ASTC22 - Lecture 9
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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 and cusps

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 and cusps

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


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