A Novel Parameterization for Rapid Cooling in Supernova Remnants, with applications to the Pa 30 nebula

We systematically study how cooling creates structural changes in supernova remnants as they evolve. Inspired by the peculiar morphology of the Pa 30 nebula, we adopt a framework in which to characterize supernova remnants under different degrees of cooling. Our cooling framework characterizes remnants with a singular parameter called $\beta$ that sets how rapidly the system's thermal energy is radiated or emitted away. A continuum of morphologies is created by the implementation of different cooling timescales. For $\beta \gtrsim 400$, or when the cooling timescale is shorter than $\approx \frac{1}{400}$ of the Sedov time, the ejecta is shaped into a filamentary structure similar to Pa 30. We explain the filament creation by the formation of Rayleigh-Taylor Instability fingers where cooling has prevented the Kelvin-Helmholtz Instability from overturning and mixing out the tips. The ejecta in these filaments have not decelerated and are moving almost completely ballistically at $\approx 95-100\%$ their free expansion speed. In this rapid cooling regime, an explosion energy $\approx 3.5 \times 10^{47}$ erg is inferred. We also propose the cooling mechanism required to create these structures necessitates removing energy at a rate of $2\%$ of $E_{\rm ej}/t$, which implies a cooling luminosity of $\approx 10^{36}$ erg/s.