Physics of Luminous Transient Light Curves: A New Relation between Peak Time and Luminosity

Simplified analytic methods are frequently used to model the light curves of supernovae and other energetic transients and to extract physical quantities, such as the ejecta mass and amount of radioactive heating. The applicability and quantitative accuracy of these models, however, have not been clearly delineated. Here we carry out a systematic study comparing certain analytic models to numerical radiation transport calculations. We show that the neglect of time-dependent diffusion limits the accuracy of common Arnett-like analytic models, and that the widely applied Arnett's rule for inferring radioactive mass does not hold in general, with an error that increases for models with longer diffusion times or more centralized heating. We present new analytic relations that accurately relate the peak time and luminosity of an observed light curve to the physical ejecta and heating parameters. We further show that recombination and spatial distribution of heating modify the peak of the light curve and that these effects can be accounted for by varying a single dimensionless parameter in the new relations. The results presented should be useful for estimating the physical properties of a wide variety of transient phenomena.