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Dense circumstellar material (CSM) is thought to play an important role inobserved luminous optical transients: if such CSM is shocked, e.g. by ejectaexpelled from the progenitor during core-collapse, then radiation produced bythe shock-heated CSM can power bright UV/optical emission. If the initial CSMhas an `outer edge' where most of the mass is contained and at which theoptical depth is large, then shock breakout -- when photons are first able toescape the shocked CSM -- occurs near this outer edge. The $\sim$thin shell ofshocked CSM subsequently expands, and in the ensuing cooling-envelope phase,radiative and adiabatic losses compete to expend the CSM thermal energy. Herewe derive an analytic solution to the bolometric light-curve produced by suchshocked CSM. For the first time, we provide an analytic solution to thecooling-envelope phase that is applicable starting from shock-breakout anduntil the expanding CSM becomes optically-thin. In particular, we account forthe planar CSM geometry that is relevant at early times and properly treatradiative losses within this planar phase. We show that these effects candramatically impact the resulting light-curves, particularly if the CSM opticaldepth is only marginally larger than $c/v_{\rm sh}$ (where $v_{\rm sh}$ is theshock velocity). This has important implications for interpreting observed fastoptical transients, which have previously been modeled using eithercomputationally-expensive numerical simulations or more simplified models thatdo not properly capture the early light-curve evolution.

Analytic Light Curves of Dense CSM Shock Breakout and Cooling