The Co-Evolution of Stellar Wind-blown Bubbles and Photoionized Gas I: Physical Principles and a Semi-Analytic Model

We propose a new framework for the simultaneous feedback of stellar winds and photo-ionizing radiation from massive stars, distinguishing the locations where forces are applied, and consequences for internal spatio-temporal evolution of the whole feedback bubble (FB). We quantify the relative dynamical importance of wind-blown bubbles (WBB) versus the photoionized region (PIR) by the ratio of the radius at which the WBB is in pressure equilibrium with the PIR, $R_{\rm eq}$, to the Str\"{o}mgren radius, $R_{\rm St}$. $\zeta \equiv R_{\rm eq}/R_{\rm St}$ quantifies the dynamical dominance of WBBs ($\zeta > 1$) or the PIR ($\zeta < 1$). We calculate $\zeta$ and find that, for momentum-driven winds, $0.1 \lesssim \zeta \lesssim 1$ for the star-forming regions in (i) typical Milky Way-like giant molecular clouds (GMCs), (ii) the most massive of individual OB stars, and (iii) dense, low-metallicity environments, relevant in the early universe. In this regime, both WBBs and the PIR are dynamically important to the expansion of the FB. We develop a semi-analytic Co-Evolution Model (CEM) that takes into account the spatial distribution of forces and the back reactions of both the WBB and PIR. In the $\zeta <1$ regime where the CEM is most relevant, the model differs in the total FB momentum by up to 25% compared to naive predictions. In the weak-wind limit of $\zeta \ll 1$, applicable to individual OB stars or low-mass clusters, the CEM has factors $\gtrsim 2$ differences in WBB properties. In a companion paper we compare these models to three-dimensional, turbulent hydro-dynamical simulations.