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

In their useful study, Landy & Szalay (1992) initiated the recent discussions on the general or "inhomogeneous" Malmquist bias and its correction, i.e. how to deal with the bias of the first kind in the case of a general space density distribution. True, they encountered some problems in the practical application of the formula derived in their paper. In fact, the basic problem was that they assumed in the beginning a distance indicator that has a zero bias of the second kind (which is unbiased at all distances) and then derived an expression for the bias of the first kind.

In Teerikorpi (1993)
the problem of the general correction was discussed with explicit reference
to direct and inverse TF relations, which served to clarify some of the
points raised by
Landy & Szalay
(1992).
It was noted that Malmquist's formula (Equation 3) was a general one, and
in modern terms is best interpreted as applicable to the direct TF relation,
for a constant value of *p* (Malmquist's "star class") and requires
that the limiting magnitude (m_{l}) =
. In that case, the
distribution of distance moduli *N*(*µ*) refers to all
moduli that could be observed without any cutoff in the magnitudes.

Interestingly,
Feast (1972)
had already given a formula quite similar to Equation 3, but now with a
quite different meaning for the distribution of *N*(*µ*):
In his formulation, *N*(*µ*) must be regarded as the
distribution of the derived distance moduli of the galaxies in the
observed sample. Inspection of
Feast's (1972)
derivation reveals the implicit assumption that the bias of the second kind
is zero, and the end result was the same as that of
Landy & Szalay
(1992).
As the inverse TF relation distance moduli have the second bias = 0, one
can conclude that Feast's and Landy & Szalay's variant of Malmquist's
formula applies to the inverse TF distance moduli.

Landy & Szalay's (1992) paper gave rise to a burst of independent discussions. Similar conclusions as in Teerikorpi's (1993) paper on the general correction and the inverse TF relation were given by Feast himself (1994), Hendry & Simmons (1994) (see also Hudson 1994, Strauss & Willick 1995).

To reiterate, Malmquist's formula (Equation 3), with
*N*(*µ*) refering to all distance moduli (m_{lim}
=
) in the considered sky
direction, applies to the direct TF moduli. In the case of the inverse
TF relation, *N*(*µ*)
is the observed distribution. Hence, for making corrections of the first
kind for the direct distance moduli, one needs information on the true space
density distribution of galaxies, and the selection function (magnitude
limit) does not enter the problem. Corrections for the inverse moduli
depend directly on the distribution of apparent magnitudes (distance
moduli) in the sample and, hence, on the selection function. In this
sense, the biases of the first and second kind for the direct and
inverse distance moduli have curious complementary properties.

One might be content with the above conclusions and try
to use the inverse relation in situations where the bias of first kind
appears:
The correction does not need the knowledge of the true space distribution.
In practice, the high inhomogeneity of the local galaxy universe requires
corrections for differing directions, which divides galaxy samples into
small subsamples where the detailed behavior *N*(*µ*) is
difficult to derive.

The general Malmquist bias for the inverse distance moduli,
even in the case of a homogeneous space distribution, can be quite
complicated. Because the observed *N*(*µ*)
usually first increases, reaches a maximum, and then decreases to zero, the
bias is first negative (too small distances), then goes through zero, and
at large derived distances is positive (i.e. the distances are too long).
Something like this is also expected simply because the average bias of the
whole sample should be zero.

An important special case of the first Malmquist bias of the direct distance moduli is that of a homogeneous space distribution, which shifts standard candles in a Hubble diagram by a constant amount in magnitude and leaves the expected slope of 0.2 intact. In this manner, Soneira (1979) argued for the local linearity of the Hubble law, in contrast to a nonlinear redshift-distance relationship espoused by Segal, where he argues against debilitating bias effects.

It has been suggested that the distribution of galaxies
at least up to some finite distance range could be fractal in nature, with
fractal dimension
*D* 2 (e.g.
Di Nella et al
1996).
In that case one might be willing to use, instead of the classical
Malmquist correction (for *D* = 3 or homogeneity), Equation 5,
which corresponds to the average density law proportional to
*r*^{D-3}. However, though the classical formula applies to
every direction in a homogeneous universe, the deformed (*D*
= 2) formula is hardly useful in any single direction of a fractal universe
because of the strong inhomogeneities. Note however that in this case the
argument by
Soneira (1979)
would be equally valid.