Self-similar champagne flows in HII regions

(Abridged) We consider the 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 is negligible for the subsequent motion of the HII region holds for distances much greater than about 100 AU and for the massive cloud cores that give rise to high-mass stars. If the initial ionization and heating is 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, 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.'' We then study the self-similar solutions of the expansion of HII regions embedded in molecular clouds characterized by more general power-law density distributions with exponent 3/2 < n < 3. In these cases, the shock velocity is an increasing function of the exponent n, and diverges as n tends to 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) nearby to the embedded ionizing source is required in order to maintain over a significant timescale the emission measure observed in champagne flows.