 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-3/14b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

At the centre, spherical symmetry and regularity require a finite central density

ρ (0) = ρ_{c}

and

\frac{d P}{d r}_{r = 0} = 0.

At the stellar surface

r = R

, the pressure vanishes:

P (R) = 0.

For a stellar polytrope these become the dimensionless conditions

θ (0) = 1

θ^{'} (0) = 0

, and the surface is the first zero

θ (ξ_{1}) = 0

; specifying

ρ_{c}

fixes the overall scale.

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