Line Expansion Opacity in Relativistically Expanding Media

Spectral lines of heavy atomic elements in the ejecta of supernovae and neutron star mergers can have important contribution to the opacity of the ejecta matter even when the abundance of the elements is very small. Under favorable conditions, the line expansion opacity arising from spectral lines and the expansion of the medium can be orders of magnitude larger than the opacity of electron scattering. In this paper we derive the formulae for evaluating the line expansion opacity and its Rosseland mean in an expanding medium in the framework of special relativity, which can be considered as a generalization of the previous work in the Newtonian approximation. Then we compare the derived relativistic formulae to the Newtonian ones to explore the relativistic effect on the opacity, and test the new formulae with the spectral lines of some heavy atomic elements. We also derive some approximation formulae for the Rosseland mean of the line expansion opacity that are easy to use in numerical works while still maintaining a high enough accuracy relative to exact solutions. The formulae derived in this paper are expected to have important applications in radiative problems related to relativistic astrophysical phenomena such as neutron star mergers, supernovae, and gamma-ray bursts where relativistic or subrelativistic expansions are involved.