and Cosmology

6.4 Scaling Relations for Clusters of Galaxies
“clusters” of low mass and temperature) are located below
the power-law fit that is obtained from higher mass
clusters. If one confines the sample to clusters with
M ≥ 5 × 10 13 M ⊙ , the best fit is described by
( )
M 500 = 3.57 × 10 13 kB T 1.58
M ⊙ , (6.49)
1keV
with an uncertainty of slightly more than 10%. This relation
is very similar to the one deduced from theoretical
considerations, M ∝ T 1.5 . With only small variations
in the parameters, the relation (6.49) is obtained both
from a cluster sample in which the mass was determined
based on an isothermal β-model, and from
a cluster sample in which the measured radial temperature
profile T(r) was utilized in the mass determination
(see Eq. 6.36). Constraining the sample to clusters
with temperatures above 3 keV, one obtains a slope
of 1.48 ± 0.1, in excellent agreement with theoretical
expectations. Considerably steeper mass–temperature
relations result from the inclusion of galaxy groups
into the sample, from which we conclude that they
do not follow the scaling argument sketched above in
detail.
257
Fig. 6.26. The top panel shows the total exposure time in the
ROSAT All Sky Survey as a function of sky position. Near the
ecliptic poles the exposure time is longest, as a consequence of
the applied observing strategy. Because of the “South Atlantic
Anomaly” (a region of enhanced cosmic ray flux over the
South Atlantic Ocean, off the coast of Brazil, caused by the
shape of the Earth’s magnetosphere), the exposure time is
generally higher in the North than in the South. The X-ray sky,
asobservedintheRASS,isshowninthelowerpanel.The
colors indicate the shape of the spectral energy distribution,
where blue indicates sources with a harder spectrum
easier to determine the mass for small radii than the
virial mass itself, the mass M 500 within the radius r 500 ,
the radius within which the average density is 500
times the critical density, has been plotted here. The
measured values clearly show a very strong correlation,
and best-fit straight lines describing power laws of
the form M = AT α are also shown in the figure. The
exact values of the two fit parameters depend on the
choice of the cluster sample; the right-hand panel of
Fig. 6.27 shows in particular that galaxy groups (thus,
The X-ray temperature of galaxy clusters apparently
provides a very precise measure for their
virial mass, better than the velocity dispersion (see
below).
With the current X-ray observatories, it will be possible
to test these mass–temperature relations with even
higher accuracy; the first preliminary results confirm the
above result, and the improved accuracy of future observations
will lead to an even smaller dispersion of the
data points around the power law.
6.4.2 Mass–Velocity Dispersion Relation
The velocity dispersion of the galaxies in a cluster also
can be related to the mass: from (6.25) we find
M vir = 3r virσv
2
G . (6.50)
Together with T ∝ σv 2 and T ∝ r2 vir , it then follows that
M vir ∝ σ 3 v . (6.51)