ASTC22 - Lecture 9 Relaxation in stellar systems

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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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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 )

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 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

the weak and strong encounters.

Random number generators help

to randomize perturbations.

The results are thus subject to

statistical noise.

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