Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 2 / b

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- Solution b

Solution

Assume Newtonian spherical hydrostatic equilibrium, neglect rotation and thermal density changes, and require continuous pressure at the layer boundary. Write

C = M_{1} - \frac{4 π}{3} ρ_{2} R_{1}^{3}, D = \frac{4 π}{3} ρ_{2},

so the mantle mass profile is

M (r) = C + D r^{3}

. Integrating

d P / d r = - G ρ_{2} M (r) / r^{2}

inward from

P (R_{p}) = P_{0}

gives the boundary pressure

P_{b} = P_{0} + G ρ_{2} [C (\frac{1}{R _{1}} - \frac{1}{R _{p}}) + \frac{D}{2} (R_{p}^{2} - R_{1}^{2})] .

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