The Calculation of Stellar Radiative Opacity

A new algorithm, suitable for large-capacity high-speed digital computers, for the evaluation of stellar radiative opacity is presented. The method is designed specifically for application to mixtures of a large number of elements in arbitrary proportional concentrations. The statistical and quantum mechanics of the stellar material are based upon the high-temperature Thomas–Fermi model of the atom. The electron density and associated electrostatic potential in the neighbourhood of an atomic nucleus is found by numerical solution of the appropriate Poisson–Fermi–Dirac equation. The numerical solution of Schrodinger's equation with the potential thus obtained, with a suitable correction for self-screening, then gives the electron wave functions for all the discrete energy levels and for a selected number of angular momenta and energy levels in the continuum. The radiative processes contributing to the opacity which are included are electron scattering, free–free, bound–free and bound–bound absorption. For scattering, the frequency dependent relativistic Klein–Nishina cross-sections are used, while those for the three absorption processes are calculated directly from the wave functions. In the case of free–free transitions, for rapid numerical convergence, dipole acceleration matrix elements are computed and used in the partial wave series at low initial electron energies and in the Born approximation at high energies. The temperature average over the distribution of initial energies, for a fixed photon frequency is performed by means of a Gauss–Laguerre quadrature. For bound–free and bound–bound transitions, dipole acceleration and dipole length matrix elements are employed. In all transitions, full allowance is made for the occupancy of the initial state and the availability of the final state, according to Fermi–Dirac statistics. The splitting of the ionization edges and of the lines because of differential configuration effects is taken fully into account, as is also the broadening of each component thus obtained, due to radiation damping, Doppler effect and pressure, including the effects of both elastic and inelastic electron collisions and of inter-ionic electric fields. Opacities are given for several astrophysical mixtures at a number of temperatures and pressures, and substantial departures from the results of other calculations are indicated.