.

Figure 8. Distributions of M/D and black hole mass with $D=16.9\,\mathrm{Mpc}$ reconstructed from the best-fit 10% of images for MAD (left panel) and SANE (right panel) models (i = 17° for ${a}_{* }\le 0$and 163° for ${a}_{* }\gt 0$) with different ${R}_{\mathrm{high}}$ and ${a}_{* }$, from the Themis (dark red, left), and GENA  (dark green, right) pipelines. The white dot and vertical black bar correspond, respectively, to the median and region between the 25th and 75th percentiles for both pipelines combined. The blue and pink horizontal bands show the range of M/D and mass at $D=16.9\,\mathrm{Mpc}$ estimated from the gas dynamical model (Walsh et al. 2013) and stellar dynamical model (Gebhardt et al. 2011), respectively. Constraints on the models based on average image scoring (Themis-AIS) are discussed in Section 5. Constraints based on radiative efficiency, X-ray luminosity, and jet power are discussed in Section 6.

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Figure 8 also shows that M/D increases with ${a}_{* }$ for SANE models. This is due to the appearance of a secondary inner ring inside the main photon ring. The former is associated with emission produced along the wall of the approaching jet. Because the emission is produced in front of the black hole, lensing is weak and it appears at small angular scale. The inner ring is absent in MAD models (see Figure 3), where the bulk of the emission comes from the midplane at all values of ${R}_{\mathrm{high}}$(Figure 4).

We now ask whether or not the PA of the jet is consistent with the orientation of the jet measured at other wavelengths. On large (~mas) scales the extended jet component has a PA of approximately 288° (e.g., Walker et al. 2018). On smaller ($\sim 100\,\mu \mathrm{as}$) scales the apparent opening angle of the jet is large (e.g., Kim et al. 2018) and the PA is therefore more difficult to measure. Also notice that the jet PA may be time dependent (e.g., Hada et al. 2016; Walker et al. 2018). In our model images the jet is relatively dim at 1.3 mm, and is not easily seen with a linear colormap. The model jet axis is, nonetheless, well defined: jets emerge perpendicular to the disk.

Figure 9 shows the distribution of best-fit PA over the same sample of snapshots from the Image Library used in Figure 7. We divide the snapshots into two groups. The first group has the black hole spin pointed away from Earth (i > 90° and ${a}_{* }\gt 0$, or i < 90° and ${a}_{* }\lt 0$). The spin-away model PA distributions are shown in the top two panels. The second group has the black hole spin pointed toward Earth (i > 90 and ${a}_{* }\lt 0$ or i > 90° and ${a}_{* }\lt 0$). These spin-toward model PA distributions are shown in the bottom two panels. The large-scale jet orientation lies on the shoulder of the spin-away distribution (the distribution can be approximated as a Gaussian with, for Themis (GENA) mean 209 (203)° and ${\sigma }_{\mathrm{PA}}\,=54\,(55)^\circ ;$ the large-scale jet PA lies $1.5{\sigma }_{\mathrm{PA}}$ from the mean) and is therefore consistent with the spin-away models. On the other hand, the large-scale jet orientation lies off the shoulder of the spin-toward distribution and is inconsistent with the spin-toward models. Evidently models in which the black hole spin is pointing away from Earth are strongly favored.

Figure 9.

Figure 9. Top: distribution of best-fit PA (in degree) scored by the Themis (left) and GENA (right) pipelines for models with black hole spin vector pointing away from Earth (i > 90° for ${a}_{* }\gt 0$ or i < 90° for ${a}_{* }\lt 0$). Bottom: images with black hole spin vector pointing toward Earth (i < 90° for ${a}_{* }\gt 0$ or i > 90° for ${a}_{* }\lt 0$). Smooth lines were drawn with a wrapped Gaussian kernel density estimator. The three lines show (1) all images in the sample (dotted line); (2) the best-fit 10% of images within each model (dashed line); and (3) the best-fit 1% of images in each model (solid line). For reference, the vertical line shows the position angle $\mathrm{PA}\sim 288^\circ $ of the large-scale (mas) jet Walker et al. (2018), with the gray area from (288 – 10)° to (288 + 10)° indicating the observed PA variation.

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The width of the spin-away and spin-toward distributions arises naturally in the models from brightness fluctuations in the ring. The distributions are relatively insensitive if split into MAD and SANE categories, although for MAD the averaged PA is $\langle \mathrm{PA}\rangle =219^\circ $, ${\sigma }_{\mathrm{PA}}=46^\circ $, while for SANE $\langle \mathrm{PA}\rangle =195^\circ $ and ${\sigma }_{\mathrm{PA}}=58^\circ $. The ${a}_{* }=0$ and ${a}_{* }\gt 0$ models have similar distributions. Again, EHT2017 data strongly favor one sense of black hole spin: either $| {a}_{* }| $ is small, or the spin vector is pointed away from Earth. If the fluctuations are such that the fitted PA for each epoch of observations is drawn from a Gaussian with ${\sigma }_{\mathrm{PA}}\simeq 55^\circ $, then a second epoch will be able to identify the true orientation with accuracy ${\sigma }_{\mathrm{PA}}/\sqrt{2}\simeq 40^\circ $ and the Nth epoch with accuracy ${\sigma }_{\mathrm{PA}}/\sqrt{N}$. If the fitted PA were drawn from a Gaussian of width ${\sigma }_{\mathrm{PA}}=54^\circ $ about $\mathrm{PA}=288^\circ $, as would be expected in a model in which the large-scale jet is aligned normal to the disk, then future epochs have a >90% chance of seeing the peak brightness counterclockwise from its position in EHT2017.

Finally, we can test the models by asking if they are consistent with the data according to Themis-AIS, as introduced in Section 4. Themis-AIS produces a probability p that the ${\chi }_{\nu }^{2}$ distance between the data and the average of the model images is drawn from the same distribution as the ${\chi }_{\nu }^{2}$ distance between synthetic data created from the model images, and the average of the model images. Table 1 takes these p values and categorizes them by magnetic flux and by spin, aggregating (averaging) results from different codes, ${R}_{\mathrm{high}}$, and i. Evidently, most of the models are formally consistent with the data by this test.

Table 1.  Average Image Scoringa Summary

Fluxb${a}_{* }$ c$\langle p\rangle $ d${N}_{\mathrm{model}}$ e$\mathrm{MIN}(p)$ f$\mathrm{MAX}(p)$ g
SANE−0.940.33240.010.88
SANE−0.50.19240.010.73
SANE00.23240.010.92
SANE0.50.51300.020.97
SANE0.750.7460.480.98
SANE0.880.6560.260.94
SANE0.940.49240.010.92
SANE0.970.1260.060.40
MAD−0.940.01180.010.04
MAD−0.50.75180.340.98
MAD00.22180.010.62
MAD0.50.17180.020.54
MAD0.750.28180.010.72
MAD0.940.21180.020.50

Notes.

aThe Average Image Scoring (Themis-AIS) is introduced in Section 4. bflux: net magnetic flux on the black hole (MAD or SANE). c ${a}_{* }$: dimensionless black hole spin. d $\langle p\rangle $: mean of the p value for the aggregated models. e ${N}_{\mathrm{model}}$: number of aggregated models. f $\mathrm{MIN}(p)$: minimum p value among the aggregated models. g $\mathrm{MAX}(p)$: maximum p value among the aggregated models.

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One group of models, however, is rejected by Themis-AIS: MAD models with ${a}_{* }=-0.94$. On average this group has p = 0.01, and all models within this group have $p\leqslant 0.04$. Snapshots from MAD models with ${a}_{* }=-0.94$ exhibit the highest morphological variability in our ensemble in the sense that the emission breaks up into transient bright clumps. These models are rejected by Themis-AIS because none of the snapshots are as similar to the average image as the data. In other words, it is unlikely that EHT2017 would have captured an ${a}_{* }=-0.94$ MAD model in a configuration as unperturbed as the data seem to be.

The remainder of the model categories contain at least some models that are consistent with the data according to the average image scoring test. That is, most models are variable and the associated snapshots lie far from the average image. These snapshots are formally inconsistent with the data, but their distance from the average image is consistent with what is expected from the models. Given the uncertainties in the model—and our lack of knowledge of the source prior to EHT2017—it is remarkable that so many of the models are acceptable. This is likely because the source structure is dominated by the photon ring, which is produced by gravitational lensing, and is therefore relatively insensitive to the details of the accretion flow and jet physics. We can further narrow the range of acceptable models, however, using additional constraints.

6. Model Constraints: EHT2017 Combined with Other Constraints

We can apply three additional arguments to further constrain the source model. (1) The model must be close to radiative equilibrium. (2) The model must be consistent with the observed broadband SED; in particular, it must not overproduce X-rays. (3) The model must produce a sufficiently powerful jet to match the measurements of the jet kinetic energy at large scales. Our discussions in this Section are based on simulation data that is provided in full detail in Appendix A.

6.1. Radiative Equilibrium

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 epsilon that is larger than the classical thin disk model. (Table 3 in Appendix A lists epsilon 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.

6.2. X-Ray Constraints

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 τT 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 τT and decrease Te. The increase in Te 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.

6.3. Jet Power

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 1042to ${10}^{45}\,\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. 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 Te 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 Te 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 Te, however, including a lower limit provided by the observed brightness temeprature. As Te declines, ne and Bincrease and that has implications for source linear polarization (Mościbrodzka et al. 2017; Jiménez-Rosales & Dexter 2018), which will be explored in future work. As Te declines and ne and ni increase there is also an increase in energy transfer from ions to electrons by Coulomb coupling, and this sets a floor on Te.

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 Te in a thermal model is unlikely to work because lower Te 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 Te 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,

Equation (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.

6.4. Constraint Summary

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${R}_{\mathrm{high}}$ cAISdepsilone${L}_{{\rm{X}}}$ f${P}_{\mathrm{jet}}$ g 
SANE−0.941FailPassPassPassFail
SANE−0.9410PassPassPassPassPass
SANE−0.9420PassPassPassPassPass
SANE−0.9440PassPassPassPassPass
SANE−0.9480PassPassPassPassPass
SANE−0.94160FailPassPassPassFail
SANE−0.51PassPassFailFailFail
SANE−0.510PassPassFailFailFail
SANE−0.520PassPassPassFailFail
SANE−0.540PassPassPassFailFail
SANE−0.580FailPassPassFailFail
SANE−0.5160PassPassPassFailFail
SANE01PassPassPassFailFail
SANE010PassPassPassFailFail
SANE020PassPassFailFailFail
SANE040PassPassPassFailFail
SANE080PassPassPassFailFail
SANE0160PassPassPassFailFail
SANE+0.51PassPassPassFailFail
SANE+0.510PassPassPassFailFail
SANE+0.520PassPassPassFailFail
SANE+0.540PassPassPassFailFail
SANE+0.580PassPassPassFailFail
SANE+0.5160PassPassPassFailFail
SANE+0.941PassFailPassFailFail
SANE+0.9410PassFailPassFailFail
SANE+0.9420PassPassPassFailFail
SANE+0.9440PassPassPassFailFail
SANE+0.9480PassPassPassPassPass
SANE+0.94160PassPassPassPassPass
MAD−0.941FailFailPassPassFail
MAD−0.9410FailPassPassPassFail
MAD−0.9420FailPassPassPassFail
MAD−0.9440FailPassPassPassFail
MAD−0.9480FailPassPassPassFail
MAD−0.94160FailPassPassPassFail
MAD−0.51PassFailPassFailFail
MAD−0.510PassPassPassFailFail
MAD−0.520PassPassPassPassPass
MAD−0.540PassPassPassPassPass
MAD−0.580PassPassPassPassPass
MAD−0.5160PassPassPassPassPass
MAD01PassFailPassFailFail
MAD010PassPassPassFailFail
MAD020PassPassPassFailFail
MAD040PassPassPassFailFail
MAD080PassPassPassFailFail
MAD0160PassPassPassFailFail
MAD+0.51PassFailPassFailFail
MAD+0.510PassPassPassPassPass
MAD+0.520PassPassPassPassPass
MAD+0.540PassPassPassPassPass
MAD+0.580PassPassPassPassPass
MAD+0.5160PassPassPassPassPass
MAD+0.941PassFailFailPassFail
MAD+0.9410PassFailPassPassFail
MAD+0.9420PassPassPassPassPass
MAD+0.9440PassPassPassPassPass
MAD+0.9480PassPassPassPassPass
MAD+0.94160PassPassPassPassPass

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. eepsilon: radiative efficiency, models are rejected if epsilon 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.

Download table as:  ASCIITypeset image

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.

7. Discussion

We have interpreted the EHT2017 data using a limited library of models with attendant limitations. Many of the limitations stem from the GRMHD model, which treats the plasma as an ideal fluid governed by equations that encode conservation laws for particle number, momentum, and energy. The eDF, in particular, is described by a number density and temperature, rather than a full distribution function, and the electron temperature Te is assumed to be a function of the local ion temperature and plasma ${\beta }_{{\rm{p}}}$. Furthermore, all models assume a Kerr black hole spacetime, but there are alternatives. Here we consider some of the model limitations and possible extensions, including to models beyond general relativity.

7.1. Radiative Effects

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 Te 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 Ti/Te, 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