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

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- Solution c

Solution

The density has the form

ρ = C r^{p - 2} Ψ^{2 p + 1}, β = 1 - \frac{p}{2} .

Insert the ansatz

σ_{r}^{2} = k Ψ

into the Spherical Jeans equation

\frac{d ( ρ σ _{r}^{2} )}{d r} + \frac{2 β}{r} ρ σ_{r}^{2} = ρ \frac{d Ψ}{d r} .

The radial-power derivative cancels the anisotropy term because

p - 2 + 2 β = 0

, leaving

k (2 p + 2) = 1

. Thus

σ_{r}^{2} = \frac{Ψ}{2 ( p + 1 )}, σ_{θ}^{2} = σ_{ϕ}^{2} = \frac{p Ψ}{4 ( p + 1 )} .

Their sum is independent of

p

:

⟨ v^{2} ⟩ = σ_{r}^{2} + σ_{θ}^{2} + σ_{ϕ}^{2} = \frac{Ψ}{2} = - \frac{Φ}{2} .

Therefore the local kinetic-energy density and gravitational potential-energy density are

K = \frac{1}{2} ρ ⟨ v^{2} ⟩ = - \frac{1}{4} ρ Φ, W = \frac{1}{2} ρ Φ,

and they satisfy the local virial relation of the hypervirial model

2 K + W = 0

at every radius. This pointwise identity is not generic. The ordinary virial theorem constrains suitable global integrals, with boundary terms when the system is truncated, but does not normally impose a virial balance shell by shell.

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