Polytropic Wind-Driven Bubbles and their Shock Structures in Radially Stratified Ambient Media

We extend the analytic expressions for polytropic wind-driven bubbles and their shock structures, formulated initially in Koo and McKee 1992(a,b), focusing on spherically symmetric configurations in astrophysical environments with $\rho\propto r^{-2}$, which arises naturally in the star-forming environment and has applications to winds flowing into a preexisting bubble. Wind luminosity is assumed to be constant, and as a result the shock velocities of these bubbles are constant in time. The ratio of specific heats is assumed to be the same in the shocked ambient medium and the shocked wind. Numerical results are presented for one selected ratio of wind density to ambient density. Exact ODEs are written for the compressed wind region and approximate solutions are found by fitting the ODE solutions. By analyzing the interactions between stellar winds and ambient media in the strong compression limit, we model the formation and evolution of spherical bubbles, highlighting their shock fronts and contact discontinuities. Our analytic method provides an intuitive approach to calculating the thickness of bubble shells, which is crucial for understanding their dynamics and observational characteristics. A numerical method explores conditions without explicitly requiring the strong compression limit, and then we compare numerical to analytical results under various conditions.