Conceptual Physics

T = satellite period (in seconds)

G = gravitational constant

M = mass of central body

a = semimajor axis

12.17 - Orbits and energy

Satellites have both kinetic energy and potential energy. The KE and PE of a satellite

in elliptical orbit both change as it moves around its orbit. This is shown in Concept 1.

Energy gauges track the satellite’s changing PE and KE. When the satellite is closer to

the body it orbits, Kepler’s second law states that it moves faster, and greater speed

means greater KE.

The PE of the system is less when the satellite is closer to the body it orbits. When

discussing gravitational potential energy, we must choose a reference point that has

zeroPE. For orbiting bodies, that reference point is usually defined as infinite

separation. As two bodies approach each other from infinity, potential energy decreases

and becomes increasingly negative as the value declines from zero.

Concept 2 shows that even while the PE and KE change continuously in an elliptical

orbit, the total energy TE stays constant. Because there are no external forces acting

on the system consisting of the satellite and the body it orbits, nor any internal

dissipative forces, its total mechanical energy must be conserved. Any increase in

kinetic energy is matched by an equivalent loss in potential energy, and vice versa.

In a circular orbit, a satellite’s speed is constant and its distance from the central body

remains the same, as shown in Concept 3. This means that both its kinetic and potential

energies are constant.

The total energy of a satellite increases with the radius (in the case of circular orbits) or

the semimajor axis (in the case of elliptical orbits). Moving a satellite into a larger orbit

requires energy; the source of that energy for a satellite might be the chemical energy

present in its rocket fuel.

Equations used to determine the potential and kinetic energies and the total energy of a

satellite in circular orbit are shown in Equation 1. These equations can be used to

determine the energy required to boost a satellite from one circular orbit to another with

a different radius. The KE equation can be derived from the equation for the velocity of

a satellite. The PE equation holds true for any two bodies, and can be derived by

calculating the work done by gravity as the satellite moves in from infinity.

The equations have an interesting relationship: The kinetic energy of the satellite equals

one-half the absolute value of the potential energy. This means that when the radius of

a satellite’s orbit increases, the total energy of the satellite increases. Its kinetic energy

decreases since it is moving more slowly in its higher orbit, but the potential energy

increases twice as much as this decrease in KE.

Equation 2 shows the total energy equation for a satellite in an elliptical orbit. This equation uses the value of the semimajor axis a instead of

the radius r.

Orbital energy

Satellites have kinetic and potential

energy

Since PE = 0 at infinite distance, PE

always negative

For a given orbit:

Total energy is constant

(^236) Copyright 2000-2007 Kinetic Books Co. Chapter 12