A Complete Analytic Inversion of Supernova Lines in the Sobolev Approximation

We show that the shape of P-Cygni line profiles of photospheric phasesupernova can be analytically inverted to extract both the optical depth andsource function of the line -- i.e. all the physical content of the model forthe case when the Sobolev approximation is valid. Under various simplifyingassumptions, we derive formulae that give S(r) and tau(r) in terms ofderivatives of the line flux with respect to wavelength. The transition regionbetween the minimum and maximum of the line profile turns out to giveespecially interesting information on the optical depth near the photosphere.The formulae give insights into the relationship between line shape andphysical quantities that may be useful in interpreting observed spectra anddetailed numerical calculations.Comment: 12 pages, 2 figures, accepted for publication in the Astrophysical Journa