Since the energy of each photon is just its momentum times the speed
of light, the radiative
energy flux is simply given by the change in the radiation pressure
(momentum flux),
$P_{\rm rad}={1\over 3}aT^4$, per mean-free-path,  
\begin{equation} 
F =-c {dP_{\rm rad}\over d\tau},
\end{equation}
where the optical-depth $\tau$ is related to the frequency-averaged
(so-called, Rosseland-mean) opacity coefficient of the gas, $\kappa$, 
\begin{equation}
\tau=\int_c^h \kappa \rho dz \approx {1\over 2} \kappa \Sigma ,
\end{equation}
where $\Sigma= 2h\rho$.
For the characteristic mass density $\rho$ and temperature $T$
encountered in the midplane of accretion disks, there are
two primary sources of opacity: {\it
electron scattering} with
\begin{equation}
\kappa_{\rm es}={\sigma_{\rm T} \over m_p},
\end{equation}
and 
{\it free-free} absorption with 
\begin{equation}
\kappa_{\rm
ff}=8\times 10^{22} {\rm cm}^2\,{\rm g}^{-1}) \left({\rho \over
{\rm
g}\,{\rm cm}^{-3}\right) \left({T\over {\rm K}\right)^{-7/2},
\end{equation}
where we assume a pure hydrogen plasma for simplicity.

%$\nu=\eta/\rho$,

\subsection{Disk Structure}

We normalize the accretion rate in the disk relative to the so-called
Eddington rate which would produce the maximum possible disk
luminosity, $L_{\rm Edd}$ (see derivation in equation \ref{LEdd}
below).  When the luminosity approaches the Eddington limit, the disk
bloates and $h$ approaches $r$, violating the thin-disk assumption.
We write
$\dot M_{\rm Edd}\equiv L_{\rm Edd}/(\epsilon {\rm c}^2)$,
where $\epsilon$ is the radiative efficiency (for converting
rest-mass to radiation near the ISCO).

In local thermodynamic equilibrium, the emergent flux from the surface of the disk 
(equation \ref{fluxdisk}) can be written
in terms of the temperature at disk midplane $T$ as $F\approx c
aT^4/kappa\Sigma$.
The surface temperature of the disk is the
roughly,
\begin{equation}
T_s\approx\left(4F\over a\right)^{1/4}=10^6~{\rm
K}\left({M\over 10^8M_\odot}\right)^{-1/4}\left({\dot{M}\over
\dot{M}_E}\right)^{1/4} \left({r\over r_{\rm
Sch}}\right)^{-3/4}\left[1-({r\over r_{\rm ISCO}})^{1/2}\right] .
\end{equation}

In characterizing the structure of the disk,
we use the following dimensionless parameters:
$r_1=R/10R_{\rm Sch}$, 
$M_8=M/(10^8 M_{\odot})$, $\dot m
= \dot M/\dot M_{\rm Edd}$, 
%$\dot m_{0.1}=\dot m/0.1$,
$\alpha_{-1}=\alpha/0.1$, $\epsilon_{-1}=\epsilon/0.1$, 
%$\tilde\kappa_{\rm es}=\kappa/\kappa_{{\rm es}}$, and
%$\tilde\kappa_{\rm ff}=\kappa/\kappa_{\rm ff}$.  

The accretion disk can be divided radially into three
distinct regions,

\begin{enumerate}

\item {\it Inner region:} Radiation pressure and electron-scattering
  opacity dominate, $P\approx P_{\rm rad}$, $\tilde\kappa_{\rm
    es}\approx 1$, valid inside $r_3\ll r_3^{{\rm gas}/{\rm rad}}$
  where $r_3^{{\rm gas}/{\rm rad}}$ is defined in
  equations~(\ref{e:gas/radb1}) and~(\ref{e:gas/radb0}) below.

\item {\it Middle region:} Gas pressure and electron-scattering
  opacity dominate, $P\approx P_{\rm gas}$, $\tilde\kappa_{\rm
  es}\approx 1$, valid between $ r_3^{{\rm gas}/{\rm rad}}\ll r_3
  \lsim r_3^{{\rm es}/{\rm ff}}$, where $r_3^{{\rm es}/{\rm ff}}$ is
  defined in equation~(\ref{e:es/ff}) below.

\item {\it Outer region:} Gas pressure and free-free opacity dominate,
  $P\approx P_{\rm gas}$, $\tilde\kappa_{\rm ff}\approx 1$, valid
  outside of $r_3 \gsim r_3^{{\rm es}/{\rm ff}}$.

\end{enumerate}

In region (1), it makes a difference whether the viscosity is
proportional to the total pressure or just the gas pressure, labeled
below by $b=0$ or 1, (i.e. $\alpha$ or $\beta$ disk) respectively. In
all cases, we assume that the disk is optically thick, i.e. $\tau \gg
1$. We obtain $\Sigma(r)$ and $H(r)$ following \citet{goodman03} or
\citet{goodmantan04},
%
\begin{eqnarray}\label{e:Sigma_definition}
 \Sigma(r) &=& \frac{2^{4/5}}{3\pi^{3/5}} \sigma_{\rm SB}^{1/5} \left(\frac{\mu_0 m_{\rm H}}{{\rm k_B}}\right)^{4/5} f_{T}^{-2}\alpha^{-4/5} \kappa^{-1/5} \dot M^{3/5} \Omega^{2/5} \beta^{-(4/5)(b-1)},\\\label{e:H_definition}
 H(r) &=& \frac{f_{T} \kappa \dot M}{2\pi {\rm c} (1-\beta)}.
\end{eqnarray}
%
where $b=0$ or 1, and the radial dependence is implicit in $\Omega$ and $\beta$.
Here, $\beta(r) \equiv P_{\rm gas}/(P_{\rm rad}+P_{\rm gas})$ which satisfies
%
\begin{eqnarray}
  \frac{\beta^{(1/2) + (1/10)(b-1)}}{1-\beta} &=& 2^{3/5}\pi^{4/5} {\rm c} \sigma_{\rm SB}^{-1/10} \left(\frac{{\rm k_B}}{\mu_0 m_{\rm H}}\right)^{2/5}\alpha^{-1/10} \kappa^{-9/10}\dot M^{-4/5} \Omega^{-7/10} \label{e:beta_definition}.
\end{eqnarray}

The asymptotic limits of equations~(\ref{e:Sigma_definition}) and
(\ref{e:H_definition}) can be obtained in regions (1--3), using
equation~(\ref{e:beta_definition}). The results are

\noindent{\em Inner region:}
%
\begin{eqnarray}\label{e:Sigma_inb1}
 \Sigma(r) &=&  (1.63 \times 10^5 {\,\rm g}{\,\rm cm}^{-2})  \mu_0^{4/5}\mu_e^{-4/5}\tilde\kappa_{\rm es}^{-1/5}  f_{T}^{-2} \alpha_{0.3}^{-4/5}  \left(\frac{\dot m}{\eff_{0.1}}\right)^{3/5} M_7^{1/5} r_3^{-3/5} \rm{~~if~} b=1,\\\label{e:Sigma_inb0}
           &=& (2.50 \times 10^4 {\,\rm g}{\,\rm cm}^{-2}) \mu_e^{-1} \tilde\kappa_{\rm es}^{-2}  f_{T}^{-2} \alpha_{0.3}^{-1} \left(\frac{\dot m}{\eff_{0.1}}\right)^{-1}  r_3^{3/2} \rm{~~~~if~} b=0, \\\label{e:H_in}
 H(r) &=&  (10.0  R_S)  f_{T} \frac{\dot m}{\epsilon_{0.1}}~~~~~{\rm for}~{\rm arbitrary}~b.
\end{eqnarray}

\noindent{\em Middle region:}
%
\begin{eqnarray}\label{e:Sigma_middle}
 \Sigma(r) &=& (1.63 \times 10^5 {\,\rm g}{\,\rm cm}^{-2})  \mu_0^{4/5}\mu_e^{-4/5} \tilde\kappa_{\rm es}^{-1/5} f_{T}^{-2} \alpha_{0.3}^{-4/5}  \left(\frac{\dot m}{\eff_{0.1}}\right)^{3/5} M_7^{1/5} r_3^{-3/5}, \\\label{e:H_middle}
 H(r) &=& (3.11  R_S)  \mu_e^{-1/10} \mu_0^{-2/5}   \tilde\kappa_{\rm es}^{1/10}f_{T} \alpha_{0.3}^{-1/10} \left(\frac{\dot m}{\epsilon_{0.1}}\right)^{1/5} M_7^{-1/10} r_3^{21/20}.
\end{eqnarray}

\noindent{\em Outer region:}
%
\begin{eqnarray}\label{e:Sigma_out}
 \Sigma(r) &=& (2.61 \times 10^5 {\,\rm g}{\,\rm cm}^{-2}) \mu_e^{-4/5} \mu_0^{3/4} \tilde\kappa_{\rm ff}^{-1/10}  f_{T}^{-143/80} \alpha_{0.3}^{-4/5} \left(\frac{\dot m}{\eff_{0.1}}\right)^{7/10} M_7^{1/5} r_3^{-3/4}, \\\label{e:H_out}
 H(r) &=& (3.08  R_S) \mu_e^{-1/10}\mu_0^{-3/8}\tilde\kappa_{\rm ff}^{1/20}  f_{T}^{143/160} \alpha_{0.3}^{-1/10}\left(\frac{\dot m}{\epsilon_{0.1}}\right)^{3/20} M_7^{-1/10} r_3^{9/8}.
\end{eqnarray}

The boundaries between the inner/middle and middle/outer regions can
be found from equations
(\ref{e:Sigma_definition})-(\ref{e:beta_definition}), by requiring
$P_{\rm gas}=P_{\rm rad}$ and $\kappa_{\rm ff}=\kappa_{\rm es}$,
respectively. Note that $\kappa_{\rm ff}(r)\propto \rho T^{7/2}$
depends on radius implicitly through the density and the
temperature. Using the (mid-plane) temperature given by
\citet{goodmantan04},
%
\begin{eqnarray}\label{e:Temperature_definition}
 T(r) &=& \left(16\pi^2\right)^{-1/5}
\left(\frac{\mu_0 m_{\rm H}}{{\rm k_B}\sigma_T}\right)^{1/5}
\alpha^{-1/5} \kappa^{1/5} \dot M^{2/5} \Omega^{3/5} \beta^{-(1/5)(b-1)},
\end{eqnarray}
%
we find that the transitions are located at the radii
%
\begin{eqnarray}
r_3^{{\rm gas}/{\rm rad}} &=& 0.482 \,\tilde\mu_0^{8/21}\tilde\mu_e^{2/21} \tilde\kappa_{\rm es}^{6/7} \alpha_{0.3}^{2/21} (\dot m_{0.1}/\epsilon_{0.1})^{16/21} M_7^{2/21}~~\rm{~~if~} b=1,\label{e:gas/radb1}\\
                          &=& 0.515 \,\tilde\mu_0^{8/21}\tilde\mu_e^{2/21} \tilde\kappa_{\rm es}^{6/7} \alpha_{0.3}^{2/21} (\dot m_{0.1}/\epsilon_{0.1})^{16/21} M_7^{2/21}~~\rm{~~if~} b=0,\label{e:gas/radb0}\\
r_3^{{\rm es}/{\rm ff}} &=& 4.10 \,\tilde\mu_0^{-1/3} \tilde f_{T}^{17/12} (\tilde \kappa_{\rm ff}/\tilde \kappa_{\rm es})^{-2/3} (\dot m_{0.1}/\epsilon_{0.1})^{2/3}\label{e:es/ff}.
\end{eqnarray}

Note that the middle and outer regions differ only in their opacity
laws, and the equations in these two regions are equivalent (this can
be seen by setting $\tilde \kappa_{\rm es}\equiv \tilde \kappa_{\rm
ff}\kappa_{\rm ff}(r)/\kappa_{\rm es}$). Since $\Sigma$, $H$, $\rho$,
and $T$ scale with a low power of $\tilde \kappa_{\rm ff}$, the radial
dependence ends up being similar in the middle and outer regions. The
distinction between these equations is nevertheless useful, since we
can assume that $\tilde \kappa_{\rm es}\rightarrow 1$ and $\tilde
\kappa_{\rm ff}\rightarrow 1$ are constants in the middle and outer
regions, respectively.

We emphasize that equations~(\ref{e:Sigma_inb1})-(\ref{e:H_out})
represent only a very non-exhaustive subset of solutions even for
radiatively efficient steady thin accretion disks.  In particular, at
large radii, there are several effects that can invalidate the disk
model described by these equations.  First, these solutions assume
that the self--gravity of the disk is negligible.  This assumption
becomes invalid at radii where the Toomre $Q$--parameter equals unity,
%
\begin{eqnarray}
r^{\rm sg}_3 &=& 12.6 \,\tilde\mu_0^{-8/9}\tilde\mu_e^{14/27} \tilde
f_T^{20/9} \tilde\kappa_{\rm es}^{2/9} \alpha_{0.3}^{8/9} \left(\dot
m_{0.1}/\eff_{0.1}\right)^{-8/27} M_7^{-26/27} \rm{~~~~if~} \tilde\kappa_{\rm es}\rightarrow1\\
r^{\rm sg}_3 &=& 30.99 \tilde\mu_0^{-1} \tilde\mu_e^{28/45} \tilde f_T^{143/60}
\tilde \kappa_{\rm ff}^{2/15} \alpha_{0.3}^{28/45} \left(\dot m_{0.1}/\eff_{0.1}\right)^{-22/45} M_7^{52/45}  \rm{~~~~if~} \tilde\kappa_{\rm ff}\rightarrow 1.
\end{eqnarray}

Beyond these radii, the disk is commonly believed to be unstable to
fragmentation.  Second, at large radii, the disk can also become
optically thin \cite[see][where solutions can be obtained by fixing
the Toomre parameter in the outermost region at $Q\equiv
1$]{sirkogoodman03}.  At these binary separations, the disks may not
actually be geometrically thin \citep{dot08,mke08}, and slim or thick
solutions might instead be relevant.  Third, beyond the radii where
the disk temperature falls below $\approx 10^4$K, the gas becomes
neutral. The corresponding change in opacity will modify the disk
structure, and the disk may become susceptible to ionization
instabilities (although see \citealt{mq01}).  Finally, at large radii
(where the orbital velocity $\gsim 100$ km/s), the gravitational
potential of the galaxy can no longer be ignored.  These regimes,
however, turn out to correspond to separations larger than we are
interested in the present paper, for BH masses above $\approx
10^5~{\rm M_\odot}$ (as will be shown in Figures~\ref{fig:tres_q1} and
\ref{fig:tres_q0.01} below).



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