5 Discussion and conclusions

We have analysed velocity curves of three white dwarfs below the photometric red edge of the instability strip to check whether the observed red edge corresponds to the physical one. For ZZ Psc, a white dwarf close to the red edge but still within the instability strip, van Kerkwijk et al. (2000) found a maximum velocity amplitude of $5.4 \pm 0.8~\rm km~s^{-1}$. We had expected larger modulations but for two objects we can exclude the presence of signals as strong as those in ZZ Psc with high confidence; for one of these two, G 67-23, we can exclude the presence of any signal greater than 3  $\rm km~s^{-1}$.

The above indicates that pulsations actually cease below the observed red edge. How secure is this conclusion? We see three weaknesses. The first is that in principle, destructive beating between different components in a multiplet might conceal a real signal in our short data sets. This, however, would be unlikely if multiple real signals were present. Furthermore, two data sets for the same object (G 126-18) taken during different observing runs are unlikely to be adversely affected. Two of the modulations found in G 126-18 (viz. 220 s and 269 s) seemed to be consistent with being at the same frequency and of the same amplitude - within the errors - for the two different observing runs. We carried out further Monte Carlo simulations to determine the likelihood of finding relatively strong peaks at the same frequency in two separate data sets. On the basis of these simulations, we conclude that these peak coincidences are almost certainly due to chance: even for the best case, the peaks at 265 s and 269 s, there is a 7% probability of a chance coincidence.

The second weakness is that our sensitivity turned out to be more marginal than we had hoped. We included one known ZZ Ceti, HL Tau 76, in the sample. Treating it as a red-edge object afforded a check on our methods. However, even for this bona-fide pulsator, we found that the velocity curve on its own was not sufficient to demonstrate velocity variations. A hint of a real signal was present, but this could only be confirmed by imposing external information from the light curve, information we lacked for the red-edge objects. In the process, HL Tau 76 was added to the select group of ZZ Cetis for which velocity variations have been detected.

The above might suggest that our experiment was simply not sensitive enough. One has to keep in mind however, that for the different objects rather different sensitivities were reached. For one object, G 67-23, we can exclude the presence of velocity amplitudes as small as 3  $\rm km~s^{-1}$ i.e. even smaller than the 4  $\rm km~s^{-1}$ peak corresponding to strongest mode in HL Tau 76. Furthermore, on theoretical grounds alone, a velocity amplitude somewhat smaller than that of ZZ Psc is expected for HL Tau 76 given that it has a slightly higher temperature and its strongest mode, a slightly shorter period. This expectation is supported by the fact that the velocity amplitude of the dominant mode is in turn larger than that found for HS 0507+0434B, which is somewhat hotter still and has an even shorter dominant periodicity. On the same grounds, one would expect, as mentioned earlier, the red-edge objects to have larger velocity amplitudes than those observed in ZZ Psc, which, if present, would have been seen in two of the three objects.

The third weakness is perhaps the most severe: G 1-7 and G 67-23 have relatively high inferred masses, which implies that these objects would have ceased to be pulsationally active at a higher effective temperature than that expected for a typical  $0.6~M_\odot$ white dwarf, and are therefore not that close anymore to the red edge. At the same time, G 126-18, which has a normal mass, is the coolest of our three observed targets. Note though that the derived effective temperatures and surface gravities depend on the treatment of convection.

In summary, while not ironclad, our results indicate that pulsations have actually ceased below the observed red edge rather than having become photometrically undectable, and that the theoretical expectation of an extended instability strip, beyond the observed red edge is flawed. In order to settle the issue observationally, more sensitive measurements on objects closer to the red edge would be worthwhile; theoretically, detailed hydrodynamic modelling might go some way towards understanding the interplay of the various physical processes in defining the red edge.

Acknowledgements

We thank the referee, Detlev Koester, for his comments, and are grateful to Y. Wu for answering our many questions. R.K. would also like to thank J. S. Vink for encouragement and H-G. Ludwig for useful discussions. We also acknowledge support for a fellowship of the Royal Netherlands Academy of Arts and Sciences (MHvK) and partial support from the Knut och Alice Wallenbergs Stiftelse (RK). This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.