Example

Example 16.1 Assume that a globular cluster has a diameter of 40 pc and contains 100,000 stars of one solar mass each.

a) Use the virial theorem to find the average velocity of the stars. You can assume that the average distance between stars equals the radius of the cluster.

b) Find the escape velocity.

c) Comparing these velocities, can you tell something about the stability of the cluster?

a) First, we have to estimate the potential energy. There are n(n — 1)/2 ~ n2/2 pairs of stars in the cluster, and the average distance of each pair is R. Thus the potential energy is about

where m = 1 Me. The kinetic energy is

where v is the root mean square velocity. According to the virial theorem we have T = —1/2U, whence

Solving for the velocity we get Gmn v2 =

_ 6.7 x 10—11 m3 kg-1 s x 2.0 x 1030 kg x 10 = 40 x 3.1 x 1016m

= 1.1 x 107m2s—2, which gives v ~ 3 km s—1. b) The escape velocity from the edge of the cluster is

2Gmn

c) No. The average velocity seems to be smaller than the escape velocity, but it was derived from the virial theorem assuming that the cluster is stable.

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