Answer to Question #89592 in Astronomy | Astrophysics for sathwik

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

In the case of a stationary circular rotation of a satellite of mass m, the Moon’s gravitational force (gravity) Fg and the centrifugal force Fc act on it, they balance each other (Fg=Fc).

$$Fc=mv^2/(R+h)$$

where v - speed of the spacecraft, m - mass of the spacecraft, R - radius of the moon, h - height of the spacecraft above the surface of the moon.

$$Fg=GMm/(R+h)^2$$

G – gravitational coVnstant, M - the mass of the Moon

Equating forces and using the formula for speed

$$v=2\pi(R+h)/T$$

(T- rotation period):

$$h=cubic root(T^2GM/(4\pi^2))-R$$

$$h=8.7*10^7 m$$

$$h>>R$$

Too far from the moon, this is impossible