Bias Properties of Extragalactic Distance Indicators.XI. Methods to Correct for Observational Selection Bias for RR Lyrae Absolute Magnitudes from Trigonometric Parallaxes Expected from the FAME Astrometric Satellite

A short history is given of the development of the correction for observation selection bias inherent in the calibration of absolute magnitudes using trigonometric parallaxes. The developments have been due to Eddington, Jeffreys, Trumpler and Weaver, Wallerstein, Ljunggren and Oja, West, Lutz and Kelker after whom the bias is named, Turon Lacarrieu and Crézé, Hanson, Smith, and many others. As a tutorial to gain an intuitive understanding of several complicated trigonometric bias problems, we study a toy bias model of a parallax catalog which incorporates assumed parallax measuring errors of various severities. The two effects of bias errors on the derived absolute magnitudes are (1) the LutzKelker correction itself that depends on the relative parallax error δπ/π and the spatial distribution, and (2) a Malmquist-like ‘incompleteness’ correction of opposite sign due to various apparent magnitude cut-offs as they are progressively imposed on the catalog. We calculate the bias properties using simulations involving 3 × 10 stars of fixed absolute magnitude using Mv = +0.6 to imitate RR Lyrae variables in the mean. These stars are spread over a spherical volume bounded by a radius 50,000 parsecs with different spatial density distributions. The bias is demonstrated by first using a fixed rms parallax uncertainty per star of 50μas , and then using a variable rms accuracy that ranges from 50μas at apparent magnitude V = 9 to 500μas at V = 15 according to the specifications for the FAME astrometric satellite to be launched in 2004. The effects of imposing magnitude limits and limits on the ‘observer’s’ error, δπ/π, are displayed. We contrast the method of calculating mean absolute magnitude directly from the parallaxes where bias corrections are mandatory, with an inverse method using maximum likelihood which is free of the Lutz-Kelker bias, although a Malmquist bias is present. Simulations show the power of the inverse method. Nevertheless,