### Some Useful Relations Governing 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.

Consider a satellite with mass Msat orbiting a central body with a mass of mass MCentral. 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

Fnet = (Msat * 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

Fgrav = (G* Msat * MCentral ) / R2

Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force are equal. Thus,

(Msat * v2) / R = (G* Msat * MCentral ) / 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 Msat. Then both sides of the equation can be multiplied by R, leaving the following equation.

v2 = (G*MCentral ) / 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 = 6.67 x 10-11 N m2/kg2, Mcentral = the mass of the central body about which the satellite orbits, and R = the radius of orbit for the satellite. (Please note that the linear velocity of any point on a rotating body is given by v = omega x r, where omega is the angular velocity and r is the distance to the center of rotation).

Note that the radius of a satellite's orbit can be found from the knowledge of the earth's radius and the height of the satellite above the earth. As shown in the diagram at the right, the radius of orbit for a satellite is equal to the sum of the earth's radius and the height above the earth. These two quantities can be added to yield the orbital radius. In this problem, the 100 km must first be converted to 100 000 m before being added to the radius of the earth. The equations needed to determine the unknown are those which are listed above. We will begin by determining the orbital speed of the satellite using the following equation:

v = SQRT [ (G*MCentral ) / R ]

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. The acceleration of gravity is given as

g = (G*Mcentral)/R2

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

where G = 6.67 x 10-11 N m2/kg2, Mcentral = the mass of the central body about which the satellite orbits, and R = 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. The period of a satellite (T) and the mean distance from the central body (R) are related by the following equation:

where T = the period of the satellite, R = the average radius of orbit for the satellite (distance from center of central planet), and G = 6.67 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 Msatellite 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 in the case of the motion of projectiles on earth, the mass of the projectile has no effect upon the acceleration towards the earth and the speed at any instant. Whenever air resistance is negligible and all forces but gravity are nonexistent, the mass of the moving object becomes a non-factor. Such is the case of orbiting satellites.