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

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7. Conclusions

The radiative heating and cooling rates of Fe I and Fe II have been calculated under various temperature, density and radiation field conditions typical for the extended atmospheres and circumstellar envelopes of cool stars. The calculations are based on large non-LTE model atoms, include a few thousands of fine-structure, forbidden, permitted and bound-free transitions, and rely on statistical equilibrium in the Sobolev limit.

In comparison to other heating/cooling agents, Fe I and Fe II have been found to be important especially at medium temperatures ([FORMULA]K) between the molecular domain and temperatures appropriate for the effective collisional excitation of the hydrogen atom. The high efficiency of iron is caused by its numerous low-excitation states in combination with its large number of spectral lines, which can easily outweigh the fairly small abundance of this element in comparison to others, e. g. C, N, O.

The calculation of the line-cooling rates requires both a non-LTE treatment of the problem (especially at small densities) and a treatment of the optical depth effects in the lines (especially at large densities). Simple formulae for the line-cooling, like [FORMULA] or the application of mean LTE-opacities, just fail for those densities ([FORMULA]), where the formation of chromospheres, the dust formation, and the driving of stellar winds is expected to occur.

A lot of the cooling by iron in fact involves fine-structure, forbidden and semi-permitted lines, as the strong permitted lines are also strongly blocked. The numerical results as well as the general considerations in Sect. 6 suggest that infrared low-excitation lines with ([FORMULA]) work best at small densities, whereas the total number of lines with ([FORMULA]) is important at large densities.

For grey LTE hydrodynamical models we cannot recommend the use of Planck means in the energy equation, as this approximation ignores the blocking of the emitted line photons completely, which leads to cooling rates being too large by 5 orders of magnitude or more. Rosseland means (scattering-free), which essentially account only for the continuum heating/cooling, are apparently the better choice in grey LTE models. However, as the assumption of LTE does not hold in stellar winds, also the application of Rosseland mean opacities is questionable. The non-LTE heating/cooling rates per mass strongly decrease with decreasing density and this behavior is usually not revealed when Rosseland means are applied. Furthermore, line cooling is definitely more important at small densities ([FORMULA]).

The calculated radiative cooling rates of iron are found to be larger in general, less temperature-dependent and less density-dependent than what is usually assumed in hydrodynamical models using pre-calculated non-LTE cooling laws (e. g. Bowen 1988, Cuntz 1990). These findings may provide new clues to chromospheric heating mechanisms and to the propagation of shock waves in the envelopes of cool stars. The radiative cooling time-scales of the gas due to iron alone are found to be much shorter as compared to these models in the important temperature-regime [FORMULA]K. Furthermore, the results of his paper may shed some new light on the driving of cool star winds by pulsation, since the overall efficiency of radiative heating/cooling can have a dramatic influence on the results of model calculations concerning this driving mechanism (Willson & Bowen 1998). The present results for iron cooling suggest a larger heating/cooling efficiency of the gas, possibly leading to smaller pulsation-driven mass loss rates.

The ionisation balance in the envelopes of AGB stars is strongly controlled by photospheric (+chromospheric) radiation. The present calculations indicate that the Fe II/Fe I-ratio is mainly a function of density and (UV) radiation field rather than a function of temperature, which favours large degrees of ionisation at small densities. This might partially explain the detection of both unexpectedly strong Fe I and Fe II fine-structure lines in M-type and C-type giants (Aoki et al. 1998). Depending on the conditions, the fine-structure lines can be strongly pumped by fluorescence which may cause excitation temperatures of several thousand degrees above the thermal level. Thus, the present paper suggests the emergence of much larger fluxes in the fine-structure lines as compared to predictions based on LTE.

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© European Southern Observatory (ESO) 1999

Online publication: June 30, 1999
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