Semi-analytic Continuum Spectra of Type II Supernovae and the Expanding Photosphere Method of Distance Determination

Abstract

We extend the approximate radiative transfer analysis of Hershkowitz, Linder, and Wagoner (1986) to a more general class of supernova model atmospheres, using a simple fit to the effective continuum opacity produced by lines (Wagoner, Perez, and Vasu 1991). At the low densities considered, the populations of the excited states of hydrogen are governed mainly by photoionization and recombination, and scattering dominates absorptive opacity. We match the asymptotic expressions for the spectral energy density $\Jnu$ at the photosphere, whose location at each frequency is determined by a first-order calculation of the deviation of $\Jnu$ from the Planck function $\Bnu$. The emergent spectral luminosity then assumes the form $ L_\nu = 4\pi^2 \r*^2 \zeta^2 \Bnu(T_p) $, where $T_p(\nu)$ is the photospheric temperature, $\zeta$ is the dilution factor, and $\r*$ is a fiducial radius [ultimately taken to be the photospheric radius $r_p (\nu)$]. The atmosphere is characterized by an effective temperature $T_e$ ($\propto L^{1/4} r_*^{-1/2}$) and hydrogen density $n_H=n_* (r_*/r)^\alpha$ ; and less strongly by the heavy element abundance and velocity gradient. Our major result is the dependence of $\zeta$ on frequency $\nu$ and the parameters $T_p$, $r_p$, and $\alpha$.