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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 346 / 1 / Solution Created 2026-09-24 Updated 2026-09-25

I_{jk} = \int ρ x_{j} x_{k} d^{3} x .

Using mass conservation to differentiate a material moment gives

\dot{I}_{jk} = \int ρ (x_{j} ⟨ v_{k} ⟩ + x_{k} ⟨ v_{j} ⟩) d^{3} x .

Multiply the Jeans equation for component

j

by

x_{k}

, integrate over space, discard the surface term of an isolated system, and symmetrize in

j, k

. The result is

\frac{1}{2} \ddot{I}_{jk} = \int ρ ⟨ v_{j} v_{k} ⟩ d^{3} x - \frac{1}{2} \int ρ (x_{k} Φ_{, j} + x_{j} Φ_{, k}) d^{3} x .

Decompose

⟨ v_{j} v_{k} ⟩ = \overset{v}{ˉ}_{j} \overset{v}{ˉ}_{k} + σ_{jk}^{2}

T_{jk} = \frac{1}{2} \int ρ \overset{v}{ˉ}_{j} \overset{v}{ˉ}_{k} d^{3} x, Π_{jk} = \int ρ σ_{jk}^{2} d^{3} x,

W_{jk} = - \frac{1}{2} \int ρ (x_{k} Φ_{, j} + x_{j} Φ_{, k}) d^{3} x .

This proves the tensor virial theorem

\frac{1}{2} \ddot{I}_{jk} = 2 T_{jk} + Π_{jk} + W_{jk} .

For a steady oblate system rotating about

z

,

T_{zz} = 0

, while symmetry gives

T_{xx} = T_{yy}

. The

zz

and

xx

Π_{zz} + W_{zz} = 0, 2 T_{xx} + Π_{xx} + W_{xx} = 0.

Using

Π_{zz} = (1 - δ) Π_{xx}

,

2 T_{xx} = M v_{0}^{2}

, and

Π_{xx} = M σ_{0}^{2}

gives

\frac{v _{0}^{2}}{σ _{0}^{2}} = (1 - δ) \frac{W _{xx}}{W _{zz}} - 1 .

With

W_{xx} / W_{zz} ≃ (1 - ϵ)^{- 0.9}

, the edge-on curves are

\frac{v _{0}}{σ _{0}} = [(1 - δ) (1 - ϵ)^{- 0.9} - 1]^{1/2}, δ = 0, 0.1, 0.2, 0.3.

At fixed ellipticity, increasing anisotropy lowers

v_{0} / σ_{0}

; the curve starts only once

(1 - ϵ)^{- 0.9} \geq (1 - δ)^{- 1}

. The

δ = 0

line is the oblate isotropic rotator reference.

At inclination

i < 9 0^{\circ}

, line-of-sight rotation is reduced approximately by

sin i

, and the projected ellipticity also decreases as the spheroid is viewed closer to its symmetry axis. A galaxy therefore moves down and left from its edge-on location, with the exact track set by intrinsic thickness and anisotropy.

Low-mass elliptical galaxies mostly occupy the fast-rotating, flattened, nearly isotropic region close to the oblate-rotator line. High-mass systems mostly occupy the slow-rotating region below it and require anisotropy or triaxiality. Gas-rich dissipative evolution retains angular momentum, forms a compact rotating stellar component, and produces low-mass fast rotators. Repeated dry major mergers randomize orbits, lower specific angular momentum, scour central cores through black-hole binaries, and build massive slow rotators.

Four characteristic contrasts are:

high-mass ellipticals rotate slowly, while low-mass ellipticals are commonly fast rotators;
high-mass systems are anisotropic and often triaxial or boxy, while low-mass systems are closer to oblate and disky;
high-mass systems commonly have shallow central cores, while low-mass systems have steep central cusps or extra central light;
high-mass systems are generally older, redder, more alpha-enhanced, and richer in hot X-ray gas, while lower-mass systems more often show younger populations, cold gas, and residual star formation.

These are population trends rather than sharp boundaries in the fast and slow rotator galaxy classification.

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 346 / 2 / Solution Created 2026-09-24 Updated 2026-09-25

A galaxy contains so many stars that its two-body relaxation time is generally much longer than its age. Individual encounters can therefore be neglected and each star moves in the smooth collective potential. Liouville conservation along these Hamiltonian trajectories gives the Collisionless Boltzmann equation

\frac{d F}{d t} = 0.

In spherical phase-space coordinates this is

F_{t} + \overset{r}{˙} F_{r} + \dot{θ} F_{θ} + \overset{φ}{˙} F_{φ} + \overset{v}{˙}_{r} F_{v_{r}} + \overset{v}{˙}_{θ} F_{v_{θ}} + \overset{v}{˙}_{φ} F_{v_{φ}} = 0 .

The spherical line element is

d s^{2} = d r^{2} + r^{2} d θ^{2} + r^{2} sin^{2} θ d φ^{2}

, so

\overset{r}{˙} = v_{r}, \dot{θ} = v_{θ} / r, \overset{φ}{˙} = v_{φ} / (r sin θ) .

For a unit-mass star in a spherical potential,

L = \frac{1}{2} (\overset{r}{˙}^{2} + r^{2} \dot{θ}^{2} + r^{2} sin^{2} θ \overset{φ}{˙}^{2}) - Φ (r) .

The Euler--Lagrange equations, followed by differentiating

v_{θ} = r \dot{θ}

and

v_{φ} = r sin θ \overset{φ}{˙}

, give

\overset{v}{˙}_{r} = \frac{v _{θ}^{2} + v _{φ}^{2}}{r} - Φ^{'} (r),

\overset{v}{˙}_{θ} = \frac{v _{φ}^{2} cot θ - v _{r} v _{θ}}{r}, \overset{v}{˙}_{φ} = - \frac{v _{φ} v _{r} + v _{φ} v _{θ} cot θ}{r} .

Integrating the Boltzmann equation over velocity space, with vanishing velocity-space boundary terms, gives spherical mass conservation:

ρ_{t} + \partial_{r} (ρ ⟨ v_{r} ⟩) + \frac{2}{r} ρ ⟨ v_{r} ⟩ = 0 .

Multiplication by

v_{r}

and integration gives the radial Jeans equation. With isotropic dispersion

⟨ v_{r}^{2} ⟩ = σ^{2} + ⟨ v_{r} ⟩^{2}, ⟨ v_{θ}^{2} ⟩ = ⟨ v_{φ}^{2} ⟩ = σ^{2},

the geometric dispersion terms cancel, and use of continuity yields

\partial_{r} (ρ σ^{2}) + ρ \partial_{t} ⟨ v_{r} ⟩ + ρ ⟨ v_{r} ⟩ \partial_{r} ⟨ v_{r} ⟩ = - ρ Φ^{'} (r) .

The factor

ρ

on the right is required dimensionally.

In a Lambda-CDM background, the local excess mass contributes

GM (r) / r^{2}

, homogeneous matter inside radius

r

contributes

4 π G ρ_{b} r /3

, and the cosmological constant contributes outward acceleration

Λ r /3

. Hence

Φ^{'} (r) = \frac{GM ( r )}{r ^{2}} + \frac{4 π G ρ _{b}}{3} r - \frac{Λ}{3} r .

Since

Ω_{m} = 8 π G ρ_{b} / (3 H^{2})

,

Ω_{Λ} = Λ/ (3 H^{2})

, and

q = Ω_{m} /2 - Ω_{Λ}

,

Φ^{'} (r) = \frac{GM ( r )}{r ^{2}} + q H^{2} r .

Write

⟨ v_{r} ⟩ = Hr + v_{p}

, where

v_{p}

is the peculiar velocity. Then

\partial_{t} ⟨ v_{r} ⟩ + ⟨ v_{r} ⟩ \partial_{r} ⟨ v_{r} ⟩ = (\dot{H} + H^{2}) r + v_{p, t} + H v_{p} + Hr v_{p, r} + v_{p} v_{p, r} .

The acceleration equation gives

\dot{H} + H^{2} = \overset{a}{¨} / a = - q H^{2}

, which cancels the background term in

Φ^{'}

. Therefore

\partial_{r} (ρ σ^{2}) = - ρ [\frac{GM ( r )}{r ^{2}} + S (r, t)],

S = v_{p} v_{p, r} + H (v_{p} + r v_{p, r}) + v_{p, t} .

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