© CAMBRIDGE UNIVERSITY PRESS 1999

2.6. Density Profile

The radial density distribution of galaxies in a rich cluster can be approximated by a bounded Emden isothermal profile (Zwicky 1957; Bahcall 1977), or by its King approximation (King 1972) in the central regions.

In the central regions, the King approximation for the galaxy distribution is

	(7)
	(8)

n_g(r) are, respectively, the space and projected profiles (of the number density of galaxies, or brightness), n⁰_g and S⁰_g are the respective central densities, and R_c is the cluster core radius [where S(R_c) = S⁰/2]. Typical central densities and core radii of clusters are listed in the following subsection. The projected and space central densities relate as

(9)

A bounded Emden isothermal profile of galaxies in clusters yields a profile slope that varies approximately as (Bahcall 1977)

	(10)
	(11)

therefore

(12)

The galaxy-cluster cross-correlation function (Peebles 1980; Lilje and Efstathiou 1988) also represents the average radial density distribution of galaxies around clusters. For R 1 clusters, and r in h^-1 Mpc, these references suggest, respectively

(13)

or

(14)

The average density distribution profile of galaxies in clusters thus follows, approximately,

	(15)
	(16)

Some substructure (subclumping) in the distribution of galaxies exists in a significant fraction of rich clusters (~ 40%) (Geller 1990).