Question

The Moon has a mass of M = 7.3 · 1022 kg, a radius of R = 1.7 · 106 m and a rotation period of
T = 27.3 days. Scientists are planning to place a satellite around the Moon that always remains
above the same position (geostationary).
(a) Calculate the distance from the Moon’s surface to this satellite.
(b) Explain if such a Moon satellite is possible in reality.

Answer 1

Gravitational force causes centripetal acceleration

$$F = \frac{GMm}{R^2} = m\omega^2 R$$

where G is gravitational constant, R is the distance from the Moon to the satellite

Using $\omega = \frac{2\pi}{T}$ ​ , we derive

$$\frac{GM}{R^3} = \frac{4\pi^2}{T^2} \Rightarrow R = \sqrt[3]{\frac{GM T^2}{4\pi^2}} = 8.82\cdot 10^7 \,\text{m} = 88\,200\,\text{km}$$

Such a satellite is impossible in reality, because be couldnt ignore gravitation of Earth, bacause it is 384 400 km away - it is comparable with R, so Earth gravitation cannot be neglected.