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SOLUTIONS TO EXERCISE 12

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(a) The spread of velocities is best explained as peculiar velocities
among the galaxies in the cluster - that is, the velocity added to the
galaxies' motions by gravity.  Peculiar velocities mimic the velocity
caused by the expansion of the universe; so if they are not properly
accounted for, the distance to each galaxy will be systematically wrong.
The large spread suggests that the galaxies in Abell 2218 are interacting
gravitationally; and some intermediate value of these peculiar velocities
would indicate how much matter is actually contained within the cluster.

(b) The escape velocity of any object from this cluster, at a distance
of 1000 kpc, would be

                         2 * 6.67 * 10^(-11) * 2.4 * 10^43 kg
  v = sqrt(2GM/r) = sqrt(------------------------------------) 
                           1000 kpc * (3.1 * 10^19 m/kpc)

    = 3.1 * 10^5 m/sec = 310 km/sec.  

(Remember that the above equation requires r in meters for that value of G.)

Since this galaxy is moving 950 km/sec, this galaxy should in fact be
able to escape the cluster's gravity, and would thus NOT be bound to Abell 2218.