# Global Electron Thermodynamics in Radiatively Inefficient Accretion Flows

Kaushik Satapathy  Department of Physics, University of Arizona, 1118 E.
Fourth Street, Tucson, AZ 85721  School of Physics, Georgia Institute of
Technology, 837 State St NW, Atlanta, GA 30332, USA  Dimitrios Psaltis  School
of Physics, Georgia Institute of Technology, 837 State St NW, Atlanta, GA
30332, USA  Feryal Ozel  School of Physics, Georgia Institute of Technology,
837 State St NW, Atlanta, GA 30332, USA

###### Abstract

In the collisionless plasmas of radiatively inefficient accretion flows,
heating and acceleration of ions and electrons is not well understood. Recent
studies in the gyrokinetic limit revealed the importance of incorporating both
the compressive and Alfvenic cascades when calculating the partition of
dissipated energy between the plasma species. In this paper, we use a
covariant analytic model of the accretion flow to explore the impact of
compressive and Alfvenic heating, Coulomb collisions, compressional heating,
and radiative cooling on the radial temperature profiles of ions and
electrons. We show that, independent of the partition of heat between the
plasma species, even a small fraction of turbulent energy dissipated to the
electrons makes their temperature scale with a virial profile and the ion-to-
electron temperature ratio smaller than in the case of pure Coulomb heating.
In contrast, the presence of compressive cascades makes this ratio larger
because compressive turbulent energy is channeled primarily into the ions. We
calculate the ion-to-electron temperature in the inner accretion flow for a
broad range of plasma properties, mass accretion rates, and black hole spins
and show that it ranges between $5\lesssim T_{i}/T_{e}\lesssim 40$. We provide
a physically motivated expression for this ratio that can be used to calculate
observables from simulations of black hole accretion flows for a wide range of
conditions.

black-hole, accretion, plasma

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##  1 Introduction

Low-luminosity accretion flows, such as the ones around the black holes at the
center of the Milky Way, Sgr A*, and of the M87 galaxy, belong to a class
known as Radiatively Inefficient Accretion Flows (Narayan & Yi, 1994a). They
are characterised by low mass-accretion rates, typically $\lesssim 10^{-3}$
times the Eddington accretion rate, and low plasma densities. The collisional
timescale between the ions and the electrons in them is significantly larger
than the accretion timescale, allowing for the ions and the electrons to co-
exist in two different temperatures. Radiative processes are also inefficient,
as a result of which the energy dissipated during the accretion process
primarily serves to increase the thermal energy carried by the plasma that, in
turn, is advected into the black hole (Narayan & Yi, 1994b, 1995a, 1995b).

Beyond this general picture of the accretion flow, heating and acceleration of
ions and electrons in these collisionless plasmas is not well understood. The
turbulence in black-hole accretion flows is thought to be driven by the
Magneto-Rotational Instability (MRI, see Balbus & Hawley, 1991). The MRI
creates a turbulent cascade of compressive and Alfven waves. These waves
cascade down to small lengthscales and undergo collisionless dissipation, in
the process, channeling a part of their energy into the ions and the rest into
the electrons. Even though the dissipation processes operate at multiple
lengthscales, they are only dominant at lengthscales comparable to the Larmor
radii of each species, i.e., ion heating is dominant at the ion Larmor radius
and electron heating dominates at the electron Larmor radius (Quataert, 1998;
Quataert & Gruzinov, 1999; Howes, 2010). For typical conditions in the
accretion flows, these radii are smaller by many orders of magnitude than the
macroscopic scales at which MRI is driven. Additional dissipation mechanisms
in weakly collisional plasmas might include scattering of particles off of
velocity-space instabilities (Sironi & Narayan, 2015; Sharma et al., 2007;
Kunz et al., 2018) and magnetic reconnection, which leads to particle
acceleration (Ball et al., 2016, 2018; Rowan et al., 2017, 2019).

Analytic studies have shown that, in the inertial range of turbulence, the
compressive and Alfvenic cascades decouple from each other at the linear order
(Schekochihin et al., 2008). These were performed in the gyrokinetic limit,
where the gyromotion of particles around the magnetic field were effectively
averaged out. The compressive and Alfvenic modes, however, mix at lengthscales
comparable to the ion Larmor radius and undergo collisionless damping, which
channels a fraction of the turbulent energy into heating the ions. The
remaining energy cascades down via a kinetic Alfven wave cascade to smaller
lengthscales and dissipates at the electron Larmor radius, heating the
electrons.

Recently, gyrokinetic simulations of collisionless plasmas driven by a
combined compressive and Alfvenic turbulent cascade showed that the ion-to-
electron heating ratio depends primarily on the plasma $\beta$ parameter and
on the ratio $P_{\rm{comp}}/P_{\rm{AW}}$ of the power in the compressive wave
cascade to the power in the Alfven wave cascade (Kawazura et al., 2020). This
work extends earlier analytic studies of particle heating in purely Alfvenic
turbulence (Quataert, 1998; Quataert & Gruzinov, 1999; Howes, 2010).

Because the compressive and Alfvenic wave cascades are decoupled in the
inertial range, the partition of turbulent energy between them is determined
at the driving scale of the turbulence. As a result, this suggests that the
partition of the dissipated energy between ions and electrons at very small
scales is determined by the mechanism that drives turbulence at macroscopic
scales. In the case of accretion flows around black holes, it is the MRI and
the resulting parasitic instability that drive turbulence in the plasma and,
therefore, determine the partition of wave energy. The partition of energy
under these conditions has not been fully explored. The one set of
calculations by Kawazura et al. (2022) were performed using reduced-MHD
simulations of an MRI-driven turbulent shearing flow with a nearly azimuthal
background magnetic field and showed that the compressive waves carry at least
twice as much power as the Alfven waves.

In this paper, we utilize these recent developments in our understanding of
the physical processes governing the heating of electrons in turbulent flows
in order to calculate the ion and electron temperatures in accretion flows and
their dependence on plasma parameters. In order to explore a broad range of
physical conditions, we employ a covariant semi-analytic model of accretion
flows around black holes that we recently developed and calibrated against
GRMHD simulations (Özel et al., 2022). We utilize the results of Kawazura et
al. (2020, 2022) to implement a realistic model of ion and electron heating
into this covariant analytic model of accretion, which allows us to calculate
the ion-to-electron temperature ratio throughout the accretion flow.

This approach differs from earlier studies that primarily relied on single-
fluid General Relativistic Magneto-HydroDynamic (GRMHD) simulations (see,
e.g., De Villiers & Hawley, 2003; Gammie et al., 2003; Porth et al., 2019) to
obtain the bulk properties of the accretion flow and prescribed temperatures
in post processing (see, e.g., Event Horizon Telescope Collaboration et al.,
2019b; Akiyama et al., 2022b). In all prescriptions, the temperature ratio
$R=T_{i}/T_{e}$ is assumed to depend only on the plasma $\beta$ and modeled
either as a step function in this parameter (Chan et al., 2015) or as a smooth
function between the ratio $R_{\rm{high}}$ in the high-$\beta$ equatorial
region and the ratio $R_{\rm{low}}$ in the low-$\beta$ funnel region
(Mościbrodzka et al., 2016). An exception is the work of Ressler et al.
(2015), who evolved the energy equations for the electrons and ions
individually in GRMHD simulations, using the model for dissipation of Alfvenic
turbulence for the partition of heating between the species. Simulations of
radiatively inefficient flows using this approach provided additional support
to the expectation that the ion-to-electron temperature ratio depends on
plasma $\beta$ (Ressler et al., 2017; Sadowski et al., 2017).

The paper is organised as follows. In §2, we describe our semi-analytical
model for the properties of a radiatively inefficient accretion flow around a
black hole, elucidating our assumptions related to plasma heating amd cooling
in constituent subsections. Following this, in §3, we compute solutions for
the ion and electron temperatures in the inner accretion flow. We specifically
examine the effects of various heating processes, radiative cooling, and our
model for the partition of turbulent heat into the ions and the electrons on
the ion-to-electron temperature ratio. We summarize our findings in §4.

##  2 Methods

Our goal is to derive a model that describes the spatial distribution of
electron and ion temperatures in a radiatively inefficient accretion flow,
taking into account particle heating due to the dissipation of Alfvenic and
compressive cascades. The key ingredients and assumptions are: (i) a covariant
semi-analytic model for the radial dependence of the density, velocity, and
internal energy of the accretion flow, (ii) an equation of state for the
magnetized plasma consisting of electrons and ions, (iii) a model of the
energy dissipation rate in the flow, (iv) macroscopic equations that capture
the partition of the dissipated energy between ions and electrons based on the
cascade of Alfven and compressive modes, and (v) expressions for the electron
cooling rate via synchrotron and Bremsstrahlung processes.

We consider the dynamics of the accretion flow in a background Kerr metric

$\displaystyle\begin{split}ds^{2}&=-\left(1-\dfrac{2r}{\Sigma}\right)\
dt^{2}-\dfrac{4ar\sin^{2}{\theta}}{\Sigma}\ dt\ d\phi+\dfrac{\Sigma}{\Delta}\
dr^{2}+\Sigma\ d\theta^{2}\\\
&+\left(r^{2}+a^{2}+\dfrac{2a^{2}r\sin^{2}{\theta}}{\Sigma}\right)\sin^{2}{\theta}\
d\phi^{2},\end{split}$ (1)

where $\Delta=r^{2}-2r+a^{2}$, $\Sigma=r^{2}+a^{2}\cos^{2}{\theta}$, and $a$
denotes the black hole spin. We also use geometrized units with $G=c=M=1$,
where G is the gravitational constant, c is the speed of light, and $M$ is the
mass of the black hole.

We describe each element of the model in the following subsections.

###  2.1 A Covariant Model for the Accretion Flow

The global dynamics of the ions and electrons in an accretion flow where ion-
electron collisions are negligible can be described by conservation of mass
and conservation of energy independently in each species (see, e.g., Ressler
et al., 2015). The mass conservation is given by

$\nabla_{\mu}\left(\rho_{(s)}u_{(s)}^{\mu}\right)=0,$ (2)

and the conservation of energy can we written as

$u_{\nu(s)}\nabla_{\mu}\left(T_{(s)}^{\mu\nu}\right)=0.$ (3)

In equations (2) and (3), $\rho$ is the density, $u^{\mu}$ is the four
velocity, $T^{\mu\nu}$ is the stress energy tensor, and the label $s$ in
parenthesis for each quantity indicates the species, $i$ (ion) or $e$
(electron).

We consider a charge-neutral hydrogen plasma such that number densities of
electrons and ions are equal, $n_{e}\approx n_{i}$, and are related to the
mass densities by $\rho_{e}=m_{e}n_{e}$ and $\rho_{i}=m_{i}n_{i}$, where
$m_{e}$ and $m_{i}$ are the electron and ion masses, respectively. We further
assume that both the ions and electrons follow the same flow four-velocity,
i.e., $u^{\mu}_{i}\approx u^{\mu}_{e}\equiv u^{\mu}$ and that all flow
quantities are azimuthally symmetric.

The stress energy tensor for each of the species in the plasma can be written
as (see Gammie & Popham, 1998; Özel et al., 2022)

$T^{\mu\nu}_{i}=\left(P_{i}+\dfrac{B^{2}}{8\pi}\right)g^{\mu\nu}+\left(\rho_{i}+\epsilon_{i}+P_{i}+\dfrac{B^{2}}{4\pi}\right)u^{\mu}u^{\nu}+t_{i}^{\mu\nu}$
(4)

for the ions and

$T^{\mu\nu}_{e}=P_{e}g^{\mu\nu}+\left(\rho_{e}+\epsilon_{e}+P_{e}\right)u^{\mu}u^{\nu}+t_{e}^{\mu\nu}$
(5)

for the electrons. Here, $B$ is the magnetic field strength, $P_{(s)}$ is the
thermal pressure, $\epsilon_{(s)}$ is the internal energy, and $\rho_{(s)}$ is
the density. The viscous stresses acting upon the plasma are accounted for in
the tensor $t^{\mu\nu}$. In the above definition of the stress energy tensors,
we assume that the effects of the magnetic field on the plasma dynamics solely
impact the ions.

We further reduce the dimensionality of our model by vertically averaging over
the scale-height of the disk in all fluid quantities so that the fluid
quantities exhibit only a radial dependence in a steady state (see Özel et
al., 2022). With this vertical integration, the equation for mass conservation
becomes

$4\pi\left(\dfrac{h}{r}\right)\sqrt{-g}\ \rho(r)\ u^{r}(r)=-\dot{M},$ (6)

where $\dot{M}$ is the mass accretion rate in the disk and $h/r$ represents
its scale height. We refer to all variables defined in this way as height-
averaged fluid quantities.

Hereafter, we will express the mass accretion rate in terms of the Eddington
critical rate

$M_{\rm E}=\frac{4\pi GMm_{p}}{\eta c\sigma_{T}},$ (7)

where $\sigma_{T}$ is the Thomson cross section and we set the radiative
efficiency of the accretion flow $\eta$ to be 0.1. We also set the scale
height by imposing local vertical pressure balance at each radius (Özel et
al., 2022), i.e.,

$\frac{h}{r}=\frac{1}{ru^{\phi}}\left(\frac{P_{i}+P_{e}}{\rho_{i}+\rho_{e}}\right)^{1/2}\;,$
(8)

where $u^{\phi}$ is the azimuthal angular velocity of the flow, assumed to be
equal to the local Keplerian value.

Using the forms of the stress energy tensors defined in Equations (4) and (5),
we simplify the conservation of energy in Equation (3) to obtain

$u^{r}\dfrac{d\epsilon^{\rm{eff}}_{i}}{dr}-u^{r}\dfrac{\epsilon^{\rm{eff}}_{i}+P_{i}+B^{2}/8\pi}{\rho_{i}}\dfrac{d\rho_{i}}{dr}=Q_{i}-Q_{ie}$
(9)

for the ions, and

$u_{e}^{r}\dfrac{d\epsilon_{e}}{dr}-u_{e}^{r}\dfrac{\epsilon_{e}+P_{e}}{\rho_{e}}\dfrac{d\rho_{e}}{dr}=Q_{e}+Q_{ie}-\Lambda_{e}$
(10)

for the electrons. Here, $Q_{i}$ is the amount of the energy dissipated per
unit volume in the plasma that goes into heating the ions, $Q_{e}$ is the
amount of heat that is channeled into the electrons, and $Q_{ie}$ is the
energy exchange rate between the ions and the electrons. We also define
$\epsilon^{\rm{eff}}_{i}$ as an effective internal energy density in the ions
that includes the contribution of the magnetic field. Because electrons in the
plasma cool through synchrotron and Bremsstrahlung radiation, we incorporate
the appropriate cooling rate $\Lambda_{e}$ for the electrons.

Following the approach in Özel et al. (2022), we write the radial velocity
outside of the innermost stable circular orbit (ISCO) as

$u^{r}=-\eta\left(\dfrac{r}{r_{\rm{ISCO}}}\right)^{-n_{R}},$ (11)

where $\eta$ and $n_{R}$ are parameters that aim to capture the efficiency of
the mechanism that drives angular momentum transport in the flow. By exploring
a range for these parameters, we allow our model to be general enough to
encompass different plausible dissipation profiles.

Using this model for the radial velocity, we can easily apply mass
conservation (equation [6]) and obtain a radial profile for density, which
yields another power law function in radial distance.

###  2.2 The equation of state of the magnetized ion-electron plasma

In order to infer the temperatures of each species, we employ an ideal gas
equation of state for the ion and electron thermal pressures,

$P_{(s)}=nk_{B}T_{(s)}\;.$ (12)

We write the internal energy of the electrons as

$\epsilon_{e}=\dfrac{nk_{B}T_{e}}{\gamma_{e}-1},$ (13)

and the effective internal energy of the ions as

$\begin{split}\epsilon^{\rm{eff}}_{i}&={nk_{B}T_{i}}\left(\dfrac{1}{\gamma_{i}-1}+\dfrac{1}{\beta}\right)\\\
&\equiv\dfrac{nk_{B}T_{i}}{\gamma_{i}^{\rm{eff}}-1}\;,\end{split}$ (14)

where we have defined the plasma parameter as $\beta=8\pi P_{i}/B^{2}$. In
equations (13) and (14), we denote the adiabatic indices of the electrons and
the ions by $\gamma_{e}$ and $\gamma_{i}$, respectively, and also define an
effective adiabatic index $\gamma_{i}^{\rm{eff}}$ for the ions to encompass
the effects of magnetic pressure.

Allowing for trans-relativistic behavior of the ions and electrons, we express
the temperature dependence of the adiabatic index of a gas as (see Hakim,
2011)

$\gamma\left(\Theta\right)=\dfrac{H-1}{H-1-\Theta},$ (15)

where $H=K_{3}(1/\Theta)/K_{2}(1/\Theta)$. In this equation, $K_{n}$ denotes
the modified Bessel function of the second kind at $n^{\rm{th}}$ order and the
dimensionless temperature for the two species is given by $\Theta_{(s)}\equiv
k_{B}T_{(s)}/m_{(s)}c^{2}$.

###  2.3 Plasma Heating

The viscous stresses present in the accretion flow channel the gravitational
potential energy into turbulent wave energy that resides as MHD waves in the
disk. These MHD waves, in principle, undergo dissipation at microscopic
length-scales and channel their energy into the thermal energies of the plasma
species (ions and electrons). Hence, the heating rates of the individual
species, $Q_{i}$ and $Q_{e}$, are determined by two factors: (i) the total
energy dissipated in the plasma due to viscous stresses (defined as $Q\equiv
Q_{i}+Q_{e}$) and (ii) the partition of the total dissipated energy into the
ions and electrons (defined as $f\equiv Q_{i}/Q_{e}$).

In addition to direct heating, there exists a channel of energy exchange
between the ions and the electrons through Coulomb collisions ($Q_{ie}$, see
equations [9] and [10]), which is given by (see Colpi et al., 1984)

$Q_{ie}=3.83\times
10^{-19}n_{e}^{2}\left(\Theta_{i}-\frac{m_{e}}{m_{i}}\Theta_{e}\right)\left(\Theta_{e}+\Theta_{i}\right)^{-3/2}.$
(16)

####  2.3.1 Energy Dissipation

Following Gammie & Popham (1998) and Özel et al. (2022), we write the total
rate of dissipation in the plasma arising from the viscous stresses
$t^{\mu\nu}$ as

$Q=-t^{\mu\nu}\sigma_{\mu\nu}\,,$ (17)

where $\sigma_{\mu\nu}$ is the covariant shear tensor. This calculation is
simplified in the local rest frame where only the $r\phi$ component of the
stress is non-negligible.

In the non-relativistic limit, the $r\phi$ component of the strain,
$\sigma_{(r)(\phi)}$, becomes (Özel et al., 2022)

$\sigma_{(r)(\phi)}\approx\frac{\sqrt{g_{\phi\phi}}}{2}\times\dfrac{d\Omega}{dr},$
(18)

where $\Omega$ is the angular velocity given by

$\Omega=\sqrt{-\dfrac{g_{tt,r}}{g_{\phi\phi,r}}}$ (19)

The same component for the viscous stress tensor, $t_{(r)(\phi)}$, is given by

$t_{(r)(\phi)}\approx\sqrt{\dfrac{g_{rr}}{g_{\phi\phi}}}t^{r}_{\phi}.$ (20)

The mixed $r\phi$ component of the viscous stress tensor, $t^{r}_{\phi}$, can
be obtained from the conservation of angular momentum

$4\pi\left(\dfrac{h}{r}\right)r^{2}t^{r}_{\phi}=\dot{M}\left(L_{z}-j\right),$
(21)

where $L_{z}$ is the angular momentum of the accreting material and $j$ is an
eigenvalue. Following Özel et al. (2022), we parametrize $j$ as a fraction of
the angular momentum at the ISCO as

$j=\lambda L_{z}(r_{\rm{ISCO}}).$ (22)

Combining Equations (18)-(22), we write the plasma heating rate $Q$ as

$Q=\dfrac{\dot{M}}{4\pi
r^{2}(h/r)}\sqrt{g_{rr}}\dfrac{d\Omega}{dr}\left\\{L_{z}-\lambda
L_{z}(r_{\rm{ISCO}})\right\\}.$ (23)

We further simplify this expression by approximating $L_{z}\approx
g_{\phi\phi}\Omega$, valid to leading order in the Schwarzschild metric.

####  2.3.2 Partition of Heat

The wave energy in the turbulent plasma is driven at macroscopic scales by the
magnetorotational instability and cascades down to much smaller scales that
are comparable to ion and electron Larmor radii, where it is channeled into
thermal energy of the particles. Gyrokinetic approaches, in which it is
possible to average out the fast gyro-motions of particles around a mean
magnetic field, have shown that the wave energy resides in compressive and
Alfven wave cascades that are decoupled in the inertial range (see discussion
in §1, Schekochihin et al., 2008). The length scales where the wave
dissipation via wave-particle resonances (such as Landau damping) dominate
correspond to the Larmor radii of the particles. Hence, at the length scale of
the ion Larmor radius, a fraction of the wave energy is dissipated into the
ions, while the remainder cascades down to the smaller electron Larmor radius,
effectively partitioning the heat between ions and electrons.

In this paper, we utilize the results of Kawazura et al. (2020, hereafter
K2020), who computed numerically the ion-to-electron heating ratio in a
turbulent cascade by means of gyrokinetic simulations, accounting for mode-
mixing in the presence of both compressive and Alfvenic cascades at the
driving scales of the turbulence (see Howes et al., 2006; Schekochihin et al.,
2008, for a rigorous description of gyrokinetic turbulent cascades). Allowing
for an arbitrary relative power $P_{\rm{comp}}/P_{\rm{AW}}$ between the
compressive and Alfvenic cascades and exploring different background plasma
conditions, they obtained an approximate empirical expression for the ion-to-
electron heating ratio given by

$\frac{Q_{i}}{Q_{e}}=\dfrac{35}{1+(\beta/15)^{-1.4}\exp\\{{-0.1/(T_{i}/T_{e})}\\}}+\dfrac{P_{\rm{comp}}}{P_{\rm{AW}}}.$
(24)

Figure 1 shows the dependence of the electron heating fraction
$Q_{e}/(Q_{e}+Q_{i})$ on plasma $\beta$, allowing for different values of
$P_{\rm{comp}}/P_{\rm{AW}}$. The dependence of the heating fraction on the
ion-to-electron temperature ratio is captured by the width of each curve, for
values in the range $1\leq T_{i}/T_{e}\leq 100$, and is very weak.

As discussed earlier, the partition of heat by means of wave-particle
interactions relies on the separation between the ion and electron Larmor
radii. The latter is set by the ratio of temperatures of the species. The
temperature ratios in radiatively inefficient flows are such that the ion
Larmor radius is always much larger than the electron Larmor radius. As a
result, the temperature ratio does not have a strong influence on the
partition of heating and $Q_{i}/Q_{e}$ is primarily determined by the plasma
$\beta$.

Figure 1: The electron heating fraction, $Q_{e}/(Q_{i}+Q_{e})$, as a function
of the plasma $\beta$, as calculated by Kawazura et al. (2020) for different
values of the relative driving power of compressive and Alfvenic waves,
$P_{\rm comp}/P_{\rm A}$. The width of each curve represents the range of
heating ratios for different ion-to-electron temperature ratios ($1\leq
T_{i}/T_{e}\leq 100$) and illustrates the weak dependence on this quantity.
The cyan curve corresponds to the earlier analytical calculation by Howes
(2011), which did not incorporate compressive waves. The typical values of
plasma $\beta$ found in the midplane regions of GRMHD simulations is shown by
the shaded region in gray.

In the limit of high plasma $\beta$, irrespective of the nature of the
turbulent cascade, strong ion Landau damping causes the majority of the heat
to be channeled into thermal energy of the ions. On the other hand, in the
limit of low plasma $\beta$, i.e., when the plasma is magnetically dominated,
the Alfven-wave speeds are relatively high. Therefore, the population of ions
available to resonate with the Alfven waves significantly drops and almost all
of the Alfven wave energy goes into heating the electrons instead. The ions,
in turn, only heat up from the turbulent energy that resides in the
compressible modes. This is the reason why, at the low-$\beta$ limit, the ion-
to-electron heating ratio becomes equal to $P_{\rm{comp}}/P_{\rm{AW}}$.

For comparison, we also consider the earlier results for the ion-to-electron
heating ratio by Howes (2010, hereafter H2010) who calculated analytically the
linear dissipation of a purely Alfvenic cascade. The ion-to-electron heating
ratio described by this model is given by

$\frac{Q_{i}}{Q_{e}}=c_{1}\frac{c_{2}+\beta^{p}}{c_{3}+\beta^{p}}\sqrt{\frac{m_{i}T_{i}}{m_{e}T_{e}}}\exp{(-1/\beta)}\;,$
(25)

where $c_{1}=0.92$, $c_{2}=1.6/(T_{i}/T_{e})$,
$c_{3}=18+5\log{(T_{i}/T_{e})}$, and $p=2-0.2\log{(T_{i}/T_{e})}$, and is also
shown in Figure 1. Because of the absence of the compressive modes, the
low-$\beta$ limits of these calculation channels all the dissipated energy to
the electrons. Moreover, the linear approximations inherent to that
calculation result in a somewhat stronger dependence of the heat partition on
the ion-to-electron temperature ratio. In the next section, we will discuss
how these assumptions impact the resulting ion-to-electron temperature ratio.

The above mentioned strong dependence of the heating fraction on the
composition of turbulent cascade ($P_{\rm{comp}}/P_{\rm{AW}}$) is quite
significant for the plasmas found in radiatively inefficient disks. GRMHD
simulations carried out in a standard and normal evolution (SANE)
configuration estimate a midplane plasma $\beta\sim 10$ in the inner accretion
flow and $\beta\lesssim 1$ in the magnetically dominated funnel (see Porth et
al., 2019). The same quantity for a simulation of a magnetically arrested disk
(MAD) lies in the range $2\leq\beta\leq 10$, even with the equatorial
accretion flow (Sadowski et al., 2015). As shown in Figure 1, in this range of
values for plasma $\beta$, the electron heating fraction shows an appreciable
dependence on the ratio of compressive to Alfven turbulent energy
$P_{\rm{comp}}/P_{\rm{AW}}$.

The ratio of the compressible to Alfvenic wave energy in MRI-driven turbulence
is not well understood. To date, there has been one set of shearing box
simulations of MRI-driven turbulence at the reduced MHD limit in a nearly
azimuthal magnetic field configuration, for which the energy injection rates
in the compressive and Alfvenic cascades have been computed (Kawazura et al.,
2022). This calculation shows that, in the regime of $0.1\leq\beta\leq 10$,
compressive waves carry twice as much energy as Alfven waves, almost
independent of plasma $\beta$ ($P_{\rm{comp}}/P_{\rm{AW}}=2$). We will use
this estimate of the composition of the cascade at driving scales as a
fiducial model but also explore the impact of this value on the thermodynamics
of the flow by allowing a variety of other $P_{\rm{comp}}/P_{\rm{AW}}$ values.

###  2.4 Electron Cooling

In a radiatively inefficient flow, the electrons cool primarily via
synchrotron and Bremsstrahlung radiation (see Yuan & Narayan, 2014). The
synchrotron emissivity ($j_{\nu}$) for temperatures above the trans-
relativistic regime can be written as (Mahadevan et al., 1996)

$j_{\nu}=\dfrac{n_{e}e^{2}}{\sqrt{3}cK_{2}(1/\Theta_{e})}\nu M(x_{M}),$ (26)

where $K_{2}$ is the modified Bessel function of the second kind and second
order, $x_{M}=2\nu/(3\nu_{b}\Theta^{2})$, and $\nu_{b}=eB/(2\pi m_{e}c)$ is
the cyclotron frequency. The functional form of $M(x_{M})$ is provided in
Mahadevan et al. (1996).

When the optical depth is low, the electron cooling rate, $\Lambda_{e}$, is
simply the emissivity integrated over all frequencies and is given by

$\begin{split}\Lambda_{e}&=\int_{0}^{\infty}j_{\nu}d\nu=\dfrac{9\sqrt{3}c\sigma_{T}}{128\pi^{3}}\left(n_{e}B^{2}\right)\left[\dfrac{\Theta_{e}^{4}}{K_{2}(1/\Theta_{e})}\right]\\\
&\qquad\qquad\times\int_{0}^{\infty}dx_{M}M(x_{M})x_{M},\end{split}$ (27)

where $\sigma_{T}$ is the Thomson cross section. We obtain the magnetic field
strength in Equation (27) from the plasma $\beta$ as $B^{2}=8\pi
nk_{B}T_{e}/\beta$. The integral in $x_{M}$ is computed numerically as

$C\equiv\int_{0}^{\infty}dx_{M}x_{M}M(x_{M})\approx 20.\;.$ (28)

The electron cooling rate via Bremsstrahlung processes has two contributions.
For ion-electron interactions, it is given by

$\Lambda_{B}^{ie}=1.48\times 10^{-22}\ n_{e}^{2}\ F(\Theta_{e}),$ (29)

where

$F(\Theta_{e})=\left\\{\begin{array}[]{ll}4\left(\dfrac{2\Theta_{e}}{\pi^{3}}\right)^{1/2}\times\left(1+1.781\
\Theta_{e}^{1.34}\right)&\quad\Theta_{e}<1\\\
\dfrac{9\Theta_{e}}{2\pi}\left\\{\ln(1.123\
\Theta_{e}+0.48)+1.5\right\\}&\quad\Theta_{e}>1\end{array}\right.,$ (30)

and for electron-electron interactions, it is written as

$\Lambda_{\rm{B}}^{\rm{ee}}(\Theta_{e})=\left\\{\begin{array}[]{ll}2.56\times
10^{-22}\ n_{e}^{2}\times\ \Theta_{e}^{3/2}\\\
\times\left(1+1.1\Theta_{e}+\Theta_{e}^{2}-1.25\Theta_{e}^{5/2}\right)&\quad\Theta_{e}<1\\\
3.40\times 10^{-22}\ n_{e}^{2}\\\ \times\
\Theta_{e}\left\\{\ln\left(1.123\Theta_{e}+1.28\right)\right\\}&\quad\Theta_{e}>1.\end{array}\right.$
(31)

We note that, while the rate of viscous heating is directly proportional to
the plasma density, the resultant energy transfer rates due to Coulomb
collisions, Bremsstrahlung cooling, and synchrotron cooling are proportional
to the square of the plasma density, introducing an overall scale which is
linear in plasma density into the equations (9) and (10). This is why the
overall mass accretion rate does not drop out of Equations (9) and (10) and
needs to be specified, unlike the case in adiabatic GRMHD simulations.

##  3 Solutions for Ion and Electron Temperatures

In this section, we solve numerically the energy equations for ions and
electrons (Eqs. 9 and 10), to obtain the radial dependence of their
temperatures. We adopt the fiducial setup for the ion-to-electron heating
ratio with $P_{\rm{comp}}/P_{\rm{AW}}=2$. In order to solve the system of
differential equations, we initialize the ions and electron temperatures at a
radial distance of $\sim 2\times 10^{5}GM_{\rm{BH}}/c^{2}$ , assigning a sub-
virial non-relativistic temperature to both species. In the following
subsections, we examine the impact on the solutions for the ion and electron
temperatures of the efficiency of turbulent and Coulomb heating, of radiative
cooling, as well as of the model parameters for the partition of energy
dissipation. We discuss the contribution of General Relativistic effects in
the Appendix.

###  3.1 Effect of Turbulent Heating

We first solve for the ion and electron temperatures in a flow with a mass
accretion rate $\dot{M}\ll 10^{-7}\dot{M}_{E}$. In this limit of extremely low
mass accretion rate, turbulent heating dominates over the effects of Coulomb
heating, while radiative cooling is negligible. We also set the strength of
the magnetic field in the plasma by choosing the fiducial value $\beta=5$ and
the radial velocity profile by setting its power-law index to $n_{R}=1.5$.

Figure 2 shows the radial profiles of the ion and electron temperatures, as
well as their ratio. As they drift inwards, both the ions and the electrons
quickly heat up from their initial sub-virial temperatures at the outer radial
boundary to the local virial temperatures. The ions then maintain a virial,
$r^{-1}$, profile all the way to the inner region of the accretion flow. On
the other hand, the electron temperature gradually shifts from one virial
profile to another with a lower normalization in the inner accretion flow.

Figure 2: Top: The ion (dashed) and electron (solid) temperatures in units of
the ion rest mass energy as a function of radial distance for a flow with
accretion rate $\dot{M}\ll 10^{-7}\dot{M}_{E}$, plasma $\beta=5$, and a radial
velocity profile index of $n_{R}=1.5$, in the absence of the radiative cooling
of the electrons. The dotted lines indicate approximate analytical solutions
for different values of the adiabatic indices $\gamma$. In the inner accretion
flow, where the electrons become relativistic, their temperature profile
shifts towards the relativistic ($\gamma_{e}=4/3$) solution. Bottom: The ion-
to-electron temperature ratio as a function of radial distance for the same
calculation.

In order to understand these solutions, it is helpful to examine an analytical
limit of the solutions to equations (9) and (10), where the effects of Coulomb
heating and radiative cooling are neglected and the turbulent heating rate is
approximated only to leading order, assuming that it is independent of the
temperature ratio $T_{i}/T_{e}$. We further simplify the equations by
eliminating the temperature dependence of the effective adiabatic indices
$\gamma_{i}$ and $\gamma_{e}$. In this limit, following the Appendix of Özel
et al. (2022), we write the ion temperature as

$\dfrac{k_{B}T_{i}}{m_{i}c^{2}}=\dfrac{\gamma_{i}-1}{1-(\gamma_{i}-1)\left(\dfrac{\beta+1}{\beta}\right)(2-n_{R})}\times\dfrac{f}{1+f}\times\dfrac{3}{2r},$
(32)

and the electron temperature as

$\dfrac{k_{B}T_{e}}{m_{i}c^{2}}=\dfrac{\gamma_{e}-1}{1-(\gamma_{e}-1)(2-n_{R})}\times\dfrac{1}{1+f}\times\dfrac{3}{2r},$
(33)

where we have introduced $f=Q_{i}/Q_{e}$ as the ratio of the ion-to-electron
heating rate due to the dissipation of turbulent energy. Because the ions
never reach relativistic temperatures even in the inner accretion flow, their
adiabatic index remains approximately $\gamma_{i}=5/3$ and their temperatures
follows a single virial profile. However, as the electrons drift towards the
black hole, their temperatures increase to relativistic values with
$k_{B}T_{e}/m_{e}c^{2}\geq 1$, and hence their adiabatic index evolves from
$\gamma_{e}=5/3$ in the outer regions to $\gamma_{e}=4/3$ inner regions of the
flow. This causes their temperatures to shift from one virial profile to
another that is approximately a factor of 2 lower. The temperature ratio of
the ion and electrons in the outer flow, where both species are non-
relativistic, is approximately equal to the ion-to-electron heating ratio,
i.e., $T_{i}/T_{e}\sim Q_{i}/Q_{e}$, whereas becomes two times larger in the
inner flow, where the electrons become relativistic. A similar result has been
noted by Sadowski et al. (2017) based on numerical simulations. We note,
however, that these authors attribute their result to the shallow density
profiles of their accretion flows, which renders adiabatic heating negligible.
In contrast, our analytic model demonstrates that the virial profiles of the
ion and electron temperatures are practically independent of the density
profile, which is controlled by the parameter $n_{R}$.

Figure 3: Top: Ion (dashed) and electron (solid) temperatures in units of the
ion rest mass energy as a function of radial distance, for plasma $\beta=5$
and $n_{R}=1.5$, at different accretion rates, neglecting the effects of
radiative cooling for the electrons. At the higher accretion rates shown,
Coulomb collisions cause the ion and electron temperatures to equilibrate at
large radii. Bottom: The ion-to-electron temperature ratio, $T_{i}/T_{e}$ as a
function of radial distance for the same calculation.

###  3.2 Coulomb Heating

Having established the scale-free virial nature of ion and electron
temperatures under turbulent heating, we now examine the effects of Coulomb
coupling between ions and electrons on the flow thermodynamics. Figure 3 shows
the temperatures at density scales corresponding to mass accretion rates of
$\dot{M}=10^{-7}\dot{M}_{E}\ \rm{and}\ 10^{-4}\dot{M}_{E}$ and also compares
with the solution previously calculated with $\dot{M}\ll 10^{-7}\dot{M}_{E}$.

At $\dot{M}=10^{-7}\dot{M}_{E}$, Coulomb collisions cause the ion and electron
temperatures to equilibrate at large radii. As the collision cross section
decreases with increasing temperature, the two species decouple inwards and
approach their individual virial solutions set by turbulent heating. Upon
further increasing the accretion rate to $\dot{M}=10^{-4}\dot{M}_{E}$, and
consequently the density scale, Coulomb coupling becomes stronger. As a
result, the ions and electrons remain in thermal equilibrium at even smaller
radii. However, even at such accretion rates, the effects of Coulomb coupling
fade away as the gas drifts inwards, with the ion and electron temperatures
approaching the limit of their scale-free solutions determined by turbulent
heating in the inner flow.

Figure 4: Top: Ion (dashed blue) and electron (solid blue) temperatures in
units of ion rest mass energy as a function of radial distance, for a mass
accretion rate of $\dot{M}=10^{-4}\dot{M}_{E}$, plasma $\beta=5$, $n_{R}=1.5$,
and neglecting radiative cooling effects. Black solid line shows the electron
temperature profile if electrons were heated purely through Coulomb collisions
with the ions. In the absence of direct heating from turbulence, the electron
temperature in the inner accretion flow increases only due to adiabatic
compressional heating, as shown in the red dashed line. Bottom: The ion-to-
electron temperature ratio, $T_{i}/T_{e}$ as a function of radial distance for
the same calculation.

The above analysis of the relative importance of turbulent and Coulomb heating
indicates that, at density scales relevant for radiatively inefficient
accretion flows, turbulent heating almost completely overwhelms the effects of
Coulomb interactions in the inner regions of the accretion flow. While we will
examine the effects of radiative cooling in a later subsection, it is
instructive to consider certain implications of the scale-free turbulent
heating of electrons.

The initial models of radiatively inefficient accretion flows (see Narayan &
Yi, 1995b) considered the electrons to be heated solely via Coulomb coupling
with the ions, without any of the turbulent energy in the plasma being
channeled directly into the electrons. We compare such a system with our set-
up of direct electron heating in Figure 4, where we show the ion and electron
temperatures for a system where the electron heating is purely through Coulomb
collisions with the ions. We also compare it with the direct heating model
described in this paper, i.e., where a fraction of the turbulent energy is
directly channeled into the electrons. The ion temperatures expectedly follow
virial profiles for both these cases since the channel for ion heating remains
the same. However, the electron temperatures are no longer virial, when
electron heating is purely through Coulomb collisions. Instead, when the
temperature increases to the point that Coulomb interactions become
negligible, electrons experience pure compressional heating ($T_{e}\sim
n_{e}^{\gamma-1}$, shown in the dotted line in Figure 4), which results in a
significantly shallower radial profile. The consequence is an ion-to-electron
temperature ratio that increases rapidly with decreasing radius to values
$\sim 100$. This comparison demonstrates that, when microphysical plasma
phenomena channel a finite amount of the turbulent wave energy into the
electrons, this introduces substantial changes to the nature of the solution
for the electron temperature compared to a setting where electrons only
receive energy via Coulomb heating.

###  3.3 Radiative Electron Cooling

Having established the relative importance of particle heating via turbulent
heating and Coulomb collisions, we now turn to examining the effects of
electron cooling on the thermodynamics of the plasma. As described in §2.4, we
include the effects of Bremsstrahlung and synchrotron cooling of the electrons
in evolving the electron thermodynamics in the accretion disk. The plasma
cooling rates due to both ion-electron and electron-electron Bremsstrahlung is
proportional the square of the number density of the plasma
($\Lambda_{\rm{b}}\propto n_{e}^{2}$). On the other hand, the synchrotron
cooling rate scale as $\Lambda_{e}\sim n_{e}B^{2}$ in our regime of interest.
Because we set the local magnetic field to be proportional to the local
thermal pressure using the plasma $\beta$ parametrization, the synchrotron
cooling rate effectively scales with $n_{e}^{2}$, which is the same as
Bremsstrahlung cooling and Coulomb heating (see appendix of Satapathy et al.,
2022).

Figure 5: Top: Ion (dashed) and Electron (solid) temperatures, in units of ion
rest mass energy, calculated including the effects of radiative cooling for
mass accretion rates of $\dot{M}=10^{-7}\dot{M}_{E}$ (red), $\dot{M}\ll
10^{-7}\dot{M}_{E}$ (black dotted), and $\dot{M}=10^{-4}\dot{M}_{E}$ (green).
The plasma $\beta$ is set to $\beta=5$ and the radial velocity profile is set
by $n_{R}=1.5$. Bottom: The ion-to-electron temperature ratio, $T_{i}/T_{e}$
for the above cases.

Figure 5 shows the effects of radiative cooling processes at different
accretion rates on the electron temperature and the ion-to-electron
temperature ratio in the inner accretion disk. At an accretion rate of
$10^{-7}\dot{M}_{E}$, radiative cooling is inefficient and the electron
temperatures are equal to the case with $\dot{M}\ll 10^{-7}\dot{M}_{E}$, for
which radiative processes have been neglected. Only at higher accretion rates,
such as $\dot{M}=10^{-4}\dot{M}_{E}$, there is a small drop in the electron
temperature. It is important to emphasize here, however, that because of the
particular dependence of the cooling rates on the square of the density, the
electron temperatures retain their virial profiles.

The effects of synchrotron cooling also depend significantly on plasma
$\beta$, which sets the strength of the magnetic field. However, since the
partition of turbulent heating between ions and electrons also depends
strongly on plasma $\beta$, we will address the combined effect in the
following subsection.

###  3.4 Heat Partition Model

In the previous subsections, we established the near-virial nature of the ion
and electron temperature profiles in the inner accretion flow and addressed
their weak dependence on the efficiency of Coulomb collisions and radiative
cooling processes. Because the ions and the electrons receive their heat
mostly from turbulent dissipation in the accretion flow, the microphysical
processes that partition the turbulent energy into the individual species are
extremely important in determining their temperature ratios. As discussed in
§2.3.2 and shown in Figure 1, the partition of heat in the regime of plasma
$\beta\leq 10$ depends strongly on the composition of the turbulent cascade,
which is not well understood for MRI-driven turbulence.

Figure 6: The ratio of the electron temperature to the total gas temperature
$T_{e}/(T_{e}+T_{i})$ in the inner region of the accretion flow (at
$r=10GM/c^{2}$) as a function of plasma $\beta$, computed using different
ratios of the compressive-to-Alfvenic power ($P_{\rm comp}/P_{A}$). The cyan
curve corresponds to the earlier analytic calculation for Alfvenic turbulence
by Howes (2010). The shaded region for each curve indicates the range of
temperature ratios predicted for different mass accretion rates in the range
$10^{-7}\leq\dot{M}/\dot{M}_{\rm E}\leq 10^{-5}$. In all calculations, we have
set the radial velocity profile to $n_{R}=1.5$. The dotted lines represent the
empirical relation by Mościbrodzka et al. (2016) that has been used in the
literature, for different values of the parameter $R_{\rm high}$.

In this subsection, we study the dependence of the electron temperatures on
the composition of the turbulent cascades, for the regime of interest in
plasma $\beta$. In particular, we will focus on the asymptotic temperature
ratio $T_{e}/(T_{e}+T_{i})$ in the innermost regions of the accretion flow,
which measures the relative contribution of the electrons to the total gas
pressure. Figure 6, shows this ratio calculated at $r=10\;GM/c^{2}$, as a
function of plasma $\beta$, for a range of mass accretion rates of
$10^{-7}\leq\dot{M}/\dot{M}_{E}\leq 10^{-5}$, and for different ratios of the
compressive-to-Alfvenic wave power.

The widths of the shaded regions in Figure 6 represent the dependence of the
temperature ratio on the accretion rate, which is determined primarily by the
efficiency of synchrotron cooling. At low plasma $\beta$, where the magnetic
field is stronger, synchrotron cooling is very efficient and introduces a
factor of $\sim 2$ range of temperature ratios for the accretion rates we
consider here, as discussed in §2.4. In the opposite limit of high plasma
$\beta$, synchrotron cooling becomes inefficient and the temperature ratio in
the inner accretion flow becomes practically independent of accretion rate.

The dependence of the temperature ratios on plasma $\beta$ simply reflects the
dependence of heat partition between ions and electrons in the turbulent
cascade shown in Figure 1. The ratio of compressive-to-Alfvenic wave power
affects primarily the temperature ratio in the low-$\beta$ regime, for which
the ion-to-electron heating ratio asymptotes to $P_{\rm comp}/P_{\rm A}$, for
reasons discussed earlier. In order for the turbulent cascades to heat
predominantly the electrons and not the ions, i.e., leading to $T_{i}<T_{e}$,
the MRI-driven turbulence would need to be purely Alfvenic.

For the latter case of purely Alfvenic turbulence, Figure 6 also compares the
temperature ratios obtained using the results of the numerical model of
Alfvenic cascades (K2020 with $P_{\rm{comp}}/P_{A}=0$) to those using the
analytic calculation of Howes (2010). The nonlinear effects captured in the
numerical calculation of the cascades lead to smaller electron temperatures,
in the intermediate plasma $\beta$ range but to larger electron temperatures
in the very high plasma $\beta$ regime compared to the earlier studies.

###  3.5 Empirical Relations for Temperature Ratios

In earlier calculations of the observational properties of radiatively
inefficient flows based on GRMHD simulations, the ratio of the ion-to-electron
temperatures have often been prescribed in post-processing, using different
empirical relations that depend on plasma $\beta$ (see, e.g., Chan et al.,
2015; Mościbrodzka et al., 2016). In particular, the empirical relation

$R\equiv\dfrac{T_{i}}{T_{e}}=R_{\rm{high}}\dfrac{\beta^{2}}{1+\beta^{2}}+R_{\rm{low}}\dfrac{1}{1+\beta^{2}},$
(34)

has been used in comparing GRMHD model predictions to EHT observations (see
Akiyama et al., 2022b). Here $R_{\rm{high}}$ is the temperature ratio in the
limit of high plasma $\beta\gg 1$, and $R_{\rm{low}}$ is the temperature ratio
in the limit of low plasma $\beta\ll 1$.

Figure 6 compares this empirical relation to our calculation of the
temperature ratio. For the empirical ratio, we have set $R_{\rm{low}}=1$, as
has been typically adopted. We find that, at the high plasma $\beta$ limit,
the asymptotic temperature ratio corresponds to $R_{\rm high}\simeq 40$.
However, the functional dependence of the temperature ratio on plasma $\beta$
is not captured by empirical relation that is typically employed. Moreover,
the low plasma $\beta$ limit given by $R_{\rm low}$ does not capture the
dependence of the temperature ratio on the composition of the turbulent
cascade.

Our model of the accretion flow allows us to generate a new, physically
motivated empirical relation for the ion-to-electron temperature ratio. Taking
the ratio of equations (32) and (33), we obtain

$\dfrac{T_{i}}{T_{e}}=\left(\dfrac{\gamma_{i}^{\rm{eff}}-1}{\gamma_{e}-1}\right)\dfrac{1-(\gamma_{e}-1)(2-n_{R})}{1-(\gamma_{i}^{\rm{eff}}-1)(2-n_{R})\left(\frac{\beta+1}{\beta}\right)}\left(\frac{Q_{i}}{Q_{e}}\right)\;.$
(35)

Suppressing the weak dependence on temperature of the partition of heat
between ion and electrons in equation (24), we write

$\frac{Q_{i}}{Q_{e}}=\dfrac{35}{1+(\beta/15)^{-1.4}}+\dfrac{P_{\rm{comp}}}{P_{\rm{AW}}}.$
(36)

We estimate the effective adiabatic index of the ions using equation (14) to
write

$\gamma_{i}^{\rm{eff}}-1=\left(\frac{1}{\gamma_{i}-1}+\dfrac{1}{\beta}\right)^{-1}\;.$
(37)

Because the ions remain subrelativistic throughout the flow, we set
$\gamma_{i}=5/3$.

As discussed above, the temperature of the electrons and, hence, their
adiabatic index, depends on the value of plasma $\beta$. At low values of
plasma $\beta$, the electrons get to sufficiently high tempeartures to become
ultra relativistic. In this limit, their adiabatic index is
$\lim_{\beta\rightarrow 0}\gamma_{e}=4/3$. In the regime of high plasma
$\beta$, the electron temperature is significantly lower and their adiabatic
index increases towards the non-relativistic value of 5/3. In principle, one
could use expression (15) for the temperature dependence of the adiabatic
index and solve the implicit equation for the electron temperature. For
computational efficiency, we have devised an empirical relation for the
adiabatic index of the electrons in the inner accretion flow using the form

$\gamma_{e}=\frac{4}{3}+\kappa\dfrac{(\beta/\beta_{\rm{break}})^{2}}{1+(\beta/\beta_{\rm{break}})^{2}}\;.$
(38)

We have then set $\kappa=0.15$ and $\beta_{\rm{break}}=10$ in order to
reproduced the behavior seen in the detailed solutions of our accretion model.

Figure 7: The ratio of the electron temperature to the total gas temperature
for $\dot{M}\leq 10^{-7}\dot{M}_{\rm E}$ in the inner region of a radiatively
inefficient accretion flow ($r=10GM/c^{2}$), as a function of plasma $\beta$,
calculated using our covariant, semi-analytic model, for different ratios of
the compressive-to-Alfvenic power ($P_{\rm comp}/P_{\rm A}$). The dashed lines
correspond to the empirical relation (35).

Figure 7 compares the ratio of the electron-to-total gas temperatures
calculated using our covariant semi-analytical model of the accretion flow to
the empirical relation (35), using expressions (36)–(38) for its parameters.
For this comparison, we have set the velocity profile parameter to $n_{R}=1.5$
and neglected the effects of radiative cooling. The latter are expected to be
relatively unimportant for the accretion flows around the two prime imaging
targets of the EHT, Sgr A* and M87.

The empirical relation (35) can be easily implemented in a calculation of
observables based on postprocessing of GRMHD simulations. The $n_{R}$ velocity
profile parameter can be inferred empirically from the time- and azimuthally-
averaged velocity (or density) profile in the simulation. Finally, the ratio
of the compressive-to-Alfvenic wave power can be either set to the value
inferred by Kawazura et al. (2022) or allowed to be a free parameter to be
inferred observationally.

##  4 Discussion

Recent studies of rarefied plasmas, such as those that exist in low-luminosity
accretion flows, have uncovered particle heating channels that affect their
thermodynamic properties. In particular, turbulent cascades of compressive and
Alfvenic waves channel heat directly into ions and electrons, with the
partition depending on the ratio of power in these waves and the plasma
$\beta$ (Kawazura et al., 2020).

In this paper, we explored the effect of turbulent heating on the temperature
profiles of ions and electrons using a covariant analytic model. We
implemented the effects of compressive and Alfvenic dissipation, the exchange
of energy between the two species through Coulomb collisions, compressional
heating, as well as radiative cooling for the electrons. We demonstrated that
both the electrons and the ions follow nearly parallel radial profiles with a
temperature ratio that is determined by three properties of the plasma: the
$\beta$ parameter, the ratio of turbulent energy residing in compressive and
Alfvenic cascades, the accretion rate that sets the efficiency of radiative
cooling. In the inner accretion flows for the conditions that are found in
radiatively inefficient accretion flows, this ratio ranges between $\sim 5$
and $\sim 40$ when these processes are considered.

There exist some additional dissipative processes not considered in the
present work that can differentially channel energy into the electrons or the
ions and, therefore, impact their temperature profiles and ratio. Magnetic
reconnection injects energy into both species (Ball et al., 2016, 2018; Rowan
et al., 2017, 2019), but the partition of heat $Q_{i}/Q_{e}$ can increase or
decrease with plasma $\beta$ depending on the strength of the guide field
(Rowan et al., 2019). Moreover, reconnection can accelerate electrons to large
Lorentz factors well above the thermal distribution. The impact of this
channel in radiatively inefficient flows depends on the fraction of internal
energy that is dissipated via magnetic reconnection, which is not known. A
second dissipative process arises from the presence of velocity-space
instabilities in weakly collisional plasmas (Sironi & Narayan, 2015; Sharma et
al., 2007; Kunz et al., 2018). In this channel, the partition can differ from
the pressure-isotropic case for plasmas that are near the threshold of mirror
and firehose instabilities. The relevance of this process for plasmas in
radiatively inefficient flows needs to be explored.

The temperature profiles in the plasma, especially those of the electrons,
connect directly to the observable properties of accreting black holes, such
as their spectra, variability, and images. Our calculations allow us to devise
an empirical relation for the ion-to-electron temperatures that depend not
only on all plasma parameters known to be relevant for particle heating but
also and on the global flow properties that determine compressional heating.
This empirical relation is easy to implement into the outputs of global GRMHD
simulations to calculate observational signatures.

A key parameter that controls the ion-to-electron temperature ratio is the
partition of turbulent energy injected into compressive and Alfvenic cascades.
Kawazura et al. (2022) provided a first estimate of the composition of a
turbulent cascade in an accretion flow driven by the MRI in the presence of a
strong azimuthal magnetic field, which we use in our calculations. Further
simulations are needed to quantify this partition under a broader range of
conditions.

As observations of radiatively inefficient flows have evolved from primarily
acquiring spectral data (see Yuan & Narayan, 2014, for a review) to horizon-
scale imaging and polarimetric observations with very long baseline
interferometry (Event Horizon Telescope Collaboration et al., 2019a; Event
Horizon Telescope Collaboration et al., 2021; Akiyama et al., 2022a), it is
becoming increasingly possible to probe plasma processes directly (see, e.g.,
Xie et al., 2023). Improved simulations that incorporate the new physical
effects will allow us to make more predictive models of black hole
environments. Conversely, we can use the observations that directly probe the
densities, temperatures, and magnetic fields of accretion flows to guide and
constrain the studies of microphysics of turbulent rarefied plasmas.

The authors acknowledge support from the NASA ATP award 80NSSC20K0521 and NSF
PIRE award OISE-1743747. We thank K. G. Klein and C.-K. Chan for useful
discussions.

##  Appendix A General Relativistic and Inner Boundary Effects

In this appendix, we examine the role of general relativistic effects related
to the black-hole spin and the viscous stresses at the innermost stable
circular orbit (ISCO) in determining the global temperature profile of the
plasma species. Our model for the radial velocity of the plasma only considers
the centrifugally supported flow outside the ISCO (eq. (11)) and is not
influenced by the black-hole spin. The only quantity affected by the spin is
the location of the ISCO, whereas the plasma heating rate, $Q$, depends on
both the location of the ISCO and the viscous stresses there (eq. (23)).

Figure 8: Top: The ion temperature in units of ion rest mass energy as a
function of radial distance, computed for a non-spinning black-hole (dashed
lines), and a black-hole with a spin $a=0.9$ (solid lines). The lines shown in
green correspond to flows with vanishing viscous stresses at the ISCO
($\lambda=1$), while the opposite extreme corresponds to $\lambda=0$. In all
calculations, we have set the plasma parameter to $\beta=5$, the radial
velocity profile to $n_{R}=1.5$, and the mass accretion rate set to
$\dot{M}\ll 10^{-7}\dot{M}_{E}$. Bottom: The ion-to-electron temperature
ratio, $T_{i}/T_{e}$.

Figure 8 shows the ion temperatures and the ion-to-electron temperature ratios
near the ISCO, for different values of the black-hole spin and the parameter
$\lambda$ that sets the stresses at the ISCO; all other model parameters are
fixed at their fiducial values, i.e., plasma $\beta=5$ and $n_{R}=1.5$. We
also ignore the contributions of electron cooling and Coulomb heating, which
are negligible in the inner accretion flow.

As the specific angular momentum of the flow at the ISCO increases towards its
local Keplerian value, stresses and the corresponding turbulent heating rate
of the plasma decreases substantially, (see eq. (23)). This causes a drop in
the overall temperature of the ions by a factor of $\sim 3$. However, the
effect on the ion-to-electron temperature ratio of changing the black-hole
spin or the stresses at the ISCO is only marginal.

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