The Luminosity Function of Magnitude and Proper - Motion Selected Samples

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. Most determinations of the white dwarf luminosity function have relied on the use of Schmidt’s 1/Vmax (Schmidt 1975) method for magnitude and proper-motion selected samples, the behavior of which has been demonstrated to follow a minimum variance maximum-likelihood pattern for large samples. Additionally, recent numerical simulations have also demonstrated that the 1/Vmax provides a reliable estimator of the true LF, even in the case of small samples (Wood and Oswalt 1998, and Garćıa-Berro et al. 1999). However, the conclusions from all these previous studies have been based on noise-free data, where errors in the derived LF have been either assumed to follow a Poisson distribution (valid only for large samples), or where other simple estimates for the uncertainties have been used. In this paper we examine the faint-end (MV > +14) behavior of the disk white dwarf luminosity function using the 1/Vmax 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, propermotions, 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.