Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 321 / 2 / a

Solution

R

from the spherical radius

r = R^{2} + z^{2}

∣ z ∣/ R ≪ 1

gives

Φ = - \frac{GM}{R} (1 + \frac{z ^{2}}{R ^{2}})^{- 1/2} = - \frac{GM}{R} + \frac{GM z ^{2}}{2 R ^{3}} + O (\frac{GM z ^{4}}{R ^{5}}) .

Φ (R, z) ≃ - \frac{GM}{R} + \frac{1}{2} Ω_{K}^{2} (R) z^{2}, Ω_{K}^{2} = \frac{GM}{R ^{3}} .

The vertical acceleration is

- \partial_{z} Φ = - Ω_{K}^{2} z

. The first term can be omitted when considering only vertical forces at fixed

R

, but must be retained for radial dynamics.

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