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Astron. Astrophys. 363, 1091-1105 (2000)

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5. Results

In the absence of ions and electrons, profiles, for a given temperature and hydrogen atom perturber density, have been computed for H[FORMULA], H[FORMULA] and H[FORMULA] using overlapping line theory. Not only are such profiles valuable in examining the contributions made by the component lines to each profile but these profiles represent a limiting real situation in the spectra of cool stars as the metallicity is reduced to zero. In Fig. 2 the profiles obtained from overlapping line theory are plotted along with profiles for the three component lines which contribute to the overlapping line profile. These three components, each weighted by the appropriate dipole matrix element, sum to give the profile.

[FIGURE] Fig. 2. Hydrogen broadened profiles for H[FORMULA] (top), H[FORMULA] (middle) and H[FORMULA] (bottom) at 8000 K and [FORMULA] perturbers per cubic cm. Shown are the complete profile from Eq. (1) (full), the p-d component (dashed), the s-p component (dotted) and the p-s component (dot-dash). The full vertical line is the line bisector of the full line profile. All profiles are area normalised. Note the different scales, and that due to the different central wavelengths of the lines the above widths should not be directly compared with each other.

It was pointed out by Lortet & Roueff (1969) that the p-d component of the Balmer lines is by far the strongest. This is clearly seen in Fig. 2, where the total profile and p-d component profile are very similar. In Fig. 2 we see that each component (all Lorentzian and therefore symmetric) has a different predicted pressure induced line shift. This leads to a very small predicted asymmetry in the total predicted line profile. However, we should comment that we expect that the shift calculations of our method are less reliable than those for widths, as shifts are perhaps more dependent on strong collisions and hence the short range interaction potential (Anstee & O'Mara 1991).

One also sees a marked difference in the relative width of the p-s component of H[FORMULA] compared to the other two lines. This component is relatively narrow, whereas in the other lines it is the broadest component. We expect the broadening of this component to be seriously overestimated in both H[FORMULA] and H[FORMULA] due to neglect of exchange effects. Fortunately however, this component has almost negligible effect on the overall line profile.

5.1. Temperature dependence

Previous theories of resonance broadening predict a line width which is independent of temperature. This is a result of the interaction decreasing with increasing separation like [FORMULA] which leads to a cross-section which is inversely proportional to the collision speed. For the 2p state (resonance interactions only) we obtain essentially the same result but for more excited p states we observe a temperature dependence which increases with increasing excitation. This temperature dependence becomes quite significant for the 5p state. This difference is a result of increasing departure in our calculations from an [FORMULA] dependence of the interaction on the interatomic separation indicating that the multipole expansion used in previous calculations is only strictly valid for the 2p state.

When one introduces the dispersive-inductive interactions the self-broadening is found to be no longer temperature independent even for the low lying states. This result is of astrophysical importance as we will discuss later.

5.2. Comparison with Ali & Griem

Fig. 3 compares line widths from our treatment of self-broadening with those of Ali & Griem (1966 , corrected), which include only the resonance broadening of the lower 2p state, for H[FORMULA] and H[FORMULA] as a function of temperature. We find that our results are in quite good agreement with the Ali & Griem (1966) theory when we only consider the resonance interactions as they did. However, we find that the effect of the dispersive-inductive interaction of other states involved in the transition is quite substantial, particularly that resulting from the d state of the upper level in Balmer lines. The dispersive contribution relative to the resonance contribution for H[FORMULA] is greater than for H[FORMULA] and this is reflected in the stronger temperature dependence of the line width.

[FIGURE] Fig. 3. Comparison of the line width (HWHM) per perturber with temperature computed in this work for the dominating 2p-3d component of H[FORMULA] (lower full) and 2p-4d component of H[FORMULA] (upper full) with that of the resonance broadening theory of Ali & Griem (dashed) for the 2p state, and our calculation of the resonance broadening (dot-dash) for this state.

Fig. 4 shows the ratio, as a function of temperature, of the line widths resulting from our treatment of resonance broadening of the 2p, 3p, 4p, and 5p states with those of Ali & Griem (1966). The difference can be attributed to a failure of the multipole expansion of the electrostatic interaction between the two atoms which is used by Ali & Griem (1966). Lortet & Roueff (1969 , Fig. 3) show calculations that suggest that the multipole expansion should breakdown for all p states but the 2p state. Our calculations suggest that the break down is seen for these states but is not severe until the 5p state for the collision speeds of interest here.

[FIGURE] Fig. 4. Comparison showing the ratio of our results for resonance broadening and the Ali & Griem (1966) theory. The full, dashed, dot-dash and dot-dot-dot-dash lines correspond to the 2p, 3p, 4p and 5p levels respectively.

5.3. Approximation by p-d component in Balmer lines

Using overlapping line theory, grids of profiles which result from self-broadening alone have been computed for a range of temperatures and hydrogen atom number densities from which one can interpolate the appropriate profile for a given set of physical conditions. However, we have already seen that these complete profiles obtained from overlapping line theory are very closely approximated by the p-d component of the relevant Balmer line. When applied to synthetic Balmer lines in the solar spectrum for H[FORMULA] the maximum difference is less than 0.2 percent of the continuum flux, less for H[FORMULA], and even less for H[FORMULA]. Errors resulting from employing the p-d approximation in the interpretation of real stellar spectra will lie in the noise associated with the observational data.

A possible objection to the p-d approximation is that it has been developed in the limit of a zero quasi-static ion field. The quasi-static ion field destroys spherical symmetry leading to [FORMULA] no longer being a good quantum number so that it is no longer strictly possible to talk about a p-d transition. This will certainly be the case in hot stars where the ions are protons produced by the almost complete ionisation of hydrogen leading to a strong quasi-static ion field. However, our interest is in cool stars where the ions are produced by thermal ionisation of metals. The quasistatic ion field is proportional to [FORMULA] where [FORMULA] is the ion number density which in cool stars is smaller than that in hot stars, where the ions are largely protons, by four orders of magnitude for a star with solar composition and by perhaps six orders of magnitude for cool stars of low metallicity. Therefore in this work we are working in the limit of very weak quasistatic ion fields. Under these circumstances the p and d states are only very weakly mixed by the weak quasistatic ion field with states of other [FORMULA] with the same n. Thus in the limit of very weak quasistatic ion fields [FORMULA] is an almost good quantum number and the p-d approximation is acceptable.

The p-d approximation neglects the line shift and asymmetry predicted using overlapping line theory. However, when overlapping line theory and the p-d approximation are used in the synthesis of Balmer lines in the solar spectrum the shifts and asymmetries predicted by overlapping line theory are not detectable due to the effects of Stark broadening and the profiles are in good agreement with those predicted using the p-d approximation. Due to the reduced ion/electron density, synthetic spectra for very cool stars (T [FORMULA] 4500 K) show some evidence of shift and asymmetry when overlapping line theory is used. However, the synthesis of the spectrum of very cool stars is complicated by impact broadening due to molecular hydrogen which is not included in our calculations.

Although the p-d approximation does not significantly reduce computing time it does permit self-broadening data to be presented in a way which is much more efficient than the publication of grids of line profiles. The data relevant to the application of the p-d approximation to the first three Balmer lines are presented in Table 3. Data in the form of grids can be obtained from the authors. In spite of the advantages of the p-d approximation grids have been used in all calculations in this paper and in Paper I.

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Online publication: December 5, 2000