ASTC22 - Lecture 9 Relaxation in stellar systems
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Transcript of 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 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