7.2.4 Malmquist Effects

There has been a great deal of discussion in the literature during the past decade concerning the significance of selection effects in the determination of extragalactic distances, particularly for those obtained via TF relations. Since the TF relations are applicable to spiral and irregular galaxies, they have until recently been the only quantitative distance indicator which had been applied to a sufficient sample of galaxies such that the effects of Malmquist ``bias'' on distance determinations can be adequately tested.

The samples in question are typically selected on the basis of apparent magnitude. In such samples, the resulting luminosity function is progressively biased toward intrinsically brighter systems with increasing mean distance. This bias is usually referred to as a ``Malmquist effect'', although this is not quite the situation Malmquist (1920) described. Tammann (1987) and Sandage (1988), for example, argue that the distances are biased, leading to an apparently small dispersion of the TF relations. The fact that the luminosity function of an apparent magnitude limited sample is progressively more biased for ever increasing distance is without doubt (e.g., Sandage et al. 1979). At issue is whether the estimated distances for such a sample are biased.

Additional confusion has arisen when analytical solutions to similar bias problems (e.g., Teerikorpi 1984, 1987; Bottinelli et al. 1986; Lynden-Bell et al. 1988) are applied to the TF samples. However, those solutions may suffer from simplifying assumptions, or they may not be general solutions. In fact, the problem described by Malmquist (1920) is only a special case of the overall problem of estimating astronomical distances.

For example, errors in predicted distance modulus can arise from both random and systematic errors. Purely random errors obviously do not result in biased distance estimates, unless they propagate into both the sample selection criteria and the estimated distance moduli. An example of this would be the use of cataloged apparent magnitudes to both select the sample and estimate individual distance moduli. Note, however, that this coupling can be broken if new photometry is obtained for a sample selected on the basis of cataloged photometry.

An example of a systematic error with distance is that described originally by Malmquist (1920). In this case, an observable (i.e., spectral type) is used to predict the mean absolute magnitude of a particular class of stars. This predicted absolute magnitude is then used to estimate distance moduli, even though the sample selection criteria have already biased the distribution in absolute magnitude, such that the mean of the parent population is not the mean of the selected sample. However, this need not be case for all apparent magnitude selected samples.

Simulations of the TF relations demonstrate that for the case in which the dispersion in inclination corrected line-width dominates the dispersion in apparent magnitude (e.g., Bothun and Mould 1987), either due to an intrinsic scatter or due to observational errors in W₂₀ and inclination, then there is essentially no bias introduced into the TF distances due to selecting the sample on the basis of apparent magnitude. The fact that the true luminosities of the sample members are biased becomes irrelevant, since the unbiased predictor of luminosity (i.e., W_Rⁱ) will produce an unbiased predicted luminosity. That is, the sample members which are at larger distance, and hence more luminous than would be expected for members of an unbiased sample will have line-widths which predict them as such. As a result, the predicted distances for such a sample are unbiased. This point was first discussed by Schechter (1980) in the context of the Faber-Jackson relations for elliptical galaxies Faber and Jackson 1976) and later by Pierce and Tully (1988), and Tully (1988) in the context of the TF relations.