Relation between PE, KE, and TE for Satellites

In the context of satellites orbiting a planet, the terms PE, KE, and TE refer to the potential energy, kinetic energy, and total mechanical energy of the satellite, respectively. Understanding the relationship between these forms of energy is crucial for analyzing satellite motion and the principles of orbital mechanics.

Potential Energy (PE)

The potential energy of a satellite is the energy due to its position in a gravitational field. For a satellite orbiting Earth, the gravitational potential energy (PE) is given by the formula:

$$ PE = -\frac{G M m}{r} $$

where:

• ( G ) is the gravitational constant ((6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2)),
• ( M ) is the mass of the Earth,
• ( m ) is the mass of the satellite, and
• ( r ) is the distance from the center of the Earth to the satellite.

The negative sign indicates that the potential energy is zero at an infinite distance from the Earth and becomes more negative as the satellite gets closer to Earth.

Kinetic Energy (KE)

The kinetic energy of a satellite is the energy due to its motion. It is given by the formula:

$$ KE = \frac{1}{2} m v^2 $$

where:

• ( m ) is the mass of the satellite, and
• ( v ) is the orbital velocity of the satellite.

Total Mechanical Energy (TE)

The total mechanical energy of a satellite is the sum of its kinetic and potential energies:

$$ TE = KE + PE $$

For a satellite in a stable orbit, the total mechanical energy is constant and negative, indicating that the satellite is bound to the planet's gravitational field.

Relation between PE, KE, and TE

For a satellite in a circular orbit, the magnitude of the kinetic energy is half that of the potential energy, but with a positive value:

$$ KE = -\frac{1}{2} PE $$

This leads to the total mechanical energy being half the potential energy but negative:

$$ TE = -\frac{1}{2} PE $$

Table: Differences and Important Points

Property	Potential Energy (PE)	Kinetic Energy (KE)	Total Mechanical Energy (TE)
Formula	( -\frac{G M m}{r} )	( \frac{1}{2} m v^2 )	( KE + PE )
Dependency	Depends on position	Depends on velocity	Sum of PE and KE
Sign	Negative	Positive	Negative
Circular Orbit	( KE = -\frac{1}{2} PE )	( KE = \frac{1}{2} m v^2 )	( TE = -\frac{1}{2} PE )

Examples

Example 1: Satellite in Circular Orbit

Consider a satellite of mass ( m ) in a circular orbit around Earth at a distance ( r ) from the center of the Earth. The gravitational force provides the necessary centripetal force for the satellite's circular motion:

$$ F_{gravity} = F_{centripetal} $$

$$ \frac{G M m}{r^2} = \frac{m v^2}{r} $$

Solving for ( v ) gives the orbital velocity:

$$ v = \sqrt{\frac{G M}{r}} $$

Substituting this into the kinetic energy formula:

$$ KE = \frac{1}{2} m \left(\sqrt{\frac{G M}{r}}\right)^2 = \frac{1}{2} m \frac{G M}{r} $$

And the potential energy:

$$ PE = -\frac{G M m}{r} $$

Thus, the total mechanical energy is:

$$ TE = KE + PE = \frac{1}{2} m \frac{G M}{r} - \frac{G M m}{r} = -\frac{1}{2} m \frac{G M}{r} $$

Example 2: Satellite in Elliptical Orbit

For a satellite in an elliptical orbit, the total mechanical energy is still the sum of the kinetic and potential energies, but the values of KE and PE vary at different points in the orbit. At the closest approach (periapsis), the satellite has the highest KE and the lowest (most negative) PE. At the farthest point (apoapsis), the satellite has the lowest KE and the highest (least negative) PE.

The total mechanical energy in an elliptical orbit is given by:

$$ TE = -\frac{G M m}{2a} $$

where ( a ) is the semi-major axis of the orbit.

In summary, the relationship between PE, KE, and TE for satellites is a fundamental aspect of orbital mechanics, determining the behavior and stability of satellite orbits. The conservation of total mechanical energy is a key principle that governs satellite motion.