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 Incoming links: Uniform-density stellar oscillation

Dynamical frequency of a uniform-density star 2026-10-06

ω_{d} = (GM / R^{3})^{1/2} = (4 π Gρ /3)^{1/2}

measures the inverse gravitational timescale of a uniform-density sphere. It is the natural normalization for uniform-density stellar oscillation and the linear restoring or destabilizing accelerations.

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 57 / 4 / b / Solution Created 2026-10-03 Updated 2026-10-06

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The spatial factor

r^{l} Y_{l}^{m}

is a regular solid harmonic. Besides the Laplace equation

\nabla^{2} F = 0

, homogeneity gives

r \cdot \nabla F = lF

. The divergence of the fluid displacement is

\nabla \cdot (U F r) \nabla \cdot (V \nabla F) = [r U^{'} + (l + 3) U] F, = \frac{l}{r} V^{'} F .

(1)

Define the scalar dilation amplitude

D_{r} = r U^{'} + (l + 3) U + (l / r) V^{'}

; this is the quantity denoted

Δ

in the question, not the notation

Δ_{L}

for a Lagrangian pressure perturbation. Then mass conservation gives

ρ = - ρ D_{r}

The equilibrium pressure gradient and homogeneity identity give

ξ \cdot \nabla p = - ρ ω_{d}^{2} (U r^{2} + l V) F .

(2)

Consequently the adiabatic equation of state yields

ρ = - ρ D_{r}, p = ρ ω_{d}^{2} (U r^{2} + l V) - γ p D_{r} .

(3)

For the force equations, the product rule gives

\nabla (p F) = p^{'} F e_{r} + p \nabla F, \nabla (Φ F) = Φ^{'} F e_{r} + Φ \nabla F .

(4)

Equating the coefficients of

F e_{r}

and

\nabla F

in the self-gravitating adiabatic displacement equations gives

ρ ω^{2} U r = ρ ω_{d}^{2} r + ρ Φ^{'} + p^{'}, ρ ω^{2} V = ρ Φ + p .

(5)

Finally, applying the Laplacian to the gravitational perturbation gives

\nabla^{2} (Φ F) = (Φ^{''} + \frac{2 ( l + 1 )}{r} Φ^{'}) F,

(6)

so the Poisson equation becomes

Φ^{''} + \frac{2 ( l + 1 )}{r} Φ^{'} = 4 π G ρ, D_{r} = r U^{'} + (l + 3) U + \frac{l}{r} V^{'} .

(7)

These are the required interior equations for uniform-density stellar oscillation.

There is a radial degeneracy at

l = 0

F

is spatially constant and

\nabla F = 0

, so the displacement is independent of

V

. The second force equation then cannot be inferred by equating independent vectors. For nonzero

ω

it may be imposed as an auxiliary definition of

V

, but it is not an additional physical radial equation. At zero frequency the radial equations should be used directly. This distinction matters for interpreting the zero factor in (c).

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Polynomial stellar-mode coefficient reduction 2026-10-06

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When the amplitudes of uniform-density stellar oscillation are finite even polynomials, their highest coefficients close algebraically. With

L = n (2 l + n + 1)

and

x = ω^{2} / ω_{d}^{2}

, the nonradial coefficient determinant is

x^{2} + (4 - γ L /2) x - l (l + 1)

. A degree balance gives a necessary frequency relation without solving every lower coefficient or imposing surface conditions.

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