next up previous
Next: Measuring the Galactic Rotation Up: No Title Previous: Binary Formation in Globular

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 ( tex2html_wrap_inline185 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 tex2html_wrap_inline197 and radius tex2html_wrap_inline199 by


The mass loss rate is given approximately by


where tex2html_wrap_inline201 , with tex2html_wrap_inline203 , is the evaporation time. The relaxation time tex2html_wrap_inline205 is given by the Spitzer-Hart formula (see lecture notes)


with tex2html_wrap_inline207 , provided that all stars in the cluster have the same mass m.

  1. Why does tex2html_wrap_inline205 change with time t as the cluster evolves? Why is k roughly constant? Please give physical, not mathematical, explanations. Be precise.
  2. Combine (1)-(4) to eliminate M(t) and obtain a first-order differential equation for how R(t) evolves with time.
  3. Assume tex2html_wrap_inline221 is constant with time; logarithms change very slowly. Integrate the equation in 7. and show that R(t) is of the form


    Express tex2html_wrap_inline225 in terms of G, m, tex2html_wrap_inline197 , and tex2html_wrap_inline199 . Sketch R(t) as a function of t.

  4. 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.
  5. Apply the virial theorem to obtain V(t), the characteristic stellar speed, in terms of R(t) and hence G, tex2html_wrap_inline197 , tex2html_wrap_inline199 and tex2html_wrap_inline225 . Sketch V(t) as a function of t. What is the value of tex2html_wrap_inline255 ? Is this realistic?
In reality, a globular cluster does not collapse to a point. Its end state is a single close binary with, say, tex2html_wrap_inline257 and tex2html_wrap_inline259 , as indicated by numerical simulations.
  1. Using 8., 10., and tex2html_wrap_inline257 , evaluate tex2html_wrap_inline263 numerically. You should get a much larger speed than tex2html_wrap_inline265 . Can you suggest two reasons why the simple theory of this problem does not predict the end state of core collapse correctly?

next up previous
Next: Measuring the Galactic Rotation Up: No Title Previous: Binary Formation in Globular

Astronomy 7
Tue Oct 19 13:15:33 PDT 1999