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Astron. Astrophys. 356, 1149-1156 (2000)

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4. Recombination line emission and energetic considerations

In H II regions all electrons captured via free-bound transitions produce a cascade of recombination lines which can be observed from radio to optical frequencies. Lines in recombination cascades contribute to the cooling of the H II region. Accurate emissivities with Gaunt factors for recombination lines depending on temperature and densities in the range of interest are tabulated in machine-readable form (Storey & Hummer 1995). The Lyc photons produced by stars heating the H II region are captured in the gas where ionization from and recombination to the ground state of H I balance each other (Rubin 1968) so that for a static ionized region

[EQUATION]

The cooling proceeds via recombination to quantum levels [FORMULA] and subsequent line emission. The quantity [FORMULA] is the rate coefficient of recombination on level n and [FORMULA] the total recombination coefficient. The value [FORMULA] can be obtained from Seaton (1959) to an accuracy of a few per cent for [FORMULA] K as

[EQUATION]

The mean energy of Lyc photons of OB-star clusters varies from [FORMULA] for B0-stars to [FORMULA] for O4-stars resulting in a mean value of [FORMULA] (Mezger et al. 1974) with [FORMULA] the energy of Ly[FORMULA] photons.

In the following, we will concentrate on ionization-bounded H II regions which - on average - are heated by

[EQUATION]

if the volume is measured in cm3 and the electron density in cm-3. For typical temperatures of [FORMULA] K, the Lyc luminosity of the stars is larger than the luminosity of free-bound emission from recombination onto ground state, which is obtained by integrating Eq. 30 for [FORMULA] over frequencies. This results in (Cooper 1966)

[EQUATION]

with the integrated Gaunt factor for free-bound emission

[EQUATION]

Numerically we find [FORMULA] for [FORMULA] K. The cooling of the H II region proceeds partly by electrons recombining on quantum levels [FORMULA]. The corresponding luminosity is easily seen to be

[EQUATION]

and amounts to 25% of [FORMULA] again for [FORMULA]K. [FORMULA] is known as the Riemann-[FORMULA]-function. The contribution of free-free emission to the cooling rate of the H II region can be neglected at the temperatures under consideration. At [FORMULA] K the free-free contribution is just 2% of the free-bound radiation. The last radiative contribution to the cooling radiation from hydrogen is provided by bound-bound transitions after recombination. As ionization-bounded regions are optically thick for Lyc photons every ionizing photon from the star ends up as a photon in the Lyman series while exited atoms decay to ground state. Here one has to distinguish between regions optically thin (case A) or thick (case B) in the Lyman lines.

In case A the most energetic photons in lines are Ly[FORMULA] photons. We approximate the bound-bound luminosity by multiplying the Ly[FORMULA] energy with the number of Lyc photons originally available, and obtain

[EQUATION]

For proton densities [FORMULA] cm-3 the [FORMULA] level of hydrogen is depopulated by two-photon decay. This produces continuum radiation which is probably optically thin, so that the energy radiated by bound-bound transitions in Case A and B is still reasonably well approximated by Eq. 39. If we take an evolved H II region, assuming it is ionization-bounded and therefore that its size is that of the Strömgren sphere, the sum of all cooling processes under discussion accounts for 70% to 75% of the cooling between 6000 and 14000 K

[EQUATION]

and can provide the total cooling of the H II region. If free-bound, free-free, and bound-bound emission from an ionized hydrogen gas would have to balance the heating of the star given in Eq. 35, the gas temperature would be as high as [FORMULA] K. A similar temperature is derived from balancing electron heating due to ionization inside the largely ionized gas derived below in Eq. 47 and free-free emission, which cools free electrons without destroying them. This gives a temperature of [FORMULA] K for the electrons independent of density and size of the ionized region. However, additional cooling is provided by forbidden line emission from collisionally excited atoms of heavier elements. These metals determine the temperature of the gas in stationary H II regions.

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© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000
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