Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 320 / 2 / c

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- Solution c

Solution

For one component with density

ρ_{k} = A_{k} r^{- 2 β_{k}} Ψ^{p_{k}}

, direct velocity integration, or the Spherical Jeans equation, gives

P_{r, k} = ρ_{k} σ_{r, k}^{2} = \frac{ρ _{k} Ψ}{p _{k} + 1} .

Each component has velocity-anisotropy parameter

β_{k}

. Therefore the combined coefficient is the radial-pressure-weighted mean

β = \frac{β _{1} P _{r, 1} + β _{2} P _{r, 2}}{P _{r, 1} + P _{r, 2}} .

Here

q (r) = \frac{P _{r, 2}}{P _{r, 1}} = \frac{3}{4} \frac{ρ _{2}}{ρ _{1}} = \frac{b r}{D ( r )},

so

β (r) = \frac{3 + 2 q ( r )}{4 [ 1 + q ( r )]} .

Since

q \sim r / b /2

at the origin and

q \sim b / r

β ⟶ \frac{3}{4} as r \to 0 and as r \to \infty .

The second, less radial component reduces the anisotropy only at intermediate radii.

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