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

**4.2. The Method of Normalized Distances for Field Galaxies**

Assume that we have two standard candle (galaxy) classes, each having
gaussian luminosity functions *G*(M_{1},
) and
*G*(M_{2},
), hence simply shifted
in M by
M = M_{1}
- M_{2}. If both are sampled up to a sharp magnitude limit
m_{lim}, it is easy to see that in the bias vs true distance
modulus *µ* diagram, the bias of the second kind suffered by
these two candles is depicted
by curves of the same form but separated horizontally by constant
*µ* =
-M (see
Figure 1*a*). The curve
of the brighter candle achieves only at larger distances
the bias suffered by the fainter candle already at smaller distances. In
this way, simultaneous inspection of two or more standard-candle classes
gives a new dimension to the problem of how to recognize a bias.
Figure 1*b* shows another
important property of the bias behavior. If one keeps
the standard candle the same but increases the limiting magnitude by
m_{lim}, the
bias curve shifts to larger distances by
*µ* =
m_{lim}.
This is the basis of what
Sandage (1988b)
calls the "adding of a fainter sample" test.

van den Bergh's morphological luminosity classes clearly showed this
effect
(Teerikorpi 1975a,
b),
and even gave evidence of the "plateau" discussed below. In the beginning
of 1980s, extensive studies started to appear where the relation
(Gouguenheim 1969,
Bottinelli et al
1971,
Tully & Fisher
1977)
between the magnitude (both B and infrared) and maximum rotational
velocity of spiral galaxies V_{max} was used as a distance
indicator. In the following, the direct regression (M against fixed log
V_{max}
*p*) form of this TF relation is written as

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There was at that time some uncertainty about which slope to use in
distance determinations to individual galaxies - direct, inverse, or
something between? Then
Bottinelli et al (1986)
argued that in order to control the Malmquist bias of the second kind, it
is best to use the direct slope, so that the regression line is derived as
M against the fixed observed value of *p*, without attempting to
correct the slope for the observational error in *p*. The observed
value of *p* devides the sample into separate standard candles
analogous to Malmquist's star classes. A similar conclusion was achieved
by Lynden-Bell et al
(1988) in connection with the first kind of bias.

The importance of the direct relation is the fact that
this particular slope allows one to generalize the mentioned example of two
standard candles to a continuum of *p* values and in this way to
recognize
and investigate how the Malmquist bias of the second kind influences the
distance determinations [that such a bias must exist also in the TF method
was suspected by
Sandage & Tammann
(1984)
and introduced on a theoretical basis in
Teerikorpi
(1984)].
If one inspects the whole sample (all *p* values clumped together)
in the diagnostic log H vs *d*_{kin}
diagram (cf Figure 1), the
bias may not be very conspicuous. On the other hand, if one divides the
sample into narrow ranges of *p*,
each will contain a small number of galaxies, which makes it difficult to
see the behavior of the bias for each separate standard candle within
*p* ± 1/2*dp*. For these reasons, it is helpful to
introduce so-called normalized distance *d*_{n}
(Teerikorpi 1984,
Bottinelli et al
1986),
which transforms the distance axis in the log H vs *d*_{kin}
diagram so that the separate *p* classes are shifted one over the
other and the bias behavior is seen in its purity (see
Figure 2 in this regard):

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In fact, one might also term this transformed
distance as the effective one. This method of normalized distances (MND)
uses as its starting point and test bench an approximative kinematical
(relative) distance scale (*d*_{kin}, e.g. as provided by
the Hubble law or Virgo-centric models) used with
observed redshifts. The method usually investigates the bias as seen in the
Hubble parameter log H, calculated from the (direct) TF distance for each
galaxy using the (corrected) radial velocity. If the Hubble law is valid,
i.e. there exists a Hubble constant H_{o}, then one expects an
unbiased plateau at small normalized distances, a horizontal part from
which the value of H_{o} may be estimated from the plot of the
apparent H_{o} vs *d*_{n}.
Bottinelli et al
(1986)
applied the method to a sample of 395 galaxies having B magnitudes and
the TF parameter *p* and could identify clearly the plateau and
determine H_{o} from it. Certain subtle points of the method
were discussed by
Bottinelli et al
(1988a),
who also presented answers to the criticism from
de Vaucouleurs &
Peters (1986),
Giraud (1986).
A somewhat developed version of it has recently been applied to the KLUN
(kinematics of the local universe) sample constructed on the basis of the
Lyon-Meudon extragalactic database and containing 5171 galaxies with
isophotal diameters *D*_{25}
(Theureau et al
1997).