Hunter-type Implosion Profiles for Energy-supercritical Polytropic Equations of State

We rigorously construct a family of smooth self-similar solutions to the isentropic gravitational Euler-Poisson system with a polytropic equation of state for polytropic indices lying in the full energy-supercritical range, $1<\gamma<\frac{6}{5}$. The result is an extension of the author's previous construction of Hunter solutions in the isothermal case, $\gamma=1$, and complements a construction of Larson-Penston-type solutions by Guo-Had\v{z}i\'c-Jang-Schrecker for the same system in the full mass-supercritical range, $1<\gamma<\frac{4}{3}$. As an ingredient in the proof, a general framework is introduced for proving local analyticity of solutions to this system in the vicinity of singular points. This framework could be used for other quasilinear self-similar blow-up constructions.