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

**8.3. Gould's Effect**

Gould (1993)
pointed out a complication present when a sample of galaxies to be used,
for example, for infrared I-mag TF relation, is constructed from a sample
originally based on selection criteria other than those of I mag,
e.g. apparent diameter. The Malmquist bias of the first kind in the
distance moduli from the I-mag TF relation does not now generally depend
on the squared dispersion
_{I}^{2}
= <_{I}^{2}> of the I-mag TF relation nor on
the squared dispersion
_{D}^{2}
= <_{D}^{2}> of the diameter relation, but on
the covariance
<_{I}
_{D}> between the
corresponding logarithmic distance errors
.
An interesting extreme case is when this covariance is zero, i.e. the
deflections about the two TF relations are independent. Then there
should be no Malmquist bias in the distance moduli from the I-mag TF
relation. This is easy to understand:
Though the original D-limited sample was selected "from the sky," the second
set of I-mag measurements produces symmetrical residuals around the TF
relation
because of the assumed independence on D residuals and because in this case
there is no I-mag limit (cf also Section 3.2 in
Landy & Szalay
1992).

In practice, it may be difficult to find such pairs of
observables that correlate with a common distance-independent parameter
(e.g. TF parameter *p*), but have independent deflections
(larger-than-average
galaxies tend to be also more luminous than average). Also, it should be
noted that the above argument is valid only if one could measure I for all
the galaxies first taken from the D-limited sample, i.e. if the I limit was
really = . In fact,
though to measure the relevant covariance is one approach,
Gould (1993)
also sees the described problem as supporting the use of the "good old"
B-band TF-relation. [For further discussions of Gould's effect, see
Willick (1994),
Strauss & Willick
(1995).]

In the KLUN project (e.g.
Paturel et al 1994),
the sample of 5174 spirals has been selected on the basis of apparent
size D_{25} (in B), and in the analysis of the diameter TF
distance moduli, Gould's effect should not appear. However, in such
cases a somewhat related problem is that
the measured diameters contain measurement error, and when one constructs
a diameter-limited sample from measured galaxies, there is a Malmquist
effect due to the dispersion
in the measurement
error: The sample contains an excess of overestimated apparent
diameters.
Ekholm &
Teerikorpi (1997)
pointed out that this may have a significant influence on the results,
especially on those from the inverse TF relation where all galaxies (in
view of the
method's supposedly unbiased nature) are used. Assume now that the apparent
sizes of such a diameter-limited sample are once more measured. Because the
first and second measurement errors are independent, their influence
vanishes from Gould's covariance, and now the second sample has correct
measured apparent
sizes, on the average. In practice, such a remeasurement of large samples
is out of the question, and one has to be aware of the problem.