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In order to solve Eq. 1, the emissivity must be known, but depends on the NLTE level
populations and therefore the NLTE rate equations must be solved
simultaneously with Eq. 1. This is further complicated by
the fact that the NLTE rate equations depend on the radiation field itself.
The NLTE rate equations have the form [11]

In Eq. 16, *n*_{i} is
the actual, non-LTE population density of a level *i*
and the symbol *n*_{i}^{*} denotes the so-called LTE population density
of the level *i*, which is given by

Here denotes the *actual*, i.e., non-LTE,
population density of the ground
state of the next higher ionization stage of the same element;
*g*_{i} and are
the statistical weights of the levels *i* and , respectively.
In Eq. 17, *E*_{i} is the excitation energy of the level *i* and
denotes the ionization energy from the ground state to
the corresponding ground state of the next higher ionization stage.
The actual, non-LTE electron density is given by *n*_{e}. The system of
rate equations is closed by the conservation equations for the nuclei and the
charge conservation equation (cf. Ref. [11]).
The sums in Eq. 16 extend only over the levels that are
included in our model atoms; for example, in singly ionized iron our
model atom
consists of 675 energy levels [3]. The weaker radiative
transitions are treated as LTE background opacity (see
Refs. [2,3]).

The rate coefficients for radiative and collisional transitions between
two levels
*i* and *j* (including transitions from and to the continuum, see below)
are given by
*R*_{ij} and *C*_{ij}, respectively.
In our notation, the upward (absorption) radiative rate coefficients *R*_{ij}
(*i*<*j*)
are given by

whereas the downward (emission) radiative rate coefficients *R*_{ji}
(*i*<*j*) are
given by

Here, *J* is the mean intensity, *T* the electron temperature, *h* and
*c* and Planck's constant and the speed of light, respectively.
For the purposes of this paper, we assume that
cross section of the transition at the
wavelength is known for both line and continuum transitions
and that it is the same for both absorption and emission processes
(complete redistribution).
Not all atomic processes fit neatly into the above scheme where the
rates are in detailed balance. Non-thermal ionization by fast
electrons, K-capture, Auger emission, and two-photon decay are
important in various stages of the evolution of novae and
supernovae. They can be included in the above formulation with
reasonable approximations, however.

** Next:** The Rate Operator
** Up:** The Problem
** Previous:** Numerical Considerations
*Peter H. Hauschildt*

*8/20/1998*