Supernovae Exploding within Dense Extended Material: Early Emission Regimes and Degeneracies in Parameter Inference from Observations

Early light curves of many core-collapse supernovae (SNe) are thought to be powered by the interaction of the shock wave with optically thick extended material, either a bound envelope or preexplosion ejected circumstellar matter (CSM). We analytically analyze the early emission produced by a shock with velocity v traversing a material of mass M_\mathrm{e} and opacity \kappa extending to radius R_\mathrm{e}, and show the emission varies qualitatively with varying \tau_\mathrm{e}=\kappa\!M_\mathrm{e}/(4\pi\!R_\mathrm{e}^2): For \tau_\mathrm{e}\gg\!c/v a shock breakout occurs near R_\mathrm{e} producing an ``edge breakout" -- a UV-dominated breakout burst followed by ``cooling emission" of the shock-heated material; for \tau_\mathrm{e}\lesssim\!c/v a ``wind breakout" occurs -- the breakout pulse is prolonged and followed by extended emission shifting from UV to X-ray as the shock becomes collisionless. We derive the dependence on \{v,\kappa,M_\mathrm{e},R_\mathrm{e}\} of the duration and luminosity characterizing the different emission phases, and show that current observations typically do not allow inference of all parameters. In particular, since the optical bands lie in the Rayleigh-Jeans tail of radiation emitted during the cooling phase, the observed cooling luminosity depends weakly on radius, \propto\!R_\mathrm{e}^{1/4}, leading to 1-2 orders of magnitude uncertainty in its inferred value. This suggests, e.g., that the common day-scale light curve features in Stripped-Envelope SNe do not necessarily imply material extending to R_\mathrm{e}\sim10^3\!R_\odot and are often consistent with low-mass R_\mathrm{e}\sim\!10^2\!R_\odot bound envelopes. Early multiband coverage (especially in UV/X-ray) can break these degeneracies; the forthcoming \emph{ULTRASAT} UV mission will allow inferring the properties of extended material around the population of SNe progenitors.