Calculation of spatial distribution of optical escape factor and its application to He I collisional-radiative model

Abstract

An integral analytical formula for a spatial distribution of the optical escape factor (OEF) in an infinite cylindrical plasma is derived as a function of an arbitrary upper state spatial density profile, the temperature ratio of the upper state to the lower state, and the optical depth of the corresponding transition. Test calculations are carried out for three different upper state profiles, i.e., uniform (rectangular), parabolic, and Gaussian upper state profiles. The OEF takes on negative values at the periphery of the parabolic and Gaussian upper state profiles. These characteristics cannot be expressed by the conventional OEF formulas derived for the center of the plasma, even though the optical depth is increased. In addition to the analytical derivation of the formula, two practical formulas are proposed: an empirical formula of the spatial distribution of the OEF for the Gaussian upper state density profile and a linear formula of the OEF distribution for upper state profiles that are expressed as linear combinations. These formulas enable us to calculate the spatial distribution of the OEF for the multiple-Gaussian upper state profile without the need for time-consuming integral calculations.