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NLTE calculations

In order to solve Eq. 1, the emissivity $\eta_{\lambda}$must be known, but $\eta_{\lambda}$ 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, ni is the actual, non-LTE population density of a level i and the symbol ni* denotes the so-called LTE population density of the level i, which is given by

Here $n_\kappa$ denotes the actual, i.e., non-LTE, population density of the ground state of the next higher ionization stage of the same element; gi and ${g_\kappa}$ are the statistical weights of the levels i and $\kappa$, respectively. In Eq. 17, Ei is the excitation energy of the level i and $E_\kappa$ 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 ne. 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 Rij and Cij, respectively. In our notation, the upward (absorption) radiative rate coefficients Rij (i<j) are given by

whereas the downward (emission) radiative rate coefficients Rji (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 $\alpha_{ij}(\lambda)$ of the transition $i\to j$ at the wavelength $\lambda$ 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 up previous
Next: The Rate Operator Up: The Problem Previous: Numerical Considerations
Peter H. Hauschildt