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

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7. Synthetic stellar spectra

The computation of synthetic stellar spectra requires the convolution of all broadenings. We convolve our self-broadened profiles with appropriate Stark profiles from Stehlé (1994) which are provided preconvolved with the Doppler profiles. The profiles are then further convolved with profiles for radiative and helium collision broadening. In these calculations we approximate the convolution in the far wings by adding the profiles (Stark, self-broadening, radiative and helium broadening) together, following the Kurucz (1993) codes.

This procedure can be justified, for cool stars, in terms of the p-d approximation which we know to be valid in the weak quasi-static ion field limit which we know to exist in such stars. In the p-d approximation in the absence of ions the line is well represented by the p-d component alone which will have a Lorentz profile due to impact broadening by electrons and hydrogen atoms. In the presence of a given weak quasi-static ion field this profile will be Stark shifted by an amount dictated by the first order Stark shift of the p-d component. The final profile can then be found by integrating this profile over the Holtsmark distribution of quasi-static ion fields which in the weak field limit will be well approximated by a Lorentzian with a width which is the sum of the electron impact width and self-broadening width somewhat enhanced by the smearing effect of the quasi-static ion field. Using this as a guide an alternative procedure is to calculate the profile in the absence of self-broadening using, for example, the profiles of Stehlé (1994). In the weak quasi-static ion field limit these profiles should be well approximated by a Lorentzian (for example Stehlé 1996) with the full impact width containing all line components (but dominated by the p-d component) somewhat enhanced by the smearing effect of the weak quasi-static ion field. As the profile is Lorentzian in the wings the absorption will be proportional to this enhanced impact width. The profile of the line in the absence of ions and electrons produced by self-broadening will also be Lorentzian with a depth in the wings proportional to the self-broadening impact width which again contains the effect of all components but dominated by the p-d component. Thus all three sources of broadening can be represented by a Lorentzian with a width which is simply the sum of the widths of the two profiles or equivalently in the line wings by simply adding the profiles. The effect of radiative broadening and broadening by helium collisions can be included in the same way.

Test calculations show this procedure for the convolution to be an excellent approximation for the cases considered here. For example in the solar synthetic profile, no difference can be seen between the profile computed in this way and that computed with a complete numerical convolution. The approximation gradually becomes worse in cool stars, and starts to break down in models of effective temperatures around 4000 K, as the lines are no longer strong enough for this approximation to be valid.

We use the spectral synthesis code of Piskunov (1992) for the radiative transfer, which assumes LTE. Radiative broadening and collisional broadening by helium are included in all calculations, though are found to be negligible in most conditions.

7.1. The impact of the self-broadening calculations on line profiles

The most interesting question, is how much difference the theory makes to predicted stellar line profiles when compared to the Ali & Griem (1966) theory, and thus the commonly used Kurucz (1993)/Peterson (1969) codes.

Fig. 6 shows computed line profiles for MARCS models (Asplund et al. 1997) for a range of effective temperatures at solar gravity and metallicity, using both our theory and the Ali & Griem (1966) resonance broadening theory. Table 4 shows the increase in the equivalent width brought about by our self-broadening theory. Although Fig. 3 indicates the effect of the new theory on the self-broadening is larger in H[FORMULA] than H[FORMULA] this is not seen in the synthetic stellar spectra in Fig. 6 due to the fact that the Stark broadening profile widths are increased by an even greater amount, as shown in Fig. 5.

[FIGURE] Fig. 6. Synthetic flux profiles for H[FORMULA] (top), H[FORMULA] (middle) and H[FORMULA] (bottom) for MARCS models of [FORMULA] = 5000, 6000 and 7000 K (top to bottom) for solar gravity and metallicity. The full lines use our line broadening theory and dashed lines use Ali & Griem's resonance broadening theory for the hydrogen broadening.


Table 4. Percentage increases in equivalent width using our self-broadening theory compared with Ali & Griem (1966) for the synthetic lower Balmer line profiles computed for MARCS models of various effective temperature, with solar gravity and metallicity.

The decline in the difference between the two theories with increasing temperature is due to the increase in the Stark broadening resulting from ionisation of hydrogen as temperature increases. For stars earlier than F type the self-broadening will become irrelevant as it will be completely overwhelmed by Stark broadening.

7.2. Predicted impact on effective temperature determinations

In Paper I we made preliminary estimates of the effect that the new calculations would have on the determination of effective temperatures. Here we present an extended analysis including H[FORMULA] calculations and covering higher effective temperatures. Also included are new estimates for [Fe/H][FORMULA]. The low metallicity calculations have also been redone on a finer model grid.

We computed a grid of MARCS models (Asplund et al. 1997) over a range of temperatures, with solar gravity, for metallicities of [Fe/H][FORMULA] and -2.0. We used the grid to estimate the difference in [FORMULA] determined from our theory and the Ali & Griem (1966) theory. For each model we computed synthetic profiles as described above using both our theory and the Ali & Griem (1966) theory. For each profile resulting from our theory we then found the best matching profile (in the line wings) using the Ali & Griem (1966) theory, and recorded the temperature difference between the models used to generate the two profiles. The results, plotted in Fig. 7, indicate that the new line broadening calculations lead to a significant lowering of the derived effective temperature.

[FIGURE] Fig. 7. The predicted difference in effective temperature determinations from our new calculations, and calculations using the resonance broadening theory of Ali & Griem (1966) for H[FORMULA], H[FORMULA] and H[FORMULA]. Plots are shown for solar metallicity (top), 1/10 solar metallicity (middle) and 1/100 solar metallicity (bottom), in both cases for solar surface gravity. The "reference temperature" is that which would be found using our broadening theory. The plot then predicts how much higher the effective temperature derived using the Ali & Griem (1966) theory is expected to be.

We see that the new results for H[FORMULA] are extremely similar to those for H[FORMULA]. This is explained by Fig. 5 where we see the relative contribution of Stark and self-broadening through the atmospheres are relatively similar in these two lines.

As pointed out in Paper I, the peak temperature "error" and the difference in location of the peak for H[FORMULA] from the other two lines is of interest. Synthetic profiles obtained using our theory are always stronger than those obtained using Ali & Griem (1966) theory. In Ali & Griem (1966) theory the H-atom broadening is resonance broadening only and is therefore temperature independent while in our theory the dispersive-inductive contribution leads to an increase with temperature. At low [FORMULA], H-atom broadening makes its greatest contribution and as [FORMULA] is raised the temperature "error" increases because of the growth in the H-atom broadening in our theory. Eventually Stark broadening begins to dominate accounting for the peak followed by a decline as Stark broadening becomes more and more dominant as [FORMULA] increases. As Stark broadening in H[FORMULA] and H[FORMULA] is greater than in H[FORMULA] the peak occurs at a lower [FORMULA]. The higher peak temperature "error" for metal poor stars reflects the higher temperature required to increase the ion/electron density sufficiently. This can be tested observationally. In agreement with this result Gardiner et al. (1999, Fig. 9), with mixing length parameter [FORMULA] and using the Ali & Griem (1966) theory, found that [FORMULA] obtained from H[FORMULA] is larger than for H[FORMULA] at [FORMULA] around 6000-7000K while the situation is reversed for stars with a lower [FORMULA] although admittedly there is only a small sample of stars in this domain. It is perhaps significant to note that Castelli et al. (1997) find, using Ali & Griem (1966) theory and the solar KOVER model, that [FORMULA] has to be raised by 100-150K (consistent with the peak of 110K for H[FORMULA] in Fig. 7) in order to fit the observed solar profiles.

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Online publication: December 5, 2000