Cooling of low-mass carbon-oxygen dwarfs from the planetary nucleus stage through the crystallization stage

The evolution of a carbon-oxygen dwarf of mass Mroughly-equal0.6 Msun has been carried all the way from an initial nuclear burning stage, when it is the central star of a planetary nebula, to the stage of complete internal crystallization, after 10/sup 10/ yr of cooling. Shell hydrogen and helium burning, neutrino losses, and the effects of liquification and crystallization have been taken into account. We show how the luminosity-time relationship may be understood in terms of balances between competing physical processes and demonstrate that, after complete crystallization, the time scale for cooling to terrestrial-like temperatures, in our approximation, is simply the optical depth of the outer, nonisothermal layer multiplied by a dimensional constant which, in years, is of the order of unity. A luminosity function based on the results covers the range -5< or approx. =log(L/Lsun)< or approx. =4 and agrees reasonably well with the observed luminosity function extending from the brighest planetary nebula nuclei to the dimmest observed white dwarfs, except perhaps for log(L/L/sub sun/)< or approx. =-4.5. Possible reasons for the apparent discrepancy at low luminosity, apart from the extreme obstacles against discovery, are discussed, one of the simplest is that the oldest dwarfs in the solar vicinitymore » are distributed over a distance from the galactic plane that is approx.5 times larger than is the case for the youngest dwarfs; another possibility is that the opacity in the outer layers of the oldest dwarf models has been overestimated (or underestimatedexclamation) by a factor of 5 or more.« less