These exciting developments have led to a renewed interest in white dwarf cooling calculations and model atmosphere calculations using upgraded input physics, and extending into the regime of very cool, evolved white dwarfs. Along with several groups in the world, my small team at Université de Montréal is currently building these ``next generation'' white dwarf models, and I would like to illustrate some representative results in this paper. My goal here is not so much to provide the reader with ``definitive answers'', but, rather, to demonstrate the potential of white dwarf cosmochronology.
White dwarf cooling theory presents many fascinating aspects from a physical point of view, but, for the present purposes, it is sufficient to study Figure 1 which illustrates the evolutionary tracks of 5 representative models of white dwarfs in the HR diagram. Except for their different mass, these models are similar in that they all consist of a pure C core, surrounded by a He mantle containing 10-2 of the total mass of the star, and an outermost H layer containing 10-4 of the total mass of the star. These values correspond roughly to the maximum amounts of H and He that can survive the previous, hot planetary nebula phase.
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| Figure 1: Evolutionary tracks (solid curves) of 5 (M = 0.4, 0.6, 0.8, 1.0, and 1.2 solar masses, from top to bottom) representative models of DA (or H-atmosphere) white dwarfs in the Hertzsprung-Russell diagram. Each model is a compositionally-stratified object made of a pure C core surrounded by a pure He envelope and an outermost pure H layer. The thick solid curves are isochrones. The number next to each isochrone gives the cooling time in units of Gyr. The small filled circles indicate the onset of crystallization at the center of each evolving model. The open circles indicate the onset of convective coupling between the surface and the thermal core. |
The detailed evolutionary tracks nearly follow curves of constant radii (straight lines with a negative slope in this log-log version of the HR diagram), particularly at low luminosities. Furthermore, these tracks indicate that the more massive white dwarfs are also those that are the smaller. This peculiar mass-radius relation is a consequence of electron degeneracy supporting the white dwarfs against gravitational collapse.
While the evolutionary paths of cooling white dwarfs in the HR diagram are extremely simple, it should be noticed that the cooling time to a given luminosity say, is not only a function of Teff, but also a stong function of the total mass of the star. This is shown in Figure 1 by the heavy curves representing isochrones. At relatively high luminosities, the plot shows that a more massive white dwarf takes longer to cool to a given Teff than a less massive object. The reason is that the more massive star has a larger energy reservoir: there are more C ions with energy kT. However, there is a dramatic reversal of behavior in the cooler phases of the evolution. Because of their larger masses and smaller radii, more massive white dwarfs have larger internal densities (for comparable temperatures) and, therefore, develop a crystallized core earlier, at higher effective temperatures. This is explicitly illustrated in Figure 1 where the small filled circle on each track indicates the onset of crystallization at the center of each evolving model. Note that crystallization is a first-order phase transition and, consequently, is accompanied by the release of latent heat. This extra source of energy produces a delay in the cooling of a crystallizing white dwarf.
More massive white dwarfs also reach earlier the state where the specific heat in the solid regime plunges to very small values, a phenomenon well explained within the framework of the simple Debye theory of solids in quantum statistical mechanics. In effect, matter under these conditions has lost its ability to store thermal energy, the energy reservoir of a white dwarf has become nearly empty, and the star must then disappear from sight in a relatively rapid and final phase sometimes referred to as ``Debye cooling''. Crystallization and subsequent Debye cooling are responsible for the ``accelerated'' evolution of the more massive models at low luminosities.
Beside crystallization, the second important event in the cooling history of a white dwarf is the onset of superficial convection. The open circles along each track in Figure 1 indicate where convective transport first becomes of importance in the evolution of the models. At high luminosities, the outer layers are fully radiative and possess the same character as the so-called ``radiative zero-like solutions'' of Schwarzschild (1958). This means that the evolution is completely insensitive to the details of the outer layers, including the atmosphere. With cooling, hydrogen, initially ionized in the outermost layers, recombines which leads to the formation of a superficial convection zone. However, it is only when the base of the H superficial convection zone reaches into the reservoir of thermal energy -and this corresponds approximately to the boundary of the degenerate core- that, for the first time, the surface becomes coupled to the reservoir. When that happens, i.e., when the envelope is fully convective from the surface down to the upper boundary of the thermal reservoir, the cooling rate then becomes coupled and strongly dependent on the details of the atmospheric layers.
The usefulness of white dwarfs as cosmochronometers has been firmly established more than a decade ago when Winget et al. (1987) first demonstrated that the white dwarf population in the solar neighborhood -a population characteristic of the galactic disk- could be used to estimate independently the age of the disk. Over the years, the method has been refined through improvements in the quality of the observational material available and improvements in the cooling models.
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| Figure 2: Comparison of the observed and theoretical luminosity functions of local white dwarfs and the age of the disk. The data points come from two separate studies, one based on a proper motion survey (Leggett et al. 1998) and the other on a colorimetric survey (Knox et al. 1999). The solid curves are theoretical luminosity functions computed on the basis of our recent cooling models with an assumed pure C core composition. These curves are normalized at a point near ~ 10-3.5 on the ascending branch corresponding to the average location of a small cluster of 4 observational points. Various ages for the white dwarf population in the disk, from 8 to 14 Gyr, are considered. |
In the spirit of this paper, Figure 2 illustrates how the age of the local disk can be estimated through a comparison of the observed and theoretical luminosity functions of local white dwarfs. On the observational side, Leggett, Ruiz, & Bergeron (1998) and Knox, Hawkins, & Hambly (1999) have recently published their studies of the luminosity function of white dwarfs in the solar neighborhood. The sample of Leggett et al. (1998) is the same one considered originally by Liebert, Dahn, & Monet (1988), except that the former authors have provided much improved estimates of effective temperatures, bolometric corrections, and absolute magnitudes. It contains 43 objects and constitutes a complete proper motion survey. In comparison, the Knox et al. (1999) is also a complete survey, but it is a colorimetric survey. It contains 58 objects.
I have rebinned the data of Leggett et al. (1998) in order to compare as closely as possible with the published data of Knox et al. (1999). This exercise has allowed me to produce 4 pairs of data points that may be compared directly, as shown in Figure 2. The first remarkable result of this comparison is that, given the usual uncertainties in this business, the data points in common between the two studies agree extremely well. The second noticeable result is that both surveys clearly suggest the existence of a bump in the luminosity function, peaking around 10-4 solar units in luminosity. Interestingly, this bump, or excess of white dwarfs, is naturally expected from theory -at least from our improved cooling models- and corresponds to the delays in cooling associated with the release of latent heat upon crystallization and the effects of convective coupling.
To show this, I plotted, in Figure 2, theoretical luminosity functions assuming various ages for the white dwarf population, from 8 to 14 Gyr, in steps of 1 Gyr. It is obvious from the figure that the theoretical expectations for the existence of a bump in the luminosity function are well borne out by the observations. This result can be taken as the first direct proof for the release of latent heat and for convective coupling in white dwarfs. This ``fine structure'' feature in the luminosity function of white dwarfs has never been discussed or even mentioned in the past, and its identification must be seen as a consequence of considerable improvements in the data and in our ability to model cooling white dwarfs.
The most important outcome of the present comparison between observed and theoretical luminosity functions, however, is the determination of the age of the disk. Although the Knox et al. study has a bright bin that happens to fall quite nicely on the ascending branch of the normalized theoretical luminosity functions, the Leggett et al. study is more useful in the present context as it provides a fainter bin that is critical for the actual comparison with the theoretical curves. Figure 2 indicates that the number density of local white dwarfs generally increases with decreasing luminosities until it reaches a maximum, followed by an important dropoff at still lower luminosities.
The simplest explanation for the observed dropoff of the density of white dwarfs at low luminosities, and the one that has been accepted quite generally, is that the first white dwarfs that were formed in the disk and that are now in our neighborhood, are still bright enough to be visible. Most of them, with representative or average masses have piled up at a luminosity ~ 10-4, while the more massive of them, much less numerous, have trickled down through Debye cooling to lower luminosities during the same time and populate the tail at the faint end of the luminosity function.
A comparison of the curves in Figure 2 with the observed points, particularly, the coolest bin, suggests an age of 11 Gyr or less for the local disk. It is important to realize here that this estimate is related to the assumption of a pure C core composition in the cooling models used in this illustrative example. A core composition containing a mixture of C and O, as is actually expected from stellar evolution theory, would lead to a smaller value than obtained here for the age of the disk. This is because the specific heat of a gram of oxygen is less than the specific heat of a gram of carbon under the fluid/solid physical conditions encountered in white dwarf interiors.
To study faint white dwarf populations in distant systems such as open and globular clusters, one often has to deal with what could be called ``minimal'' or 2-band photometry that produces a single color-magnitude diagram (CMD). Cooling theory can be used in conjunction with model atmospheres to compute the evolutionary tracks and plot isochrones in the CMD. However, the photometric scatter in such CMD's is generally large for white dwarfs and only qualitative results can be obtained from the direct comparison of isochrones with observational points in the CMD's. Clearly, the proper way to exploit the information contained in these diagrams is through actual stellar counts and the construction of observed luminosity functions.
The observational signature of the finite age of the white dwarf population in a cluster is the maximum in white dwarf density, the expected pile-up at some luminosity characteristic of the cluster, followed by a dropoff in number density at lower luminosities. Obviously, the sensitivity of the observations must be large enough to reveal this pile-up; otherwise, white dwarf cosmochronology can be used only to provide lower limits to the age of a cluster.
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| Figure 3: Comparison of the observed and theoretical white dwarf luminosity functions in the old open cluster M67. The data points are from Richer et al. (1998). The solid curves are theoretical cluster luminosity functions computed on the basis of our recent cooling models with an assumed pure C core composition. The curves are normalized to the second observed bin. Various ages for the white dwarf population in M67, from 2 to 6 Gyr, are considered. |
Figure 3 illustrates a nice example of a comparison between the observed and theoretical luminosity functions of white dwarfs in a cluster. The data points were taken from the impressive CFHT study of the old open cluster M67 by Richer et al. (1998). I have regrouped their data into wider bins to improve on the statistics (88 stars distributed in 4 luminosity bins), but, otherwise, these results come straight from Richer et al. (1998). The solid curves are expected luminosity functions for various assumed ages for the white dwarf population obtained on the basis of our recent models. These curves match very well the 3 observational points on the ascending branch, and the observed dropoff in the last bin can be used, as in the case of the disk, to estimate an age for the white dwarf population in M67. Here, the figure suggests an age slightly less than 6 Gyr.
As in the case of the disk discussed in a previous section, this latest result must be qualified as it is based on the assumption of pure C core white dwarfs. More realistic C/O core models would lead to a younger age for M67. I note also that if the turnoff age of M67 were to be known with great accuracy, then the problem could be turned around and one could adjust the core composition of the white dwarf models until the white dwarf age of M67 would match the turnoff age. This would be a wonderful result: the first determination of the mean core composition of white dwarfs in a given population. Given that the exact proportions of C and O in the cores of white dwarfs are still unknown (because of existing uncertainties in the rates of He thermonuclear burning), such a result could shed much needed light on this gray zone of thermonuclear astrophysics.
In the last few years, there has been mounting evidence in favor of the existence of a very old white dwarf population in the galactic halo, to the point where, today, there can be no doubt that halo white dwarfs do exist. In the context of white dwarf cosmochronology, this development is particularly exciting because such a population could be used, in principle, to obtain an independent estimate of the age of the halo. In addition, these old white dwarfs could contribute significantly -perhaps even in a dominant way- to baryonic dark matter in our galaxy, and, by extension, in other galaxies as well.
The first piece of evidence is the documented presence of fast movers in our neighborhood, white dwarfs that have very large space velocities and that are best interpreted as interlopers from the halo. In their proper motion survey, Liebert, Dahn, & Monet (1989) had identified 5 halo candidates, white dwarfs with tangential velocities larger than 250 km/s. I carried out a critical analysis of that sample in the light of the results of Leggett et al. (1998) which provide much improved estimates of the effective temperatures of these objects, and was left with two possible candidates: WD1022+009 and WD2316-064. With ~ 5000 K for both stars, those could belong to an ancient halo white dwarf population provided they are on the low side of the mass distribution, i.e., with masses around ~ 0.5 solar masses. This is certainly possible, but it would be nice to have reliable parallaxes for these two objects in order to determine their actual masses.
In a related effort, Ibata et al. (2000) have reported the discovery of two cool high proper motion white dwarfs in the solar neighborhood which they interpret also as interlopers from the halo. Amazingly, one of the Ibata et al. objects is a rediscovery of WD2316-064, one of the original ``halo sample'' of Liebert et al. (1989) and one of the two stars, at ~ 4740 K, that I retained as discussed just above. The other star, named F351-50, appears to be a genuine addition to the putative family of local halo white dwarfs. Our model atmosphere fits to the available photometry published in Ibata et al. (2000) suggest that it is a very cool DA white dwarf with ~ 3140 K. A parallax measurement would be necessary to estimate its mass and, ultimately, its age.
In a very significant series of papers (Hambly, Smartt, & Hodgkin 1997; Hambly et al. 1999; Hodgkin et al. 2000), a British group has presented a study of WD0346+246, another nearby, cool white dwarf with clear halo kinematic characteristics. Quite interestingly, these authors have measured a parallax for that star, and have published multiband photometry extending into the infrared domain. A preliminary model atmosphere fit to their data suggests that WD0346+246 has ~ 3820 K and ~ 0.8 solar masses. On the basis of the pure C core models used in this paper, this leads to an age of ~ 12.7 Gyr for WD0346+246, significantly larger than the estimated age of 11 Gyr for the local disk, as discussed above.
The second piece of evidence in favor of an old population of white dwarfs in the galactic halo is more indirect in nature and comes from the MACHO microlensing experiment. I reproduce here a key paragraph taken from one paper published less than a year ago: ``The most straightforward interpretation of the results is that MACHOS make up between 20% and 100% of the dark matter in the halo, and that these objects weight about 0.5 solar masses. Objects of substellar mass do not comprise much of the dark matter.'' (Alcock et al. 1999). Cool white dwarfs are, of course, the most likely candidates for subluminous objects that ``weight'' ~ 0.5 solar masses.
The third development is due to Hansen (1998) who published a key paper pointing out that some of the so-called blue unidentified objects in the Hubble Deep Field (HDF) could be associated with very old DA white dwarfs in the halo. The presence of such objects there would be in line with the results of the MACHO experiment, but a proof was required to be certain that some of the blue unidentified objects in the HDF are truly halo white dwarfs, namely, they should show small, but measurable proper motions. Ibata et al. (1999) followed up on this idea and, using second-epoch HDF exposures, they reported the exciting discovery of detectable proper motions in up to 5 ``blue unidentified objects'' in the HDF. Our analysis of the available photometry of these extremely faint objects (I ~ 28) indicates that two of the Ibata et al. (1999) high proper motion objects have energy distributions compatible with those of very cool DA white dwarfs. We find that 4-492 has Teff ~ 2600 K and is located at ~ 1.6 kpc, while 4-551 has Teff ~ 2300 K and is located at a distance of ~ 0.8 kpc. The masses of these stars cannot be estimated on the basis of the observations currently available, so their ages cannot be inferred at the moment. Nevertheless, the case for 4-492 and 4-551 as genuine halo white dwarfs appears quite strong.
White dwarf cosmochronology is still in its infancy, but it already shows its potential as a powerful tool for estimating the ages of the various components of the Galaxy. Models of cooling white dwarfs, particularly the more recent ones extending into the interesting but difficult regime of very low effective temperatures, need to be improved. Nevertheless, the method already confines the age of the local disk to the 9-11 Gyr interval, with the largest source of uncertainties coming from the unknown exact proportions of C and O in the cores of field white dwarfs. A very preliminary estimate of the age of the halo based on white dwarf cosmochronology, as first presented here, suggests a significantly larger age than the disk, about 12.7 Gyr for the halo compared to ~ 11 Gyr for the disk (using the same pure C core models). Finally, in the era of giant 8-10 m ground-based telescopes, the potential of the method for dating open and globular clusters is immense! We are just beginning...
It is a pleasure for me to thank my young colleagues Pierre Brassard and Pierre Bergeron for their continued and essential help in this exciting venture of white dwarf cosmochronology.
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Gille Fontaine Département de Physique, Université de Montréal Electronic-mail: <fontaine@astro.umontreal.ca> I have been able to pursue a career in astronomy because of the unfailing support of Hugh Van Horn (U. of Rochester), John Landstreet (U. of Western Ontario), and Georges Michaud (U. de Montreal) who have been my mentors and models. After some 4 years as a postdoctoral fellow, during which I acquired the basic skills of observational astronomy to complement my formal training in theoretical stellar astrophysics, I started my academic career at Université de Montréal in 1977. In collaboration with my colleague Francois Wesemael who joined me in 1981, I founded a small research group interested mostly in the late phases of stellar evolution (the hot subdwarf and white dwarf stars) and in asteroseismology. Over the years, we have been very fortunate to attract excellent students, several of which being simply outstanding. With their help, we have been able to contribute to all aspects, observational and theoretical, of subdwarf and white dwarf research. This research team is the achievement that I am the most proud of; and the adventure continues... On the more personal side, I have been married for almost 31 years to ...the same woman (an obviously very patient and supportive person!). I still play hockey twice a week -this is truly my "religion"-, and I ride a tandem bike with my wife during the summer months. I play in a blues band with some ex-students, Pierre Bergeron, Alain Beauchamp, and, formally, Pierre Chayer (now at FUSE in Baltimore). I also thoroughly enjoy playing old irish, scottish, and french canadian reels and gigs on my accordeon, a style that my bluesman brothers do not approve of! Once in a while, I play in old folks retirement homes where I find sympathetic (if captive!) audiences for my accordeon playing. |
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