Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 320 / 1 / b

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- Solution b

Solution

At fixed energy, the circular orbit has

L_{c}^{2} = GM a

, while

L = L_{c} 1 - e^{2} .

An isotropic galactic distribution function has conditional angular-momentum density

n (L) d L \propto L d L

. Therefore

n (e) d e \propto L \frac{d L}{d e} d e = L_{c}^{2} e d e .

Normalization on

0 \leq e < 1

gives the thermal eccentricity distribution

n (e) = 2 e .

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