Computation of $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$

The formal solution of the SSRTE is performed along the characteristic rays on a mesh $\{r_i\}$, $i=1,\ldots,N_s$ of discrete shells using the short-characteristic (SC) method of Olson and Kunasz [24] with piece-wise parabolic or linear interpolation. The characteristic rays are curved in the case of the SSRTE and have to be calculated before the solution of the radiative transfer equation proceeds (see [1] for details). Improvements in this method [1,25] include an improved angle integration using generalized Simpson-quadrature and a generalization of the approximate $\Lambda$-operator to an arbitrary number of bands below and above the main diagonal (up to the full $\Lambda$-operator).

We describe here the general procedure of calculating the $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ with arbitrary bandwidth, up to the full $\Lambda$-operator, for the SC method in spherical symmetry [25]. Although we consider the SSRTE as given in the previous section, the same procedure applies for radiative transfer problems in static media or in (static or moving) media with plane-parallel symmetry. The specialization of the formulae given in this section is straightforward.

The formal solution along a characteristic of the SSRTE (hereafter, a ``ray'') is done using a polynomial interpolation of the source function, S, along the ray. For reasons of numerical stability, we use linear or quadratic interpolation of S along each ray, although this is not required by the method. This leads to the following expressions for the specific intensity $I(\tau_i)$ along a ray (cf. Ref. [24] for a derivation of the formulae):

where the superscript k labels the ray; $\tau^k_i$ denotes the optical depth along the ray k with $\tau^k_1\equiv 0$ and $\tau^k_{i-1} \le \tau^k_i$while $\tau^k$ is calculated, e.g., using piecewise linear interpolation of $\hat\chi$ along the ray, viz.

and

where i is the ``running'' index along the ray and |s^k_i-1-s^k_i| is the geometrical path length between points i and i-1. The expressions for the coefficients $\alpha^k_i$, $\beta^k_i$ and $\gamma^k_i$are given in Ref. [24] (see also Ref. [1]).

We describe the construction of $\ifmmode{\Lambda^*}\else\hbox{$\Lambda^*$}\fi$ for arbitrary bandwidth using the example of a characteristic that is tangential to an arbitrary shell: Ray k is the ray that is tangent to shell k+1 The intersection points (including the point of tangency) are labeled from left to right, the direction in which the formal solution proceeds. Ray k has 2k+1 points of intersection with discrete shells $1\ldots k+1$. To compute row j of the discrete $\Lambda$-operator (or $\Lambda$-matrix), $\Lambda_{ij}$,we sequentially label the intersection points of the ray k with the shell i, and define auxiliary quantities $\lambda_{ij}^k$ and $\hat\lambda_{ij}^k$ as follows:

For the calculation of $\hat\lambda^k_{i,j}$, we obtain:

Using the $\lambda^k_{ij}$ and $\lambda^k_{ij}$, we can now write the $\Lambda$-Matrix as

where w^k_i,j are the angular quadrature weights, $\{l\}$ is the set $\{i \le k+1\}$ and $\{l'\}$ is the set $\{i \gt k+1\}$.This expression gives the full $\Lambda$-matrix, it can easily be specialized to compute only certain bands of the $\Lambda$-matrix. In that case, not all of the $\lambda^k_{i,j}$ and $\hat\lambda^k_{i,j}$ have to be computed, reducing the CPU time from that required for the computation of the full $\Lambda$-matrix.