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\appendix
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\section {Appendix}
%------------------
%--------------------------------
\subsection {Numerical algorithm}
%--------------------------------

Here, we briefly describe the iterative scheme which we use for the
solution of the radiative transfer equation. As everywhere in this
paper, the indices ``m'', ``b'', and ``s'' refer to the quantities
defined at the midplane, at the upper or lower boundary, and in the
equivalent spherical envelope, correspondingly. For clarity, we omit
here the indices $i$ and $k$ labeling different dust components,
as well as any dependence on other quantities than frequency. We
refer to the equations which are solved at the corresponding steps
of the numerical procedure.

\begin {enumerate}

   \item Initial distributions of all quantities.

   \item Ray tracing for each frequency $\nu,$ Eqs.~(2),\,(3).

      {\em Input:}
      \begin {enumerate}
         \item frequency-dependent mean intensities
         \bea
         \nonumber
             (J_\nu)_{\rm m, b, s}\,,
         \eea
         \item dust temperatures
         \bea
         \nonumber
             T_{\rm m},\, T_{\rm b},\, T_{\rm s}\,,
         \eea
         \item absorption and scattering coefficients
         \bea
         \nonumber
             \sigma^{\rm a}_{\nu},\;
             \sigma^{\rm s}_{\nu}\,.
         \eea
      \end {enumerate}

      {\em Output:}
      \begin {enumerate}
         \item frequency-dependent moments
         \bea
         \nonumber
             (J_\nu,\, H_\nu,\, K_\nu)_{\rm m, b, s}\,.
         \eea
      \end {enumerate}

   \item Frequency integration and averaging, Eqs.~(6)\,-\,(8),\,(11).

      {\em Input:}
      \begin {enumerate}
         \item frequency-dependent moments
         \bea
         \nonumber
             (J_\nu,\, H_\nu,\, K_\nu)_{\rm m, b, s}\,.
         \eea
      \end {enumerate}

      {\em Output:}
      \begin {enumerate}
         \item frequency-integrated moments
         \bea
         \nonumber
             (J,\, H,\, K)_{\rm m, b, s}\,,
         \eea
         \item Eddington factors
         \bea
         \nonumber
             f_{\rm m}\!&=&\! K_{\rm m} / J_{\rm m}\,,\\
         \nonumber
             f_{\rm b}\!&=&\! K_{\rm b} / J_{\rm b}\,,\\
         \nonumber
             f_{\rm s}\!&=&\! K_{\rm s} / J_{\rm s}\,,
         \eea
         \item geometric factors
         \bea
         \nonumber
             X_{\rm m}\!&=&\! J_{\rm m} / J_{\rm s}\,,\\
         \nonumber
             X_{\rm b}\!&=&\! J_{\rm b} / J_{\rm s}\,,
         \eea
         \item averaged absorption and extinction coefficients
         \bea
         \nonumber
             \sigma_{J_{\rm m}}\,,\,\,
             \sigma_{J_{\rm b}}\,,\,\,
             \sigma_{J_{\rm s}}\,,\,\,
             \sigma_{H_{\rm s}}\,.
         \eea
      \end {enumerate}

   \item Solution of the moment equations, Eqs.~(9),\,(10).

      {\em Input:}
      \begin {enumerate}
         \item averaged extinction coefficients
         \bea
         \nonumber
             \sigma_{H_{\rm s}}\,,
         \eea
         \item Eddington factors
         \bea
         \nonumber
             f_{\rm s} = K_{\rm s}/J_{\rm s}\,,
         \eea
         \item geometric factors
         \bea
         \nonumber
             X_{\rm m}\!&=&\! J_{\rm m}/J_{\rm s},\\
         \nonumber
             X_{\rm b}\!&=&\! J_{\rm b}/J_{\rm s}\,.
         \eea
      \end {enumerate}

      {\em Output:}
      \begin {enumerate}
         \item equivalent sphere mean intensities
         \bea
         \nonumber
             \tilde J_{\rm s}\,,
         \eea
         \item disk mean intensities
         \bea
         \nonumber
             \tilde J_{\rm m}\!&=&\! X_{\rm m}\,\tilde J_{\rm s}\,,\\
         \nonumber
             \tilde J_{\rm b}\!&=&\! X_{\rm b}\,\tilde J_{\rm s}\,.
         \eea
      \end {enumerate}

   \item Calculation of dust temperatures, Eq.~(13).

      {\em Input:}
      \begin {enumerate}
         \item moment equation mean intensities
         \bea
         \nonumber
             \tilde J_{\rm m}\,,\,
             \tilde J_{\rm b}\,,\,
             \tilde J_{\rm s}\,,
         \eea
         \item averaged absorption coefficients
         \bea
         \nonumber
             \sigma_{J_{\rm m}}\,,\,
             \sigma_{J_{\rm b}}\,,\,
             \sigma_{J_{\rm s}}\,.
         \eea
      \end {enumerate}

      {\em Output:}
      \begin {enumerate}
         \item dust temperatures
         \bea
         \nonumber
             T_{\rm m} \!&=&\!{\cal F}^{-1} (\sigma_
                            {J_{\rm m}} \tilde J_{\rm m})\,,\\
         \nonumber
             T_{\rm b}\,\!&=&\!{\cal F}^{-1} (\sigma_
                            {J_{\rm b}} \, \tilde J_{\rm b})\,,\\
         \nonumber
             T_{\rm s}\,\!&=&\!{\cal F}^{-1} (\sigma_
                            {J_{\rm s}} \, \tilde J_{\rm s})\,.
         \eea
      \end {enumerate}

   \item Convergence check (Sect.~2.6); if necessary, go to step 2.

   \item Ray tracing for each $\nu$ and viewing angles
         $\theta_{\rm v},$ Eqs.~(2),\,(3).

      {\em Input:}
      \begin {enumerate}
         \item frequency-dependent mean intensities
         \bea
         \nonumber
             (J_\nu)_{\rm m, b, s}\,,
         \eea
         \item dust temperatures
         \bea
         \nonumber
             T_{\rm m},\, T_{\rm b},\, T_{\rm s}\,,
         \eea
         \item absorption and scattering coefficients
         \bea
         \nonumber
             \sigma^{\rm a}_{\nu},\;
             \sigma^{\rm s}_{\nu}\,.
         \eea
      \end {enumerate}

      {\em Output:}
      \begin {enumerate}
         \item emergent intensity distribution
         \bea
         \nonumber
             I_\nu^{+}\,,
         \eea
         \item spectral energy distribution
         \bea
         \nonumber
             \nu\,F_\nu\,.
         \eea
      \end {enumerate}

\end {enumerate}

In the scheme, we implicitly assume that the appropriate boundary
conditions should be applied, if necessary. If we solve the
radiative transfer equation together with a system of hydrodynamical
equations, the step 4 includes a solution of all those equations.
The function ${\cal F}$ appearing at the step 5 results from the
assumption of radiative equilibrium (Sect.~2). At the step 7, the source
function is calculated using the midplane temperatures everywhere
in the disk, i.e. assuming that $T\,(r,\vartheta)=T_{\rm m}\,(r).$
The scattering term of the source function is computed linearly
interpolating the mean intensity between the midplane $J_{\nu, \rm m}$
and the boundary $J_{\nu, \rm b}$ values.

\end {document}
