Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 315 / 3 / b / Solution

Assemble a uniform-density sphere from shells. Since

m (r) = M (r / R)^{3}

and

d m = 3 M r^{2} d r / R^{3}

,

U = - \int_{0}^{R} \frac{G m ( r )}{r} d m = - \frac{3 G M ^{2}}{R ^{6}} \int_{0}^{R} r^{4} d r = - \frac{3 G M ^{2}}{5 R} .

Hydrostatic equilibrium gives its central pressure

P_{c} = \frac{3 G M ^{2}}{8 π R ^{4}} .

Thus, relative to the same uniform-density estimate for Earth,

\frac{P _{c, p}}{P _{c, \oplus}} = (\frac{M _{p}}{M _{\oplus}})^{2} (\frac{R _{p}}{R _{\oplus}})^{- 4} .

Using

(M, R)_{N} ≃ (17.1, 3.88)

gives

P_{c, N} / P_{c, \oplus} ≃ 1.3

, while

(M, R)_{J} ≃ (318, 11.2)

gives

P_{c, J} / P_{c, \oplus} ≃ 6.4

. Real central pressures differ because all three planets are compressible and compositionally stratified.

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