Annu. Rev. Astron. Astrophys. 1997. 35: 101-136 Copyright © 1997 by Annual Reviews. 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 <Rⁿ> 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 dµ. 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).