Lecture #4a-- Stromgren spheres and H II regions



1) Degree of ionization considering only direct transport of Lyman continuum photons from the star, with no light attenuation (a sink) and no light diffusion (a source).
Balancing the radiative ionization rate with the radiative recombination rate gives that the neutral fraction, when small, is proportional to density and inversely proportional to the dilution factor (W) of the radiation field. The latter effect makes it increase like distance-squared, and that would continue all the way out to about 100 parsecs (where the neutral fraction would no longer be small). However, that neglects both attenuation and photoionization due to Lyman continuum photons emitted by nearby recombinations (the latter being thought of as diffusion of Lyman continuum photons, the "stumbling drunk").
2) Problem with the above:
Long before you get to 100 parsecs, you will run out of ionizing photons from the star, so you cannot neglect attenuation. The crucial process here is that about 60% of recombinations are to excited states of H, not the ground state, and these cascade down to the ground state by emitting line photons that escape from the nebula and do not cause additional ionization. So 60% of the time that a Lyman continuum photon is used to ionize H, say within about 1 parsec of the star, it will not be usable again for further ionizations in the region from 1 parsec to 100 parsecs, as needed by the above approach.
2) "On the spot" approximation to the diffuse component:
This approximation attempts to include both the attenuation of Lyman continuum photons by the above process, and the diffusive Lyman continuum radiation field coming from nearby recombinations, by simply asserting that every recombination to the ground state creates a Lyman continuum photon that causes an ionization so nearby to the recombining atom that we may as well treat that ionization as happening in the recombining atom. In short, recombinations to the ground state "don't count" as recombinations at all, and we simply neglect that contribution to the recombination rate. That handles the effects of the diffusive radiation field as a source of ionization.
3) The Stromgren-sphere treatment of the attenuation:
We also handle the attenuation by noting that as soon as we begin to see appreciable attenuation, the neutral fraction will rise rapidly, causing even more rapid attenuation-- and this leveraging effect will mean that we make a transition from a region of essentially no attenuation to one of essentially complete attenuation very rapidly. The former region defines the "Stromgren sphere", so as long as we stay within that sphere, we neglect attenuation altogether.
4) The size of the Stromgren sphere:
We may find the volume of Stromgren sphere simply by equating the incident flux of ionizing photons from the star with the exiting flux of photons made by recombination to levels other than the ground state (so we are applying approximations from (2) and (3) above). The exiting flux is proportional to the volume times the density squared, because essentially all the density is in free electrons/protons and the total recombination rate will scale with the square of that density times the volume (you should be able to see why that is). Thus the volume of the ionized sphere is proportional to the ionizing photon flux (given by the type of central star) and inversely proportional to the square of the density in the sphere (determined by the attributes of the pre-existing interstellar medium). The weakest approximation in this approach is the use of a constant density, which will be violated by both clumps in the ISM and by the likelihood of a spherically diverging wind coming from the star itself, but one may hope that over the ~1 parsec scale of the sphere, these details are averaged against the prevailing features of the ISM in that region. You can look at pictures of H II regions on the astronomy picture of the day site to satisfy yourself of the relative strengths and weaknesses of the Stromgren-sphere idealization of H II regions (such as here).
5) The mass of an H II region
If we take a typical density of about 10^3 particles per cc and convert that to a mass, we find the mass inside a 1 parsec sphere is about 100 solar masses. Thus an O star of about 50 solar masses is capable of ionizing an amount of mass even larger than its own mass. As density drops, this ionized mass increases, because the ionized volume drops like the square of the density.
6) Ultracompact and ultra-extended H II regions
Just to show how complicated the real world actually is, when massive stars first form they are generally shrouded in very dense molecular clouds (say 10^4 particles per cc, not the putative 10^3 used above) that also include a lot of dust. This makes their H II regions initially about 10 or more time smaller in radius, called "ultracompact HII regions", which are hard to even see outside of the IR or X-ray wavelengths. As the star ages, it tends to blow away much that dense cloud, via photodissociation of dust and molecules in a fashion that is out of equilibrium so temporally evolves. There is also the impact of its wind, gradually blowing a bubble that dissipates the ultracompact H II region into a more spread out "normal" H II region. Also, when there is a whole cluster of O stars (and they do tend to form in clusters), then you can have a much larger H II region, as you can see here.