Satellite Orbit Dynamics
Any object that orbits another body must obey Newton's Law of Universal Gravitation. Which states that every object that has mass has a pull on every other object that has mass inversely proportional to the square of the distance between the objects. Therefore, the more massive the object the greater the gravitational pull between the objects and the farther apart the objects the less the pull. If you double the distance between two objects, the pull would be one-fourth as much. The equation is:
Note: mE is the mass of the earth, m is the mass of the other object and r is the distance between the objects. Any object that travels in a curved path must have a force acting on it. So for a satellite to go around the earth the gravitational pull of the earth must cause the satellite to move in the curved path. If there was no gravitational pull on the satellite it would travel in a straight line. All objects traveling in a curved path must obey the centripetal force equation. A greater force is required for an object to move faster, Therefore, if you double the speed of the object four times more force is required. The Further the object is from the center point of the curve the less force required to make it travel in a circular path. The equation for centripetal force is:
Note: v is the velocity of the object. Therefore, a force is required to travel in a curve, which must obey the centripetal force equation. The force that keeps a satellite orbiting the earth is caused by gravitational force which must obey Newton's Law of Universal Gravitation. If we set these two equations equal to each other then we have the relationship that can be solved for different parameters, such as velocity (magnitude) or distance from the center of the earth. Note; this is not the height above the earth because we can think of all the mass of the earth as acting out of the center of the earth.
For example, if we want to place a satellite so that stays in a synchronous orbit (remains over the same location of the earth), we know the earth turns once on its axis every 24 hours and we can determine the circumference of the orbit, if we know the radius. Therefore, the circumference of the orbit divided by the period (24 hours converted to seconds, 86,400) will equal the velocity of the satellite. This relationship can be used for any orbital period known.
The U.S. GPS Satellites are not in synchronous orbits; they orbit the earth twice a day. Since these satellites orbit the earth twice a day, they must be in a fixed radius orbit which could be calculated. Remember, the orbit radius is from the center of the earth and would need to have the radius of the earth subtracted off the calculation to determine the height above the surface of the earth. Intelligence gathering satellites and weather satellites are in synchronous orbits. You will not be calculating orbital distance in this course, but it is important to understand how all satellites must be placed into orbit to function properly. Some satellites are placed into polar orbits and others in equatorial orbits and still others orbit in a fashion that crosses over the earth's equator.