A practical method to improve luminosity calibrations from trigonometric parallaxes

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) p^-4|⁠, corrections | M = Mₜrue-Mₒbserved|⁠. The corrections are large (∼ 0. 5 mag), and require exceptionally precise parallax data | (ₚ/p1/5) |⁠. 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 |M_RR=+0. 400. 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.