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Astron. Astrophys. 347, 617-629 (1999)

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6. Discussion: why are the forbidden and fine-structure lines so important?

It seems to be a paradox: the permitted lines typically have radiative transition probabilities 4 ... 10 orders of magnitudes larger than the forbidden and fine-structure lines, but do not dominate the radiative heating and cooling according to the results of this paper 9.

In the following, we will briefly discuss the radiative cooling by spectral lines in two important limiting cases. We neglect bound-free transitions, which enables us to evaluate the bound-bound radiative heating/cooling rate either from Eq. (6) or, alternatively, from Eq. (12):

[EQUATION]

1) Cooling in the low-density limit: When the density of the gas is very small, the upper levels mostly de-populate radiatively, because the radiative lifetimes are much shorter than the characteristic time-scales for collisional de-excitation ([FORMULA]10. Consequently, the upper levels are much less populated with respect to LTE: [FORMULA]. This relation can be used straightforwardly in Eq. (12), so we choose to calculate [FORMULA]. The condition that the upper levels are less populated than in LTE also implies that the collisional excitation occurs mainly from the ground level, i. e. we can approximately truncate the first sum in Eq. (12) after [FORMULA]. Furthermore, for simplicity, we put [FORMULA] and roughly assume that the collisional de-excitation rates (electron collisions only) [FORMULA], which are in fact slowly varying temperature-functions, are given by a universal constant [FORMULA]. Eq. (12) then becomes

[EQUATION]

Eq. (26) states that the cooling rates of spectral lines are not at all dependent on the number or the properties of the lines but do only depend on the collisional properties of its carrier in this case 11. Since the function [FORMULA] has its relative maximum at [FORMULA], we conclude that in the low-density limiting case

  • the cooling rates linearly depend on [FORMULA] (roughly [FORMULA]), favouring abundant elements (e. g. C, N, O),

  • most important contributions come from those upper levels u which have [FORMULA] (implying [FORMULA]). Therefore, the most efficient cooling lines are expected to be low-excitation lines in the infrared (e. g. at [FORMULA] µm for [FORMULA]).

2) Cooling in the optically thick LTE case: When the gas is hot and dense enough to keep the level populations close to LTE, we use Eq. (6). The escape probabilities in the optically thick limit in Sobolev-approximation are given by [FORMULA]. We neglect stimulated emission ([FORMULA]) and put [FORMULA] as we are interested in cooling and not in heating. Furthermore, we apply the Boltzmann distribution [FORMULA]. Eq. (6) then becomes

[EQUATION]

Eq. (27) states that the line-cooling does not depend on the density of the carrier, nor on the "strength" of the lines (the Einstein coefficients [FORMULA]12. Since the function [FORMULA] has its relative maximum at [FORMULA], we conclude that in the optically thick LTE limiting case

  • the cooling rates depend linearly on [FORMULA],

  • the cooling rates are roughly density-independent, which means that the cooling rate per mass decreases linearly with increasing density (hence, trace elements may be as important as abundant elements),

  • the cooling rates depend linearly on the number of lines in the spectral region [FORMULA]. The most efficient cooling lines in this case are therefore expected to lie in the optical spectral region (e. g. at [FORMULA] nm for [FORMULA]).

In both considered limiting cases, the radiative cooling by spectral lines does not depend on the strength of the lines, e. g. permitted lines are in fact not favoured. In contrast, the other listed criteria (i. e. the number of lines in case 2 and especially the low excitation energy in case 1) in fact suggest that forbidden and fine-structure lines are more important in both limiting cases.

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© European Southern Observatory (ESO) 1999

Online publication: June 30, 1999
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