Virially Bound Systems 201in stars, the halo mass and the length scale. A frequently used radial density profile
parametrization is
휌DM(푟)=휌 0 ∕[(푟∕푟s)훼( 1 +푟∕푟s)^3 −훼], (9.7)where휌 0 is a normalization constant and 0⩽훼⩽ 3 ∕2. A standard choice is훼= 1
for the Navarro–Frenk–White profile (NFW) [1] This presents a singularity at푟→ 0
although the total integrated mass is finite. The sharp rise of the density at the halo
center forms a “cusp”. Another cusped profile is that of Mooreet al.[2] with훼= 3 ∕2.
The Einasto profile [3 and earlier references therein] is defined as
휌DM(푟)=휌푒exp{−푑푛[(푟∕푟푒)^1 ∕푛− 1 ]}, (9.8)
where the term푑푛is a function of푛such that휌푒is the density at푟푒, which defines a
volume containing half of the total mass. At푟=0 the density is then finite and the
profile iscored.
The Burkert profile [4] has a constant density core
휌DM(푟)=휌 0 ∕[( 1 +푟∕푟푠)( 1 +(푟∕푟푠)^2 )], (9.9)which appears to fit dwarf galaxy halos well.
Some clusters are not well fitted by any spherical approximation. The halo may
exhibit a strong ellipticity or triaxiality in which case none of the above profiles is
good, or the cluster (like Coma) has a binary center.
The dependence of the physical size of clusters on the mass, characterized by the
mass concentration index푐≡푟vir∕푟푠, has been studied in훬CDM simulations. At inter-
mediate radii푐is a crucial quantity in determining the density profile.
The Local Group. The Local Group (LG) is a very small virial system, dominated
by two large galaxies, the M31 or Andromeda galaxy, and the Milky Way. The M31
exhibits blueshift, falling in towards us. Evidently our Galaxy and M31 form a bound
system together with all or most of the minor galaxies in the Local Group. The Local
Group extends to about 3 Mpc and the velocity dispersions of its members is about
200km s−^1. The virial mass is푀vir= 8. 2 +−^21 ..^56 푀⊙.
In this group the two large galaxies dominate the dynamics, so that it is not mean-
ingful to define a statistically average pairwise separation between galaxies, nor an
average mass nor an average orbital velocity. The total kinetic energy퐸is still given
by the sum of all the group members, and the potential energy푈by the sum of all the
galaxy pairs, but here the pair formed by the M31 and the Milky Way dominates, and
the pairings of the smaller members with each other are negligible.
Observations of galaxies in the LG can be used to constrain the nature of DM. The
number of low mass satellites of MW and M31 as compared with CDM predictions can
be explained through the effects of gas-dynamics on baryons, making them invisible,
or they can simply not exist, if DM particles have a mass in the keV scale.
Small Galaxy Groups. There are other examples of groups formed by a small num-
ber of galaxies which are enveloped in a large cloud of hot gas, visible by its X-ray