| © CAMBRIDGE UNIVERSITY PRESS 1999
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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) |
ng(r) are, respectively, the space and projected profiles (of the number density of galaxies, or brightness), n0g and S0g are the respective central densities, and Rc is the cluster core radius [where S(Rc) = 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).