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

**2.2. The Classical Malmquist Bias**

In his work, "A study of the stars of spectral type A,"
Malmquist (1920)
investigated how to derive the luminosity function of stars from their
proper motions, provided that it is gaussian and one knows the
distribution of apparent magnitudes up to some limiting magnitude. This
led Malmquist to investigate the question of what is the average value
(and other moments) of the quantity *R*, or the reduced distance,
as earlier introduced by Charlier:

(2) |

Malmquist made three assumptions: 1. There
is no absorption in space. 2. The frequency function of the absolute
magnitudes is gaussian (M_{o},
). 3. This function is
the same at all distances. The third assumption is the principle of
uniformity as implied in
Lundmark's (1946)
definition of distance indicators.

Using the fundamental equation of stellar statistics, Malmquist derived
<*R*^{n}> and showed that it may be expressed in
terms of the luminosity function constants M_{o} and
and the distribution
*a*(m) of apparent magnitudes, connected with the stellar space
density law. Especially interesting for Malmquist was the case *n*
= -1,
or the mean value of the "reduced parallax," that appears in the analysis
of proper motions. However, for distance determination, the case *n*
= 1 is relevant because it allows one to calculate, from the mean value of
the reduced distance, the average value of the distance <r> for
the stars that have their apparent magnitude in the range m ± 1/2
*dm* or their distance modulus *µ* = m - M_{o} in
the range *µ* ± 1/2 d*µ*. The result is,
written here directly in terms of the distance modulus distribution
*N*(*µ*) instead of *a*(m),

(3) |

where *b* = 0.2 ^{.} ln 10.
This equation is encountered in connection with the general Malmquist
correction
in Section 6. Naturally, in Malmquist's
paper one also finds his formula
for the mean value of M for a given apparent magnitude m:

(4) |

The term including the distribution of apparent
magnitudes (or distance moduli in Equation 3) reduces to a simple form when
one assumes that the space density distribution *d*(r)
r:

(5) |

With = 0, one finally obtains the celebrated Malmquist's formula valid for a uniform space distribution:

(6) |

Hubble (1936)
used Malmquist's formula (Equation 6) when, from the brightest stars of
field
galaxies (and from the magnitudes of those galaxies), he derived the value
of the Hubble constant. Hubble derived from a local calibrator sample the
average (volume-limited) absolute photographic magnitude and its dispersion
for the brightest stars. As "the large-scale distribution of nebulae and,
consequently, of brightest stars is approximately uniform," he derived the
expected value for the mean absolute magnitude of the brightest stars
<M_{s}>
for a fixed apparent magnitude. His field galaxies were selected, Hubble
maintained, on the basis of the apparent magnitudes of the brightest stars,
which justified the calculation and use of <_{s}>. In the
end, he compared the mean apparent magnitudes of the brightest stars in
the sample galaxies with <M_{s}>, calculated the average
distance <r>, and derived the value of the Hubble constant (526
km/s/Mpc). Hence, it is important to recognize that this old value of
H_{o}, canonical for some time, already includes an attempt to
correct for the Malmquist bias. Also, it illustrates the role of
assumptions in this type of correction
(what is the space density law, what is the mode of selection of the sample,
etc).