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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: threebody encounters, and tidal capture.
(Hint: many of the calculations in this question are
orderofmagnitude; do not be afraid to approximate when
you cannot calculate something exactly.)
(A) ThreeBody 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

Explain why two point masses that are unbound initially
cannot become bound after an encounter unless a third body
is simultaneously nearby.

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 orderofmagnitude approach.)

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.

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.

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.

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?

Using your answers to 3., 5., and 6.,
show that the time between triple encounters is roughly
.

Evaluate numerically for typical globular
cluster parameters (i) in the core and (ii) in the halo.
Is binary creation by threebody 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

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.

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.)
Next: Core Collapse by Evaporation
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Astronomy 7
Tue Oct 19 13:15:33 PDT 1999