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We extend the approximate radiative transfer analysis of Hershkowitz, Linder, and Wagoner [Ap. J. 303, 800 (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; Ap. J. 377, 639 (1991)]. 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 $J_\nu$ at the photosphere, whose location at each frequency is determined by a first-order calculation of the deviation of $J_\nu$ from the Planck function $B_\nu$. The emergent spectral luminosity then assumes the form $ L_\nu = 4\pi^2 r_*^2 \zeta^2 B_\nu(T_p) $, where $T_p(\nu)$ is the photospheric temperature, $\zeta$ is the dilution factor, and $r_*$ is a fiducial radius.
The atmosphere is characterized by the 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 $Z$ and velocity gradient $v/r = (t-t_0)^{-1}$. We obtain the dependences of $\zeta$ and $T_p$ on frequency $\nu$ and the parameters $T_e$, $n_*$, $r_*$, and $\alpha$.
The resulting understanding of the dependence of the spectral luminosity on the parameters which characterize the relevant physical conditions will be of particular use in assessing the reliability of the expanding photosphere method of distance determination. This is particularly important at cosmological distances, where no information about the progenitor star will be available. This technique can also be applied to other low-density photospheres, such as those in accretion disks.
This research was supported by a grant to the Supernova Intensive Study group by the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. M.J.M. was supported by the NASA Graduate Researchers Program through grant NGT 70194-52.
