2002AJ....123.2047S

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 & Weaver, Wallerstein, Ljunggren & Oja, West, Lutz & Kelker, after whom the bias is named, Turon Lacarrieu & 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 that 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, which 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 cutoffs as they are progressively imposed on the catalog. We calculate the bias properties using simulations involving 3x10<sup>6</sup> stars of fixed absolute magnitude using M<sub>v</sub>=+0.6 to imitate RR Lyrae variables in the mean. These stars are spread over a spherical volume bounded by a radius 50,000 pc with different spatial density distributions. The bias is demonstrated by first using a fixed rms parallax uncertainty per star of 50 µas and then using a variable rms accuracy that ranges from 50 µas at apparent magnitude V=9 to 500 µas at V=15 according to the specifications for the Full-Sky Astrometric Mapping Explorer (FAME) 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 that is free of the Lutz-Kelker bias, although a Malmquist bias is present. Simulations show the power of the inverse method. Nevertheless, we recommend reduction of the data using both methods. Each must give the same answer if each is freed from systematic error. Although the maximum likelihood method will, in theory, eliminate many of the bias problems of the direct method, nevertheless the bias corrections required by the direct method can be determined empirically via Spaenhauer diagrams immediately from the data, as discussed in the earlier papers of this series. Any correlation of the absolute (trigonometric) magnitudes with the (trigonometric) distances is the bias. 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 Lyrae stars closer than 1.5 kpc. Of course, use will also be made of the entire FAME database for the RR Lyrae stars over the complete range of distances that can be used to empirically determine the random and systematic errors from the FAME parallax catalog, using correlations of derived absolute magnitude with distance and position in the sky. These bias corrections are expected to be much more complicated than only a function of apparent magnitude because of various restrictions due to orbital constraints on the spacecraft.