Question

3. Congratulationsl! You have just been selected as prospective commander of the Mars Orbital Mission (MOM). Your task is to place your spacecraft in circular orbit about Mars with an orbital period of 8 hours and 40 minutes. The mass of Mars is 6.45 x 1023 kg, and the radius of Mars is 3394 km. What will be the radius of your circular orbit?

Answer 1

Consider a planet with mass M_planet to orbit in nearly circular motion about the sun of mass M_Sun. The net centripetal force acting upon this orbiting planet is given by the relationship:

F_net = (M_planet * v²) / R

This net centripetal force is the result of the gravitational force that attracts the planet towards the sun, and can be represented as

F_grav = (G* M_planet * M_Sun ) / R²

Since F_grav = F_net, the above expressions for centripetal force and gravitational force are equal. Thus,

(M_planet * v²) / R = (G* M_planet * M_Sun ) / R²

Since the velocity of an object in nearly circular orbit can be approximated as v = (2*pi*R) / T,

v² = (4 * pi² * R²) / T²

Substitution of the expression for v² into the equation above yields,

(M_planet * 4 * pi² * R²) / (R • T²) = (G* M_planet * M_Sun ) / R²

By cross-multiplication and simplification, the equation can be transformed into

T² / R³ = (M_planet * 4 * pi²) / (G* M_planet * M_Sun )

The mass of the planet can then be canceled from the numerator and the denominator of the equation's right-side, yielding

T² / R³ = (4 * pi²) / (G * M_Sun ) ------------------(1)

This is Kepler's third law.

Now, according to the question,

T = 8h40mins = 31200s

M_mars = 6.45 x 10²³ kg

G = 6.67259 x 10^-11 N m²/kg²

R_orbit = ?

Using (1)

R³ = (T² *G * M_Mars ) / (4 * pi²)

R = [ (T² *G * M_Mars ) / (4 * pi²) ]^1/3

R = 10200024.2 m = 10200.02 km