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\documentstyle[12pt,aasms4]{article}
\def\Cs{{C_{\rm s}}}
\def\etal{et~al.~}
\def\e{\epsilon}
\def\ellin{\ell_{\rm in}}
\def\Ld{{L_{\rm d}}}
\def\MBH{{M_{\rm BH}}}
\def\Mdot{{\dot M}}
\def\Mdotc{{\dot M_{\rm c}}}
\def\Ms{{M_\odot}}
\def\OmegaK{{\Omega_{\rm K}}}
\def\Qrad{{Q_{\rm rad}^-}}
\def\Qvis{{Q_{\rm vis}^+}}
\def\Qadv{{Q_{\rm adv}^-}}
\def\rin{{r_{\rm in}}}
\def\rout{{r_{\rm out}}}
\def\rS{{r_{\rm g}}}
\def\rg{{r_{\rm g}}}
\def\tff{{\tau_{\rm ff}}}
\def\Tvir{{T_{\rm vir}}}
\def\vp{{v_\varphi}}
\def\vr{{v_{\rm r}}}
\received{4 April 1997}
\accepted{6 June 1997}
\lefthead{MANMOTO, MINESHIGE & KUSUNOSE}
\righthead{SPECTRUM OF ADVECTION DOMINATED ACCRETION FLOW}
\begin{document}
\baselineskip 5mm
\title{SPECTRUM OF OPTICALLY THIN\\
ADVECTION-DOMINATED ACCRETION FLOW AROUND A BLACK HOLE:\\
APPLICATION TO SAGITTARIUS A*}
\author{T. Manmoto and S. Mineshige}
\affil{Department of Astronomy, Faculty of Science,
Kyoto University, Sakyo-ku, Kyoto 606-01, Japan}
\and
\author{M. Kusunose}
\affil{Department of Physics, School of Science
Kwansei Gakuin Univeristy, 1-155 Uegahara Ichibancho
Nishinomiya 662, Japan}
\begin{abstract}
The global structure of optically thin advection dominated accretion flows
which are composed of two-temperature plasma around black holes is
calculated.
We adopt the full set of basic equations including
the advective energy transport in the energy equation for
the electrons. The spectra emitted by the optically thin accretion flows
are also investigated. The radiation mechanisms which are taken into accout
are bremsstrahlung, synchrotron emission, and Comptonization.
The calculation of the spectra and that of the structure
of the accretion flows are made to be completely consistent by calculating
the radiative cooling rate at each radius.
As a result of the advection domination for the
ions, the heat transport from the ions to the electrons becomes practically
zero and the radiative cooling balances with the advective {\it heating}
in the energy equation of
the electrons. Following up on the successful work of Narayan et al.
(1995), we applied our model to the spectrum of Sgr A*. We find that the
spectrum of Sgr A* is explained by the optically thin advection dominated
accretion flow around a black hole of the mass $\MBH=10^{6}\Ms$. The
parameter dependence of the spectrum and the structure of the accretion
flows is also discussed.
\end{abstract}
\keywords{accretion, accretion disks --- black hole physics ---
radiation mechanisms: non-thermal --- Galaxy: center}
\section{Introduction}
Ever since the pioneering studies of steady thin accretion disks by Shakura \&
Sunyaev (1973, hereafter SS),
the model of thin accretion disks has been applied successfully to
low energy emission from astrophysical objects, such as
dwarf novie and pre-main-sequence stars.
However, the model has been less successful in
modeling of high energy emission from Galactic black hole candidates and
active
galactic nuclei which are considered to be powered by accreting black
holes. The main problem lies in
the fact that the thin accretion disks can not reproduce
the observed spectra of such systems.
The thin accretion disk model which SS originally proposed
assumes that the disk is optically thick in the vertical direction
and radiates the energy generated by the viscosity locally.
This model predicts that the generated spectrum is
multi-colored black body, which cannot explain the observed power-law
spectra of X-rays from AGNs
and Galactic black hole candidates
even though it can explain the UV bump of the AGN or the soft
state spectrum of the Galactic black hole candidates.
More fundamentally,
although it is generally believed that QSOs and Seyfert galaxies are powered
by the gas
accretion onto a super-massive black hole of the mass $\MBH\sim 10^8\Ms$,
the thin accretion disks onto such super-massive
black holes are too cool to generate high energy photons
which are observed in many QSOs and Sayfert galaxies.
The model which is investigated by Shapiro, Lightman \& Eardley (1976,
hereafter SLE)
is quite attractive in that the accreted gas is optically thin and is much
hotter than that in the SS solution
and is hot enough to produce high energy photons.
SLE considered two-temperature plasma with ions being much
hotter than electrons, which enabled quantitative studies of the
non-blackbody spectra.
SLE model has been applied to explain the spectrum of X-ray binaries
and active galactic nuclei successfully.
However, it is known that the optically thin hot accretion disk
which is considered in SLE is thermally unstable (Piran 1978).
If the accretion disk is heated up,
then the disk expands and the density decreases, so that the
bremsstrahlung cooling rate decreases. The reduced cooling then causes
the gas to become even hotter, leading to a runaway thermal instability.
For this reason it is not likely that such hot accretion disks exist in real
systems for much longer than the thermal timescale.
The models introduced so far are local solutions in the sense that the heat
generated via viscosity is locally radiated away efficiently, which
corresponds to neglecting the advective energy transport
in the energy equations. When
plasma cannot emit radiation efficiently, the heat generated via viscosity
is advected inwards as the internal energy of the plasma.
Abramowicz et al. (1988) investigated the effect of advection term in
their ``slim disk'' model in the optically thick case,
and made it clear
that there exists advection dominated branch where the viscous heating is
balanced with the advection term rather than the radiative cooling term at
high mass accretion rates.
The optically thin advection dominated solution at low mass accretion
rates is studied intensively by
Abramowicz et al. (1995) and Narayan \&Yi
(1995) (see also Ichimaru 1977, Matsumoto et al. 1985).
Although they claimed that optically thin advection dominated solution is
thermally stable for the long wavelength perturbations, Kato et
al. (1996) showed the possibilities of the instability against the short
wavelength perturbations. Manmoto et al. (1996) demonstrated that such an
instability is favored for explaining rapid X-ray fluctuations from
Galactic black hole candidates and does not affect the global stability of
the accretion flows.
The optically thin advection dominated accretion flows are applied to
explain the observed spectra of accreting black holes.
Narayan \& Yi (1995) investigated the self-similar solution
which is used later to calculate the spectrum from several
low-luminosity accreting systems with great success.
As a next step,
the calculations of full global steady solutions were awaited
for further investigations.
Chen et al. (1997) solved optically thin
advection dominated solution globally, but their solutions are those of
one-temperature plasma and do not include detailed radiation mechanisms.
Narayan et al. (1997) derived global solution for optically thin advection
dominated accretion flows and showed that the self-similar solution is
a good approximation at the radius far enough from the outer and the inner
boundaries. This means that the spectra which are derived by using the
self-similar solution
may be modified, because considerable amount of photons may come from the
hot region near the inner boundary. Narayan et al. (1997)
did calculate the spectra from two-temperature accretion flows
with their global solutions, but they
treated the electron energy equations locally,
neglecting the effect of the electron advection.
Nakamura et al. (1996) were the first to solve the energy equations for
ions and electrons
and obtained the global two-temperature advection dominated solutions,
and showed that the temperature profiles which are crucial to the
generated spectra are largely modified when
the effect of electron advection is taken into account.
However, Nakamura et al. (1996) focused their attentions on the structure
of the optically thin
accretion disks and did not investigate the spectra from the disks.
The study on the spectrum from optically thin advection dominated
accretion flows with full global treatment of the basic equations is yet to
be done.
Thus we are motivated to consistently solve full set of equations including
the energy equation for ions and electrons with detailed radiation
mechanisms and obtain the spectra from the optically thin accretion flows.
In section 2, we present the physical assumptions and the basic equations
of our model. We show the results of our
calculations in section 3. We
then apply our model to Sgr A* (the central core of our Galaxy) in section
4. We conclude in section 5 with a summary and discussion.
\section{Accretion Flow Model}
\subsection{Physical Assumptions}
We consider an optically thin, steady axisymmetric accretion flow around
a black hole.
To investigate the spectra generated by the optically thin gas flows, we
discuss gas dynamics in the context of two-temperature plasma.
Assuming the existence of randomly oriented magnetic fields which possibly
originate from the turbulence in the gas flow, we take total pressure
$p$ to be
\begin{equation}
p=p_{\rm gas}+p_{\rm mag},
\label{eq2.1}
\end{equation}
where $p_{\rm gas}$ is the gas pressure, $p_{\rm mag}$ is the magnetic
pressure.
We neglect the radiation pressure in this paper because the optically thin
accretion
flows we consider is always gas pressure dominated.
We take the ratio of
gas pressure to the total
pressure as a global parameter which we designate as $\beta$. Technically we
need to solve the magnetic field self-consistently with the gas dynamics,
but it is beyond the scope of this paper to treat full
magneto-hydrodynamics equations.
Due to the two-temperature assumption, we write $p_{\rm gas}$ as
\begin{equation}
p_{\rm gas}=\beta p=p_{i}+p_{e}={\rho \over {\mu _{i}}}{k
\over {m_{\rm H}}}T_{i}+{\rho
\over {\mu _{e}}}{k \over {m_{\rm H}}}T_{e}.
\label{eq2.2}
\end{equation}
Here and hereafter subscripts $i$ and $e$ indicate the quantities for
ions and electrons, respectively. In eq. (\ref{eq2.2}), $T$ is the
temperature, $\rho$ is the density, and $\mu$ is the mean molecular weight
which is given by
\begin{equation}
\mu _{i}=1.23,\quad\mu _{e}=1.14,
\label{eq2.3}
\end{equation}
where numerical values correspond to the cosmic abundance.
We estimate magnetic field $B$ via magnetic pressure by the following equation:
\begin{equation}
p_{\rm mag}=\left( {1-\beta } \right)p={{B^2} \over {8\pi }}.
\label{eq2.4}
\end{equation}
\subsection{Basic Equations}
We adopt cylindrical coordinate system ($r$,$\varphi$,$z$) to describe
axisymmetric (
${\partial \over {\partial \varphi }}=0$
) accretion flows. Basic equations which describe the
dynamics of the accretion disks are those of mass conservation, Euler
equations which comprises three spatial components, and the energy
equations.
Mass conservation gives
\begin{equation}
{\partial \over {\partial r}}\left( {r\rho v_{r}} \right)+r{\partial
\over
{\partial z}}\left( {\rho v_{z}} \right)=0,
\label{eq2.5}
\end{equation}
where $v_{r}$ and $v_{z}$ are the
radial and the vertical velocity respectively.
The radial component of Euler equation is
\begin{equation}
{\partial \over {\partial r}}\left( {r\rho v_{r}^2} \right)+r{\partial
\over {\partial z}}\left( {\rho v_{r}v_{z}} \right)=-\rho \left(
{r{{\partial
\psi } \over {\partial r}}-v_{\varphi} ^2} \right)-r{{\partial p}
\over
{\partial r}},
\label{eq2.6}
\end{equation}
where $v_{\varphi}$ is the azimuthal
velocity and $\psi$ is the potential energy. To simulate general relativistic
effects, we adopt pseudo-Newtonian potential
\begin{equation}
\psi =-G\MBH/ (R-\rg)
\label{eq2.7}
\end{equation}
with $\MBH$ being the mass of the black hole, $\rg$ the
Schwarzschild radius, and $R=(r^{2}+z^{2})^{1/2}$ the distance from
the central black hole.
This potential is known to
represent the dynamical aspects of general relativistic effects quite well
for $r>2\rg$
(Paczy\'nski \& Wiita 1980) and greatly simplifies the basic equations.
Technically we need to solve the basic equations in the Schwarzschild
or Kerr metric especially for the photon propagation. However solving
the equations including the photon propagation in the full
relativisitc metric is out of the scope of current paper.
Azimuthal component of Euler equation is the conservation of angular momentum
which gives
\begin{equation}
{\partial \over {\partial r}}\left( {r^2\rho v_rv_\varphi} \right)+
r{\partial \over {\partial z}}\left( {r\rho v_\varphi v_z} \right)=
{\partial \over {\partial r}}\left( {r^2\tau _{r\varphi }} \right).
\label{eq2.8}
\end{equation}
Here $\tau_{r \varphi}$ is the $r \varphi$-component of the stress tensor.
According to the conventional $alpha$-prescription of sheer viscosity,
$\tau_{r \varphi}$ can be written as
\begin{equation}
\tau _{r\varphi }=\alpha p{{d\ln \Omega } \over {d\ln r}}
{\Omega \over {\Omega _k}},
\label{eq2.8.5}
\end{equation}
where $\Omega$ is the angular velocity and $\OmegaK$ is the Keplerian
angular velocity
on the equatorial plane.
In our paper, we take $\tau_{r \varphi}$ to be simply proportional to the
local pressure
$p$:
\begin{equation}
\tau _{r\varphi }=-\alpha p,
\label{eq2.9}
\end{equation}
where $\alpha$ is the dimensionless viscosity parameter which in
general is
considered to be around $0.01\sim0.1$.
Two-temperature assumption requires two energy equations i.e. one for ions and
one for electrons, both of which contain advective energy transport terms.
The energy
equation for each species is
\begin{equation}
\rho T_i\left( {v_r{{\partial s_i} \over {\partial r}}+v_z{{\partial
s_i}
\over {\partial z}}} \right)=\tau _{r\varphi }r{{\partial \Omega }
\over
{\partial r}}-\lambda _{ie},
\label{eq2.10}
\end{equation}
\begin{equation}
\rho T_e\left( {v_r{{\partial s_e} \over {\partial r}}+v_z{{\partial
s_e}
\over {\partial z}}} \right)=\lambda _{ie}-q_{\rm rad}^-,
\label{eq2.11}
\end{equation}
where $s$ is the specific entropy,
$\lambda_{ie}$ is the volume energy transfer rate from ions to electrons,
and $q_{\rm rad}^{-}$ is the volume radiative cooling rate.
We assume that the energy is transferred
from ions to electrons via Coulomb collisions only. Stepney \& Guilbert
(1983) give an explicit expression:
\begin{equation}
\begin{array}{l}
\lambda _{ie}=1.25\times {3 \over 2}{{m_e} \over
{m_p}}n_en_i\sigma _{\rm T}c{{\left( {kT_i-kT_e} \right)} \over
{K_2\left( {1/
\theta _e} \right)K_2\left( {1/ \theta _i} \right)}}\ln \Lambda\\
\quad\times \left[ {{{2\left( {\theta _e+\theta _i} \right)^2+1} \over
{\left( {\theta _e+\theta _i} \right)}}K_1\left( {{{\theta _e+\theta
_i}
\over {\theta _e\theta _i}}} \right)+2K_0\left( {{{\theta _e+\theta
_i}
\over {\theta _e\theta _i}}} \right)} \right]
\end{array}
\label{eq2.12}
\end{equation}
where the $K$'s are modified Bessel functions, $\ln \Lambda=20$ is the
Coulomb logarithm, and
$\theta \equiv kT/ m_ec^2$
is the dimensionless temperature.
Note that the energy
equations assume that the ions and the electrons are in thermal equilibrium by
some mechanisms. The thermodynamic relations are
\begin{equation}
\rho T_ids_i={1 \over {\gamma -1}}\left( {dp_i-\gamma {{p_i} \over
\rho
}d\rho } \right),
\label{eq2.13}
\end{equation}
\begin{equation}
\rho T_eds_e={1 \over {\gamma -1}}\left( {dp_e-\gamma {{p_e} \over
\rho
}d\rho } \right),
\label{eq2.14}
\end{equation}
where $\gamma=5/3$ is the adiabatic index.
In the energy equations we have assumed viscous heating acts only on ions
because an ion particle is much heavier than an electron particle.
Now we have full set of basic equations which describe the steady
accretion flow around a black hole.
However, it is rather a difficult problem to solve above partial differential
equations with respect to $r$ and $z$ plus radiative transfer equations to
obtain global steady solutions. To lessen the number of variables,
we fix the vertical structure before we solve the full spatial
equations, assuming the accretion flow is geometrically thin.
The validity of the assumption that the disk is
geometrically thin will be discussed later.
As a first order approximation, we adopt isothermal structure in the
vertical direction, which means that the sound velocity
$c_{\rm s}\equiv \left( {p/ \rho } \right)^{1/ 2}$
is independent
of $z$. Furthermore, we also assume radial and azimuthal velocities are
independent of $z$.
The vertical structure is obtained analytically by solving the remaining
vertical component of Euler equation which represents hydrostatic balance
in the vertical direction:
\begin{equation}
{{\partial p} \over {\partial z}}=-\rho {{\partial \psi } \over
{\partial
z}}=-\rho \OmegaK^2z.
\label{eq2.15}
\end{equation}
The last equation of eq. (\ref{eq2.15}) assumes that the disk is geometrically
thin.
The density distribution in the vertical
direction is
\begin{equation}
\rho \left( {r,z} \right)=\rho \left( {r,0} \right)\exp \left(
{-{{z^2}
\over {2H^2}}} \right).
\label{eq2.16}
\end{equation}
Here, H is the vertical scale height defined by
\begin{equation}
H\equiv c_s/ \Omega _{\rm K}.
\label{eq2.17}
\end{equation}
With the vertical structure given above, we can now integrate above basic
equations in the vertical direction and rewrite them as follows.
Mass conservation now gives
\begin{equation}
\dot M=2\pi r\Sigma v_r,
\label{eq2.18}
\end{equation}
where
$\Sigma \left( r \right)\equiv\int_{-\infty }^\infty {\rho \left( {r,z}
\right)}dz=2\sqrt {\pi / 2}\rho \left( {r,0} \right)H$
is the surface density. We have taken $v_{r}$ to be positive for the inward
flow so that
the mass accretion rate $\dot M$ takes positive value.
The height-integrated version of the radial component of eq. (\ref{eq2.6}) is
\begin{equation}
v_r{{dv_r} \over {dr}}+{1 \over \Sigma }{{dW} \over {dr}}=r\left(
{\Omega
^2-\OmegaK^2} \right)-{W \over \Sigma }{{d\ln \OmegaK} \over
{dr}},
\label{eq2.19}
\end{equation}
where
$W\left( r \right)\equiv\int_{-\infty }^\infty {p\left( {r,z}
\right)}dz=2\sqrt
{\pi / 2}p\left( {r,0} \right)H$
is the height-integrated pressure. The last
term of the right hand side of eq. (\ref{eq2.19}) corresponds
to the correction for the decrease
of the radial component of gravitational force away from the equatorial
plane (Matsumoto et al. 1984).
We can integrate the conservation of angular momentum (eq.
[\ref{eq2.8}]) with respect to $r$ as
well as $z$ and obtain following simple equation:
\begin{equation}
\dot M\left( {l-l_{\rm in}} \right)=2\pi r^2\alpha W,
\label{eq2.20}
\end{equation}
where $l_{\rm in}$ is the
specific angular momentum swallowed by the central black hole.
The energy equations (eq. [\ref{eq2.10}], [\ref{eq2.11}]) are vertically
integrated
together with the thermodynamic relations (eq. [\ref{eq2.13}],
[\ref{eq2.14}]) and now can
be written as
\begin{equation}
\dot M{{W_i} \over \Sigma }\left( {{{\gamma +1} \over {2\left( {\gamma
-1} \right)}}{{d\ln W_i} \over {dr}}-{{3\gamma -1} \over {2\left(
{\gamma
-1} \right)}}{{d\ln \Sigma } \over {dr}}-{{d\ln \Omega _k} \over
{dr}}}
\right)=\dot M\left( {l-l_{\rm in}} \right){{d\Omega } \over
{dr}}+2\pi
r\Lambda _{ie},
\label{eq2.21}
\end{equation}
\begin{equation}
\dot M{{W_e} \over \Sigma }\left( {{{\gamma +1} \over {2\left( {\gamma
-1} \right)}}{{d\ln W_e} \over {dr}}-{{3\gamma -1} \over {2\left(
{\gamma
-1} \right)}}{{d\ln \Sigma } \over {dr}}-{{d\ln \Omega _k} \over
{dr}}}
\right)=2\pi r\left( {Q_{\rm rad}^--\Lambda _{ie}} \right),
\label{eq2.22}
\end{equation}
where $\Lambda _{ie}\equiv\int_{-\infty }^\infty {\lambda _{ie}
\left( z \right)}dz=\sqrt \pi H\lambda _{ie}\left( 0 \right)$
is the energy transfer rate from ions to
electrons per unit surface area. $Q_{\rm rad}^-$ is the
radiative cooling rate per unit surface area which will be explicitly
given in the next subsection.
The energy equations can be written compactly as follows:
\begin{equation}
Q_{{\rm adv},i}^-=Q_{\rm vis}^+-\Lambda _{ie},
\label{eq2.22.5}
\end{equation}
\begin{equation}
Q_{{\rm adv},e}^-=\Lambda _{ie}-Q_{\rm rad}^-.
\label{eq2.23}
\end{equation}
\subsection{Radiation Mechanism}
Following the work of Narayan \& Yi (1995), we consider three processes for
radiative cooling: bremsstrahlung, synchrotron radiation, and Comptonization
of soft
photons.
In order to obtain the spectrum generated by the accretion flow and the
global structure of the accretion disk, we need to solve global radiation
transfer problem in the radial and the vertical direction which involves
self-absorption and incoherent scatterings. We treat such complicated
problem in a rather simplified way: 1) we assume locally plane parallel gas
configuration at each radius and 2) we separate the Compton scattering process
from the rest of radiation processes, i.e. emission and absorption. This
will not make serious error because the gas is so tenuous that generated
photons rarely experience multiple scatterings before they escape. Note
that at such low frequencies as radio frequencies,
the effect of free-free and synchrotron self-absorption
is so large that the disk can not be considered optically thin, while the
optical depth for scatterings is constant at all frequencies.
We first estimate the spectrum of unscattered photons at given radius by
solving radiative diffusion equation in the vertical direction. For the
isothermal plane parallel gas atmosphere where density configuration is
given by eq. (\ref{eq2.16}) the radiative diffusion equation can be solved
and gives
the energy flux $F_{\nu}$ of the unscattered photons at
given radius (see Appendix B):
\begin{equation}
F_\nu ={{2\pi } \over {\sqrt 3}}B_\nu \left[ {1-\exp \left( {-2\sqrt
3\tau _\nu ^*} \right)} \right],
\label{eq2.24}
\end{equation}
where $\tau _\nu ^*\equiv{{\sqrt \pi } \over 2}\kappa _\nu (0)H$ is the
optical depth for
absorption of the accretion flow in the vertical direction
with $\kappa_{\nu}(0)$ being the absorption coefficient on the
equatorial plane. Assuming LTE, we can write
$\kappa _\nu =\chi _\nu / \left( {4\pi B_\nu } \right)$
where $\chi _\nu =\chi _{\nu ,{\rm brems}}+\chi _{\nu ,{\rm synch}}$
is the emissivity. Note
that equation (\ref{eq2.24}) includes the effects of free-free absorption and
synchrotron self-absorption at low frequencies.
\subsubsection{Bremsstrahlung Emission}
In our model, electron temperature exceeds rest mass energy of an electron in
some cases where electron-electron bremsstrahlung is important as well as
electron-proton bremsstrahlung which dominates electron-electron
bremsstrahlung in the classical temperature regime. Thus the
cooling rate per unit volume for bremsstrahlung is
\begin{equation}
q_{\rm br}^-=q_{ei}^-+q_{ee}^-.
\label{eq2.25}
\end{equation}
Following Narayan \& Yi (1995), we employ
\begin{equation}
q_{ei}^-=1.25n_e^2\sigma _{\rm T}c\alpha _fm_ec^2F_{ei}
\left( {\theta _e} \right),
\label{eq2.26}
\end{equation}
where $\alpha_{f}$ is the fine-structure constant and $F_{ei}$ is
given by
\begin{equation}
F_{ei}\left( {\theta _e} \right)=4\left( {{{2\theta _e} \over {\pi
^3}}}
\right)^{1/ 2}\left( {1+1.781\theta _e^{1.34}} \right)\quad
for\;\theta
_e<1,
\label{eq2.26.5}
\end{equation}
\begin{equation}
F_{ei}\left( {\theta _e} \right)={{9\theta _e} \over {2\pi }}\left[
{\ln
\left( {1.123\theta _e+0.48} \right)+1.5} \right]\quad for\;\theta
_e>1,
\label{eq2.27}
\end{equation}
and
\begin{equation}
\begin{array}{l}
q_{ee}^-=n_e^2cr_e^2\alpha _fc^2{{20} \over {9\pi
^{1/ 2}}}\left( {44-3\pi ^2} \right)\theta _e^{3/ 2}\\
\quad\times
\left( {1+1.1\theta _e+\theta _e^2-1.25\theta _e^{5/ 2}}
\right)\quad for\;\theta _e<1,
\end{array}
\label{eq2.27.5}
\end{equation}
\begin{equation}
q_{ee}^-=n_e^2cr_e^2\alpha _fc^224\theta _e\left( {\ln 1.1232 \theta
_e+1.28} \right)\quad for\;\theta _e>1.
\label{eq2.28}
\end{equation}
With the cooling rate given above, we can write the emissivity
$\chi _{\nu ,{\rm brems}}$ as
\begin{equation}
\chi _{\nu ,{\rm brems}}=q_{\rm br}^-\bar G \exp \left( {{{h\nu }
\over {kT_e}}} \right),
\label{eq2.29}
\end{equation}
where $\bar G$ is the Gaunt factor which is given by (see Rybicki \&
Lightman 1979)
\begin{equation}
\bar G={h \over {kT_{e}}}\left( {{3 \over \pi }{{kT_{e}} \over {h\nu
}}} \right)^
{1/ 2}\quad for\;{{kT_{e}} \over {h\nu }}<1,
\label{eq2.29.5}
\end{equation}
\begin{equation}
\bar G={h \over {kT_{e}}}{{\sqrt 3} \over \pi }\ln \left( {{4 \over
\zeta }{{kT_{e}}
\over {h\nu }}} \right)\quad for\;{{kT_{e}} \over {h\nu }}>1.
\label{eq2.30}
\end{equation}
\subsubsection{Synchrotron Emission}
For the relativistic temperatures for the electrons, and in the presence of
magnetic field which is of the same order as equipartition magnetic field,
synchrotron emission is also very important.
Following Narayan \& Yi (1995), the optically thin
synchrotron emissivity by a relativistic Maxwellian distribution of
electrons is
\begin{equation}
\chi _{\nu ,{\rm synch}}=4.43\times 10^{-30}{{4\pi n_e\nu } \over
{K_2\left(
{1/ \theta _e} \right)}}I'\left( {{{4\pi m_ec\nu } \over {3eB\theta
_e^2}}} \right),
\label{eq2.31}
\end{equation}
where $I'(x)$ is given by
\begin{equation}
I'\left( x \right)={{4.0505} \over {x^{1/ 6}}}\left( {1+{{0.4} \over
{x^{1/ 4}}}+{{0.5316} \over {x^{1/ 2}}}} \right)\exp \left(
{-1.8899x^{1/ 3}} \right).
\label{eq2.32}
\end{equation}
\subsubsection{Compton Scattering}
We proceed to consider the effect of Compton scattering. We make use
of the idea of energy enhancement factor which is derived by Dermer, Liang, \&
Canfield (1991) and modified in
part by Esin et al. (1996).
The energy enhancement factor $\eta$ is defined as the
average energy boost of a photon. The prescription for $\eta$ is
\begin{equation}
\eta =\exp \left( {s\left( {A-1} \right)} \right)\left[ {1-P\left(
{j_{\rm m}+1,As} \right)} \right]+\eta _{\rm max}P
\left( {j_{\rm m}+1,s} \right),
\label{eq2.33}
\end{equation}
where $P$ is the incomplete gamma function and
\begin{equation}
\begin{array}{l}
A=1+4\theta _e+16\theta _e^2,\\
s=\tau _{\rm es}+\tau _{\rm es}^2,\\
\eta _{\rm max}={{3kT_e} \over {h\nu }},\\
j_{\rm m}={{\ln \left( {\eta _{\rm max}} \right)}
\over {\ln \left( A \right)}}.
\end{array}
\label{eq2.34}
\end{equation}
$\tau_{es}$ is the optical depth for scattering:
\begin{equation}
\tau _{\rm es}=2n_e\sigma _{\rm T}H\times \max \left( {1,{1\over\tau
_{\rm eff}}} \right),
\label{eq2.35}
\end{equation}
where $\tau _{\rm eff}\equiv\tau _\nu \left( {1+n_e\sigma _{\rm T}/ \kappa
_\nu } \right)^{1/ 2}$
is the effective optical depth (Rybicki \& Lightman 1979).
Equation (\ref{eq2.32}) gives the correct estimate for the optical depth for
scattering in the presence of the absorption.
Note that the simple treatment given above assumes that the cross-section
is Thomson cross-section rather than exact Klein-Nishina cross-section.
With the energy enhancement factor $\eta$, the local radiative
cooling rate $Q_{\rm rad}^{-}$ is given by
\begin{equation}
Q_{\rm rad}^-=\int {d\nu \eta \left( \nu \right)2F_\nu }.
\label{eq2.36}
\end{equation}
We calculate the radiative cooling rate numerically and use it for the
energy equation for electrons at each radius to calculate the global
solution of the accretion flows.
\subsubsection{Calculation of the Spectrum}
We need more detailed specification for the calculation of the spectrum.
Knowing the spectrum of unscattered photons and the probability that a
photon will suffer scattering given times, we can calculate the Compton
scattered spectrum if we know the information about how the original
spectrum is
modified after experiencing one scattering. Note that the average energy
boost given in (\ref{eq2.31}) is the energy boost {\it averaged} over the
Maxwellian
distribution of electrons and can not be used directly for the spectrum
calculations. Such problem was precisely discussed first by
Jones (1968) and corrected
afterwards by Coppi \& Blandford (1990).
We make use of the formula given by Coppi \& Blandford (1990) and calculate
the
resulting spectrum.
The remaining problem concerning Compton
scattering is how to treat the spectrum of the saturated Comptonized
photons. We assume that photons which are scattered more than $j_{m}$ times
saturate and obey the Wien distribution
$\propto \nu ^3\exp \left( {-h\nu / kT_e} \right)$.
Since large fraction of the emitted radiation is generated at the radius
fairly close to the black holes in our model, we cannot ignore the effect
of redshift due to the gravity and the gas motion. We include the
gravitational
redshift by simply taking the ratio of the energy of a photon when observed to
its energy emitted at radius r to be $\sqrt {1-\rg/ r}$.
To treat redshift due to relativistic gas motion in a simple way,
we concentrate on the face-on case where the optically thin assumption is most
adequate. Thus we simply take the ratio of energy change for the redshift
due to
gas motion to be $1/ \sqrt {1-(v/ c)^2}$.
\subsection{Numerical Procedure}
We solve numerically the set of equations given so far with the boundary
conditions. The outer boundary conditions imposed are
\begin{equation}
\Omega =0.8\OmegaK,
\label{eq2.36.5}
\end{equation}
\begin{equation}
T_{\rm gas}\equiv
\mu \left( {{{T_i} \over {\mu _i}}+{{T_e} \over {\mu _e}}} \right)=
0.1T_{\rm vir},
\label{eq2.36.75}
\end{equation}
\begin{equation}
Q_{\rm rad}^-=\Lambda _{ie}
\label{eq2.37}
\end{equation}
at $r_{\rm out}=10000\rg$. Here $T_{\rm vir}$ is the virial temperature
defined by
\begin{equation}
T_{\rm vir}\equiv\left( {\gamma -1} \right){{G\MBH m_{\rm H}} \over
{kr}}
\label{eq2.38}
\end{equation}
We did not set the
angular velocity to be the Keplerian angular velocity itself at the outer
boundary for simply technical reasons.
We need to set somewhat sub-Keplerian disk at the outer boundary so
that the viscous heating term take positive value with
the simple viscosity prescription adopted in this paper [eq.
(\ref{eq2.9})].
We confirmed that the outer boundary condition have little effect on
the structure of the inner advection-dominated flows.
There have been many suggestions about how outer Keplerian disks are
connected to the advection dominated disks. For instance, Honma
(1996) took into account the effect of thermal conductivity and obtained
the global solutions where the outer cool Keplerian disk is connected to
the inner hot optically thin disk. However the mechanism of the transition
is not yet clear.
The free parameters in the set of equations are $\alpha$, $\beta$,
$\MBH$, $\dot M$, and $l_{\rm in}$.
These five parameters are not independent because of the
transonic nature of the equations. $l_{\rm in}$ can be determined uniquely
so that the
solution should satisfy the transonic condition when the remaining parameters
are given. We have to adjust $l_{\rm in}$ recursively to obtain smooth
transonic
solutions, since the location of the transonic point is not known
until we obtain global transonic solutions.
\section{Results}
To discuss the general properties of our model,
we set $m\equiv\MBH/\Ms=10^8$ and $\dot m\equiv\Mdot/\Mdotc=10^{-4}$
where $\Mdotc\equiv32\pi c\rg/ \kappa _{\rm es}$
and assign typical values for other parameters:
$\alpha=0.1$, $\beta=0.5$ (case of equipartition).
To demonstrate how the heatings and coolings balance in the energy equations,
we show Figure 1 the $Q$'s for the ions (upper panel), and for the electrons
(lower panel); (see eq. [\ref{eq2.23}]).
The solid line in the upper panel shows the ratio of advective cooling of ions
to the viscous heating, which is commonly denoted by $f$. We have $f=1$ for
the
purely advection dominated accretion flows and $f=0$ for the purely radiative
cooling dominated accretion flows. We see that $f$ asymptotically approaches
unity as the radius decreases.
We use the region $r<2000 \rg$ to investigate the
spectrum from the optically thin advection dominated accretion flows.
We see from Figure 1 that the accretion flows becomes highly
advection dominated in the region $r<100 \rg$ where the heat transport from
ions to
electrons by the Coulomb coupling practically becomes zero. The interesting
point
is that it is electron advective {\it heating},
rather than the heat supplied from the ions, that balances with the
radiative cooling of electrons, which means the electron accretion
flow becomes cooling flow near the central black hole where
the electrons are cooled by consuming the stored internal energy
to radiate before being heated up by the ions (cf. Nakamura et al. 1997).
Many analyses concerning two-temperature accretion flows have
adopted simplified energy equation for electrons: $\Lambda
_{ie}=Q_{\rm rad}^-$.
Our results shows that above simplified energy equation is inadequate for the
analysis of two-temperature advection dominated flows and one should adopt
$Q_{{\rm adv},e}^-=-Q_{\rm rad}^-$ instead.
Figure 2 shows the ion and electron temperature profiles. We see that the
electron temperature is heated up to $T_{e}\sim10^{10}{\rm K}$.
Although we need to include the effect of electron-positron pair production
and annihilation for such hot accretion flows, we make simple assumption
that the pair density is very low (see discussion in Esin et al. 1996,
Bjornsson et al. 1996, Kusunose \& Mineshige 1996).
Figure 3 shows the aspect ratio $h/r$ of the accretion flow.
We have used height-integrated equations for our calculations which neglect
the higher order of $h/r$. We may conclude that the value of $h/r$ shown in
Figure 3 is marginally
safe for the height integration. However, of course, the vertical structure
of the advection dominated accretion flows is an important issue and we
will investigate it in future papers.
Figure 4 shows the angular momentum and the various velocities as a function
of the
radius $r$. Here $c_{\rm s}^{*}$ is defined as
\begin{equation}
c_{\rm s}^{*}\equiv\left( {{{\left( {3\gamma -1} \right)+
2\left( {\gamma -1} \right)\alpha ^2} \over {\gamma +1}}
{W \over \Sigma }} \right)^{1/ 2},
\label{eq3.1}
\end{equation}
so that $v_{r}=c_{\rm s}^{*}$ at the critical point.
We see from the Figure 4 that the radial dependence of the radial and the
azimuthal velocities are different from that of the self-similar
solutions, in which all the velocities are proportional to $r^{-1/2}$,
especially in the super-sonic region.
We see that for the parameters given above,
the azimuthal velocity $v_{\varphi}$ is fairly
sub-Keplerian and even sub-sonic and of the
same order as the radial velocity $v_{r}$ unlike the standard model,
which is known to be a common feature for $\alpha=0.1$.
(In the standard model, we have $v_\varphi \sim r \OmegaK = c_{\rm
s}^{*}h/ r\gg c_{\rm s}^{*}\gg v_{r}$.)
Note that we have `sub-sonic' azimuthal velocity when the rotation is
highly sub-Keplerian ($v_\varphi < r \OmegaK$) and the disk is hot and
geometrically thick ($h/r\sim 1$).
For the reason that the radial velocity $v_{r}$ is the same order as the
azimuthal velocity $v_{\varphi}$,
we call the accreting gas ``accretion flow'' rather
than to call ``accretion disk'' in this paper.
Figure 5 illustrates the luminosity distribution as a function of $r$. The
contributions from respective radiation mechanisms (i.e. bremsstrahlung,
synchrotron, Comptonization) are also shown. We see that the bremsstrahlung
emission is important and the effect of the
synchrtron emmision and the
Comptonization is negligibly small in the outer region of the accretion
flow ($r>10\rg$), while the synchrotron emission and the Comptonization
dominate in the
hot inner region ($r<10\rg$).
The contribution of the Comptonization rapidly increases as the
radius decreases because the amount of the synchrotron soft photons
increases and not because the Compton $y$-parameter $y\equiv\tau_{\rm
es}kT_{e}/m_{e}c^{2}$ increases. Note that in the innermost region
the electron temperature $T_{e}$ is approximately constant and the
surface density decreases and thereby $y$ actually decreases.
Note that almost all emission comes from the hot inner
region ($r<10\rg$) and the fraction of the emission from the super-sonic
region, which the self-similar solutions fail to describe accurately,
is fairly large. As far as the
bremsstrahlung is concerned, the contribution of the emission from the
large radii is not negligible, which make the slope of the
bremsstrahlung spectrum less steep, since what we observe is the
superposition of the bremsstrahlung peaks from different radii.
In Figure 6, we shows the spectrum generated by the optically thin
accretion flows.
Parameters are $m=10^8$, $\dot m=10^{-4}$,
$\alpha=0.1$, $\beta=0.5$.
S indicates the synchrotron
peak which is composed of Rayleigh-Jeans slope and the optically thin
synchrotron emission. C1 and C2 indicate the once and twice Compton
scattered photons, respectively. B indicates the bremsstrahlung
emission plus
photons suffering multiple Compton scattering. W indicates saturated
Comptonized photons which form Wien tail.
The upper panel of Figure 7 shows the surface density of the accretion flow.
The dashed line
corresponds to the case with the central black
hole mass of $\MBH=10\Ms$. We find that the structure of the accretion
flow is nearly the same when
the radius and the mass accretion rate are
scaled with the Schwarzschild radius and the critical
mass accretion rate, respectively. However we see in the lower panel
of the Figure 7 that the electron temperature at the innermost region
is varied slightly.
This is due to the fact that the accretion flow is highly
advection dominated.
We remind the readers that the viscous heating is balanced almost entirely
with the advective cooling in the ion energy equation and there is little
coupling between the ions and the electrons. Thus the structure of the
accretion flow is governed by ion energy equation while the electron energy
equation including the radiative cooling is decoupled and determines the
electron temperature. Although the height-integrated quantities are the
same when the radius is scaled with the Schwarzschild radius, the amount of
radiative
cooling, which is the function of the density distribution rather
than the height-integrated quantities, is
different. Thus we have different electron temperatures for different black
hole masses.
We show the spectrum from the accretion flow for the $10\Ms$ case in
Figure 8. The shape of the spectrum is essentially the same as the
$10^{8}\Ms$ case, but the position of the synchrotron peak and the
absolute luminosity differs considerably.
\section{Application to Sagittarius A*}
Following up on the successful work of Narayan et al. (1995) which applied
the
advection dominated model to the Sgr A* (the central core of our Galaxy),
we improve their model
by fully solving the basic equations to explain the observed emission from
radio
frequencies to Gamma-rays.
Figure 9 shows the model which explains observed radio and X-ray data
quite well.
The parameters assigned are given in the
figure. The points and the short lines are the observational data,
which assume interstellar absorption ${\rm N}_{H}=6\times10^{22}cm^{-2}$
and a distance $d=8.5{\rm kpc}$, both typical for the Galactic Center,
compiled by Narayan et al. (1995) (references therein).
The various lines correspond to the
different values of mass accretion rate which varies by factor of 2.
As Narayan et al. noticed, the position of the Rayleigh-Jeans slope is
determined solely by the mass of the central black hole.
Thus to give a good fit to the
observed radio emission with our model, we have no choice but to fix the
mass of the
central black hole to be $\MBH=10^{6}\Ms$.
Considering the inaccuracy of the potential at
the innermost region of the accretion flow, this value is consistent with
the value which is derived from the gas and stellar dynamics
(Genzel \& Townes 1987). We have $\Mdotc=3.5\times10^{-2}\Ms/{\rm yr}$
for the black hole of the mass $\MBH=10^{6}\Ms$,
hence the predicted mass accretion rate is $\sim 2-5 \times 10^{-6}\Ms/{\rm
yr}$.
Figure 9 also illustrates the $\Mdot$ dependence of our model. The surface
density sensitively depends on $\Mdot$, while the temperature is fairly
insensitive to $\Mdot$. Thus the luminosity at all frequencies decreases
when we reduce the mass accretion rate.
We find that if we change the mass
accretion rate by a factor of $\sim2$, X-ray luminosity varies by the same
factor
while the synchrotron peak varies little. We suggest that the various
X-ray data seemingly inconsistent with each other are purely due to the
change of the mass accretion rate of the accretion flow.
There have been an argument that the X-ray luminosity of Sgr A* is variable
on the timescale of half a year, which is consistent with our suggestion.
Figure 10 illustrates
how the spectrum changes with different black hole masses. As we mentioned
before, the structure of the accretion flow is the same with proper
scalings, but the electron temperature is slightly different because of the
difference of the emissivity.
We see clearly that the position of the Rayleigh-Jeans slope is
determined by the mass of the central black hole.
We show in Figure 11 the $\beta$-dependence of the structure of the accretion
flow and the spectrum. We remind the readers that the parameter $\beta$ is
defined as the ratio of the gas pressure to the total pressure. We have
the stronger magnetic field for the smaller value of $\beta$.
Unlike the other parameter dependence, the ion temperature as well as the
electron temperature decreases when we lower the value of $\beta$. This is
reasonable because for the lower value of $\beta$, the stronger becomes the
magnetic
pressure and the gas pressure can be smaller to support the accretion flow.
This temperature change has a large effect on the bremsstrahlung spectrum but
has little effect on the synchrotron peak, which is because the effects of
the stronger (weaker) magnetic field and the lower (higher) temperature
roughly cancel out.
We set $T_{\rm gas}=0.2T_{\rm vir}$ at the outer boundary for
$\beta=0.95$ since we
could not find advection dominated solution for $\beta=0.95$ with
$T_{\rm gas}=0.1T_{\rm vir}$.
Narayan et al. (1995) did not investigate the parameter dependence of the
viscosity
parameter $\alpha$ and the mass accretion rate $\Mdot$ independently, since
they used simplified disk model, in which $\alpha$ and $\Mdot$
cannot be chosen independently.
Thus it is meaningful to study the $\alpha$ dependence of the spectrum. We
show in Figure 12 the $\alpha$-dependence of our model.
When $\alpha$ is large, the angular momentum is
extracted efficiently and the surface density decreases, which makes the
bremsstrahlung emission weak.
However, the increase of the electron temperature at the innermost region
makes the synchrotron emission stronger. We find that the width of the
synchrotron peak is the most sensitive to the value of $\alpha$. The upper
limits in the radio frequency band imposes strong restriction upon the
value of $\alpha$ for the case of Sgr A*. For instance, we can not fit the
entire spectrum with $\alpha=0.1$. We have to set $\alpha<0.025$ to explain
entire spectrum of Sgr A* with our model. This fact is important
because it is claimed that $\alpha\sim0.1$ in advection-dominated
accretion flows on various grounds (see discussions in Narayan 1996).
If we are to set $\alpha=0.1$, we have to reduce the mass accretion
rate by factor of 4 so as to give a good fit to the observed radio and
IR spectrum (see Figure 12).
In that case, we cannot explain the X-ray emission from
the Galactic Center. Various X-ray observations have found the X-ray
sources at the Galactic Center but the angular resolution has been
insufficient to identify a source with the radio source Sgr A*.
There remains the possibility that X-rays do not come from Sgr A* at all
(e.g. see Duschl et al. 1996). If that is the case, all the X-ray
data are merely upper limits and the accretion flows with $\alpha=0.1$
do not conflict with the observations.
There exist some data points in near IR and radio frequencies
which cannot be accounted for. However, it is natural to think
that there must be dust region or stellar contamination or non-thermal
objects like jets in central region of our Galaxy overlapping the
accreting black hole which in total we observe as a
point source Sgr A*. In that sense, one should consider the observational
data as upper limits.
A point worth emphasizing here is that the entire spectrum of Sgr A* is
basically explained with an optically thin accretion flow around a black
hole of the mass $\MBH=10^{6}\Ms$. Moreover, we have solved globally the
basic
equations including the gas dynamics and the radiation processes
consistently rather than to consider more primitive models like isothermal
gas complex of certain size.
\section{Summary and Discussion}
In this paper, we have calculated the global structure of optically
thin advection dominated accretion flows in the context of two-temperature
plasma, adopting the full set of basic equations including the energy
equation for the electrons. We have also calculated the spectra
emitted by the optically thin accretion flows which we calculated.
We have made the calculation of the spectra and that of the structure of the
accretion flows to be completely consistent by calculating the
radiative cooling rate at each radius by numerically integrating
the whole spectrum emitted at the radius.
As a result of the advection domination for the ions, the heat
transport from the ions to the electrons becomes practically zero and
the radiative cooling balances with the advective {\it heating} of
the electrons. This means that the electron cools itself by releasing
the stored internal energy as a radiation. Hence the energy equation
for the electrons play an important role for the calculation of the
spectra, where the temperature profile of the electron is the
important factor.
An interesting feature of the advection dominated flow,
which is known already, is that the
azimuthal velocity becomes
highly sub-Keplerian and of the same order as
the radial velocity and the sound velocity.
The point worth noting is that in the innermost hot luminous region,
the divergence of the velocities from those in the self-similar solution
is fairly large.
The accreting gas becomes very hot. The electron temperature even exceeds
the rest mass energy of an electron. We have not taken into account
the effect of the pair production and the annihilation, which is
an important issue. For such hot accretion flows, the synchrotron
emission and the Compton scattering are very important.
The spectrum is composed by 1) the synchrotron peak which comprises
optically thin synchrotron emission and the self-absorbed
Rayleigh-Jeans slope and 2) the unsaturated Comptonized photons which
forms some bumps and 3) the bremsstrahlung emission and 4)the
saturated Comptonized photons. The dependence of the each component on
the model parameters is complex. Among them, the position of the
Rayleigh-Jeans slope is almost solely determined by the mass of the
central black hole. When we make the magnetic field stronger, the
temperature of the entire flow decreases, which has significant effect
on the Comptonization and the bremsstrahlung emission, but has little
effect on the synchrotron emission. When we make the viscosity
smaller, the surface density increases and the bremsstrahlung emission
increases, but the synchrotron emission and the Comptonization
decreases. The simplest relation is the dependence on the mass
accretion rate. When we reduce the mass accretion rate, the entire
emission is reduced. However the bremsstrahlung emission is much more
sensitive to the change of the mass accretion rate than the
synchrotron emission.
We find that the spectrum of Sgr A* is explained by the optically thin
advection dominated accretion flow around a black hole of the mass
$\MBH=10^{6}\Ms$. Narayan et al. (1995) also calculated the spectrum of
Sgr A* using an optically thin advection dominated accretion flow model.
Their best fit parameters are different from ours.
For instance the mass of the central black hole is $\MBH=7.0\times
10^{5}\Ms$ according to their model.
The different points in our model are 1) full global treatment of the
basic equations and 2) inclusion of the electron energy equation and
3) calculation of the innermost region where the flow is supersonic
and the effects of the redshift are important. 2) and 3) have very
important effect on the emitted spectrum, which is not considered in
Narayan et al. (1995).
We conclude that the X-ray data obtained by various satellite
observations are explained by the variation of the mass accretion
rate by a factor of $\sim 2$, if we allow $\alpha$ to have small value.
If we set $\alpha\sim0.1$, which is considered to be a standard value
for the advection dominated accretion flows, it is not likely that
the X-rays come from Sgr A*.
We have computed the model using height-integrated equations with fixed
structure in the vertical direction. We have also adopted simplified
form of equations for the radiation field. Our immediate goal is to
solve the basic equations including equations for the radiation field
in at least two dimensional space.
However, our successful result presented in this paper tells us that the
basic idea is correct.
The full treatment of Schwarzschild or Kerr metric is also an
important issue.
\acknowledgments
We thank Professor Shoji Kato for useful discussions,
and Ann Esin and Jun Fukue for useful comments. We are very grateful to
R. Narayan for providing us with a code to calculate
inverse-Compton spectra. We thank the referee J. P. Lasota for
many valuable comments and suggestions.
\appendix
\section{Height Integrations of the Basic Equations}
In this paper, we have adopted the height-integrated equations to
obtain the structure of the optically thin advection dominated accretion
flows. We have seen that the temperature of the accretion flows is
very high and the aspect ratio $h/r$ is $\sim 0.5$.
Thus the height-integration may not be an excellent approximation since
the operation of
the height-integration omits the higher orders of $h/r$.
Thus we consider it useful to describe in detail the operation of the
height-integration to discuss the reliability of our calculations.
The potential we adopted is spherically symmetric, while we adopt the
cylindrical coordinate system. Thus we need to expand it around the
equatorial plane ($z=0$) in order to integrate the basic equations
in the vertical direction.
In the case of geometrically thin accretion flows, we can neglect
the terms containing the higher orders of $z/r$ and the operation of
the height-integration becomes very simple.
The potential $\psi$ is expanded as
\begin{equation}
\psi \left( {r,z} \right)=\psi \left( {r,0} \right)\left[ {1+{1 \over
2}\left. {{{\partial \ln \psi } \over {\partial \ln r}}}
\right|_{z=0}{{z^2} \over {r^2}}+O\left( {{{z^4} \over {r^4}}}
\right)}
\right].
\label{eqA.1}
\end{equation}
Thus we approximate $\psi$ as
\begin{equation}
\psi \left( {r,z} \right)=\psi \left( {r,0} \right)+{1 \over 2}
\Omega _k^2z^2,
\label{eqA.2}
\end{equation}
where $\Omega_{k}$ is defined as
\begin{equation}
\Omega _k\equiv \left. {\left( {{1 \over r}{{\partial \psi } \over
{\partial r}}} \right)^{1/ 2}} \right|_{z=0}.
\label{eqA.3}
\end{equation}
Using Equation (A.1), the derivatives of $\psi$ are also expanded as
follows:
\begin{equation}
{{\partial \psi } \over {\partial r}}=\left. {{{\partial \psi } \over
{\partial r}}} \right|_{z=0}\left[ {1+{{d\ln \Omega _k} \over {d\ln
r}}{{z^2} \over {r^2}}+O\left( {{{z^4} \over {r^4}}} \right)} \right],
\label{eqA.4}
\end{equation}
\begin{equation}
{{\partial \psi } \over {\partial z}}=\left. {{{\partial \psi } \over
{\partial r}}} \right|_{z=0}{z \over r}\left[ {1+O\left( {{{z^3} \over
{r^3}}} \right)} \right].
\label{eqA.5}
\end{equation}
We approximate the derivatives of $\psi$ as follows:
\begin{equation}
{{\partial \psi } \over {\partial r}}=r\Omega _k^2\left( {1+{{d\ln
\Omega
_k} \over {d\ln r}}{{z^2} \over {r^2}}} \right),
\label{eqA.6}
\end{equation}
\begin{equation}
{{\partial \psi } \over {\partial z}}=\Omega _k^2z.
\label{eqA.7}
\end{equation}
We have used eq. (\ref{eqA.7}) to determine the vertical structure of the
accretion flows [see eq. (\ref{eq2.15})].
We then integrate the basic equations in the vertical direction. The
integrations of the continuity equation (eq. [\ref{eq2.5}]) and the azimuthal
component of the Euler equation (eq. [\ref{eq2.8}]) are straightforward. In
the process of the integration of the radial component of the Euler
equation, we encounter the following integration:
\begin{equation}
\int_{-\infty }^\infty {\rho {{\partial \psi } \over {\partial
r}}}dz=\Sigma r\Omega _k^2+W{{d\ln \Omega _k} \over {dr}}.
\label{eqA.8}
\end{equation}
The last term in the right hand side of eq. (\ref{eqA.8}) is sometimes
omitted in the height-integrated equations which appear in the papers
on the accretion disks, which is not proper, since it can be
the same order as the other terms in the Euler equation.
The left hand sides of the energy equations (eq. [\ref{eq2.10}], eq.
[\ref{eq2.11}]) together with the
thermodynamic relations (eq. [\ref{eq2.13}], eq. [\ref{eq2.14}])
are transformed to give
\begin{equation}
\rho T\left( {v_r{{\partial s} \over {\partial r}}+v_z{{\partial s}
\over
{\partial z}}} \right)={\gamma \over {\gamma -1}}\left[ {{1 \over
r}{\partial \over {\partial r}}\left( {rv_rp} \right)+{\partial \over
{\partial z}}\left( {v_zp} \right)} \right]-v_r{{\partial p} \over
{\partial r}}-v_z{{\partial p} \over {\partial z}}.
\label{eqA.9}
\end{equation}
We can integrate Equation (\ref{eqA.9}) with the integration
\begin{equation}
\int_{-\infty }^\infty {v_z{{\partial p} \over {\partial
z}}}dz=-Wv_r{{d\ln H} \over {dr}},
\label{eqA.10}
\end{equation}
to obtain the height-integrated version of the Energy equations (eq.
[\ref{eq2.21}], [\ref{eq2.22}]).
\section{Calculation of The Flux of Unscattered Photons}
Consider the isothermal plane parallel atmosphere with the density
distribution being
\begin{equation}
\rho \left( z \right)=\rho \left( 0 \right)\exp \left( {-{{z^2} \over
{2H^2}}} \right).
\label{eqB.1}
\end{equation}
Assuming the Eddington Approximation which is valid for isotropic
radiation fields (and even for slightly nonisotropic fields, see
Rybicki \& Lightman 1979), the radiation field in the vertical
direction is described by the radiative diffusion equation:
\begin{equation}
{1 \over 3}{{\partial ^2J_\nu } \over {\partial \tau _\nu ^2}}=J_\nu
-B_\nu ,
\label{eqB.2}
\end{equation}
where $\tau _\nu$ is the optical depth from the surface of the
accretion flow. There is no well-defined surface of the accretion
flow since the density tends to zero
when we increase $z$ but will never equals to zero [see eq.
(\ref{eqB.1})]. However, the
optical depth $\tau_{\nu}$ of the accretion flow in the vertical
direction is finite and hence we can define the surface when the vertical
height is measured with the optical depth.
Note that $\tau=\tau _\nu ^*={{\sqrt \pi } \over 2}\kappa _\nu (0)H$
at the equatorial plane and $\tau=2\tau _\nu ^*$ at the other surface.
We solve eq. (\ref{eqB.2}) with boundary conditions. We take
\begin{equation}
\begin{array}{l}
{1 \over {\sqrt 3}}{{\partial J_\nu } \over {\partial \tau _\nu
}}=J_\nu \quad(\tau _\nu =0),\\
{{\partial J_\nu } \over {\partial \tau
_\nu }}=0\quad(\tau _\nu =\tau _\nu ^*).
\end{array}
\label{eqB.3}
\end{equation}
The boundary condition at the surface assumes there is no irradiation
onto the surface of the accretion flow and is derived by adopting
two-stream approximation (see Rybicki \& Lightman 1979).
The solution for eq. (\ref{eqB.2}) which satisfies the boundary conditions
[eq. (\ref{eqB.3})] is
\begin{equation}
J_\nu =B_\nu \left\{ {1-{{e^{-\sqrt 3\tau _\nu }} \over 2}\left[
{e^{-2\sqrt 3\left( {\tau _\nu ^*-\tau _\nu } \right)}+1} \right]}
\right\},
\label{eqB.4}
\end{equation}
The energy flux $F_{\nu}$ on the surface of the accretion flow is
given by
\begin{equation}
F_\nu \left( 0 \right)={{4\pi } \over 3}\left. {{{\partial J_\nu }
\over
{\partial \tau _\nu }}} \right|_{\tau _\nu =0}={{2\pi } \over {\sqrt
3}}B_\nu \left[ {1-\exp \left( {-2\sqrt 3\tau _\nu ^*} \right)}
\right].
\label{eqB.5}
\end{equation}
Note that
\begin{equation}
\begin{array}{l}
F_\nu \left( 0 \right)={{2\pi } \over {\sqrt 3}}B_\nu
\quad(\tau _\nu ^*\gg1),\\
F_\nu \left( 0 \right)={{\sqrt \pi } \over
2}\chi _\nu \left( {z=0} \right)H\quad(\tau _\nu ^*\ll1).
\end{array}
\label{eqB.6}
\end{equation}
\clearpage
\begin{thebibliography}{}
\bibitem{abr95} Abramowicz, M. A., Chen, X., Kato, S., Lasota, J.-P., \&
Regev, O. 1995,
\apj, 438, L37
\bibitem{abr88} Abramowicz, M. A., Czerny, B., Lasota, J. P., \&
Szuszkiewicz, E. 1988,
\apj, 332, 646
\bibitem{bjo96} Bjornsson, G., Abramowicz, M. A., Chen, X., \& Lasota,
J. P. 1996, \apj, 467, 99
%\bibitem{che95} Chen, X. 1995, \mnras, 275, 641
\bibitem{che97} Chen, X., Abramowicz, M. A., \& Lasota, J. P. 1997, \apj,
476, 61
\bibitem{cop90} Coppi, P. S. \& Blandford, R. D. 1990, \mnras, 245, 453
\bibitem{dar91} Dermer, C. D., Liang, E. P., \& Canfield, E. 1991, \apj,
369, 410
\bibitem{dus96} Duschl, W. J., Beckert, T., Mezger, P. G., \& Robert,
Z. 1996, in Physics of Accretion Disks, ed.
Kato, S., Inagaki, S., Mineshige, S., Fukue, J. (Gordon and Breach Science
Publishers), p. 91
\bibitem{esi96} Esin, A. A., Narayan, R., Ostriker, E., \& Yi, I. 1996, \apj
465, 312
\bibitem{gan87} Genzel, R., \& Townes, C. H. 1987, Ann. Rev. Astron.
Astrophys., 25, 377
\bibitem{hon96} Honma., F. 1996, PASJ 48, 77
\bibitem{ich77} Ichimaru, S. 1977, \apj, 214, 840
\bibitem{jon68} Jones, F. C.,1968, Phys. Rev., 167, 1159
\bibitem{kat96} Kato, S., Abramowicz, M. A., \& Chen, X. 1996, PASJ, 48, 67
\bibitem{kus96} Kusunose, M., \& Mineshige, S. 1996, \apj, 468, 330
\bibitem{man96} Manmoto, T., Takeuchi, M., Mineshige, S., Matsumoto, R.,
\& Negoro, H. 1996, \apj, 464, L135
\bibitem{mat84} Matsumoto, R., Fukue, J., Kato, S., \& Okazaki, A. S. 1984,
PASJ, 36, 71
\bibitem{mat85} Matsumoto, R., Kato, S., \& Fukue, J. 1985, in Theoretical
Aspects on
Structure, Activity and Evolution of Galaxies III, ed. Aoki, S.,
Iye, M., \& Yoshii, Y. (Tokyo Astronomical Observatory, Tokyo), p.102
\bibitem{nak96} Nakamura, E. K., Matsumoto, R., Kusunose, M., \& Kato, S.,
1996, PASJ
48, 761
\bibitem{nak97} Nakamura, E. K., Matsumoto, R., Kusunose, M., \& Kato, S.,
1997, PASJ submitted
\bibitem{nar96a} Narayan, R. 1996, in Physics of Accretion Disks, ed.
Kato, S., Inagaki, S., Mineshige, S., Fukue, J. (Gordon and Breach Science
Publishers), p. 15
\bibitem{nar97} Narayan, R., Kato, S., \& Honma, F. 1997, \apj, in press
%\bibitem{nar96b} Narayan, R., McClintock, J., \& Yi, I. 1996, \apj, 457, 821
\bibitem{nar95} Narayan, R., \& Yi, I. 1995, \apj, 452, 710
\bibitem{nar95b} Narayan, R., \& YI, I., Mahadevan, R. 1995, Nature 374, 623
%\bibitem{nov73} Novikov, I. D., \& Thorne, K. S. 1973, in Blackholes, ed.
C. DeWitt
%\& B. DeWitt (New York: Gordon \& Breach), 343
%\bibitem{pac70} Pacholczyk, A. G. 1970, Radio Astrophysics (San Francisco:
Freeman)
\bibitem{pac80} Paczy\'nski, B., \& Wiita, P.J. 1980, \aap 88, 23
\bibitem{pir78} Piran, T. 1978, \apj, 221, 652
\bibitem{ryb79} Rybicki, G. B. \& Lightman, A. P. 1979, Radiative Processes in
Astrophysics (New York: John Wiley \& Sons)
\bibitem{sha73} Shakura, N. I., \& Sunyaev, R. A. 1973, \aap, 24, 337
\bibitem{sha76} Shapiro, S. L., Lightman, A. P., \& Eardley, D. M. 1976,
\apj, 204, 187
\bibitem{ste83} Stepney, S., \& Guilbert, P. W. 1983, \mnras 204, 1269
\end{thebibliography}
\clearpage
\figcaption{The amount of the heating and the cooling rates for the ions
(upper panel) and for the electrons (lower panel) in the units of
$2 \pi r^{2}/(\Mdot W_{i}/\Sigma)$ and $2 \pi r^{2}/(\Mdot W_{e}/\Sigma)$,
respectively.
The ratio of advective cooling
of ions to the viscous heating $f$ is also shown in the upper panel. The
accretion flow becomes advection dominated ($f=1$) as the radius
decreases and the energy transport from the ions to the electrons
becomes practically zero. The radiative cooling is balanced with the
electron advective {\it heating} in the innermost region. \label{fig1}}
\figcaption{The temperature of the ions and the electrons as a
function of $r$. The electrons attain the temperature of $10^{10}{\rm
K}$. \label{fig2}}
\figcaption{The aspect ratio $h/r$ of the accretion flow. \label{fig3}}
\figcaption{The angular momentum (upper panel) and the various
velocities (lower panel) of the accretion flow. The azimuthal
velocity $v_{\varphi}$ is highly sub-Keplerian. In the hot luminous
region near the central black hole, the divergence of the velocities from
those in the
self-similar solution is fairly large. \label{fig4}}
\figcaption{The luminosity distribution as a function of $r$. The
contributions from respective radiation mechanisms (i.e. bremsstrahlung,
synchrotron, Comptonization) are also shown. The synchrotron emission
and the Compton scattering dominate in the innermost region. The
distribution of the bremsstrahlung emission is relatively flat.
\label{fig5}}
\figcaption{The spectrum generated by the optically thin
accretion flow around the central black hole of the mass
$\MBH=10^{8}\Ms$. S indicates the synchrotron
peak which comprises Rayleigh-Jeans slope and the optically thin
synchrotron emission. C1 and C2 indicate the once and twice Compton
scattered photons, respectively and B indicates the bremsstrahlung
emission plus
photons suffering multiple Compton scattering. W indicates saturated
Comptonized photons. \label{fig6}}
\figcaption{The surface density (upper panel) and
the temperature (lower panel) of the accretion flow. The dashed line
corresponds to the case where $\MBH=10\Ms$. We find that the structure of
the accretion
flow is nearly the same when
the radius and the mass accretion rate are
scaled with the Schwarzschild radius and the critical
mass accretion rate, respectively, while the electron temperature at the
innermost region is varied
slightly. \label{fig7}}
\figcaption{The spectrum generated by the optically thin
accretion flow around the central black hole of the mass
$\MBH=10\Ms$. The letters indicate the same meaning as Fig. 6.
The shape of the spectrum is essentially the same. However, the
position of the synchrotron peak and the total luminosity differ
significantly. \label{fig8}}
\figcaption{The spectrum of Sgr A* (upper panel). The lines correspond to the
spectrum calculated with our model presented in this paper.
The $\Mdot$-dependence of the spectrum and the
temperature and the surface density (lower panel) are illustrated.
The spectra behaves in the simplest way. When $\Mdot$ is reduced,
the surface density and the entire emission are also reduced.
The bremsstrahlung emission is more sensitive to the $\Mdot$ change
than the synchrotron emission.
\label{fig9}}
\figcaption{The $\MBH$-dependence of the spectrum (upper panel) the
temperature and the surface density (lower panel).
The position of the Rayleigh-Jeans slope determines the mass of the
central black hole. The surface density is not varied when $\MBH$ is
changed. \label{fig10}}
\figcaption{The $\beta$-dependence of the spectrum (upper panel) the
temperature and the surface density (lower panel).
When $\beta$ is lowered, the temperature decreases and the
bremsstrahlung emission is weakened. However, the synchrotron emission
is fairly insensitive to the change of $\beta$. \label{fig11}}
\figcaption{The $\alpha$-dependence of the spectrum (upper panel) the
temperature and the surface density (lower panel).
When $\alpha$ is lowered, the surface density increases and so the
bremsstrahlung emission. However, the synchrotron emission weakened
due to the decrease of the electron temperature in the innermost
region. \label{fig12}}
\end{document}
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