200 Dark Matter
9.1 Virially Bound Systems
The planets move around the Sun along their orbits with orbital velocities balanced
by the total gravity of the Solar system. Similarly, stars move in galaxies in orbits with
orbital velocities푣determined by the gravitational field of the galaxy, or they move
with velocity dispersion휎. Galaxies in turn move with velocity dispersion휎under
the influence of the gravitational field of their environment, which may be a galaxy
group, a cluster or a supercluster. Dark matter forms halos which extend far beyond
the luminous matter.
Dynamics. In the simplest dynamical framework one treats massive systems (galax-
ies, groups and clusters) as statistically steady, spherical, self-gravitating systems of푁
objects with average mass푚and average velocity푣or velocity dispersion휎.Thetotal
kinetic energy퐸of such a system is then (we now use휎rather than푣)
퐸=( 1 ∕ 2 )Nm휎^2. (9.1)If the average separation is푟, the potential energy of푁(푁− 1 )∕2 pairings is
푈=−( 1 ∕ 2 )푁(푁− 1 )퐺푚^2 ∕푟. (9.2)Thevirial theoremstates that for such a system
퐸=−푈∕ 2. (9.3)The total dynamic mass푀dyncan then be estimated from휎and푟
푀dyn=Nm= 2 푟휎^2 ∕퐺. (9.4)This can also be written
휎^2 ∝(푀dyn∕퐿)IR, (9.5)where퐼is a surface luminosity,푅is a scale, and푀dyn∕퐿is themass to light ratio.
Choosing the scale to be the half light radius푅푒, this implies a relationship between
the observed central velocity dispersion휎 0 ,퐼푒and푅푒called theFundamental Plane.
of the form
푅푒∝(휎 0 )푎(퐼푒)푏. (9.6)The virial theorem predicts the values푎=2,푏=1 for the coefficients. This relationship
is found in ellipticals and in some other types of stellar populations with somewhat
different coefficients.
Halo Density Profiles. Dark matter halos in galaxies and clusters can of course not
be observed by their radiation, but there is an excellent tool to determine their sizes
and weights: lensing. This we described in full detail in Section 4.3.
The shapes of DM halos in galaxies and clusters need to be simulated or fitted by
empirical formulae. Mostly the shape is taken to be spherically symmetric so that the
total gravitating mass profile푀(푟)depends on three parameters: the mass proportion