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., $v(r)\propto r$), is analytic, and even takes account of occultation by a pseudo-photosphere. Previous inversion methods have been developed which are restricted to optically thin lines, but the particular case of homologous expansion permits an analytic result for lines of {\it arbitrary} optical depth. In fact, we show that the quantity that is generically retrieved is the run of line intensity $I_{\lambda }$ 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 ${\tau }_{\lambda }$ with radius in the case of a pure resonance scattering emission line.