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 timescales involved in the chronometry and star formation history of the galactic disk; therefore, intrinsically faint (and 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/V-max (Schmidt) 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/V-max provides a reliable estimator of the true LF, even in the case of small samples (Wood & Oswalt; et al.). However, the conclusions from all these previous studies Garcia-Berro 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 behavior of the disk white dwarf (M-V +14) luminosity function using the 1/V-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 prespecified 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 oversimulations, rather than the mean. When using the latter, an age of 10 Gyr or older can not be ruled out from the sample of Leggett, Ruiz, & Bergeron.