Core Collapse by Evaporation

Evaporation of stars from a globular cluster causes the cluster to shrink over time.

1. Write a paragraph explaining how this happens physically. Start from first principles and include every step in your reasoning. At the same time, be concise ( page including any diagrams); aim for clarity through economy of expression. (This exercise is intended to help you practise writing `essay paragraphs' in exams.)
2. Suppose evaporation occurs sufficiently slowly for the cluster to remain in equilibrium at every stage of the collapse. Using the virial theorem, express the total mechanical energy of the cluster, E(t), in terms of its mass M(t) and radius R(t). Your answer will be approximate; ignore factors of order unity.
3. Each evaporating star carries away very little total mechanical energy as it escapes the cluster. Explain why, physically, this is so.
4. Use 2. and 3. to show that the rate of change of the cluster radius is related to the mass loss rate by

   

5. Also use 2. and 3. to relate M(t) and R(t) to the initial mass and radius by

   

The mass loss rate is given approximately by

where , with , is the evaporation time. The relaxation time is given by the Spitzer-Hart formula (see lecture notes)

with , provided that all stars in the cluster have the same mass m.

6. Why does change with time t as the cluster evolves? Why is k roughly constant? Please give physical, not mathematical, explanations. Be precise.
7. Combine (1)-(4) to eliminate M(t) and obtain a first-order differential equation for how R(t) evolves with time.
8. Assume is constant with time; logarithms change very slowly. Integrate the equation in 7. and show that R(t) is of the form

   

   Express in terms of G, m, , and . Sketch R(t) as a function of t.

9. Evaluate numerically how long it takes for a typical globular cluster to collapse to a point. Then repeat your estimate for just the core of a cluster. Use typical numbers such as those in the lecture notes. Comment.
10. Apply the virial theorem to obtain V(t), the characteristic stellar speed, in terms of R(t) and hence G, , and . Sketch V(t) as a function of t. What is the value of ? Is this realistic?

In reality, a globular cluster does not collapse to a point. Its end state is a single close binary with, say, and , as indicated by numerical simulations.

11. Using 8., 10., and , evaluate numerically. You should get a much larger speed than . Can you suggest two reasons why the simple theory of this problem does not predict the end state of core collapse correctly?