###### Abstract

In the past few years, the Event Horizon Telescope (EHT) has provided the
first-ever event horizon-scale images of the supermassive black holes (BHs)
M87∗ and Sagittarius A∗ (Sgr A∗). The next-generation EHT project is an
extension of the EHT array that promises larger angular resolution and higher
sensitivity to the dim, extended flux around the central ring-like structure,
possibly connecting the accretion flow and the jet. The ngEHT Analysis
Challenges aim to understand the science extractability from synthetic images
and movies to inform the ngEHT array design and analysis algorithm
development. In this work, we compare the accretion flow structure and
dynamics in numerical fluid simulations that specifically target M87∗ and Sgr
A∗, and were used to construct the source models in the challenge set. We
consider (1) a steady-state axisymmetric radiatively inefficient accretion
flow model with a time-dependent shearing hotspot, (2) two time-dependent
single fluid general relativistic magnetohydrodynamic (GRMHD) simulations from
the H-AMR code, (3) a two-temperature GRMHD simulation from the BHAC code, and
(4) a two-temperature radiative GRMHD simulation from the KORAL code. We find
that the different models exhibit remarkably similar temporal and spatial
properties, except for the electron temperature, since radiative losses
substantially cool down electrons near the BH and the jet sheath, signaling
the importance of radiative cooling even for slowly accreting BHs such as
M87∗. We restrict ourselves to standard torus accretion flows, and leave
larger explorations of alternate accretion models to future work.

###### keywords:

black holes; general relativity; accretion; relativistic jets; very-long-
baseline interferometry

1 1 0 Academic Editor: Dimitris M. Christodoulou 4 December 2022 14 February
2023 20 February 2023 https://doi.org/ Accretion Flow Morphology in Numerical
Simulations of Black Holes from the ngEHT Model Library: The Impact of
Radiation Physics Accretion Flow Morphology in Numerical Simulations of Black
Holes from the ngEHT Model Library: The Impact of Radiation Physics Koushik
Chatterjee 1,2,*, Andrew Chael 3, Paul Tiede 1,2, Yosuke Mizuno 4,5,6, Razieh
Emami 2, Christian Fromm 6,7,8, Angelo Ricarte 1,2, Lindy Blackburn 1,2, Freek
Roelofs 1,2, Michael D. Johnson 1,2, Sheperd S. Doeleman 1,2, Philipp Arras
9,10, Antonio Fuentes 11, Jakob Knollmüller 10,12, Nikita Kosogorov 13,14,
Greg Lindahl 2, Hendrik Müller 8, Nimesh Patel 2, Alexander Raymond 1,2,
Efthalia Traianou 11 and Justin Vega 2,15 Koushik Chatterjee, Andrew Chael,
Paul Tiede, Yosuke Mizuno, Razieh Emami, Christian Fromm, Angelo Ricarte,
Lindy Blackburn, Freek Roelofs, Michael D. Johnson, Sheperd S. Doeleman,
Philipp Arras, Antonio Fuentes, Jakob Knollmüller, Nikita Kosogorov, Greg
Lindahl, Hendrik Müller, Nimesh Patel, Alexander Raymond, Efthalia Traianou
and Justin Vega Chatterjee, K.; Chael, A.; Tiede, P.; Mizuno, Y.; Emami, R.;
Fromm, C.; Ricarte, A.; Blackburn, L.; Roelofs, F.; Johnson, M.D.; et al.
Correspondence<EMAIL_ADDRESS>(K.C.);
<EMAIL_ADDRESS>(A.C<EMAIL_ADDRESS>(P.T.);
<EMAIL_ADDRESS>(Y.M<EMAIL_ADDRESS>(R.E.);
<EMAIL_ADDRESS>(C.F<EMAIL_ADDRESS>(A.R<EMAIL_ADDRESS>(L.B<EMAIL_ADDRESS>(F.R<EMAIL_ADDRESS>(M.D.J<EMAIL_ADDRESS>(S.S.D.);
<EMAIL_ADDRESS>(P.A<EMAIL_ADDRESS>(A.F.); jakob@mpa-
garching.mpg.de (J.K<EMAIL_ADDRESS>(N.K<EMAIL_ADDRESS>(G.L.);
<EMAIL_ADDRESS>(H.M<EMAIL_ADDRESS>(N.P.);
<EMAIL_ADDRESS>(A.R<EMAIL_ADDRESS>(E.T<EMAIL_ADDRESS>(J.V.)

##  1 Introduction

With the advent of the Event Horizon Telescope ([EHT; Event Horizon Telescope
Collaboration et al. (2019, 2022)), imaging the near-horizon structure of
supermassive black holes (BHs) is now a reality. The primary targets of the
EHT and the future next-generation EHT (or ngEHThttps://www.ngeht.org/
(accessed on 22 February 2023)) are M87∗ (the supermassive BH in the
elliptical galaxy M87; Event Horizon Telescope Collaboration et al. (2019))
and Sgr A∗ in the Galactic Center (Event Horizon Telescope Collaboration et
al., 2022), which are two of the most well-studied low-luminosity active
galactic nuclei. Extracting information about the event horizon-scale
accretion flows in these two sources using the EHT’s enormous resolving power
is an active area of research. With the ngEHT, we will achieve unprecedented
levels of angular resolution and sensitivity to low-flux regions, with the
dynamic range in flux expected to increase to $\sim$1000 compared to the EHT’s
current dynamic range of $\sim$10 (e.g., Doeleman et al. (2019)). This would
enable us to investigate the BH shadow shape with higher precision as well as
provide a crucial connection between the accretion flow and the jet launching
region. The expected advances in sensitivity require deeper investigations of
feature extraction from simulated synthetic reconstructions of BH systems.
Hence, we designed the ngEHT analysis challenges https://challenge.ngeht.org/
(accessed on 22 February 2023) (Roelofs et al., ) to test our ability to
capture the complex dynamics of gas and magnetic fields around M87∗ and Sgr A∗
using the ngEHT reference array (e.g., Raymond et al. (2021)) with various
analysis methods.

Black hole accretion and jet physics have been intensively studied over the
past few decades (e.g., Shakura and Sunyaev (1973); Rees et al. (1982);
Narayan et al. (1995); Quataert and Gruzinov (2000); Narayan et al. (2003);
McKinney (2006); Komissarov et al. (2007); Tchekhovskoy et al. (2011); Narayan
et al. (2012); Tchekhovskoy and Bromberg (2016)). In the context of M87∗ and
Sgr A∗, we expect the accretion flow to be highly sub-Eddington, radiatively
inefficient, and geometrically thick, popularly known as radiatively
inefficient accretion flows (RIAFs). This accretion flow solution has been
used to successfully model the multiwavelength spectrum of Sgr A∗ (e.g., Yuan
et al. (2003)). On the other hand, semi-analytical models of jets are
preferred to explain the spectrum of M87∗ (e.g., Lucchini et al. (2019)).
Thus, these two sources already provide a means to probe two different
components of BH accretion, namely, the inner accretion flow structure and
turbulence in Sgr A∗ and the prominent jet feature in M87∗. The first three
EHT numerical simulation papers (Event Horizon Telescope Collaboration et al.,
2019, 2021, 2022) already provide us with important clues about the horizon-
scale conditions of these BH systems based on numerical simulations: (1) these
BHs probably have non-zero spin, (2) the accretion disk is expected to have
colder electrons than the jet sheath, and (3) the observations favor the
presence of dynamically important magnetic fields close to the BH. All of
these results point us toward the magnetically arrested disk (MAD;
Igumenshchev et al. (2003); Narayan et al. (2003)) state, an accretion mode
where the BH magnetosphere becomes over-saturated with magnetic flux and
exhibits quasi-periodic explosions of vertical magnetic field bundles. MAD
flows around spinning BHs also have powerful relativistic jets, where the jet
power can exceed the input accretion power (Tchekhovskoy et al., 2011), which
is a definite signature of BH spin energy extraction via the Blandford and
Znajek (1977) process.

Building on the semi-analytical RIAF models, the time-dependent general
relativistic magneto-hydrodynamic (GRMHD) simulations have become important
tools for deciphering BH accretion physics in a variety of astrophysical
systems (e.g., Gammie et al. (2003); McKinney (2006); Fragile et al. (2007);
Tchekhovskoy et al. (2011); Chael et al. (2019); Porth et al. (2019); Narayan
et al. (2022)). Indeed, the EHT regularly uses the libraries of GRMHD
simulations to model the observed horizon-scale BH images of M87∗ and Sgr A∗
as well as larger-scale jet images (such as for Centaurus A; Janssen et al.
(2021)) in order to constrain the time-variable plasma properties. In
designing the ngEHT reference array, it is therefore crucial to use GRMHD
simulations for understanding the attainability of specific science goals,
such as resolving the photon ring and the disk-jet connection region as well
as tracing out time-variable features via the ngEHT analysis challenges.

In this work, we discuss the numerical fluid simulations that were used as
source models for the ngEHT analysis challenges. In particular, our objective
is to compare between the models that incorporate increasingly complicated
levels of accretion and electron physics, focusing on M87∗ and Sgr A∗. Our
model set consists of a time-dependent shearing hotspot stitched to a steady-
state RIAF solution, two standard GRMHD simulations of MAD accretion flows, a
GRMHD MAD simulation with electron heating via incorporating two-temperature
physics, and a fully radiative, two-temperature, GRMHD MAD simulation. We
describe the equations and setup of our numerical models in Section 2, show
our comparison results in Section 3, and, finally, conclude in Section 4.

##  2 Numerical Simulations

In this section, we provide a brief description of the semi-analytical
stationary RIAF and shearing hotspot model as well as the (two-
temperature/radiative) GRMHD simulations used for the ngEHT analysis
challenges.

###  2.1 RIAF + Hotspot Solutions

RIAF models attempt to describe the time and azimuthally averaged appearance
of accretion flows. This is performed using a set of building blocks. The
specific RIAF models used in the challenges are based on Yuan et al. (2003);
Broderick and Loeb (2006); Broderick and McKinney (2010) approach. We
decompose the accretion flow into a set of phenomenological models that
describe the electron density, temperature, magnetic field, and velocity
profile. We have a cylindrical coordinate system $x^{\mu}\equiv(R_{\rm
cyl},\varphi,z)$. The electron density profile is defined in terms of the
cylindrical radius $R_{\rm cyl}=r|\sin(\theta)|$ and vertical displacement
$z=r\cos(\theta)$, and is provided by

$n_{e,\rm{X}}(R_{\rm cyl},z)=n_{e,\rm X,0}R_{\rm
cyl}^{p_{X}}\exp\left(-\frac{z^{2}}{2h^{2}R_{\rm cyl}^{2}}\right)$ (1)

where $X$ denotes the population of electrons. The disk height $h$ is set to
unity for this work. For the challenge dataset, we included both thermal
synchrotron emitting ($X\equiv{\rm th})$, and non-thermal synchrotron
($X\equiv{\rm nth}$) emitting electrons. The thermal electrons have $n_{e,{\rm
th},0}=1.3\times 10^{8}$ and $p_{\rm th}=-1.1$, while the non-thermal
electrons are provided by $n_{e,{\rm nth},0}=1.3\times 10^{5}$, $p_{\rm
nth}=-2.02$. These numbers are from Tiede et al. (2020) and are set to match
the best-fit parameters for the Sgr A∗ spectrum from Broderick et al. (2011)
and Broderick et al. (2016).

The temperature profile of the thermal electrons is also provided by a radial
power law with a Gaussian envelope describing the height:

$T_{e}(t,r,\theta,\varphi)=T_{e,0}R_{\rm
cyl}^{-0.84}\exp\left(-\frac{z^{2}}{2h^{2}R_{\rm cyl}^{2}}\right),$ (2)

where, for the challenge data, we set $T_{e,0}=6.3\times 10^{10}$ K.

Following Pu et al. (2016), we define the gas pressure by assuming that the
ultra-relavistic protons are in roughly virial equilibrium with the
gravitational force providing:

$p_{\rm gas}(x^{\mu})=\frac{m_{p}n_{e,th}(x^{\mu})}{6}\frac{GM_{\rm BH}}{r},$
(3)

where $m_{p}$, $G$, and $M_{\rm BH}$ are the proton mass, gravitational
constant, and the black hole mass.

For the local magnetic field, we then assume a constant $\beta=p_{\rm
gas}/p_{\rm mag}=10$ plasma, which, combined with Equation (3), provide us the
magnetic field strength. The orientation of the magnetic field is then
provided by a purely toroidal configuration, relative to the plasma observer.
Finally, we extract the emission and absorption coefficients from the
synchrotron self-absorption model in Broderick and Blandford (2004) for both
the thermal and non-thermal synchrotron electrons. For the non-thermal
electrons, we use a power law prescription with radiation coefficients from
Jones and O’Dell (1977), and a photon spectral power law index of 1.25 (see
Tiede et al. (2020) for more details).

We follow the velocity field prescription from Pu et al. (2016) to describe
the accretion flow dynamics. Using the notation from Tiede et al. (2020), the
velocity field $u^{\mu}=u^{t}v^{\mu}$ is provided by

$\displaystyle v^{r}$ $\displaystyle=v^{r}_{K}+\alpha_{\rm
vel}(v^{r}_{ff}-v^{r}_{K})$ (4) $\displaystyle v^{\theta}$ $\displaystyle=0$
$\displaystyle v^{\varphi}$ $\displaystyle=v^{\varphi}_{K}+(1-\kappa_{\rm
vel})(v^{\varphi}_{ff}-v^{\varphi}_{K}),$

where $v^{\mu}_{K}$ denotes the Keplerian velocity field, $v^{\mu}_{ff}$ is
the free-fall velocity field, and $\alpha_{\rm vel}=0.01$ and $\kappa_{\rm
vel}=1.0$, i.e., the hotspot rotates with Keplerian angular velocity and is
weakly free-falling. For $\alpha_{\rm vel}=0,\kappa_{\rm vel}=1$, we have a
Keplerian orbit, whereas, for $\alpha_{\rm vel}=1,\kappa_{\rm vel}=0$, we have
free-fall. The remaining component $u^{t}$ is provided by the normalization
condition $u^{\mu}u_{\mu}=-1$. The radial component outside the inner stable
circular orbit (ISCO) is $v^{r}_{K}=0$ as the disk is in steady-state.
However, inside the ISCO, we use plunging geodesics that are specified by
matching the angular momentum and energy at the ISCO.

The hotspot evolution follows the model from Tiede et al. (2020) (also see
Broderick and Loeb (2006)), where we assume that the hotspot travels passively
along the accretion flow velocity field Equation (4). This implies that the
equation of motion is provided by the conservation of particle number Equation
(6). For the emission, we assume a non-thermal synchrotron hotspot, with an
initial Gaussian density profile

$n_{e}(x^{\mu})=n_{0}e^{-\Delta r^{\mu}r_{\mu}+(\Delta
r_{\mu}v^{\mu})^{2}/2R_{s}^{2}},$ (5)

where we have set $n_{0}=6\times 10^{6}$, hotspot size $R_{s}=0.5\,r_{\rm
g}=0.5GM_{\rm BH}/c^{2}$, and $\Delta r^{\mu}$ is the displacement from the
hotspot center.

###  2.2 GRMHD Simulations

Over the previous two decades, multiple GRMHD codes have been developed and
utilized to model black hole accretion and jet launching physics over long
dynamical timescales. The wide usage of GRMHD simulations is particularly
encouraging since this allows for verification of code-specific numerical
choices that users usually have to make even while solving the same base set
of GRMHD equations. Indeed, recently, there was a community-wide effort to
benchmark these codes against each other for a standard problem: evolving a
weakly magnetized torus of gas around a spinning black hole (Porth et al.,
2019). It was found that these codes largely provide similar solutions, though
some disk quantities remain unconverged with increasing grid resolutions,
suggesting more investigation is required. For this work, we employ three
different GRMHD codes to probe black hole accretion, increasing the complexity
of the equations solved at each step: (1) single fluid GRMHD simulations from
the H-AMR code (Liska et al., 2022), (2) a two-temperature single fluid GRMHD
simulation from the BHAC code (Porth et al., 2017), and (3) a two-temperature
radiative GRMHD simulation from the KORAL code (Sądowski et al., 2013).

First, we describe the set of GRMHD Equations (e.g., from Gammie et al.
(2003); Porth et al. (2019)). We have the conservation of particle number and
energy-momentum:

$\displaystyle\partial_{t}(\sqrt{-g}\rho u^{t})$ $\displaystyle=$
$\displaystyle-\partial_{i}(\sqrt{-g}\rho u^{i})$ (6)
$\displaystyle\partial_{t}(\sqrt{-g}T^{t}_{\nu})$ $\displaystyle=$
$\displaystyle-\partial_{i}(\sqrt{-g}T^{i}_{\nu})+\sqrt{-g}T^{\alpha}_{\beta}\Gamma^{\beta}_{\nu\alpha}.$
(7)

Here, $\rho$ is the rest-mass gas density and can also be written in the form
of $\rho=mn$, where $m$ is the mean rest–mass per particle and $n$ is the
particle number density. We also have the four-velocity $u^{\mu}$,
stress–energy tensor $T^{\mu}_{\nu}$, metric determinant $g\equiv
det(g_{\mu\nu})$, and the metric connection $\Gamma^{\beta}_{\nu\alpha}$. Note
that the index $t$ refers to the temporal component of the vector or tensor
and $i$ denotes the spatial indices.

The stress-energy tensor $T^{\mu}_{\nu}$ is provided as

$T^{\mu}_{\nu}=(\rho+U_{\rm gas}+p_{\rm gas}+2p_{\rm
mag})u^{\mu}u_{\nu}+(p_{\rm gas}+p_{\rm
mag})\delta^{\mu}_{\nu}-b^{\mu}b_{\nu},$ (8)

where $U_{\rm gas}$ and $p_{\rm gas}$ are the gas internal energy and
pressure, related by the ideal gas equation: $p_{\rm gas}=(\Gamma_{\rm
gas}-1)U_{\rm gas}$ assuming a gas adiabatic index $\Gamma_{\rm gas}$. We also
have the magnetic pressure $p_{\rm mag}=b^{2}/2$ and the magnetic field
4-vector $b^{\mu}$, which can be defined in terms of the magnetic field
3-vector $B^{i}$:

$\displaystyle b^{t}$ $\displaystyle=$ $\displaystyle B^{i}u^{\mu}g_{i\mu}$
(9) $\displaystyle b^{i}$ $\displaystyle=$
$\displaystyle(B^{i}+b^{t}u^{i})/u^{t}.$ (10)

Here, we included a factor of $\sqrt{4\pi}$ into the definition of $B^{i}$. We
evolve the magnetic field $B^{i}$ using the spatial components of the
induction equation,

$\partial_{t}(\sqrt{-g}B^{i})=-\partial_{j}(\sqrt{-g}(b^{j}u^{i}-b^{i}u^{j})),$
(11)

while the temporal component provides the no monopoles constraint,

$\frac{1}{\sqrt{-g}}\partial_{i}(\sqrt{-g}B^{i})=0.$ (12)

These equations are numerically integrated in the conservative form (Gammie et
al., 2003) to obtain the physically relevant quantities $\rho$, $U_{\rm gas}$,
$u^{\mu}$, and $B^{i}$. We refer the reader to the corresponding code papers
for more information on the numerical techniques used to evolve these
equations over space and time.

In this work, we use two GRMHD simulations performed with the H-AMR code, one
targeting M87∗ and the other Sgr A∗. These simulations employ logarithmic
Kerr–Schild coordinates and the grid resolutions are $N_{r}\times
N_{\theta}\times N_{\varphi}=580\times 288\times 512$ for the M87∗ simulation
and $348\times 192\times 192$ for the Sgr A∗ simulation. All the simulations
in this work adopt the geometrical unit convention, $G=c=1$ and using $M_{\rm
BH}=1$, normalizing the length scale to the gravitational radius $r_{\rm
g}=GM_{\rm BH}/c^{2}$. The M87∗ GRMHD simulation evolves a MAD flow around a
black hole with spin $a=0.9375$. The Sgr A∗ model also simulates a MAD flow
but around a black hole with spin $a=1/2$. H-AMR uses outflowing radial
boundary conditions (BCs), transmissive polar BCs, and periodic azimuthal BCs
(for more details, see Liska et al. (2018)).

Since GRMHD simulations are scale-free, we determine the gas density code-to-
CGS units conversion factor (hereafter, “density scaling”) by raytracing the
simulation at 230 GHz for a target source and flux. We use the general
relativistic ray-tracing codes BHOSS (Younsi et al., 2020) and IPOLE
(Mościbrodzka and Gammie, 2018) to compute images at 230 GHz and set the
compact flux to be approximately 0.5 Jy for M87∗ (Event Horizon Telescope
Collaboration et al., 2019) and 2.4 Jy for Sgr A∗ (Event Horizon Telescope
Collaboration et al., 2022). We use the black hole masses and distances of
$M_{\rm BH}=6.2\times 10^{9}M_{\odot}$ and $D_{\rm BH}=16.9$ Mpc for M87∗
(e.g., Gebhardt et al. (2011); Event Horizon Telescope Collaboration et al.
(2019) and references therein) and $M_{\rm BH}=4.14\times 10^{6}M_{\odot}$ and
$D_{\rm BH}=8.127$ kpc for Sgr A∗ (Do et al., 2019; GRAVITY Collaboration et
al., 2019; Event Horizon Telescope Collaboration et al., 2022).

GRMHD simulations evolve a single temperature fluid. At the low accretion
rates seen in M87∗ and Sgr A∗, Coulomb coupling between ions and electrons is
inefficient and, therefore, the two particle species are not in thermal
equilibrium. Since ions are much heavier than electrons, the ions dominate the
single fluid thermodynamics evolved in GRMHD simulations. Hence, to calculate
the radiative output from GRMHD simulations, we calculate the electron
temperature $T_{\rm e}$ using sub-grid models such as the $R-\beta$
prescription (Mościbrodzka et al., 2016) based on local gas plasma-$\beta$
($\equiv p_{\rm gas}/p_{\rm mag}$):

$\displaystyle T_{\rm e}$ $\displaystyle=$ $\displaystyle\frac{2m_{\rm
p}U_{\rm gas}}{3k_{\rm B}\rho(2+R)},$ (13) $\displaystyle{\rm where,\,\,}R$
$\displaystyle=$ $\displaystyle\frac{1+R_{\rm high}\beta^{2}}{1+\beta^{2}}.$
(14)

For the ngEHT analysis challenges, we select $R_{\rm high}$ values of 160 and
40 and source inclinations of $163^{\circ}$ and $50^{\circ}$ for the M87∗ and
Sgr A∗ simulations, respectively. We assume a thermal relativistic
Maxwell–Jüttner distribution for describing the electron energy distribution
in the Sgr A∗ model, and a hybrid thermal$+$non-thermal $\kappa-$distribution
(e.g., Xiao (2006); Davelaar et al. (2018)) for the M87∗ model. The model
images are shown in Roelofs et al. .

###  2.3 Two-Temperature Physics

Two-temperature GRMHD (or 2t-GRMHD) simulations (e.g., Ressler et al. (2015);
Dexter et al. (2020)) evolve the ion and electron entropy equation separately
and, hence, provide the ion and electron temperature in a self-consistent
manner. The main advantage of this method is that we remove the electron
temperature as a free parameter when constructing images. However, we do have
to make a choice about the sub-grid prescription that determines the fraction
of local dissipative energy that heats the electrons. There are two heating
mechanisms that are thought to be applicable to global accretion flows:
turbulent heating (Howes, 2010; Kawazura et al., 2019), and magnetic
reconnection (Werner et al., 2018; Rowan et al., 2017). For the ngEHT analysis
challenges, we focus only on one simulation with reconnection heating, from
Mizuno et al. (2021), as heating via magnetic reconnection in equatorial
current sheets formed in magnetically arrested flows, captured by the Rowan et
al. (2017) heating model, is arguably more important than heating via small-
scale turbulent eddies that are more prevalent in weakly magnetized disks.

We assume that the number densities and velocities of ions and electrons are
equal, i.e., $n_{\rm i}=n_{\rm e}=n$ and $u^{\mu}_{\rm i}=u^{\mu}_{\rm
e}=u^{\mu}$, thus maintaining charge neutrality. The electron entropy equation
is provided as

$\partial_{\mu}(\sqrt{-g}\rho u^{\mu}s_{\rm e})=\frac{\sqrt{-g}(\Gamma_{\rm
e}-1)}{\rho^{\Gamma_{\rm e}-1}}f_{\rm e}Q,$ (15)

where the electron entropy is $s_{\rm e}=p_{\rm e}/\rho^{\Gamma_{\rm e}}$,
with $p_{\rm e}$ and $\Gamma_{\rm e}$ as the electron pressure and adiabatic
index. The total heating rate $Q$ is calculated by comparing the total
internal energy of the gas and the internal energy obtained from the electron
entropy conservation Equation (see Ressler et al. (2015) for more details).
The fraction of dissipative heating that contributes to electron heating is
provided by $f_{\rm e}$. For this particular simulation, $f_{\rm e}$ is
designed to capture electron/ion heating via magnetic reconnection from Rowan
et al. (2017):

$f_{\rm e}=\frac{1}{2}\exp\Big{[}-\frac{1-\beta/\beta_{\rm
max}}{0.8+\sigma_{\rm h}^{0.5}}\Big{]},$ (16)

where $\beta_{\rm max}=1/4\sigma_{\rm h}$, defined using the hot gas
magnetization $\sigma_{\rm h}=b^{2}/\rho E_{\rm th}$ and the specific gas
enthalpy $E_{\rm th}=1+\Gamma_{\rm gas}p_{\rm g}/[\rho(\Gamma_{\rm gas}-1)]$.

The 2t-GRMHD simulation from Mizuno et al. (2021) assumes modified Kerr–Schild
coordinates and a black hole spin of $0.9375$. The grid resolution is
$384\times 192\times 192$. The accretion mode is of a magnetically arrested
flow and the simulation is raytraced (using BHOSS) once the near-horizon flow
has reached steady state. The target source is M87∗, assuming a black hole
mass of $M_{\rm BH}=6.5\times 10^{9}M_{\odot}$ and distance of 16.9 Mpc (Event
Horizon Telescope Collaboration et al., 2019). The accretion rate is
normalized such that the 230 GHz compact flux density is 0.8 Jy. We assume a
thermal electron distribution everywhere except in the jet sheath where we
adopt a $\kappa-$distribution. More details about the image are provided in
Roelofs et al. .

###  2.4 Radiative GRMHD

Two temperature GRMHD simulations do not include radiative cooling and, hence,
are thought to be appropriate for low luminosity supermassive black holes such
as M87∗ and Sgr A∗. To verify this assumption, we consider a two-temperature
radiative GRMHD (2t-GRRMHD hereafter) simulations from Chael et al. (2019).
This simulation accounts for self-consistent radiation physics, incorporating
both particle heating via magnetic reconnection (as in Section 2.3) and
radiative cooling via bremsstrahlung, synchrotron, Compton, and Coulomb
losses. The simulation is run using the 2t-GRRMHD code KORAL (Sądowski et al.,
2013; Sądowski and Narayan, 2015; Chael et al., 2018), which evolves a two-
temperature magnetized fluid and treats the radiation field as a second fluid
(Sądowski et al., 2017). The conservation equations solved in 2t-GRRMHD are
different from that of GRMHD:

$(T^{\mu}_{\nu}+R^{\mu}_{\nu})_{;\,\,\,\mu}=0,$ (17)

where $R^{\mu}_{\nu}$ is the frequency-integrated radiation field, defined as,

$R^{\mu}_{\nu}=\frac{4}{3}\overline{E}u^{\mu}_{\rm R}u_{\nu\rm
R}+\frac{1}{3}\overline{E}\delta^{\mu}_{\nu}.$ (18)

Here, the radiation field is described by its rest-frame energy density
$\overline{E}$ and four-velocity $u^{\mu}_{\rm R}$ following the M1 closure
scheme. The ion and electron entropy equations are

$\displaystyle T_{\rm e}(ns_{\rm e}u^{\mu})_{;\mu}$ $\displaystyle=$
$\displaystyle f_{\rm e}q^{\rm v}+q^{\rm C}-G,$ (19) $\displaystyle T_{\rm
i}(ns_{\rm i}u^{\mu})_{;\mu}$ $\displaystyle=$ $\displaystyle(1-f_{\rm
e})q^{\rm v}-q^{\rm C},$ (20)

where $q^{\rm v}$ is the dissipative heating rate and $q^{\rm C}$ is the
Coulomb coupling rate that captures the exchange of energy between ions and
electrons. The heating fraction $f_{\rm e}$ is from Rowan et al. (2017) (see
Chael et al. (2018) for more details), same as the 2t-GRMHD simulation.
Finally, $G$ is the radiative cooling rate (Sądowski et al., 2013). For
further details about the equations, see Sądowski et al. (2013, 2017); Chael
et al. (2018).

The simulation assumes a black hole spin of $a=0.9375$ and mass $M_{\rm
BH}=6.2\times 10^{9}M_{\odot}$, targeting M87∗. The gas density is scaled to
physical CGS units such that the compact emission at 230 GHz is roughly 0.98
Jy (Doeleman et al., 2012; Akiyama et al., 2015). The simulation uses modified
Kerr–Schild coordinates with a grid resolution of $N_{r}\times
N_{\theta}\times N_{\varphi}=288\times 224\times 128$. See Chael et al. (2019)
for more details about the simulation, while not utilized for the ngEHT
analysis challenges, we included this simulation in this work since this model
captures the coupling between gas and radiation, necessary for black holes
accreting close the Eddington limit. Further, this model has been used in
previous ngEHT reference array papers (Doeleman et al., 2019; Raymond et al.,
2021).

##  3 Results

We perform a series of comparisons focused on the time-evolution of horizon-
scale quantities and radial dependence of disk and jet properties. The
diagnostics are chosen such that any trends we find can inform EHT/ngEHT
science applications, such as horizon-scale morphology and variability of the
accretion flow. Further, the quantities are similar to those reported in the
GRMHD code comparison project (Porth et al., 2019) and, thus, can be directly
compared. There is a total of five models: three (2t-radiative) GRMHD
simulations targeting M87 *, and one RIAF solution and one GRMHD simulation
for Sgr A *. We further note that all three numerical simulations of M87 *
have the same BH spin, favoring direct comparisons of the horizon-scale gas
properties.

###  3.1 Temporal Behavior of Horizon Fluxes

We calculate the mass, magnetic, angular momentum and energy fluxes in the
radial direction as follows:

$\displaystyle{\rm Mass:\,\,}$ $\displaystyle\dot{M}$
$\displaystyle=\int^{2\pi}_{0}\int^{\pi}_{0}\,(-\rho\,u^{r})\,\sqrt{-g}\,d\theta\,d\varphi\,,$
(21) $\displaystyle{\rm Magnetic:\,\,}$ $\displaystyle\Phi$
$\displaystyle=\frac{\sqrt{4\pi}}{2}\int^{2\pi}_{0}\int^{\pi}_{0}\,|B^{r}|\,\sqrt{-g}\,d\theta\,d\varphi\,,$
(22) $\displaystyle{\rm Ang.Mom.:\,\,}$ $\displaystyle\dot{J}$
$\displaystyle=\int^{2\pi}_{0}\int^{\pi}_{0}\,T^{r}_{\varphi}\,\sqrt{-g}\,d\theta\,d\varphi\,,$
(23) $\displaystyle{\rm Energy:\,\,}$ $\displaystyle\dot{E}$
$\displaystyle=\int^{2\pi}_{0}\int^{\pi}_{0}\,(T^{r}_{t})\,\sqrt{-g}\,d\theta\,d\varphi\,,$
(24)

where all quantities are calculated at the event horizon radius $r_{\rm
hor}=r_{\rm g}(1+\sqrt{1-a^{2}})$. We note that there could be a substantial
contribution of density floors when calculating the mass accretion rate for
MAD systems. However, this radius was chosen for simplicity when comparing to
previous simulations in the literature. Figure 1 shows the mass accretion rate
$\dot{M}$ in units of solar masses per year ($M_{\odot}$/yr), the
dimensionless magnetic flux $\phi=\Phi/\sqrt{\dot{M}r_{\rm g}^{2}c}$, the
outflow power $P_{\rm out}=\dot{M}c^{2}-\dot{E}$, and the specific angular
momentum flux $\dot{J}/\dot{M}$ for simulations targeting M87 * and Sgr A *.
The RIAF solution being a steady-state solution is excluded from this section
(though the hotspot evolves with time). The quantities from the 2t-GRRMHD
simulation are only shown for $(11-16)\times 10^{3}\,\,r_{\rm g}/c$, i.e., the
time period over which the simulation was raytraced in Chael et al. (2019).
Remarkably, despite the difference in electron the physics complexity, the
simulations behave very similarly. The factor of 2 difference in $\dot{M}$
between the M87∗ non-radiative simulations and the 2t-GRRMHD simulation can be
explained by the lower electron temperatures in the near-horizon accretion
flow due to radiative cooling (see Section 3.2) as well as the higher 230 GHz
flux normalization used for the radiative model.

The accretion rate in all simulations show large variation with quasi-periodic
sharp drops. These drops in $\dot{M}$ occur due to the emergence of magnetic
flux eruptions, a characteristic feature of the magnetically arrested disks
(Porth et al., 2021; Begelman et al., 2022; Ripperda et al., 2022; Chatterjee
and Narayan, 2022). These eruptions also lower the value of $\phi$ since
magnetic flux bundles escape from the vicinity of the BH, carrying away the
magnetic flux accumulated in the BH magnetosphere. We see that $\phi$ often
crosses the magnetic flux saturation value of 50 (Tchekhovskoy et al., 2011),
overwhelming the BH magnetosphere with strong magnetic fields that eventually
reconnect and trigger flux eruptions (see Ripperda et al. (2022) for the
detailed mechanism). Figure 2 shows a series of equatorial snapshots from the
M87∗ 2t-GRMHD simulation where we follow a particular magnetic flux eruption
by the rotating gas that is hot ($T_{e}\gtrsim 10^{12}$ K), has low density
($\rho<-2$ in code units), and primarily consists of vertical field lines. As
these field line bundles move out and interact with the disk, they (1) hinder
accretion, lowering $\dot{M}$, (2) remove magnetic flux from near the BH,
lowering the jet power, and (3) push gas outwards, reducing the inward angular
momentum flux. Curiously, we see larger drops in the specific angular momentum
flux for the Sgr A∗ GRMHD model. This is possibly due to the smaller BH spin
($a=0.5$ as opposed to $0.9375$ for the M87∗ models) as the weakly powered jet
does not carry away angular momentum as efficiently as the higher BH spin
models and flux eruptions play a bigger role in regulating disk angular
momentum transport. Additionally, the reconnection events that trigger these
eruptions accelerate electrons to higher energies, and are, thus, crucial for
understanding flare activity in BH sources.

Figure 1: We show the mass accretion rate $\dot{M}$, dimensionless magnetic flux $\phi\equiv\Phi/\sqrt{\dot{M}}$, the outflow efficiency $P_{\rm out}/\dot{M}c^{2}=1-\dot{E}/\dot{M}c^{2}$, and specific radial flux of the angular momentum $\dot{J}/\dot{M}$ over time. Values are calculated at the event horizon. Table 1: GRMHD simulations considered in this work. Simulation grid resolution, simulation time period over which raytracing was performed, modulation index (MI) of the mass accretion rate $\dot{M}$, dimensionless magnetic flux $\phi$, outflow efficiency $P_{\rm out}/\dot{M}c^{2}$, and the specific angular momentum flux $\dot{J}/\dot{M}$ for each GRMHD model. The MI is calculated over the final $5000\,\,r_{\rm g}/c$ in runtime and at the event horizon (see Figure 1). Model | Grid Resolution | Sim. Time | MI($\dot{M}$) | MI($\phi$) | MI | MI
---|---|---|---|---|---|---
Name | ($N_{r}\times N_{\theta}\times N_{\varphi}$) | ($\times 10^{3}\,\,r_{\rm g}/c$) |  |  | ($P_{\rm out}/\dot{M}c^{2}$) | ($\dot{J}/\dot{M}$)
M87∗ GRMHD | $580\times 288\times 512$ | 5.6–10.6 | 0.27 | 0.15 | 0.26 | 0.33
M87∗ 2t-GRMHD | $348\times 192\times 192$ | 10–15 | 0.29 | 0.14 | 0.25 | 0.31
M87∗ 2t-GRRMHD | $288\times 224\times 128$ | 11–16 | 0.28 | 0.14 | 0.14 | 0.31
Sgr A∗ GRMHD | $348\times 192\times 192$ | 30–35 | 0.23 | 0.21 | 0.39 | 0.57

To quantify the time-variability of the horizon-fluxes, we calculate the
modulation index MI, which is defined as the ratio of the standard deviation
and the mean of the quantity over time (Event Horizon Telescope Collaboration
et al., 2022). We show the MI for the different fluxes in Table 1. The
MI($\dot{M}$) is usually a good proxy for the variability of the sub-
millimeter (sub-mm) emission in these slowly accreting optically thin black
hole sources (e.g., Chatterjee et al. (2021)). The MI($\dot{M}$) values we see
from the simulations are $\sim$0.23–0.29 and are larger than expected from Sgr
A∗ 230 GHz lightcurves (where MI $\sim$ 0.1; Wielgus et al. (2022)). This
suggests that careful analysis of the electron distribution function is needed
to understand if we are substantially over-predicting the 230 GHz lightcurve
variability. Further, in general, weakly magnetized accretion flows exhibit
lower MI($\dot{M}$) values due to the absence of flux eruptions, which
suggests that further study of the accretion mode in Sgr A∗ is also necessary.
It is encouraging to note that our MI values for $\dot{M}$ and $\phi$ are
consistent with the MI values from longer time-evolved GRMHD simulations of
$a=0.9$ BHs in Narayan et al. (2022), indicating that our simulations are
sufficiently converged with respect to horizon-scale quantities.

Figure 2: We show equatorial cross-sections of gas density $\rho$ (in
arbitrary code units) and electron temperature $T_{e}$ (in Kelvin) during a
magnetic flux eruption in the 2t-GRMHD simulation of M87∗. Magnetic flux
eruptions, a characteristic feature of magnetically arrested disks, eject
bundles of vertical magnetic fields filled with relativistically hot, low-
density plasma. These flux bundles spiral around the black hole and may be
responsible for high-energy flares (Ripperda et al., 2022). The time unit $M$
is equivalent to $r_{g}/c$.

###  3.2 Disk-Averaged Quantities

Here, we calculate the disk-averaged properties of each model, namely gas
density $\rho$, thermal pressure $p_{\rm gas}$, magnetic pressure $p_{\rm
mag}$, radial velocity $|v_{r}|$, azimuthal velocity $|v_{\varphi}|$, angular
momentum $u_{\varphi}$, disk scale height $h/r$, ion temperature $T_{i}$, and
the electron temperature $T_{e}$. We define disk-averaging of a quantity $q$
as

$\langle
q\rangle(r,t)=\frac{\int^{2\pi}_{0}\int^{\pi}_{0}\,\rho\,q\,\sqrt{-g}\,d\theta\,d\varphi}{\int^{2\pi}_{0}\int^{\pi}_{0}\,\rho\,\sqrt{-g}\,d\theta\,d\varphi},$
(25)

where $q\in\\{\rho,p_{\rm gas},p_{\rm
mag},|v_{r}|,|v_{\varphi}|,u_{\varphi},h/r,T_{i}[\rm Kelvin],T_{e}[\rm
Kelvin]\\}$. Further definitions follow:

$\displaystyle p_{\rm gas}$ $\displaystyle=$ $\displaystyle(\Gamma_{\rm
ad}-1)u,$ (26) $\displaystyle p_{\rm mag}$ $\displaystyle=$ $\displaystyle
b^{\mu}b_{\mu}/2,$ (27) $\displaystyle|v_{i}|$ $\displaystyle=$
$\displaystyle\sqrt{v^{i}v^{i}g_{ii}},$ (28) $\displaystyle{\rm
where,\,\,}v^{i}$ $\displaystyle=$ $\displaystyle u^{i}/u^{t},$ $\displaystyle
h/r$ $\displaystyle=$ $\displaystyle|\theta-\pi/2|,$ (29)

where $\Gamma_{\rm ad}$ and $u$ are the adiabatic index and the internal
energy of the gas.

Figure 3: We show the radial profiles of gas density $\rho$, plasma$-\beta$,
proton temperature $T_{\rm p}$, and electron temperature $T_{\rm e}$.
Quantities are disk-averaged and time-averaged over the raytracing period.

Figure 3 and 4 show the respective disk-averaged radial profiles for each
model, including the Sgr A∗ RIAF solution. The density profiles in the inner
few tens of $r_{\rm g}$ converge roughly to a $n_{\rm e}\propto r^{-1}$
profile, matching the RIAF density profile as well as the longer time-evolved
MAD simulations (Chatterjee and Narayan, 2022). The M87∗ 2t-GRRMHD density is
larger by a factor of $\approx 2$ from the GRMHD/2t-GRMHD, as is expected from
the difference in the mass accretion rate (Figure 1). The 2t-GRRMHD simulation
exhibits a slightly more magnetized inflow within the inner $2\,\,r_{\rm g}$,
but, overall, the GRMHD simulations have a similar plasma-$\beta\equiv p_{\rm
gas}/p_{\rm mag}$ disk profile. The stronger magnetic field seen in the
2t-GRRMHD model could explain the higher values of the horizon magnetic flux
seen in Figure 1. The RIAF model assumes a constant disk plasma-$\beta=10$,
(see Section 2.1), which is substantially higher when compared to the MAD
GRMHD models. This value of plasma-$\beta$ is chosen in order to match the
observed 230 GHz flux density of Sgr A∗. As we see from the disk scale height
in Figure 4, the RIAF model has a much thicker disk than the GRMHD models,
and, therefore, produces a lot more sub-mm emission even with a low electron
temperature and weak magnetic field strength.

Figure 4: We show the radial profiles of disk scale height $h/r$, radial
velocity $|v_{r}|$, angular velocity $\Omega$, and specific angular momentum
$u_{\varphi}$. Quantities are disk-averaged and time-averaged over the
raytracing period.

Next, we see that the disk-averaged electron temperature $T_{\rm e}$ in the
2t-GRRMHD M87∗ model is more than an order of magnitude lower than the other
GRMHD models within the inner $10\,\,r_{\rm g}$, but actually matches the Sgr
A∗ RIAF $T_{\rm e}$ profile and has a shallower profile $T_{\rm e}\propto
r^{-1}$ instead of $r^{-3/2}$. It is also interesting to note that the disk
ion temperatures $T_{\rm i}$ are very similar in all the GRMHD simulations
shown here. Therefore, despite the same reconnection-driven heating mechanism
captured in both the 2t-GRMHD and the 2t-GRRMHD models, the radiative cooling
of hot electrons plays a crucial role in determining the eventual $T_{\rm e}$.
Due to the low $T_{\rm e}$, the required accretion rate normalization is
higher in the 2t-GRRMHD, as we noted in the previous subsection.

In Figure 4, we show the average disk scale height $h/r$, the radial and
angular velocities ($v_{r}$ and $\Omega$ and the specific angular momentum
$u_{\varphi}$). The MAD simulations all show very similar disk properties. The
$\langle h/r\rangle\approx$ 0.1–0.3 with a sharp increase within $3\,\,r_{\rm
g}$ where the inflow becomes vertically supported by strong poloidal magnetic
fields. The radial velocity has a profile of $r^{-1/2}$, similar to the
scaling relation found in ADAF solutions assuming a constant viscosity
parameter $\alpha$ (Narayan and Yi, 1994; Narayan et al., 1998). The $\alpha$
parameter profile depends on how magnetized the accretion flow is, with
$\alpha\propto r^{-1}$ for the weakly magnetized flows and close to constant
for the MAD-like flows (e.g., Liska et al. (2020); Chatterjee and Narayan
(2022)). We also see highly sub-Keplerian angular velocity profiles in the
GRMHD models, typical for magnetically supported disks. For the RIAF model,
the RIAF disk is not infalling and has a constant Keplerian angular velocity.
Instead, the hotspot, added to the RIAF solution, undergoes shearing and
disappears into the BH with a radial velocity similar to the values found in
the GRMHD MAD disks. This occurs because the hotspot is designed to travel
along plunging geodesics (see Section 2.1), similar to rapid gas infall close
to the BH in the GRMHD models. The angular momentum in the GRMHD models looks
sub-Keplerian as expected for MADs.

Figure 5: We show the jet radius $R_{\rm jet}$ and the jet Lorentz factor
$\gamma$ from the M87∗ GRMHD and 2t-GRRMHD models, and the Sgr A∗ GRMHD model.
The gray circles indicates the deprojected jet radius of the M87 jet assuming
a BH mass of $6.2\times 10^{9}M_{\odot}$ and a source inclination of
$14^{\circ}$ (Nakamura et al., 2018). The data points are a compilation of
various papers (Doeleman et al., 2012; Asada and Nakamura, 2012; Hada et al.,
2013; Nakamura and Asada, 2013; Akiyama et al., 2015; Hada et al., 2016).

###  3.3 Jet Properties

Here, we calculate the radial profiles of the jet half width $R_{\rm jet}$ and
Lorentz factor $\gamma$:

$R_{\rm
jet}(r,t)=\sqrt{\frac{\int^{2\pi}_{0}\int^{\pi}_{0}\,\sqrt{-g/g_{rr}}(\mu>2)\,d\theta\,d\varphi}{2\pi}}\,,$
(30) $\gamma(r,t)=\frac{\int^{2\pi}_{0}\int^{\pi}_{0}\,(\mu>2)\,\alpha
u^{t}\,\sqrt{-g}\,d\theta\,d\varphi}{\int^{2\pi}_{0}\int^{\pi}_{0}\,(\mu>2)\,\sqrt{-g}\,d\theta\,d\varphi},$
(31)

where $\mu=-T^{r}_{t}/(\rho u^{r})$ is the specific radial energy density and
$\alpha=1/\sqrt{-g^{tt}}$. We calculate the average jet radius $R_{\rm jet}$
by integrating over the total surface area covered by the two jets over a
shell at a particular radius, and assume that the total surface area is
approximately $2\pi R_{\rm jet}^{2}$. We define the jet boundary as $\mu>2$,
i.e., the region over which the jet still remains highly energized. This
definition of the jet boundary is quite similar to the condition $\sigma=1$,
which is widely used in the literature (e.g., Narayan et al. (2022)). Since
$\mu=\gamma(\sigma+E_{\rm th})$, our condition $\mu>2$ also incorporates
regions where the jet might not be magnetically dominated but is
relativistically hot or fast. Since we restrict our jet profiles to within
$r\lesssim 10^{3}\,r_{\rm g}$, the jet radius is primarily determined by the
jet magnetization. Specific enthalpy includes the rest-mass energy
contribution in our definition from Section 2.3.

Figure 5 shows the jet radius $R_{\rm jet}$ and Lorentz factor $\gamma$ as a
function of the radial distance from the BH for the M87∗ GRMHD and 2t-GRRMHD
models as well as the Sgr A∗ GRMHD model. The M87∗ jet radius from our models
matches the observed jet width from M87 (gray circles) quite well, with the
radial profile roughly proportional to $r^{0.625}$, which is the fitted
powerlaw for the M87 jet (Nakamura et al., 2018), though the index value has
also been reported to be slightly smaller in some works (0.57; Asada and
Nakamura (2012); Nokhrina et al. (2019)). The power law index of 0.625 is
larger than that found using the $\sigma=1$ condition from Narayan et al.
(2022), where the authors found a power law index of 0.428 for their MAD spin
$a=0.9$ GRMHD model. It is possible that we find larger jet radii as we
incorporate a part of the hot jet sheath region within our definition of
$R_{\rm jet}$ (as suggested by Figure 7 in Chatterjee et al. (2019)). For the
Sgr A∗ model, we also find a similar $R_{\rm jet}$ profile. While there are no
detections of an extended jet in Sgr A∗ (e.g., Issaoun et al. (2019)), semi-
analytical and GRMHD models of Sgr A∗ largely favor a jet component from a
spinning BH (e.g., Markoff et al. (2007); Event Horizon Telescope
Collaboration et al. (2022)).

We also show the Lorentz factor $\gamma$ in Figure 5. Mostly, the jets
accelerate to $\gamma\approx$ 3–4 by $10^{3}\,r_{\rm g}$ in all of our GRMHD
models. It is difficult to compare our $\gamma$ profiles with values inferred
from observations of the M87 jet (e.g., Mertens et al. (2016)) since our
$\gamma$ values are biased toward the jet spine while the observations
generally capture the velocities of localized features in the sub-relativistic
jet sheath/disk wind, especially at small distances from the BH. Indeed, both
simulations and observations show that the jet Lorentz factor varies greatly
as a function of jet radius (e.g., see Chatterjee et al. (2019)). We speculate
that a better approach might be to calculate emissivity-weighted Lorentz
factors in order to compare to the measured $\gamma$ from M87. Since our focus
is on the comparison between GRMHD simulations, we leave direct comparisons to
observational data to future work.

###  3.4 Axisymmetrized Profiles

In the previous sections, we found that the largest differences between the
GRMHD models occur in electron temperature distribution. Figure 6 shows the
time and azimuthally averaged 2D vertical plots of gas density $n_{\rm e}$ and
electron temperature $T_{\rm e}$. We show the normalized $n_{\rm e}$ so as to
capture the relative change in the disk/wind density distribution, which would
provide us information about the disk structure. The large difference in disk
scale height is immediately apparent between the RIAF and the MAD GRMHD models
(also see Figure 4). The presence of a prominent wide jet component in MADs
squeezes the inner disk and pushes against the accretion flow, a feature which
is not captured in the constant $h/r$ RIAF model. However, the RIAF model
roughly reproduces the density profile of the disk midplane region, suggesting
that the RIAF model could represent non/weakly jetted, quasi-spherical
accretion flows quite well. For sources such as M87∗, where we see a prominent
jet component, the density gradient in the vertical direction is expected to
be steeper as strong magnetic stresses power winds carry gas away from the
disk (e.g., Chatterjee and Narayan (2022)).

Overall, the disk/wind density distribution among the GRMHD models look
similar with small differences in the lateral extension of the wind region and
the steepness of the vertical gradient in density. For example, if we compare
the 2t-GRRMHD model with the other two simulations, the density in the wind
region is larger in the radiative model. The reason for the shallow vertical
density profile in the 2t-GRRMHD model is unclear since the weakly magnetized
thick disk simulations tell us that radiative cooling would lead to the loss
of gas pressure in the disk and would result in the disk collapsing to a
relatively dense structure in the midplane (e.g., Fragile and Meier (2009);
Yoon et al. (2020)). However, in the presence of strong poloidal magnetic
fields, i.e., in the MAD state, the plasma-$\beta$ decreases to $\beta\approx$
0.2–1 in the disk midplane (see Figure 3, third row, left panel), and can
reach even lower values in the upper layers of the accretion flow. The high
magnetic pressure could help support the disk against collapse while
sufficiently strong magnetic stresses could power disk winds. Such behavior is
also seen in recent GRMHD simulations of near-Eddington, geometrically thin,
strongly magnetized disks, where the inner disk (or corona) has a larger $h/r$
than the outer disk due to magnetic pressure support (Sadowski, 2016; Liska et
al., 2022; Mishra et al., 2022). To verify how radiative cooling affects the
inner disk/wind structure in highly sub-Eddington accretion flows such as M87∗
and Sgr A∗, we require longer 2t-GRRMHD simulations such that the disk is in
an inflow–outflow equilibrium out to a radius of at least $50\,r_{\rm g}$.

Figure 6: We show t- and $\varphi$-averaged data: electron number density
$n_{\rm e}$ (top row) and temperature $T_{\rm e}$ (bottom row). We also denote
the jet boundary with $\sigma=1$ (black lines). The time-averaging is
performed over the $5000\,r_{\rm g}/c$ for each model. RIAF plots are for Sgr
A∗ while the rest are for M87. The Sgr A∗ GRMHD model produces similar plots
of $n_{\rm e}$ and $T_{\rm e}$ as the M87∗ model, and, hence, we do not show
it here.

The 2D temperature plot of the RIAF model also looks vastly different in the
inner disk ($r\lesssim 20\,r_{\rm g}$) when compared to the GRMHD and 2t-GRMHD
simulations, but is similar to the temperature distribution in the 2t-GRRMHD
disk midplane (also seen in the $T_{\rm e}$ plot of Figure 3). The RIAF model
does not capture gas heating in the jet sheath region (the region just outside
of the jet boundary indicated by the $\sigma=1$ dashed line) and, therefore,
$T_{\rm e}$ drops as we move away from the midplane toward the poles. In the
GRMHD models, the jet sheath is as hot, if not hotter, than the inner
accretion flow as temperatures reach $T_{\rm e}>10^{11}$ K. For the GRMHD
simulation, the electron temperature is provided as a fraction of the fluid
temperature, where the fraction depends on how magnetized the gas is in the
region, as per the $R-\beta$ prescription from Equation (14). For the M87∗
model, we chose a $R_{\rm high}$ value of 160 to have a jet-dominated sub-mm
image. This choice of $R_{\rm high}$ suppresses the electron temperature in
the disk, focusing higher temperatures in the jet sheath. Comparing the GRMHD
model with the 2t-GRMHD model, the jet sheath region exhibits very similar
$T_{\rm e}$ values, but the disk midplane is hotter by a factor of a few in
the 2t-GRMHD model. We note that this difference in $T_{\rm e}$ in the
midplane is more noticeable in the 2D plot rather than in the disk-averaged
$T_{\rm e}$ profile shown in Figure 3 as the upper layers of the disk become
substantially hotter in the GRMHD model.

For the radiative 2t-GRRMHD model, the inner regions of the disk are cooler as
electrons heated by magnetic reconnection quickly cool via synchrotron and
Compton losses. From Figure 3, the drop in $T_{\rm e}$ for the 2t-GRRMHD model
is shown to be as large as an order of magnitude when compared to the (2t-)
GRMHD models. Another interesting feature is that the hot region ($T_{\rm
e}>10^{11}$ K) in the jet sheath is much narrower in the 2t-GRRMHD model,
which could have a significant bearing on the ray-traced image, possibly
producing a thinner jet sheath. Finally, the difference in $T_{\rm e}$ in the
jet body between the GRMHD models is due to the different density/internal
energy floor setups used by the corresponding codes. Since the gas in the jet
sheath and the jet body undergo mixing due to boundary instabilities (e.g.,
Chatterjee et al. (2019); Wong et al. (2021)), it is possible that the choice
of floors could affect the overall electron temperature in the jet sheath.
Such a study is outside the scope of our paper and is left to future work.

Figure 7: We show the $\varphi-$averaged hotspot electron number density as a
function of radius and time. The hotspot falls into the BH and becomes sheared
over time.

###  3.5 Orbiting Hotspot in a RIAF Model

High-energy flares are commonly observed in AGNs, with GeV and MeV flares seen
in M87∗ (e.g., Aharonian et al. (2006); Acciari et al. (2010)) and quasi-daily
nIR and X-ray flares in Sgr A∗ (e.g., Baganoff et al. (2001); Eckart et al.
(2006); Hornstein et al. (2007); Nowak et al. (2012); Neilsen et al. (2013);
Witzel et al. (2018); Do et al. (2019); Haggard et al. (2019)). A number of
attempts have been performed to explain the origin of flaring, such as
magnetic reconnection in turbulent gas flows in the disk and the jet (Dodds-
Eden et al., 2010; Dibi et al., 2014; Chatterjee et al., 2019; Nathanail et
al., 2020; Chatterjee et al., 2021) and magnetic flux eruptions (Dexter et
al., 2020; Porth et al., 2021; Scepi et al., 2022; Ripperda et al., 2022). For
Sgr A∗, the semi-analytical models found that we require high-energy
electrons, assumed to be accelerated via an ad hoc process such as a large
magnetic reconnection event or shocks, to describe the large flaring events
(Markoff et al., 2001; Dibi et al., 2016; Gutiérrez et al., 2020). The near-
infrared observations from the GRAVITY telescope provided further evidence for
orbiting hotspot features in the accretion flow (Gravity Collaboration et al.,
2020) that may be linked to acceleration events. It has also been recently
shown that orbiting hotspots can be used to model double-peaked X-ray flares
(Haggard et al., 2019; Ball et al., 2021) and prominent Stokes Q-U loops in
the sub-mm emission of Sgr A∗ (Wielgus et al., 2022). These results provide us
with considerable motivation to test the capability of the ngEHT to detect
hotspot formation in accretion flows around black holes.

Instead of isolating a particular magnetic flux eruption event in our
simulations, we added a shearing hotspot to the RIAF solution as detailed in
Section 2.1. Figure 7 shows the temporal evolution of the azimuthally averaged
electron number density of the hotspot. We begin with a Gaussian distribution
of gas that undergoes shearing as the gas falls in closer to the BH. The
overall density normalization is much lower than in the RIAF disk since the
optically thin hotspot gas produces a large enough non-thermal synchrotron
emissivity. The hotspot is evolved over $800\,r_{\rm g}/c$, but the gas
distribution comes to a near-steady-state profile within the first
$200\,r_{\rm g}/c$, which is roughly one hour for Sgr A∗. The shearing of the
hotspot gas has a significant impact on the evolution of the 230 GHz image
(Tiede et al., 2020; Roelofs et al., ). From Figure 4 (right column), we see
that the radial velocity matches the disk-averaged gas velocity from the GRMHD
model, showing nearly free-fall speeds, while the azimuthal velocity becomes
highly sub-Keplerian. The velocity profiles show that our hotspot model should
be able to reproduce the expected hotspot motion from the GRMHD models, and is
ideal for investigating multiwavelength flare lightcurves. Emami et al.
(2022), a companion paper, goes into further details about how current
dynamical reconstruction techniques can be used to trace out the motion and
morphology of the shearing hotspot in the context of ngEHT observations. These
hotspot models and reconstruction methods would be integral in deciphering the
more complex gas dynamics of magnetic flux eruption events in MADs, which have
been shown to produce a significant variation in the image structure of M87∗
at 230 GHz (e.g., Gelles et al. (2022)).

##  4 Conclusions

In this work, for the first time, we compare a series of numerical solutions
with increasing complexity that were specifically constructed to understand
the accretion flow around the supermassive black holes M87∗ and Sgr A∗. We
include a time-independent radiatively inefficient accretion flow (RIAF) model
as well as fully 3D GRMHD simulations of accreting black holes, incorporating
the effects of electron heating and cooling losses via two-temperature and
radiation physics. In addition, each of our simulations are run with different
GRMHD codes, which is similar to the approach of another community-wide code
comparison effort (Porth et al., 2019). We found that the simulations exhibit
remarkably similar properties given that the simulations incorporate varying
levels of complexity in electron physics. The notable exception is the
electron temperature, where radiative cooling decreases the temperature by a
factor of $\lesssim 10$ within the inner 10 gravitational radii, the region
that produces the bulk of the 230 GHz emission in M87∗, one of the two primary
targets of the EHT and the ngEHT (the other being Sgr A∗). The main goal of
this work is to understand the variation in the underlying accretion flow and
jet properties in our models since synthetic ray-traced images constructed
from these models are used as “truth” images of M87∗ and Sgr A∗ for the ngEHT
Analysis Challenges (Roelofs et al., ). The ngEHT Analysis Challenges are an
effort to determine how much information about the accretion flow and jet
dynamics we can glean from the proposed ngEHT reference array, and what
modifications in the image reconstruction tools are necessarily required to
decode future ngEHT observational data.

Our paper deals with numerical models designed to investigate hotspot
evolution, turbulent inspiralling gas flows, and extended powerful jets,
targeting M87∗ and Sgr A∗. We restricted our model set to the community-
standard setup: a rotating, geometrically thick, optically thin torus of
magnetized gas around a spinning black hole, which is the fiducial model
choice of the EHT (Event Horizon Telescope Collaboration et al., 2019, 2021,
2022). This model choice leaves out the exploration of multiple new setups of
black hole accretion, such as quasi-spherical wind-fed inflows (e.g., Ressler
et al. (2020); Lalakos et al. (2022)), strongly wind-fed accretion (e.g.,
Cruz-Osorio et al. (2017); Kaaz et al. (2022)), geometrically thin accretion
disks (e.g., Avara et al. (2016); Liska et al. (2022); Mishra et al. (2022)),
puffy radiation-dominated super-Eddington disks (e.g., Sadowski and Narayan
(2016); Curd and Narayan (2019)), and misaligned accretion disks (e.g.,
Fragile et al. (2007); Liska et al. (2018); White et al. (2019); Chatterjee et
al. (2020)). Apart from varying the accretion mode, the high resolution of the
images from EHT and ngEHT could potentially help distinguish between different
space–time metrics (Event Horizon Telescope Collaboration et al., 2022). To
date, only a limited number of non-Kerr GRMHD simulations have only been
performed (e.g., Mizuno et al. (2018); Olivares et al. (2020); Nampalliwar et
al. (2022)). The future of numerical studies is bright, given their rising
popularity in the astrophysics community and the increase in computational
resources. The breadth of the current investigations in accretion physics
would result in a plethora of variable structures that should be thoroughly
studied keeping the observational capabilities of the ngEHT in mind.

Conceptualization, Koushik Chatterjee; Methodology, Koushik Chatterjee, Paul
Tiede, Yosuke Mizuno and Freek Roelofs; Software, Koushik Chatterjee, Andrew
A. Chael, Paul Tiede and Yosuke Mizuno; Formal analysis, Koushik Chatterjee
and Paul Tiede; Investigation, Koushik Chatterjee, Yosuke Mizuno, Razieh
Emami, Christian Fromm, Angelo Ricarte, Lindy Blackburn, Freek Roelofs,
Philipp Arras, Antonio Fuentes, Jakob Knollmüller, Nikita Kosogorov, Greg
Lindahl, Hendrik Müller, Nimesh Patel, Alexander Raymond, Efthalia Traianou
and Justin Vega; Writing – original draft, Koushik Chatterjee and Paul Tiede;
Writing – review & editing, Koushik Chatterjee, Andrew A. Chael and Angelo
Ricarte; Visualization, Koushik Chatterjee; Supervision, Michael D. Johnson
and Sheperd Doeleman.

###### Acknowledgements.

We thank the anonymous referees for their suggestions that greatly improved
the text. This research was enabled by support provided by grant no. NSF
PHY-1125915 along with INCITE and ASCR Leadership Computing Challenge (ALCC)
programs under the award PHY129, using resources from the Oak Ridge Leadership
Computing Facility, Summit, which is a US Department of Energy office of
Science User Facility supported under contract DE-AC05- 00OR22725, as well as
Calcul Quebec and Compute Canada under award xsp-772-ab (PI: D. Haggard). We
thank the National Science Foundation (AST-1716536, AST-1935980, AST-1935980,
and AST-2034306) and the Gordon and Betty Moore Foundation (GBMF-10423) for
financially supporting this work. This work was supported in part by the Black
Hole Initiative, which is funded by grants from the John Templeton Foundation
(JTF-61497) and the Gordon and Betty Moore Foundation (GBMF-8273) to Harvard
University. K.C. is also supported in part by the Black Hole PIRE program (NSF
grant OISE-1743747). R.E. acknowledges the support by the Institute for Theory
and Computation at the Center for Astrophysics as well as grant numbers
21-atp21-0077, NSF AST-1816420, and HST-GO-16173.001-A for very generous
supports. H.M. received financial support for this research from the
International Max Planck Research School (IMPRS) for Astronomy and
Astrophysics at the Universities of Bonn and Cologne. This research is
supported by the DFG research grant “Jet physics on horizon scales and beyond”
(Grant No. FR 4069/2-1), the ERC synergy grant “BlackHoleCam: Imaging the
Event Horizon of Black Holes” (Grant No. 610058), and ERC advanced grant
“JETSET: Launching, propagation and emission of relativistic jets from binary
mergers and across mass scales” (Grant No. 884631). J.K. acknowledges funding
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany´s Excellence Strategy—EXC 2094—390783311. Y.M. is supported by the
National Natural Science Foundation of China (No. 12273022) and the Shanghai
pilot program of international scientists for basic research (No.
22JC1410600). The data shown in this study has been generated using the
following codes: BHAC (Porth et al., 2017), BHOSS (Younsi et al., 2020), H-AMR
(Liska et al., 2022), IPOLE (Mościbrodzka and Gammie, 2018), KORAL (Sądowski
et al., 2013), with access provided upon request to the corresponding authors.
The authors declare no conflict of interest. The funders had no role in the
design of the studyl in the collection, analyses, or interpretation of data;
in the writing of the manuscript, or in the decision to publish the results.
[custom] References

## References

  * Event Horizon Telescope Collaboration et al. (2019) Event Horizon Telescope Collaboration; Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; et al. First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. Astrophys. J. Lett. 2019, 875, L1. https://doi.org/10.3847/2041-8213/ab0ec7.
  * Event Horizon Telescope Collaboration et al. (2022) Event Horizon Telescope Collaboration; Akiyama, K.; Alberdi, A.; Alef, W.; Algaba, J.C.; Anantua, R.; Asada, K.; Azulay, R.; Bach, U.; Baczko, A.K.; et al.  First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way.  Astrophys. J. Lett. 2022, 930, L12. https://doi.org/10.3847/2041-8213/ac6674.
  * Doeleman et al. (2019) Doeleman, S.; Blackburn, L.; Dexter, J.; Gomez, J.L.; Johnson, M.D.; Palumbo, D.C.; Weintroub, J.; Farah, J.R.; Fish, V.; Loinard, L.; et al.  Studying Black Holes on Horizon Scales with VLBI Ground Arrays.  Bulletin of the AAS 2019, 51(7), retrieved from https://baas.aas.org/pub/2020n7i256.
  * (4) Roelofs, F.; Blackburn, L.; Lindahl, G.; Doeleman, S.S.; Johnson, M.D.; Arras, P.; Chatterjee, K.; Emami, R.; Fromm, C.; Fuentes, A.; et al.  The ngEHT Analysis Challenges.  Galaxies, 2022, 11(1), 12. https://doi.org/10.3390/galaxies11010012.
  * Raymond et al. (2021) Raymond, A.W.; Palumbo, D.; Paine, S.N.; Blackburn, L.; Córdova Rosado, R.; Doeleman, S.S.; Farah, J.R.; Johnson, M.D.; Roelofs, F.; Tilanus, R.P.J.; et al.  Evaluation of New Submillimeter VLBI Sites for the Event Horizon Telescope.  Astrophys. J. Suppl. Ser. 2021, 253, 5 https://doi.org/10.3847/1538-3881/abc3c3.
  * Shakura and Sunyaev (1973) Shakura, N.I.; Sunyaev, R.A.  Reprint of 1973A&A….24..337S. Black holes in binary systems. Observational appearance.  Astron. Astrophys. 1973, 500, 33–51.
  * Rees et al. (1982) Rees, M.J.; Phinney, E.S.; Begelman, M.C.; Blandford, R.D.  Ion-supported tori and the origin of radio jets.  Nature 1982, 295, 17–21.
  * Narayan et al. (1995) Narayan, R.; Yi, I.; Mahadevan, R.  Explaining the spectrum of Sagittarius A∗ with a model of an accreting black hole.  Nature 1995, 374, 623–625. https://doi.org/10.1038/374623a0.
  * Quataert and Gruzinov (2000) Quataert, E.; Gruzinov, A.  Convection-dominated Accretion Flows.  Astrophys. J. 2000, 539, 809–814. https://doi.org/10.1086/309267.
  * Narayan et al. (2003) Narayan, R.; Igumenshchev, I.V.; Abramowicz, M.A.  Magnetically Arrested Disk: An Energetically Efficient Accretion Flow.  PASJ 2003, 55, L69–L72. https://doi.org/10.1093/pasj/55.6.L69.
  * McKinney (2006) McKinney, J.C.  General relativistic magnetohydrodynamic simulations of the jet formation and large-scale propagation from black hole accretion systems.  Mon. Not. R. Astron. Soc. 2006, 368, 1561–1582, https://doi.org/10.1111/j.1365-2966.2006.10256.x.
  * Komissarov et al. (2007) Komissarov, S.S.; Barkov, M.V.; Vlahakis, N.; Königl, A.  Magnetic acceleration of relativistic active galactic nucleus jets.  Mon. Not. R. Astron. Soc. 2007, 380, 51–70. https://doi.org/10.1111/j.1365-2966.2007.12050.x.
  * Tchekhovskoy et al. (2011) Tchekhovskoy, A.; Narayan, R.; McKinney, J.C.  Efficient generation of jets from magnetically arrested accretion on a rapidly spinning black hole.  Mon. Not. R. Astron. Soc. 2011, 418, L79–L83. https://doi.org/10.1111/j.1745-3933.2011.01147.x.
  * Narayan et al. (2012) Narayan, R.; Sa̧dowski, A.; Penna, R.F.; Kulkarni, A.K.  GRMHD simulations of magnetized advection-dominated accretion on a non-spinning black hole: Role of outflows.  Mon. Not. R. Astron. Soc. 2012, 426, 3241–3259. https://doi.org/10.1111/j.1365-2966.2012.22002.x.
  * Tchekhovskoy and Bromberg (2016) Tchekhovskoy, A.; Bromberg, O.  Three-dimensional relativistic MHD simulations of active galactic nuclei jets: Magnetic kink instability and Fanaroff–Riley dichotomy.  Mon. Not. R. Astron. Soc. 2016, 461, L46–L50. https://doi.org/10.1093/mnrasl/slw064.
  * Yuan et al. (2003) Yuan, F.; Quataert, E.; Narayan, R. Nonthermal Electrons in Radiatively Inefficient Accretion Flow Models of Sagittarius A*.  Astrophys. J. 2003, 598, 301–312. https://doi.org/10.1086/378716.
  * Lucchini et al. (2019) Lucchini, M.; Krauß, F.; Markoff, S.  The unique case of the active galactic nucleus core of M87: A misaligned low-power blazar?  Mon. Not. R. Astron. Soc. 2019, 489, 1633–1643. https://doi.org/10.1093/mnras/stz2125.
  * Event Horizon Telescope Collaboration et al. (2019) Event Horizon Telescope Collaboration; Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; et al.  First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring.  Astrophys. J. Lett. 2019, 875, L5. https://doi.org/10.3847/2041-8213/ab0f43.
  * Event Horizon Telescope Collaboration et al. (2021) Event Horizon Telescope Collaboration; Akiyama, K.; Algaba, J.C.; Alberdi, A.; Alef, W.; Anantua, R.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; et al.  First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon.  Astrophys. J. Lett. 2021, 910, L13. https://doi.org/10.3847/2041-8213/abe4de.
  * Event Horizon Telescope Collaboration et al. (2022) Event Horizon Telescope Collaboration; Akiyama, K.; Alberdi, A.; Alef, W.; Carlos Algaba, J.; Anantua, R.; Asada, K.; Azulay, R.; Bach, U.; Baczko, A.K.; et al.  First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole.  Astrophys. J. Lett. 2022, 930, L16. https://doi.org/10.3847/2041-8213/ac6672.
  * Igumenshchev et al. (2003) Igumenshchev, I.V.; Narayan, R.; Abramowicz, M.A.  Three-dimensional Magnetohydrodynamic Simulations of Radiatively Inefficient Accretion Flows.  Astrophys. J. 2003, 592, 1042–1059. https://doi.org/10.1086/375769.
  * Blandford and Znajek (1977) Blandford, R.D.; Znajek, R.L.  Electromagnetic extraction of energy from Kerr black holes.  Mon. Not. R. Astron. Soc. 1977, 179, 433–456.
  * Gammie et al. (2003) Gammie, C.F.; McKinney, J.C.; Tóth, G. HARM: A Numerical Scheme for General Relativistic Magnetohydrodynamics. Astrophys. J. 2003, 589, 444–457. https://doi.org/10.1086/374594.
  * Fragile et al. (2007) Fragile, P.C.; Blaes, O.M.; Anninos, P.; Salmonson, J.D.  Global General Relativistic Magnetohydrodynamic Simulation of a Tilted Black Hole Accretion Disk.  Astrophys. J. 2007, 668, 417–429. https://doi.org/10.1086/521092.
  * Chael et al. (2019) Chael, A.; Narayan, R.; Johnson, M.D.  Two-temperature, Magnetically Arrested Disc simulations of the jet from the supermassive black hole in M87.  Mon. Not. R. Astron. Soc. 2019, 486, 2873–2895. https://doi.org/10.1093/mnras/stz988.
  * Porth et al. (2019) Porth, O.; Chatterjee, K.; Narayan, R.; Gammie, C.F.; Mizuno, Y.; Anninos, P.; Baker, J.G.; Bugli, M.; Chan, C.k.; Davelaar, J.; et al.  The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project.  Astrophys. J. Suppl. Ser. 2019, 243, 26. https://doi.org/10.3847/1538-4365/ab29fd.
  * Narayan et al. (2022) Narayan, R.; Chael, A.; Chatterjee, K.; Ricarte, A.; Curd, B.  Jets in magnetically arrested hot accretion flows: Geometry, power, and black hole spin-down.  Mon. Not. R. Astron. Soc. 2022, 511, 3795–3813. https://doi.org/10.1093/mnras/stac285.
  * Janssen et al. (2021) Janssen, M.; Falcke, H.; Kadler, M.; Ros, E.; Wielgus, M.; Akiyama, K.; Baloković, M.; Blackburn, L.; Bouman, K.L.; Chael, A.; et al.  Event Horizon Telescope observations of the jet launching and collimation in Centaurus A.  Nat. Astron. 2021, 5, 1017–1028. https://doi.org/10.1038/s41550-021-01417-w.
  * Broderick and Loeb (2006) Broderick, A.E.; Loeb, A.  Imaging optically thin hotspots near the black hole horizon of Sgr A* at radio and near-infrared wavelengths.  Mon. Not. R. Astron. Soc. 2006, 367, 905–916. https://doi.org/10.1111/j.1365-2966.2006.10152.x.
  * Broderick and McKinney (2010) Broderick, A.E.; McKinney, J.C.  Parsec-scale Faraday Rotation Measures from General Relativistic Magnetohydrodynamic Simulations of Active Galactic Nucleus Jets.  Astrophys. J. 2010, 725, 750–773. https://doi.org/10.1088/0004-637X/725/1/750.
  * Tiede et al. (2020) Tiede, P.; Pu, H.Y.; Broderick, A.E.; Gold, R.; Karami, M.; Preciado-López, J.A.  Spacetime Tomography Using the Event Horizon Telescope.  Astrophys. J. 2020, 892, 132. https://doi.org/10.3847/1538-4357/ab744c.
  * Broderick et al. (2011) Broderick, A.E.; Fish, V.L.; Doeleman, S.S.; Loeb, A.  Evidence for Low Black Hole Spin and Physically Motivated Accretion Models from Millimeter-VLBI Observations of Sagittarius A*.  Astrophys. J. 2011, 735, 110. https://doi.org/10.1088/0004-637X/735/2/110.
  * Broderick et al. (2016) Broderick, A.E.; Fish, V.L.; Johnson, M.D.; Rosenfeld, K.; Wang, C.; Doeleman, S.S.; Akiyama, K.; Johannsen, T.; Roy, A.L.  Modeling Seven Years of Event Horizon Telescope Observations with Radiatively Inefficient Accretion Flow Models.  Astrophys. J. 2016, 820, 137. https://doi.org/10.3847/0004-637X/820/2/137.
  * Pu et al. (2016) Pu, H.Y.; Akiyama, K.; Asada, K.  The Effects of Accretion Flow Dynamics on the Black Hole Shadow of Sagittarius A*.  Astrophys. J. 2016, 831, 4. https://doi.org/10.3847/0004-637X/831/1/4.
  * Broderick and Blandford (2004) Broderick, A.; Blandford, R.  Covariant magnetoionic theory - II. Radiative transfer.  Mon. Not. R. Astron. Soc. 2004, 349, 994–1008. https://doi.org/10.1111/j.1365-2966.2004.07582.x.
  * Jones and O’Dell (1977) Jones, T.W.; O’Dell, S.L.  Transfer of polarized radiation in self-absorbed synchrotron sources. I. Results for a homogeneous source.  Astrophys. J. 1977, 214, 522–539. https://doi.org/10.1086/155278.
  * Liska et al. (2022) Liska, M.T.P.; Chatterjee, K.; Issa, D.; Yoon, D.; Kaaz, N.; Tchekhovskoy, A.; van Eijnatten, D.; Musoke, G.; Hesp, C.; Rohoza, V.; et al.  H-AMR: A New GPU-accelerated GRMHD Code for Exascale Computing with 3D Adaptive Mesh Refinement and Local Adaptive Time Stepping.  Astrophys. J. Suppl. Ser. 2022, 263, 26. https://doi.org/10.3847/1538-4365/ac9966.
  * Porth et al. (2017) Porth, O.; Olivares, H.; Mizuno, Y.; Younsi, Z.; Rezzolla, L.; Moscibrodzka, M.; Falcke, H.; Kramer, M.  The black hole accretion code.  Comput. Astrophys. Cosmol. 2017, 4, 1. https://doi.org/10.1186/s40668-017-0020-2.
  * Sądowski et al. (2013) Sądowski, A.; Narayan, R.; Tchekhovskoy, A.; Zhu, Y.  Semi-implicit scheme for treating radiation under M1 closure in general relativistic conservative fluid dynamics codes.  Mon. Not. R. Astron. Soc. 2013, 429, 3533–3550. https://doi.org/10.1093/mnras/sts632.
  * Liska et al. (2018) Liska, M.; Hesp, C.; Tchekhovskoy, A.; Ingram, A.; van der Klis, M.; Markoff, S.  Formation of precessing jets by tilted black hole discs in 3D general relativistic MHD simulations.  Mon. Not. R. Astron. Soc. 2018, 474, L81–L85. https://doi.org/ 10.1093/mnrasl/slx174.
  * Younsi et al. (2020) Younsi, Z.; Porth, O.; Mizuno, Y.; Fromm, C.M.; Olivares, H.  Modelling the polarised emission from black holes on event horizon-scales.  Proceedings of the IAU 2020, 14(S342), 9–12. Published by Cambridge University Press as part of Perseus in Sicily: From Black Hole to Cluster Outskirts, eds. Asada, K., de Gouveia Dal Pino, E., Giroletti, M., Nagai, H., Nemmen, R., Noto, Italy, 14–18 May 2018. https://doi.org/10.1017/S1743921318007263.
  * Mościbrodzka and Gammie (2018) Mościbrodzka, M.; Gammie, C.F.  IPOLE - semi-analytic scheme for relativistic polarized radiative transport.  Mon. Not. R. Astron. Soc. 2018, 475, 43–54. https://doi.org/10.1093/mnras/stx3162.
  * Gebhardt et al. (2011) Gebhardt, K.; Adams, J.; Richstone, D.; Lauer, T.R.; Faber, S.M.; Gültekin, K.; Murphy, J.; Tremaine, S.  The Black Hole Mass in M87 from Gemini/NIFS Adaptive Optics Observations.  Astrophys. J. 2011, 729, 119. https://doi.org/10.1088/0004-637X/729/2/119.
  * Do et al. (2019) Do, T.; Witzel, G.; Gautam, A.K.; Chen, Z.; Ghez, A.M.; Morris, M.R.; Becklin, E.E.; Ciurlo, A.; Hosek, Matthew, J.; Martinez, G.D.; et al.  Unprecedented Near-infrared Brightness and Variability of Sgr A *.  Astrophys. J. Lett. 2019, 882, L27. https://doi.org/ 10.3847/2041-8213/ab38c3.
  * GRAVITY Collaboration et al. (2019) GRAVITY Collaboration; Abuter, R.; Amorim, A.; Bauböck, M.; Berger, J.P.; Bonnet, H.; Brandner, W.; Clénet, Y.; Coudé Du Foresto, V.; de Zeeuw, P.T.; et al.  A geometric distance measurement to the Galactic center black hole with 0.3% uncertainty.  Astron. Astrophys. 2019, 625, L10. https://doi.org/10.1051/0004-6361/201935656.
  * Mościbrodzka et al. (2016) Mościbrodzka, M.; Falcke, H.; Shiokawa, H.  General relativistic magnetohydrodynamical simulations of the jet in M87.  A&A 2016, 586, A38. https://doi.org/10.1051/0004-6361/201526630.
  * Xiao (2006) Xiao, F.  Modelling energetic particles by a relativistic kappa-loss-cone distribution function in plasmas.  Plasma Phys. Control. Fusion 2006, 48, 203–213. https://doi.org/10.1088/0741-3335/48/2/003.
  * Davelaar et al. (2018) Davelaar, J.; Mościbrodzka, M.; Bronzwaer, T.; Falcke, H.  General relativistic magnetohydrodynamical $\kappa$-jet models for Sagittarius A*.  Astron. Astrophys. 2018, 612, A34. https://doi.org/10.1051/0004-6361/201732025.
  * Ressler et al. (2015) Ressler, S.M.; Tchekhovskoy, A.; Quataert, E.; Chandra, M.; Gammie, C.F.  Electron thermodynamics in GRMHD simulations of low-luminosity black hole accretion.  Mon. Not. R. Astron. Soc. 2015, 454, 1848–1870. https://doi.org/10.1093/mnras/stv2084.
  * Dexter et al. (2020) Dexter, J.; Jiménez-Rosales, A.; Ressler, S.M.; Tchekhovskoy, A.; Bauböck, M.; de Zeeuw, P.T.; Eisenhauer, F.; von Fellenberg, S.; Gao, F.; Genzel, R.; et al.  A parameter survey of Sgr A * radiative models from GRMHD simulations with self-consistent electron heating.  Mon. Not. R. Astron. Soc. 2020, 494, 4168–4186. https://doi.org/10.1093/mnras/staa922.
  * Howes (2010) Howes, G.G.  A prescription for the turbulent heating of astrophysical plasmas.  Mon. Not. R. Astron. Soc. 2010, 409, L104–L108. https://doi.org/10.1111/j.1745-3933.2010.00958.x.
  * Kawazura et al. (2019) Kawazura, Y.; Barnes, M.; Schekochihin, A.A.  Thermal disequilibration of ions and electrons by collisionless plasma turbulence.  Proc. Natl. Acad. Sci. USA 2019, 116, 771–776. https://doi.org/10.1073/pnas.1812491116.
  * Werner et al. (2018) Werner, G.R.; Uzdensky, D.A.; Begelman, M.C.; Cerutti, B.; Nalewajko, K.  Non-thermal particle acceleration in collisionless relativistic electron-proton reconnection.  Mon. Not. R. Astron. Soc. 2018, 473, 4840–4861. https://doi.org/10.1093/mnras/stx2530.
  * Rowan et al. (2017) Rowan, M.E.; Sironi, L.; Narayan, R.  Electron and Proton Heating in Transrelativistic Magnetic Reconnection.  Astrophys. J. 2017, 850, 29. https://doi.org/10.3847/1538-4357/aa9380.
  * Mizuno et al. (2021) Mizuno, Y.; Fromm, C.M.; Younsi, Z.; Porth, O.; Olivares, H.; Rezzolla, L.  Comparison of the ion-to-electron temperature ratio prescription: GRMHD simulations with electron thermodynamics.  Mon. Not. R. Astron. Soc. 2021, 506(1), 741–758. https://doi.org/ 10.1093/mnras/stab1753.
  * Sądowski and Narayan (2015) Sądowski, A.; Narayan, R.  Photon-conserving Comptonization in simulations of accretion discs around black holes.  Mon. Not. R. Astron. Soc. 2015, 454, 2372–2380. https://doi.org/10.1093/mnras/stv2022.
  * Chael et al. (2018) Chael, A.; Rowan, M.; Narayan, R.; Johnson, M.; Sironi, L.  The role of electron heating physics in images and variability of the Galactic Centre black hole Sagittarius A*.  Mon. Not. R. Astron. Soc. 2018, 478, 5209–5229. https://doi.org/10.1093/mnras/sty1261.
  * Sądowski et al. (2017) Sądowski, A.; Wielgus, M.; Narayan, R.; Abarca, D.; McKinney, J.C.; Chael, A.  Radiative, two-temperature simulations of low-luminosity black hole accretion flows in general relativity.  Mon. Not. R. Astron. Soc. 2017, 466, 705–725. https://doi.org/ 10.1093/mnras/stw3116.
  * Doeleman et al. (2012) Doeleman, S.S.; Fish, V.L.; Schenck, D.E.; Beaudoin, C.; Blundell, R.; Bower, G.C.; Broderick, A.E.; Chamberlin, R.; Freund, R.; Friberg, P.; et al.  Jet-Launching Structure Resolved Near the Supermassive Black Hole in M87.  Science 2012, 338, 355. https://doi.org/10.1126/science.1224768.
  * Akiyama et al. (2015) Akiyama, K.; Lu, Ru.; Fish, V.L.; Doeleman, S.S.; Broderick, A.E.; Dexter, J.; Hada, K.; Kino, M.; Nagai, H.; Honma, M.; et al.  230 GHz VLBI Observations of M87: Event-horizon-scale Structure during an Enhanced Very-high-energy $\gamma$-Ray State in 2012.  Astrophys. J. 2015, 807, 150. https://doi.org/10.1088/0004-637X/807/2/150.
  * Porth et al. (2021) Porth, O.; Mizuno, Y.; Younsi, Z.; Fromm, C.M.  Flares in the Galactic Centre - I. Orbiting flux tubes in magnetically arrested black hole accretion discs.  Mon. Not. R. Astron. Soc. 2021, 502, 2023–2032. https://doi.org/10.1093/mnras/stab163.
  * Begelman et al. (2022) Begelman, M.C.; Scepi, N.; Dexter, J.  What really makes an accretion disc MAD.  Mon. Not. R. Astron. Soc. 2022, 511, 2040–2051. https://doi.org/10.1093/mnras/stab3790.
  * Ripperda et al. (2022) Ripperda, B.; Liska, M.; Chatterjee, K.; Musoke, G.; Philippov, A.A.; Markoff, S.B.; Tchekhovskoy, A.; Younsi, Z.  Black Hole Flares: Ejection of Accreted Magnetic Flux through 3D Plasmoid-mediated Reconnection.  Astrophys. J. Lett. 2022, 924, L32. https://doi.org/10.3847/2041-8213/ac46a1.
  * Chatterjee and Narayan (2022) Chatterjee, K.; Narayan, R.  Flux Eruption Events Drive Angular Momentum Transport in Magnetically Arrested Accretion Flows.  Astrophys. J. 2022, 941(1), 30. https://doi.org/10.3847/1538-4357/ac9d97
  * Chatterjee et al. (2021) Chatterjee, K.; Markoff, S.; Neilsen, J.; Younsi, Z.; Witzel, G.; Tchekhovskoy, A.; Yoon, D.; Ingram, A.; van der Klis, M.; Boyce, H.; et al.  General relativistic MHD simulations of non-thermal flaring in Sagittarius A*.  Mon. Not. R. Astron. Soc. 2021, 507, 5281–5302. https://doi.org/10.1093/mnras/stab2466.
  * Wielgus et al. (2022) Wielgus, M.; Marchili, N.; Martí-Vidal, I.; Keating, G.K.; Ramakrishnan, V.; Tiede, P.; Fomalont, E.; Issaoun, S.; Neilsen, J.; Nowak, M.A.; et al.  Millimeter Light Curves of Sagittarius A* Observed during the 2017 Event Horizon Telescope Campaign.  Astrophys. J. Lett. 2022, 930, L19. https://doi.org/10.3847/2041-8213/ac6428.
  * Narayan and Yi (1994) Narayan, R.; Yi, I.  Advection-dominated accretion: A self-similar solution.  Astrophys. J. Lett. 1994, 428, L13–L16. https://doi.org/ 10.1086/187381.
  * Narayan et al. (1998) Narayan, R.; Mahadevan, R.; Quataert, E.  Advection-dominated accretion around black holes.  Proceedings of the Theory of Black Hole Accretion Disks 1998 p. 138–182, eds. Abramowicz, M.A., Bjornsson, G.; Pringle, J.E., published by Cambridge University Press. https://doi.org/10.48550/arXiv.astro-ph/9803141
  * Liska et al. (2020) Liska, M.; Tchekhovskoy, A.; Quataert, E.  Large-scale poloidal magnetic field dynamo leads to powerful jets in GRMHD simulations of black hole accretion with toroidal field.  Mon. Not. R. Astron. Soc. 2020, 494, 3656–3662. https://doi.org/ 10.1093/mnras/staa955.
  * Nakamura et al. (2018) Nakamura, M.; Asada, K.; Hada, K.; Pu, H.Y.; Noble, S.; Tseng, C.; Toma, K.; Kino, M.; Nagai, H.; Takahashi, K.; et al.  Parabolic Jets from the Spinning Black Hole in M87.  Astrophys. J. 2018, 868, 146. https://doi.org/10.3847/1538-4357/aaeb2d.
  * Asada and Nakamura (2012) Asada, K.; Nakamura, M.  The Structure of the M87 Jet: A Transition from Parabolic to Conical Streamlines.  ApJL 2012, 745, 5.
  * Nokhrina et al. (2019) Nokhrina, E.E.; Gurvits, L.I.; Beskin, V.S.; Nakamura, M.; Asada, K.; Hada, K.  M87 black hole mass and spin estimate through the position of the jet boundary shape break.  Mon. Not. R. Astron. Soc. 2019, 489, 1197–1205. https://doi.org/10.1093/mnras/stz2116.
  * Chatterjee et al. (2019) Chatterjee, K.; Liska, M.; Tchekhovskoy, A.; Markoff, S.B.  Accelerating AGN jets to parsec scales using general relativistic MHD simulations.  Mon. Not. R. Astron. Soc. 2019, 490, 2200–2218. https://doi.org/10.1093/mnras/stz2626.
  * Issaoun et al. (2019) Issaoun, S.; Johnson, M.D.; Blackburn, L.; et al.  The Size, Shape, and Scattering of Sagittarius A* at 86 GHz: First VLBI with ALMA.  Astrophys. J. 2019, 871, 30. https://doi.org/10.3847/1538-4357/aaf732.
  * Markoff et al. (2007) Markoff, S.; Bower, G.C.; Falcke, H.  How to hide large-scale outflows: Size constraints on the jets of Sgr A *.  Mon. Not. R. Astron. Soc. 2007, 379, 1519–1532. https://doi.org/10.1111/j.1365-2966.2007.12071.x.
  * Hada et al. (2013) Hada, K.; Kino, M.; Doi, A.; Nagai, H.; Honma, M.; Hagiwara, Y.; Giroletti, M.; Giovannini, G.; Kawaguchi, N.  The Innermost Collimation Structure of the M87 Jet Down to 10 Schwarzschild Radii.  Astrophys. J. 2013, 775, 70.
  * Nakamura and Asada (2013) Nakamura, M.; Asada, K.  The Parabolic Jet Structure in M87 as a Magnetohydrodynamic Nozzle.  Astrophys. J. 2013, 775, 118.
  * Hada et al. (2016) Hada, K.; Kino, M.; Doi, A.; Nagai, H.; Honma, M.; Akiyama, K.; Tazaki, F.; Lico, R.; Giroletti, M.; Giovannini, G.; et al.  High-sensitivity 86 GHz (3.5 mm) VLBI Observations of M87: Deep Imaging of the Jet Base at a Resolution of 10 Schwarzschild Radii.  Astrophys. J. 2016, 817, 131.
  * Mertens et al. (2016) Mertens, F.; Lobanov, A.P.; Walker, R.C.; Hardee, P.E.  Kinematics of the jet in M87 on scales of 100–1000 Schwarzschild radii.  Astron. Astrophys. 2016, 595, A54. https://doi.org/10.1051/0004-6361/201628829.
  * Fragile and Meier (2009) Fragile, P.C.; Meier, D.L.  General Relativistic Magnetohydrodynamic Simulations of the Hard State as a Magnetically Dominated Accretion Flow.  Astrophys. J. 2009, 693, 771–783. https://doi.org/10.1088/0004-637X/693/1/771.
  * Yoon et al. (2020) Yoon, D.; Chatterjee, K.; Markoff, S.B.; van Eijnatten, D.; Younsi, Z.; Liska, M.; Tchekhovskoy, A.  Spectral and imaging properties of Sgr A * from high-resolution 3D GRMHD simulations with radiative cooling.  Mon. Not. R. Astron. Soc. 2020, 499, 3178–3192. https://doi.org/10.1093/mnras/staa3031.
  * Sadowski (2016) Sadowski, A.  Magnetic flux stabilizing thin accretion discs.  Mon. Not. R. Astron. Soc. 2016, 462, 960–965. https://doi.org/10.1093/ mnras/stw1852.
  * Liska et al. (2022) Liska, M.T.P.; Musoke, G.; Tchekhovskoy, A.; Porth, O.; Beloborodov, A.M.  Formation of Magnetically Truncated Accretion Disks in 3D Radiation-Transport Two-Temperature GRMHD Simulations.  arXiv 2022, arXiv:2201.03526.
  * Mishra et al. (2022) Mishra, B.; Fragile, P.C.; Anderson, J.; Blankenship, A.; Li, H.; Nalewajko, K.  The Role of Strong Magnetic Fields in Stabilizing Highly Luminous Thin Disks.  Astrophys. J. 2022, 939, 31. https://doi.org/10.3847/1538-4357/ac938b.
  * Wong et al. (2021) Wong, G.N.; Du, Y.; Prather, B.S.; Gammie, C.F.  The Jet-disk Boundary Layer in Black Hole Accretion.  Astrophys. J. 2021, 914, 55. https://doi.org/10.3847/1538-4357/abf8b8.
  * Aharonian et al. (2006) Aharonian, F.; Akhperjanian, A.G.; Bazer-Bachi, A.R.; Beilicke, M.; Benbow, W.; Berge, D.; Bernlöhr, K.; Boisson, C.; Bolz, O.; Borrel, V.; et al.  Fast Variability of Tera-Electron Volt $\gamma$ Rays from the Radio Galaxy M87.  Science 2006, 314, 1424–1427. https://doi.org/10.1126/science.1134408.
  * Acciari et al. (2010) Acciari, V.A.; Aliu, E.; Arlen, T.; Aune, T.; Beilicke, M.; Benbow, W.; Boltuch, D.; Bradbury, S.M.; Buckley, J.H.; Bugaev, V.; et al.  Veritas 2008-2009 Monitoring of the Variable Gamma-ray Source M 87.  Astrophys. J. 2010, 716, 819–824. https://doi.org/10.1088/0004-637X/716/1/819.
  * Baganoff et al. (2001) Baganoff, F.K.; Bautz, M.W.; Brandt, W.N.; Chartas, G.; Feigelson, E.D.; Garmire, G.P.; Maeda, Y.; Morris, M.; Ricker, G.R.; Townsley, L.K.; et al.  Rapid X-ray flaring from the direction of the supermassive black hole at the Galactic Centre.  Nature 2001, 413, 45–48. https://doi.org/10.1038/35092510.
  * Eckart et al. (2006) Eckart, A.; Baganoff, F.K.; Schödel, R.; Morris, M.; Genzel, R.; Bower, G.C.; Marrone, D.; Moran, J.M.; Viehmann, T.; Bautz, M.W.; et al.  The flare activity of Sagittarius A*. New coordinated mm to X-ray observations.  Astron. Astrophys. 2006, 450, 535–555. https://doi.org/10.1051/0004-6361:20054418.
  * Hornstein et al. (2007) Hornstein, S.D.; Matthews, K.; Ghez, A.M.; Lu, J.R.; Morris, M.; Becklin, E.E.; Rafelski, M.; Baganoff, F.K.  A Constant Spectral Index for Sagittarius A* during Infrared/X-Ray Intensity Variations.  Astrophys. J. 2007, 667, 900–910. https://doi.org/10.1086/520762.
  * Nowak et al. (2012) Nowak, M.A.; Neilsen, J.; Markoff, S.B.; Baganoff, F.K.; Porquet, D.; Grosso, N.; Levin, Y.; Houck, J.; Eckart, A.; Falcke, H.; et al.  Chandra/HETGS Observations of the Brightest Flare Seen from Sgr A *.  Astrophys. J. 2012, 759, 95. https://doi.org/10.1088/0004-637X/759/2/95.
  * Neilsen et al. (2013) Neilsen, J.; Nowak, M.A.; Gammie, C.; Dexter, J.; Markoff, S.; Haggard, D.; Nayakshin, S.; Wang, Q.D.; Grosso, N.; Porquet, D.; et al.  A Chandra/HETGS Census of X-Ray Variability from Sgr A * during 2012\.  Astrophys. J. 2013, 774, 42. https://doi.org/10.1088/0004-637X/774/1/42.
  * Witzel et al. (2018) Witzel, G.; Martinez, G.; Hora, J.; Willner, S.P.; Morris, M.R.; Gammie, C.; Becklin, E.E.; Ashby, M.L.N.; Baganoff, F.; Carey, S.; et al.  Variability Timescale and Spectral Index of Sgr A * in the Near Infrared: Approximate Bayesian Computation Analysis of the Variability of the Closest Supermassive Black Hole.  Astrophys. J. 2018, 863, 15. https://doi.org/10.3847/1538-4357/aace62.
  * Haggard et al. (2019) Haggard, D.; Nynka, M.; Mon, B.; de la Cruz Hernandez, N.; Nowak, M.; Heinke, C.; Neilsen, J.; Dexter, J.; Fragile, P.C.; Baganoff, F.; et al.  Chandra Spectral and Timing Analysis of Sgr A *’s Brightest X-Ray Flares.  Astrophys. J. 2019, 886, 96. https://doi.org/10.3847/1538-4357/ab4a7f.
  * Dodds-Eden et al. (2010) Dodds-Eden, K.; Sharma, P.; Quataert, E.; Genzel, R.; Gillessen, S.; Eisenhauer, F.; Porquet, D.  Time-Dependent Models of Flares from Sagittarius A*.  Astrophys. J. 2010, 725, 450–465. https://doi.org/10.1088/0004-637X/725/1/450.
  * Dibi et al. (2014) Dibi, S.; Markoff, S.; Belmont, R.; Malzac, J.; Barrière, N.M.; Tomsick, J.A.  Exploring plasma evolution during Sagittarius A* flares.  Mon. Not. R. Astron. Soc. 2014, 441, 1005–1016. https://doi.org/10.1093/mnras/stu599.
  * Nathanail et al. (2020) Nathanail, A.; Fromm, C.M.; Porth, O.; Olivares, H.; Younsi, Z.; Mizuno, Y.; Rezzolla, L.  Plasmoid formation in global GRMHD simulations and AGN flares.  Mon. Not. R. Astron. Soc. 2020, 495, 1549–1565. https://doi.org/10.1093/mnras/staa1165.
  * Dexter et al. (2020) Dexter, J.; Tchekhovskoy, A.; Jiménez-Rosales, A.; Ressler, S.M.; Bauböck, M.; Dallilar, Y.; de Zeeuw, P.T.; Eisenhauer, F.; von Fellenberg, S.; Gao, F.; et al.  Sgr A * near-infrared flares from reconnection events in a magnetically arrested disc.  Mon. Not. R. Astron. Soc. 2020, 497, 4999–5007. https://doi.org/10.1093/mnras/staa2288.
  * Scepi et al. (2022) Scepi, N.; Dexter, J.; Begelman, M.C.  Sgr A * X-ray flares from non-thermal particle acceleration in a magnetically arrested disc.  Mon. Not. R. Astron. Soc. 2022, 511, 3536–3547. https://doi.org/10.1093/mnras/stac337.
  * Markoff et al. (2001) Markoff, S.; Falcke, H.; Yuan, F.; Biermann, P.L.  The Nature of the 10 kilosecond X-ray flare in Sgr A *.  Astron. Astrophys. 2001, 379, L13–L16. https://doi.org/10.1051/0004-6361:20011346.
  * Dibi et al. (2016) Dibi, S.; Markoff, S.; Belmont, R.; Malzac, J.; Neilsen, J.; Witzel, G.  Using infrared/X-ray flare statistics to probe the emission regions near the event horizon of Sgr A *.  Mon. Not. R. Astron. Soc. 2016, 461, 552–559. https://doi.org/10.1093/mnras/stw1353.
  * Gutiérrez et al. (2020) Gutiérrez, E.M.; Nemmen, R.; Cafardo, F.  A Nonthermal Bomb Explains the Near-infrared Superflare of Sgr A *.  Astrophys. J. Lett. 2020, 891, L36. https://doi.org/10.3847/2041-8213/ab7998.
  * Gravity Collaboration et al. (2020) Gravity Collaboration; Bauböck, M.; Dexter, J.; Abuter, R.; Amorim, A.; Berger, J.P.; Bonnet, H.; Brandner, W.; Clénet, Y.; Coudé Du Foresto, V.; et al.  Modeling the orbital motion of Sgr A *’s near-infrared flares.  Astron. Astrophys. 2020, 635, A143. https://doi.org/10.1051/0004-6361/201937233.
  * Ball et al. (2021) Ball, D.; Özel, F.; Christian, P.; Chan, C.K.; Psaltis, D.  A Plasmoid model for the Sgr A * Flares Observed With Gravity and CHANDRA.  Astrophys. J. 2021, 917, 8. https://doi.org/10.3847/1538-4357/abf8ae.
  * Wielgus et al. (2022) Wielgus, M.; Moscibrodzka, M.; Vos, J.; Gelles, Z.; Martí-Vidal, I.; Farah, J.; Marchili, N.; Goddi, C.; Messias, H.  Orbital motion near Sagittarius A∗. Constraints from polarimetric ALMA observations.  Astron. Astrophys. 2022, 665, L6. https://doi.org/10.1051/0004-6361/202244493.
  * Emami et al. (2022) Emami, R.; Tiede, P.; Doeleman, S.S.; Roelofs, F.; Wielgus, M.; Blackburn, L.; Liska, M.; Chatterjee, K.; Ripperda, B.; Fuentes, A.; et al.  Tracing Hot Spot Motion in Sagittarius A* Using the Next-Generation Event Horizon Telescope (ngEHT).  Galaxies 2023, 11(1), 23. https://doi.org/10.3390/galaxies11010023.
  * Gelles et al. (2022) Gelles, Z.; Chatterjee, K.; Johnson, M.; Ripperda, B.; Liska, M.  Relativistic Signatures of Flux Eruption Events near Black Holes.  Galaxies 2022, 10(6), 107. https://doi.org/10.3390/galaxies10060107
  * Ressler et al. (2020) Ressler, S.M.; White, C.J.; Quataert, E.; Stone, J.M.  Ab Initio Horizon-scale Simulations of Magnetically Arrested Accretion in Sagittarius A* Fed by Stellar Winds.  Astrophys. J. Lett. 2020, 896, L6. https://doi.org/10.3847/2041-8213/ab9532.
  * Lalakos et al. (2022) Lalakos, A.; Gottlieb, O.; Kaaz, N.; Chatterjee, K.; Liska, M.; Christie, I.M.; Tchekhovskoy, A.; Zhuravleva, I.; Nokhrina, E.  Bridging the Bondi and Event Horizon Scales: 3D GRMHD Simulations Reveal X-shaped Radio Galaxy Morphology.  Astrophys. J. Lett. 2022, 936(1), L5. https://doi.org/10.3847/2041-8213/ac7bed
  * Cruz-Osorio et al. (2017) Cruz-Osorio, A.; Sánchez-Salcedo, F.J.; Lora-Clavijo, F.D.  Relativistic Bondi-Hoyle-Lyttleton accretion in the presence of small rigid bodies around a black hole.  Mon. Not. R. Astron. Soc. 2017, 471, 3127–3134. https://doi.org/10.1093/mnras/stx1815.
  * Kaaz et al. (2022) Kaaz, N.; Murguia-Berthier, A.; Chatterjee, K.; Liska, M.; Tchekhovskoy, A.  Jet Formation in 3D GRMHD Simulations of Bondi-Hoyle-Lyttleton Accretion.  arXiv 2022, arXiv:2201.11753.
  * Avara et al. (2016) Avara, M.J.; McKinney, J.C.; Reynolds, C.S.  Efficiency of thin magnetically arrested discs around black holes.  Mon. Not. R. Astron. Soc. 2016, 462, 636–648. https://doi.org/10.1093/mnras/stw1643.
  * Sadowski and Narayan (2016) Sadowski, A.; Narayan, R.  Three-dimensional simulations of supercritical black hole accretion discs - luminosities, photon trapping and variability.  Mon. Not. R. Astron. Soc. 2016, 456, 3929–3947. https://doi.org/10.1093/mnras/stv2941.
  * Curd and Narayan (2019) Curd, B.; Narayan, R.  GRRMHD simulations of tidal disruption event accretion discs around supermassive black holes: Jet formation, spectra, and detectability.  Mon. Not. R. Astron. Soc. 2019, 483, 565–592. https://doi.org/10.1093/mnras/sty3134.
  * White et al. (2019) White, C.J.; Quataert, E.; Blaes, O.  Tilted Disks around Black Holes: A Numerical Parameter Survey for Spin and Inclination Angle.  Astrophys. J. 2019, 878, 51. https://doi.org/10.3847/1538-4357/ab089e.
  * Chatterjee et al. (2020) Chatterjee, K.; Younsi, Z.; Liska, M.; Tchekhovskoy, A.; Markoff, S.B.; Yoon, D.; van Eijnatten, D.; Hesp, C.; Ingram, A.; van der Klis, M.B.M.  Observational signatures of disc and jet misalignment in images of accreting black holes.  Mon. Not. R. Astron. Soc. 2020, 499, 362–378. https://doi.org/10.1093/mnras/staa2718.
  * Event Horizon Telescope Collaboration et al. (2022) Event Horizon Telescope Collaboration; Akiyama, K.; Alberdi, A.; Alef, W.; Algaba, J.C.; Anantua, R.; Asada, K.; Azulay, R.; Bach, U.; Baczko, A.K.; et al.  First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric.  Astrophys. J. Lett. 2022, 930, L17. https://doi.org/10.3847/2041-8213/ac6756.
  * Mizuno et al. (2018) Mizuno, Y.; Younsi, Z.; Fromm, C.M.; Porth, O.; De Laurentis, M.; Olivares, H.; Falcke, H.; Kramer, M.; Rezzolla, L.  The current ability to test theories of gravity with black hole shadows.  Nat. Astron. 2018, 2, 585–590. https://doi.org/10.1038/s41550-018-0449-5.
  * Olivares et al. (2020) Olivares, H.; Younsi, Z.; Fromm, C.M.; De Laurentis, M.; Porth, O.; Mizuno, Y.; Falcke, H.; Kramer, M.; Rezzolla, L.  How to tell an accreting boson star from a black hole.  Mon. Not. R. Astron. Soc. 2020, 497, 521–535. https://doi.org/10.1093/mnras/staa1878.
  * Nampalliwar et al. (2022) Nampalliwar, S.; Yfantis, A.I.; Kokkotas, K.D.  Extending GRMHD for thin disks to non-Kerr spacetimes.  Phys. Rev. D 2022, 106, 063009. https://doi.org/10.1103/PhysRevD.106.063009.