# Observational Constraints on Direct Electron Heating in the Hot Accretion
Flows in Sgr A* and M87*

Fu-Guo Xie Key Laboratory for Research in Galaxies and Cosmology, Shanghai
Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road,
Shanghai 200030, People’s Republic of China  Ramesh Narayan Center for
Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138,
USA  Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge,
MA 02138, USA  Feng Yuan Key Laboratory for Research in Galaxies and
Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80
Nandan Road,
Shanghai 200030, People’s Republic of China  University of Chinese Academy of
Sciences, No. 19A Yuquan Road, Beijing 100049, People’s Republic of China

###### Abstract

An important parameter in the theory of hot accretion flows around black holes
is $\delta$, which describes the fraction of “viscously” dissipated energy in
the accretion flow that goes directly into heating electrons. For a given mass
accretion rate, the radiative efficiency of a hot accretion flow is determined
by $\delta$. Unfortunately, the value of $\delta$ is hard to determine from
first principles. The recent Event Horizon Telescope Collaboration (EHTC)
results on M87* and Sgr A* provide us with a different way of constraining
$\delta$. By combining the mass accretion rates in M87* and Sgr A* estimated
by the EHTC with the measured bolometric luminosities of the two sources, we
derive good constraints on the radiative efficiencies of the respective
accretion flows. In parallel, we use a theoretical model of hot magnetically
arrested disks (MAD) to calculate the expected radiative efficiency as a
function of $\delta$ (and accretion rate). By comparing the EHTC-derived
radiative efficiencies with the theoretical results from MAD models, we find
that Sgr A* requires $\delta\gtrsim 0.3$. A similar comparison in the case of
M87* gives inconclusive results as there is still a large uncertainty in the
accretion rate in this source.

accretion, accretion disks — Astrophysical black holes — low-luminosity active
galactic nuclei (individual: Sgr A*, M87*)

##  1 Introduction

Accretion flows around black holes (BHs) can be divided into two types: hot
accretion flows, which occur at accretion rates below about $1\%$ of Eddington
(Narayan & Yi 1994, 1995; Abramowicz et al. 1995, see Yuan & Narayan 2014 for
a review)111Note that the $1\%$ limit quoted here is a typical value for low-
luminosity active galactic nuclei (Ho, 2008). In the case of BH X-ray binaries
in the hard state (Remillard & McClintock, 2006; Done et al., 2007), where
again hot accretion flows are present (Esin et al., 1997), the luminosity can
reach much higher values, $\gtrsim 10-20\%$ of the Eddington luminosity
${L_{\rm Edd}}$ (e.g., Dunn et al. 2010). Here ${L_{\rm Edd}}=4\pi
Gcm_{p}{M_{\rm BH}}/\sigma_{T}=1.26\times 10^{44}\,({M_{\rm
BH}}/10^{6}{M_{\odot}})\,{\rm erg\,s^{-1}}$ for accretion onto a BH with mass
${M_{\rm BH}}$., and cold accretion disks which are found at accretion rates
closer to Eddington (Shakura & Sunyaev, 1973; Novikov & Thorne, 1973; Pringle,
1981). Most supermassive BHs in the nearby universe have low accretion rates
and are believed to have hot accretion flows (Ho, 2008).

The plasma in a hot accretion flow is two-temperature (Shapiro, Lightman, &
Eardley, 1976), with electrons and ions having different temperatures. Hence
the two species need to be treated with separate energy equations (Narayan &
Yi, 1995; Yuan & Narayan, 2014). The energy equation of electrons takes the
form

$\rho V_{r}\left({de_{\rm e}\over dR}-{p_{e}\over\rho^{2}}{d\rho\over
dR}\right)=q_{\rm ie}+\delta q_{\rm vis}-q^{-},$ (1)

where $\rho$ is the gas density, $V_{r}$ is the radial velocity, $R$ is the
radius, $e_{\rm e}$ and $p_{\rm e}$ are, respectively, the specific internal
energy and pressure of electrons, and $q^{-}$ is the radiative cooling rate
per unit volume. The first two terms in the right hand side of the above
equation correspond to two different mechanisms by which electrons are heated.
One mechanism is via Coulomb collisions between ions and electrons, denoted by
$q_{\rm ie}$, the other is through direct “viscous heating,” which is a catch-
all term describing several physical dissipation processes in the plasma as we
explain in the next paragraph. We assume that, out of the total viscous
heating rate per unit volume $q_{\rm vis}$, a fraction $\delta q_{\rm vis}$
heats the electrons, and the rest $(1-\delta)q_{\rm vis}$ heats the ions.
Since almost all the radiation of the accretion flow is emitted by electrons,
therefore, when other model parameters such as the mass accretion rate are
given, the value of $\delta$ determines the radiative efficiency $\epsilon$ of
the accretion flow, which we define as

$\epsilon={L_{\rm bol}\over\dot{M}_{\rm BH}c^{2}}=10\%\,{L_{\rm bol}/{L_{\rm
Edd}}\over\dot{M}_{\rm BH}/{\dot{M}_{\rm Edd}}}.$ (2)

Here $L_{\rm bol}$ is the bolometric luminosity of the accretion flow and
$\dot{M}_{\rm BH}$ is the mass accretion rate at the BH horizon. In the second
expression, the luminosity and the accretion rate are normalized to,
respectively, the Eddington luminosity ${L_{\rm Edd}}$, and the Eddington
accretion rate ${\dot{M}_{\rm Edd}}={L_{\rm Edd}}/(0.1c^{2})=2.21\times
10^{-2}\,({M_{\rm BH}}/10^{6}{M_{\odot}})\,{M_{\odot}}\,{\rm yr}^{-1}$.

The viscous heating rate of electrons $\delta q_{\rm vis}$ includes several
microphysical processes. Many attempts have been made to estimate $\delta$
from first-principle calculations of these processes, including magnetic
reconnection (Bisnovatyi-Kogan & Lovelace, 1997; Quataert & Gruzinov, 1999;
Ding et al., 2010; Hoshino, 2012, 2013; Numata & Loureiro, 2015; Sironi &
Narayan, 2015; Rowan et al., 2017, 2019; Ball et al., 2018),
magnetohydrodynamic (MHD) turbulence (Quataert, 1998; Quataert & Gruzinov,
1999; Blackman, 1999; Medvedev, 2000; Lehe et al., 2009; Howes, 2010; Ressler
et al., 2015; Ryan et al., 2017), dissipation of pressure anisotropy in a
collisionless plasma (Sharma et al., 2007), or low Mach number shocks (Guo et
al., 2017, 2018). Unfortunately, there is no consensus, and the value of
$\delta$ remains poorly determined.

Yuan et al. (2003) considered an alternative approach to constrain the value
of $\delta$. By calculating the dynamics and radiation of the accretion flow
using a height-integrated model, and comparing the results with observations
of the supermassive black hole Sagittarius A* (Sgr A*) at the center of our
Galaxy, they estimated $\delta\sim 0.5$.

In this work, we follow the approach of Yuan et al. (2003) and estimate the
value of $\delta$ using new and updated observational data and a more modern
theoretical model. Specifically, we make use of results from the Event Horizon
Telescope (EHT), an Earth-size sub-millimeter radio interferometer, which has
recently obtained 230 GHz images of the innermost horizon-scale regions of the
low-luminosity supermassive BHs, M87* (EHTC, 2019, 2021) and Sgr A* (EHTC,
2022a). Key results from the work of the EHT Collaboration (EHTC) are
estimates of the mass accretion rates $\dot{M}_{\rm BH}$ for the two sources
to within a factor of several, see Sec. 2 below. By combining these estimates
with a reliable measurement of the bolometric luminosities of the soruces from
their broad-band spectra, we are able to evaluate the radiative efficiencies
$\epsilon$ (cf. Equation 2) of the two sources. We compare these measurements
with the predictions of the model described in Xie & Zdziarski (2019) and
thereby evaluate the value of $\delta$.

Before going into details, we briefly introduce additional background
information on hot accretion flows. In the last decade or two, it has become
clear that the degree of magnetization of the accretion flow plays an
important role in the dynamics and observational characteristics of hot
accretion flows. Broadly, one distinguishes between two kinds of models. One
has relatively weak magnetic fields and is referred to as the standard and
normal evolution (SANE) model, while the other has the maximum saturated level
of magnetic field and is called the magnetically arrested disk (MAD) model
(Narayan et al., 2003; Igumenshchev et al., 2003; Bisnovatyi-Kogan &
Ruzmaikin, 1974, 1976; Tchekhovskoy et al., 2011; McKinney et al., 2012; Liska
et al., 2020). The dynamical differences between SANE and MAD are investigated
in detail in Narayan et al. (2012, see also ).

The accumulation of a considerable quantity of poloidal magnetic flux around
the BH in the MAD state has the attractive feature that it provides the ideal
magnetic field configuration to launch a powerful relativistic jet
(Tchekhovskoy et al., 2011; Liska et al., 2020; Narayan et al., 2022). Indeed,
there is accumulating evidence that systems with strong relativistic jets are
generally MAD, e.g., Zamaninasab et al. (2014); Ghisellini et al. (2014);
Zdziarski et al. (2015) for a sample of blazars. Meanwhile, the EHT has
provided evidence that M87* and Sgr A* may also both be MAD (EHTC, 2021,
2022a). The MAD nature of M87* has been confirmed by an analysis of the
observed rotation measure (Yuan et al., 2022). These results lead one to
speculate that perhaps the MAD configuration is the inevitable final state of
most hot accretion flows in nature (e.g., Narayan et al. 2022).

The radiative properties of MAD flows are found to be fairly similar to those
of SANE flows, except that, for a given accretion rate, MAD is brighter by
about a factor of $\sim 3$ (Xie & Zdziarski, 2019). This motivates us to use a
model appropriate to MAD systems when attempting to estimate the electron
heating parameter $\delta$.

This paper is organized as follows. In Sec. 2, we provide the observational
and modeling results for Sgr A* and M87*, with an emphasis on the accretion
rate $\dot{M}_{\rm BH}$, the bolometric lumonosity $L_{\rm bol}$, and the
radiative efficiency $\epsilon$. In Sec. 3, we present theoretical
calculations of the radiative efficiency of MAD flows corresponding to
different values of $\delta$. We then compare these theoretical results with
the values of $\epsilon$ inferred by the EHTC for Sgr A* and M87*, and we
thereby constrain the value of $\delta$. The final section is devoted to a
brief summary.

##  2 Observational Constraints on the Radiative Efficiency

Both M87* and Sgr A* are low-luminosity systems with hot accretion flows. They
are the main targets of the EHT project. Below we provide basic properties of
the two sources and derive their radiative efficiencies.

###  2.1 M87*

M87* is located at a distance of $d=16.9$ Mpc, and the BH mass is measured to
be ${M_{\rm BH}}=6.2\times 10^{9}{M_{\odot}}$ (Gebhardt et al., 2011; EHTC,
2019). By comparing the values of four physical quantities obtained from
theoretical predictions and those derived from the reconstructed EHT image of
M87* and ALMA-only (Atacama Large Millimeter/submillimeter Array)
measurements, the EHTC concluded that the accretion flow in M87* is very
likely in a MAD state (EHTC, 2021). This result has been confirmed recently by
Yuan et al. (2022) by comparing the predicted Faraday Rotation Measure (RM) in
M87* with that measured along the jet. The advantage of the latter work is
that it does not suffer from uncertainties in the electron temperature and the
contribution of nonthermal electrons in the accretion flow, which are a
challenge for the EHTC analysis. The spin of the BH in M87* is hard to
determine (EHTC, 2021); but some recent works have begun to constrain its
value by comparing observed images of the M87* jet with theoretical
predictions (Cruz-Osorio et al., 2022; Yang et al., 2022).

Broad-band quasi-simultaneous observations of M87* indicate that the turnover
frequency of the sub-mm bump in the $\nu L_{\nu}$ plot is located at $\sim
230$ GHz (Algaba et al. 2021, see also Hada et al. 2011), and that the
bolometric luminosity is approximately $L_{\rm bol}\approx 8.5\times
10^{41}\,{\rm erg\,s^{-1}}\approx 1.1\times 10^{-6}\,{L_{\rm Edd}}$. In
estimating the bolometric luminosity, we only include nuclear emission, cf.
Fig. 16 and Model 1a of Algaba et al. (2021)222Note that Prieto et al. (2016)
derived a larger value of the bolometric luminosity, $L_{\rm bol}\approx
3\times 10^{42}{\rm erg\,s^{-1}}$. The main difference from Algaba et al.
(2021) is the treatment of data in optical/UV.. We adopt a $20\%$ uncertainty
in $L_{\rm bol}$.

The second piece of information we need is the mass accretion rate in M87*.
From the RM measurements of the nucleus of M87 at sub-millimeter wavelengths,
and using an argument previously developed by Agol (2000) and Quataert &
Gruzinov (2000), Kuo et al. (2014) derived an upper limit on the accretion
rate: $\dot{M}_{\rm BH}<9.2\times 10^{-4}{{M_{\odot}}\,{\rm yr}^{-1}}$. EHT
observations now allow a more careful analysis.

The one-zone model discussed in EHTC (2019) provides a first approximate
estimate of $\dot{M}_{\rm BH}$. The emission size (radius $R$) of the
accretion flow is directly measured from the diameter of the ring in the EHT
image. Combining this with the sub-mm flux, brightness temperature, and
synchrotron peak frequency, one estimates the density as well as the magnetic
field strength at radius $R$. This, along with an estimate of the radial
velocity of the gas, then gives an approximate estimate of the mass accretion
rate, $\dot{M}_{\rm BH}\sim 25\times 10^{-4}{{M_{\odot}}\,{\rm yr}^{-1}}$.

The above estimate is further refined in EHTC (2021), where EHT polarimetric
data are included as important additional constraints, and elaborate
theoretical models are developed based on GRMHD simulations. The conclusion of
this work is that the mass accretion rate near the BH horizon in M87* lies in
the range $\dot{M}_{\rm BH}\approx(3-20)\times 10^{-4}{{M_{\odot}}\,{\rm
yr}^{-1}}$ (EHTC, 2021), or equivalently, $\dot{M}_{\rm BH}/{\dot{M}_{\rm
Edd}}=(2.2-15)\times 10^{-6}$. This is the range we use in the present work.
Note that the EHT result is generally consistent with the initial estimate of
Kuo et al. (2014) based on polarimetric data.

Combining the estimates of the bolometric luminosity and mass accretion rate
discussed above, we derive the radiative efficiency $\epsilon$ (cf. Equation
2) of M87* to be

$7.5\times 10^{-3}\lesssim\epsilon\lesssim 5.0\times 10^{-2},$ (3)

where the range is driven almost entirely by the uncertainty in $\dot{M}_{\rm
BH}$. The two extreme values of the radiative efficiency and their geometric
mean are shown in Figure 1 as green filled circles. The gray solid curve
connecting the three points represents the allowed combinations of $\epsilon$
and $\dot{M}_{\rm BH}$.

Figure 1: Radiative efficiencies $\epsilon$ vs mass accretion rates
$\dot{M}_{\rm BH}/{\dot{M}_{\rm Edd}}$ of MAD systems. Observational
constraints obtained from M87* and Sgr A* (see Sec. 2) are shown,
respectively, by the green circles and cyan diamonds, connected by gray solid
curves. The colored lines correspond to numerical results from a theoretical,
pseudo-Newtonian (Paczyński-Wiita) potential version of MAD. From top to
bottom, the curves correspond to different values of the electron heating
fraction: $\delta=0.5$ (solid red), 0.3 (dotted purple), 0.1 (dot-dashed
blue), $10^{-3}$ (dashed black). For a given $\delta$, the radiative
efficiency in the models scales as $\epsilon\propto\dot{M}_{\rm BH}^{0.92}$.
All calculations use a viscosity parameter $\alpha=0.08$ (see Sec. 3.2).

We note that the above estimates of $\dot{M}_{\rm BH}$ are based on the
assumption that the electrons have a thermal distribution. Although electrons
and ions in a hot accretion flow are poorly coupled via Coulomb collisions,
electron-electron collisions are more effective and will maintain a thermal
distribution, at least for moderately large accretion rates. Mahadevan &
Quataert (1997) have shown that synchrotron self-absorption couples electrons
efficiently at even lower accretion rates. They estimate that electrons remain
thermal so long as

$\frac{\dot{M}_{\rm BH}}{{\dot{M}_{\rm
Edd}}}>10^{-4}\,\frac{\alpha^{2}}{\Theta_{e}},$ (4)

where $\alpha\sim 0.1$ is the effective viscosity parameter of the accretion
flow (Shakura & Sunyaev 1973, see also Sec. 3.2) and
$\Theta_{e}=kT_{e}/m_{e}c^{2}\sim$ few is the dimensionless electron
temperature in the hot accretion flow. M87* satisfies the above constraint, so
the thermal assumption is probably reasonable.

A weak nonthermal tail in the electron distribution is partially equivalent to
a thermal model with a higher electron temperature (e.g., EHTC 2022a).
However, models that include a significant non-thermal component are quite
different from thermal models (EHTC, 2019). For a given radio flux, including
non-thermal electrons generally leads to a lower estimate for the accretion
rate (lower optical depth), and thus to a larger radiative efficiency.

###  2.2 Sgr A*

The second object we consider is Sgr A*. It is at a distance of $d=8.13$ kpc
and has a BH mass of ${M_{\rm BH}}=4.14\times 10^{6}{M_{\odot}}$ (both are
average values, as adopted in EHTC 2022a). According to the EHT work, the
accretion flow in Sgr A* is probably in a MAD state; specifically, EHTC
(2022a) showed that MAD models are more consistent with observational
constraints in Sgr A* compared to SANE models, although the favored models are
generally too variable compared to observations (a presently unsolved
problem).

The spectrum of Sgr A* peaks at the sub-millimeter waveband (von Fellenberg et
al., 2018; Bower et al., 2019; EHTC, 2022b), and the luminosity of this sub-mm
bump is $\approx 5.0\times 10^{35}\,{\rm erg\,s^{-1}}$ (cf. Bower et al. 2019;
EHTC 2022b). The near-infrared emission (Witzel et al., 2018), which also
originates from the inner accretion flow, should be included when we estimate
the bolometric luminosity. Adopting a spectrum that takes the form of a power-
law with exponential cutoff near $10^{13}$ Hz (i.e.,
$F_{\nu}\propto\nu^{\alpha}\exp(-\nu/10^{13}\,{\rm Hz})$, Bower et al. 2019),
we empirically derive a luminosity of $\approx 1.9\times 10^{35}\,{\rm
erg\,s^{-1}}$ for the infrared emission (we take the 50th percentile of Witzel
et al. (2018) as representative, see their Figure 19). Combining the sub-
millimeter and infrared contributions, the bolometric luminosity of Sgr A* is
estimated to be $L_{\rm bol}\approx 6.9\times 10^{35}\,{\rm
erg\,s^{-1}}=1.5\times 10^{-9}\,{L_{\rm Edd}}$. This model-independent
measurement of $L_{\rm bol}$ agrees with values derived from MAD models (EHTC,
2022a), which are in the range $L_{\rm bol}=(6.8-9.2)\times 10^{35}\,{\rm
erg\,s^{-1}}$. The X-ray emission of Sgr A* during the quiescent state mainly
originates from the region $R>10^{4}R_{\rm g}$ (here $R_{\rm g}=G{M_{\rm
BH}}/c^{2}$ is the gravitational radius of the BH). Detailed analysis of high-
resolution Chandra observations indicates that nuclear ($R<10^{3}R_{\rm g}$)
X-ray emission has only $\nu L_{\nu}\approx 1.0\times 10^{32}\,{\rm
erg\,s^{-1}}$ at 5 keV (Roberts et al., 2017). It is three orders of magnitude
fainter than the sub-millimeter bump, thus we ignore this contribution. For
the uncertainty in our $L_{\rm bol}$ measurement of Sgr A*, we again assume
that it is $20\%$ of $L_{\rm bol}$.

Early hot accretion flow models of Sgr A* invoked mass accretion rates at the
BH comparable to the Bondi accretion rate, but such large rates were thrown
into doubt when linear polarization was reported in Sgr A* at millimeter
wavelengths by Aitken et al. (2000, later confirmed by ). In two influential
papers, Agol (2000) argued that this detection implied that RM must be low and
hence $\dot{M}_{\rm BH}$ must be $<10^{-8}{{M_{\odot}}\,{\rm yr}^{-1}}$, while
Quataert & Gruzinov (2000) used a similar argument to estimate $\dot{M}_{\rm
BH}\approx 10^{-8}{{M_{\odot}}\,{\rm yr}^{-1}}$.

Recently, EHTC (2022a, see references therein for earlier
theoretical/observational studies) combined observational constraints from the
230 GHz EHT image of Sgr A* with detailed simulations-based theoretical
models, to come up with a tight constraint on the mass accretion rate in Sgr
A*: $\dot{M}_{\rm BH}=(5.2-9.5)\times 10^{-9}\,{{M_{\odot}}\,{\rm yr}^{-1}}$,
or equivalently $\dot{M}_{\rm BH}/{\dot{M}_{\rm Edd}}=(5.7-10.4)\times
10^{-8}$. Although these models do not include any spatially resolved
polarization constraints from the EHT (none have been published yet), the
estimated $\dot{M}_{\rm BH}$ is perfectly consistent with the original
estimates from Agol (2000) and Quataert & Gruzinov (2000), and subsequent work
following the same lines. We use the quoted EHT-derived range in the present
work.

Using the above estimates of the bolometric luminosity and the mass accretion
rate, we estimate the radiative efficiency $\epsilon$ of Sgr A* to be in the
range

$1.3\times 10^{-3}\lesssim\epsilon\lesssim 2.3\times 10^{-3}.$ (5)

This range is shown in Figure 1 as cyan diamonds connected by a gray solid
curve. The constraint on $\epsilon$ in the case of Sgr A* is much tighter than
in M87*. This is because of the factor of $<2$ uncertainty in $\dot{M}_{\rm
BH}$, which is perhaps a little optimistic. However, none of our conclusions
will be affected even if we allow another factor of 2 uncertainty.

###  2.3 Comparing the Two Objects

The radiative efficiencies of M87* and Sgr A*, together with the large
difference in their Eddington-scaled accretion rates, confirm an important
theoretical prediction of hot accretion flow theory. All models, however much
they may differ in physics details or parameter choices, agree that the
radiative efficiency of hot accretion flows should decrease with decreasing
Eddington-scaled mass accretion rate $\dot{M}_{\rm BH}/{\dot{M}_{\rm Edd}}$
(e.g., Narayan & Yi 1995; Narayan et al. 1998; Xie & Yuan 2012; Xie &
Zdziarski 2019, and Figure 1 here). This is clearly borne out by the EHT-
derived estimates for M87* and Sgr A*. However, due to the large uncertainty
in $\dot{M}_{\rm BH}/{\dot{M}_{\rm Edd}}$ (especially the factor of $\sim 7$
uncertainty in the case of M87*), it is not possible to reliably estimate the
slope of the $\epsilon-\dot{M}_{\rm BH}/{\dot{M}_{\rm Edd}}$ relation from
current observational data.

##  3 Radiative Efficiency of the MAD Model

###  3.1 The MAD model

In our numerical model of hot accretion flows in the MAD state, we assume for
simplicity a non-spinning BH and adopt a pseudo-Newtonian Paczyński-Wiita
gravitational potential (Paczyński & Wiita, 1980) to mimic the potential of
the BH. Since relativity is not explicitly taken into account in the dynamics
of our model, the radial velocity can unphysically exceed the speed of light
$c$ near the BH horizon. To correct for this, we follow Xie et al. (2010) and
interpret the velocity $V_{r,{\rm PW}}$ in our Paczyński-Wiita potential model
as $\gamma V_{r,{\rm real}}$, where $\gamma$ is the bulk Lorentz factor of the
accreting gas and $V_{r,{\rm real}}$ is the corrected radial velocity.

The neglect of BH spin in the model is reasonable. Although the jet power
depends sensitively on spin (e.g., Tchekhovskoy et al. 2010), the emission
from the main body of the accretion flow has only a weak dependence. This is
because, unlike the cold accretion disk model (Shakura & Sunyaev, 1973) where
the location of the disk inner edge varies substantially with BH spin, a hot
accretion flow extends down to the BH horizon $R_{\rm horizon}$ which is much
less sensitive to BH spin. Dynamical properties of the flow outside
$(3-5)R_{\rm horizon}$ are mostly unaffected by BH spin, while emission from
regions inside of $\sim 2R_{\rm horizon}$ (e.g., radius of marginally bound
orbit) is mostly beamed toward the BH and lost through the horizon.

The details of our MAD model and its radiation calculation can be found in Xie
& Zdziarski (2019). Below we highlight the main points. Guided by numerical
simulations, Xie & Zdziarski (2019) solve in cylindrical coordinates the
height-integrated equations of a steady MAD system, paying attention to the
effects of non-axisymmetric gas streams/spirals, which are a dominant feature
of hot accretion flows in the MAD regime (e.g., Narayan et al. 2003;
Tchekhovskoy et al. 2011; McKinney et al. 2012; White et al. 2019; Chatterjee
& Narayan 2022). We follow the GRMHD simulations of McKinney et al. (2012) to
set a $R^{-s_{bz}}$ profile of the global vertical component of the magnetic
field $B_{z}$, where $s_{bz}\approx 1.1$. The field strength is determined by
the azimuthally-averaged plasma $\beta$ (gas-to-magnetic pressure ratio)
parameter at the outer boundary $200R_{\rm g}$, i.e. $\bar{\beta}_{z0}=1.5$
(see Xie & Zdziarski 2019 for the definition of $\bar{\beta}_{z0}$). Other
components are calculated numerically according to the gas dynamics, where the
magnetic Prandtl number $\mathcal{P}_{\rm m}=2$ and a parameter
$\kappa_{\phi}=-0.5$ are introduced respectively, for the calculation of
$B_{r}/B_{z}$ and $B_{\phi}/B_{z}$. With this setup (strength and profile) of
global magnetic fields, the vertical magnetic flux threading the accretion
flow inside $10\,R_{\rm g}$ is about $50\,(\dot{M}_{\rm BH}R_{\rm
g}^{2}c)^{1/2}$, in agreement with that threading the BH in numerical
simulations of MAD systems (Tchekhovskoy et al. 2011; McKinney et al. 2012;
Davis & Tchekhovskoy 2020; Narayan et al. 2022, see their reported values of
the parameter $\phi_{\rm BH}$). The stress of global ordered magnetic fields,
approximately proportional to $B_{z}B_{\phi}$ (Xie & Zdziarski, 2019), can
then be derived. Besides the global field, we also include a turbulent
magnetic field, by setting the gas-to-turbulent-magnetic pressure ratio
$\beta_{\rm t}=10$. Most importantly, unlike most MHD simulations where a
single-fluid equation is adopted, we use separate energy equations for the
electrons and ions (Narayan & Yi, 1995; Yuan & Narayan, 2014; Xie & Zdziarski,
2019). As usual, in our model we mimic the turbulent stress term, which is
automatically present in MHD simulations, via a radius-independent viscosity
parameter $\alpha$ (see Sec. 3.2 below for the determination of its value).

Numerical simulations suggest that hot accretion flows, including MADs, have a
strong mass outflow (e.g., Narayan et al. 2012; Yuan et al. 2012, 2015; Yang
et al. 2021) outside of a certain radius $R_{\rm flat}$ ($\sim 6-10R_{\rm g}$;
refer to Fig. 5 in Yuan et al. 2015 and Fig. 6 in Yang et al. 2021). Inside
$R_{\rm flat}$, the outflow is highly suppressed and the accretion rate is
nearly a constant. Instead of the conventional broken power-law fit, we adopt
a smooth function for $\dot{M}(R)$, viz., $\dot{M}(R)=\dot{M}_{\rm
BH}\left[1+({R-R_{\rm BH})^{2}/R^{2}_{\rm flat}}\right]^{s/2}$, where $s$
measures the strength of the mass outflow. This expression ensures that
$\dot{M}(R)\propto R^{s}$ for $R\gg R_{\rm flat}$ and
$\dot{M}(R)\approx\dot{M}_{\rm BH}$ for $R\ll R_{\rm flat}$. In this work we
follow the BH spin $a=0$ MAD case of Yang et al. (2021) and set $R_{\rm
flat}\approx 6R_{\rm g}$, $s\approx 0.2$. Since the outflow strength we adopt
is fairly weak (e.g., $\dot{M}(200R_{\rm g})\approx 2\dot{M}_{\rm BH}$), it
has only a minor impact on the radiative efficiency of the model.

###  3.2 Viscous parameter $\alpha$ for the turbulent stress term

The “viscous parameter” $\alpha$ (Shakura & Sunyaev, 1973), which describes
the strength of the turbulent stress, is a free parameter in our model. We
note that MHD turbulence can be automatically generated and sustained in MHD
simulations of accretion flows (e.g., Stone & Pringle 2001; Igumenshchev et
al. 2003; White et al. 2019), thus $\alpha$ is implicitly included there. In
our model, $\alpha$ has a strong effect on the predicted radiative output of
the model. The radiation emitted by a hot accretion flow is largely determined
by the gas density $\rho$ which, for a given mass accretion rate, is
determined by the efficiency of angular momentum transport. The latter is
proportional to $\alpha$ in the SANE case (note that in our MAD model the
stress by global ordered magnetic fields is handled separately, based on
numerical simulation results, cf. Sec. 3.1). For a SANE model, we typically
have $\rho\propto\dot{M}_{\rm BH}V_{r}^{-1}\propto\alpha^{-1}\dot{M}_{\rm BH}$
(Narayan & Yi, 1994). A similar expression can also be applied for the
magnetic field strength.333We note that for the low-$\dot{M}/{\dot{M}_{\rm
Edd}}$ regime appropriate to a hot accretion flow, most radiation is in the
form of synchrotron, and for optical depth $\tau\lesssim 10^{-6}$ inverse
Compton scattering only plays a negligible role. In this case, the radiative
luminosity scales as $L_{\rm bol}\propto\rho
B^{2}\propto\rho^{2}\propto\alpha^{-2}\dot{M}_{\rm BH}^{2}$ (assuming that the
electron temperature is un-affected if $\epsilon\ll 1$, cf. Narayan et al.
1998 and Figure 1). In addition, the total dissipated energy is determined by
the strength of turbulence, i.e. it is linearly proportional to $\alpha$ in
our model, cf. Xie & Zdziarski (2019). Since part of the dissipated energy
goes into electrons (through the $\delta$ parameter), the value of $\alpha$
has a direct and strong impact on the radiative luminosity.

As shown in Figure 2, we use the radial velocity measured in a GRMHD MAD
simulation (Narayan et al., 2022) of a non-spinning BH to calibrate the value
of $\alpha$ in our model. We focus on the region $R\approx(3-6)R_{g}$ from
which most of the observed synchrotron radiation in the EHT band (230 GHz)
comes, shown by the shadowed region in Figure 2.

The calculated radial profile of the radial velocity as obtained from our
height-integrated MAD model with $\alpha=0.08$ is shown as the purple dashed
line in Figure 2. In comparison, the black solid curve shows the velocity
profile seen in a long-duration GRMHD MAD simulation with BH spin $a=0$
(Narayan et al., 2022). The two profiles agree very well, especially in the
shaded region which produces most of the radiation observed by the EHT.
Therefore, we use $\alpha=0.08$ in computing all our model results. For
completeness, we also show by the blue dotted curve the GRMHD simulation
result for a SANE model with $a=0$ (Ricarte et al., 2022). There is a large
difference, suggesting that the global magnetic field in MAD plays an
important role in transferring angular momentum and speeding up the radial
velocity.

Figure 2: Estimation of the viscosity parameter $\alpha$ corresponding to the
turbulent stress. The black solid and blue dotted curves show the density-
weighted radial velocity in GRMHD simulations of MAD and SANE respectively,
around a non-spinning BH (Narayan et al., 2022; Ricarte et al., 2022). The
purple dashed curve is derived from our height-integrated MAD model with
$\alpha=0.08$. The shadowed region ($3R_{\rm g}<R<6R_{\rm g}$) shows where
most of the observed radiation comes from.

###  3.3 Radiative efficiency

We now calculate the radiative efficiency of our height-integrated MAD model
using various values of the electron heating fraction parameter $\delta$. As
mentioned earlier, there are two source of heating for electrons in a hot
accretion flow: one is through energy transfer from ions via electron-ion
Coulomb collisions, and the other is through direct viscous heating at a rate
proportional to $\delta$ (see equation 1). The Coulomb heating rate is
$\propto\rho^{2}$, whereas the viscous heating rate is $\propto\rho$ (e.g.,
Narayan & Yi, 1995). Since $\rho\propto\dot{M}_{\rm BH}$, we expect the impact
of $\delta$ to be most evident at low values of $\dot{M}_{\rm
BH}/{\dot{M}_{\rm Edd}}$. Thus, among the two sources we are considering, Sgr
A* is the most likely to give a useful constraint on $\delta$.

For our calculations, we do not focus on detailed microphysics of viscous
dissipation (see Introduction). Instead, we take a “model-independent”
approach in which we assume a constant electron heating fraction $\delta$ that
is independent of radius and explore the observational constraint on its
value. We adopt several values: $\delta=10^{-3},~{}0.1,~{}0.3$, and $0.5$. The
calculated radiative efficiencies corresponding to these values of $\delta$
are shown as a function of $\dot{M}_{\rm BH}/{\dot{M}_{\rm Edd}}$ in Figure 1.
In our calculations, we limit ourselves to the “advection-dominated regime”
which corresponds to low accretion rates (Yuan & Narayan, 2014). Both M87* and
Sgr A* belong to this regime. The maximal accretion rate at the horizon, above
which the radiation is so strong that the flow is no longer advection-
dominated but enters into the “luminous hot accretion flow regime” (Yuan,
2001; Yuan & Narayan, 2014), is determined numerically by having the advection
term $f_{\rm adv}=1-q^{-}/q_{\rm vis}$ (Narayan & Yi, 1994) to be zero at any
radius. In our model, this limit is located at $\sim(1-3)\times
10^{-5}{\dot{M}_{\rm Edd}}$.

Because the strong poloidal field in a MAD system breaks axisymmetry and
causes the accretion to occur in the form of dense gas streams surrounded by
dilute magnetic voids (Narayan et al., 2003; Tchekhovskoy et al., 2011;
McKinney et al., 2012; White et al., 2019; Chatterjee & Narayan, 2022), at a
given accretion rate, the density of the gas in the accreting streams in MAD
will be higher than that of the more uniform density in SANE. Consequently,
the ion-electron Coulomb heating becomes comparable to the turbulent heating
at a lower value of $\dot{M}_{\rm BH}/{\dot{M}_{\rm Edd}}$ in a MAD model
compared to a SANE model (compare the results shown in Fig. 1 with those in
Xie & Yuan 2012).

As seen in Figure 1, the maximal accretion rate of the ADAF regime in MAD
models (the upper limits of the curves) decreases with increasing $\delta$.
This is naturally expected. At a given accretion rate, a higher $\delta$ means
a larger fraction of the viscously dissipated energy goes to electrons. This
makes the electrons hotter, and radiatively more luminous (higher in radiative
cooling $q^{-}$). Consequently, it will reach a lower advection term $f_{\rm
adv}$.

In addition, for a typical ADAF regime with $\dot{M}_{\rm BH}/{\dot{M}_{\rm
Edd}}\lesssim 6\times 10^{-6}$, we find that the radiative efficiency has a
roughly linear dependence on the accretion rate: $\epsilon\propto\dot{M}_{\rm
BH}^{0.92}$ (for a similar result, see also Xie & Zdziarski 2019).
Equivalently, the bolometric luminosity follows $L_{\rm
bol}\propto\dot{M}_{\rm BH}^{1.92}$. We note that an estimation of $L_{\rm
bol}\propto\dot{M}_{\rm BH}^{2}$ was derived previously in Narayan et al.
(1998) for the SANE case.

We now explore the $\delta$-dependence of $\epsilon$ based on our
calculations. We find that, at a given accretion rate, the radiative
efficiency differs by a factor of $\sim 5$ for different values of $\delta$.
Besides, when $\delta\lesssim 0.05$, the radiative efficiency becomes
insensitive to the value of $\delta$, because electron heating by Coulomb
collisions with ions is dominant. For the typical ADAF accretion rate regime
with $\dot{M}_{\rm BH}/{\dot{M}_{\rm Edd}}\lesssim 6\times 10^{-6}$, we have
$\epsilon\propto{\rm max}(\delta^{0.7},0.1)$, i.e. $\delta=0.5$ and
$\delta=0.3$ cases are brighter by a factor of $\approx 3$ and $\approx 2.2$
respectively, than the $\delta=0.1$ case. Combining the dependence on $\delta$
and $\dot{M}_{\rm BH}$, our model results can be summarized as,

$\displaystyle\epsilon$ $\displaystyle\approx$ $\displaystyle
1.9\%\left({\dot{M}_{\rm BH}\over 10^{-6}{\dot{M}_{\rm
Edd}}}\right)^{0.92}\,{\rm max}(\delta^{0.7},0.1),$ (6)
$\displaystyle\quad{\rm where}\,\,\,\left({\dot{M}_{\rm BH}\over
10^{-6}{\dot{M}_{\rm Edd}}}\right)<6.$

The dependence of $\epsilon$ on $\delta$ becomes weaker when $\dot{M}_{\rm
BH}/{\dot{M}_{\rm Edd}}\gtrsim 6\times 10^{-6}$, with
$\epsilon\approx(2-4)\%$. This is because of the increasing importance of
Coulomb collision heating of electrons at high accretion rates.

###  3.4 Constraint on $\delta$ from Sgr A* and M87*

Finally, we apply the above theoretical calculations to the observational
constraints from Sgr A* and M87*. From Figure 1 we find that in the case of
Sgr A*, low values of $\delta$ (i.e. $\delta<0.3$) are ruled out, and that
$\delta\sim 0.5$ is preferred, given the uncertainties in $\dot{M}_{\rm BH}$.
On the other hand, we cannot put any constraint on $\delta$ based on the
current observational data from M87*. A useful constraint at a similar level
to Sgr A* will require the mass accretion rate in M87* to be further
constrained to $\dot{M}_{\rm BH}/{\dot{M}_{\rm Edd}}\lesssim 8\times 10^{-6}$
(equivalently, $\dot{M}_{\rm BH}\lesssim 10^{-3}{M_{\odot}}\,{\rm yr}^{-1}$),
i.e., the upper limit on $\dot{M}_{\rm BH}$ will need to be reduced by a
factor $\sim 2$ compared to what the EHT has achieved so far.

##  4 Summary

In the theory of hot accretion flows around black holes, an important but
poorly understood parameter is $\delta$, which describes the fraction of
viscous energy that directly heats electrons (refer to Equation 1). The value
of $\delta$ determines the thermal energy of electrons and therefore the
radiative efficiency of the accretion flow for a given accretion rate. While
the underlying microphysics is complicated and the value of $\delta$ is poorly
determined from theory, in this paper we try to constrain its value by using
the most recent EHTC observational and modeling results on M87* and Sgr A*.

EHTC papers have provided good constraints on the mass accretion rates at the
BH horizon for the two sources. These results, combined with the measured
bolometric luminosities of the two objects, lead to useful constraints on the
radiative efficiencies, as presented in Equations (3) and (5) and shown in
Figure 1. Meanwhile, we can analytically solve the height-integrated dynamical
equations of a hot accretion flow in the MAD regime and calculate the
theoretically expected radiative efficiency. For this calculation, we need to
assume a value for the viscous parameter $\alpha$. By comparing the radial
velocity profile in the height-integrated model with the velocity profile
obtained in numerical GRMHD simulations, we estimate that $\alpha\sim 0.08$.
Using this $\alpha$, we calculate the radiative efficiency in our model as a
function of the accretion rate and the electron heating fraction parameter
$\delta$. By comparing these theoretical results with the observational
constraints on Sgr A* and M87* from EHTC, we are able to estimate the value of
$\delta$.

In the case of Sgr A*, we find that we can rule out $\delta\lesssim 0.3$, and
that the most likely value is $\delta\sim 0.5$. This result is in excellent
agreement with that obtained by Yuan et al. (2003) who modeled the spectral
energy distribution of Sgr A*. The analysis in the present paper was made
possible because the EHTC reported a tight constraint on the mass accretion
rate in Sgr A*. Even if we doubled their uncertainty range, our results would
still be largely unchanged. However, we caution that our height-integrated
model of MAD accretion makes several approximations whose effects are
difficult to quantify.

In the case of M87*, we are unable to obtain a useful constraint on $\delta$
because EHTC does not provide a sufficiently strict constraint on the mass
accretion rate.

FGX is supported in part by National SKA Program of China (No.
2020SKA0110102), the National Natural Science Foundation of China (NSFC;
grants 11873074, 12192220, and 12192223), and the Youth Innovation Promotion
Association of CAS (Y202064). FY is supported in part by NSFC (grants
12133008, 12192220, and 12192223). RN is supported in part by the National
Science Foundation under grants OISE-1743747 and AST1816420. RN’s work was
carried out at the Black Hole Initiative, Harvard University, which is funded
by grants from the John Templeton Foundation and the Gordon and Betty Moore
Foundation.

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