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

TL;DR

This paper analytically derives and extends the peak luminosity and time relation for supernova light-curves, confirming its validity and limitations through mathematical solutions and comparison with previous numerical results.

Contribution

It provides an exact analytical solution for homologous expansion models, clarifying the universality and limitations of the peak luminosity-time relation introduced by Khatami & Kasen (2019).

Findings

Analytical solutions reproduce previous numerical results.

Clarifies the conditions for the universality of the peak relation.

Identifies limitations of the relation in certain cases.

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.