7 Discussion: Comparison with previous studies

We now compare our work with the results of previous studies; in particular the most recent studies of Iben et al. (1997, ITY97) and Han (1998, HAN98).

  
7.1 Birth rates

In Table 2 we show the birth rates of close double white dwarfs for the different models. We also include numbers from HAN98 (model 1) and a set of numbers computed with the same code as used in ITY97, but for an age of the galactic disk of 10 Gyr, as in our models. The numbers of HAN98 are for an age of the disk of 15 Gyr. Our model D is the closest to the models of ITY97 and HAN98, assuming a constant SFR and 100% binaries. To estimate the influence of the binary evolution models only in comparing the different models we correct for their different normalisations.

In the recomputed ITY97 model the formation rate of interacting binaries in which the primary evolves within the age of the Galaxy is 0.35 yr^-1. In our model D this number is 0.25 yr^-1. In the following we therefore multiply the formation rates of ITY97 as given in Table 2 with 0.71.

In the model of HAN98 one binary with a primary mass above 0.8 $M_\odot$ is formed in the Galaxy annually with $\log a_{\rm i} < 6.76$, i.e. 0.9 binary with $\log a_{\rm i} < 6$, which is our limit to $a_{\rm i}$. Correcting for the different assumed age of the Galaxy we estimate this number to be 0.81; in our model this number is 0.73. We thus multiply the the formation rates of HAN98 as given in Table 2 with 0.9.

Applying these corrections to the normalisation, we find that some interesting differences remain. The birth rate of double white dwarfs is 0.029, 0.053 and 0.062 per year for HAN98, model D and ITY98 respectively. At the same time the ratio of the merger rate to the birth rate decreases: 0.97, 0.53 and 0.28 for these models. This can probably be attributed to the different treatment of the common envelope. HAN98 uses a common envelope spiral-in efficiency of 1 in Webbinks (1984) formalism, while we use 4 (for $\lambda = 0.5$, see De Kool et al. 1987). ITY97 use the formalism proposed by Tutukov & Yungelson (1979) with an efficiency of 1. This is comparable to an efficiency of 4-8 in the Webbink formalism. This means that in the model of HAN98, more systems merge in a common envelope, which yields a low formation rate of double white dwarfs. The ones that form (in general) have short periods for the same reason, so the ratio of merger to birth rate is high. In the ITY model the efficiency is higher, so more systems will survive both common envelopes and have generally wider orbits, leading to a much lower ratio of merger to birth rate. Our model D is somewhat in between, but also has the different treatment of the first mass transfer phase (Sect. 2), in which a strong spiral-in is avoided.

The difference between the models in the SN Ia rate ( $\nu_{\rm SN~Ia}$) is related both to the total merger rate and to the masses of the white dwarfs. The former varies within a factor $\sim$1.5: 0.017, 0.028, and 0.028 ${\rm yr}^{-1}$ for ITY97, model D, and HAN98, while $\nu_{\rm SN~Ia}$ is higher by a factor 2-3 in model D compared to the other models. This is caused by the initial-final mass relation in our models, which is derived from stellar models with core overshooting, producing higher final masses.

The difference in the birthrate of interacting white dwarfs ( $\nu_{\rm AM CVn}$) is mainly a consequence of our treatment of the first mass transfer, which gives for model D a mass ratio distribution which is peaked to 1 (Sect. 6.4), while in ITY97 and HAN98 the mass ratio is in general different from 1 (Sect. 7.2), favouring stable mass transfer and the formation of AM CVn systems. An additional factor, which reduces the number of AM CVn systems is the assumption in model D and ITY97 that the mass transfer rate is limited by the Eddington rate. The formation and evolution of AM CVn stars is discussed in more detail in Tutukov & Yungelson (1996) and Nelemans et al. (2001).

  
7.2 Periods, masses and mass ratios

Comparing our Fig. 2 with the corresponding figure in Saffer et al. (1998), we find the same trend of higher white dwarf masses at longer periods. However, in our model the masses are higher than in the model of Saffer et al. (1998) at the same period. This is a consequence of the absence of a strong spiral-in in the first mass transfer phase in our model, contrary to the conventional common envelope model, as discussed in 2.2.

In our model the mass ratio distribution is peaked at $q \approx 1$. This is different from the models of ITY97 and Saffer et al. (1998) which predict a strong concentration to $q \sim 0.5{-}0.7$ and from HAN98 who finds typical values of $q \sim 0.5$, with a tail to $q \sim 2$. The difference between these two latter groups of models may be understood as a consequence of enhanced wind in Han's model (see also Tout & Eggleton 1988), which allows wider separations before the second common envelope. The mass ratio distribution of our model, peaked at $q \simeq 1$, appears to be more consistent with the observed mass ratio distribution.

  
7.3 Cooling

To explain the lack of observed white dwarfs with masses below 0.3 $M_\odot$ we had to assume that these white dwarfs cool faster than predicted by the models of DSBH98.

The same assumption was required by Kerkwijk et al. (2000), to bring the cooling age of the white dwarf that accompanies PSR B1855+09 into agreement with the pulsar spin-down age; and to obtain cooling ages shorter than the age of the Galaxy for the white dwarfs accompanying PSR J0034-0534 and PSR J1713+0747.

The absence of the lowest mass white dwarfs could also be explained by the fact that a common envelope involving a giant with a low mass helium core ( $M_{\rm c} < 0.2$- $0.25\,M_\odot$) always leads to a complete merger, according to Sandquist et al. (2000). However it can not explain the absence of the systems with $0.25 < M < 0.3\,M_\odot$, which would form the majority of the observed systems using the full DSBH98 cooling (model A1; see Fig. 2).