6 Inhomogeneities and the plane parallel model

For perhaps a century we have known that the spectrum of the solar photosphere varies from one point on the disk to another. The first high-resolution spectra obtained from the McMath-Hulbert Observatory showed striking spatial variations that came to be known as "wiggley lines''. The solar line profiles vary markedly, both in time and space, and while we have understood the general the nature and cause of these variations for decades, recent numerical calculations by Nordlund, Stein, and their collaborators have provided a detailed description (cf. Nordlund & Stein 2001).

In spite of its origin in a turbulent roil, the average line spectrum of the sun is remarkably constant. This is particularly surprising in the case of the Balmer lines, where the large Boltzmann factor ( $\theta\chi_{\rm lower} \approx 10$) suggests huge local non-linear effects. Naively, one would not expect them to average out, and the extent to which they do average out remains to be fixed.

In the 1950's, de Jager (1952) attempted to fix the temperature fluctuations in the solar atmosphere by making use of the putative nonlinearities of the Balmer lines. His conclusions, of temperature differences of a thousand degrees from hot to cool columns agrees remarkably with modern numerical models. Surely, he was guided by physical insight into what the answer needed to be. The Stark-broadening theory of that time was rudimentary, and the influence of collisions with neutral hydrogen were entirely neglected.

We have found that reasonable matches to the four lower Balmer lines can be achieved using modern Stark profiles provided recent parameters for broadening by neutral hydrogen by BPO and the HM model are used. In fact, the fits illustrated in Figs. 7 and  8, were all based on the empirical plane-parallel Holweger-Müller model, and include no attempts to improve the fits by plausible adjustments of the line-broadening parameters. Other studies have explored the sensitivity of the Balmer lines to different theoretical model atmospheres and to variations in the convective mixing length to the pressure scale height (l/H).

We remark here on the surprising linearity of the Balmer profiles with the temperature of plane-parallel models. This may be illustrated in several ways. In Fig. 1 we can see that for $T_{\rm eff}$ about 4000 K to 6250 K the wing strengths plot nearly linearly with temperature for the three higher gravities. This near linearity holds for most points on the line profiles, apart from the most central portions. If one takes an equally weighted average of H$\alpha$ fluxes from Kurucz models with $T_{\rm eff}$ = 5500 K and 6500 K, the resulting mean differs imperceptibly from that for a $T_{\rm eff}$ = 6000 K model. Means for $T_{\rm eff}$ = 5000 K and 7000 K models differ only by 2% from the $T_{\rm eff}$ = 6000 K model beyond 3 Å from the line center. Even for the mean of $T_{\rm eff}$ = 4500 K and 7500 K models the difference is of the order of 5% (see Fig. 9).

Figure 9: Percentage differences in H$\alpha$ profiles for 6000 K model and average profiles for three pairs of models as indicated (H$_{\alpha }$ profiles from Kurucz 1993a).

The same effect may be seen in the left panel of Fig. 3 of Fuhrmann et al. (1993). They show a series of Balmer profiles from H$\alpha$ through H$\delta$for $\log~g$ = 4, with effective temperatures running from 5000 K to 6700 K, in steps of 100 K. It can be seen that the different profiles are, for the most part, quite evenly spaced.

The simple means of Fig. 9 are certainly not equivalent to the detailed calculation performed, for example, by Asplund et al. (1999), based on the 3-dimensional numerical models of the solar convection zone. Nevertheless, they demonstrate that the non-linearities that one might expect from the very large Boltzmann factors of the n = 2 level are not realized in the resultant Balmer profiles of cool stars. This, in turn, supports endeavors to use theoretical profiles from simplified stellar models to help fix fundamental stellar parameters.