3. Model atmospheres
The surface gravities of the programme stars were known to be low (Heber et al. 1986), placing their atmospheres close to the Eddington limit. As has been usual for our studies of extreme helium stars, a large number of model atmospheres had to be calculated in order to analyse their surface abundances. It quickly became apparent that the ionization equilibria predicted by these models were almost independent of effective temperature, and so considerable effort was spent on verifying the models.
A well-known feature of hydrogen-deficient model atmospheres at low surface gravity is their poor convergence properties. As implemented, the model atmosphere code STERNE solved the radiative transfer equations using the scheme originally proposed by Avrett & Loeser (1963), and calculated the temperature correction following the Lucy-Unsöld procedure (Lucy 1964), accelerated according to the method proposed by Ng (1974). Due to computational time considerations, convergence has customarily been accepted when the mean square relative temperature correction has fallen below some value, normally . This is usually achieved after 40 iterations of the code. It was believed that the Avrett-Loeser solution of the transfer equation may be prone to instabilities, particularly at very small optical depths, so it has been replaced by a Feautrier scheme (Feautrier 1964). At present this assumes coherent scattering, although partial and complete redistribution can also be treated. It was also found that the Ng acclerator did not have a major effect on the convergence, indeed it was prone to delay convergence and was switched off. Other pertinent features of STERNE were described by Jeffery & Heber (1992). Line-blanketing is treated through fixed composition opacity distribution functions calculated for a hydrogen-deficient mixture by Möller (1990).
With these changes, convergence was found to be excellent at optical depths where the temperature corrections converged to zero within approximately 30 iterations. However at small optical depths, the corrections remain significant and always negative. After many iterations, their amplitude decreases, but never reaches zero or changes sign. The effect is that the temperature of the outermost layers of the model atmosphere decreases asymptotically. Even after 300 iterations, K, for .
Possible causes for the poor convergence were investigated. Although changing a boundary condition changes the final model, it does not alter the convergence behaviour. The omission of line blanketing allows the models to converge more rapidly and successfully with . Our conclusion is that the use of the Lucy-Unsöld temperature-correction procedure is at fault. Temperature correction procedures leave the temperature structure of the outermost layers essentially undetermined particularly when the radiation field is only weakly coupled to the local thermal pool, as is the case here where scattering dominates (see Mihalas 1978, p175). A resolution awaits the implementation of a more sophisticated correction procedure.
Since most of the model atmosphere, including the region where most of the spectral lines are formed, has converged successfully, it remained likely that these models could be used for our analyses. Synthetic spectra including lines formed at a large range of optical depths were calculated, using models converged after 50, 100 and 300 iterations. The absence of convergence at small optical depths had no effect on these spectra, apart from the cores of very strong lines, including He i, , and C II 4267 Å. All of these lines were already known to show discrepancies between theory and observation which have been previously attributed to departures from local thermal equilibrium (LTE) (Heber 1983, Jeffery & Heber 1992). The current result suggests that at least part of the resolution will be achieved by constructing model atmospheres for low-gravity helium stars which are fully self-consistent at small optical depths.
The present study relies on the dual approximations of plane-parallel geometry and local thermodynamic equilibrium; departures from both become increasingly important in the atmospheres of low gravity stars. The atmospheres of the current sample are only "slightly extended" according to the definition of Schmid-Burgk & Scholz (1975). The latter found that, for the low-gravity halo star Barnard 29, the differences between plane parallel and spherical model atmospheres amounted to a few per cent in the upper atmosphere. However they doubted that abundance discrepancies relative to Peg could be explained by extended atmosphere effects. A similar argument applies here. We do not observe any other phenomena associated with extended atmospheres in the current sample. For example, there is no evidence of emission lines, such as those observed in the low-gravity helium stars DY Cen (Jeffery & Heber 1993) and BD (Jeffery & Heber 1992) and which have been attributed to a circumstellar shell.
The approximation of local thermodynamic equilibrium is frequently violated in low gravity stars, with the combined effects of modifying both the global structure of the atmosphere and the profiles of individual absorption lines. The correct approach is to calculate both model atmospheres and synthetic spectra without this approximation. To date, the only successful attempt to compute NLTE spectra for non-expanding low-gravity hydrogen-deficient model atmospheres with K found extremely slow convergence in a model containing only H, He and C (Rauch 1996). At lower temperatures, the neglect of line blanketing caused by the omission of other species will have consequences for the global structure of the atmosphere which are far more profound than the LTE approximation (Dudley & Jeffery 1993, Anderson & Grigsby 1991).
Until the problem of non-LTE model atmospheres has been solved, line-blanketed plane-parallel LTE models remain the most appropriate (and only) choice for the analysis of the present sample. This is not a poor choice, since it allows us to make a direct comparison with other extreme helium stars (and also RCrB stars, Asplund 1997) analyzed using similar methods; many of these have L/M ratios similar to or higher than the present sample. The assumption of LTE does not necessarily lead to errors in the derived abundances. Dufton (1993) has noted, for example, that non-LTE analyses of the B star Sco (Becker & Butler 1988, 1989, Becker 1991) arrived at the same elemental abundances as those obtained nearly half a century earlier by less sophisticated means (e.g. Unsöld 1942).
A sequence of low-g model atmospheres was constructed for between 14 000 and 19 000 K, with , supplemented by a grid of higher gravity models at the same temperatures. At the lowest effective temperatures, these models constitute the lowest gravity models which could be calculated. On the basis of our preliminary analysis, the chemical composition was given by hydrogen and carbon abundances, by number, solar abundances for other metals, and helium making up the residue. In the event, a second grid with was also constructed.
© European Southern Observatory (ESO) 1998
Online publication: November 9, 1998