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Astron. Astrophys. 357, 767-776 (2000)

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6. Conclusions

The escape probability approach in conjunction with a simple stratified atmosphere model is effective in describing the limb brightening and ratio variation curves of the C II and C III data considered here. The most effective model is found to be a layer of density that varies exponentially with height, agreeing qualitatively with the findings of Mariska et al. (1978). Optimal density scale heights were found to be 1.2 arc sec for C II and 2.5 arc sec for C III although the result for C III is felt to be an overestimation given the models' underestimate of the extent of the dip in the ratios upon crossing the limb. Cutoffs were included to ensure the onset of scattered light dominance at 974 and 969 arc sec for C II and C III respectively. A physical interpretation of this is that the exponential fall off of density reflects the reduction of the number of spicules intersected by the line of sight whereas the cutoff represents the actual fall off of density at the top of a spicule. The fact that the cutoff is greater for the lower temperature emission is possibly due to the overestimate of the C III scale height. However, it could be linked with the apparent evidence in the observed fluxes of unresolved optically thick structures extending to greater heights.

The escape probability approach has been further simplified since the optical depths relevant here are moderate. Thus the population modification leads to negligible distortion of the upper level population distribution relative to the lower level distribution and so it may be assumed that they differ by a constant. This assumption allows the layer averaged escape probability, [FORMULA], to be used.

Due to the approximately linear dependence of intensity on upper level column density, even the average limb brightening curves display a sensitivity to atmospheric structure. In contrast, the ratios reflect more the dependence of the escape probability on optical depth and are thus less sensitive. This suggests that, since [FORMULA] is similarly insensitive to optical depth for optical depths greater than [FORMULA] 1, and, as Eq.  6 shows, the general quantity [FORMULA] is a function of the radiation field - a quantity that represents an integral over space and thus a degree of averaging over inhomogeneity - the population modification itself is relatively insensitive to inhomogeneity.

A principle motivation for this work was to assess the potential of the simple escape probability techniques for use within dynamic atmosphere models. In a fully resolved picture the dynamic evaluation of [FORMULA] point to point is arduous. However, the insensitivity to structure suggests that the absorption factors may be well approximated using a stratified (averaged) atmosphere model, or even using [FORMULA]. Since the emergent intensities are more sensitive to structure the fully resolved model must be used in Eqs. 12 or 27 (whichever is appropriate). This may be done with no increase in computational effort.

Line blending, when significant, has a marked influence on both the emergent intensities and the population structure. Blending may be easily included within all the escape probability and absorption factor expressions.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000