A Temperature of HII Gas: Intuitive Explanation

[zviz]

I have a question about the temperature of HII gas. According to Table 1.3 in Draine's book (physics of the interstellar and intergalactic medium) this temperature is T~10^4 K.
I get a similar value by using the Saha equation for a hydrogen gas.
However, a naive estimate of the same temperature could be obtained by demanding k*T = 13.6eV, and this would give T>10^5K.

My question is what produces this discrepancy? I am looking for an intuitive answer, if possible, of why the estimate using kT=ionization energy is higher by more than an order of magnitude from the actual temperature where most of the hydrogen gas becomes ionized.

 

[snorkack]

I have a question about the temperature of HII gas. According to Table 1.3 in Draine's book (physics of the interstellar and intergalactic medium) this temperature is T~10^4 K.
I get a similar value by using the Saha equation for a hydrogen gas.
However, a naive estimate of the same temperature could be obtained by demanding k*T = 13.6eV, and this would give T>10^5K.

My question is what produces this discrepancy? I am looking for an intuitive answer, if possible, of why the estimate using kT=ionization energy is higher by more than an order of magnitude from the actual temperature where most of the hydrogen gas becomes ionized.

The intuitive answer is that ion+electron is a state which, at a low density, has far bigger probability and entropy than the state where electron is bound to the ion. Or looking at the process side, atom can be ionized at any time because the electron and ion are together, whereas recombination requires the ion and electron to find one another.
Therefore, at kT=ionization energy, only the tiny fraction corresponding to the tiny volume fraction actually taken up by ions is not ionized. In order to recombine most of the ions, you need to lower the temperature by an order of magnitude till the exponential component outweighs the density component.

 

[Hyperfine]

However, a naive estimate of the same temperature could be obtained by demanding k*T = 13.6eV, and this would give T>10^5K.

My question is what produces this discrepancy? I am looking for an intuitive answer, if possible, of why the estimate using kT=ionization energy is higher by more than an order of magnitude from the actual temperature where most of the hydrogen gas becomes ionized.

The discrepancy is the result of thinking that the ionization process is thermal. It is not.

The ionization process is photoionization. The HI is ionized by interaction with the radiation field of one or more hot stars--type O for example-- typically embedded within the gas cloud. The kinetic energy of the resultant free electrons that determines the temperature is given by the difference between the energy of the ionizing photon and the ionization potential of HI.

You have also not taken account of cooling mechanisms. HII regions are effectively cooled by radiative decay of the collisionally excited, low lying metastable states of ions such as OII, OIII, NII, and SII. The relevant transitions, which are within the visible spectral domain, are electric-dipole forbidden. However, the lifetimes of those metastable states are such that radiative decay via electric-quagrupole or magnetic-dipole allowed transitions is fast relative to collisional decay channels. Being forbidden transitions, the emitted photons are not reabsorbed by the material within the HII region allowing energy loss.

References:
1. E. J. Chaisson, "Gaseous Nebula and Their Interstellar Environment" (Chapter 7) in Frontiers of Astrophysics, E. H. Avrett, Ed., Harvard University Press 1976.
2. D. E. Osterbrock and G. J. Ferland, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei 2nd Ed., University Science Books 2006

 

[Hyperfine]

First, an editorial correction to Reference 2 above. The correct title of the book by Osterbrock and Ferland is Astrophysics of Gaseous Nebulae and Active Galactic Nuclei.

[Mentor Note -- update made to Reference 2 in above post]

Second, it may be of interest to point out a pragmatic aspect of the electric-dipole forbidden transitions associated with the metastable states of OIII and NII. Both species emit within the visible spectral domain: OIII at 436.3, 495.9, and 500.7 nm; NII at 575.5, 654.8, and 658.3 nm. Those are rest-frame wavelengths. Those spectral lines can be very prominent in the observed spectra of HII regions.

Assuming effective local thermodynamic equilibrium, a free electron density that is sufficiently low to preclude collisional decay of the metastable states, and further assuming that the metastable states are not populated by radiative decay from higher excited states, the ratios of the line amplitudes provides a direct, observable measure of the nebular electron temperature.