Properties of Circular Orbits
Learning Goal: To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth.

The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit--a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass .

For all parts of this problem, where appropriate, use for the universal gravitational constant.

Part A
Find the orbital speed for a satellite in a circular orbit of radius .
Part A.1 Find the force Part not displayed
Part A.2 A basic kinematic relation Part not displayed
Hint A.3 Newton's 2nd law Hint not displayed
Express the orbital speed in terms of , , and .
 = sqrt(G*M/R)
Part B
Find the kinetic energy of a satellite with mass in a circular orbit with radius .
 = G*M*m/(2*R)
Part C
Express the kinetic energy in terms of the potential energy .
Part C.1 Potential energy
What is the potential energy of the satellite in this orbit?
 = -G*M*m/R
 = -1/2*U
This is an example of a powerful theorem, known as the Virial Theorem. For any system whose motion is periodic or remains forever bounded, and whose potential behaves as

,

Rudolf Clausius proved that

,

where the brackets denote the temporal (time) average.
Part D
Find the orbital period .
Hint D.1 Hint not displayed
 = 2*pi*R^(3/2)*(G*M)^(-1/2)
Part E
Find an expression for the square of the orbital period.
 = (2*pi)^2/(G*M)* R^3
This shows that the square of the period is proportional to the cube of the semi-major axis. This is Kepler's Third Law, in the case of a circular orbit where the semi-major axis is equal to the radius, .
Part F
Find , the magnitude of the angular momentum of the satellite with respect to the center of the planet.
Hint F.1 Definition of angular momentum Recall that , where is the momentum of the object and is the vector from the pivot point. Here the pivot point is the center of the planet, and since the object is moving in a circular orbit, is perpendicular to .