The Rate Operator

As described above, a simple fixed point iteration scheme for the solution of the rate equations will converge much too slowly to be useful for most cases of practical interest. Therefore, we use an extension of the operator splitting idea for the solution of the rate equations.

We rewrite the rate equations in the form of an ``operator equation.'' This equation is then used to introduce an ``approximate rate operator'' in analogy to the approximate $\Lambda$-operator which can then be used to iteratively solve the rate and statistical equations by an operator splitting method, details of the approach are given in [2].

We introduce first the ``rate operator'' [R_ij] for upward transitions in analogy to the $\Lambda$-operator. [R_ij] is defined so that

Here, [n] denotes the ``population density operator'', which can be considered as the vector of the population densities of all levels at all points in the medium under consideration. The radiative rates are (linear) functions of the mean intensity J, which is given by $J(\lambda) = \Lambda(\lambda) S(\lambda)$, where $S=\eta(\lambda)/\chi(\lambda)$ is the source function. Using the $\Lambda$-operator, we can write [R_ij][n] as:

This can be brought into the form (see [2] for details)

The corresponding expression for the emission rate-operator [R_ji] is given by

where we have used the definition

and $[E(\lambda)]$ is a linear operator such that $[E(\lambda)][n]$gives the emissivity $\eta(\lambda)$.

Using the rate operator, we can write the rate equations in the form

This form shows, explicitly, the non-linearity of the rate equations with respect to the population densities. Note that in addition, the rate equations are non-linear with respect to the electron density via the collisional rates. Furthermore, the charge conservation constraint condition directly couples the electron densities and the population densities of all level of all atoms and ions with each other.

In analogy to the operator splitting method discusses above, we split the rate operator, by writing $\ifmmode{[R_{ij}]}\else\hbox{$[R_{ij}]$}\fi=\ifmmode{[R_{ij}^*]}\else\hbox{$[R_... ...hbox{$[R_{ij}^*]$}\fi+\ifmmode{[\Delta R_{ij}]}\else\hbox{$[\Delta R_{ij}]$}\fi$(analog for the downward radiative rates), where $\ifmmode{[R_{ij}^*]}\else\hbox{$[R_{ij}^*]$}\fi$ is the ``approximate rate-operator''. We then rewrite the rate R_ij as

and analogously for the downward radiative rates. In Eq. 26, $[n_{\rm old}]$denotes the current (old) population densities, whereas $[n_{\rm new}]$ are the updated (new) population densities to be calculated. The $\ifmmode{[R_{ij}^*]}\else\hbox{$[R_{ij}^*]$}\fi$ and $\ifmmode{[R_{ji}^*]}\else\hbox{$[R_{ji}^*]$}\fi$ are linear functions of the population density operator [n_k] of any level k, due to the linearity of $\eta$ and the usage of the $\Psi$-operator instead of the $\Lambda$-operator.

If we insert Eq. 26 into Eq. 16, we obtain the following system for the new population densities:

Due to its construction, the [R_ij^*][R_ij^*]-operator contains information about the influence of a particular level on all radiative transitions. Therefore, we are able to treat the complete multi-level non-LTE radiative transfer problem including active continua and overlapping lines. The $[E(\lambda)]$-operator, at the same time, gives us information about the strength of the coupling of a radiative transition to all levels that are considered. This information may be used to include or neglect certain couplings dynamically during the iterative solution of Eq. 27. Furthermore, we have not yet specified either a method for the formal solution of the radiative transfer equation or a method for the construction of the approximate $\Lambda$-operator (and, correspondingly, the $\ifmmode{[R_{ij}^*]}\else\hbox{$[R_{ij}^*]$}\fi$-operator). We proceed by considering rapidly expanding spherically symmetric media and use the tri-diagonal ALO given by Hauschildt [1]. However, any method for the formal solution of the radiative transfer equation and the construction of the ALO may be used, including multi-dimensional and/or time dependent methods.