Self-similar Champagne Flows in H II Regions

We consider the idealized expansion of an initially self-gravitating, static, singular, isothermal cloud core. For t ≥ 0, the gas is ionized and heated to a higher uniform temperature by the formation of a luminous but massless star in its center. The approximation that the mass and gravity of the central star are negligible for the subsequent motion of the H II region holds for distances r much greater than ~100 AU and for the massive cloud cores that give rise to high-mass stars. If the initial ionization and heating are approximated to occur instantaneously at t = 0, then the subsequent flow (for r ≫ 100 AU) caused by the resulting imbalance between self-gravity and thermal pressure is self-similar. Because of the steep density profile (ρ ∝ r^-2), pressure gradients produce a shock front that travels into the cloud, accelerating the gas to supersonic velocities in what has been called the "champagne phase." The expansion of the inner region at t > 0 is connected to the outer envelope of the now ionized cloud core through this shock, whose strength depends on the temperature of the H II gas. In particular, we find a modified Larson-Penston (L-P) type of solution as part of the linear sequence of self-similar champagne outflows. The modification involves the proper insertion of a shock and produces the right behavior at infinity (v → 0) for an outflow of finite duration, reconciling the long-standing conflict on the correct (inflow or outflow) interpretation for the original L-P solution. For realistic heating due to a massive young central star that ionizes and heats the gas to ~10⁴ K, we show that even the self-gravity of the ionized gas of the massive molecular cloud core can be neglected. We then study the self-similar solutions of the expansion of H II regions embedded in molecular clouds characterized by more general power-law density distributions: ρ ∝ r^-n with 3/2 < n < 3. In these cases, the shock velocity is an increasing function of the exponent n and diverges as n → 3. We show that this happens because the model includes an origin where the pressure driving the shock diverges because the enclosed heated mass is infinite. Our results imply that the continued photoevaporation of massive reservoirs of neutral gas (e.g., surrounding disks and/or globules) near the embedded ionizing source is required in order to maintain over a significant timescale the emission measure observed in champagne flows.