Relaxation and evolution of globular clusters

The virial theorem and the negative heat capacity of gravitational systems

Mass segregation, evaporation of clustersMonte Carlo, N-body and other simulation methods

(The Kepler problem )

See errata

Histogram of trelax

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

The virial theorem (p. 105 of textbook)

KE = total kinetic energy = sum of (1/2)m*v^2PE = total potential energy E = total mechanical energy = KE + PE

2<KE> + <PE> = 0

2(E - <PE>) + <PE> = 0 => <PE> = 2 Ealso <KE> = - E

Mnemonic: circular Keplerian motionKE = (1/2) * v^2 = GM/(2r) (per unit mass)PE = -GM/rE = -GM/(2r) thus PE = 2E, KE=-E, and 2*KE = -PE

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 thesystem hotter!

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

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

BT p.520

Evolution of a globular cluster

Initial profile was a Plummer sphere.

Comparison of results of a exactN-body simulations (symbols), usually with N=150-350, with semi-analytic Mte Carlo method (line).In Mte Carlo, stellar orbits in a smooth potential are followed withoccasional 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.

= Initial densityprofile of type ~ 1/[1+ (r/b)^2 ] or a similar King model

Evolution of a globular cluster

Results of a semi-analyticmethod using Fokker-Planck equation. Unlike the Mte Carlomethod, these results are notsubject to statistical noise.

Typical time of evolution before core collapse is 20 trelax

BT p.528

dt=trelax