A practical method to improve luminosity calibrations from trigonometric parallaxes

Abstract

A major problem in using trigonometric parallaxes is the systematic error in luminosity calibrations due to the combination of accidental errors of observation with the steeply sloping true parallax distribution. Lutz & Kelker have evaluated, for constant space density $$P(p)\propto p^{-4}$$, corrections $$\Delta M = M_\text{true}-M_\text{observed}$$. The corrections are large (∼ 0.5 mag), and require exceptionally precise parallax data $$(\sigma_p/p\lt1/5)$$. Detailed assessment of magnitude, motion and spectroscopic selection effects is required before the corrections can be applied. These difficulties may drastically reduce the utility of trigonometric parallaxes. A practical method to resolve the problem has been devised, using the observed distribution N(μ) of the proper motions. Given generally valid kinematical assumptions, N(μ) bears a simple power-law relation to P(p). The theoretical relation is verified from proper motion catalogue data. Thus the selection effects on P(p) can be determined from N(μ) and valid Lutz–Kelker ΔM corrections calculated. A simplified procedure for evaluating ΔM is presented. The N(μ) method is applied, as an example, to Sandage's globular cluster luminosity calibration from subdwarf parallaxes. Sandage's distance moduli are increased on average by 0.4 mag, and $$\langle{M}_\upsilon\rangle_\text{RR}=+0.40\pm0.2 $$ mag results from the improved calibration. Generally, the usefulness of parallax data, and the accuracy of luminosity calibrations, can be substantially improved through the N(μ) method.