Binary Formation in Globular Clusters

Binaries are key elements of a globular cluster because, as discussed in lectures, they store a large amount of gravitational binding energy. When this energy is released, it significantly affects the evolution of the cluster.

Some of the binaries in a globular cluster are primordial, i.e. they formed at the same time as the cluster. Other binaries are created later through stellar encounters of different kinds. In this problem, we look at two creation mechanisms: three-body encounters, and tidal capture.

(Hint: many of the calculations in this question are order-of-magnitude; do not be afraid to approximate when you cannot calculate something exactly.)

(A) Three-Body Encounters. Three stars approach close enough together to capture each other gravitationally and form a binary or triple system, as in Fig. 1. We assume the stars are point masses, so that there are zero tidal losses.

Figure: 1

1. Explain why two point masses that are unbound initially cannot become bound after an encounter unless a third body is simultaneously nearby.
2. Suppose all three stars have mass m and speed v. Show that the velocity perturbation of each star is given by , where b is the approximate distance of closest approach of the stars. (Hint: take an order-of-magnitude approach.)
3. Use the result 2. to argue that the condition must hold if two or more of the stars are to become bound after the encounter. Explain your reasoning carefully.
4. Compare to the characteristic (i) radius and (ii) separation of stars in a globular cluster. Use typical numbers such as those in the lecture notes. Comment on what you find.
5. Let n be the number of stars per unit volume in the cluster. Obtain an expression, in terms of n, b, and v, for how long it takes before two stars approach within a distance b of each other.
6. What is the probability, in terms of n and b, that a third star will be in the vicinity when the other two stars meet?
7. Using your answers to 3., 5., and 6., show that the time between triple encounters is roughly .
8. Evaluate numerically for typical globular cluster parameters (i) in the core and (ii) in the halo. Is binary creation by three-body encounters a significant process in globular clusters?

(B) Tidal Capture. Two stars, initially unbound, approach close together, each raising a tidal bulge on the other. As the tidal bulges change orientation during the encounter, orbital energy is dissipated as heat and the two stars bind to form a binary, as in Fig. 2. In this case, the stars are not point masses.

Figure: 2

9. Explain why two stars that are unbound initially can become bound after an encounter if tidal losses are present. Compare with the situation in 1.
10. A tidal capture binary is formed as a result of a close encounter of two stars of equal mass m. The minimum separation during the encounter is , and the orbital energy dissipated in the encounter satisfies . After the binary has formed, with an elliptical orbit in general, more energy is dissipated in each successive orbit until the orbit is circularised. Ignoring the spin of the two stars completely, show that the radius of the final circular orbit is . (Hint: this part has nothing to do with any of the previous parts; think afresh.)