Analytic Inversion of Emission Lines of Arbitrary Optical Depth for the Structure of Supernova Ejecta

We derive a method for inverting emission-line profiles formed in supernova ejecta. The derivation assumes spherical symmetry and homologous expansion [i.e., v(r) ∝ r], is analytic, and even takes account of occultation by a pseudophotosphere. Previous inversion methods have been developed that are restricted to optically thin lines, but the particular case of homologous expansion permits an analytic result for lines of arbitrary optical depth. In fact, we show that the quantity that is generically retrieved is the run of line intensity Iλ with radius in the ejecta. This result is quite general and so could be applied to resonance lines, recombination lines, etc. As a specific example, we show how to derive the run of (Sobolev) optical depth τλ with radius in the case of a pure resonance scattering emission line.