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The masses of clusters of galaxies within a given radius,
M( r),
can be determined from three independent methods:
The first two methods were discussed in Sections 2 and 3; the last (and newest) method is discussed in this book by Narayan. (See also Tyson et al. 1990, 1996; Kaiser and Squires 1993; Smail et al. 1995, and references therein).
The galaxy and hot gas methods yield cluster masses that are
consistent with each other as discussed in
Section 3 (with
2r
kT/µmp). Gravitational lensing
results, which
provide direct information about cluster masses, are available only
for a small number of clusters so far (with data rapidly increasing).
For all but one of the clusters the masses determined from lensing are
consistent, within the uncertainties, with the masses determined from
the galaxies and hot gas methods (see
Bahcall 1995
for a summary).
Some differences in masses between the different methods are expected
for individual clusters due to anisotropic velocities, cluster
orientation (yielding larger lensing surface mass densities for
clusters elongated in the line-of-sight, and vice versa), and
sub-structure in clusters. On average, however, all three independent
methods yield cluster masses that are consistent with each other.
This triple check on cluster masses provides strong support for the
important cluster mass determinations.
The masses of rich clusters range from ~ 1014 to
~ 1015h-1
M within
1.5h-1 Mpc radius of the cluster center.
When normalized by the cluster luminosity, a robust mass-to-light
ratio is determined for nearby clusters, with only small variations
from cluster to cluster (Section 2.10)
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This result is similar to the one obtained from the baryon fraction in Section 4.
If, as desired by theoretical arguments, the universe has critical
mass density
m = 1,
than most of the mass in the universe
cannot be associated with galaxies, groups, and clusters; the mass
distribution in this case would be strongly biased (i.e., mass does
not follow light, with the mass distributed considerably more
diffusely than the light).