The cooling of oxygen-neon white dwarfs

Transcription

1 Mon. Not. R. Astron. Soc. 289, (1997) The cooling of oxygen-neon white dwarfs Enrique Garcia-Berro, I Jordi Isem 2 and Margarida Hemanz 2 1 Departament de Flsica Aplicada, Universitat Politecnica de Catalunya, lordi Girona Salgado sin, Mbdul B-5, Campus Nord, Barcelona, Spain 2Institutejor Space Studies oj Catalonia (CSIC Research Unit), Edifici Nexus-l04, Gran Capita 2-4,08034 Barcelona, Spain Accepted 1997 Apri128. Received 1997 March 20; in original form 1997 January 13 1 INTRODUCTION White dwarfs are the most common endpoint of stellar evolution. The vast majority of single white dwarfs have progenitors with masses between 0.8 and 7.8 M0 on the main sequence, which corresponds to white dwarf masses in the range M0, and their internal composition is almost exclusively carbon and oxygen. Massive white dwarfs have progenitors with masses M :2: 8 M0 and are made of oxygen and neon. Because white dwarfs have exhausted all their thermonuclear fuel their evolution can be explained in terms of a cooling process. The physics of cooling white dwarfs has been extensively studied during recent years by several groups (see for instance Segretain et al. 1994, and references therein) and can be considered as relatively firmly established for luminosities 10g(UL0 ) > This allows to use the luminosity function of white dwarfs as a tool to determine some aspects of Galactic evolution, such as the age of the disc (see for instance Winget et al. 1987, Garcia-Berro et al and Hernanz et al for a more detailed discussion), the star formation history of the disc (Dfaz-Pinto et al. 1994; Isern et al. 1995a,b) or the properties of the halo (Mochkovitch et al. 1990; Isern et al. 1997a; Adams & Laughlin 1996; Chabrier, Segretain & Mera 1996). However, most of the information about the birth and evolution of the Galactic disc - or at least of the solar neighbourhood - is concentrated in the faint end of the white dwarf luminosity function which is dominated by the contribution of massive white dwarfs (Diaz-Pinto et al. 1994). Unfortunately, these white dwarfs have ABSTRACT Recent observations of very cool white dwarfs suggest that the low-luminosity portion of the white dwarf luminosity function is indeed dominated by the contribution of massive objects. These objects should predominantly be made of oxygen, neon and other trace isotopes. In this paper we compute cooling sequences for massive oxygen-neon (ONe) white dwarfs and we examine the question of whether massive ONe white dwarfs could represent a significant contribution to the white dwarf luminosity function at very low luminosities. We find that massive ONe white dwarfs cool much faster than carbonoxygen (CO) white dwarfs and that, consequently, their population effectively detaches from the bulk of CO white dwarfs leading to the formation of a tail in the low-luminosity portion of the white dwarf luminosity function. Nevertheless, the position of the reported cut-off in the white dwarf luminosity function is not affected and, therefore, the age estimates of the solar neighbourhood inferred from the position of the drop-off of the white dwarf luminosity function remain unaffected. Key words: stars: interiors - stars: luminosity function, mass function - stars: statistics - white dwarfs - Galaxy: stellar content. proved to be very elusive until now because their cooling timescales are short when compared to the bulk of carbon-oxygen (CO) white dwarfs and, therefore, they are intrinsically dim. The interest in very low-luminosity white dwarfs has increased since the recent discovery by Ruiz et al. (1995) of the dimmest single white dwarf known (L = 10-5 L0). This white dwarf probably belongs to the old disc population and has a reported mass M = 1.2M0. Therefore, it should be a typical ONe white dwarf. Moreover, the MACHO collaboration in their first season reported a microlensing event with a duration of nod towards the Galactic bulge (Alcock et al. 1995). For this particular event a parallax could be obtained from the shape of the light curve, from which a mass of 1.3:::6:~ M0 was derived, indicating that probably the gravitational lens was a massive old ONe white dwarf or a neutron star. The evolution of single stars in the initial mass range 8 :::s M :::s 11 M0, which leads to the formation of ONe-rich cores, had remained almost unexplored until very recently, in spite of the fact that this mass interval contains about one half of the stars that burn carbon under non-degenerate and partially degenerate conditions and could be able to form massive white dwarfs. It has been argued (Nomoto 1984) that the evolution of isolated stars needs a fine tuning of the mass-loss rate in order to produce massive ONe white dwarfs, so the expected number of these should be small. Another suggested scenario is the merger of two white dwarfs in a close binary system (Yungelson et al. 1994). However, it should be noted that a major drawback of these scenarios is that no detailed and self-consistent calculation of the evolution of stars in this mass 1997 RAS

2 974 E. Garda-Berro, 1. [sern and M. Hemanz range existed in the literature. In several studies, the thermal instability causing the pulses was inhibited in order to follow the growth of the helium-exhausted core to larger masses and eventually to a mass close to the Chandrasekhar limit (Nomoto 1984, 1987), and little attention was paid to the properties of the thermal pulses themselves. In other studies (Then 1977) the suppression mechanism was invoked at different stages of the evolution and a couple of thermal pulses were studied, at most. One of the probable reasons for the lack of such models in the literature is that the conversion of a CO core into an ONe core involves following the evolution of an unstable nuclear burning flame as it propagates towards the centre. Very recently Garcia-Berro & Then (1994) and Ritossa, Garcia-Berro & Iben (1996) studied the evolution from the main sequence of a 1O-M0 star, demonstrating that the hydrogen shell is eventually reignited, thus leading to the thermally pulsing AGB phase. Garcia-Berro, Ritossa & Then (1997) studied the formation and composition of the ONe degenerate core of a 9-M0 star and its early thermal pulsing phase and Then et al. (in preparation) computed the evolution of a 1O.5-M0 star. A real counterpart of these kinds of models is expected to experience a rapid radiative wind during the thermally pulsing AGB phase, eventually leading to the ejection of its hydrogen-rich envelope and most of its helium-rich buffer and converting the central star into a PG 1159 object which would finally evolve into a cooling ONe white dwarf. One probable observational counterpart is EUVE J (Dupuis & Vermes 1997). One of the most important conclusions of all these works is that stars initially more massive than -8 M0 form ONe degenerate cores of masses larger than -1 M0. Therefore, white dwarfs with masses larger than -1 M0 are expected to be ONe white dwarfs. The exact mass of the resulting white dwarf depends somehow on the details of the calculations during the rapid radiative wind phase - which is not well known - but, under reasonable assumptions (Ritossa et al. 1996; Garcia-Berro et al. 1997), a 9-M0 star would produce a white dwarf with an ONe core of mass MONe = M0 surrounded by a CO buffer of mass ilmco = 0.05 M0, leading to a total mass of the white dwarf Mwo = 1.12M0. The corresponding values for the 10- and the 1O.5-M0 models are respectively MONe = 1.186M0, ilmco = 0.02M0 and MONe = 1.263M0, ilmco = M0 and, thus, the corresponding white'dwarf masses are Mwo = 1.21 and = 1.27 M0, respectively. Stars with masses larger than -11 M0 on the main sequence ultimately reach the threshold for electron captures on 24Mg and then on 20Ne in their ONe cores and most probably undergo gravitational collapse of their degenerate cores (Gutierrez et al. 1996). Regarding the chemical composition of the ONe cores of massive white dwarfs, the most abundant element is 160, with a mass fraction X in all three models, followed by 20Ne (X ) and other minor chemical species, the most important being 23Na, 24Mg, 22Ne and some unburnt 12C. The abundance of the remaining 12C is X l for the 9-M0 model, for the 1O-M0 model and negligible for the 1O.5-M0 model. The precise amount of carbon left unburnt is very sensitive both to mass zoning during the carbon-burning phase and to the exact values of the cross-sections involved in the nuclear network. In this paper we compute cooling sequences for massive ONe white dwarfs. The paper is organized as follows: in Section 2 we present the cooling sequences, in Section 3 we present the white dwarf luminosity functions computed on the basis of these cooling sequences, in Section 4 we compute the associated discovery functions, and in Section 5 we provide a summary and our conclusions. 2 THE COOLING SEQUENCES We have computed cooling sequences for white dwarfs of masses 1.01, 1.1 and 1.2 M0 following the method described in Diaz-Pinto et al. (1994). In summary, with the method used here the cooling times are derived from the binding energy of the white dwarf and a relationship between the temperature of its core, which is assumed to be isothermal, and the mass and luminosity of the white dwarf, which we have adopted from detailed evolutionary sequences (Wood & Winget 1989). This relationship essentially depends on the chemical composition and thickness of the envelope. For the relationship considered here a pure-helium envelope of thickness 1O-4Mwo was adopted (see, however, Section 3 below). Our cooling sequences start at core temperatures of 5 x 10 7 K, which roughly correspond to luminosities _10-1 L0 for which the core can be considered to be isothermal to a good degree of approximation. We have used the equation of state described in Segretain et al. (1994) which includes accurately all the relevant contributions to the thermodynamical quantities both in the liquid and in the solid phases. Neutrino losses, which are important at relatively high temperatures, were also taken into account following Garcia-Berro et al. (1996). Because the chemical composition of the degenerate ONe cores is not extremely dependent on the mass of the cores (Ritossa et al. 1996; Garcia-Berro et al. 1997; Then et al., in preparation) we have assumed the following (average) chemical composition: X Ne = 0.36, Xo = In essence all trace isotopes were considered to be neon, the ultimate reason for that being simplicity. The influence of those trace isotopes in the cooling times is expected to be small since their mass fractions are small as well. No phase separation upon crystallization was assumed due to the small impact of chemical differentiation on the cooling times of a crystallizing ONe white dwarf. In fact, the ratio of nuclear charges of neon to oxygen is very similar to that of oxygen to carbon (1.25 and 1.33 respectively) and, therefore, the phase diagram of a binary ONe mixture is very similar to that of a CO mixture (Segretain & Chabrier 1993). However, the average charge of the ONe mixture is larger than that of the CO mixture and, consequently, the crystallization temperature is larger. As a result, the energy released by chemical sedimentation upon crystallization has smaller effects on the cooling times since this energy is released at larger luminosities (nevertheless, see below for a quantitative estimate). It can be argued as well that the ratio of nuclear charges of both oxygen and neon to carbon - which is one of the trace elements - is relatively large, but the exact amount of carbon left unburnt during the carbonburning phase is so sensitive to the details of the calculations and to the mass of the progenitor of the white dwarf that we prefer to ignore its effect on the cooling sequences. Moreover, the exact shape of the phase diagrams of ternary mixtures (and, in particular, of an OfNeC mixture) is still very uncertain. In Fig. 1 we show the cooling sequences for the three values of the white dwarf mass mentioned above as solid lines (the cooling times for selected luminosities and the aforementioned masses are given in Table 1 as well!). For comparison the cooling sequences for CO white dwarfs of the same mass (Segretain et al. 1994) are also shown as dotted lines. These cooling sequences include phase separation of carbon and oxygen upon crystallization and the initial chemical profiles were a flat 50:50 mixture (by mass) of carbon and oxygen. The most obvious feature of this figure is that, on average, for a fixed mass, ONe white dwarfs cool down to a given luminosity 1 Detailed cooling sequences are available upon request to the authors.

3 -2.0 Cooling of oxygen-neon white dwaifs J 3 bii B !og(t eoo1 ) Figure 1. Luminosity as a function of the cooling times (in yr) for ONe white dwarf models of masses 1.01, 1.1 and 1.2 Mo (solid lines), compared to models with the same masses made of carbon and oxygen (dotted lines). Table 1. Cooling times (in Gyr) for 1.01-, 1.1- and 1.2-Mo ONe white dwarfs. -log(ulo ) tcool (Gyr) 1.0Mo 1.1 Mo 1.2Mo faster than their corresponding CO counterparts. The reason for this is threefold. First, the average charge of an ONe mixture is larger than that of a CO mixture, and therefore crystallization sets in at larger temperatures (luminosities) in an ONe mixture. Consequently, the latent heat of crystallization is released at higher temperatures and has smaller effects on the evolution of ONe white dwarfs. Secondly, there is no release of gravitational energy due to chemical differentiation since there is no difference between the chemical compositions of the liquid and the solid phases. The 8.0? ~ll.0~~--~~_~2-.0~~~--~_~3.-0~~~~--_4l.0~~~ log(ll,;,) Figure 2. Characteristic cooling time-scales as a function of the luminosity for ONe dwarf models of masses 1.01, 1.1 and 1.2Mo (solid lines), compared to models with the same masses made of carbon and oxygen (dotted lines). combined effect of these two results is quite apparent in Fig. 1, where the onset of crystallization is clearly marked by a sudden change in the slope of the cooling curves for CO white dwarfs, while it is difficult to see the same feature in the ONe cooling sequences. Thirdly, the core has a reduced heat capacity due to the larger average atomic mass of the ONe mixture and, therefore, the cooling time-scales are shorter. The net result of all three effects is that ONe white dwarfs cool down much faster than CO white dwarfs. In fact, the time necessary to reach a luminosity of log(ll 0 ) = -4.5 is, roughly speaking, 7 Gyr for ONe white dwarfs whereas the time necessary for CO white dwarfs to reach the same luminosity is about 9 Gyr. For the sake of completeness we have computed as well the delay introduced in the cooling times by the inclusion of phase separation of the ONe mixture upon crystallization by following closely Isern et al. (1997b). For a typical l.l-m0 ONe white dwarf the cooling times are increased by only Gyr. This is a direct consequence of the high average charge of the ONe mixture, as stated previously. In Fig. 2 the characteristic cooling time-scales (defined as Tcool = dtcoodmbol) of ONe white dwarfs are shown as a function of the luminosity. Once again, the corresponding characteristic cooling time-scales of CO white dwarfs of the same mass are shown for comparison purposes. This quantity is important in evaluating the white dwarf luminosity function (see Section 3 below). The same features seen in Fig. I are present here as well: namely, solidification starts at higher luminosities (temperatures) for the ONe cooling sequences - the sharp increase in the characteristic cooling time-scales marks the onset of solidification - and the energy released upon crystallization is smaller for these sequences - the amplitude of the bump that follows the sharp increase in the characteristic cooling times is larger for the CO cooling sequences mostly because of the sedimentation process.

4 976 E. Garcia-Berro, J. Isern and M. Hernanz -3.0 C bii s Liebert, Dahn & Monet (1988) t di k =9.3 Gyr! -5.0 i -6.0 '--'~~~-'---~~~--'--'---~L-L-'---'-~~L-L-'---'-~~ log(ll0) Figure 3. White dwarf luminosity functions computed with the cooling sequences described in Section 2. The observational data have been taken from Liebert et al (1988). See text for details. 3 THE WHITE DWARF LUMINOSITY FUNCTION The white dwarf luminosity function is defined as the number of white dwarfs per unit volume and per unit of bolo metric magnitude: where M is the mass on the main sequence of the white dwarf progenitor, <I> is the initial mass function, which we adopt to be the Salpeter mass function (Salpeter 1961), ir is the star formation rate per unit volume, which we adopt to be constant, tdisc is the age of the disc, tms is the main-sequence lifetime (Salaris et al. 1997), tcool is the time necessary to cool down to a given luminosity, and 7cool is the characteristic cooling time-scale previously defined in Section 2. Of course, a relationship between the mass of the white dwarf progenitor and the mass of the white dwarf itself is needed for evaluating the previous expression. We have adopted that of Then & Laughlin (1989). The upper limit of the integral is given by the maximum mass of the progenitor of a white dwarf, M ~ 11 Mo, and the lower limit depends on the luminosity at which we are evaluating the white dwarf luminosity function and is obtained solving the equation tms(mi ) + tcool(l,mi ) = tdisc' We have computed the white dwarf luminosity function with the following cooling sequences: for CO white dwarfs (M WD :S 1.0 Mo) we have adopted the cooling sequences computed in Salaris et al. (1997). These cooling sequences take into account the latest results on the 12C(a, '}')160 cross-section to derive detailed CO profiles. For more massive white dwarfs the cooling sequences described in Section 2 were adopted. The result for a disc age of tdisc = 9.3 Gyr is shown in Fig. 3 as a dotted line. The observational data have been taken from Liebert, Dahn & Monet (1988) and the theoretical luminosity function was normalized to the point log(ulo ) = -2.9 because of its smaller error bar. For comparison purposes one of the white dwarf luminosity functions derived in Hernanz et al. (1994) for a disc age of 9.S Gyr is also shown as (1) a dot-dashed line. We have chosen for this purpose the cooling sequences that we consider to be more realistic; that is, a previously stratified CO profile (Mazzitelli & D' Antona 1986a,b, 1987), and phase separation of carbon and oxygen (Segretain et al. 1994). All the other inputs required to compute the white dwarf luminosity function were the same. There are two prominent features in Fig. 3 that deserve to be commented on. The first one is the larger value of the white dwarf luminosity function at high luminosities [log(ulo ) 2: -2.0] obtained using the cooling sequences of Segretain et al. (1994) for CO white dwarfs when compared to the cooling sequences used here. The reason is simply that in the aforementioned paper neutrino losses were neglected, while in the present calculation neutrino losses have been properly taken into account. The second important feature is the presence of an extended tail beyond log(lilo) = -4.S, which extends up to log(lilo) = -S.6, and which is only present when the cooling sequences for ONe white dwarfs are taken into account. The existence of this tail can be well understood in terms of the cooling sequences described in the previous section. First, ONe white dwarfs have shorter cooling time-scales than their CO counterparts, as shown previously. Therefore it is expected that ONe white dwarfs will populate the low-luminosity portion of the white dwarf luminosity function. Secondly, the progenitors of ONe white dwarfs are very massive and, thus, their main-sequence lifetimes are much shorter on average than those of CO white dwarfs. The net result of these two effects is that ONe white dwarfs evolve much faster than the bulk of CO white dwarfs and effectively detach from them. Besides, since we have assumed a constant star formation rate, massive white dwarfs are continously formed and pumped to the low-luminosity tail. The tail stops at log(lilo) = -S.6, which is the real cut-off of the white dwarf luminosity function, although there is an abrupt drop at 10g(LlLo) = -4.S, which corresponds to the observed falloff of the disc white dwarf luminosity function, and is caused by the transition between CO white dwarfs and ONe white dwarfs. One important consequence of this is that the age of the disc derived from the white dwarf luminosity is not significantly affected by the inclusion of massive ONe white dwarfs. In fact the age difference between the luminosity function computed here and that ofhernanz et al. (1994) results from a more realistic treatment of the crystallization process - see Salaris et al. (1997) for a detailed discussion. Since the shape of the tail of the disc white dwarf luminosity function depends on the adopted internal temperature-luminosity relationship, and this, in tum, depends basically on the characteristics of the insulating envelope, the properties of which are poorly known at very low luminosities, we have computed another set of cooling sequences and luminosity functions adopting a 'classical' L ex T~ relation between the luminosity and the core temperature for very low luminosities log(lilo) = We have adopted 'Y = -1.62, which fits quite well the last computed points of the results of Wood & Winget (1989) and is far more conservative than the standard 'Y ~ -3 law. Using this core temperature-luminosity relationship, white dwarfs of the same core temperature are brighter than if the 'classical' core temperature-luminosity is used. The corresponding white dwarf luminosity function is shown in Fig. 3 as a solid line. It is evident that the exact shape of the tail strongly depends on the properties of the envelope. However, the important result is that the tail is present independently of the model envelope adopted. Very recently, Oswalt et al. (1996) have carried out a series of new observations, which include the data from ~SOO white dwarfs in wide binary systems, and have identified some new very cool

5 -3.0 Oswalt, Smith, Wood & Hintzen (1996) t di k = 11.0 Gyr :. Cooling of oxygen-neon white dwaifs <I 2.0 'bii t dlsk = 11.0 Gyr Oswalt, Smith, Wood & Hintzen (1996) "2 bii )' l...l--w'-'-"--...jl...l~--'--"--...jl...l~--'--"--l...l~--'--"--l...l-'--'--'-'--' log(lle) Figure 4. White dwarf luminosity functions computed with the cooling sequences described in Section 2. The observational data have been taken from Oswalt et ai. (1996). See text for details. white dwarfs. The main result of this work is a new observational white dwarf luminosity function which shows a less pronounced drop-off located at lower luminosities than that of Liebert et al. (1988). Therefore, it is important to analyse the impact of our ONe cooling sequences on this new white dwarf luminosity function. We have computed a series of white dwarf luminosity functions with the same input as described above, varying only the age of the disc. The age of the disc that best fits these new observational data is 11 Gyr, and our best fits are shown in Fig. 4. Again the presence of a tail beyond the cut-off is notorious, but the new result is that now the tail appears in all three cases. The reason for the presence of an extended tail in the white dwarf luminosity function computed with CO cooling sequences holds some common points with the presence of a similar tail when ONe cooling sequences are taken into account. First, massive CO white dwarfs have progenitors with large masses on the main sequence and, therefore, most of the age of the Galactic disc is available for pure cooling; and, secondly, the geometry of the cut-off is sharp for small ages of the disc and shallower for larger ages. 4 THE DISCOVERY FUNCTIONS The presence of a tail in the computed white dwarf luminosity functions can help us to understand the existence of extremely cool white dwarfs such as ESO , which have luminosities beyond that of the cut-off of the white dwarf luminosity function. This star, discovered by Ruiz et al. (1995), has a luminosity of log(ul0 ) = and a reported mass of -1.2 M0. Since this is a massive white dwarf, its mere existence does not pose any problem to the standard scenario, except for the space density of such stars, which is unknown. First, it should be noted that this star most probably is not a halo white dwarf because for reasonable halo ages (- 15 Gyr), its luminosity would be even smaller than its reported luminosity [in fact its luminosity would be log(ul0 ) :5-6]. In order to provide numerical estimates of the expected number of <I 2.0 'bii t disk =9.3 Gyr Liebert, Dahn & Monet (1988) log(lle) Figure 5. White dwarf discovery functions for the white dwarf luminosity functions of Figs 3 and 4: see text for details. such dim objects it is convenient to compute the discovery functions associated with the previously computed white dwarf luminosity functions. The discovery function gives the number of white dwarfs per interval of magnitude that can be detected in a survey limited to a given apparent magnitude (Mochkovitch et al. 1990). If we limit ourselves to nearby white dwarfs, this volume can be considered spherical and the discovery function,..:l(mbol) is easily obtained from its corresponding luminosity function: 471' 3..:l(Mbol) = 3d (Mbol)n(Mbol), (2) where d(m bol) is the distance at which a white dwarf of bolometric magnitude Mbol has an apparent magnitude rna (in a photometric band centred at A), which is given by d(m ) = 114 (L loo.2[m,+(v-m,)+bc] (3) bol. YL;;, where (V - rna) is a colour index, BC is the bolometric correction and d is in pc. Since white dwarfs with luminosities log(ul0) - -5 have effective temperatures of K and radiate most of their energy in the red or infrared, we have computed the discovery functions for the I band assuming that white dwarfs behave as blackbodies. Since ESO has a reported V magnitude of20.52 mag and a colour index V - I = 1.11 we have assumed a limiting magnitude for the survey of I = 20 mag. We show the results in Fig. 5 for the three cases studied here and the two different data sets described in the previous section: that of Liebert et al. (1988) - lower panel- and that of Oswalt et al. (1996) - upper panel. As would be expected the number of detectable white dwarfs in such a survey drops precipitously at luminosities beyond the observed cut-off in the white dwarf luminosity function, which is located at log(ul0 ) = for the data set of Liebert et al. (1988) and at log(ul0 ) = -4.5 for the data set of Oswalt et al. (1996). On the other hand, it should be noted that except for the white dwarf luminosity function computed with the CO cooling sequences and a small age of the disc - which corresponds to the

6 978 E. Garda-Berro, J. Isern and M. Hernanz data set of Liebert et al. (1988) - the tail of the white dwarf discovery function extends well beyond 10g(ULo ) Therefore, except for this case, it is possible to explain the detection of a white dwarf as dim as reported by Ruiz et al. (1995). The expected number of white dwarfs at a luminosity of 10g(ULo ) = -5 is -1 (per unit bolometric magnitude) for the rest of the white dwarf luminosity functions. If the survey is extended to fainter magnitudes, say I = 21 mag, the expected number of white dwarfs at 10g(ULo) = -5 increases to -1.5 for all the cases where the white dwarf luminosity function shows an extended tail. Finally, if the survey is performed in the V band the expected number of white dwarfs at 10g(ULo ) = -5 is -0.1 for a limiting magnitude of V = 20 mag and -0.5 for a limiting magnitude of V = 21 mag. Note, however, that it would be difficult to reconcile this scheme with the discovery of new white dwarfs at the same luminosities. Also note that the same would happen if the high densities (1 star per 1000 pc 3 ) suggested by Ruiz & Takarniya (1995) turned out to be correct. 5 CONCLUSIONS We have computed cooling sequences for ONe white dwarfs and we have analysed their impact on the luminosity function of disc white dwarfs. Our results indicate that the age of the disc derived from the observed white dwarf luminosity function is not modified significantly given a set of observational data, but an extended tail at very low luminosities appears. The existence of this tail is not excluded by observations and can help to explain why there are some disc white dwarfs at very low luminosities. The presence of this tail at very low luminosities is due to the very different cooling times of CO and ONe white dwarfs. The shape of the tail and, therefore, of the white dwarf luminosity function at low luminosities depends sensitively on the details of the white dwarf cooling theory and, consequently, requires more detailed,and realistic calculations of very cool white dwarf models. We emphasize the need for further improvements in our understanding of the properties of white dwarf envelopes at very low luminosities. ACKNOWLEDGMENTS This work has been supported by DGICYT grants PB and PB C02-02, by the CIRIT grant GRQ , by the AIHF 237-B and by the C 4 Consortium. We are indebted to Robert Mochkovitch for his very useful comments. REFERENCES Adams F., Laughlin G., 1996, ApJ, 468, 586 Alcock C. et al., 1995, ApJ, Ll25 Chabrier G., Segretain L., Mera D., 1996, ApJ, 468, L21 Diaz-Pinto A, Garda-Berro E., Hemanz M., Isem J., Mochkovitch K, 1994, A&A, 282, 86 Dupuis J., Vennes S., 1997, ApJ, 475, Ll31 Garda-Berro E., Then I., 1994, ApJ, 434, 306 Garda-Berro E., Hemanz M., Isem J., Mochkovitch K, 1988, Nat, 333, 642 Garda-Berro E., Hemanz M., Isem J., Chabrier G., Segretain L., Mochkovitch R., 1996, A&AS, 117, 13 Garda-Berro E., Ritossa C., Then I., Jr, 1997, ApJ, in press Gutierrez J., Garda-Berro E., Iben I., Jr, Isem J., Labay J., Canal K, 1996, ApJ, 459, 701 Hemanz M., Garda-Berro E., Isem J., Mochkovitch K, Segretain L., Chabrier G., 1994, ApJ, 434, 652 Then I., Jr, 1977, ApJ, 217, 788 Then I., Laughlin G., 1989, ApJ, 341,312 Isem J., Garda-Berro E., Hemanz M., Mochkovitch K, Burkert A, 1995a, in Koester D., Werner K., eds, White Dwarfs. Springer Verlag, Heidelberg, p. 19 Isem J., Hemanz M., Mochkovitch K, Burkert A., Garda-Berro E., 1995b, in Alfaro E., Delgado A., eds, The formation of the Milky Way. Cambridge Univ. Press, Cambridge, p. 179 Isem J., Hemanz M., Garda-Berro E., Mochkovitch K, Burkert A, 1997a, in Burkert A, Hartmann D. H., Majewski S. R., eds, ASP Conf. Ser. Vol. 112, The History of the Milky Way and its Satellite System. Astron. Soc. Pac., San Francisco, p. 181 Isem J., Mochkovitch K, Garda-Berro E., Hemanz M., 1997b, ApJ, in press (astro-phl ) Liebert J., Dahn C. c., Monet D. G., 1988, ApJ, 332, 891 Mazzitelli I., D' Antona F., 1986a, ApJ, 308, 706 Mazzitelli I., D'Antona F., 1986b, ApJ, 311, 762 Mazzitelli I., D' AntonaF., 1987, in Philip A G. D., Hayes S., Liebert J., eds, Proc. lau Colloq. 95, The Second Conference on Faint Blue Stars. L. Davis Press, Schenectady, p. 351 Mochkovitch K, Garcia-Berro E., Hemanz M., Isem J., Panis J. F., 1990, A&A, 233, 456 Nomoto K., 1984, ApJ, 277, 791 Nomoto K., 1987, ApJ, 322, 206 Oswalt T. D., Smith J. A, Wood M. A., Hintzen P., 1996, Nat, 382, 692 Ritossa C., Garda-Berro E., Then I., 1996, ApJ, 460, 489 Ruiz M. T., Takarniya M., 1995, AJ, 109,2817 Ruiz M. T" Bergeron P., Leggett S. K., Anguita C., 1995, ApJ, 455, Ll59 Salaris M., Dominguez I., Garcia-Berro E., Hemanz M., Isem J., Mochkovitch K, 1997, ApJ, in press (astro-phl ) Salpeter E. E., 1961, ApJ, 134,669 Segretain L., Chabrier G., 1993, A&A, 271, L13 Segretain L., Chabrier G., Hemanz M., Garda-Berro E., Isem J., Mochkovitch K, 1994, ApJ, 434, 641 Winget D. E., Hansen C. J., Liebert J., Van Hom H. M., Fontaine G., Nather K E., Kepler S. 0., Lamb D. Q., 1987, ApJ, 315, L77 Wood M. A., Winget D. E., 1989, in Wegner G., ed., Proc. lau Colloq. 114, White Dwarfs. Springer Verlag, Berlin, p. 282 Yungelson L. K, Livio M., Tutukov A. V., Satter R. A, 1994, ApJ, 420, 336 This paper has been typeset from ate XL ATE X file prepared by the author.