Efficiently Cooled Stellar Wind Bubbles in Turbulent Clouds. I. Fractal Theory and Application to Star-forming Clouds

Winds from massive stars have velocities of 1000 km s⁻¹ or more and produce hot, high-pressure gas when they shock. We develop a theory for the evolution of bubbles driven by the collective winds from star clusters early in their lifetimes, which involves interaction with the turbulent, dense interstellar medium of the surrounding natal molecular cloud. A key feature is the fractal nature of the hot bubble's surface. The large area of this interface with surrounding denser gas strongly enhances energy losses from the hot interior, enabled by turbulent mixing and subsequent cooling at temperatures T ∼ 10⁴–10⁵ K, where radiation is maximally efficient. Due to the extreme cooling, the bubble radius scales differently ($R_b\propto t^{1/2}$) from the classical Weaver et al. solution and has expansion velocity and momentum lower by factors of 10–10² at given $R_b$, with pressure lower by factors of 10²–10³. Our theory explains the weak X-ray emission and low shell expansion velocities of observed sources. We discuss further implications of our theory for observations of the hot bubbles and cooled expanding shells created by stellar winds and for predictions of feedback-regulated star formation in a range of environments. In a companion paper, we validate our theory with a suite of hydrodynamic simulations.