Covariance function analysis and the clustering of galaxies

Abstract

Data on a statistic derived from the angular covariance function show that (contrary to the claim of Peebles that galaxies are distributed continuously with no distinct scales), superclusters and the maximum size of clusters are probably defined at scales of 15 and 2.0h ⁻¹ Mpc. This suggests some stepped-density profile like the idealized models of de Vaucouleurs and Wertz: consideration is therefore given to a semi-continuous hierarchy in which there are galaxies outside clusters, clusters outside superclusters etc. Theories of the origin of clustering by gravitational clumping and the escape of galaxies from clusters suggests the hypothesis that the average mass (m _g) of galaxies outside clusters is smaller than that of those inside (=fractionf of the total), a hypothesis supported by results on the continuity of the angular and spatial covariance functions. In a semi-continuous hierarchy, the overall packing fractionf _e and the fraction (1-f) of galaxies outside clusters both appear to increase as the distancer from a local origin increases, because a line-of-sight to greater depths intersects systems of the hierarchy of continually greater size (R _i). If the hypothesis is valid thatm _g inside clusters is slightly larger thanm _g outside, the apparent effect is to makem _g systematically distance-dependent from a local origin with$m_g = m_g (r)\alpha r^{ - \eta _1 }$ and η₁≃0.3. No direct data on galaxy masses exist to refute such a small trend, but since the absolute magnitudes of galaxies are known to be correlated (very weakly) with their masses, a semi-continuous hierarchy has a location-dependent luminosity function, ϕ(M). Within uncertainties as to the steepness of ϕ(M) at the bright end, the model is consistent with optical number counts to a limiting photographic magnitudem _pg (isotropic slope,q′=0.6; semicontinuous modelq′=0.64; observation,q′=0.67±0.03, standard error.) this removes the discrepancy between the determinations by de Vaucouleurs and Sandageet al. of the thinning factor ⊝(≃1.7). Predictions of the semi-continuous model are made which are at present observationally feasible to carry out. In particular, it is predicted thatq′(20<|M|<22)/q′(14<|M|<19)≃2(±0.2).