# Accretion around black holes: The geometry and spectra

B.F. Liu<EMAIL_ADDRESS>Erlin Qiao<EMAIL_ADDRESS>Key Laboratory of
Space Astronomy and Technology, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100012, China  School of Astronomy and Space
Sciences, University of Chinese Academy of Sciences, 19A Yuquan Road, Beijing
100049, China

###### Abstract

Observations of black hole X-ray binaries and active galactic nuclei indicate
that the accretion flows around black holes are composed of hot and cold gas,
which have been theoretically described in terms of either a hot geometrically
thick corona lying above and below a cold geometrically thin disk or an inner
advection dominated accretion flow connected to an outer thin disk. This
article reviews the accretion flows around black holes, with an emphasis on
the physics that determines the configuration of hot and cold accreting gas,
and how the configuration varies with the accretion rate and thereby produces
various luminosity and spectra.

###### keywords:

accretion, accretion disks , black hole physics , galaxies: active , X-rays:
binaries , X-rays: galaxies

##  1 Introduction: solutions of accretion flow

It has been widely accepted that gas accretion onto black holes provides the
principal power of radiations from black hole X-ray binaries (BHXRBs) and
active galactic nuclei (AGNs). With the action of viscosity, gas captured by
the black hole spirals toward the central black hole, drops off its initial
angular momentum outwards, releases gravitational potential energy into heat.
A fraction or all of the viscous heat is radiated, yielding a spectrum in
dependence on specific radiation processes. The accretion flows are described
by four basic solutions, that is, the standard thin disk [e.g. Shakura and
Sunyaev, 1973], the optically thin two-temperature disk Shapiro et al. [1976],
the slim disk [e.g. Katz, 1977, Begelman, 1978, Abramowicz et al., 1988] and
the advection-dominated accretion flow [e.g. Ichimaru, 1977, Rees et al.,
1982, Narayan and Yi, 1994, 1995a, 1995b, Abramowicz et al., 1995, 1996]. When
these solutions are applied to black holes, a composite of them are commonly
adopted.

The most famous solution is the standard thin disk model developed in 1970s by
Shakura and Sunyaev [1973], Novikov and Thorne [1973], Lynden-Bell and Pringle
[1974][for textbooks see Frank et al., 2002, Kato et al., 2008]. The thin disk
is a geometrically thin, optically thick accretion flow, characterized by
multi-color blackbody radiation with effective temperatures ranging from
$10^{5}$K to $10^{7}$K from supermassive black holes to stellar-mass black
holes. The thin disk model has been the common paradigm in accretion theory
and successfully used to model a large number of astrophysical systems. The
most successful applications are the steady state and time-dependent nature of
thin disk in dwarf novae and soft X-ray transients.

The second solution was presented by Shapiro et al. [1976], in which the
accreting gas is optically thin, hot plasma with temperature near virial in
ions and $\sim 10^{9}$K in electrons. Such a solution is, however, thermal
unstable Piran [1978], and is therefore not considered as applicable to black
holes.

In the above two solutions, the energy released by viscous heating is assumed
to be emitted locally. In contrast, a third solution was proposed Ichimaru
[1977], Rees et al. [1982], Narayan and Yi [1994] and extensively investigated
[e.g. Narayan and Yi, 1995a, b, Abramowicz et al., 1995] [see the review Yuan
and Narayan, 2014] by taking into account the advection of viscous heating. In
this solution, the accreting gas has a very low density and the Coulomb
coupling between the ions and electrons is weak. The viscous energy can not be
radiated efficiently, instead, part of it is stored in the gas as thermal
energy and eventually advected onto the black hole. Therefore, the gas forms
an optically thin, two-temperature accretion flow with the ion temperature
close to virial temperature and electron temperature $T_{\rm e}\sim 10^{9}$K,
referred to as advection dominated accretion flow (ADAF). Apparently, the ADAF
solution is only valid at low accretion rates so that the advection plays an
important role in ”digesting” the viscous heating. The radiation is,
therefore, inefficient, however, contributing to hard X-ray by synchrotron
self-Compton process.

The fourth solution is developed for the high accretion rate, in which the gas
density is very large [e.g. Katz, 1977, Begelman, 1978, Abramowicz et al.,
1988, Honma, 1996, Chen and Taam, 1993]. While the viscous heating is
efficiently transferred to emissions locally, most of photons are trapped in
the disk, not able to escape from the disk surface during the accretion time.
The energy is largely advected into the black hole in the form of photons
rather than thermal energy in the ADAF solution. Therefore, the energy
conversion into radiation is inefficient too. The temperature in this solution
is higher than that in a standard thin disk as a result of larger accretion
rate, while it is lower than that in an ADAF since ions can easily transfer
viscous heat to electrons through Coulomb collisions. The disk is radiation-
pressure supported, thereby the vertical scale height is larger than a
standard disk, but not so extensive as an ADAF. The solution is, thus, called
slim disk, sometimes referred to as an optically thick, advection-dominated
accretion flow.

Other solutions of the accretion flows, the adiabatic inflow-outflow solution
(ADIOS), convection-dominated accretion flow (CDAF), and the luminous hot
accretion flow (LHAF) are the variants of ADAF, emphasizing the roles of
outflows, convection and radiation efficiency, respectively. The ADIOS, CADF,
and LHAF are not discussed as particular solutions in addition to the ADAF.

Application of the four accretion models to black holes is constrained by both
the theoretical assumption of specific models and observational luminosity and
spectrum. The thin disk model applies for objects with luminosity below the
Eddington luminosity, $L_{\rm Edd}={4\pi GMm_{p}c\over\sigma_{T}}$, the
critical luminosity for the outward radiation force (through Thomson
scatterings) to balance the central gravity. For objects with high luminosity,
slim disk model is often adopted. Examples are the ultra-luminous X-ray
sources (ULXs), the narrow-line Seyfert 1 galaxies (NLS1s), and the tidal
disruption events. On the other hand, the ADAF model, characterized by its
inefficient radiation and high temperature, successfully applies to the
quiescent and low state BHXRBs, the Galactic center black hole, Sgr A∗, and
the low-luminosity AGNs.

The luminosity $L$ is translated to the accretion rate $\dot{M}$ with
$\dot{M}=L/\eta c^{2}$, where $\eta$ is the radiation efficiency. Accordingly,
the Eddington ratio $L/L_{\rm Edd}$ is related to the Eddington scaled
accretion rate $\dot{m}$, $L/L_{\rm Edd}=(\eta/0.1)\dot{m}$, with
$\dot{m}\equiv\dot{M}/\dot{M}_{\rm Edd}$ and $\dot{M}_{\rm Edd}\equiv L_{\rm
Edd}/0.1c^{2}$. By this definition, the thin disk model applies for
$\dot{m}\leq 0.3$. For higher accretion rates, $\dot{m}>0.3$, the accretion is
via a slim disk, while for lower accretion rates,
$\dot{m}<(0.1-0.3)\alpha^{2}$ [e.g. Narayan and Yi, 1995b, Yuan and Narayan,
2014], the accretion is via an ADAF according to the ”strong ADAF principle”
Narayan and Yi [1995b], though the thin disk is valid in this regime.

##  2 Theory confronted with observation: configuration of accretion flows
around black holes

Black hole X-ray transient binaries can be used as a probe of the accretion
process over a wide range in luminosity. The X-ray spectral behavior of these
systems exhibits different states and transitions, in particular, the X-ray
spectral state transition between the well-known high/soft state and low/hard
state [see the reviews Tanaka and Shibazaki, 1996, Remillard and McClintock,
2006]. Combined with the optical/UV observations, the X-ray observations
reveal that these two spectral states originate from different accretion
modes. At low luminosities the systems are characterized by a power law
spectrum in hard X-rays, accompanied with a weak thermal optical component.
The X-ray emission is commonly thought to be produced by the Compton
scattering of soft photons off the hot electrons in an inner ADAF and the
optical component by an outer truncated thin disk [e.g. Lasota et al., 1996,
Dubus et al., 2001, Poutanen et al., 2018]. At high luminosities the systems
are characterized by a multicolor blackbody component that dominates at about
1 keV, accompanied with a weak hard tail sometimes. This has been interpreted
as arising from a geometrically thin, optically thick accretion disk, covered
by a weak, hot corona [e.g. Poutanen and Coppi, 1998, Gierliński et al.,
1999]. The transition occurs at an Eddington ratio around $\sim 0.02$ [e.g.
Maccarone, 2003, Gierliński and Newton, 2006], though there is large
difference between individual sources.

Broadband spectral features of AGNs also provide strong evidence for hot gas
coexist with the thin disk in the neighborhood of the supermassive black holes
[e.g. Mushotzky et al., 1993, Zdziarski et al., 1999]. The big blue bump and
the power-law X-ray radiation are commonly thought to be the disk thermal
emission and inverse Compton scattering of the disk photons with the high-
temperature electrons in the hot corona, respectively [e.g. Yuan et al.,
2010]. The fluorescent iron lines are explained by reflection of the corona
radiation by the underlying disk [e.g. Fabian et al., 2000, Reeves et al.,
2001], supporting the cold disk and hot corona scenario.

Observational data of black holes in recent years display more complicated
spectral features, indicating a combination of various accretion flows [e.g.
You et al., 2021, Kara et al., 2019, Ruan et al., 2019, Jin et al., 2017a, b],
accompanied by wind/outflows in some cases [for observational and theoretical
studies of the wind see e.g. Tombesi et al., 2010, Gofford et al., 2015, Zhu
and Stone, 2018, Nomura et al., 2016, Yang et al., 2018, 2019, Wang et al.,
2013, Shi et al., 2021, Yuan et al., 2015, Bu et al., 2016, Bu and Gan, 2018].
A common feature of the accretion geometry in all these observations is the
co-existence of hot and cool accretion flows, of which the well-known
composites are either an inner ADAF connecting to a truncated disk, or an
accreting corona lying above a standard thin disk which extends inward to the
innermost stable circular orbit (ISCO). The former configuration was proposed
by Esin et al. [1997] and applied to both quiescent BHXRBs and low-luminosity
AGNs, and the latter configuration was proposed to explain the power-law X-ray
emission and lower frequency blackbody component observed in both BHXRBs and
AGNs Bisnovatyi-Kogan and Blinnikov [1976], Liang and Price [1977], Haardt and
Maraschi [1991, 1993], Nakamura and Osaki [1993], Svensson and Zdziarski
[1994], Zdziarski et al. [1999].

The physical mechanism triggering the transition between such configurations
has been studied since different accretion solutions were found. The “strong
ADAF principle” suggests that, whenever the accreting gas has a choice between
a standard thin disk and an ADAF, the ADAF configuration is chosen Narayan and
Yi [1995b]. With the radius-dependent critical accretion rate for existence of
the ADAF, this rule provides a transition radius for a given accretion rate
[e.g. Esin et al., 1997]. In a different approach Honma [1996] determines the
truncation radius by considering radial diffusive transport of heat between
the inner hot accretion flows and the outer cool disk . The thermal
instability in the radiation pressure-supported disk can also trigger a
transition from a thin disk to an ADAF in the inner region Gu and Lu [2000],
Lu et al. [2004]. Among various possibilities, the disk and corona interaction
model provides a promising explanation for the disk truncation and the
spectral state transition Liu et al. [1999], Meyer et al. [2000b, a], RóżaŃska
and Czerny [2000], Różańska and Czerny [2000], Meyer-Hofmeister and Meyer
[2001], Meyer and Meyer-Hofmeister [2002], Liu et al. [2002], Spruit and
Deufel [2002], Meyer-Hofmeister and Meyer [2003], Liu et al. [2003], Dullemond
and Spruit [2005], Liu et al. [2005, 2006, 2007], Meyer et al. [2007], Taam et
al. [2008], Meyer-Hofmeister et al. [2009], Liu and Taam [2009], Liu et al.
[2011, 2012], Qiao and Liu [2012], Taam et al. [2012], Qiao and Liu [2013],
Qiao et al. [2013], Liu and Taam [2013], Liu [2013], Liu et al. [2015, 2016,
2017], Meyer-Hofmeister et al. [2017], Qiao and Liu [2017, 2018], Taam et al.
[2018], Cheng et al. [2020]. Specifically, the interaction between the disk
and corona causes disk gas evaporating to the corona or coronal gas condensing
to the disk, depending on the gas supply rate and how the gas feeds to the
accretion. In the case that the accreting gas is supplied to the thin disk, as
in low-mass X-ray binaries of black hole systems (LMXBs), gas evaporates from
the disk to the corona. Thus, the disk is truncated when the gas supply is
insufficient for evaporation. However, it extends to the ISCO when the gas
supply is more than the evaporation [e.g. Meyer et al., 2000b, a, Liu et al.,
2002]). In the case that the accreting gas is hot, as the wind in high-mass
X-ray binaries of black hole systems (HMXBs) or Bondi flows in AGNs, the hot
accretion flow condenses partially to the disk if the supply rate is high,
while it keeps the pure ADAF form if the gas supply rate is low [e.g. Liu et
al., 2015, Qiao and Liu, 2018, Taam et al., 2018].

The coexistence of different accretion flows has been found by
magnetohydrodynamic (MHD) simulations [e.g. Miller and Stone, 2000, Hawley and
Krolik, 2001, McKinney et al., 2014, Ohsuga et al., 2005, Sadowski and
Narayan, 2016]. In particular, the two-phase accretion flow, either in a form
of a corona overlapping a thin disk or an inner corona connecting outer thin
disk, has been confirmed by the global radiation MHD simulations [e.g. Jiang
et al., 2019, and references therein].

It is now widely accepted that the accretion flows around black holes are
composed of hot and cold flows. The relative strength of the two flows, either
the disk and the corona at high states, or the inner ADAF and the outer disk
at low states is constrained by observational spectral energy distribution
(SED). For instance, in modeling the soft and hard components in BHXRBs, a
proper value of the truncated radius is adopted [e.g. Poutanen et al., 2018,
and references therein]; While in interpreting the strong X-ray emission in
luminous AGN, a certain fraction of accretion energy is assumed to release in
the corona [e.g. Haardt and Maraschi, 1991, 1993]. Then, what is the physical
mechanism that drives the formation of the hot and cold flows? How does the
relative strength change? Can one determine the truncation radius consistently
with observations? We’ll give a detailed review aiming to answer such
questions.

In the following sections, we describe the interaction between the hot and
cold flows, elucidate how the interaction leads to disk truncation and
consequently determines the relative strength of hot and cold accretion flows
as a function of black hole mass, accretion rate and viscosity parameter,
thereby producing spectra comparable with observations. Specifically, the disk
evaporation model applicable to the accretion of Roche-lobe overflow (RLOF) is
presented in Sect.3.1, and the corona condensation model applicable for
accretion of wind is presented in Sect.3.2. The geometry of two-phase
accretion flows as a consequence of disk-corona interaction is displayed in
Sect.3.3 . The radiative properties of the disk-corona system are shown and
compared with observations in Sect.3.4. The conclusions are in Sect.4.

##  3 Formation of two-phase accretion flows around black holes

The two-phase accretion flows refer to the coexistent cold and hot accretion
flows, which are, respectively, the standard thin disk and the ADAF/corona.
The corona is physically similar to the ADAF excepted that a thin disk is
sandwiched in. Thus, the properties of the corona, such as temperature and the
optical depth, can be strongly affected by the cold thin disk via heat
conduction, radiation coupling and the consequent mass exchange at a dynamic
equilibrium between the disk and the corona.

The concept of the evaporation of matter from an accretion disk was originally
proposed in the pioneering work by Meyer and Meyer-Hofmeister [1994] to
explain the UV lag observed in dwarf novae. The model was developed in a more
detailed form for application to accretion flows surrounding black holes Liu
et al. [1999], Meyer et al. [2000b, a], Meyer-Hofmeister and Meyer [2001], Liu
et al. [2002], which demonstrated the possible truncation of an outer
optically thick disk and the consequent formation of an inner optically thin
ADAF. The incorporation of various elements was included and the effects were
investigated Meyer and Meyer-Hofmeister [2002], Liu et al. [2002], Meyer-
Hofmeister and Meyer [2003], Liu et al. [2005], Meyer-Hofmeister et al.
[2005], Meyer-Hofmeister and Meyer [2006], Qian et al. [2007], Qiao and Liu
[2009]. Similar conceptual models, but with differences in detail were
proposed in a semi-analytical model proposed Dullemond and Spruit [2005] and
in a vertically stratified model RóżaŃska and Czerny [2000], Różańska and
Czerny [2000]. The coronal condensation, instead of disk evaporation, was
found in the innermost region at intermediate accretion rates, leading to a
weak inner disk at intermediate state Liu et al. [2006, 2007], Meyer et al.
[2007], Taam et al. [2008], Meyer-Hofmeister et al. [2009], Liu et al. [2011],
Meyer-Hofmeister and Meyer [2011], Meyer-Hofmeister et al. [2012]. The
radiation spectra from such interacting disk and corona were calculated and
compared with observations in BHXRBs [e.g. Qiao and Liu, 2012, 2013, Meyer-
Hofmeister and Meyer, 2014, Qiao and Liu, 2015] and AGNs [e.g. Liu and Taam,
2009, Liu et al., 2012, Taam et al., 2012, Qiao et al., 2013, Liu and Taam,
2013, Liu, 2013]. The condensation model was extended to the case of wind
accretion and applied to HMXBs and AGN Liu et al. [2015, 2017], Meyer-
Hofmeister et al. [2017], Qiao and Liu [2017, 2018], Taam et al. [2018], Qiao
and Liu [2019, 2020], Meyer-Hofmeister et al. [2020].

###  3.1 The disk evaporation model

As occurs in LMXBs, the accretion is assumed to take place via a cold thin
disk as gas transferred from the donor star is constrained to the orbital
plane. A corona forms above the thin disk either by processes similar to those
operating in the surface of the sun, or by a thermal instability in the
uppermost layers of the disk [e.g. Shaviv and Wehrse, 1986]. The disk and
corona are individually powered by the release of gravitational energy
associated with the accretion of matter, and the interaction between the disk
and corona is an important and distinctive process. Specifically, the corona
is an ADAF-like accretion flow modified by the vertical heat conduction and
inverse Compton scattering of disk photons, which play a key role in cooling
the electrons as a consequence of the existence of an underlying disk. The
ions in the corona are directly heated by viscous dissipation, partially
transferring their energy to the electrons by means of Coulomb collisions. The
energy gained by electrons can be effectively conducted to the lower, cooler,
and denser layers by electron-electron collisions. In the lower, transition
layer, the conductive heat flux is radiated away through bremsstrahlung only
if the number density in this layer reaches a critical value. If the density
is too low to efficiently radiate the energy, a certain amount of lower,
cooler gas is heated up so that an energy equilibrium is established between
the conduction, radiation and heating of the cool gas. The transfer of gas
from the disk to the corona, to establish the equilibrium, is called
evaporation. These processes are schematically described in Figure 1. An
opposite process could also take place, that is, if the density in the
transition layer is too high, a certain amount of gas is over-cooled by
radiation, and condenses to the disk (which is described in next subsection).

Figure 1: Schematic description of the interaction between the disk and the
corona. The left part shows the vertical conduction, the direct and Comption
radiations in the corona and the right part shows the gas flows.

The gas evaporating into the corona retains angular momentum and
differentially rotates around the black hole. By frictional stresses, the gas
loses angular momentum and drifts inward in such a way that the corona
continuously drains gas towards the black hole. The coronal accretion flow is
re-supplied by continuous evaporation and, therefore, steady flows are
established in the disk and corona with the mass exchange between the two
flows. The coronal accretion is supplied by evaporating disk matter. Thus, the
accretion rate in the corona at a given distance is the sum of the mass
evaporation from the outer disk inward to the given distance, hence it
increases toward the black hole as a result of accumulation of the mass
evaporation. When the accretion rate in the corona reaches a critical value,
Coulomb coupling between electrons and ions is efficient so that radiation in
the corona is efficient. The increased radiative cooling and decreased
conductive heating to the transition layer reduce the evaporation, thereby
evaporation ceases at some radius and condensation takes place in the inner
region. Therefore, there exist a most efficient evaporation region where the
evaporation rate reaches a maximum. If the gas supply rate to the disk is too
low to replenish the disk for the evaporation, the corresponding disk region
will be completely evaporated and filled by hot gas, that is, the disk is
truncated.

The corona temperature, density and mass flowing rate supplied by the
evaporation can be determined by solving the equations describing the dynamics
of the corona [e.g. Meyer et al., 2000b, Liu et al., 2002, Meyer-Hofmeister
and Meyer, 2003, Qian et al., 2007, Qiao and Liu, 2009], as listed below.

Equation of state

$P={\Re\rho\over 2\mu}(T_{i}+T_{e}).$ (1)

Equation of continuity

$\centering{d\over dz}(\rho v_{z})={2\over R}\rho v_{R}-{2z\over
R^{2}+z^{2}}\rho v_{z},\@add@centering$ (2)

where $2z/(R^{2}+z^{2})$ describes the gradual expansion of the vertical flow
channel with height as the geometry changes from cylindrical to spherical.

Equation of the $z$-component of momentum

$\rho v_{z}{dv_{z}\over dz}=-{dP\over dz}-\rho{GMz\over(R^{2}+z^{2})^{3/2}}.$
(3)

The energy equation of the ions

$\begin{array}[]{l}{d\over dz}\left\\{\rho_{i}v_{z}\left[{v^{2}\over
2}+{\gamma\over\gamma-1}{P_{i}\over\rho_{i}}-{GM\over(R^{2}+z^{2})^{1\over
2}}\right]\right\\}\\\ ={3\over 2}\alpha P\Omega-q_{ie}\\\ +{}{2\over
R}\rho_{i}v_{R}\left[{v^{2}\over
2}+{\gamma\over\gamma-1}{P_{i}\over\rho_{i}}-{GM\over(R^{2}+z^{2})^{1\over
2}}\right]\\\ -{2z\over{R^{2}+z^{2}}}\left\\{\rho_{i}v_{z}\left[{v^{2}\over
2}+{\gamma\over\gamma-1}{P_{i}\over\rho_{i}}-{GM\over(R^{2}+z^{2})^{1\over
2}}\right]\right\\},\end{array}$ (4)

where $q_{ie}$ is the energy exchange rate between the electrons and the ions
through Coulomb collisions Stepney [1983], Liu et al. [2002],

${q_{ie}}={\bigg{(}{2\over\pi}\bigg{)}}^{1\over 2}{3\over 2}{m_{e}\over
m_{p}}{\ln\Lambda}{\sigma_{T}cn_{e}n_{i}}(\kappa T_{i}-\kappa
T_{e}){{1+{T_{*}}^{1\over 2}}\over{{T_{*}}^{3\over 2}}}$ (5)

with

$T_{*}={{\kappa T_{e}}\over{m_{e}c^{2}}}\bigg{(}1+{m_{e}\over
m_{p}}{T_{i}\over T_{e}}\bigg{)}.$ (6)

The energy equation for both the ions and the electrons is

$\begin{array}[]{l}{\frac{d}{dz}\left\\{\rho{v}_{z}\left[{v^{2}\over
2}+{\gamma\over\gamma-1}{P\over\rho}-{GM\over\left(R^{2}+z^{2}\right)^{1/2}}\right]+F_{c}\right\\}}\\\
=\frac{3}{2}\alpha P{\mit\Omega}-n_{e}n_{i}L(T_{e})-q_{\rm Comp}\\\ +{2\over
R}\rho v_{R}\left[{v^{2}\over
2}+{\gamma\over\gamma-1}{P\over\rho}-{GM\over\left(R^{2}+z^{2}\right]^{1/2}}\right]\\\
-{2z\over R^{2}+z^{2}}\left\\{\rho v_{z}\left[{v^{2}\over
2}+{\gamma\over\gamma-1}{P\over\rho}-{GM\over\left(R^{2}+z^{2}\right)^{1/2}}\right]+F_{c}\right\\},\end{array}$
(7)

where $n_{e}n_{i}L(T_{e})$ is the bremsstrahlung cooling rate, of which the
radiative cooling function $L(T_{e})$ is taken from Raymond et al. [1976];
${q_{\rm{Comp}}}$ is the cooling rate through inverse Compton scattering Liu
et al. [2002],

${q_{\rm Comp}}={4\kappa T_{e}\over m_{e}c^{2}}n_{e}\sigma_{T}cu,$ (8)

with $u$ the energy density of the disk radiation; $F_{c}$ is the thermal
conduction flux given by Spitzer [1962]

$F_{c}=-\kappa_{0}T_{e}^{5/2}{dT_{e}\over dz}$ (9)

with $\kappa_{0}=10^{-6}{\rm erg\,s^{-1}cm^{-1}K^{-7/2}}$ for a fully ionized
plasma.

In the above equations, parameters $M$, $P$, $\rho$, $T_{i}$ and $T_{e}$ are
the mass of black hole, the pressure, density, ion temperature and electron
temperature; $n_{i}$ and $n_{e}$ are the number density of ions and electrons;
$v_{z}$ and $v_{R}$ are the vertical and radial speed of the coronal flow;
$\mu=0.62$ is the mean molecular weight assuming a standard chemical
composition (mass fractions of hydrogen and helium are $X=0.75,Y=0.25$) for
the corona. Constants $G$, $m_{p}$ and $m_{e}$ are respectively is the
gravitational constant, the mass of the proton and the electron; $\kappa$ is
the Boltzmann constant, $\Re$ the gas constant, $c$ the speed of light,
$\sigma$ Stefan-Boltzmann constant, $\sigma_{T}$ the Thomson scattering cross
section, $\gamma=5/3$ is the ratio of specific heats, and $\ln\Lambda=20$ is
the Coulomb logarithm.

The five differential equations, Eqs.(2), (3), (4), (7), and (9), which
contain five variables $P(z)$, $T_{i}(z)$, $T_{e}(z)$ , $F_{c}(z)$, and
$\dot{m}_{z}(\equiv\rho(z)v_{z})$, can be solved with five boundary
conditions. At the upper boundary, there is no artificial confinement and
hence no pressure, which means a sonic transition at some height $z=H$. As
there is no heat flux from/to the boundary, the upper boundary conditions are,

$F_{c}=0\ {\rm and}\ v_{z}^{2}=V_{s}^{2}\equiv P/\rho={\Re\over
2\mu}(T_{i}+T_{e})\ {\rm at}\ z=H.$ (10)

At the lower boundary of the interface between the disk and corona, the
conductive flux is exactly radiated away and there is no downward heat flux.
The temperature of the gas should be the effective temperature of the
accretion disk. Detailed investigations Liu et al. [1995] show that the
temperature increases from the effective temperature to $10^{6.5}$K in a very
thin layer and that the conductive flux can be expressed as a function of
pressure at this temperature. Thus, the lower boundary conditions is
reasonably approximated as Meyer et al. [2000b]

$T_{i}=T_{e}=10^{6.5}K,\ {\rm and}\ F_{c}=-2.73\times 10^{6}P\ {\rm at}\
z=z_{0}.$ (11)

Therefore, the coronal parameter $P(z)$, $\rho(z)$, $T_{i}(z)$, $T_{e}(z)$,
evaporation rate $\dot{m}_{z}$ can be determined by integrating the
differential equation for given mass of black hole, $m\equiv M/M_{\odot}$, the
mass supply rate to the disk, $\dot{m}\equiv\dot{M}/\dot{M}_{\rm Edd}$, and
viscosity parameter $\alpha$. Numerical calculations of the coronal vertical
structure revealed that the corona can be approximately divided into two
layers, the very thin transition layer with steep variation in temperature and
density along $z$, and the geometrically thick, hot corona with decoupled ion
and electron temperature. This justifies the configuration of accretion flows
as dominated by a cold disk and a hot corona connected by a very thin
transition layer.

###  3.2 The corona condensation model

The evaporation occurs when the corona is a tenuous flow. If the density is
sufficiently high so that the radiative cooling in the transition layer is
more efficient than the conductive heating, a certain amount of gas is over-
cooled and condenses onto the disk until energy equilibrium is reached.
Efficient cooling processes in a high-density corona, such as strong inverse
Compton scattering of the disk photons, can also facilitate such condensation.
In the low-mass black hole X-ray binaries, condensation is weak and only
occurs in the innermost region since the corona density built up by
evaporation can not be very large. In the high mass black hole X-ray binaries,
wind captured by the black hole can form a strong corona and condenses to the
disk on the way of accretion to the black holes. Such condensation can also
occur in AGN accretion flows, accompanied by strong X-ray emission from the
efficient comptonization process.

For convenient to model the radiation spectrum, the vertical structure of the
disk and corona is simplified to three layers, i.e., the thin disk, the
transition layer, and the corona. The parameters in the corona, such as
$T_{i}$, $T_{e}$, $\rho$, can be determined similar to that of an ADAF, except
that heat conduction and Compton scattering of disk photons are added to the
energy balance equation for electrons. Thus, the disk radiation is involved in
the corona equations, which depends on the accretion rate in the disk (and
coronal radiation if illumination is included). Thus, given the gas supply
rate in the outer boundary, the mass condensation rate can be derived as the
coronal temperatures and density from the energy balance in the transition
layer. This re-allocates the mass accretion rates in the disk and the corona,
and in turn, affects the coronal temperatures and density. The complete set of
equations is list as below.

The density and pressure in the corona are,

$\begin{array}[]{l}\rho=3.79\times
10^{-5}\alpha^{-1}c_{1}^{-1}c_{3}^{-1/2}m^{-1}\dot{m}_{c}r^{-3/2}~{}{\rm
g~{}cm^{-3}},\\\ n_{\rm e}=2.00\times
10^{19}\alpha^{-1}c_{1}^{-1}c_{3}^{-1/2}m^{-1}\dot{m}_{c}r^{-3/2}{\ \rm
cm^{-3}},\\\ p=1.71\times
10^{16}\alpha^{-1}c_{1}^{-1}c_{3}^{1/2}m^{-1}\dot{m}_{c}r^{-5/2}{\ \rm
g\,cm^{-1}\,s^{-2}},\end{array}$ (12)

where $\dot{m}_{c}$ denotes the Eddington-scaled accretion rate in the corona,
$m$ is the black hole mass in units of solar mass, $r$ is the radius in units
of Schwarzschild radius. $c_{1}\approx 0.6$ and $c_{3}\approx 0.4$ are
insensitive to the chosen parameters (precisely,
$c_{1}=\frac{5+2\epsilon^{\prime}}{3\alpha^{2}}g(\alpha,\epsilon^{\prime})$,
$\epsilon^{\prime}={1\over f}\frac{5/3-\gamma}{\gamma-1}$,
$\gamma=\frac{8-3\beta}{6-3\beta}$,
$g(\alpha,\epsilon^{\prime})=\left[1+\frac{18\alpha^{2}}{(5+2\epsilon^{\prime})^{2}}\right]^{1/2}-1$,
and $c_{3}=(2/3)c_{1}$. $f$ is the fraction of viscously dissipated energy
which is advected, $\beta$ is the ratio of gas pressure and total pressure).

The temperatures are determined by the self-similar solution of the sound
speed (${p\over\rho}=c_{s}^{2}=c_{3}\Omega_{K}^{2}R^{2}$, see Narayan and Yi
[1995b])

$T_{\rm i}+1.08T_{\rm e}=6.66\times 10^{12}\beta c_{3}r^{-1}{\rm K}$ (13)

and the energy balance in electrons

$q_{\rm ie}(\rho,T_{i},T_{e})=q_{\rm rad}(\rho,T_{e})+{\Delta F_{c}/H},$ (14)

where $q_{ie}$ is the heating rate to electrons (see Eq.5), $\Delta F_{\rm
c}/H\approx k_{0}T_{\rm e}^{7/2}/H^{2}$ is an approximation of the conductive
cooling rate, $q_{\rm rad}$ is the radiative cooling rate including the
bremsstrahlung, synchrotron, self-Compton scattering and external Compton
scattering (see Eq.8), which are the functions of $T_{e}$, $\rho$, and $H$
([e.g. Narayan and Yi, 1995b, Manmoto et al., 1997]) and the disk emission.

The disk is heated up by the viscous stress in the accretion and illumination
from the corona, producing radiation with an effective temperature Liu et al.
[2011], $T_{\rm eff}(R)$,

$\sigma T_{\rm eff}^{4}(R)={3GM\dot{M}_{\rm d}(R)\over 8\pi
R^{3}}\left[1-\left({3R_{S}\over R}\right)^{1/2}\right]+{(1-a)L_{\rm
c,int}\over 8\pi}{H_{s}\over(R^{2}+H_{s}^{2})^{3/2}}$ (15)

where $\dot{M}_{\rm d}(R)$ denotes the accretion rate in the disk, $L_{\rm
c,int}=2\int q_{\rm rad}2\pi RHdR$ is volume-integrated radiation luminosity
from the corona on both sides of the disk, $H_{s}\sim(3-10)R_{s}$ is the
height of the coronal illumination approximated as a lamppost, and $a$ is
albedo of the disk. Therefore, the soft photon involved in the inverse Compton
scattering in the corona is dominated by the local disk radiation in the inner
region with an energy density $u=(2/c)\sigma T_{\rm eff}^{4}$; While in the
outer region, photons emitted from the inner disk could overwhelm the local
disk emission. Thus, the energy density $u(R)$ for Compton scattering can be
approximated by

$u(R)\approx{2\over c}\max\left[\sigma T_{\rm
eff}^{4}(R),(1-1/\sqrt{2}){L_{d}\over 8\pi R^{2}}\right],$ (16)

where the second term represents vertically averaged irradiation of disk
emission to the outer corona ($L_{d}$ the total disk luminosity) Liu et al.
[2003].

From Eqs.(12), (13),(14), (16) the temperatures ($T_{i}$, $T_{e}$) and the
density ($\rho$ or $n_{e}$) in the corona, and the effective temperature in
the disk can be calculated numerically for given $\alpha$, $\beta$, $a$, $m$,
$R$, provided that the accretion rate in the corona ($\dot{m}_{c}$), and in
the disk ($\dot{M}_{d}$) are known.

Given the corona temperature and density, the evaporation rate at unit surface
area can be derived from the transition layer, that is [e.g. Liu et al., 2007,
Meyer et al., 2007],

$\dot{m_{\rm z}}=\frac{\gamma-1}{\gamma}\beta\frac{{-F_{\rm c}}}{kT_{\rm
i}/\mu_{i}m_{p}}(1-\sqrt{C})$ (17)

with

$C=\kappa_{0}b\left({\frac{\beta^{2}p^{2}}{\pi
k^{2}}}\right)\left(\frac{T_{\rm cpl}}{{F_{\rm c}}}\right)^{2},$ (18)

where $b=10^{-26.56}{\rm g}\,{\rm cm^{5}}{\rm s^{-3}}{\rm K^{-1/2}}$, $F_{\rm
c}\approx-k_{0}T_{\rm e}^{7/2}/H$, and $T_{\rm cpl}$ the coupling temperature
in the transition layer determined by the energy balance ($q^{+}+q^{c}=q_{\rm
ie}$ with compressive heating rate $q^{c}={f\over 1-\beta}q^{+}$) in this
layer. $T_{i}$ and $p$ are the ion temperature and pressure of the corona.

The positive value of $\dot{m_{\rm z}}$ means that gas evaporates from the
disk to the corona, while negative $\dot{m_{\rm z}}$ means that gas condenses
from the corona to the disk. As a consequence, the accretion rates in the disk
and in the corona vary with the distance. The integrated
evaporation/condensation rate (in unit of Eddington rate $\dot{M}_{\rm Edd}$)
from $R$ to the outer boundary $R_{b}$ is

$\dot{m}_{\rm evap}(R)=\int_{R}^{R_{b}}{4\pi R\over\dot{M}_{\rm
Edd}}\dot{m}_{z}dR.$ (19)

If the gas captured by the black hole is supplied to disk at a rate of
$\dot{m}$, as the Roche-lobe overflow in the LMXBs, disk gas will evaporate to
the corona when it accretes toward the black hole. The accretion rates at
distance $R$ via the disk and the corona are, respectively,

$\dot{m}_{\rm d}(R)=\dot{m}-\dot{m}_{\rm evap}(R){\ \ \rm and\ \ }\dot{m}_{\rm
c}(R)=\dot{m}_{\rm evap}(R).$ (20)

If the gas is supplied to the corona at a rate of $\dot{m}$, gas can condense
to the disk when the gas supply rate is sufficiently high. We then have

$\dot{m}_{\rm d}(R)=-\dot{m}_{\rm evap}(R){\ \ \rm and\ \ }\dot{m}_{\rm
c}(R)=\dot{m}+\dot{m}_{\rm evap}(R).$ (21)

As the $\dot{m}_{\rm evap}(R)$ is an implicit function of $T_{i}$, $T_{e}$,
$\rho$, and $T_{\rm eff}$ for given $\alpha$, $\beta$, $a$, $m$, $R$,
$\dot{m}$, the accretion rates in disk and in corona are the implicit function
of $T_{i}$, $T_{e}$, $\rho$, and $T_{\rm eff}$. Therefore, Eqs.(12),
(13),(14), (15) are a complete set of equations when the accretion rates in
disk and corona are expressed by Eq.(20) or Eq.(21). Solving the equations we
determine the the disk and corona characteristic parameters $T_{i}$, $T_{e}$,
$\rho$, and $T_{\rm eff}$. Other quantities, such as the pressure, viscosity
heating rate and its advection fraction, the evaporation/condensation rate,
the radiation cooling functions by different processes, are also functions of
$T_{i}$, $T_{e}$, $\rho$, $T_{\rm eff}$ and involved in calculating the
characteristic parameters.

###  3.3 Consequence of disk-corona interaction: the two-phase accretion flows

Interaction of the disk and corona results in gas exchange between the two
accretion flows, with an evaporation/condensation rate in dependence on the
property of the two flows. The final steady geometry of the accretion flows
depends not only on the mass-supply rate, but also on how the gas is supplied
to the accretion, the RLOF or the wind.

In the case of RLOF supply to the thin disk, the coronal flow increases during
its accretion toward the black hole as the evaporation gas continuously joins;
Meanwhile the radiation efficiency increases, which depresses the evaporation
from some distance inwards. Thus, the evaporation rate is expected to reach a
maximum value and decreases in the innermost region, as shown by numerical
computation. While the evaporation process diverts the accretion flow from the
disk to the corona, the disk is completely evaporated within a specific region
for lower gas supply rates. This region is filled with coronal gas, and the
accretion takes place in a geometrically thick, hot flow. On the other hand,
for gas supply rate higher than the maximal evaporation rate, the optically
thick disk cannot be significantly depleted at any distance. Thus, the disk
extends to the ISCO with an overlying accreting coronal flow coexisting with
the disk, where continuous evaporation feeds the corona accretion. The typical
accretion geometry is displayed in Figure 2.

Figure 2: Schematic description of disk truncation and the spectral transition
illustrating its dependence on the mass accretion rate. The evaporation rate
(in units of Eddington rate) is illustrated as a function of the distance (in
units of Schwarzschild radius) in the left panel. At low accretion rates
supplied by RLOF, the disk is truncated by evaporation due to insufficient
mass supply at distances determined by the evaporation curve, When the RLOF
rate exceeds the maximum, evaporation cannot evacuate any disk region, thus,
the disk extends to the ISCO and dominates the accretion flow. Figure from Liu
and Taam [2009]

The evolution of accretion geometry during an outburst of LMXBs is illustrated
in Figure 3. On the rise of an outburst the accretion rate increases, pushing
the truncated disk inwards. When the critical accretion rate (of $\sim 0.02$)
is reached, the thin disk extends to the ISCO. Further increase of accretion
rate in the disk leads to efficient Compton cooling and hence condensation of
coronal gas. Then the corona becomes very tenuous. During the decay, the disk
weakens with decrease of accretion rate, is truncated at the most efficient
evaporation region when the accretion rate decreases to a lower critical
value, leaving an inner disk at the intermediate state; The inner disk
eventually disappears with further decrease of accretion rate.

Figure 3: Variation of the accretion flow geometry and spectral states with
the accretion rate of gas supplied to the thin disk, as the Roche-lobe
overflow in low mass X-ray binaries. The disk truncation radius moves inwards
with increase of gas supply rate as a sequence of equilibrium of evaporation
and inflow of gas in the disk; When the gas supply rate is larger than the
maximal evaporation rate, the disk cannot be truncated anymore, in stead, the
evaporation-fed corona becomes weak as the increased Compton cooling to the
corona restrains the evaporation. An inner disk occurs during the decay of an
outburst since the disk is first depleted at the most efficiently evaporating
region around 200 Schwarzschild radii. The inner disk becomes small with
decrease of gas supply rate and eventually disappears.

In the case of wind supply, the spatial extent and thermal state of the
accretion flow in the outer regions can significantly differ from the RLOF. As
revealed by the 3D hydrodynamical simulations Walder et al. [2014], the mass
captured from a stellar wind is heated at the bow shock to temperatures of
$\sim 10^{7}$K, supplied to the accretion in a hot rather than in a cool
physical state. Such gas is not necessarily constrained to lie in the disk
plane, and hence can form a hot accretion flow as a consequence of inefficient
radiative cooling in a geometrically thick flow, i.e., ADAF. By studying such
an ADAF interacting with a pre-existing disk, it is found that the accretion
geometry depends on the accretion rate supplied by the wind or interstellar
medium. At low accretion rates, the gas evaporates from the thin disk, quickly
evacuating the thin disk, if existing initially. Thus, the accretion is via
ADAF when it reaches a steady state. This result is similar to that of cold
gas supply to a thin disk, e.g. the RLOF supply. At high accretion rate, it is
found that the hot gas partially condenses to the underlying cool disk as it
flows toward the black hole. Such a thin accretion disk can be maintained in a
steady state with gas continuously condensing to the disk. Since the
condensation efficiency depends on the gas supply rate, the radial
distribution of mass flow rates in the hot and cool components varies with the
accretion rate supplied by winds, in particular, the thin disk can be
maintained exist only in a limited radial scale in the neighborhood of the
black hole. Therefore, the disk size and its strength relative to the corona
depends on the wind supply rate. The variation of accretion flow geometry with
wind supply rate is displayed in Figure 4.

Figure 4: Variation of the accretion flow geometry with accretion rate of gas
supplied by wind, as that in high mass X-ray binaries or AGNs. The weak wind
captured by the black hole forms an ADAF; When the wind supply rate increases
to a critical value, part of the corona gas can collapse to form a weak disk
in the inner region as a result of over-cooling; Further increase of wind
supply leads to strong condensation and a large disk.

It is displayed above that the accretion flow is a combination of hot and cold
flows, dominated by the ADAF at low accretion rates, while by the thin disk at
high accretion rates. Such a conclusion is valid no matter the accretion gas
is supplied by RLOF or by wind/interstellar medium. This is essential as the
hot gas inevitably interacts with the cold gas during the accretion. The
variation of the gas supply rate changes the relative strength of the thin
disk and the hot accretion flow, leading to a change in the spectrum and even
state transition, as discussed in the following section.

###  3.4 The radiative properties as compared with observations

The emission of accretion flows comes dominantly from the inner region, as a
results of release of gravitational energy into radiation. Given the fact that
the radiation efficiency of hot gas increases toward the black hole (as a
consequence of efficient Coulomb coupling at inner region), the corona (or
ADAF) emission is even more dominated by the inner region Liu et al. [2017],
no matter the corona is compact or extended. Therefore, the total radiation
and SED from the disk and corona are not significantly affected by the style
of gas supply, but are determined by accretion rates flowing respectively in
the inner disk and inner corona, which are eventually determined by the the
gas supply rate. An example of spectral variation with the gas supply rate
from the accretion flows is displayed in Figure 5 for the case of wind
accretion.

Figure 5: Mass flow rate in the disk and corona (right panel) and the
corresponding spectra (left panel). Left panel: Spectra emitted from a disk
and corona with a hot mass supply for a black hole of $10^{8}M_{\odot}$,
viscosity parameter $\alpha=0.3$, magnetic field parameter $\beta=0.95$,
albedo $a=0.15$. Curves from bottom to top refer to mass supply rates of 0.02,
0.03, 0.05, and 0.1. Right panel: The corresponding radial distribution of the
mass flow rate in the disk and corona caused by condensation. Curves in red
(black) refer to the mass flow rate in the disk (corona). Figure adapted from
Qiao and Liu [2017]

The left panel of the figure shows that the spectra are soft at high accretion
rate, with strong disk emission and steep X-ray spectrum. This is caused by
the strong condensation of coronal gas to the disk. When the accretion rate
decreases, condensation becomes weak, thereby the accretion rate in the disk
decreases, as shown in the right panel of Figure 5. Condensation stops when
the gas supply rate decreases to $\sim 0.02$. Correspondingly, the spectrum
undergoes transition from soft state to hard state.

We note that the relative strength of disk and corona, and hence the emission
spectrum, also depend on the viscosity parameter ($\alpha$), the magnetic
field (parameterized in $\beta$ ), and the albedo ($a$) of the disk Qiao and
Liu [2018]. A larger value of $\alpha$ means more heating, leading to a
stronger corona and harder X-ray spectrum; The spectrum is not sensitive to
the magnetic field if the magnetic energy is no larger than the energy of
equipartition (i.e. $0.5<\beta<1$); The effect of albedo is also not very
important since it varies only in a limit range, for example, $a=0.1-0.2$ for
a disk around AGN Haardt and Maraschi [1993]. Nevertheless, a change in the
combination of the three parameters can, to some extent, change the geometry
of the two-phase accretion flows, thereby change the overall spectral shape.
The effects of viscosity and magnetic field on the spectra for $a=0.15$ are
shown in Figure 6.

Figure 6: Spectra dependent on the viscosity (Left panel) and magnetic field
(Right Panel). Figures from Qiao and Liu [2018]

Therefore, the two-phase accretion flows can produce various types of spectra,
which are mainly determined by the gas supply rates, modified by the value of
viscosity as well as magnetic filed and albedo. The transition of spectral
states can occur at an accretion rate depending on the viscous parameter and
evolution history (from hard to soft state or from soft to hard state). The
model naturally interprets the diversity of spectra and state transition of
black holes.

While the general features of the stationary accretion flows are similar for
the accretion supplied by RLOF and wind, there exist some differences. In
contrast to the extensive thin disk formed from RLOF, the disk is much smaller
as fed by the condensation of hot wind because the radiative cooling at outer
region is insufficient to cause condensation. The absence of an outer disk
precludes the possibility of thermal instabilities in the hydrogen ionization
zone and therefore the large outbursts. This interprets why Cyg X-1 is a
persistent source with no large outburst, unlike the transient sources.
Luminosity fluctuations in the wind-fed sources would be a consequence of
turbulence in the accretion wake region, which are likely smaller than that
caused by thermal instabilities. Thus, the wind-fed sources could be described
as persistent, varying occasionally between soft and hard spectral states with
a relatively small amplitude.

The second difference is the presence/absence of hysteresis between the
transition luminosities at hard-to-soft and soft-to-hard transitions. The
hysteresis in the light curve of transient sources [e.g. Maccarone and Coppi,
2003, Zdziarski and Gierliński, 2004] is thought to be caused by the large
difference in Compton cooling rate before the state transitions Liu et al.
[2005], Meyer-Hofmeister et al. [2005]. When the transient sources are in a
deep quiescent state before outbursts are triggered, the evaporation is
efficient as there is no/weak Compton cooling in the corona. Only when the
accretion rate exceeds the maximal evaporation rate can the disk is fully
filled in and spectral state transits. During the decay of an outburst the
strong disk radiation involved in Compton scattering depresses the disk
evaporation, which leads to the disk to be truncated at a lower accretion
rate. However, this effect is insignificant in persistent sources because of a
small difference in the variation of accretion rate in the hard and soft
states Taam et al. [2018]. Therefore, the light curve during the transition
between states is likely to be more symmetrical in persistent systems, as it
is observed in Cyg X-1.

The third distinction is that the accretion rates remained in the innermost
corona are similar ($\dot{m}_{\rm c}\sim 0.02$) for different rates of the hot
gas supply ($\dot{m}>0.02$). This is in contrast to the conclusion from RLOF
that very little gas accretes via a corona at a high gas supply rates, i.e.
the higher gas supply rate, the lower rate in the corona. Therefore, the X-ray
emission in wind accretion systems, which originates from the liberation of
gravitational energy of the coronal gas and enthalpy of the condensed gas, can
be larger than that with only cold gas supply to the disk (e.g. RLOF). Hence,
the wind accrretion model provides insight to the generation of strong X-ray
radiation in luminous AGNs without necessarily invoking artificially heating
to the corona.

Finally, there should be difference in timing properties between the accretion
of RLOF and wind, if spatial region of hot and cold gas is involved in
producing the noise component.

##  4 Conclusion

This paper reviews the physics of formation of multi-phase accretion flows
around black holes. Of importance is the fact that the coupling between the
cold and hot gas leads to gas evaporation or condensation, thereby the
formation of various configurations of accretion flows as a function of
accretion rate and system parameters. In particular, it shows how the gas
supply affects the accretion configuration. For the Roche-lobe overflow
supply, as in low mass X-ray binaries, evaporation from the disk leads to
accretion via a thin disk sandwiched in a corona at high accretion rates, and
an inner ADAF connecting to an outer disk at low accretion rates; For the wind
or interstellar medium supply, an ADAF forms and extends to ISCO at low
accretion rates, while a thin disk forms in the inner region owing to hot gas
condensation, which is coexisted with the left corona at high accretion rates.
The different configuration caused by different gas supply does not
significantly affect the radiation spectrum, except for the relatively strong
coronal emission in the case of wind supply. However, it does cause different
variability features, such as the presence/absence of hysteresis in transition
luminosity, the transient behavior in LMXBs and persistent behavior in Cyg
X-1.

Acknowledgments: Financial support for this work is provided by the National
Natural Science Foundation of China (grants 12073037 and 11773037) and NAOC
Nebula Talents Program.

## References

  * Abramowicz et al. [1995] Abramowicz, M.A., Chen, X., Kato, S., Lasota, J.P., Regev, O., 1995.  Thermal Equilibria of Accretion Disks.  ApJL 438, L37.  doi:10.1086/187709, arXiv:astro-ph/9409018.
  * Abramowicz et al. [1996] Abramowicz, M.A., Chen, X.M., Granath, M., Lasota, J.P., 1996\.  Advection-dominated Accretion Flows around Kerr Black Holes.  ApJ 471, 762.  doi:10.1086/178004, arXiv:astro-ph/9607021.
  * Abramowicz et al. [1988] Abramowicz, M.A., Czerny, B., Lasota, J.P., Szuszkiewicz, E., 1988\.  Slim Accretion Disks.  ApJ 332, 646.  doi:10.1086/166683.
  * Begelman [1978] Begelman, M.C., 1978.  Black holes in radiation-dominated gas: an analogue of the Bondi accretion problem.  MNRAS 184, 53--67.  doi:10.1093/mnras/184.1.53.
  * Bisnovatyi-Kogan and Blinnikov [1976] Bisnovatyi-Kogan, G.S., Blinnikov, S.I., 1976.  A hot corona around a black-hole accretion disk as a model for CYG X-1.  Soviet Astronomy Letters 2, 191--193.  arXiv:astro-ph/0003275.
  * Bu and Gan [2018] Bu, D.F., Gan, Z.M., 2018\.  On the wind production from hot accretion flows with different accretion rates.  MNRAS 474, 1206--1213.  doi:10.1093/mnras/stx2894, arXiv:1711.02238.
  * Bu et al. [2016] Bu, D.F., Yuan, F., Gan, Z.M., Yang, X.H., 2016\.  Hydrodynamical Numerical Simulation of Wind Production from Black Hole Hot Accretion Flows at Very Large Radii.  ApJ 818, 83.  doi:10.3847/0004-637X/818/1/83, arXiv:1510.03124.
  * Chen and Taam [1993] Chen, X., Taam, R.E., 1993\.  The Structure and Stability of Transonic Accretion Disks Surrounding Black Holes.  ApJ 412, 254.  doi:10.1086/172916.
  * Cheng et al. [2020] Cheng, H., Liu, B.F., Liu, J., Liu, Z., Qiao, E., Yuan, W., 2020\.  Magnetic-reconnection-heated corona in active galactic nuclei: refined disc-corona model and application to broad-band radiation.  MNRAS 495, 1158--1171.  doi:10.1093/mnras/staa1250, arXiv:2006.02665.
  * Dubus et al. [2001] Dubus, G., Hameury, J.M., Lasota, J.P., 2001.  The disc instability model for X-ray transients: Evidence for truncation and irradiation.  A&A 373, 251--271.  doi:10.1051/0004-6361:20010632, arXiv:astro-ph/0102237.
  * Dullemond and Spruit [2005] Dullemond, C.P., Spruit, H.C., 2005\.  Evaporation of ion-irradiated disks.  A&A 434, 415--422.  doi:10.1051/0004-6361:20042517, arXiv:astro-ph/0501463.
  * Esin et al. [1997] Esin, A.A., McClintock, J.E., Narayan, R., 1997.  Advection-Dominated Accretion and the Spectral States of Black Hole X-Ray Binaries: Application to Nova Muscae 1991.  ApJ 489, 865--889.  doi:10.1086/304829, arXiv:astro-ph/9705237.
  * Fabian et al. [2000] Fabian, A.C., Iwasawa, K., Reynolds, C.S., Young, A.J., 2000\.  Broad Iron Lines in Active Galactic Nuclei.  PASP 112, 1145--1161.  doi:10.1086/316610, arXiv:astro-ph/0004366.
  * Frank et al. [2002] Frank, J., King, A., Raine, D.J., 2002.  Accretion Power in Astrophysics: Third Edition.
  * Gierliński and Newton [2006] Gierliński, M., Newton, J., 2006\.  X-ray spectral transitions of black holes from RXTE All-Sky Monitor.  MNRAS 370, 837--844.  doi:10.1111/j.1365-2966.2006.10514.x, arXiv:astro-ph/0601676.
  * Gierliński et al. [1999] Gierliński, M., Zdziarski, A.A., Poutanen, J., Coppi, P.S., Ebisawa, K., Johnson, W.N., 1999\.  Radiation mechanisms and geometry of Cygnus X-1 in the soft state.  MNRAS 309, 496--512.  doi:10.1046/j.1365-8711.1999.02875.x, arXiv:astro-ph/9905146.
  * Gofford et al. [2015] Gofford, J., Reeves, J.N., McLaughlin, D.E., Braito, V., Turner, T.J., Tombesi, F., Cappi, M., 2015.  The Suzaku view of highly ionized outflows in AGN - II. Location, energetics and scalings with bolometric luminosity.  MNRAS 451, 4169--4182.  doi:10.1093/mnras/stv1207, arXiv:1506.00614.
  * Gu and Lu [2000] Gu, W.M., Lu, J.F., 2000\.  Bimodal Accretion Disks: Shakura-Sunyaev Disk-Advection-dominated Accretion Flow Transitions.  ApJL 540, L33--L36.  doi:10.1086/312864.
  * Haardt and Maraschi [1991] Haardt, F., Maraschi, L., 1991\.  A two-phase model for the X-ray emission from Seyfert galaxies.  ApJL 380, L51--L54.  doi:10.1086/186171.
  * Haardt and Maraschi [1993] Haardt, F., Maraschi, L., 1993\.  X-ray spectra from two-phase accretion disks.  ApJ 413, 507--517.  doi:10.1086/173020.
  * Hawley and Krolik [2001] Hawley, J.F., Krolik, J.H., 2001\.  Global MHD Simulation of the Inner Accretion Disk in a Pseudo-Newtonian Potential.  ApJ 548, 348--367.  doi:10.1086/318678, arXiv:astro-ph/0006456.
  * Honma [1996] Honma, F., 1996.  Global Structure of Bimodal Accretion Disks around a Black Hole.  PASJ 48, 77--87.  doi:10.1093/pasj/48.1.77.
  * Ichimaru [1977] Ichimaru, S., 1977.  Bimodal behavior of accretion disks: theory and application to Cygnus X-1 transitions.  ApJ 214, 840--855.  doi:10.1086/155314.
  * Jiang et al. [2019] Jiang, Y.F., Blaes, O., Stone, J.M., Davis, S.W., 2019\.  Global Radiation Magnetohydrodynamic Simulations of sub-Eddington Accretion Disks around Supermassive Black Holes.  ApJ 885, 144.  doi:10.3847/1538-4357/ab4a00, arXiv:1904.01674.
  * Jin et al. [2017a] Jin, C., Done, C., Ward, M., 2017a.  Super-Eddington QSO RX J0439.6-5311 - I. Origin of the soft X-ray excess and structure of the inner accretion flow.  MNRAS 468, 3663--3681.  doi:10.1093/mnras/stx718, arXiv:1703.07118.
  * Jin et al. [2017b] Jin, C., Done, C., Ward, M., Gardner, E., 2017b.  Super-Eddington QSO RX J0439.6-5311 - II. Multiwavelength constraints on the global structure of the accretion flow.  MNRAS 471, 706--721.  doi:10.1093/mnras/stx1634, arXiv:1706.08125.
  * Kara et al. [2019] Kara, E., Steiner, J.F., Fabian, A.C., Cackett, E.M., Uttley, P., Remillard, R.A., Gendreau, K.C., Arzoumanian, Z., Altamirano, D., Eikenberry, S., Enoto, T., Homan, J., Neilsen, J., Stevens, A.L., 2019\.  The corona contracts in a black-hole transient.  Nature 565, 198--201.  doi:10.1038/s41586-018-0803-x, arXiv:1901.03877.
  * Kato et al. [2008] Kato, S., Fukue, J., Mineshige, S., 2008.  Black-Hole Accretion Disks --- Towards a New Paradigm ---.
  * Katz [1977] Katz, J.I., 1977.  X-rays from spherical accretion onto degenerate dwarfs.  ApJ 215, 265--275.  doi:10.1086/155355.
  * Lasota et al. [1996] Lasota, J.P., Narayan, R., Yi, I., 1996.  Mechanisms for the outbursts of soft X-ray transients.  A&A 314, 813--820.  arXiv:astro-ph/9605011.
  * Liang and Price [1977] Liang, E.P.T., Price, R.H., 1977\.  Accretion disk coronae and Cygnus X-1.  ApJ 218, 247--252.  doi:10.1086/155677.
  * Liu [2013] Liu, B.F., 2013.  Coupling of the accretion disk and corona around black holes, in: Zhang, C.M., Belloni, T., Méndez, M., Zhang, S.N. (Eds.), Feeding Compact Objects: Accretion on All Scales, pp. 62--65.  doi:10.1017/S1743921312019217.
  * Liu et al. [2011] Liu, B.F., Done, C., Taam, R.E., 2011.  The Effect of Coronal Radiation on a Residual Inner Disk in the Low/Hard Spectral State of Black Hole X-ray Binary Systems.  ApJ 726, 10.  doi:10.1088/0004-637X/726/1/10, arXiv:1011.2580.
  * Liu et al. [2005] Liu, B.F., Meyer, F., Meyer-Hofmeister, E., 2005.  Spectral state transitions in low-mass X-ray binaries - the effect of hard and soft irradiation.  A&A 442, 555--562.  doi:10.1051/0004-6361:20053207, arXiv:astro-ph/0506444.
  * Liu et al. [2006] Liu, B.F., Meyer, F., Meyer-Hofmeister, E., 2006.  An inner disk below the ADAF: the intermediate spectral state of black hole accretion.  A&A 454, L9--L12.  doi:10.1051/0004-6361:20065430, arXiv:astro-ph/0606217.
  * Liu et al. [2002] Liu, B.F., Mineshige, S., Meyer, F., Meyer-Hofmeister, E., Kawaguchi, T., 2002.  Two-Temperature Coronal Flow above a Thin Disk.  ApJ 575, 117--126.  doi:10.1086/341138, arXiv:astro-ph/0204174.
  * Liu et al. [2003] Liu, B.F., Mineshige, S., Ohsuga, K., 2003.  Spectra from a Magnetic Reconnection-heated Corona in Active Galactic Nuclei.  ApJ 587, 571--579.  doi:10.1086/368282, arXiv:astro-ph/0301142.
  * Liu and Taam [2009] Liu, B.F., Taam, R.E., 2009\.  Application of the Disk Evaporation Model to Active Galactic Nuclei.  ApJ 707, 233--242.  doi:10.1088/0004-637X/707/1/233, arXiv:0910.3725.
  * Liu and Taam [2013] Liu, B.F., Taam, R.E., 2013\.  Constraints on the Viscosity and Magnetic Field in Hot Accretion Flows around Black Holes.  ApJS 207, 17.  doi:10.1088/0067-0049/207/1/17, arXiv:1306.5881.
  * Liu et al. [2007] Liu, B.F., Taam, R.E., Meyer-Hofmeister, E., Meyer, F., 2007\.  The Existence of Inner Cool Disks in the Low/Hard State of Accreting Black Holes.  ApJ 671, 695--705.  doi:10.1086/522619, arXiv:0709.0143.
  * Liu et al. [2015] Liu, B.F., Taam, R.E., Qiao, E., Yuan, W., 2015\.  A Hybrid Two Component Accretion Flow Surrounding Supermassive Black Holes in AGNs.  ApJ 806, 223.  doi:10.1088/0004-637X/806/2/223, arXiv:1505.00551.
  * Liu et al. [2017] Liu, B.F., Taam, R.E., Qiao, E., Yuan, W., 2017\.  Centrally Concentrated X-Ray Radiation from an Extended Accreting Corona in Active Galactic Nuclei.  ApJ 847, 96.  doi:10.3847/1538-4357/aa894c, arXiv:1709.09799.
  * Liu et al. [1999] Liu, B.F., Yuan, W., Meyer, F., Meyer-Hofmeister, E., Xie, G.Z., 1999.  Evaporation of Accretion Disks around Black Holes: The Disk-Corona Transition and the Connection to the Advection-dominated Accretion Flow.  ApJL 527, L17--L20.  doi:10.1086/312383, arXiv:astro-ph/9911051.
  * Liu et al. [1995] Liu, F.K., Meyer, F., Meyer-Hofmeister, E., 1995.  Dwarf novae in quiescence: the relationship between disk evaporation and accretion onto a white dwarf.  A&A 300, 823.
  * Liu et al. [2012] Liu, J.Y., Liu, B.F., Qiao, E.L., Mineshige, S., 2012\.  The Structure and Spectral Features of a Thin Disk and Evaporation-fed Corona in High-luminosity Active Galactic Nuclei.  ApJ 754, 81.  doi:10.1088/0004-637X/754/2/81, arXiv:1205.6958.
  * Liu et al. [2016] Liu, J.Y., Qiao, E.L., Liu, B.F., 2016.  Revisiting the Structure and Spectrum of the Magnetic-reconnection-heated Corona in Luminous AGNs.  ApJ 833, 35.  doi:10.3847/1538-4357/833/1/35, arXiv:1611.07149.
  * Lu et al. [2004] Lu, J.F., Lin, Y.Q., Gu, W.M., 2004.  The Shakura-Sunyaev Disk Can Smoothly Match an Advection-dominated Accretion Flow.  ApJL 602, L37--L40.  doi:10.1086/382209, arXiv:astro-ph/0401018.
  * Lynden-Bell and Pringle [1974] Lynden-Bell, D., Pringle, J.E., 1974\.  The evolution of viscous discs and the origin of the nebular variables.  MNRAS 168, 603--637.  doi:10.1093/mnras/168.3.603.
  * Maccarone [2003] Maccarone, T.J., 2003.  Do X-ray binary spectral state transition luminosities vary?  A&A 409, 697--706.  doi:10.1051/0004-6361:20031146, arXiv:astro-ph/0308036.
  * Maccarone and Coppi [2003] Maccarone, T.J., Coppi, P.S., 2003\.  Hysteresis in the light curves of soft X-ray transients.  MNRAS 338, 189--196.  doi:10.1046/j.1365-8711.2003.06040.x, arXiv:astro-ph/0209116.
  * Manmoto et al. [1997] Manmoto, T., Mineshige, S., Kusunose, M., 1997.  Spectrum of Optically Thin Advection-dominated Accretion Flow around a Black Hole: Application to Sagittarius A*.  ApJ 489, 791--803.  doi:10.1086/304817, arXiv:astro-ph/9708234.
  * McKinney et al. [2014] McKinney, J.C., Tchekhovskoy, A., Sadowski, A., Narayan, R., 2014\.  Three-dimensional general relativistic radiation magnetohydrodynamical simulation of super-Eddington accretion, using a new code HARMRAD with M1 closure.  MNRAS 441, 3177--3208.  doi:10.1093/mnras/stu762, arXiv:1312.6127.
  * Meyer et al. [2000a] Meyer, F., Liu, B.F., Meyer-Hofmeister, E., 2000a.  Black hole X-ray binaries: a new view on soft-hard spectral transitions.  A&A 354, L67--L70.  arXiv:astro-ph/0002053.
  * Meyer et al. [2000b] Meyer, F., Liu, B.F., Meyer-Hofmeister, E., 2000b.  Evaporation: The change from accretion via a thin disk to a coronal flow.  A&A 361, 175--188.  arXiv:astro-ph/0007091.
  * Meyer et al. [2007] Meyer, F., Liu, B.F., Meyer-Hofmeister, E., 2007.  Re-condensation from an ADAF into an inner disk: the intermediate state of black hole accretion?  A&A 463, 1--9.  doi:10.1051/0004-6361:20066203, arXiv:astro-ph/0611487.
  * Meyer and Meyer-Hofmeister [1994] Meyer, F., Meyer-Hofmeister, E., 1994\.  Accretion disk evaporation by a coronal siphon flow.  A&A 288, 175--182.
  * Meyer and Meyer-Hofmeister [2002] Meyer, F., Meyer-Hofmeister, E., 2002\.  The effect of disk magnetic fields on the truncation of geometrically thin disks in AGN.  A&A 392, L5--L8.  doi:10.1051/0004-6361:20021075, arXiv:astro-ph/0207573.
  * Meyer-Hofmeister et al. [2005] Meyer-Hofmeister, E., Liu, B.F., Meyer, F., 2005.  Hysteresis in spectral state transitions - a challenge for theoretical modeling.  A&A 432, 181--187.  doi:10.1051/0004-6361:20041631, arXiv:astro-ph/0411145.
  * Meyer-Hofmeister et al. [2009] Meyer-Hofmeister, E., Liu, B.F., Meyer, F., 2009.  The hard to soft spectral transition in LMXBs-affected by recondensation of gas into an inner disk.  A&A 508, 329--337.  doi:10.1051/0004-6361/200913044, arXiv:0910.4273.
  * Meyer-Hofmeister et al. [2012] Meyer-Hofmeister, E., Liu, B.F., Meyer, F., 2012.  Coronae above accretion disks around black holes: the effect of Compton cooling.  A&A 544, A87.  doi:10.1051/0004-6361/201219245, arXiv:1208.0265.
  * Meyer-Hofmeister et al. [2017] Meyer-Hofmeister, E., Liu, B.F., Qiao, E., 2017.  Interaction of the accretion flows in corona and disk near the black hole in active galactic nuclei.  A&A 607, A94.  doi:10.1051/0004-6361/201731105, arXiv:1712.02031.
  * Meyer-Hofmeister et al. [2020] Meyer-Hofmeister, E., Liu, B.F., Qiao, E., Taam, R.E., 2020\.  Wind accretion in Cygnus X-1.  A&A 637, A66.  doi:10.1051/0004-6361/202037561, arXiv:2004.13241.
  * Meyer-Hofmeister and Meyer [2001] Meyer-Hofmeister, E., Meyer, F., 2001\.  The change from accretion via a thin disk to a coronal flow: Dependence on the viscosity of the hot gas.  A&A 380, 739--744.  doi:10.1051/0004-6361:20011449, arXiv:astro-ph/0112431.
  * Meyer-Hofmeister and Meyer [2003] Meyer-Hofmeister, E., Meyer, F., 2003\.  The formation of the coronal flow/ADAF.  A&A 402, 1013--1019.  doi:10.1051/0004-6361:20030320, arXiv:astro-ph/0303525.
  * Meyer-Hofmeister and Meyer [2006] Meyer-Hofmeister, E., Meyer, F., 2006\.  The effect of heat conduction on the interaction of disk and corona around black holes.  A&A 449, 443--447.  doi:10.1051/0004-6361:20053997, arXiv:astro-ph/0512528.
  * Meyer-Hofmeister and Meyer [2011] Meyer-Hofmeister, E., Meyer, F., 2011\.  Broad iron emission lines in Seyfert galaxies - re-condensation of gas onto an inner disk below the ADAF?  A&A 527, A127.  doi:10.1051/0004-6361/201015478, arXiv:1101.4854.
  * Meyer-Hofmeister and Meyer [2014] Meyer-Hofmeister, E., Meyer, F., 2014\.  The relation between radio and X-ray luminosity of black hole binaries: affected by inner cool disks?  A&A 562, A142.  doi:10.1051/0004-6361/201322423, arXiv:1401.7525.
  * Miller and Stone [2000] Miller, K.A., Stone, J.M., 2000\.  The Formation and Structure of a Strongly Magnetized Corona above a Weakly Magnetized Accretion Disk.  ApJ 534, 398--419.  doi:10.1086/308736, arXiv:astro-ph/9912135.
  * Mushotzky et al. [1993] Mushotzky, R.F., Done, C., Pounds, K.A., 1993.  X-ray spectra and time variability of active galactic nuclei.  ARA&A 31, 717--717.  doi:10.1146/annurev.aa.31.090193.003441.
  * Nakamura and Osaki [1993] Nakamura, K., Osaki, Y., 1993\.  Self-consistent accretion disk-coronal model for active galactic nuclei.  PASJ 45, 775--787.
  * Narayan and Yi [1994] Narayan, R., Yi, I., 1994\.  Advection-dominated Accretion: A Self-similar Solution.  ApJL 428, L13.  doi:10.1086/187381, arXiv:astro-ph/9403052.
  * Narayan and Yi [1995a] Narayan, R., Yi, I., 1995a.  Advection-dominated Accretion: Self-Similarity and Bipolar Outflows.  ApJ 444, 231.  doi:10.1086/175599, arXiv:astro-ph/9411058.
  * Narayan and Yi [1995b] Narayan, R., Yi, I., 1995b.  Advection-dominated Accretion: Underfed Black Holes and Neutron Stars.  ApJ 452, 710.  doi:10.1086/176343, arXiv:astro-ph/9411059.
  * Nomura et al. [2016] Nomura, M., Ohsuga, K., Takahashi, H.R., Wada, K., Yoshida, T., 2016.  Radiation hydrodynamic simulations of line-driven disk winds for ultra-fast outflows.  PASJ 68, 16.  doi:10.1093/pasj/psv124, arXiv:1511.08815.
  * Novikov and Thorne [1973] Novikov, I.D., Thorne, K.S., 1973\.  Astrophysics of black holes., in: Black Holes (Les Astres Occlus), pp. 343--450.
  * Ohsuga et al. [2005] Ohsuga, K., Mori, M., Nakamoto, T., Mineshige, S., 2005\.  Supercritical Accretion Flows around Black Holes: Two-dimensional, Radiation Pressure-dominated Disks with Photon Trapping.  ApJ 628, 368--381.  doi:10.1086/430728, arXiv:astro-ph/0504168.
  * Piran [1978] Piran, T., 1978.  The role of viscosity and cooling mechanisms in the stability of accretion disks.  ApJ 221, 652--660.  doi:10.1086/156069.
  * Poutanen and Coppi [1998] Poutanen, J., Coppi, P.S., 1998\.  Unification of Spectral States of Accreting Black Holes.  Physica Scripta Volume T 77, 57\.  arXiv:astro-ph/9711316.
  * Poutanen et al. [2018] Poutanen, J., Veledina, A., Zdziarski, A.A., 2018.  Doughnut strikes sandwich: the geometry of hot medium in accreting black hole X-ray binaries.  A&A 614, A79.  doi:10.1051/0004-6361/201732345, arXiv:1711.08509.
  * Qian et al. [2007] Qian, L., Liu, B.F., Wu, X.B., 2007.  Disk Evaporation-Fed Corona: Structure and Evaporation Features with Magnetic Field.  ApJ 668, 1145--1153.  doi:10.1086/521388, arXiv:0707.0408.
  * Qiao and Liu [2009] Qiao, E., Liu, B.F., 2009\.  Dependence of Spectral State Transition and Disk Truncation on Viscosity Parameter $\alpha$.  PASJ 61, 403.  doi:10.1093/pasj/61.2.403, arXiv:0901.0475.
  * Qiao and Liu [2012] Qiao, E., Liu, B.F., 2012\.  The Emission from an Inner Disk and a Corona in the Low and Intermediate Spectral States of Black Hole X-Ray Binaries.  ApJ 744, 145.  doi:10.1088/0004-637X/744/2/145, arXiv:1109.5824.
  * Qiao and Liu [2013] Qiao, E., Liu, B.F., 2013\.  A Model for the Correlation of Hard X-Ray Index with Eddington Ratio in Black Hole X-Ray Binaries.  ApJ 764, 2.  doi:10.1088/0004-637X/764/1/2, arXiv:1212.1770.
  * Qiao and Liu [2015] Qiao, E., Liu, B.F., 2015\.  A disc corona-jet model for the radio/X-ray correlation in black hole X-ray binaries.  MNRAS 448, 1099--1106.  doi:10.1093/mnras/stv085, arXiv:1501.03565.
  * Qiao and Liu [2017] Qiao, E., Liu, B.F., 2017\.  The condensation of the corona for the correlation between the hard X-ray photon index $\Gamma$ and the reflection scaling factor R in active galactic nuclei.  MNRAS 467, 898--905.  doi:10.1093/mnras/stx121, arXiv:1701.04211.
  * Qiao and Liu [2018] Qiao, E., Liu, B.F., 2018\.  A systematic study of the condensation of the corona and the application for $\Gamma$ 2-10keV-Lbol/LEdd correlation in luminous active galactic nuclei.  MNRAS 477, 210--218.  doi:10.1093/mnras/sty652, arXiv:1803.03375.
  * Qiao and Liu [2019] Qiao, E., Liu, B.F., 2019\.  A model for the radio/X-ray correlation in three neutron star low-mass X-ray binaries 4U 1728-34, Aql X-1ad, and EXO 1745-248.  MNRAS 487, 1626--1633.  doi:10.1093/mnras/stz1365, arXiv:1905.05996.
  * Qiao and Liu [2020] Qiao, E., Liu, B.F., 2020\.  A systematic study of the advection-dominated accretion flow for the origin of the X-ray emission in weakly magnetized low-level accreting neutron stars.  MNRAS 492, 615--627.  doi:10.1093/mnras/stz3510, arXiv:1912.05762.
  * Qiao et al. [2013] Qiao, E., Liu, B.F., Panessa, F., Liu, J.Y., 2013\.  The Disk Evaporation Model for the Spectral Features of Low-luminosity Active Galactic Nuclei.  ApJ 777, 102.  doi:10.1088/0004-637X/777/2/102, arXiv:1309.0090.
  * Raymond et al. [1976] Raymond, J.C., Cox, D.P., Smith, B.W., 1976.  Radiative cooling of a low-density plasma.  ApJ 204, 290--292.  doi:10.1086/154170.
  * Rees et al. [1982] Rees, M.J., Begelman, M.C., Blandford, R.D., Phinney, E.S., 1982\.  Ion-supported tori and the origin of radio jets.  Nature 295, 17--21.  doi:10.1038/295017a0.
  * Reeves et al. [2001] Reeves, J.N., Turner, M.J.L., Pounds, K.A., O’Brien, P.T., Boller, T., Ferrando, P., Kendziorra, E., Vercellone, S., 2001\.  XMM-Newton observation of an unusual iron line in the quasar Markarian 205.  A&A 365, L134--L139.  doi:10.1051/0004-6361:20000424, arXiv:astro-ph/0010490.
  * Remillard and McClintock [2006] Remillard, R.A., McClintock, J.E., 2006\.  X-Ray Properties of Black-Hole Binaries.  ARA&A 44, 49--92.  doi:10.1146/annurev.astro.44.051905.092532, arXiv:astro-ph/0606352.
  * Różańska and Czerny [2000] Różańska, A., Czerny, B., 2000\.  Two-phase radiative/conductive equilibrium in active galactic nuclei and galactic black holes.  MNRAS 316, 473--478.  doi:10.1046/j.1365-8711.2000.03429.x, arXiv:astro-ph/9906101.
  * RóżaŃska and Czerny [2000] RóżaŃska, A., Czerny, B., 2000\.  Vertical structure of the accreting two-temperature corona and the transition to an ADAF.  A&A 360, 1170--1186.  arXiv:astro-ph/0004158.
  * Ruan et al. [2019] Ruan, J.J., Anderson, S.F., Eracleous, M., Green, P.J., Haggard, D., MacLeod, C.L., Runnoe, J.C., Sobolewska, M.A., 2019\.  The Analogous Structure of Accretion Flows in Supermassive and Stellar Mass Black Holes: New Insights from Faded Changing-look Quasars.  ApJ 883, 76.  doi:10.3847/1538-4357/ab3c1a, arXiv:1903.02553.
  * Sadowski and Narayan [2016] Sadowski, A., Narayan, R., 2016\.  Three-dimensional simulations of supercritical black hole accretion discs - luminosities, photon trapping and variability.  MNRAS 456, 3929--3947.  doi:10.1093/mnras/stv2941, arXiv:1509.03168.
  * Shakura and Sunyaev [1973] Shakura, N.I., Sunyaev, R.A., 1973\.  Reprint of 1973A&A....24..337S. Black holes in binary systems. Observational appearance.  A&A 500, 33--51.
  * Shapiro et al. [1976] Shapiro, S.L., Lightman, A.P., Eardley, D.M., 1976.  A two-temperature accretion disk model for Cygnus X-1: structure and spectrum.  ApJ 204, 187--199.  doi:10.1086/154162.
  * Shaviv and Wehrse [1986] Shaviv, G., Wehrse, R., 1986\.  The vertical temperature stratification and corona formation of accretion disc atmospheres.  A&A 159, L5--L7.
  * Shi et al. [2021] Shi, F., Li, Z., Yuan, F., Zhu, B., 2021\.  An energetic hot wind from the low-luminosity active galactic nucleus M81*.  Nature Astronomy 5, 928--935.  doi:10.1038/s41550-021-01394-0, arXiv:2106.04041.
  * Spitzer [1962] Spitzer, L., 1962.  Physics of Fully Ionized Gases.
  * Spruit and Deufel [2002] Spruit, H.C., Deufel, B., 2002\.  The transition from a cool disk to an ion supported flow.  A&A 387, 918--930.  doi:10.1051/0004-6361:20020294, arXiv:astro-ph/0108497.
  * Stepney [1983] Stepney, S., 1983.  Two-body relaxation in relativistic thermal plasmas.  MNRAS 202, 467--481.  doi:10.1093/mnras/202.2.467.
  * Svensson and Zdziarski [1994] Svensson, R., Zdziarski, A.A., 1994\.  Black hole accretion disks with coronae.  ApJ 436, 599--606.  doi:10.1086/174934.
  * Taam et al. [2008] Taam, R.E., Liu, B.F., Meyer, F., Meyer-Hofmeister, E., 2008\.  On the Properties of Inner Cool Disks in the Hard State of Black Hole X-Ray Transient Systems.  ApJ 688, 527--536.  doi:10.1086/591901, arXiv:0807.3402.
  * Taam et al. [2012] Taam, R.E., Liu, B.F., Yuan, W., Qiao, E., 2012\.  Disk Corona Interaction: Mechanism for the Disk Truncation and Spectrum Change in Low-luminosity Active Galactic Nuclei.  ApJ 759, 65.  doi:10.1088/0004-637X/759/1/65, arXiv:1209.4961.
  * Taam et al. [2018] Taam, R.E., Qiao, E., Liu, B.F., Meyer-Hofmeister, E., 2018\.  A Model for Spectral States and Their Transition in Cyg X-1.  ApJ 860, 166.  doi:10.3847/1538-4357/aac50d, arXiv:1805.06076.
  * Tanaka and Shibazaki [1996] Tanaka, Y., Shibazaki, N., 1996\.  X-ray Novae.  ARA&A 34, 607--644.  doi:10.1146/annurev.astro.34.1.607.
  * Tombesi et al. [2010] Tombesi, F., Cappi, M., Reeves, J.N., Palumbo, G.G.C., Yaqoob, T., Braito, V., Dadina, M., 2010.  Evidence for ultra-fast outflows in radio-quiet AGNs. I. Detection and statistical incidence of Fe K-shell absorption lines.  A&A 521, A57.  doi:10.1051/0004-6361/200913440, arXiv:1006.2858.
  * Walder et al. [2014] Walder, R., Melzani, M., Folini, D., Winisdoerffer, C., Favre, J.M., 2014.  Simulation of Microquasars: The Challenge of Scales, in: Pogorelov, N.V., Audit, E., Zank, G.P. (Eds.), 8th International Conference of Numerical Modeling of Space Plasma Flows (ASTRONUM 2013), p. 141.  arXiv:1405.0600.
  * Wang et al. [2013] Wang, Q.D., Nowak, M.A., Markoff, S.B., Baganoff, F.K., Nayakshin, S., Yuan, F., Cuadra, J., Davis, J., Dexter, J., Fabian, A.C., Grosso, N., Haggard, D., Houck, J., Ji, L., Li, Z., Neilsen, J., Porquet, D., Ripple, F., Shcherbakov, R.V., 2013.  Dissecting X-ray-Emitting Gas Around the Center of Our Galaxy.  Science 341, 981--983.  doi:10.1126/science.1240755, arXiv:1307.5845.
  * Yang et al. [2018] Yang, X.H., Bu, D.F., Li, Q.X., 2018.  Numerical Simulations of Winds Driven by Radiation Force from the Corona above a Thin Disk.  ApJ 867, 100.  doi:10.3847/1538-4357/aae4e2, arXiv:1809.09778.
  * Yang et al. [2019] Yang, X.H., Bu, D.F., Li, Q.X., 2019.  Magnetohydrodynamic Numerical Simulation of the Outflows Driven by Magnetic Field and Radiation Force from the Corona above a Thin Disk.  ApJ 881, 34.  doi:10.3847/1538-4357/ab2b47, arXiv:1906.07364.
  * You et al. [2021] You, B., Tuo, Y., Li, C., Wang, W., Zhang, S.N., Zhang, S., Ge, M., Luo, C., Liu, B., Yuan, W., Dai, Z., Liu, J., Qiao, E., Jin, C., Liu, Z., Czerny, B., Wu, Q., Bu, Q., Cai, C., Cao, X., Chang, Z., Chen, G., Chen, L., Chen, T., Chen, Y., Chen, Y., Chen, Y., Cui, W., Cui, W., Deng, J., Dong, Y., Du, Y., Fu, M., Gao, G., Gao, H., Gao, M., Gu, Y., Guan, J., Guo, C., Han, D., Huang, Y., Huo, J., Jia, S., Jiang, L., Jiang, W., Jin, J., Jin, Y., Kong, L., Li, B., Li, C., Li, G., Li, M., Li, T., Li, W., Li, X., Li, X., Li, X., Li, Y., Li, Z., Liang, X., Liao, J., Liu, C., Liu, G., Liu, H., Liu, X., Liu, Y., Lu, B., Lu, F., Lu, X., Luo, Q., Luo, T., Ma, X., Meng, B., Nang, Y., Nie, J., Ou, G., Qu, J., Sai, N., Shang, R., Song, L., Song, X., Sun, L., Tan, Y., Tao, L., Wang, C., Wang, G., Wang, J., Wang, L., Wang, W., Wang, Y., Wen, X., Wu, B., Wu, B., Wu, M., Xiao, G., Xiao, S., Xiong, S., Xu, Y., Yang, J., Yang, S., Yang, Y., Yi, Q., Yin, Q., You, Y., Zhang, A., Zhang, C., Zhang, F., Zhang, H., Zhang, J., Zhang, T., Zhang, W., Zhang, W., Zhang, W., Zhang, Y., Zhang, Y., Zhang, Y., Zhang, Y., Zhang, Z., Zhang, Z., Zhao, H., Zhao, X., Zheng, S., Zhou, D., Zhou, J., Zhu, Y., Zhu, Y., 2021\.  Insight-HXMT observations of jet-like corona in a black hole X-ray binary MAXI J1820+070.  Nature Communications 12, 1025\.  doi:10.1038/s41467-021-21169-5, arXiv:2102.07602.
  * Yuan et al. [2015] Yuan, F., Gan, Z., Narayan, R., Sadowski, A., Bu, D., Bai, X.N., 2015\.  Numerical Simulation of Hot Accretion Flows. III. Revisiting Wind Properties Using the Trajectory Approach.  ApJ 804, 101.  doi:10.1088/0004-637X/804/2/101, arXiv:1501.01197.
  * Yuan and Narayan [2014] Yuan, F., Narayan, R., 2014\.  Hot Accretion Flows Around Black Holes.  ARA&A 52, 529--588.  doi:10.1146/annurev-astro-082812-141003, arXiv:1401.0586.
  * Yuan et al. [2010] Yuan, W., Liu, B.F., Zhou, H., Wang, T.G., 2010\.  X-ray Observational Signature of a Black Hole Accretion Disk in an Active Galactic Nucleus RX J1633+4718.  ApJ 723, 508--513.  doi:10.1088/0004-637X/723/1/508, arXiv:1009.2808.
  * Zdziarski and Gierliński [2004] Zdziarski, A.A., Gierliński, M., 2004\.  Radiative Processes, Spectral States and Variability of Black-Hole Binaries.  Progress of Theoretical Physics Supplement 155, 99--119.  doi:10.1143/PTPS.155.99, arXiv:astro-ph/0403683.
  * Zdziarski et al. [1999] Zdziarski, A.A., Lubiński, P., Smith, D.A., 1999.  Correlation between Compton reflection and X-ray slope in Seyferts and X-ray binaries.  MNRAS 303, L11--L15.  doi:10.1046/j.1365-8711.1999.02343.x, arXiv:astro-ph/9812215.
  * Zhu and Stone [2018] Zhu, Z., Stone, J.M., 2018\.  Global Evolution of an Accretion Disk with a Net Vertical Field: Coronal Accretion, Flux Transport, and Disk Winds.  ApJ 857, 34.  doi:10.3847/1538-4357/aaafc9, arXiv:1701.04627.