Photoionization equilibrium and H II regions

Let neutral density be n HI, proton density n p and electron density n e, all per cubic metre. The photoionization rate per neutral atom is an integral of photon intensity times the atomic cross section over frequencies above threshold. If J includes both direct and diffuse photons, steady balance uses the total recombination coefficient alpha A. That coefficient has cubic metres per second units, so the recombination rate per volume has inverse cubic metres per second units. Collisional ionization is neglected in this photoionization-dominated example.

The temperature must be supplied or found from a separate heating and cooling calculation. It controls recombination coefficients but does not by itself fix the external ionizing spectrum. A Maxwellian electron distribution can coexist with strongly non-LTE hydrogen populations. Density structure and shielding also alter the neutral fraction even when the ionized volume is nearly fully ionized. Thus an ionization equilibrium is a rate balance with declared radiation and atomic processes; it is not synonymous with thermodynamic equilibrium or a guarantee of uniform temperature.

A recombination directly to the hydrogen ground state emits an ionizing photon. In an optically thick ionizing medium, the on-the-spot approximation assumes that photon causes another nearby ionization. These two local events cancel from the budget of stellar photons. The remaining effective recombination coefficient is alpha B, excluding direct ground-state recombinations. Correspondingly the reduced local equation uses the external stellar photoionization rate, rather than the total direct-plus-diffuse rate. This approximation is inappropriate when the recombination ionizing photons escape freely.

For a static, uniform, pure-hydrogen sphere with a sharp ionization front, set n e and n p approximately equal to the hydrogen density inside. If all emitted source ionizing photons are absorbed by hydrogen, integrating the Case B recombination rate gives the Strömgren radius. The source quantity Q H is photons per second, not its bolometric energy luminosity. Dust absorption, leakage, helium, clumps and a nonuniform density change this budget. Optical thickness to ionizing photons also does not require an optically thick radio continuum or visible recombination line.

After a source switches off, pure hydrogen supplies its own electrons. At fixed total density and temperature, with the same effective Case B recycling approximation maintained, the ionized fraction satisfies a loss rate proportional to its square. Its solution is a reciprocal linear function of time, not an exponential with constant electron density. An exponential can be appropriate for a trace ion in an externally fixed electron bath, but that is a different physical population assumption. The distinction matters when translating fading emission into an elapsed time.

Recombination-line emissivity usually weights electron times proton density with a line-specific effective coefficient. That coefficient is not automatically the total alpha B coefficient. For fixed geometry, temperature and line coefficient, brightness therefore falls as the ionized fraction squared in the simple recombining model. A real heated ionized region can expand as pressure accelerates the surrounding gas; its eventual radius need not equal a static initial Strömgren estimate. Comparing recombination and sound-crossing times helps decide when a fixed-density ionization calculation is a useful approximation.