5. Comparison of atmosphere and evolutionary models of WN b stars
Our aims in this section are: to compare the predictions of the evolutionary models with the observed stars (interpreted by atmosphere models); to highlight the existing problems with such a comparison; and to indicate directions in which improvement may be most readily forthcoming.
Fig. 11 (cf. Maeder & Meynet 1994) shows the overall prediction of evolutionary models: WN stars, once free of their hydrogen envelope, lie along a band in the temperature-luminosity diagram. This band has been previously designated WNE by model builders. However, since some WNE (meaning WN 2-6 stars) still have hydrogen, this band (if our hypothesis is correct) should be called the WN b band or phase. When a star (of sufficient mass) looses its last hydrogen shell, it settles onto the WN b band. Then, as a result of continued mass loss, the luminosity decreases and the temperature increases, so that the star moves along the band. Its position on the WN b band depends only on its mass (see also Sect. 6). When the products of He-burning appear at the surface, the star leaves the WN b band and settles onto the WC band, which is slightly cooler.
Placing observed WR stars into the theoretical HR diagram (log vs. log L) presents several problems of which the most acute are that the definition of is ambiguous and that the B.C. determined from the atmospheric models appear to be underestimated.
We compare the results of atmospheric models for WN b stars (HKW95) to evolutionary Co-star models by Schaerer (1995; 1996) and Schaerer et al. (1996ab). Parameters available for comparison are, in principle: luminosity L; effective temperature T; and radius R. However, these three are redundant since they are uniquely related via the black body relationship. In WR stars, the definition and determination of T and R are complicated by the presence of the expanding atmospheres. This complication is handled differently by atmospheric and evolutionary models.
Atmosphere models fit the strength (and profiles in a "tailored analysis") of the helium lines to determine at = 10 or 20. The luminosity L comes from the observed or assumed plus the B.C. (from the models) and the radius R follows from the black body relationship.
Evolutionary models (mostly concerned with the interiors) add a parameterised atmosphere to the hydrostatic interior model. (The atmosphere model is similar to that used by the atmosphere calculations but assumes a dependence of the wind parameters, and , on parameters defined by the interior model.) The luminosity L comes from the interior model, R comes from the assumed velocity law and the superposed model atmosphere and T follows from the black body relationship.
It has been known for some time (e.g. Howarth & Schmutz 1992) that, for stars of known mass and , the luminosities derived using B.C.'s from atmosphere models fall short of those required by the interior models. Since there appears to be no uncertainty in the mass-luminosity relationship for WR stars, the conclusion must be that the B.C. are underestimated by the pure helium atmosphere models. Various explanations have been suggested. Heger & Langer (1997) suggest that some of the discrepancy may be due to the mechanical energy that leaves the star in the wind. Crowther et al. (1995) found that He-N models for two weak line stars yield higher values of than do H-He models; however, Hamann et al. (1994) find no significant difference. Schmutz (1996) shows that inclusion of photon losses and their effect on the ionisation balance lead to higher values of .
Bearing in mind that the "evolutionary" L, and therefore , may be overestimated (if mechanical energy is significant) and the "atmosphere" L, and therefore , are probably underestimated, we now compare the temperature-radius diagrams that result from HKW95's pure helium atmosphere models applied to individual stars with predictions of the Co-star models by Schaerer (loc. cit.). We choose the - diagram rather than the L- diagram since it appears to better illustrate the problem we wish to highlight.
Fig. 12 shows the log vs. log diagram for atmospheric models of WN b stars from HKW95; has been modified to allow for the improved - relationship. The data shows a scatter of about in log , which is a reasonable "observational uncertainty". The lines are from Schaerer's models for various optical depths and opacities: and 1.0 with OPAL opacities (Iglesias et al. 1992, Iglesias & Rogers 1993) and with opacities (as described by Schaller et al., 1992) based on electron scattering and a force multiplier to allow for the contribution of spectral lines (cf. Kudritzki et al. 1989, Schmutz 1991).
The atmosphere models lie neatly between the two lines (solid lines) of the evolutionary models. The underestimate of and by the atmosphere models (see above) would bring the points into relatively good agreement with the (solid) line representing the OPAL opacity models for . However, the apparent agreement is illusory since of HKW95 were derived at not at . HKW's models at give values of near 4.5 for all stars. Conversely, Schaerer's models at or 20 give values of log T near 5.0, or 5.1, respectively, for all stars.
Thus, the primary result of our comparison of atmospheric to interior models is a large difference between the temperatures and/or the optical depths derived by HKW and by Schaerer (loc. cit.).
The disagreement appears to hinge on the connection between R and which is, in principle, defined by the assumed velocity law. Different choices of the exponent beta in the wind velocity law (HKW use ; Schaerer uses beta = 2.1) will (according to Schaerer) give a difference of 0.23 dex in which is in the right direction but insufficient to account for the disagreement between the two sets of models. However, since the velocity law probably does not correspond to reality, the result may be acute sensitivity to boundary conditions such as the core radius.
We conclude that there is a qualitative correspondence between the evolutionary models and the observed stars interpreted by atmosphere models; in particular that the WN b stars appear to follow a simple relationship of their basic stellar properties indicating a one-parameter family. However, there appear to be systematic errors that make in Schaerer's treatment occur at approximately the same radius as , in HKW's treatment of the atmospheres.
© European Southern Observatory (ESO) 1998
Online publication: June 2, 1998