Annu. Rev. Astron. Astrophys. 1997. 35:
101-136
Copyright © 1997 by . 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 *p*_{1} and *p*_{2}. 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
M_{1} and M_{2} for *p*_{1} and
*p*_{2}, respectively.

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 *p*_{2} class by the factor 0.2(M_{2} -
M_{1}) = 0.2*a*(*p*_{2} -
*p*_{1}). This normalization
puts the bias curve for *p*_{2} on top of the bias curve
for *p*_{1}, 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 1*b*).