Shock cooling emission from explosions of red super-giants: I. A numerically calibrated analytic model

Supernova light curves are dominated at early time, hours to days, by the escape of photons from the expanding shock heated envelope. We provide a simple analytic description of the time dependent luminosity, $L$, and color temperature, $T_{\rm col}$, for explosions of red supergiants (with convective polytropic envelopes), valid up to H recombination ($T\approx0.7$ eV). The analytic description is based on an interpolation between earlier analytic expressions valid at different (initial planar and later spherical) stages of the expansion, calibrated against the results of numerical hydrodynamic diffusion calculations for a wide range of progenitor parameters (mass, radius, core/envelope mass and radius ratios, metalicity), and explosion energies. The numerically derived $L$ and $T_{\rm col}$ are described by the analytic expressions with 10\% and 5\% accuracy respectively. $T_{\rm col}$ is inferred from the hydrodynamic profiles using (time and space dependent) effective "gray" (frequency independent) opacity, based on opacity tables that we have constructed for this purpose (and will be made publicly available) including the contributions of bound-bound and bound-free transitions. In an accompanying paper (Paper II) we show, using a large set of multi-group photon diffusion calculations, that the spectral energy distribution is well described by a Planck spectrum with $T=T_{\rm col}$, except at UV frequencies (beyond the spectral peak at $3T_{\rm col}$), where the flux is significantly suppressed due to the presence of strong line absorption. We defer the full discussion of the multi-group results to paper II, but provide here for completeness an analytic description also of the UV suppression. Our analytic results are a useful tool for inferring progenitor properties, explosion velocity, and also relative extinction based on early multi-band shock cooling observations of supernovae.