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 )

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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^2 PE = total potential energy E = total mechanical energy = KE + PE 2<KE> + <PE> = 0 2(E - <PE>) + <PE> = 0 => <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

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!

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

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

Evolution of a globular cluster Initial profile was a Plummer sphere. Comparison of results of a exact N-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 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

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