Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 1 / vi

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- Solution vi

Solution

The limiting trajectory grazes the planet at periapsis

R_{p}

. Equating the asymptotic and periapsis values of specific angular momentum and using the vis-viva equation gives

b_{m i n}^{2} Δ v_{p c}^{2} = R_{p}^{2} (Δ v_{p c}^{2} + \frac{2 G M _{p}}{R _{p}}) .

Hence the comet avoids the planet when

b > b_{m i n} = R_{p} 1 + \frac{2 G M _{p}}{R _{p} Δ v _{p c}^{2}} .

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