Analytic Light Curves of Dense CSM Shock Breakout and Cooling

Dense circumstellar material (CSM) is thought to play an important role in observed luminous optical transients: if such CSM is shocked, e.g. by ejecta expelled from the progenitor during core-collapse, then radiation produced by the shock-heated CSM can power bright UV/optical emission. If the initial CSM has an `outer edge where most of the mass is contained and at which the optical depth is large, then shock breakout -- when photons are first able to escape the shocked CSM -- occurs near this outer edge. The $sim$thin shell of shocked CSM subsequently expands, and in the ensuing cooling-envelope phase, radiative and adiabatic losses compete to expend the CSM thermal energy. Here we derive an analytic solution to the bolometric light-curve produced by such shocked CSM. For the first time, we provide a solution to the cooling-envelope phase that is applicable already starting from shock breakout. In particular, we account for the planar CSM geometry that is relevant at early times and impose physically-motivated initial conditions. We show that these effects can dramatically impact the resulting light-curves, particularly if the CSM optical depth is only marginally larger than $c/v_{rm sh}$ (where $v_{rm sh}$ is the shock velocity). This has important implications for interpreting observed fast optical transients, which have previously been modeled using either computationally-expensive numerical simulations or more simplified models that do not properly capture the early light-curve evolution.