Answer to Question #89569 in Astronomy | Astrophysics for Barnabé

Question

The Moon has a mass of M = 7.3*10²² kg, a radius of R = 1.7*10^6 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

(a) The rotation period of the Moon T₁ is equal to rotation period of a satellite T₂:

$$T_1=T_2 (1)$$

Centripetal acceleration of satellite:

$$a=\frac { 4 \pi^2 r } {T_2^2} (2)$$

where r is the distance from the Moon's surface to this satellite

The satellite moves according to the second law of Newton and under the action of the gravitational attraction of the Moon:

$$ma= G\frac { mM} {r^2} (3)$$

Put (3) in (4):

$$m\frac { 4 \pi^2 r } {T_2^2} =G\frac { mM} {r^2} (4)$$

Using (1) and (4) we got:

$$r=\sqrt[3]{ \frac { G M T_1^2} {4 \pi^2}} (5)$$

Using (5) we got:

r=1.9×10⁷ m

(b) The distance between the Earth and the Moon is equal to 3.84×10⁸ m. The mass of the Earth is 81 times greater than the mass of the Moon. In this case, the gravitational force of the Earth will be greater than that of the Moon. This Moon satellite is not possible in reality.