Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 320 / 3 / a

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- Solution a

Solution

At fixed radius, use spherical coordinates in velocity space with polar angle

α

measured from the radial direction:

v_{r} = v cos α, v_{t} = v sin α, L = r v sin α .

The galactic distribution function and volume element give angular weight

L^{p - 2} d^{3} v \propto sin^{p - 1} α d α d φ .

All dependence on

v

,

r

, and

g (E)

cancels from ratios of second moments. Symmetry in the tangential plane gives

\frac{σ _{θ}^{2}}{σ _{r}^{2}} = \frac{σ _{ϕ}^{2}}{σ _{r}^{2}} = \frac{\frac{1}{2} \int _{0}^{π} s i n ^{p + 1} α d α}{\int _{0}^{π} c o s ^{2} α s i n ^{p - 1} α d α} .

Writing the angular integrals as beta function integrals and using the Gamma function recurrence,

\frac{1}{2} \frac{B (( p + 2 ) /2 , 1/2 )}{B ( p /2 , 3/2 )} = \frac{p}{2} .

Therefore, for every admissible energy factor

g

,

σ_{θ}^{2} = σ_{ϕ}^{2} = \frac{p}{2} σ_{r}^{2} .

Equivalently, the velocity-anisotropy parameter is the constant

β = 1 - p /2

.

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