6 Results

 

 

Table 2: Birth and event rates and numbers for the different models. All birth and event rates ($\nu$) are in units of yr^-1 in the Galaxy. All numbers (#) are total numbers in the Galaxy. Close double white dwarfs are represented with (wd, wd). See Sect. 6.1 for a discussion of these rates. For comparison (in Sect. 7.1) we also include numbers computed by the code from ITY97 but using an age of the galactic disk of 10 Gyr instead of the 15 Gyr used by ITY97; and numbers of model 1 of HAN98

Model	SFH	% bin	$\nu_{\rm (wd, wd)}$	$\nu_{\rm merge}$	SN Ia	$\nu_{\rm AM CVn}$	#(wd, wd)
			(10-2)	(10-2)	(10-3)	(10-3)	(108)
A	Exp	50	4.8	2.2	3.2	4.6	2.5
B	Exp	100	8.1	3.6	5.4	7.8	4.1
C	Cnst	50	3.2	1.6	3.4	3.1	1.2
D	Cnst	100	5.3	2.8	5.8	5.2	1.9
ITY971	Cnst	100	8.7	2.4	2.7	12.0	3.5
HAN981	Cnst	100	3.2	3.1	2.9	26	1.0
1 Note that ITY97 and HAN98 used a normalisation that is higher than we use
for model D by factors $\sim$1.4 and $\sim$1.1 respectively (see Sect. 7.1).

Our results are presented in the next subsections. In Sect. 6.1 we give the birth rates and total number of double white dwarfs in the Galaxy. These numbers allow a detailed comparison with results of earlier studies, which we defer to Sect. 7. They cannot be compared with observations directly, with the exception of the SN Ia rate. For comparison with the observed sample, described in Sect. 6.2, we compute magnitude limited samples in the remaining sections. In Sect. 6.3 the distribution over periods and masses is compared with the observations, which constrains the cooling models. Comparison of the mass ratio distribution with the observations gives further support for our new description of a common envelope without spiral-in (Sect. 6.4). In Sect. 6.5 we compare our model with the total population of single and binary white dwarfs and in Sect. 6.6 we compare models that differ in the assumed star formation history with the observed rate of PN formation and the local space density of white dwarfs.

  
6.1 Birth rates and numbers

In Table 2 the birth rates for all models are given. According to Eq. (1) the mass of a binary is on average 1.5 times the mass of a single star. For each binary in models A and C we also form a single star, i.e. per binary a total of 2.5 times the mass of a single star is formed (1.5 for the binary, 1 for the single star). For models B and D only 1.5 times the mass of a single star is formed per binary. Thus for the same SFR in $M_\odot$ yr^-1 the frequency of each process involving a binary of the models A and C is 0.6 times that in models B and D.

For model A the current birth rate for close double white dwarfs is 4.8 10^-2 yr^-1 in the Galaxy. The expected total population of close binary white dwarfs in the galactic disk is $\sim$2.5 10⁸ (see Table 2).

The double white dwarfs are of the following types: 53% contains two helium white dwarfs; 25% two CO white dwarfs; in 14% a CO white dwarf is formed first and a helium white dwarf later and in 6% a helium white dwarf is formed followed by the formation of a CO white dwarf. The remaining 1% of the double white dwarfs contains an ONeMg white dwarf. The CO white dwarfs can be so called hybrid white dwarfs; having CO cores and thick helium envelopes (Iben & Tutukov 1985, 1987). Of the double CO white dwarfs, 6% contains one and 5% two hybrid white dwarfs. In the mixed pairs the CO white dwarf is a hybrid in 20% of the cases.

Forty eight percent of all systems are close enough to be brought into contact within a Hubble time. Most are expected to merge. The estimated current merger rate of white dwarfs is 2.2 10^-2 yr^-1. The current merger rate of pairs that have a total mass larger than the Chandrasekhar limit ( $M_{\rm Ch}$ = 1.44$M_\odot$) is 3.2 10^-3yr^-1. Since the merging of binary CO white dwarfs with a combined mass in excess of $M_{\rm Ch}$ is a viable model for type Ia SNe (see Livio 1999, for the most recent review), our model rate can be compared with the SN Ia rate of $\sim$($4 \pm 1$) 10^-3yr^-1 for Sbc type galaxies like our own (Cappellaro et al. 1999). In 19% of the systems that come into contact the ensuing mass transfer is stable and an interacting double white dwarf (identified with AM CVn stars) is formed. The model birth rate of AM CVn systems is 4.6 10^-3 yr^-1 (see Table 2).

  
6.2 Observed sample of double white dwarfs

 

 

Table 3: Parameters of known close double white dwarfs (first 14 entries) and subdwarfs with white dwarf companions. m denotes the mass of the visible white dwarf or subdwarf. The mass ratio q is defined as the mass of the brighter star of the pair over the mass of the companion. For references see Maxted & Marsh (1999); Moran et al. (1999); Marsh (1999); and Maxted et al. (2000). The mass of 0136+768 is corrected for a misprint in Maxted & Marsh (1999), for 0135+052 the new mass given in Bergeron et al. (1997) is taken. Data for the sdB star KPD 0422+5421 are from Oroz & Wade (1999) and for KPD 1930+2752 from Maxted et al. (2000). The remaining sdB stars do not have reliable mass estimates

WD/sdB	P(d)	q	m	sdB	P(d)
0135-052	1.556	0.90	0.25	0101+039	0.570
0136+768	1.407	1.31	0.44	0940+068	8.33
0957-666	0.061	1.14	0.37	1101+249	0.354
1022+050	1.157		0.35	1432+159	0.225
1101+364	0.145	0.87	0.31	1538+269	2.50
1202+608	1.493		0.40	2345+318	0.241
1204+450	1.603	1.00	0.51
1241-010	3.347		0.31
1317+453	4.872		0.33
1704+481A	0.145	0.7	0.39
1713+332	1.123		0.38
1824+040	6.266		0.39
2032+188	5.084		0.36
2331+290	0.167		0.39
KPD 0422+5421	0.090	0.96	0.51
KPD 1930+2752	0.095	0.52	0.5

The properties of the observed double white dwarfs with which we will compare our models are summarised in Table 3. Only WD 1204+450 and WD 1704+481 are likely to contain CO white dwarfs, having components with masses higher than 0.46$M_\odot$; the limiting mass to form a helium white dwarf (Sweigart et al. 1990). The remaining systems are probably helium white dwarfs. In principle in the mass range $M \simeq 0.35 - 0.45\,M_\odot$ white dwarfs could also be hybrid; however in this range the probability for a white dwarf to be hybrid is 4-5 times lower than to be a helium white dwarf, because hybrid white dwarfs originate from more massive stars which fill their Roche lobe in a narrow period range (see, however, an example of such a scenario for WD  0957-666 in Nelemans et al. 2000). We assume $0.05\,M_\odot$ for the uncertainty in the estimates of the masses of white dwarfs, which may be somewhat optimistic.

Table 3 also includes data on subdwarf B stars with suspected white dwarf companions. Subdwarf B (sdB) stars are hot, helium rich objects which are thought to be helium burning remnants of stars which lost their hydrogen envelope. When their helium burning has stopped they will become white dwarfs. Of special interest are KPD 0422+5421 (Koen et al. 1998; Orosz & Wade 1999) and KPD1930+2752 (Maxted et al. 2000). With orbital periods as short as 0.09 and 0.095 days, respectively, their components will inevitably merge. In both systems the sdB components will become white dwarfs before the stars merge. In KPD 1930+2752 the total mass of the components is close to the Chandrasekhar mass or even exceeds it. That makes this system the only currently known candidate progenitor for a SN Ia.

   
6.3 Period-mass distribution; constraints on cooling models

Figure 2: Model population of double white dwarfs as function of orbital period and mass of the brighter white dwarf of the pair. Top left: distribution of the double white dwarfs that are currently born for models A. This is independent of cooling. In the remaining three plots we show the currently visible population of double white dwarfs for different cooling models: (top right) cooling according to DSBH98 and Blöcker (1995, model A1); (bottom right) cooling according to DSBH98, but with faster cooling for WD with masses below 0.3$M_\odot$ (model A2). Both plots are for a limiting magnitude $V_{\rm lim} = 15$; (bottom left) with constant cooling time of 100 Myr (model A3, note that in this case we only obtain the total number of potentially visible double white dwarfs in the Galaxy and we cannot construct a magnitude limited sample). For comparison, we also plot the observed binary white dwarfs

The observed quantities that are determined for all double white dwarfs are the orbital period and the mass of the brighter white dwarf. Following Saffer et al. (1998), we plot in Fig. 2 the $P_{\rm orb} - m$ distributions of the frequency of occurrence for the white dwarfs which are born at this moment and for the simulated magnitude limited sample for the models with different cooling prescriptions, (models A1, A2 and A3; see Table 1), where we assume $V_{\rm lim} = 15$ as the limiting magnitude of the sample. For m we always use the mass of the brighter white dwarf. In general the brighter white dwarf is the one that was formed last, but occasionally, it is the one that was formed first as explained in Sect. 2.3.3. For comparison, we also plot the observed binary white dwarfs in Fig. 2.

There is a clear correlation between the mass of new-born low-mass (He) white dwarf and the orbital period of the pair. This can be understood as a consequence of the existence of a steep core mass-radius relation for giants with degenerate helium cores (Refsdal & Weigert 1970). Giants with more massive cores (forming more massive white dwarfs) have much larger radii and thus smaller binding energies. To expell the envelope in the common envelope, less orbital energy has to be used, leading to a larger orbital period. The spread in the distribution is caused by the difference in the masses of the progenitors and different companion masses.

In the simulated population of binary white dwarfs there are three distinct groups of stars: He dwarfs with masses below 0.45$M_\odot$, hybrid white dwarfs with masses in majority between 0.4 and 0.5 $M_\odot$ and periods around a few hours, and CO ones with masses above 0.5$M_\odot$. The last groups are clearly dominated by the lowest mass objects. The lowest mass CO white dwarfs are descendants of most numerous initial binaries with masses of components 1-2$M_\odot$.

The different cooling models result in very different predicted observable distributions. Model A1 where the cooling curves of DSBH98 are applied favours low mass white dwarfs to such an extent that almost all observed white dwarfs are expected to have masses below 0.3$M_\odot$. This is in clear contrast with the observations, in which all but one white dwarf have a mass above 0.3$M_\odot$. Reduced cooling times for white dwarfs with masses below 0.3$M_\odot$ (model A2) improves this situation. Model A3, with a constant cooling time (so essentially only affected by merging due to GWR), seems to fit all observed systems also nicely. However, a complementary comparison with the observations as given by cumulative distributions of the periods (Fig. 3), shows that model A2 fits the data best, and that model A3 predicts too many short period systems.

The observed period distribution for double white dwarfs shows a gap between 0.5 and 1 day, which is not present in our models. If we include also sdB binaries, the gap is partially filled in. More systems must be found to determine whether the gap is real.

The comparison of our models with observations suggests that white dwarfs with masses below 0.3$M_\odot$ cool faster than predicted by DSBH98. Mass loss in thermal flashes and a stellar wind may be the cause of this.

Figure 3: Cumulative distribution of periods. Solid line for our best model (A2); DSBH98 cooling, but with lower luminosity due to thermal flashes for white dwarfs with masses below 0.3$M_\odot$. Dashed line for DSBH98 without modifications (model A1) and dash dotted line for constant cooling time of 100 Myr (model A3). Open squares for the observed double white dwarfs, filled circles give the observed systems including the sdB binaries (Table 3)

The model sample of detectable systems is totally dominated by He white dwarfs with long cooling times. Given our model birth rates and the cooling curves we apply, we estimate the number of double white dwarfs to be detected in a sample limited by $V_{\rm lim}$ = 15 as 220 of which only 10 are CO white dwarfs for model A2. Roughly one of these is expected to merge within a Hubble time having a total mass above $M_{\rm Ch}$. For future observations we give in Table 4 a list of expected number of systems for different limiting magnitudes.

It should be noted that these numbers are uncertain. This is illustrated by the range in birth rates for the different models (Table 2) and by the differences with previous studies (see Sect. 7.1). Additional uncertainties are introduced by our limited knowledge of the initial distributions (Eq. (1)) and the uncertainties in the cooling and the Galactic model (Eq. (2)). For example Yungelson et al. (1994) compare models with two different $q_{\rm i}$distributions (one peaked towards $q_{\rm i} \sim 1$) and show that the birth rates differ by a factor $\sim$1.7. In general the relative statistics of the model is more reliable than the absolute statistics.

Figure 4: Cumulative distribution of periods. Solid line for model A2 as in Fig. 3, dashed line for the same model but with $\alpha _{\rm ce} \lambda = 1$, dash-dotted line for a model with $\gamma = 1.5$ and finally the dotted line for model C (constant SFR)

 

 

Table 4: Number of observable white dwarfs, close double white dwarfs and SN Ia progenitors as function of the limiting magnitude of the sample for model A2

$V_{\rm lim}$	#wd	#wdwd	#SN Ia prog
15.0	855	220	0.9
15.5	1789	421	1.7
16.0	3661	789	3.2
17.0	12155	2551	11.2

Before turning to the mass ratio distribution, we illustrate the influence of the model parameters we choose. We do this by showing cumulative period distributions for some models with different parameters in Fig. 4; $\alpha _{\rm ce} \lambda = 1$ (dashed line) and $\gamma = 1.5$ (dash-dotted line). It shows that the change in parameters influences the distributions less than the different cooling models discussed above, although the observations favour a higher $\alpha_{\rm ce}\lambda$. We also included the cumulative distribution for model C (with a constant SFR; dotted line) which differs from that for model A2 in that it has fewer long period systems. This is a consequence of the larger relative importance of old, low-mass progenitor binaries in model A2, which lose less mass and thus shrink less in the first phase of mass transfer (see Eq. (A.16)).

   
6.4 Period-mass ratio distribution

Figure 5: Top: current population of double white dwarfs as function of orbital period and mass ratio, for model A2, a limiting magnitude of 15 and a maximal ratio of luminosities of 5. Bottom: the same for a run in which the first phase of mass transfer is treated as a standard common envelope, as is done by ITY97 and HAN98. For comparison, we also plot theobserved binary white dwarfs

Figure 6: Cumulative mass ratio distributions for the models A2 (solid line) and A$^\prime$ (dotted line) as explained in Sect. 6.4. The observed mass ratio's are plotted as the open squares

Our assumption that a common envelope can be avoided in the first phase of mass transfer between a giant and a main-sequence star, is reflected in the mass ratios of the model systems. A clear prediction of the model is that close binary white dwarfs must concentrate to $q = m/M \sim 1.$ For the observed systems, the mass ratio can only be determined if both components can be seen, which in practice requires that the luminosity of the fainter component is more than 20% of that of the brighter component (Moran et al. 2000). Applying this selection criterium to the theoretical model, we obtain the distribution shown in Fig. 5 for the magnitude limited sample. Note that since lower mass white dwarfs cool slower this selection criterion favours systems with mass ratios above unity. In the same figure we also show the observed systems. For comparison we also computed a run (A$^\prime$) in which we used the standard common envelope treatment for the first phase of mass transfer, which is done by ITY97 and HAN98. The fraction of double white dwarfs for which the mass ratio can be determined according to the selection criterium of a luminosity ratio greater than 0.2, is 27% for model A2 and 24% for model A$^\prime$. In a total of 14 systems one thus expects $4 \pm 2$ and $3 \pm 2$ systems of which the mass ratio can be determined. Model A2 fits the observed number better, but the numbers are too small to draw conclusions. The distribution of mass ratios in model A$^\prime$ (Fig. 5, bottom) however clearly does not describe the observations as well as our model A2, as illustrated in more detail in a plot where the cumulative mass ratio distributions of the two models and the observations are shown (Fig. 6).

  
6.5 Mass spectrum of the white dwarf population; constraints on the binary fraction

Figures 7 and 8 show the model spectrum of white dwarf masses for models B and A2, including both single and double white dwarfs for a limiting magnitude $V_{\rm lim} = 15$. For this plot we consider as "single'' white dwarfs all objects that were born in initially wide pairs, single merger products, white dwarfs that became single as a result of binary disruption by SN explosions, white dwarfs in close pairs which are brighter than their main-sequence companions and genuine single white dwarfs for the models with an initial binary fraction smaller than 100%.

These model spectra can be compared to the observed mass spectrum of DA white dwarfs studied by Bergeron et al. (1992) and Bragaglia et al. (1995), shown in Fig. 9. The latter distribution may have to be shifted to higher masses by about 0.05$M_\odot$, if one uses models of white dwarfs with thick hydrogen envelopes for mass estimates (Napiwotzki et al. 1999). Clearly, a binary fraction of 50% fits the observed sample better, if indeed helium white dwarfs cool much slower than CO white dwarfs. We can also compare the absolute numbers. Maxted & Marsh (1999) conclude that the fraction of close double white dwarfs among DA white dwarfs is between 1.7 and 19% with 95% confidence. For model B the fraction of close white dwarfs is $\sim$43% (853 white dwarfs of which 368 are close pairs), for model A2 is is $\sim$26% (855 white dwarfs and 220 close pairs). Note that this fraction slightly decreases for higher limiting magnitudes because the single white dwarfs are more massive and thus generally dimmer, sampling a different fraction of Galaxy. An even lower binary fraction apparently would fit the data better, but is in conflict with the estimated fraction of binaries among normal main sequence binaries (Abt 1983; Duquennoy & Mayor 1991). However this number highly depends on uncertain selection effects.

There are some features in the model mass spectrum in model A2 that appear to be in conflict with observations. The first is the clear trend that with the cooling models of DSBH98, even with our modifications, there should be an increasing number of helium white dwarfs towards lower masses. The observed distribution is flat. A very simple numerical experiment in which we assign a cooling curve to all helium white dwarfs as the one for a 0.414$M_\odot$ white dwarf according to DSBH98 and a cooling curve as for a 0.605$M_\odot$ white dwarf according to Blöcker (1995) for all CO white dwarfs (Fig. 10), shows that an equal cooling time for all helium white dwarfs seems to be in better agreement with the observations. It has a fraction of double white dwarfs of 18%. Another feature is the absence of stars with 0.45 0.5 in the model distributions. This is a consequence of the fact that in this interval in our models only hybrid white dwarfs can be present, which have a low formation probability (see Sect. 6.2).

We conclude that an initial binary fraction of 50% can explain the observed close binary fraction in the white dwarf population. The shape of the mass spectrum, especially for the helium white dwarfs is a challenge for detailed mass determinations and cooling models.

  
6.6 Birth rate of PN and local WD space density; constraints on the star formation history

Figure 7: Mass spectrum of all white dwarfs for model B (100% binaries). Members of close double white dwarfs are in grey. The cumulative distribution is shown as the solid black line. For comparison, the grey line shows the cumulative distribution of the observed systems (Fig. 9)

Figure 8: Mass spectrum of all white dwarfs for model A2 (initial binary fraction of 50%) Double white dwarfs are in grey. The cumulative distribution is shown as solid black line and cumulative distribution of observed systems as the grey line

Figure 9: Mass spectrum of observed white dwarfs. Data are taken from Bergeron et al. (1992) and Bragaglia et al. (1995). The solid line is the cumulative distribution

Figure 10: Mass spectrum of all white dwarfs as in Fig. 8 in a model in which all helium white dwarfs cool like a 0.4$M_\odot$ dwarf and all CO white dwarfs cool like a 0.6$M_\odot$ white dwarf. Lines are cumulative distributions for the model (black) and the observations (grey)

 

 

Table 5: Galactic number and local space density of white dwarfs; and Galactic and local PN formation rate for the models A and C. Unit of the PN formation rates is yr^-1; unit for $\rho _{\rm wd,\odot }$ is pc^-3. The ranges of observed values are given forcomparison. For references and discussion see Sect. 6.6

Model	SFH	% bin	#wd	$\nu_{\rm PN}$	$\rho _{\rm wd,\odot }$	$\nu_{\rm PN, \odot}$
			109		(10-3)	(10-12)
A	Exp	50	9.2	1.1	19	2.3
C	Const	50	4.1	0.8	8.5	1.7
Obs					4-20	3

Finally, we compare models A and C (see Table 1), which differ only by the assumed star formation history. The star formation rate was probably higher in the past than at present and some (double) white dwarfs descend from stars that are formed just after the galactic disk was formed.

Table 5 gives the formation rates of PN and the total number of white dwarfs in the Galaxy for models A and C. The total number of white dwarfs is computed by excluding all white dwarfs in binaries where the companion is brighter. The local density of white dwarfs and PN rate are computed with Eq. (3) as described in Sect. 4.4.

We can compare these numbers with the observational estimates for the local PN formation rate of 3 10^-12 pc^-3 yr^-1(Pottasch 1996) and the local space density of white dwarfs, which range from e.g. 4.2 10^-3 pc^-3 (Knox et al. 1999) through 7.6^+3.7_-0.7 10^-3 pc^-3 (Oswalt et al. 1995) and 10 10^-3 pc^-3 (Ruiz & Takamiya 1995) to $20 \pm 7$ 10^-3 pc^-3 (Festin 1998).

This list shows the large uncertainty in the observed local space density of white dwarfs. It appears that the lower values are somewhat favoured in the literature. Both models A and C appear for the moment to be consistent with the observed local white dwarf space density and with the PN formation rate. However, we prefer model A2 since it fits the period distribution better (see Fig. 4).

The ratio of the local space density of white dwarfs to the current local PN formation rate could in principle serve as a diagnostic for the star formation history of the Galaxy, given better knowledge of $\rho _{\rm wd,\odot }$, which critically depends on the estimates of the incompleteness of the observed white dwarf samples and the applied cooling curves.