Analytical calculation of the numerical results of Khatami and Kasen for transient peak time and luminosity

Abstract

The diffusion approximation is often used to study supernovae light-curves around peak light, where it is applicable. By analytic arguments and numerical studies of toy models, Khatami & Kasen (2019) recently argued for a new approximate relation between peak bolometric Luminosity, $L_p$, and the time of peak since explosion, $t_p$, for transients involving homologous expansion: $L_p=2/(\beta t_p)^2\int_0 ^{\beta t_{p}} t'Q(t')dt'$, where $Q(t)$ is the heating rate of the ejecta, and $\beta$ is an order unity parameter that is calibrated from numerical calculations. Khatami & Kasen (2019) demonstrated its validity using Monte-Carlo radiation transfer simulations of ejecta with homogenous density and (for most cases considered) constant opacity. Interestingly, constant values of $\beta$ accurately reproduce the numerical calculations for different heating distributions and over a wide range of energy release times. Here we show that the diffusion and the adiabatic loss of energy in homologous expansion is equivalent to a static diffusion equation and provide an analytic solution for the case of uniform density and opacity (extending the results of Pinto & Eastman 2000). Our accurate analytical solutions reproduce and extend the results of Khatami & Kasen (2019) for this case, allowing clarification for the universality of their peak time-luminosity relation as well as new limitations to its use.