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\begin{document}

\section{Appendix A: free-free emission}

In this appendix the treatment of free-free emission is described
taking into account the emission of the central star and optical depth
effects.  The geometry of the problem is sketched in Fig. 11. A
central star of effective temperature $T_{\rm eff}$ and radius
$R_{\star}$ is surrounded by a spherical shell of radius $R_{\rm
out}$.  Free electrons are present in this region, but it is assumed
there is no dust.  This latter assumption is valid since $R_{\rm out}$
(identified with the extent of the `chromosphere') can reasonably be
assumed to be smaller than the inner dust radius.



For each impact parameter, $p$, the radiative transfer equation is
solved along the z-axis for the specific intensity $I$ for a given
wavelength
\begin{displaymath}
I_{\lambda}(p) = I_0 \; e^{-\int_{z_0}^{z_{\rm m}} \; {\alpha}(r') dz'} 
\end{displaymath}
\begin{equation}
\hspace{1.5cm} + \int_{z_0}^{z_{\rm m}} \;{\alpha}(r')\; B_{\lambda}(T(r')) \;  
e^{ -\int_{z'}^{z_{\rm m}} \; {\alpha}(r'') dz''} dz'
\end{equation}
where $z_{\rm m}$ = +$\sqrt{R_{\rm out}^2-p^2}$, $B_{\lambda}(T(r))$
the Planck function with the electron temperature and ${\alpha}$ the
absorption coefficient.  The quantities $I_0$ and $z_0$ are 0 and
$-\sqrt{R_{\rm out}^2 -p^2}$ when $p > R_{\star}$ and
$B_{\lambda}(T_{\rm eff})$ and +$\sqrt{R_{\star}^2-p^2}$ when $p \le
R_{\star}$. Dust extinction in the wavelength region of interest in
entirely negligible, so the intensity at $z$ = $z_{\rm m}$ equals the
intensity at $z$ = $+\infty$ (the observer).

The absorption coefficient (in cm$^{-1}$) is given by
\begin{displaymath}
\alpha = 1.3703 \times 10^{-35}\, g \, {\lambda}^3\, Z^2 \, n_e \, n_i 
\end{displaymath}
\begin{equation} 
\hspace{3cm} \star\ \,(1- \exp (-14387.7/\lambda/T)) /T^{0.5}
\end{equation}
where $\lambda$ is the wavelength in micron, $g$ the Gaunt factor, $Z$
the charge of the ions (assumed 1), $T$ the electron temperature and
$n_e$ and $n_i$ the electron and ion density. The H-density is
calculated from an assumed mass loss rate and gas expansion velocity
and follows an $r^{-2}$ law.  The ion density is calculated from
the H-density by multiplying by a constant `ionisation fraction'. The
electron density is assumed to equal the ion density. The electron
temperature is assumed to follow $T_0 \; (R_{\star}/r)^{\gamma}$.
These assumptions are untested in the sense that the density and
temperature structure close to the star are not known. The Gaunt
factor has been calculated according to subroutines kindly provided by
Dr. Alex de Koter (Goddard Space Flight Center, USA).

The flux at earth is calculated from
\begin{equation}
F_{\lambda} = \frac{2\, \pi}{D^2} \; \int_{0}^{R_{\rm out}} I_{\lambda}(p) 
\; p  \; dp
\end{equation}
where $D$ is the distance to earth. The model also calculates the FWHM
values of the surface brightness distribution $I_{\lambda}(p)$, and the flux
distribution $p\,I_{\lambda}(p)$.  Model parameters are $R_{\rm out}$, the
ionisation fraction and electron temperature ($T_0$ and $\gamma$).
Other parameters that are constrained from the dust modeling are the
mass loss rate and gas velocity to calculate the H-density, the
effective temperature and radius of, and the distance to, the star.

The integrals in Eqs. (A1, A3) are solved using the trapezium
rule. Tests show that for reasonable values for $R_{\rm out}$ (a few
$R_{\star}$) 100 equidistant values for $p$ and a similar number of
points along the z-axis (except in extremely optically thick cases)
give very accurate results.
\end{document}
