Mathematics of Satellite Motion

Mathematics of Satellite Motion

The motion of objects are governed by Newton's laws. The same simple laws which govern the motion of objects on earth also extend to the heavens to govern the motion of planets, moons, and other satellites. The mathematics which describes a satellite's motion are the same mathematics presented for circular motion. In this part we will be concerned with the variety of mathematical equations which describe the motion of satellites.

Consider a satellite with mass M_sat orbiting a central body with a mass of mass M_Central. The central body could be a planet, the sun or some other large mass capable of causing sufficient acceleration on a less massive nearby object. If the satellite moves in circular motion, then the net centripetal force acting upon this orbiting satellite is given by the relationship

F_net = ( M_sat • v² ) / R

This net centripetal force is the result of the gravitational force which attracts the satellite towards the central body and can be represented as

F_grav = ( G • M_sat • M_Central ) / R²

Since F_grav = F_net, the above expressions for centripetal force and gravitational force can be set equal to each other. Thus,

(M_sat • v²) / R = (G • M_sat • M_Central ) / R²

Observe that the mass of the satellite is present on both sides of the equation; thus it can be canceled by dividing through by M_sat. Then both sides of the equation can be multiplied by R, leaving the following equation.

v² = (G • M_Central ) / R

Taking the square root of each side, leaves the following equation for the velocity of a satellite moving about a central body in circular motion

where G is 6.673 x 10^-11 N•m²/kg², M_central is the mass of the central body about which the satellite orbits, and R is the radius of orbit for the satellite.

Similar reasoning can be used to determine an equation for the acceleration of our satellite that is expressed in terms of masses and radius of orbit. The acceleration value of a satellite is equal to the acceleration of gravity of the satellite at whatever location which it is orbiting. In below, the equation for the acceleration of gravity was given as

g = (G • M_central)/R²

Thus, the acceleration of a satellite in circular motion about some central body is given by the following equation

where G is 6.673 x 10^-11 N•m²/kg², M_central is the mass of the central body about which the satellite orbits, and R is the average radius of orbit for the satellite.

The final equation which is useful in describing the motion of satellites is Newton's form of Kepler's third law. Since the logic behind the development of the equation has been presented elsewhere, only the equation will be presented here. The period of a satellite (T) and the mean distance from the central body (R) are related by the following equation:

where T is the period of the satellite, R is the average radius of orbit for the satellite (distance from center of central planet), and G is 6.673 x 10^-11 N•m²/kg².

There is an important concept evident in all three of these equations - the period, speed and the acceleration of an orbiting satellite are not dependent upon the mass of the satellite.

None of these three equations has the variable M_satellite in them. The period, speed and acceleration of a satellite is only dependent upon the radius of orbit and the mass of the central body which the satellite is orbiting. Just as in the case of the motion of projectiles on earth, the mass of the projectile has no affect upon the acceleration towards the earth and the speed at any instant. When air resistance is negligible and only gravity is present, the mass of the moving object becomes a non-factor. Such is the case of orbiting satellites.