The Luminosity Function of Magnitude and Proper-Motion Selected Samples. The case of White-Dwarfs

The luminosity function of white dwarfs is a powerful tool for studies of the evolution and formation of the Milky Way. The (theoretical) white dwarf cooling sequence provides a useful indicator of the evolutionary time scales involved in the chronometry and star formation history of the galactic disk, therefore, intrinsically faint (& old) white dwarfs in the immediate solar neighborhood can be used to determine an upper limit for the age of the galactic disk. In this paper we examine the faint-end ($M_V > +14$) behavior of the disk white dwarf luminosity function using the $1/V_{\rm max}$ method, but fully including the effects of realistic observational errors in the derived luminosity function. We employ a Monte Carlo approach to produce many different realizations of the luminosity function from a given data set with pre-specified and reasonable errors in apparent magnitude, proper-motions, parallaxes and bolometric corrections. These realizations allow us to compute both a mean and an expected range in the luminosity function that is compatible with the observational errors. We find that current state-of-the art observational errors, mostly in the bolometric corrections and trigonometric parallaxes, play a major role in obliterating (real or artificial) small scale fluctuations in the luminosity function. We also find that a better estimator of the true luminosity function seems to be the median over simulations, rather than the mean. When using the latter, an age for the disk of 10 Gyr or older can not be ruled out from the sample of Leggett, Ruiz, and Bergeron (1998).