A relay satellite beyond the Moon

Adapted from IOAA 2019 (Keszthely), theory problem 9. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

On 3 January 2019 the Chinese spacecraft Chang'e-4 landed on the far side of the Moon. Earth is always below the horizon there, so a relay satellite, Queqiao, was placed in an orbit round the Earth–Moon L2 point, beyond the far side of the Moon. Find the height $h$ of L2 above the Moon's surface. Take the Moon's orbit as a circle of radius $R=384400$ km, the Moon's mass as 0.0123 of Earth's and its radius as 1737 km, and ignore the Sun and planets. You may use $1/(1+x{)}^2\approx 1-2x$ for small $x$.

AnswerThe height is about 59 700 km above the Moon's surface. Solving the equation exactly, without the approximation, puts L2 at 0.1678 of the Earth–Moon distance, 64 500 km from the Moon's centre and 62 800 km above its surface, so the approximation is about 5 per cent short.