Annu. Rev. Astron. Astrophys. 1997. 35: 101-136
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7. MALMQUIST BIAS AND THE GREAT ATTRACTOR

A good example of the difficulties in establishing the distance scale within the local space is provided by the attempts to determine whether there is a differential infall velocity field around the Great Attractor at about Vcosm approx 4500 km/s (Lynden-Bell et al 1988), or alternatively, a bulk flow of galaxies in this same direction of the Hydra-Centaurus supercluster, as earlier proposed by Tammann & Sandage (1985). In principle, with good distances to galaxies in this direction, extending beyond the putative Great Attractor, one would be able to construct a Hubble diagram, velocity vs distance, which in the former case should show both the foreground and backside infall deviations from the linear Hubble law. Such deviations have been detected in the direction of the Virgo cluster (Tully & Shaya 1984, Teerikorpi et al 1992). When aiming at the Great Attractor or the Hydra-Centaurus complex, one must look four to five times farther, hampered by the additional complication of the Zone of Avoidance at low galactic latitudes.

Though different studies agree that there is a flow towards the Great Attractor region, the question of the backside infall remains controversial, after some evidence supporting its detection (Dressler & Faber 1990a, b). Landy & Szalay (1992) suggested that the Malmquist bias (of the first kind) could cause an apparent backside infall signal behind a concentration of galaxies: In the velocity-vs-distance diagram, the distances in front of the concentration, where the density increases, will come out smaller, whereas those calculated behind the concentration, where the density decreases, will be larger on the average than predicted by the classical Malmquist formula.

Mathewson et al (1992a) concluded on the basis of a sample of 1332 southern spirals with TF I-mag distances (Mathewson et al 1992b) that the Hubble diagram does not show any backside infall into the Great Attractor. They thus questioned its very existence. This and Landy & Szalay's (1992) suggestion led Ekholm & Teerikorpi (1994) to investigate the outlook of the velocity-distance and distance-velocity diagrams in the presence of a mass concentration, utilizing a synthetic spherically symmetric supercluster and making comparisons with the data of Mathewson et al (1992a, b). The background method was given by a discussion of four alternative approaches (direct and inverse TF, velocity vs distance, and distance vs velocity) (Teerikorpi 1993). We concluded there that the most promising approach for the detection of the background infall is the analysis of the data in the sense "distance vs velocity." The other way via the classical Hubble diagram needs uncertain Malmquist corrections of the first kind. On the other hand, a p-class analysis (related to MND and TEC) showed that the available sample is not deep enough for the background infall to be detectable using the direct TF relation (Ekholm & Teerikorpi 1994). However, the inverse TF relation is capable of revealing the backside infall, if it exists. For this, it is crucial to know the relevant inverse slope and to resolve the other problems hampering the inverse relation (see Section 5.2).

Federspiel et al (1994) made an extensive analysis of the Mathewson et al (1992a, b) sample using the Spaenhauer diagrams with their "triple entry corrections" (TEC). They identified clearly the Malmquist bias of the second kind and concluded that the dispersion in the I-mag TF relation is relatively large (sigma = 0.4-0.7 mag), depending on the TF parameter p. After making first order bias corrections, based on an underlying Hubble law, they concluded that there is no detectable backside infall, though there is in the foreground a bulk flow of about 500 km/s, apparently dying out before the putative Great Attractor is reached. A question is whether the first order correction, based on the assumed Hubble law, could lead to a null result or a vicious circle. A demonstration against such a vicious circle was made by Ekholm (1996) using the synthetic supercluster. The Hubble law and the TEC formalism for "fast rotators" should reveal the backside infall, if it exists.

Hudson (1994) calculated inhomogeneous Malmquist corrections to the direct Dn - sigma distance moduli in the Mark II sample of E and SO galaxies: He used the density field that he had derived from another, larger sample of galaxies with redshifts known. As a result, Hudson concluded that the apparent backside infall, visible in the Hubble diagram when the homogeneous Malmquist correction was made, was significantly reduced after the new corrections. This is in agreement with the conclusion by da Costa et al (1996).

A special problem is posed by galactic extinction. For instance, the center of the putative Great Attractor, at l approx 307°, b approx 9°, is situated behind the Zone of Avoidance, and the Local Group apex of Rubin et al's (1976) motion observations, at l approx 163°, b approx -11°, was also at low latitudes. Whatever the role of extinction is in those particular cases, in general its influence is subtle and needs special attention. One is not only concerned with good estimates of extinction in the directions of sample galaxies. The variable galactic extinction also creeps in indirect ways into the analyses of galaxy samples. An extinction term must be added into the formula of normalized distances in MND, because galaxies behind an enhanced extinction look fainter in the sky; hence, their "effective" limiting magnitude is brighter (the normalized distance is shorter). In the methods where one uses the limiting magnitude to calculate the importance of the bias of the second kind, one should take into account the extra factor due to extinction (also the inclination effect of internal extinction; Bottinelli et al 1995). Change in the effective limiting magnitude means that other things being equal, the galaxies in the direction of the enhanced extinction are influenced by the second bias already at smaller true distances than are their counterparts in more transparent parts of the sky. This effect is not allowed for just by making normal corrections to the photometric quantities in the TF relation.

Influence of the extinction is also felt when one uses infrared magnitudes where the individual extinction corrections are small. Such samples are usually based on B-mag or diameter-limited samples that have been influenced by variable B extinction, which changes the effective limiting magnitude in different directions of the sky. Depending on the correlation between B and the magnitude in question, this influence of extinction propagates as a kind of Gould's effect (Section 8.3), into the bias properties of the latter.

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