Four reasons combine to make the white dwarf stars excellent chronometers for measuring the age of the various stellar populations of the galaxy. The first reason is that the white dwarf stars are representative of the general population of stars; most stars are, or will become, white dwarf stars. Second they are a homogenous class of objects with a very narrow distribution of total stellar masses between $0.4
\lesssim M_{wd}/M_\odot \lesssim 1.1$ and a mean mass of $\left< M/M_\odot
\right> = 0.590 \pm 0.134$ (Bergeron et al. 1995). Further, on theoretical grounds (see Iben 1991 for an excellent summary), we expect all of the objects with masses in the range $0.5 - 1.1\ M/M_\odot$ to have C/O cores, with thin overlying He-layers, and most an additional surface H-layer. Third, high surface gravity $\log\ g \sim 8$, typically slow rotation rates and low magnetic fields, nuclear energy and gravitational energy generation rates near zero, argue that these objects are physically simple. Fourth, the unavailability of energy sources other than the residual thermal energy of the ions implies that the evolution of white dwarf stars is primarily a simple cooling problem; there is a straightforward relation between the age of a white dwarf and its luminosity. These factors combine to imply that the white dwarf stars provide an archeological record of the history of star formation in the galaxy.

Two excellent recent reviews of this subject are Chabrier (1998) and Isern (1999). I refer you to these sources for a different perspective, and more complete treatment of the problems of chemical fractionation in the cool white dwarf stars.

White dwarf evolution as a cooling problem

Many years ago, Mestel (1952) demonstrated the simplicity of the evolution of a star supported by electron degeneracy pressure and without significant nuclear energy sources. He constructed an analytical model for white dwarf evolution by treating the structure as if it consisted of only two layers. The inner layer contains most of the mass of the star, and is assumed to be isothermal because of strong e--degeneracy. For the same reason, the electrons don't contribute significantly to the heat capacity; it comes almost entirely from the ions, which are assumed to behave as a classical ideal gas. The thin non-degenerate outer layer forms an insulating blanket and controls the rate at which the energy from the ion reservoir is leaked out into space. The specific rate is controlled by the radiative opacity at the boundary between these two layers, and is assumed to obey Kramer's law, $\kappa = \kappa_o\rho
T^{-3.5}$. The Mestel theory has been nicely summarized by Van Horn (1971), Wood (1990) and most recently by Kepler and Bradley (1995). As expected it gives a simple form for the age-luminosity relation,

\begin{displaymath}log( \tau_{cool}) \approx Const. - \frac{5}{7}\ log(L/L_\odot).\end{displaymath}

This analytical model for white dwarf cooling is both a good picture to have in your head, and as shown by Iben and Tutukov (1984), gives surprisingly good agreement with the predictions of detailed numerical models. We can think of the key physical effects not included in the Mestel theory as perturbations to Mestel theory to a good approximation. Or, we can even construct a modified Mestel theory including them (see for example Wood 1990).

As discussed by Lamb & Van Horn (1975), we can expect five key physical effects to produce deviations from the predictions of classical Mestel theory. These are neutrino energy loss, possible residual nuclear burning in the H-layer (dismissed by Lamb & Van Horn and previous authors, but see Iben & Tutukov 1984, and Iben & MacDonald 1986), gravitational contraction, surface convection, and crystallization. In the latter we include the closely related phase separation as well as the subsequent Debye cooling (Cvion is proportional to T3).

The white dwarf luminosity function and the age and history of the galactic disk

The idea of using white dwarf stars as chronometers has a long history beginning with Mestel (1952). Schwarzshild (1958) wrote, "We see that the cooling times we have derived in this manner are of just the right order of magnitude. The cooling times are sufficiently short so that many stars have had time enough to reach those values of the internal temperature which... give the observed low-luminosities of the white dwarfs. At the same time, the cooling times are sufficiently long so that a cooling sta r will not become unobservably faint too quickly." This made clear that if we could observationally determine the population of the coolest white dwarf stars, we would have the oldest white dwarf stars (allowing for mass effects), and thereby a constraint on the ages of the oldest stars. This idea was further developed by Schmidt (1959) in a galactic context, and later by D'Antona & Mazitelli (1978).

This concept was challenged by Ostriker and Axel (1969), which we recall were the first to consider the effects of Debye cooling on white dwarf evolution. They pointed out that the onset of Debye cooling would produce a sharp decrease in the number of cool white dwarf stars observed below $M_v \approx 13 \rightarrow 14$. It is important to note that this value was based on calculations for white dwarfs of $1 M_\odot$, which was believed at the time to be a typical white dwarf mass (Greenstein & Trimble 1967). We thought that Greenstein (1969) had identified the coolest population of white dwarf stars. This later turned out to be an observational selection effect, but it was an extremely important result for the field in that it stimulated the future search by Liebert and collaborators (Liebert 1979, Liebert, Dahn & Monet 1988), which did find a downturn in the white dwarf luminosity function.

At the 1979 meeting in Rochester, N.Y. (I.A.U. Colloquium No. 53 entitled "White Dwarfs and Variable Degenerate Stars") Jim Liebert (Liebert 1979) announced the new evidence for the downturn in the white dwarf luminosity function as a paucity of cool degenerates. He pointed out that this could be the result of accelerated cooling, or because of the finite age of the disk, but it was not an observational selection effect. Recently completed evolutionary calculations by Van Horn, Lamb, and Winget indicated a cooling age in the vicinity of this downturn to be 9-10 Gyrs. During the course of the meeting I drew this to the attention of Jim Liebert, Gilles Fontaine, and my thesis advisor, Hugh Van Horn. We realized that this could reflect the finite age of the disk, and give us a meaningful measure of its age. Other investigators, however, were finding much younger ages for white dwarf stars at the luminosity where Liebert observed the downturn (D'Antona & Mazzitelli 1978 Shaviv & Kovetz 1976) and so the prevailing opinion at the meeting was the paucity was not due to the age of the disk, but due instead to Debye cooling (D'Antona & Mazzitelli 1978, and the later work of Mazzitelli & D'Antona 1986).

Improved observational results (Liebert, Dahn, and Monet 1988, 1989 hereafter both papers are referred to as LDM), and further calculations of theoretical white dwarf luminosity functions incorporating a distribution of masses and an estimate of the average main-sequence evolutionary time, convinced us that the turndown Liebert had found was indeed a measure of the age of the galactic disk. Later, at the urging of Gerry Wasserberg, we published this method of cosmochronometry along with a preliminary result of $\tau_{disk} \approx 9.3 \pm 2.0$ (Winget et al. 1987). This work also demonstrated the possibility of using the shape of the luminosity function prior to the turndown to constrain constitutive physics in the models and the star formation history of our galaxy. Icko Iben refereed this paper, and after carefully examining the methods and results, he concluded that this was correct. He published (Iben & Laughlin 1989) a careful re-examination of this method of obtaining a similar age for the galactic disk using the evolutionary models of Winget et al. as well as Mestel theory. This work was more sophisticated than Winget et al. in that they modeled the initial-final mass relation and main-sequence lifetimes as a function of mass.

There was still a problem in this method of obtaining an age for the galactic disk, in that there is a broad dispersion in ages for the coolest white dwarf stars depending on which group carried out the evolutionary calculations. Winget and Van Horn (1987) demonstrated that this variation was due to very different treatments of the constitutive physics (for example inclusion of crystallization, treatment of convection), or different choices of model parameters (stellar mass, or choice of surface layer masses). When these effects were taken into account they demonstrated remarkable agreement between the results of various investigators. Some still believed, however, that the downturn could still be the result of short cooling times and subsequent Debye cooling, as indicated by the comments of D'Antona following the paper by Winget and Van Horn (1987,p. 376).

Further theoretical calculations employing more sophisticated treatments were carried out by Weidemann & Yuan (1989), Noh & Scalo (1990) and culminated in the work of Wood (1990, 1992, and 1995). By this time there was a consensus that the white dwarf luminosity function gave one of the most accurate measures for the age of the galactic disk.

With all this effort, and enormous improvements in the treatment of the evolutionary models and the constitutive physics it is important to see how far we have come since the beginning of this field more than 40 years ago. To establish that, it is amusing to compare the current results with those published for the age of the lowest luminosity white dwarfs by Schwarzschild. For the most current results, we quote the results of Leggett, Ruiz, & Bergeron (1998; hereafter LRB). They used the LDM sample supplemented with new optical and infrared data, and derived temperatures from the dramatically improved cool model atmospheres of Bergeron, Saumon, & Wesemael (1995). Using Wood's theoretical models they obtain

\begin{displaymath}\tau_{disk}(1988) = 8 \pm 1.5\ {\rm Gyr},\end{displaymath}

while Schwarzschild, armed with Mestel theory, obtained,

\begin{displaymath}\tau_{disk} (1958) = 8 \times 10^9\ {\rm yr}.\end{displaymath}

With tongue in cheek, we note that in 40 years we have gained both error bars, and a new expression for units. In a more serious vein we must point out that with the same LRB data, and the phase-separation models of Montgomery (1998) or Salaris et al. (1997) the value for the age becomes 9 Gyr. Also, an entirely independent sample of cool white dwarfs in wide binaries reported by Oswalt et al. (1996) gives an age of 9.5 Gyr with the Wood (1995) models, or 10.5 Gyr with the Montgomery (1998) or Salaris et al. (1997) phase separation models.

Clearly there is much more to the story but this remarkable consistency underscores why the white dwarf stars are such useful chronometers: they are simple!

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