On the computation of moments in the Super-Transition-Arrays model for radiative opacity calculations

TL;DR

This paper improves the computation of moments in the Super-Transition-Arrays model for radiative opacity, offering faster, more accurate formulas and addressing numerical challenges in the process.

Contribution

It introduces new formulas and recursion relations for partition functions and moments, enhancing computational efficiency and stability in the Super-Transition-Arrays method.

Findings

Derived a recursion relation and explicit formula for supershell partition functions.

Proposed a positive-definite, non-alternating formula for moments of any order.

Presented methods to accelerate calculations and avoid numerical difficulties.

Abstract

In the Super-Transition-Array statistical method for the computation of radiative opacity of hot dense matter, the moments of the absorption or emission features involve partition functions with reduced degeneracies, occurring through the calculation of averages of products of subshell populations. In the present work, we discuss several aspects of the computation of such peculiar partition functions, insisting on the precautions that must be taken in order to avoid numerical difficulties. In a previous work, we derived a formula for supershell partition functions, which takes the form of a functional of the distribution of energies within the supershell and allows for fast and accurate computations, truncating the number of terms in the expansion. The latter involves coefficients for which we obtained a recursion relation and an explicit formula. We show that such an expansion can be combined with the recurrence relation for shifted partition functions. We also propose, neglecting the effect of fine structure as a first step, a positive-definite formula for the Super-Transition-Array moments of any order, providing an insight into the asymmetry and sharpness of the latter. The corresponding formulas are free of alternating sums. Several ways to speed up the calculations are also presented.