Equations and Problem Description

The co-moving frame radiative transfer equation for spherically symmetric flows can be written as :

We set c=1; $\beta$ is the velocity; and $\gamma = (1-\beta^2)^{-1/2}$ is the usual Lorentz factor. Equation  is a integro-differential equation, since the emissivity $\eta_\nu$ contains J_$J_{\nu}$ , the zeroth angular moment of I_$I_{\nu}$ :

$$\eta_\nu = \kappa_\nu \ifmmode{S_{\nu}}\else{\hbox{$S_{\nu}$} }\fi+ \sigma_\nu \ifmmode{J_{\nu}}\else{\hbox{$J_{\nu}$} }\fi,$$

and

$$\ifmmode{J_{\nu}}\else{\hbox{$J_{\nu}$} }\fi= 1/2 \int_{-1}^{1} d\mu\, \ifmmode{I_{\nu}}\else{\hbox{$I_{\nu}$} }\fi,$$

where $\ifmmode{S_{\nu}}\else{\hbox{$S_{\nu}$} }\fi$ is the source function, $\kappa_\nu$ is the absorption opacity, and $\sigma_\nu$ is the scattering opacity. With the assumption of time-independence $\frac{\partial\ifmmode{I_{\nu}}\else{\hbox{$I_{\nu}$} }\fi}{\partial t} = 0$ and a monotonic velocity field Eq. 1 becomes a boundary-value problem in the spatial coordinate and an initial value problem in the frequency or wavelength coordinate. The equation can be written in operator form as:

where $\Lambda$ is the lambda-operator.

Implicit in the solution of these equations is obtaining correct expressions for the opacity and the source function, both of which depend on the level populations of the material at each spatial point. Thus, one is forced to include the auxiliary equations which include the steady state rate equations (transitions into a given level are balanced by those out of that level), the NLTE equation of state which enforces charge and mass conservation, and the radiative equilibrium equation which enforces energy conservation. These auxiliary equations of course involve the radiation field, which makes the problem highly non-linear.