6 Conclusion

In this paper we provide new non-linear radial pulsation models computed within the iron-group instability strip where low-mass extreme-helium stars are located. We attempt to reproduce the observed light and velocity curves of V652 Her and BX Cir and, hence, study their possible masses, luminosities and radii.

All the models have temperatures and masses close to those measured for V652 Her and BX Cir. Their luminosities are chosen to ensure the pulsation period matches that observed. The chemical composition was initially chosen to be similar to the observed surface composition, but this produces stable models. Metallicity was gradually increased to find unstable (overstable) models, which provide a lower limit to the iron abundance of around $n_{\rm Fe}\sim 4.17 \times 10^{-5}$. This indicates that the recently published iron abundance BX Cir may be too small, although a more detailed study of the dependence of the instability boundaries on the iron abundance using a linear analysis is recommended. It is evident that the pulsation properties (amplitude, light curve shape) are very sensitive to the chemical composition.

We have found that several models can reproduce with good accuracy the observed velocity and luminosity curves for these two stars.

The parameters of the best fit model for V652 Her are close to those previously provided by Fadeyev & Lynas-Gray (1996) but with lower helium, carbon and iron abundances (model 20) more similar to those reported in a recent spectroscopic analysis (Jeffery et al. 2001). An attempt to reproduce the velocity and light curves with a lower mass ( $0.59{-}0.67~\mbox{$M_{\odot}$ }$), closer to that provided by recent observations, within the range of temperatures 21 900-23 400 $~\mbox{K}$ was not successful. For $\mbox{~\em$T_{\rm eff}$ }\sim 23~400 ~~\mbox{K}$, a mass of at least  $0.70~\mbox{$M_{\odot}$ }$ is necessary to reproduce the velocity and luminosity curves. This value would be slightly higher for lower temperatures. The white dwarf merger model (Saio & Jeffery 2000) introduced to account for the origin of V652 Her requires a final mass between 0.6 and  $0.7~\mbox{$M_{\odot}$ }$, which is in satisfactory agreement with the results of our pulsation analysis.

As suggested by Jeffery et al. (2001), it remains a problem to reconcile fully the optical line spectrum, the overall flux distribution and, as here, the pulsation properties of V652 Her.

The pulsation period and velocity and luminosity amplitudes of BX Cir could be satisfactorily reproduced by a model (25) with a mass and temperature within the observational errors. A minimum mass of $0.38~\mbox{$M_{\odot}$ }$ for 23 900 $~\mbox{K}$ and a maximum mass of $0.50~\mbox{$M_{\odot}$ }$ for 22 300 $~\mbox{K}$ were found. Between these limits, an increase in stellar mass requires a corresponding decrease in effective temperature in order to reproduce the observed velocity and luminosity amplitudes.

It is not possible to match the observed luminosity for BX Cir, which seems too large by a factor of $\sim$2. The reason is far from clear because both temperature and radius are well defined by observation (Woolf & Jeffery 2000). Pulsation theory imposes a very tight relation between pulsation period and mean density - or mass-to-radius ratio. Small departures from this ratio do not significantly alter the shapes of light and velocity curves. Thus we believe that fixing the period is the most appropriate way to approach the problem. There are several other possibilities. One is that the radius of Woolf & Jeffery (2000) is too big, but their data do not seem to allow this. Another is that systematic errors in the model atmospheres overestimate the effective temperature and a smaller radius results. A third is that similar errors led to an underestimate of the surface gravity of Drilling et al. (1996) so that a higher mass would result. Finally, systematic errors in the pulsation models could yield incorrect amplitudes. Such errors would have to be selective in order to modify the BX Cir result more than the V652 Her result, and could occur if the composition adopted for either star were incorrect. Further observational work might yield evidence for a solution with a higher mass. This would be easier to reconcile with white dwarf merger models (Saio & Jeffery 2000), which require masses in excess of ${\sim} 0.5~M_{\odot}$. However the conclusion of this paper is that the current pulsation models favour a low-mass solution for BX Cir. Otherwise the theoretical velocity amplitude is too large compared with that observed.

Acknowledgements

This research is supported by a grant to the Armagh Observatory from the Northern Ireland Department of Culture, Arts and Leisure and by the British Council through Collaborative Research Grant TOK/880/41/4. We are grateful to Prof. H. Saio (Tohoku University, Sendai) for valuable advice, to Prof. Y. A. Fadeyev (Institute of Astronomy of the RAS, Moscow) for his comments, to Dr. A. Bridger (UK Astronomy Technology Centre, Edinburgh) for his pulsation code and to the referee Dr. T. Metcalfe for his review of the paper.