## 5. Metal-free H II regionsAn ionization-bounded H II region with primordial abundancies of hydrogen and helium, but without heavier elements, must be much hotter than present day H II regions as shown in the previous section. Even then, not all ionizing photons will be absorbed and converted to soft photons, but reach the actual boundary of the H II region and drive an ionization front into the surrounding neutral gas. Some fraction of the stellar Lyc luminosity is consumed at the front for ionization and heating of the gas. We approximate the ionizing spectrum of the star and the radiation at the front by a Wien-law of temperature . The gas heating at the front is then which also suffers from losses inside the ionized region by an opacity factor . If we set up the energy balance for the ionizing photons injected by the star, we find a minimal temperature such that is real, but can balance the energy budget for any temperature above that. The number of ionizing photons is reduced in the H II region only by free-bound transitions, which equal the number of recombinations determined by Eq. 34 in ionization equilibrium. The total energy balance is then with the recombination rate given by Eq. 34 and the cooling from Eqs. 39 and 38. The free-free luminosity includes the spectral averaged Gaunt factor
. The balance Eq. 42 has two
solutions: One at high temperatures of several
K, where the cooling is dominated
by free-free emission, and another one between
and a few
K, where the cooling is due to
free-bound and bound-bound transitions. In both cases the equilibrium
temperature is very sensitive to density and radius or the product
of the H II region.
The high temperature solution faces two severe problems in our
context. The expansion time for the ionization front driven by a
massive star is of the order of
years while the recombination time scale for reaching ionization
balance in a region of mean density
cm with the energy of the ionization edge from Eq. 31 and the exponential integral (e.g., Abramowitz & Stegun 1972). The approximation in Eq. 45 holds only for low effective stellar temperatures . The inside heating of H II regions is and we get an approximate electron energy balance which is only satisfied to within a factor of two for present in Eq. 41 in case of the lower temperature solution. The electron energy balance Eq. 48 cannot hold for the high energy solution. In that case the free-free cooling rises with temperature while the recombination rate declines and the gas heating at the ionization front is more or less independent of in the H II region. The opacity due to conversion of photons inside the ionized region becomes less important with increasing temperature. For illustration we have performed a simplified 1D radiative transfer and ionization balance calculation including hydrogen and helium. The local calculation finds only the lower temperature solution. We have taken a density distribution with a central cusp with cm
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