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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ii/paper-4/9c/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 9C / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-05

For a thin spherical shell, the radial pressure force per unit volume is

- d P / d r

. By the shell theorem, the inward gravitational acceleration is

GM (r) / r^{2}

, because exterior spherical shells contribute no force. Balancing force densities in hydrostatic equilibrium gives

\frac{d P}{d r} = - \frac{GM ( r ) ρ ( r )}{r ^{2}}, \frac{d M}{d r} = 4 π r^{2} ρ (r) .

The second equation is mass conservation; it closes the pressure-support equation once an equation of state is supplied.

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