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

Abstract

Please note that this abstract has been shortened from that found in the paper. A brief history is given of the development of the correction for observation selection bias inherent in the calibration of absolute magnitudes using trigonometric parallaxes. 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 Lutz-Kelker correction itself that depends on the fractional parallax error 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. The simulations involve 3 million stars spread with varying density distributions in a volume bounded by a radius of 50,000 pc. A fixed absolute magnitude of M_V = +0.6 is used to imitate RR Lyrae variables in the mean. The bias is demonstrated for various assumed spatial densities and parallax error models, including one that is projected for the FAME satellite. The effects of imposing magnit ude limits and limits on the `observer's' error 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. We discuss the level of accuracy that can be expected in a calibration of RR Lyrae absolute magnitudes from the FAME data over the metallicity range of [Fe/H] from 0 to -2, given the known frequency of the local RR Lyraes closer than 1.5 kpc.