A Complete Analytic Inversion of Supernova Lines in the Sobolev Approximation

Abstract

We show that the shape of P-Cygni line profiles of photospheric phase supernova can be analytically inverted to extract both the optical depth and source function of the line -- i.e. all the physical content of the model for the case when the Sobolev approximation is valid. Under various simplifying assumptions, we derive formulae that give S(r) and tau(r) in terms of derivatives of the line flux with respect to wavelength. The transition region between the minimum and maximum of the line profile turns out to give especially interesting information on the optical depth near the photosphere. The formulae give insights into the relationship between line shape and physical quantities that may be useful in interpreting observed spectra and detailed numerical calculations.