1. Relaxation in stellar systems
ASTC22 - Lecture 9
Relaxation in stellar systems
Relaxation and evolution of globular clusters
The virial theorem and the negative heat capacity of
gravitational systems
Mass segregation, evaporation of clusters
Monte Carlo, N-body and other simulation methods
(The Kepler problem )

5. See errata

6. Histogram of trelax

7. Relaxation effects in real systems: globular clusters N=1e5 - 1e6

8. The virial theorem (p. 105 of textbook)
KE = total kinetic energy = sum of (1/2)m*v^2
PE = total potential energy
E = total mechanical energy = KE + PE
2<KE> + <PE> = 0
2(E - <PE>) + <PE> = => <PE> = 2 E
also <KE> = - E
Mnemonic: circular Keplerian motion
KE = (1/2) * v^2 = GM/(2r) (per unit mass)
PE = -GM/r
E = -GM/(2r)
thus PE = 2E, KE=-E, and 2*KE = -PE

9. Consequence of the virial theorem:
(1/2) m < v^2 > = (3/2) kB T (from Maxwell’s distribution)
dE/dT = -(3/2) N kB < 0
negative specific heat: removing energy makes the
system hotter!

10. In a globular cluster, star
exceeding the escape speed
ve leave the system, or
“evaporate”.

11. Evolution of globular clusters leads to small, dense cores and cusps

15. Evolution of a globular cluster
Initial profile was a Plummer sphere.
Comparison of results of a exact
N-body simulations (symbols),
usually with N= , with
semi-analytic Mte Carlo method (line).
In Mte Carlo, stellar orbits in a
smooth potential are followed with
occasional added jolts simulating
the weak and strong encounters.
Random number generators help
to randomize perturbations.
The results are thus subject to
statistical noise.
Upper lines show radius enclosing
90% of mass, middle - 50%, and
the lower 10% of totl mass.
BT p.520

16. Evolution of a globular cluster
Results of a semi-analytic
method using Fokker-Planck
equation. Unlike the Mte Carlo
method, these results are not
subject to statistical noise.
Typical time of evolution before
core collapse is 20 trelax
dt=trelax
= Initial density
profile of type ~ 1/[1+ (r/b)^2 ] or a similar King model
BT p.528