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

**3.3. Situations Where the Second Malmquist Bias Is Important**

A simple way to see the second kind
of Malmquist bias in action is to take the sample above, calculate log H
for each galaxy, and plot log H against redshift. As demonstrated in several
studies during the last twenty or so years, log H stays first roughly
constant
and then starts to increase. So clear and dramatic is this phenomenon, and
also so much expected from simple reasoning as due to bias, that it is
appropriate to repeat the words of
Tammann et al
(1980): "If an author finds H_{o}
to increase with distance he proves in the first place only one thing, i.e.,
he has neglected the Malmquist effect! This suspicion remains until he has
proved the contrary." To this statement, one should add that here "distance"
means the true distance or at least a true relative distance (e.g. redshift
in the case of the Hubble law). If "distance" is the measured or inferred
distance, H_{o} does not necessarily change with the distance, though it may have a wrong average value - we
have come back to the first kind of bias. This also serves as a warning that
simple comparison of the "linearity" of photometric distances from two
methods may hide a common bias, as seen in the comparisons of two or
more distances by separate indicators discussed by
de Vaucouleurs
(1979).
Each of the indicators suffers the same type of bias properties.

It seems that
Behr (1951)
was the first to point out, after comparison of the width of the Local Group
luminosity function to that of the field galaxies, that application of the
standard candle method may lead to systematically short distances at large
true distances. As we discussed above, this idea was transformed into a
quantitative model by
Scott (1957)
for the selection of brightest cluster galaxies. It was then reinvented by
Sandage & Tammann
(1975a),
Teerikorpi (1975a,
b),
in connection with concrete field samples of spiral galaxies with
van den Bergh's
(1960a,
b)
luminosity classes. The basic reasoning is illustrated by the formula that
connects the derived distance modulus to apparent magnitude, the assumed
standard absolute magnitude M_{p}, and to the magnitude limit
m_{lim}:

(10) |

Clearly, there is a maximum derived distance *µ*_{max}
= m_{lim} - M_{p}. However, because the standard candle
actually has a dispersion
in absolute magnitude,
some galaxies can be seen from true distances beyond
*µ*_{max} that necessarily become underestimated.

Under the assumption of a gaussian luminosity function
*G*(M_{p}, )
and a sharp magnitude limit m_{lim}, it is a straightforward
task to calculate the magnitude of the second Malmquist bias
<*µ*> - *µ*_{true} at each true
distance modulus
(Teerikorpi 1975b,
Sandage 1994a),
or how much <log H>
will increase due to the magnitude limit cutting galaxies away from the
fainter
wing of the luminosity function. Such an analytical calculation needs the
correct distance scale and the relevant dispersion
as input, which restricts its application in practice. However, it gives
the general behavior of the bias, which is quite similar to the observed
behavior of <log H> vs kinematic distance, and forms the basis for
recognition and correction methods that are independent of H_{o}
(Section 4).

A fundamental property of the theoretical bias curves is that the curves
for standard candles with different means, M_{p}, will show a
shift along the axis of true distance (redshift) (see
Figure 1*a*). This, in the first place, led
to the recognition of the bias by
Teerikorpi (1975a),
where different van den Bergh's luminosity classes were inspected, and by
analogy, to the proposal that de Vaucouleurs's parabolic velocity-distance
relation
(de Vaucouleurs
1972) was caused by a related effect.

An earlier example of the second Malmquist bias at work, is found in
Hawkins (1962),
where the composite Hubble diagram for field galaxies (using the magnitudes
as distance indicators) was interpreted as supporting the quadratic law z
= k ^{.} r^{2}.
This was suggested to be expected from the gravitational redshift in a
static uniform universe, contrary to the linear law expected for the
expansion redshift
in a homogeneous and isotropic universe, described by the Robertson-Walker
line element
(Robertson 1955).
If written formally as a velocity law, the quadratic law assumes the form
log H = 0.5log V + const, which roughly describes the run of the data points
between the unbiased region (log H = 0 ^{.} log V +
const) and the strongly biased one (log H = 1.0 ^{.} log
V + const). One sometimes sees, though not quite appropriately, "Lundmark's
law" mentioned in connection with the quadratic law. True,
Lundmark (1924,
1925)
produced the first diagram where a dependence between redshift and distance
was discernable, and he suggested the representation z = *a* + *b*
^{.} r - *c* ^{.}
r^{2}. In Lundmark's formal solution, the negative quadratic
term does not seem to be related to any selection or other real effect -
the scatter from galaxy diameters as distance indicator was large, and
he might have easily suggested a linear law. Perhaps he was motivated by the
interesting prediction that redshifts have a maximum value (3000 km/s),
which was soon to be contradicted by Humason's measurements.

A special case of the second kind of Malmquist
bias is related to Kapteyn's Problem I and sometimes called cluster
population
incompleteness bias. Here the galaxies are at the same true distance, and
the bias in the derived distance modulus is caused and calculated similarly
as above. This bias makes the distances to clusters of galaxies, calculated
by the TF method, progressively too short, and as it does for field
galaxies,
making log H increase with true distance. Assertions in literature (e.g.
Aaronson et al 1980
de Vaucouleurs & Corwin 1985, Bothun et al 1992) have stated that there
is no Malmquist bias in clusters (because the galaxies in clusters have no
volume effect like the field galaxies), implying that one may utilize the
TF method without bias. However, there were indications in the 1980s that
the clusters, in comparison with Malmquist corrected field galaxies, give
H_{o} values that are too large, which led to recognition of the
cluster incompleteness bias by
Teerikorpi
(1987),
Bottinelli et al
(1987),
Kraan-Korteweg et al
(1988).