If the star ionizing an HII region suddenly doubled the radiation emitted, would that increase or decrease the temperature? I feel like the expansion of the region would cancel out the increased energy, but I don't know.
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My impression is that the temperature does indeed increase. I think that the growth in radius of a Strömgren sphere due to increase in radiation of the star is not an "expansion" of the sphere, but an shift of the limit at which the properties of the gas change. Taking this into account I would proceed as follows to make some calculations.
For most of the ionizing stars, a high fraction of the ionizing photons will have energies greater than the ionization energy of hydrogen. The excess of energy of the ionizing photons will become kinetic energy of the electron gas. Knowing the temperature of the star and considering this excess of energy, you should be able to calculate or make some assumption about the average energy of a photo-ejected electron.
On the other hand, the heating mechanisms will be balanced by cooling and the gas will tend to local thermodynamic equilibrium. I guess that cooling of the electron gas will be mainly due to recombination. You should be able to express the average energy that is loss in the gas for a single recombined electron. Probably there are other cooling processes, but this would make things too complex.
Since in equilibrium the recombination rate is equal to the ionization rate in the sphere, the energetic balance between those two quantities for an average electron should provide the equilibrium temperature. As you see, at least according to my understanding, the radius of the sphere does not have any impact on the determination of the temperature.
Knowing the temperature, the recombination rate [itex]\beta[/itex] is known which provides the necessary information to calculate the radius of the Strömgren sphere. Putting things together you should come to an relation between the radius and the increase in radiation. This would be a nice exercise.