Determine the time it takes for a satellite to orbit the Mar

Determine the time it takes for a satellite to orbit the Mars in a circular \"near-Mars\" orbit. A \"near-Mars\" orbit is at a height above the surface of the Mars that is very small compared to the radius of the Mars. [Hint: You may take the acceleration due to gravity as essentially the same as that on the surface.] Does your result depend on the mass of the satellite? (y/n)

Solution

The gravitational force on the satellite is GMm/R²

Where G is the universal gravitational constant, M the mass of the mars, m is the mass of the satellite, and R is the radius of the Mars.

This equates with the centripetal force acting on the satellite which is mv²/R

Equating the two forces:

GMm/R² = mv²/R

v² = GM/R

=> v = sqrt[GM/R] ................. I

The period needed to orbit around the Mars is determined from: Distance = speed x time

Where, distance is the circumference of an orbit = 2 pi R

The speed can be calculated from I

The time is the period (T) that we have to find.

T = Distance / speed = 2 pi R/sqrt[GM/R]

T = 2 piR^(3/2) / Sqrt[GM] .................II

As, we can see from the above expression that time DOES NOT depend on the mass of the satellite. It has no \'m\' term in it.

To find that T, we need mass of Mars and radius of Mars.

M = 6.39 × 10²³

Radius of mars = 3,390 km = 3,390,000 = 3.39 * 10⁶ m

Substituting in the expression II

T = 2 pi * (3.39 * 10⁶)^(3/2) / Sqrt [6.67408 × 10^-11 * 6.39 × 10²³]

T = 6005.28 seconds = 100.088 minutes

Therefore, the time it takes for a satellite to orbit the Mars in a circular is 100.088 minutes