Annu. Rev. Astron. Astrophys. 1997. 35: 101-136
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The solution, Equation 1, proposed by Kapteyn for his Problem I, gives the distance to a cluster or to any subsample of galaxies known, e.g. from the Hubble law or other velocity field model, to be at the same distance. However, this presupposes that (a) there is a sharp magnitude limit, with data complete up to mlim, (b) the standard candle has a gaussian distribution G(Mp, sigma), (c) the mean Mp is known from local considerations (calibration), and (d) the dispersion sigma is known. These are rather strict conditions. Fortunately, there are situations and aims that do not necessarily require a complete knowledge of all these factors. For instance, study of the linearity of the Hubble law does not need an absolute calibration of the standard candle, and knowledge of sigma is not necessary in all methods for deriving the Hubble constant.

4.1. The Bias-Free Redshift Range

Sandage & Tammann (1975a) introduced the concept of the bias-free distance (redshift) range in their Hubble parameter H-vs-log Vo diagram for luminosity classified spiral galaxies, which showed an increase of H with redshift. They explained this increase as caused by the truncation effect of the limiting magnitude, which makes the derived distances too small.

One expects an unbiased region at small true distances (redshifts) because the sample can be distance-limited rather than flux-limited, hence no part of the faint end of the luminosity function is truncated at appropriately small distances. It is only here that the Hubble constant can be derived without correction for bias. Such a region of about constant H was clearly visible in the H-vs-log Vo diagram of Sandage & Tammann (1975). As a first approximation, they cut away galaxies more distant than a fixed Vo (approx 2000 km/s), independently of the morphological luminosity class. Actually, each luminosity class has its own limiting distance, as Sandage & Tammann recognized. Using one fixed Vo, one (a) loses high-Vo data, part of it possibly unbiased, and (b) allows a remaining bias due to intrinsically faint luminosity classes with their proper limit <V>o. This was inspected by Teerikorpi (1976), where from the ST data (and their luminosity class calibration) the low value of Ho = 41 was derived, in comparison with Ho = 57 by Sandage & Tammann (1975). I give this reference because it was a step toward the method of normalized distances, later applied to samples of galaxies with TF measurements. The similarity with Ho approx 43 by Sandage (1993) is probably not just a coincidence; both determinations relied on M101. An up-to-date discussion of the bias in the luminosity class method was given by Sandage (1996a).

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