# Nonthermal emission from the plunging region: a model for the high-energy
tail of black hole X-ray binary soft states

Amelia M. Hankla,1,2 Nicolas Scepi,1 Jason Dexter1,3
1JILA, University of Colorado and National Institute of Standards and
Technology, 440 UCB, Boulder, CO 80309-0440, USA
2Department of Physics, University of Colorado, 390 UCB, Boulder, CO
80309-0390, USA
3Department of Astrophysical and Planetary Sciences, University of Colorado,
391 UCB, Boulder, CO 80309, USA E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Science Foundation Graduate Research Fellow

(Last updated ; in original form )

###### Abstract

X-ray binaries exhibit a soft spectral state comprising thermal blackbody
emission at 1 keV and a power-law tail above 10 keV. Empirical models fit the
high-energy power-law tail to radiation from a nonthermal electron
distribution, but the physical location of the nonthermal electrons and the
reason for their power-law index and high-energy cut-off are still largely
unknown. Here, we propose that the nonthermal electrons originate from within
the black hole’s innermost stable circular orbit (the “plunging region”).
Using an analytic model for the plunging region dynamics and electron
distribution function properties from particle-in-cell simulations, we outline
a steady-state model that can reproduce the observed spectral features. In
particular, our model reproduces photon indices of $\Gamma\gtrsim 2$ and
power-law luminosities on the order of a few percent of the disk luminosity
for strong magnetic fields, consistent with observations of the soft state.
Because the emission originates so close to the black hole, we predict that
the power-law luminosity should strongly depend on the system inclination
angle and black hole spin. This model could be extended to the power-law tails
observed above 400 keV in the hard state of X-ray binaries.

###### keywords:

X-rays: binaries – accretion – black hole physics – acceleration of particles

††pubyear: 2022††pagerange: Nonthermal emission from the plunging region: a
model for the high-energy tail of black hole X-ray binary soft
states–Nonthermal emission from the plunging region: a model for the high-
energy tail of black hole X-ray binary soft states

##  1 Introduction

Observations of accreting black hole X-ray binary systems (XRBs) show two main
X-ray spectral states. The hard state has an inverted power-law photon
spectrum $N(E)\propto E^{-\Gamma}$ with a spectral index $\Gamma\lesssim 2$ at
10 keV, peaking in the $100-200$ keV range. The soft state has a thermal
blackbody component peaking around 1 keV and a high-energy power-law tail
component from 10 keV to 1 MeV. This hard X-ray to soft gamma-ray power law
has a spectral index $\Gamma\gtrsim 2$ and no observed cut-off (McConnell et
al., 2002; Zdziarski & Gierliński, 2004; Remillard & McClintock, 2006).

Models of the soft spectral state usually involve a thin accretion disk that
produces the thermal emission and a hot, optically-thin gas that produces the
high-energy power-law tail. The thin accretion disk (Shakura & Sunyaev, 1973;
Novikov & Thorne, 1973) models the soft state’s blackbody emission well.
However, the origin of the power-law tail is less well-understood. Most models
for the power-law tail assume a population of nonthermal electrons that are
somehow accelerated to high energies, presumably by shocks or magnetic
reconnection. The hybrid thermal/nonthermal electron distribution function
fits the soft state spectrum up to 600 keV well (Zdziarski et al., 2001;
Gierliński et al., 1999), but the electron distribution parameters are
determined by the best fit to observational data rather than ab initio theory.

The spatial location of the hot, optically-thin gas and thus the nonthermal
electrons is unknown. Hybrid electron distribution models usually assume that
the nonthermal electrons originate from a jet or corona above a thin accretion
disk extending down to the innermost stable circular orbit (ISCO; Zdziarski &
Gierliński, 2004). Alternatively, the nonthermal electrons could come from gas
within the ISCO. In the “plunging region,” particle orbits transition from
predominantly circular to predominantly radial. The ISCO serves as an
artificial boundary condition in early thin disk models, which assume that the
stress there goes to zero and no light is emitted from within the plunging
region (Shakura & Sunyaev, 1973; Novikov & Thorne, 1973). Later thin disk
models find that stress at the ISCO can lead to extra dissipation, which is
often parameterized as an additional accretion efficiency $\Delta\epsilon$
(Agol & Krolik, 2000; Gammie, 1999). Magnetohydrodynamic (MHD) and general
relavitistic MHD (GRMHD) simulations have measured additional dissipation
within the ISCO on the order of $\Delta\epsilon\gtrsim 0.2$ for weakly
magnetized disks and up to twice the Novikov-Thorne efficiency for strongly
magnetized disks (Armitage et al., 2001; Shafee et al., 2008; Noble et al.,
2009; Penna et al., 2010; Avara et al., 2016). This extra dissipation could
come from magnetic reconnection and turbulence occurring in the plunging
region.

Extra dissipation from within the plunging region could cause the soft state’s
high-energy power-law tail. In a purely thermal model, the gas temperature
rises so rapidly towards the event horizon that the Wien peaks sum to a power
law (Zhu et al., 2012). This model could also explain high-frequency quasi-
periodic oscillations in the steep power law state (Dexter & Blaes, 2014).
These models assume that the electron and proton gas are in thermal
equilibrium. However, shorter infall times in the plunging region can mean
electrons and protons no longer have time to thermalize. Estimates of the
electron-proton thermalization time in GRMHD simulations become shorter than
the infall time close to the black hole, suggesting that electrons and protons
may decouple into a two-temperature plasma within the ISCO (Zhu et al., 2012).

In this work, we present a model for the soft spectral state’s high-energy
power-law tail using nonthermal electrons in the plunging region. We outline
the parameter space where electron-proton decoupling should occur and
demonstrate that the plunging region conditions are conducive to accelerating
electrons to nonthermal energies (Section 2). We then propose a model for the
hybrid thermal/nonthermal electron distribution in the plunging region based
on particle acceleration prescriptions from particle-in-cell simulations
(Section 3). We show that our model can reproduce observational
characteristics of a high-energy power-law tail for highly-magnetized
parameters, recover the transition to thermal-dominated spectra at lower
magnetizations, and predict trends with inclination angle and black hole spin
that are consistent with current observations (Section 4). We end by
discussing our model assumptions and its general applicability to XRBs
(Section 5) and summarizing our findings (Section 6).

##  2 Dynamical Model for the Plunging Region

Within the ISCO, the accretion flow cannot be described as a thin disk.
Particle orbits transition from mostly circular outside the ISCO to mostly
radial inside the ISCO. The fast infall times in the plunging region mean that
the usual assumptions of a collisional, single-temperature plasma need to be
revisited. In this section, we will motivate our treatment of the plunging
region as a two-temperature, highly-magnetized region with the potential to
continuously accelerate particles to nonthermal energies.

We will use Gammie (1999)’s plunging region model as a backdrop for our model.
Gammie (1999) provides radial profiles of the gas number density $n_{0}$,
four-velocity $u^{\mu}$, and radial magnetic field strength $B_{r}$ that agree
well with numerical simulations (Penna et al., 2010). The model assumes a one-
dimensional, steady-state MHD inflow in the plunging region, parameterized by
black hole spin $a$, mass $M$, and the $F_{\theta\phi}$ component of the
Maxwell tensor, which reduces to $-r^{2}B_{r}$ at the midplane in the
nonrelativistic limit. Throughout, we assume an Eddington factor
$\eta=0.1=L_{\rm Edd}/(\dot{M}_{\rm Edd}c^{2})$, where $L_{\rm Edd}$ and
$\dot{M}_{\rm Edd}$ are the Eddington luminosity and accretion rate,
respectively. We avoid numerical artefacts due to the boundary condition
$v^{r}=0$ at the ISCO by starting our solutions at $r_{\rm ISCO}-0.1r_{g}$. We
scale the model to physical units using a black hole mass of $10M_{\odot}$ and
a density scale height $H/r=1$, the latter choice motivated by Section 2.1’s
observation that ions will not collide with electrons in the plunging region.

For our fiducial model, we use values $a=0.95$ and $F_{\theta\phi}=6.0$, which
corresponds to an accretion efficiency $\Delta\epsilon=1.04$. For all models,
we assume that $\Delta\epsilon$ is uniformly distributed throughout the volume
within the ISCO and that this heating comes from magnetic reconnection
processes. Representative quantities of the fiducial background are plotted in
Figure 1. For these parameters, the magnetic field has a strength on the order
of $10^{8}$ G. We assume a virial ion temperature $T_{i}=2GMm_{p}/5k_{B}r$,
which gives mildly relativistic ions, and calculate the ion beta
$\beta_{i}=8\pi n_{0}k_{B}T_{i}/B_{r}^{2}$. The electron scattering optical
depth $\tau_{\rm es}=n_{0}\sigma_{T}H$ is always slightly greater than 1. The
values of $B_{r}$, $\tau$, and $\beta_{i}$ vary by a factor of $\sim 3$ from
the ISCO to the event horizon.

Figure 1: Properties of the plunging region calculated from the Gammie (1999)
background for fiducial parameters $a=0.95$ and $\Delta\epsilon=1.04$. From
left to right: radial magnetic field strength $B_{r}$ (left axis) and the
fluid’s Lorentz factor (right axis), dimensionless ion temperature
$\theta_{i}=k_{B}T_{i}/m_{p}c^{2}$ (left axis) and ratio of gas-to-magnetic
pressure $\beta_{i}=8\pi n_{0}k_{B}T_{i}/B_{r}^{2}$ (right axis), electron
scattering optical depth $\tau_{\rm es}$, and cold ion magnetization
$\sigma_{i}$. Here, $B_{0}=10^{8}$ G. The plunging region extends from the
ISCO at $1.94r_{g}$ to the event horizon at $1.31r_{g}$ in the midplane.

###  2.1 Development of a Two-temperature Plasma

Analytic models of a thin disk that permit non-zero stress at the ISCO predict
a factor of ten increase in the gas temperature between the ISCO and the event
horizon (Agol & Krolik, 2000). Single-temperature GRMHD simulations find a
similar increase (Zhu et al., 2012) due to the flow becoming effectively
optically thin. The rapid rise of temperature means that the timescale for
electrons and protons to reach thermal equilibrium increases as well. This
electron-ion thermalization timescale $t_{\rm th}^{ei}$ is given by

$\frac{dT_{e}}{dt}=\frac{T_{i}-T_{e}}{t_{\rm th}^{ei}}$ (1)

where $t_{\rm th}^{ei}\equiv 1/\nu_{\rm th}$. For hot protons heating
electrons, the thermalization frequency from Stepney & Guilbert (1983) reads:

$\displaystyle\nu_{\rm th}$
$\displaystyle=\frac{m_{e}}{m_{p}}\frac{n_{0}\sigma_{T}c\ln\Lambda}{K_{2}(1/\theta_{e})K_{2}(1/\theta_{i})}$
(2)
$\displaystyle\left[\frac{2(\theta_{e}+\theta_{i})^{2}+1}{\theta_{e}+\theta_{i}}K_{1}\left(\frac{\theta_{e}+\theta_{i}}{\theta_{e}\theta_{i}}\right)+2K_{0}\left(\frac{\theta_{e}+\theta_{i}}{\theta_{e}\theta_{i}}\right)\right]~{}{\rm
s^{-1}}$ (3) $\displaystyle=3.6\times
10^{-37}\frac{\left(m_{e}m_{p}\right)^{1/2}n_{0}\ln\Lambda}{\left(m_{p}T_{e}+m_{e}T_{i}\right)^{3/2}}~{}{\rm
s^{-1}},$ (4)

where $\theta_{e}=k_{B}T_{e}/m_{e}c^{2}$ and
$\theta_{i}=k_{B}T_{i}/m_{p}c^{2}$, $m_{e}$ and $m_{p}$ are electron and
proton masses, respectively. The last line takes the nonrelativistic limit
assuming $\theta_{e}\ll 1$ and $\theta_{i}\ll 1$ (Spitzer, 1956). Here,
$K_{0}(x)$, $K_{1}(x)$, and $K_{2}(x)$ are modified Bessel functions of the
second kind. Throughout, we set the Coulomb logarithm $\ln\Lambda=10$ and
employ cgs units.

In the plunging region, the thermalization timescale can become longer than
the accretion timescale. Zhu et al. (2012) finds that a weakly magnetized flow
cannot equilibrate within $3~{}r_{g}$ for black hole spin $a=0.7$. Therefore,
the assumption that electrons and protons couple efficiently no longer applies
within the ISCO. The inefficient Coulomb coupling means that protons cannot
cool, and should heat due to viscous and magnetic dissipation. Such increased
temperature and thus increased ion pressure support will puff up the plunging
region beyond a thin disk. The formation of a thick inner disk and a thin
outer disk was recently simulated for the first time (Liska et al., 2022). For
those model parameters, the thin outer disk with $H/r\sim 10^{-2}$ expanded to
$H/r\sim 0.3$ within the ISCO.

This decoupling motivates our subsequent treatment of the electrons and
protons in the plunging region as completely independent, with different
temperatures, as well as our assumption that $H/r\sim 1$. Indeed, Figure 2
shows that in our model, the thermalization time (black solid line) is longer
than the infall time (dashed horizontal line) for a wide range of accretion
efficiencies. The same hierarchy holds for all black hole spins discussed in
this work.

###  2.2 Nonthermal Particle Acceleration

Particle-in-cell (PIC) simulations of magnetic reconnection have shown that
efficient particle acceleration in an electron-ion plasma occurs for ion
magnetizations $\sigma_{i}\gtrsim 1$ (Werner et al., 2018). Using background
profiles for magnetic field and gas density from Gammie (1999), we will
demonstrate that efficient particle acceleration is possible inside the ISCO.
We calculate the ion magnetization

$\sigma_{i}=\frac{B_{0}^{2}}{4\pi n_{0}m_{p}c^{2}}$ (5)

using the definition from Werner et al. (2018). We set $B_{0}=B_{r}$ and
$n_{0}=\rho/m_{p}$, with the radial magnetic field $B_{r}(r)$ and mass density
$\rho(r)$ obtained from Gammie (1999). For fiducial model parameters, the ion
magnetization ranges from 0.2 to 3.0 (Figure 1). In PIC simulations of
magnetic reconnection with similar ion magnetizations, Werner et al. (2018)
finds that electrons are accelerated into power-law distribution with high-
energy cut-off $\gamma_{c}\sim 4\sigma_{e}$ (rising slowly with $\sigma_{i}$)
and power-law index $p$ prescribed by

$p=1.9+0.7/\sqrt{\sigma_{i}}.$ (6)

Werner et al. (2018) fixes $\theta_{i}=\sigma_{i}/200$ (i.e.
$\beta_{i}=4\theta_{i}/\sigma_{i}\to 0.02$), a factor of 10 to 100 times
colder than the equipartition ion temperature calculated above. PIC studies of
electron-ion turbulence (rather than magnetic reconnection) that set
$\beta_{i}=2/3$ and scanned $\theta_{i}\in[1/256,10]$ with
$\sigma_{i}\in[1/256,10]$ have found broadly similar power-law indices
(Zhdankin et al., 2019). The turbulent simulations in general find more
efficient particle acceleration, measuring $p\approx 2.8$ for
$\theta_{i}=1/16$ and $\sigma_{i}=0.2$ when Equation 6 would predict $p=3.5$.

With ion magnetizations close to unity, the plunging region can plausibly
accelerate electrons to nonthermal energies via turbulence or reconnection.
This acceleration could happen in a two-stage process, with sharp increases in
particle Lorentz factor when the particle crosses a current sheet and slower,
second-order Fermi-like acceleration that continues to increase particle
energy over time (Comisso & Sironi, 2019). For a magnetically-dominated pair
plasma, the current sheet acceleration time is on the order of $t_{\rm
acc}\sim(300/\sigma)\omega_{pe}^{-1}$ (Comisso & Sironi, 2019), where
$\omega_{pe}=\sqrt{4\pi e^{2}n_{0}/m_{e}}$ is the electron plasma frequency.
For the fiducial parameters $\Delta\epsilon=1.04$ and $a=0.95$, the Gammie
(1999) model yields $100\omega_{pe}^{-1}\sim 10^{-11}~{}{\rm s}$, much shorter
than infall times on the order of $10^{-4}~{}{\rm s}$ for a ten solar-mass
black hole and electron-ion thermalization times (Figure 2 orange line).
Therefore, we argue that the plunging region can accelerate electrons locally,
forming a power-law distribution that will not have time to be thermalized by
protons.

###  2.3 Electron-electron collisions

A pure power-law distribution of electrons will interact via Coulomb
collisions, eventually forming a pure thermal distribution if allowed to
evolve freely (Nayakshin & Melia, 1998). Lower-energy electrons thermalize
first, forming a hybrid thermal/nonthermal population at intermediate
evolution times. If nonthermal electrons are continuously injected, as Section
2.2 showed is possible in the plunging region, the electron distribution will
reach some equilibrium of nonthermal and thermal populations. We model this
hybrid distribution as a thermal Maxwell-Jüttner distribution up to a
transition Lorentz factor $\gamma_{1}$, where the distribution becomes a power
law. A nonthermal electron with Lorentz factor $\gamma$ will interact with the
thermal bulk on the energy loss timescale $t_{\rm coll}^{ee}(\gamma)\equiv
1/\nu_{\epsilon}^{ee}(\gamma)$. The energy loss rate is given by

$\nu_{\epsilon}^{ee}(\gamma)=\left[\psi(x^{ee})-\psi^{\prime}(x^{ee})\right]\frac{16\sqrt{\pi}e^{4}\ln\Lambda~{}n_{0}}{m_{e}^{2}c^{3}\beta^{3}},$
(7)

where $\beta=\sqrt{1-1/\gamma^{2}}$ and $\psi(x)$ is the lower incomplete
Gamma function (Spitzer, 1956). The kinetic ratio
$x^{ee}=\beta^{2}/(2\theta_{e})$. For an electron temperature of $10^{9}$ K, a
particle with a Lorentz factor of 2.0 has an energy loss timescale of
$10^{-6}$ s, about 0.002 times the infall timescale (Figure 2 purple). We
therefore cannot neglect electron-electron collisions, especially at lower
electron energies.

Though nonthermal electrons will also collide with thermal protons over the
corresponding electron-proton energy loss timescale, this timescale will be
larger than the thermalization timescale discussed in Section 2.1. We will
therefore ignore interactions between nonthermal electrons and thermal ions.

###  2.4 Importance of Radiative Cooling

High-energy electrons cool rapidly due to either synchrotron or inverse
Compton cooling. In our model, the cooling time $t_{\rm cool}(\gamma)$ is set
by either synchrotron or inverse Compton cooling, depending on the ratio
$U_{B}/U_{ph}$, where $U_{B}=B^{2}_{r}/8\pi$ is the magnetic energy density
and $U_{ph}=L/(4\pi c~{}r_{\rm ISCO}^{2})$ is the energy density in the
ambient photons, assumed to come from the disk such that $L=0.1~{}L_{\rm
Edd}$, where $L_{\rm Edd}=4\pi GMm_{p}c/\sigma_{T}=1.26\times 10^{39}~{}{\rm
erg/s}$. If $U_{B}/U_{ph}>1$, then the cooling time is set to the time it
takes for an electron to lose half its energy due to synchrotron radiation
(Rybicki & Lightman, 1979):

$t_{\rm sync}(\gamma)=\frac{5.1\times 10^{8}}{\gamma B_{r}^{2}}.$ (8)

In using the above equation, we assume that the particle is moving in gyro-
orbits around the dominant magnetic field $B_{r}$, with a pitch-angle
$\alpha=\pi/2$. If on the other hand $U_{B}/U_{ph}<1$, the cooling time is set
to the inverse Compton cooling time, assumed to take the form

$t_{\rm IC}(\gamma)=\frac{U_{B}}{U_{ph}}t_{\rm sync}(\gamma).$ (9)

Notice that Equation 8 and 9 are valid only for ultra-relativistic particles.
As such, we ignore cooling for electrons below a minimum Lorentz factor
$\gamma_{\rm min}=2$, since below this Lorentz factor the cooling rate
decreases rapidly.

The cooling times decrease for larger Lorentz factors, with values at the
minimum Lorentz factor two orders of magnitude smaller than the electron-
electron collision time, and five orders of magnitude smaller than the infall
timescale (Figure 2 magenta). The electrons in the plunging region therefore
have time to radiate before disappearing beyond the event horizon.

We show the relevant timescales of the plunging region in Figure 2 for models
with different black hole spin $a$ and accretion efficiencies $\Delta\epsilon$
resulting from different magnetic field strengths. At a radius of
$r=1.58~{}r_{g}$, all models satisfy the ordering

$t_{\rm accel}\ll t_{\rm cool}(\gamma)\ll t_{\rm coll}^{ee}(\gamma)\ll t_{\rm
infall}\lesssim t_{\rm therm}^{ei}$ (10)

for $\gamma\geq\gamma_{1}$. We will use this ordering to motivate our model
for the plunging region’s hybrid thermal/nonthermal electron distribution.

Figure 2: Timescales in the plunging region show that all physical processes
except for electron-ion collisions are fast compared to the infall time for a
wide range of accretion efficiencies $\Delta\epsilon$, consistent with the
hierarchy $t_{\rm accel}\ll t_{\rm cool}(\gamma)\ll t_{\rm
coll}^{ee}(\gamma)\ll t_{\rm infall}\lesssim t_{\rm therm}^{ei}$. Timescales
are shown at a radius $r=1.58r_{g}$ for black hole spin $a=0.95$. These
timescales are calculated for thermal ion and hybrid electron distributions
determined by the steady-state model described in Section 3.

##  3 Multizone Equilibrium Model

In this section, we present a model for the electron distribution in the
plunging region, motivated by the properties outlined in Section 2. We will
assume that nonthermal electrons are continuously injected by local processes
such as magnetic reconnection or turbulence, which then partially thermalize
via electron-electron collisions. We further assume that the hybrid
thermal/nonthermal electron distribution reaches a steady state at each
radius, matching the losses due to radiation with the heating due to magnetic
dissipation. Below, we describe the assumptions that allow us to fully
constrain this hybrid electron distribution’s form.

###  3.1 Initial Electron Distribution

Because the advection timescales are much longer than the particle
acceleration timescales (Figure 2), we assume that the initial (pre-
thermalization) electron population at each radius is a pure power-law
distribution: $f_{\rm PL,0}(\gamma)=A_{0}\gamma^{-p}$ for
$1\leq\gamma\leq\gamma_{2}$, where $\gamma_{2}$ is a high-energy cut-off. The
high-energy cut-off at each radius is set to

$\gamma_{2}(r)=\gamma_{c}(r)=4\sigma_{e}(r),$ (11)

where $\sigma_{e}=B_{r}^{2}(r)/(4\pi n_{0}m_{e}c^{2})$ is the (cold) electron
magnetization (Werner et al., 2018). The power-law index $p$ is also set by
Werner et al. (2018)’s prescription (Equation 6), increased by 1 to account
for cooling (Blumenthal & Gould, 1970). Requiring the distribution to be
normalized to $n_{0}$ sets the constant $A_{0}$. All together, the initial
distribution reads:

$\displaystyle f_{\rm PL,0}(\gamma)$
$\displaystyle=\frac{n_{0}(p-1)}{1-\gamma_{2}^{1-p}}~{}\gamma^{-p}$
$\displaystyle 1\leq\gamma\leq\gamma_{2}.$ (12)

Electron-electron Coulomb collisions will partially thermalize this initial
power-law distribution. Radial profiles for the high-energy cut-off
$\gamma_{2}$ and the power-law index $p$ in the fiducial model are given in
Figure 3.

Figure 3: Radial profiles of the electron distribution function model
properties for fiducial parameters $a=0.95$ and $\Delta\epsilon=1.04$. From
left to right: the high-energy Lorentz factor cut-off of the power law
$\gamma_{2}$, the electron power-law index, the thermal electron temperature
$\theta_{e}=k_{B}T_{e}/m_{e}c^{2}$, and the fraction of the total cooling rate
$Q_{-}$ from the power-law and thermal component. Vertical dashed line shows
the location of the half-light radius.

###  3.2 Steady-state Electron Distribution

Some of the injected power-law electrons will interact with each other via
Coulomb collisions, creating a thermal distribution below some Lorentz factor
$\gamma_{1}$ (Section 2.3). The resulting hybrid thermal/nonthermal electron
distribution continuously gains more nonthermal electrons due to the injection
process and loses nonthermal electrons to the thermal bulk, radiating energy.
We approximate these rapid processes with a steady-state electron distribution
comprising a Maxwell-Jüttner thermal distribution $f_{\rm MJ}$ and a power-law
distribution $f_{\rm PL}(\gamma)$ for $\gamma_{1}\leq\gamma\leq\gamma_{2}$.

The thermal electron distribution $f_{\rm MJ}$ is fully described by two
variables: its normalization and its temperature $T_{e}$. We fix the
normalization of the thermal distribution to the number density $n_{0}$,
assuming that the number of nonthermal electrons is negligible compared to the
thermal electrons. Explicitly, the thermal distribution reads

$f_{\rm
MJ}(\gamma,~{}\theta_{e},~{}n_{0})=n_{0}\frac{\gamma^{2}\beta}{\theta_{e}K_{2}(1/\theta_{e})}e^{-\gamma/\theta_{e}},$
(13)

where $K_{2}$ is the modified Bessel function of the second kind. The electron
temperature $T_{e}=\theta_{e}m_{e}c^{2}/k_{B}$ is set by the model assumptions
outlined below.

The nonthermal electrons are described as a power-law between a minimum
Lorentz factor $\gamma_{1}$ and a high-energy cut-off $\gamma_{2}$, with a
power-law index $p_{\rm eff}$ and a normalization $A_{\rm PL}$:

$\displaystyle f_{\rm PL}(\gamma)$ $\displaystyle=A_{\rm PL}\gamma^{-p_{\rm
eff}}$ $\displaystyle\gamma_{1}\leq\gamma\leq\gamma_{2}.$ (14)

We will now outline how our model determines the variables $\gamma_{1}$,
$A_{\rm PL}$, $p_{\rm eff}$, and $T_{e}$.

####  3.2.1 Determining $\gamma_{1}$

The minimum Lorentz factor $\gamma_{1}$ of the nonthermal distribution is set
at the energy where collisions and cooling balance. At $\gamma_{1}$, the
energy loss timescale $t_{\rm coll}^{ee}$ (Equation 7) equals the cooling time
$t_{\rm cool}$:

$t_{\rm coll}^{ee}(\gamma_{1})=t_{\rm cool}(\gamma_{1}).$ (15)

Here, the cooling time $t_{\rm cool}$ is set to the minimum of the synchrotron
and inverse Compton cooling times (Equations 8 and 9). We enforce the lower
bound $\gamma_{1,\rm min}=\gamma_{\rm min}=2.0$ to model the inefficiency of
cooling at low energies. We also require that $\gamma_{1}>\gamma_{\rm
threshold}(T_{e})$, where the threshold Lorentz factor $\gamma_{\rm
threshold}$ is where the energy loss frequency (Equation 7) is zero for a
given temperature. A power-law electron with a Lorentz factor below
$\gamma_{\rm threshold}$ would increase in energy due to collisions with the
thermal bulk. Imposing the requirement that the power-law start at Lorentz
factors greater than the threshold Lorentz factor means that $\gamma_{1}$ will
always be to the right of the thermal peak, thereby avoiding getting trapped
at solutions with $\gamma_{1}\to 1$.

Often, the cooling times are shorter than the energy loss times for all values
of Lorentz factor. In this case, collisions are never important for the
nonthermal electrons (Figure 2). However, it is still important to maintain a
thermal population of electrons to provide the bulk of the density in the
plunging region. When no solution to Equation 15 exists, we set
$\gamma_{1}=\max\left(\gamma_{1,{\rm threshold}},\gamma_{\rm min}\right)$.

####  3.2.2 Determining $A_{\rm PL}$

The power-law normalization $A_{\rm PL}$ is set by requiring continuity of the
thermal and nonthermal electrons at $\gamma_{1}$:

$f_{\rm MJ}(\gamma_{1})=f_{\rm PL}(\gamma_{1}).$ (16)

####  3.2.3 Determining $p_{\rm eff}$

The power-law index $p_{\rm eff}$ of the steady-state nonthermal distribution
is calculated by assuming that none of the highest-energy particles are
thermalized by Coulomb collisions; that is,

$\displaystyle f_{\rm PL,0}(\gamma_{2})=f_{\rm PL}(\gamma_{2}).$ (17)

We use an effective power-law index because high-energy particles collide more
slowly than lower-energy particles, resulting in a power-law index that
decreases over time (Nayakshin & Melia, 1998). Using the power-law index from
PIC simulations would underestimate the total power law. Our formulation
assumes that the initial power-law is continually replenished by in-situ
particle acceleration, so the impact of cooling is primarily in the radiation
rather than the dynamics. The difference between $p_{\rm eff}$ and $p$ is
largest close to the event horizon (Figure 3). Figure 4 shows how the final
power-law distribution (blue solid line) differs from the initial power-law
distribution (purple dashed line).

Figure 4: The effective power-law index $p_{\rm eff}$ is calculated from the
initial power-law distribution $f_{\rm PL,0}$ by assuming that both power laws
have the same value at the high-energy cut-off $\gamma_{2}$ (Equation 17).
This sample electron distribution was calculated for fiducial parameters close
to the event horizon at a radius $r=1.31~{}r_{g}$, where $p=3.3$ (purple
dashed line) and $p_{\rm eff}=0.6$ (blue solid line). For radii closer to the
ISCO, the difference between $f_{\rm PL,0}$ and $f_{\rm PL}$ is not so
pronounced. The thermal distribution is also shown for reference (green dotted
line).

Combining Eqns. 12 and 17 yields an expression for $p_{\rm eff}$:

$p_{\rm eff}=\frac{\log\left[f_{\rm PL,0}(\gamma_{2})/f_{\rm
MJ}(\gamma_{1})\right]}{\log\left[\gamma_{1}/\gamma_{2}\right]}.$ (18)

####  3.2.4 Determining $T_{e}$

The temperature $T_{e}$ of the thermal electrons is set by requiring a steady
state that balances energy lost to radiation with energy dissipated by
magnetic torques. We obtain the volume heating rate $Q_{+}$ from the accretion
efficiency $\Delta\epsilon$ given by the Gammie (1999) model, assuming that
dissipation occurs evenly throughout the plunging region’s volume:

$Q_{+}=\delta_{e}\frac{\Delta\epsilon~{}L_{\rm Edd}}{V_{\rm PR}},$ (19)

where $V_{\rm PR}=4\pi(r_{\rm ISCO}^{3}-r_{\rm EH}^{3})/3$ is the (Newtonian)
volume of the plunging region. The fraction of dissipated energy $\delta_{e}$
that goes into electrons is set at a constant value of 0.5, as motivated by
PIC simulations of the energy partition in an electron-ion plasma with
magnetizations close to 1 (Werner et al., 2018; Zhdankin et al., 2019). The
remainder $1-\delta_{e}$ fraction of dissipated energy heats the ions, which
do not radiate.

The total cooling rate $Q_{-}$ sums the bremsstrahlung cooling rate $Q_{-,\rm
MJ}$ from the thermal electrons with the inverse Compton or synchrotron
cooling rate from the nonthermal electrons. To account for additional
scattering, the nonthermal cooling rate $Q_{-,\rm PL}$ is set to the
nonthermal emitted power $\mathcal{P}_{-,\rm PL}$ decreased by a factor of
$\tau_{e}=n_{0}\sigma_{T}H=n_{0}\sigma_{T}(H/r)r$, such that $Q_{-,\rm
PL}=\mathcal{P}_{-,\rm PL}/\tau_{e}$. This additional factor estimates how
many of the synchrotron/IC photons would escape the plunging region without
additional scattering, which is important since $\tau_{e}\gtrsim 1$. This
choice is motivated by Monte Carlo simulations of an isotropic sphere of gas
with photons packets initialized evenly throughout the sphere. These
simulations show that about $1/\tau_{e}$ of the electrons escape with no
scattering. The remaining $1-1/\tau_{e}$ fraction of photons will Compton
scatter off of the thermal/nonthermal electrons, heat the gas, and re-energize
nonthermal electrons. The equation for steady-state thus reads:

$Q+=Q_{-}=Q_{-,\rm MJ}+Q_{-,\rm PL}.$ (20)

The nonthermal emissivity $\mathcal{P}_{-,\rm PL}$ is calculated by
integrating the Larmor formula for power emitted per electron over frequency
$\omega$, the power-law electron distribution $f_{\rm PL}(\gamma)$, and solid
angle. In integrating over solid angle $\Omega$, we assume an isotropic
distribution of pitch-angles $\alpha$. The resulting power lost per unit
volume is given by:

$\displaystyle\mathcal{P}_{-,\rm PL}^{\rm sync}$
$\displaystyle=\int~{}d\Omega~{}d\omega~{}d\gamma~{}f_{\rm
PL}(\gamma)\frac{2e^{4}B_{r}^{2}\gamma^{2}\beta^{2}\sin^{2}\alpha}{3m_{e}^{2}c^{3}}$
(21) $\displaystyle=\frac{16\pi
e^{4}B_{r}^{2}}{9m_{e}^{2}c^{3}}\int_{\gamma_{1}}^{\gamma_{2}}(\gamma^{2}-1)A_{\rm
PL}n_{0}\gamma^{-p_{\rm eff}}d\gamma.$ (22)

If inverse Compton dominates, the emissivity is set to $\mathcal{P}_{-,\rm
PL}^{\rm IC}=(U_{ph}/U_{B})\mathcal{P}_{-,\rm PL}^{\rm sync}$.

We describe the cooling due to thermal particles in the plunging region as
saturated Comptonized bremsstrahlung. Because the optical depth is of order 1
and the Compton $y$ parameter is much greater than 1, the bremsstrahlung is
amplified by repeated inverse Compton scatterings. The resulting emissivity is

$Q_{-,\rm MJ}=A(n_{0},T_{e})\epsilon_{\rm ff}(n_{0},T)$ (23)

where $A(n_{0},T_{e})=0.74\left[\ln(2.25/x_{coh})\right]^{2}$ is an
approximate amplification factor due to repeated inverse Compton scattering,
$x_{coh}=2.4\times 10^{17}(m_{e}n_{0})^{1/2}T_{e}^{-9/4}$ and
$\epsilon_{ff}(n_{0},T)=1.68\times 10^{-27}T_{e}^{1/2}n_{0}^{2}$ is the
standard bremsstrahlung emissivity (Rybicki & Lightman, 1979).

Because the energy loss collision times (Equation 7) assume a nonrelativistic
thermal bulk, we put an upper limit on $T_{e}$ at the temperature $T_{\rm
max}$ where setting $\nu_{\epsilon}^{ee}(x^{ee}_{c})=0$ would require a
particle velocity greater than the speed of light. This maximum temperature is
$T_{\rm max}=3\times 10^{9}$ K.

Equations 15, 17, 16, and 20 are solved iteratively at each radius using
Brent’s method to obtain $\gamma_{1}(r)$, $p_{\rm eff}(r)$, $A_{\rm PL}(r)$,
and $T_{e}(r)$.

##  4 Results

###  4.1 High-spin, highly-magnetized case

To illustrate the model, we examine the fiducial case $a=0.95$ and
$\Delta\epsilon=1.04$, which corresponds to $F_{\theta\phi}=6.0$ and magnetic
flux a factor of 2 below the saturation value for a magnetically-arrested disk
(Avara et al., 2016). The hierarchy of timescales for this fiducial model
justifies the assumptions outlined in Section 2. For this set of parameters,
$\gamma_{1}$ always hits the lower bound $\gamma_{1,\rm min}=2.0$. The
equilibrium electron temperature $\theta_{e}$ decreases from 0.14 ($8.9\times
10^{8}$ K) at the ISCO to 0.03 ($1.9\times 10^{8}$) K at the event horizon
(Figure 3). These temperatures are roughly consistent with the inner disk
temperatures in two-temperature GRMHD simulations (Liska et al., 2022). The
temperature decreases towards the event horizon because of the increase in
$\gamma_{2}$, decreasing the power law’s overall normalization and shifting
the temperature lower because of the continuity requirement (Equation 16). The
decoupling of electrons and protons happens very close to the ISCO, at
$1.82~{}r_{g}$, meaning that almost the entire plunging region has decoupled.
The cooling rate is dominated by nonthermal synchrotron emission, with thermal
cooling representing $\lesssim 1\%$ of the cooling rate at the ISCO and
$0.1\%$ at the event horizon (Figure 3). This increase in the fraction of
cooling from nonthermal electrons is again driven by the increase in
$\gamma_{2}$, which is set by the background magnetic field and gas density.

The total spectrum, shown in Figure 5, consists of three parts: the plunging
region’s nonthermal electrons (blue solid line), the plunging region’s thermal
electrons (green dotted line), and the thin disk’s spectrum (orange dashed
line). The spectrum is calculated with redshifts obtained by ray tracing the
plunging region-disk system with the code geokerr to a camera inclined at
$i=60^{\circ}$ to the black hole and disk angular momentum vector (Dexter &
Agol, 2009). The gas is assumed to have a four-velocity given by the Gammie
(1999) model within the plunging region and a purely azimuthal, Keplerian
velocity in the disk body. The power law intensities $I_{\nu,{\rm
PL},ij}=P_{\nu,{\rm PL},ij}H/(4\pi)$ at a camera pixel with position $i$ and
$j$ in the $x-$ and $y-$directions are calculated by assuming a hybrid
electron distribution at the radius $r_{ij}$ where the ray-traced photon hits
the plunging region midplane. The power-law distribution is $P_{\nu,{\rm
PL},ij}=A_{ij}\nu^{-s_{ij}}$ between the characteristic synchrotron
frequencies $\nu_{1,ij}$ and $\nu_{2,ij}$ for electrons with Lorentz factors
$\gamma_{1,ij}$ and $\gamma_{2,ij}$. That is,
$\nu_{1}=\nu_{0}\gamma_{1,ij}^{2}$ and $\nu_{2}=\nu_{0}\gamma_{2,ij}^{2}$,
where $\nu_{0}=3e|B_{r}|/(4\pi m_{e}c)$. The power-law slope is
$s_{ij}=(p_{\rm eff,ij}-1)/2$. The constant $A_{ij}$ is set by requiring that
$\int P_{\nu,{\rm PL},ij}d\nu=Q_{-,\rm PL,ij}$. The emission from the thermal
plunging region electrons is in the Comptonized Wien regime, such that
$I_{\nu,{\rm MJ},ij}=I_{\nu}^{W}(T_{ij})e^{-\alpha_{ij}}$, where $T_{ij}$ is
the thermal electron temperature at $r_{ij}$,
$I_{\nu}^{W}(T_{ij})=2h\nu^{3}/c^{2}e^{-kT_{ij}/h\nu}$ is the Wien intensity,
and $e^{-\alpha_{ij}}$ is the Compton saturation factor. The Compton
saturation factor is determined by setting the frequency-integrated intensity
equal to the isotropic thermal Comptonized bremsstrahlung emissivity:
$I^{W}(T_{ij})e^{-\alpha_{ij}}=Q_{-,\rm MJ}H/4\pi$. Luminosities are obtained
by summing over $64\times 64$ camera pixels:
$L_{\nu}=4\pi\Delta\alpha\Delta\beta\sum_{ij}I_{\nu,ij}$, where
$\Delta\alpha=\Delta\beta=0.125r_{g}$ are the pixel widths. The disk spectrum
is calculated by assuming a blackbody with a radially-dependent temperature
extending from the ISCO to $10^{10}$ cm (Frank et al., 2002):

$T(r)=\left(\frac{3GM\dot{M}}{8\pi\sigma
r^{3}}\frac{\mathscr{Q}}{\mathscr{B}\mathscr{C}^{1/2}}\right)^{1/4}$ (24)

where $\mathscr{B}$, $\mathscr{C}$, and $\mathscr{Q}$ are relativistic
corrections (Novikov & Thorne, 1973; Page & Thorne, 1974).

Several features of the spectrum in Figure 5 stand out. First, emission from
the nonthermal plunging region electrons dominates the emission above 10 keV
and produces an observable high-energy power-law tail. The power law has a
relatively flat slope past 200 keV, a feature that is difficult to produce
through thermal Comptonization (Coppi, 1999). This power law cuts off at a
little less than 100 MeV, though above 1 MeV pair production could become
important (gray region). The total luminosity from the power law across all
frequencies is 6.1% of the disk luminosity. This fraction decreases to 1.7% if
non-X-ray frequencies below 1keV are excluded. Excluding luminosity from the
pair-emitting frequency range above 1 MeV further decreases the fraction to
1.3%. The nonthermal power-law electrons dominate emission from the plunging
region, with a luminosity 63, 15, or 12 times that of the thermal plunging
region electrons for all frequencies, frequencies above 1 keV, and frequencies
between 1keV and 1 MeV, respectively.

Most of the observed luminosity comes from radii close to the ISCO, as shown
in Figure 6. This figure calculates the amount of observed luminosity that
originates from between the event horizon and a radius $r$. We define the
half-light radius $r_{1/2}$ as the location within which 50% of the light from
the plunging region has been emitted; analogously, $r_{1/10}$ is where 10% of
the light has been emitted. For this fiducial case with the ISCO at
$r=1.94r_{g}$, 50% of the light comes from $r>1.77r_{g}$ and 90% comes from
$r>1.66r_{g}$.

The emission depends on the spin and the inclination of the disk-plunging
region system. The inclination of the disk with respect to an observer changes
the effective observed area, an effect that goes as $\cos i$, where $i$ is the
angle between the black hole spin axis and the line-of-sight. Relativistic
beaming effects have the opposite trend with inclination angle; radiation from
relativistically-moving gas is beamed into the plane of motion, leading to a
higher luminosity for larger inclination angles. In the plunging region,
beaming effects dominate, resulting in a factor of 100 increase in power-law
luminosity from $i=0^{\circ}$ ($L_{\rm PL}=5\times 10^{35}$ erg/s) to
$i=90^{\circ}$ ($L_{\rm PL}=6\times 10^{37}$ erg/s) for $a=0.95$. Power-law
emission for $a=0.5$ is roughly constant over inclination angle, with $L_{\rm
PL}\sim 2-3\times 10^{36}$ erg/s. Beaming effects no longer dominate for the
smaller spin because the fluid Lorentz factor is smaller due to a larger
$r_{\rm ISCO}$. The blackbody disk’s peak temperature depends on the location
of the ISCO, which is set by the black hole spin and decreases by a factor of
about 3 from a non-spinning to a maximally-spinning black hole (Cunningham,
1975; Li et al., 2005). The disk emission is non-monotonic as a function of
inclination angle, peaking around $60^{\circ}$, where area projection effects
start to dominate over Doppler effects. The peak disk emission is a factor of
two larger than at $i=0^{\circ}$ for $a=0.95$, and a factor of 1.7 larger for
$a=0.5$. As a result, $L_{\rm PL}/L_{\rm disk}({\rm 1keV}<\nu<{\rm 1MeV})$ for
$a=0.95$ increases from 0.002 at $i=0^{\circ}$ to 0.17 at $i=85^{\circ}$,
whereas for $a=0.5$ it increases from 0.055 at $i=0^{\circ}$ to 0.09 at
$i=85^{\circ}$, with a minimum of 0.042 at $i=45^{\circ}$.

Figure 5: Spectrum for the fiducial model parameters $a=0.95$ and
$F_{\theta\phi}=6.0$ at an inclination of $60^{\circ}$. The thin disk emission
(dashed orange line) peaks around a few keV. The nonthermal electrons’
radiation (solid blue line) extends from 10 eV to 0.1 GeV, although emission
beyond 1 MeV (gray region) may be impacted by pair creation. The thermal
electrons in the plunging region create a small excess at around 10 keV
(dotted green line). Figure 6: Portion of the fiducial model’s observed
frequency-integrated luminosity originating between the event horizon and a
radius $r$ for an inclination angle of $60^{\circ}$. Emission from the
plunging region’s nonthermal electrons (blue) dominates over the plunging
region’s thermal electrons (green). 10% of the luminosity comes from
$r<r_{1/10}=1.66r_{g}$ (dotted line), while 50% comes from
$r<r_{1/2}=1.77r_{g}$ (dashed line). Vertical dash-dot line shows the
decoupling radius.

###  4.2 Parameter Space

Figure 7 shows the power law to disk luminosity fraction as a function of
plunging region dissipation and black hole spin. For $i=60^{\circ}$, the
plunging region can emit up to 1% of the accretion disk luminosity for a high-
spin black hole, and even higher for lower spin black holes. At constant spin,
disks with less magnetic torque at the ISCO and hence a lower accretion
efficiency $\Delta\epsilon$ have less nonthermal luminosity because weaker
magnetic fields lead to less efficient cooling of high-energy particles,
whereas the thin disk emission is constant with $\Delta\epsilon$. At fixed
$\Delta\epsilon$, lower black hole spins lead to a more visible power law
because the plunging region extends to larger radii, increasing the magnitude
of the nonthermal plunging region luminosity and decreasing the disk
luminosity. For $\Delta\epsilon\approx 0.4$, the power law’s X-ray luminosity
increases by a factor of 400, from $10^{33}$ erg/s for $a=0.95$ to $4\times
10^{35}$ erg/s for $a=0.70$. This increase is due to a combination of a larger
area radiating 90% of the emission and redshifts closer to unity for smaller
spins. The thin disk’s X-ray luminosity decreases for decreasing spin by a
factor of ten, from $10^{38}$ to $10^{37}$ erg/s. These combined effects
explain why $L_{\rm PL}/L_{\rm disk}$ is a factor of 4000 different for the
same magnetization.

Our model demonstrates two regimes for the plunging region X-ray emission: one
where nonthermal electrons dominate and one where thermal electrons dominate.
For fixed spin, weaker magnetic fields lead to less cooling by nonthermal
electrons and thus require more thermal cooling and higher thermal
temperatures to balance the total heating rate. Higher thermal temperatures
and a less prominent power-law lead to a significant decrease in $L_{\rm
PL}/L_{\rm MJ}$ with decreasing accretion efficiency. Nonthermal electrons
dominate only at the highest disk magnetizations because particle acceleration
requires strong magnetic fields. For less strongly magnetized disks, thermal
electrons dominate the plunging region emission; in this regime, we might
expect the observed power law to come from thermal electrons with a rapidly-
increasing temperature rather than nonthermal electrons (Zhu et al., 2012).
Figure 7 marks the regime where nonthermal electrons dominate with black
circles around the colored markers.

To facilitate comparison with observations, we show our model’s predictions
for the power-law fraction (PLF) as a function of inclination angle and spin
in Figure 8. The PLF is defined as

${\rm PLF}=\frac{\mathcal{L}_{\rm PL}}{\mathcal{L}_{\rm PL}+\mathcal{L}_{\rm
disk}}$ (25)

where $\mathcal{L}_{\rm PL}=L_{\rm PL}({\rm 1~{}keV}<\nu<{\rm 100~{}keV})$ is
the power-law luminosity in the X-rays and $\mathcal{L}_{\rm disk}=L_{\rm
disk}({\rm 0.001~{}keV}<\nu<{\rm 100~{}keV})$ is the disk luminosity over a
wide range of frequencies, to avoid cutting off low-spin blackbody emission
that peaks at $\sim$1 keV (Dunn et al., 2010; Muñoz-Darias et al., 2013). For
the high-spin case, the PLF is almost the same as $L_{\rm PL}/L_{\rm
disk}({\rm 1~{}keV}<\nu<{\rm 1~{}MeV})$ because $L_{\rm PL}\ll L_{\rm disk}$.
For lower spins, the PLF is lower than $L_{\rm PL}/L_{\rm disk}({\rm
1~{}keV}<\nu<{\rm 1~{}MeV})$ by a factor of 2 - 3 because of the change in
denominator. The parameters in Figure 8 are the most strongly magnetized disks
where the nonthermal emission dominates thermal emission from the plunging
region; for the least magnetized cases, the PLF is less than $10^{-3}$ (not
shown) and thermal plunging region electrons would significantly change the
shape of the emission above 10 keV. Overall, the PLF calculated from the model
lies between $10^{-3}$ and 0.3, values consistent with observations of the
soft state (Dunn et al., 2010).

Figure 7: Frequency-integrated luminosity in X-rays ($1~{}{\rm
keV}<\nu<1~{}{\rm MeV}$) in predicted power law compared to the thermal
blackbody disk for different model parameters, assuming an inclination angle
$i=60^{\circ}$. Models where nonthermal electron emission dominates over
thermal electron emission, satisfying $L_{\rm PL}>L_{\rm MJ}$, are marked with
large black circles. Figure 8: Power-law fraction (Equation 25) as a function
of inclination angle. Calculated from the most strongly magnetized plunging
region parameters such that nonthermal plunging region emission dominates
thermal plunging region emission.

##  5 Discussion

###  5.1 Application to Astrophysical Systems

Our model broadly agrees with observations of X-ray binaries in the soft
state. The soft state usually exhibits a photon index $\Gamma\gtrsim 2$
(Remillard & McClintock, 2006), corresponding to model fits with an electron
power-law index $p\gtrsim 3$. The model’s most strongly magnetized parameters
give an electron power-law index of $3-4$ (Figure 3), which increases for
decreasing magnetization. The soft state’s gamma-ray tail has luminosities on
the order of $10^{35}-10^{37}$ erg/s above 50 keV (Grove et al., 1998). Our
model finds power-law luminosities above 50 keV between $10^{34}-10^{37}$
erg/s, within the same order of magnitude. Calculations from observations find
that the PLF varies between $10^{-3}$ and 0.2 in the soft state of XRBs (Dunn
et al., 2010), which agrees well with the values found from our model (Figure
8). As yet, trends of the PLF with inclination angle and spin cannot be found
from observations (Muñoz-Darias et al., 2013). Testing our predictions
concretely against observations will require less uncertainty in the PLF.

The power-law emission in our model comes from synchrotron emission, whereas
typical models for the power law assume inverse Compton scattering of
blackbody disk photons off nonthermal particles (McConnell et al., 2002;
Gierliński et al., 1999). Our model therefore predicts more strongly polarized
emission for the soft state’s emission above 10 keV than other models where
inverse Compton scattering dominates. Current polarization measurements set an
upper limit of 70% for Cyg X-1’s 0.4 - 2 MeV emission in the soft state
(Rodriguez et al., 2015), which does not constrain the radiation mechanism.

The hard state of XRBs such as Cyg X-1 shows an excess above 400 keV that fits
well to a hybrid thermal/nonthermal model with a power-law electron
distribution index between 3.5 and 5 (McConnell et al., 2000). The nonthermal
electrons in that model could also be accelerated by turbulence or magnetic
reconnection. If the hierarchy of timescales satisfies Equation 10, the
multizone equilibrium model presented in Section 3 could be adapted to a
geometry relevant to the hard state.

###  5.2 Model Limitations

Many of our model assumptions were motivated by GRMHD simulations. Our finding
that ions and electrons decouple in the plunging region is consistent with
estimates of the thermalization time from single-temperature, ideal GRMHD
simulations (Zhu et al., 2012). In fact, the decoupling region could extend
even outside the ISCO. The Gammie (1999) model artificially sets the radial
velocity to zero at the ISCO, forcing a thin disk at $r>r_{\rm ISCO}$ whereas
the thin disk could in principle thicken at larger radii depending on the
dominance of Coulomb collisions. For simplicity, in our model, the ion
temperature and thus the ratio $H/R$ was discontinuous at the ISCO. This
discontinuity is not present in full 3D GRMHD simulations; instead, a
transition region forms between a thin disk and the hot flow (Hogg & Reynolds,
2018; Liska et al., 2022). Such a transition region would likely result in a
less distinct bump from the thermal electrons in the plunging region. GRMHD
simulations also motivate magnetic reconnection as an important dissipation
mechanism, demonstrating hallmarks of reconnection such as current sheets and
plasmoids throughout a magnetically-dominated disk (Ripperda et al., 2022;
Scepi et al., 2022). Our prescription for $\gamma_{2}$ and $p$ does not
include the impact of particle anisotropy or magnetic guide field on the
electron distribution (Comisso et al., 2020; Werner & Uzdensky, 2021), which
could alter the observed spectrum. Including a guide field tends to steepen
$p$ and decrease $\gamma_{2}$ for a pair plasma (Werner & Uzdensky, 2017,
2021). Pitch-angle anisotropy in a pair plasma can also lead to temperature
anisotropy for the thermal bulk (Comisso & Sironi, 2019; Comisso et al.,
2020), which could affect radiation from the thermal plunging region
electrons. Unlike GRMHD simulations, our model does not include vertical
structure and we assumed a constant $H/R=1$ and a uniform distribution of the
dissipation throughout the plunging region volume. Vertical structure would
presumably lead to a dense equatorial region with more optically thin upper
parts of the plunging region. We also assume uniform dissipation such that
$Q_{+}\propto\rho$, as commonly done in disk atmosphere models (Davis &
Hubeny, 2006).

For simplicity, we did not include full radiative transport in our model.
Instead, we assumed that the thermal plunging region electrons cooled solely
through thermal bremsstrahlung, neglecting thermal synchrotron radiation
because the average Lorentz factor of the thermal particles was less than 2
(i.e. temperatures less than $\sim 10^{9}$ K). We also neglect synchrotron
self-absorption, since this effect is only important for frequencies below
$\sim 10^{15}$ Hz (Wardziński & Zdziarski, 2000), outside the range we
consider. We neglect cyclotron radiation for the same reason. We restrict the
observed radiation from the nonthermal electrons to be either from synchrotron
radiation or from inverse Comptonization of soft disk photons, i.e. we neglect
synchrotron self-Compton emission (SSC). Neglecting SSC does not result in
large errors for small optical depth and spectral index $>1$, where
synchrotron emission dominates over SSC (Wardziński & Zdziarski, 2001). We
also do not consider synchrotron and inverse Compton processes simultaneously;
instead, we assume one or the other dominates. We also assume a simple thermal
+ power-law electron distribution function, whereas Fokker-Planck models
evolving a power-law electron distribution thermalizing under electron-
electron collisions show a more complicated structure (Nayakshin & Melia,
1998). Although pair production could become important in this regime, it
remains unclear how pair-regulated reconnection would change when synchrotron
losses are included (Mehlhaff et al., 2021). We note that Compton reflection
off the cold disk could increase the luminosity above 10 keV by order unity
for sandwich-type models (Zdziarski & Gierliński, 2004). Including Compton
reflection of the plunging region radiation off the cold disk could also
increase the high-energy luminosity and lead to higher power-law fractions.
More detailed modelling of reflection features such as the Fe K$\alpha$
emission line is beyond the scope of this work.

##  6 Conclusions

We have presented a semi-analytic model for the soft state of XRBs where the
high-energy power-law tail is produced by nonthermal electrons in the plunging
region. We demonstrated the feasibility of having nonthermal electrons in the
plunging region by examining the hierarchy between electron-proton
thermalization time, electron-electron collision time, electron cooling time,
electron acceleration time, and the infall time (Section 2, Figure 2). Using
an analytic dynamical background and the results of particle-in-cell
simulations of magnetic reconnection, we constructed a steady-state model for
the electron distribution function at each radius in the plunging region
(Section 3). The nonthermal electrons in this model produce an observable
power law from $10-1000$ keV with a photon spectral index $\Gamma\gtrsim 2$
(Figure 5). By exploring the model parameter space, we show that plunging
region emission from nonthermal electrons dominates over thermal electrons for
strongly-magnetized models and vice-versa for less strongly-magnetized models,
suggesting a Zhu et al. (2012)-type model. We found fractions of power law to
total X-ray luminosity (power-law fraction, PLF) consistent with observations
for all values of spin (Figure 7). We predict an increase in the observed PLF
with inclination angle (Figure 8). Although the PLF in our model is consistent
with observational values, testing the trend with inclination angle and spin
will require more observations.

Future work using numerical simulations could explore how and where the
decoupling happens, as well as its possible role in spectral state transitions
and jet/wind launching.

## Acknowledgements

The authors thank Mitch Begelman, Dmitri Uzdensky, and the referee, Luca
Comisso, for helpful comments. AMH acknowledges support from the National
Science Foundation Graduate Research Fellowship Program under Grant No. DGE
1650115. This work was supported in part by NASA Astrophysics Theory Program
grants NNX16AI40G, NNX17AK55G, and 80NSSC20K0527 and by an Alfred P. Sloan
Research Fellowship (JD).

## Data Availability

The simulation data underlying this article will be shared on reasonable
request to the corresponding author.

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