Annu. Rev. Astron. Astrophys. 1997. 35: 101-136 Copyright © 1997 by Annual Reviews. All rights reserved

4.3. Spaenhauer Diagrams and the Triple-Entry Correction by Sandage

The triple-entry correction (TEC) of Sandage (1994a, b) is an approach and method for bias recognition, derivation of unbiased TF relations, and calculation of unbiased Hubble constant, which in some respects differs from the method of normalized distances though is based on similar basic reasoning. The theory of the TEC method is given in a clear manner in the two articles by Sandage (1994a, b), and it is applied to the Mathewson et al (1992a, b) sample of 1355 galaxies in Federspiel et al (1994). The method, originally suggested by Sandage (1988b, 1988c), is based on Spaenhauer (1978) log V_o-M diagrams, which reveal how the average absolute magnitude M of a standard candle changes with increasing kinematical distance V_o when the magnitude limit cuts away a progressively larger part of the luminosity function.

Because these two approaches are the most developed ways of deriving the direct TF relation and the value of H_o from large field spiral samples, it is important to see clearly how they are connected. This is best done with the aid of the absolute magnitude M-vs-log V_o diagram (Figure 2, upper panel), similar to Figure 3 by Sandage (1994b), though with only two inserted Spaenhauer patterns corresponding to TF parameters p₁ and p₂. The inclined straight line corresponds to the magnitude limit, cutting away everything to the right of the line. The tips, i.e. the apex <M>, of the Spaenhauer patterns give the average TF magnitudes M₁ and M₂ for p₁ and p₂, respectively.

Figure 3. Schematic explanation of how the inverse TF relation (p vs M) may under ideal conditions overcome the Malmquist bias of the second kind. The nearby calibrator sample is made to glide over the distant sample so that the regression lines overlap. The condition for this is given by Equation 14, with µ = µest.

Of course, the data used in the MND can be given exactly the same Spaenhauer representation as in Figure 2. Then what happens when the data are tranformed into the form used in the MND? First, calculate the Hubble parameter log H for each galaxy, using an approximate TF relation. This shifts the tips of the Spaenhauer patterns to about the same vertical level in the log H-vs-log V_o diagram. Second, shift the kinematical distances (log V_o) of the p₂ class by the factor 0.2(M₂ - M₁) = 0.2a(p₂ - p₁). This normalization puts the bias curve for p₂ on top of the bias curve for p₁, and the Spaenhauer patterns lie one over the other (Figure 2, lower panel). The individual unbiased parts of the patterns thus amalgamate together to form the common unbiased plateau. Refering to Figure 2, the MND means letting the upper Spaenhauer pattern glide down along the limiting magnitude line until it settles over the lower pattern.

In both methods, the TF relation M = a ^. p + b is derived from the unbiased data: in the TEC using the undistorted part of the Spaenhauer pattern and in the MND using the unbiased plateau in an iterative manner. A difference is that in the MND, the dispersion is generally assumed to be a constant for all p, whereas the TEC allows different s for different p classes. The same basic idea lies behind the two, with TEC leaving the p classes separated where their individual behavior is better inspected, whereas MND unites them into one ensemble, so that the collective and common bias behavior is more easily seen. With the large KLUN sample now available, it is also possible with MND to conveniently cut the sample into many subsamples according to type, inclination, etc, in order to investigate in detail their influence. In this manner, the inclination correction and the type dependence have been studied by Bottinelli et al (1995), Theureau et al (1996), respectively. An indication of the close relationship between the MND and TEC are the similar values of H_o that were derived from field samples by Theureau et al (1997), Sandage (1994b), of 55 ± 5 and 48 ± 5 km/s/Mpc.

Finally, it should be noted that MND does not depend on any assumption on the space density distribution, as sometimes has been suspected. An advantage of both MND and TEC is that they are relatively empirical in essence: A minimum of assumptions is needed. More analytical methods, though needed in some situations, rely on ideal mathematical functions and assumed behavior of the selection function (Staveley-Smith & Davies 1989, Willick 1994) that often are not met.

Sandage (1994a) subtitled his paper "The Hubble Constant Does Not Increase Outward." He emphasized that inspection of the Spaenhauer diagrams of M_cal vs log for samples with different limiting magnitudes and different TF parameter p values allows one to exclude any such significant systematic deviation from the linear Hubble law as repeatedly proposed by (cf Segal & Nicoll 1996) with the quadratic redshift-distance law of his chronogeometric cosmology. Especially, addition of information in the form of several standard candles breaks the vicious circle, which allows one to interpret the data either in terms of the linear Hubble law plus selection bias or the quadratic law plus little selection bias (meaning very small dispersion of the luminosity function). The same thing can be said of the method of normalized distances: One would not expect the constant plateau for H, built by galaxies of widely different p values and widely different redshifts, if H_o actually does not exist. The method is also the same as adding a fainter sample of the same type of indicator and testing if the bias properties of the extended sample moves toward fainter magnitudes by the difference in the magnitude limits of the two lists Sandage (1988b) (see Figure 1b).