The calibration of luminosity criteria
Date
1968
Authors
Clutton-Brock, Martin
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Abstract
All the information about the distance of a star given by the observational data is contained in the likelihood function. Efficient statistical estimation must, therefore, start with the likelihood. The meaning of the likelihood, and its application to estimation, is reviewed. A Bayesian approach is adopted, and it is shown how nuisance parameters can be eliminated using Bayes's theorem.
The calculation of the likelihood of the distance, given the parallax and the proper motion, is described. The parallax sets a lower limit to the distance, but no upper limit; the proper motion is essential if an upper limit is to be found. The proper motion contribution to the likelihood is obtained from the velocity distribution of stars relative to the sun, using the substitution.
Tangential velocity = proper motion x distance. The presence of errors in the proper motion is unimportant for relatively nearby stars with reasonably large proper motions. It is shown how to calculate the likelihood for more distant stars for which the effect of errors in the proper motions is not negligible.
The quoted probable errors of the parallaxes bear little relation to the external errors, judged from comparisons of independent parallaxes for the same star. Estimates are made of the external standard deviations of the parallaxes of nine observatories, based on data given by Dahlgren and Vasilevskis. The results, in units of 0.001, are:
Allegheny, 12.59 ± 0.45; Cape, 20.39 ± 0.66; Greenwich, 14.65 ± 1.26; McCormick, 16.39 ± 0.43; Mt. Wilson, 24.56 ± 1.01; Sproul, 16.12 ± 0.78; Van Vleck, 13.72 ± 1.16; Yale, 14.70 ± 0.60; Yerkes, 18.51 ± 0.78.
The practice of selecting stars for observation with large parallaxes leads to a systematic bias, which may be overcome by proper normalization of the likelihood.
The calibration of a luminosity criterion essentially involves the estimation of the parameters of the distribution of the luminosity criterion, conditional on the absolute magnitude and the color. The likelihood of the parameters, based on observations of the luminosity criterion, color, and parallax and proper motion, must be found by eliminating the true values of the absolute magnitude as "nuisance parameters". The estimates of the parameters are found by locating the maximum of the likelihood. This leads to a set of non-linear equations, which can be solved by iteration.
The method is applied to the calibration of the H and K emission line widths and the Fe λ 5250 triplet. The results for the H and K lines are compared to those obtained by a least squares calculation, and it is shown that the least squares result contains a systematic error, so that the luminosity of giant stars is underestimated by 0.6 magnitudes. This supports the suggestion of Hodge and Wallerstein that the distance modulus of the Hyades cluster is in error.