8 Conclusions

We computed a model of the population of close binary white dwarfs and found good agreement between our model and the observed double white dwarf sample. A better agreement with observations compared to earlier studies is found due to two modifications.

The first is a different treatment of unstable mass transfer from a giant to a main sequence star of comparable mass. The second is a more detailed modelling of the cooling of low mass white dwarfs which became possible because detailed evolutionary models for such white dwarfs became available. Our main conclusions can be summarised as follows.

1.

Comparing the mass distribution of the white dwarfs in close pairs with the observations, we find a lack of observed white dwarfs with masses below 0.3 $M_\odot$. This discrepancy can be removed with the assumption that low-mass white dwarfs cool faster than computed by Driebe et al. (1998). The same assumption removes discrepancies between observed and derived ages of low-mass white dwarfs that accompany recycled pulsars, as shown by van Kerkwijk et al. (2000). Faster cooling is expected if the hydrogen envelopes around low-mass white dwarfs are partially expelled by thermal flashes or a stellar wind;

2.

Our models predict that the distribution of mass ratios of double white dwarfs, when corrected for observational selection effects as described by Moran et al. (2000), peaks at a mass ratio of unity, consistent with observations. The distributions predicted in the models by Iben et al. (1997) and Han (1998) peak at mass ratios of about 0.7 and above 1.5 and agree worse with the observations even after applying selection effects;

3.

Our models predict a distribution of orbital periods and masses of close double white dwarfs in satisfactory agreement with the observed distribution;

4.

Amongst the observed white dwarfs only a small fraction are members of a close pair. To bring our models into agreement with this, we have to assume an initial binary fraction of 50% (i.e. as many single stars as binaries);

5.

In our models the ratio of the local number density of white dwarfs and the planetary nebula formation rate is a sensitive function of the star formation history of the Galaxy. Our predicted numbers are consistent with the observations;

6.

Using detailed cooling models we predict that an observed sample of white dwarfs near the Sun, limited at the magnitude V=15, contains 855 white dwarfs of which 220 are close pairs. Of these pairs only 10 are double CO white dwarfs and only one is expected to merge having a combined mass above the Chandrasekhar mass. The predicted merger rate in the Galaxy of double white dwarfs with a mass that exceeds the Chandrasekhar mass is consistent with the inferred SN Ia rate. ITY97 estimated, depending on $\alpha_{\rm ce}$, to find one such pair in a sample of $\sim$200 to $\sim$600 white dwarfs. Reversing this argument, when the statistics become more reliable, the observed number of systems with different types of white dwarfs could provide constraints on the cooling models for these white dwarfs.

Acknowledgements

We thank the referee A. Gould for valuable comments. LRY and SPZ acknowledge the warm hospitality of the Astronomical Institute "Anton Pannekoek''. This work was supported by NWO Spinoza grant 08-0 to E. P. J. van den Heuvel, the Russian Federal Program "Astronomy'' and RFBR grant 99-02-16037 and by NASA through Hubble Fellowship grant HF-01112.01-98A awarded (to SPZ) by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA under contract NAS 5-26555.