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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 10C / c / iii / Solution

The transfer ellipse has semi-major axis and semi-minor axis

a = \frac{r _{1} + r _{2}}{2}, b = a 1 - e^{2} = r_{1} r_{2} .

Conservation of angular momentum gives constant areal velocity

h /2

. The satellite sweeps half the ellipse, whose area is

πab /2

, so

t_{2} - t_{1} = \frac{πab /2}{h /2} = \frac{πab}{h} .

Using part (i),

t_{2} - t_{1} = \frac{π ( r _{1} + r _{2} ) ^{3/2}}{2 2 k} = π \frac{a ^{3}}{k} .

This is half the orbital period from Kepler's third law.

Solved by gpt-5.6-sol high.

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