White Dwarf Cosmochronometry. I. Monte Carlo Simulations of Proper-Motion ̶ and Magnitude-Limited Samples Using Schmidt’s 1/Vmax Estimator

Observationally, white dwarf stars are a remarkably homogeneous class with a minimum observed T _eff ~4000 K. Theoretically, the physics that determines their cooling timescales is relatively more straightforward than that which determines main-sequence evolutionary timescales. As a result, the white dwarf luminosity function has for the last decade been used as a probe of the age and star formation rate of the Galactic disk, providing an estimated local disk age of ~10 Gyr with estimated total uncertainties of roughly 20%. A long-standing criticism of the technique is that the reality of the reported downturn in the luminosity function (LF) hinges on just a handful of stars and on statistical arguments that fainter (older) objects would have been observed were they present. Indeed, the likely statistical variations of these small-number samples represent one of the primary uncertainties in the derived Galactic age, and the behavior of Schmidt's 1/ V _max estimator in this limit is not well understood.

In this work, we explore these uncertainties numerically by means of a Monte Carlo population synthesis code that simulates the kinematics and relative numbers of cooling white dwarfs. The “observationally selected” subsamples are drawn using typical proper motion and V -magnitude limits. The corresponding 1/ V _max LFs are then computed and compared to the input-integrated LFs. The results from our (noise-free) data suggest that (1) Schmidt’s 1/ V _max technique is a reliable and well-behaved estimator of the true space density with typical uncertainties of ~50% for 50 point samples and 25% for 200 point samples; (2) the age uncertainties quoted in previously published observational studies of the LF are consistent with uncertainties in the Monte Carlo results ̶ specifically, there is a ~15% and ≤10% observational uncertainty in the ages inferred from 50 point and 200 point samples, respectively; and (3) the large statistical variations in the bright end of these LFs ̶ even in the large- N limit ̶ preclude using the white dwarf LF to obtain an estimate of the recent star formation rate as a function of time.