We consider the idealized expansion of an initially self-gravitating , static , singular , isothermal cloud core .
For t \geq 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 H ii region holds for distances r much greater than \sim 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 \gg 100 AU ) caused by the resulting imbalance between self-gravity and thermal pressure is self-similar .
Because of the steep density profile ( \rho \propto 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 \rightarrow 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 which ionizes and heats the gas to \sim 10 ^ { 4 } 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 : \rho \propto 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 \rightarrow 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 .