The first step is to rewrite the dynamical equation
and thus
is the relation we want. Note, however, that in a flat universe with the critical density that
so we can write our differential simply as
The integrand becomes
which integrates to
which you can check by taking the derivative. Substitution of
gives us the expression
we wanted.
As z goes to infinity, we get the current horizon distance
which gives 10000 Mpc for h=0.6 (H0 = 60 km/s/Mpc).
Simple substitution gives
which tends toward the limiting angular diameter
and our fiducial cluster of D = 1.8 h-1 Mpc tends to the limiting diameter of 61.9 arcseconds (approximately 1 arcminute) at infinite redshift.
Upon substitution we find
which now diverges at z = 0 and infinity (try it). Thus, it must be a minimum somewhere in between. If we make the substitution a = (1+z)
and the derivative is
which when set to zero, and dividing out common factors, gives
or
Our solution is then
thus the angular size reaches a minimum at z = 1.25.
Note: you could also have computed the redshift for which the angular diameter distance dA is a maximum, which is actually more straightforward. Again, with our variable change
we set the derivative
equal to zero gives
as before.
At the minimum a = 9/4 and thus
versus the comoving angular diamter at infinte redshift (calculated above).
Our fiducial galaxy has a size of
so its minimum apparent angular size is 23.2".
Using
the current critical density is
or 2.76 × 1011 h2 Msun Mpc-3. The critical energy density is therefore
The radiation constant a (not the scale factor a in the last problem) has the value
which means for T0 = 2.73 K
for the radiation energy density. Thus,
is the fraction of the critical density currently contributed by radiation.
Since
then we will have matter-radiation equality at
and thus the redshift of equality zeq ~ 14500 for h=0.6.
The radial velocity dispersion reprents a 1-D projection of the true 3-D velocities of the galaxies, so
as the squares of the velocities add in quadrature. For Coma
The virial relation 2K = -U gives
for mass M and virial radius Rvir. Thus,
which for radial velocity dispersion 977 km/s and
gives us
for the virial mass.
We use the usual relation between luminosity and absolute magnitude
for the Sun's absolute visual magnitude of +4.83. Technically, though it wasn't stated in the problem, that since the absolute magnitude is determined from the apparent magnitude and luminosity distance dL (in h-1 Mpc), then the luminosity will be proportional to h2, which gives
for Coma. Thus,
for Coma cluster, which is significantly larger than that for the centers of spiral and elliptical galaxies ( ~10 h) and the extended halos of giant elliptical galaxies ( ~100 h). (Note: if you missed the factor of h^2 from the luminosity, thats OK, because I missed it also the first time.)
If we count up the mass associated with the light which we see, which comes from the galaxies in the cluster, then we would estimate
which represents a fraction of the total virial (gravitational) mass
and thus 94.8% of the gravitational mass is unaccounted for by the galaxies, and must be dark matter in the extended cluster halo!
The density of the Coma cluster within the virial radius is
which when compared with
gives
and indeed the Coma cluster is extremely overdense compared to the critical density, and thus should behave like a part of a
closed universe. Since the density
which for R = 1.8 h-1 Mpc gives RL = 12.0 h-1 Mpc. Thus, since the early Universe had a density close to the critical density (as all FRW models do), the Coma cluster has collected up all the matter from a region equivalent to a sphere of radius 12 h^-1 Mpc today!
smyers@nrao.edu Steven T. Myers