Annu. Rev. Astron. Astrophys. 1996. 34: 461-510 Copyright © 1996 by Annual Reviews. All rights reserved

2.5. Tests of Stellar Models

The interior structures of low-mass, main-sequence stars are believed to be much simpler, and therefore (presumably) better understood, than those of their higher-mass (M 1.15 M) counterparts because, in part, they do not contain convective cores and so are unaffected by the uncertainties (e.g. the extent of overshooting) associated with them. Perhaps the main evidence for possible inadequacies in the theory has been the longstanding failure of canonical models to reproduce the observed flux of neutrinos from the Sun, but solar oscillation studies have considerably diminished that concern. As Dziembowski et al (1994, 1995) have concluded, the inferred structure of the Sun from helioseismology is now so close to that predicted by the standard model, throughout its interior, that there is little room left for an astrophysical solution to the solar neutrino problem. Certainly, there are many examples in the scientific literature demonstrating how well current stellar evolutionary theory can match superb observational data. One of the nicest of these is the study of the Hyades by Swenson et al (1994). They obtained a self-consistent fit to the CMD, to the mass-luminosity relation defined by the cluster binaries, and to the Li abundances in the G stars, using opacities for the observed [Fe/H] value and without applying any ad hoc adjustments of any kind. Indeed, the best-observed binaries (e.g. AI Phe - see Andersen et al 1988) and the mass-luminosity relations derived from them appear to agree rather well with the predictions of standard models (cf Andersen 1991).

Considering the more evolved, post-turnoff phases, the main challenge to the theory would appear to be the observed chemical abundance variations among bright GC giants (already summarized in Section 2.2.2) and some anomalies in the luminosity function (LF) data for a few clusters (see below). These difficulties will probably be resolved once rotation is accurately treated (which, admittedly, is not easily done). Otherwise, as extensively reviewed by Renzini & Fusi Pecci (1988), there appears to be little basis for believing that the sorts of models that have been computed for the past 25 years or so are seriously in error. Although many discrepancies between theory and observation can be identified, it is much more likely that they are due to deficiencies in, for instance, the opacity or convection theory, than to a problem with the basic stellar structure equations themselves. But it may be the case that the LF anomalies have a different origin.

Relatively little work has been done on the luminosity functions of GCs, in spite of the fact that they provide a superior test of stellar models compared with the fitting of CMDs and despite some tantalizing results from early studies. For instance, Simoda & Kimura (1968) suggested that the LFs of M3 and M13 differed from one another - which might be an important clue (yet to be followed up) as to the cause of the differences in the HB morphologies of these common-[Fe/H] clusters. Also, making use of the fact that LFs provide one of the few ways to infer the helium abundances in GCs (see the recent study by Ratcliff 1987), Hartwick (1970) derived Y ~ 0.35 from such an analysis of M92. However, not until Bergbusch's (1990) study of the latter cluster was a possible inconsistency between an observed LF and theoretical predictions identified. Depending on how the synthetic and observed LFs were matched, the M92 data showed either a broad dip between 19 < V < 20 or a bump near V = 18.6 that was not present in the models. Stetson's (1991) combined LF for M68, M92, and NGC 6397, based on new CCD observations of these clusters, confirmed the existence of these features. In addition, it revealed that, when theoretical LFs for the appropriate Y and Z were normalized to the turnoff data, the observed RGB had a significant excess of stars compared with the number predicted. These anomalies are not readily explained in terms of variations in any of the usual parameters, but they have turned out to be the anticipated signature of WIMPs (as already recounted in Section 2.3).

The lower panel in Figure 5 illustrates Bolte's (1994) LF for M30, and it too poses the same problems for the theory as the data for the other [Fe/H] ~ -2.1 GCs. But note that the M5 LF (in the upper panel) shows no such anomalies; indeed, it conforms remarkably well to the theoretical predictions. Similarly, Bergbusch & VandenBerg (1992) have not found any obvious difficulties in fitting the available luminosity function data for 47 Tuc (though their matching of the brighter to the fainter data is somewhat uncertain). And Stetson & VandenBerg's (1996) Canada-France-Hawaii Telescope photometry for a sample of ~ 10⁵ stars in M13 shows no evidence of a subgiant bump either. Curiously, their very preliminary analysis suggests that the RGB in M13 may be underpopulated relative to the turnoff; i.e. the opposite to what is seen in the more metal-poor clusters.

Figure 5. Comparisons of the observed luminosity functions for M5 (upper panel) and M30 (lower panel) with Bergbusch & VandenBerg (1992) isochrones, on the assumption of the indicated ages, [Fe/H] values, and distance moduli. The M5 data are from Sandquist et al (1996); the M30 results are as reported by Bolte (1994).

There is clearly much to be learned from such LF studies, but from these first results, one has the impression that the anomalous subgiant bump is characteristic of only the extremely metal-poor clusters - and hence it can hardly be due to WIMPs, which should not show a preference for a particular [Fe/H] value. Is that feature somehow connected with the deep-mixing phenomenon? We do not know. The differences in the relative RGB-to-turnoff populations might be due to differences in helium abundance. Alternatively, it may be an indication of differences in core rotation. Using the simplest possible treatment of rotation (cf Mengel & Gross 1976), Larson, VandenBerg & De Propris (1995) have found that the number of giants relative to the number of turnoff stars is larger if the stars have significant internal rotation. Perhaps the main point to be made here is that, unless and until the LF data are satisfactorily explained, one should be wary of trusting the application of standard, nonrotating, unmixed-envelope models to those clusters (apparently the most metal-poor ones) whose luminosity functions cannot be reproduced by such models.

We conclude that, although there remain unexplained observations of evolved stars in globular clusters, the stellar models for GC stars at the main-sequence turnoff and probably to a few magnitudes down the presently observed main sequence are reliable. In particular, the agreement between predicted and observed mass-luminosity relations suggests that the theory is basically correct and essentially complete. The main outstanding issue for models (in the context of predicting the main-sequence lifetimes of low-mass stars) is the extent to which helium diffusion may reduce cluster age estimates. Our best estimate for the maximum reduction in ages due to this effect is 1 Gyr. Although it could be postulated that some currently unconsidered physics will in the future reduce measured cluster ages, there currently does not appear to be any viable candidate physical processes, nor is there any clear motivation for seeking them other than to try to reduce GC ages.