Motion of Orbiting Satellites (College Board AP® Physics 1: Algebra-Based): Study Guide

Motion of orbiting satellites

• When a system consists only of a satellite orbiting a massive central object:

  • the mass of the orbiting satellite is considered negligible (compared to the mass of the central object)

  • both bodies orbit around a single center of mass, but the central object is considered stationary

• Satellites can be categorized as natural satellites or artificial satellites

• Examples of satellite-central-object systems include:

  • planets or comets orbiting the Sun

  • moons or spacecraft orbiting a planet

Gravitational potential energy of a system

• Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field

• The gravitational potential energy of a system is defined as:

The work done to assemble the system from infinite separation of the components of the system

• Gravitational potential energy is calculated using the following equation:

$U_g=-G\frac{m_1{m}_2}{r}$

• Where:

  • $U_g$ = gravitational potential energy of the system, in $\mathrm{J}$

  • $G$ = universal gravitational constant $\left(6.67\times {10}^{-11}\mathrm{N}·{\mathrm{m}}^2/{\mathrm{kg}}^2\right)$

  • $m_1$ = mass of the central body, in $\mathrm{kg}$

  • $m_2$ = mass of the orbiting satellite, in $\mathrm{kg}$

  • $r$ = distance between the centers of the masses, in $\mathrm{m}$

• For a satellite-central-object system:

  • potential energy is defined as zero when the satellite is an infinite distance from the central object

  • potential energy increases when the distance of the satellite from the central object increases

  • potential energy decreases when the distance of the satellite from the central object decreases

Energy of an orbiting satellite

• Gravity is a conservative force, therefore, the total energy of a satellite-central-object system is constant

• The total mechanical energy is equal to the sum of the system's kinetic and potential energies:

Total mechanical energy = kinetic energy + gravitational potential energy

$E_{total}=K+\hspace{0.17em}U$

• A satellite moving in a circular orbit of fixed orbital radius has a constant orbital speed

• Therefore, the satellite's kinetic and potential energies are also constant for a particular orbital radius

• To change the orbital radius of a satellite, work must be done, which means that:

  • if the orbital radius of the satellite decreases then $E_{total}$ decreases, $K$ increases, and $U$ decreases

  • if the orbital radius of the satellite increases then $E_{total}$ increases, $K$ decreases, and $U$ increases

Variation of energy with orbital radius

Comparing energies of satellites in different orbits

Circular and elliptical orbits

• The path of an orbiting satellite around a central object may be circular or elliptical

• In both types of orbit, the total energy and the total angular momentum of the system are constant throughout the satellite's motion

Analyzing circular orbits

• In a circular orbit, the satellite's speed and radius are constant, which means:

  • the system's total mechanical energy and the gravitational potential energy are constant

  • the satellite's angular momentum, kinetic energy, and speed are constant

Satellite in a circular orbit

Orbital speed of a satellite

• A satellite’s orbit is maintained by the force of gravitational attraction, $F_g$, between the satellite and the central object

• In a circular orbit, this force acts towards the center of the circular path as a centripetal force, $F_c$

$F_g=F_c$

$\frac{GMm}{r^2}=\frac{m{v}^2}{r}$

• Where:

  • $G$ = universal gravitational constant $\left(6.67\times {10}^{-11}\mathrm{N}·{\mathrm{m}}^2/{\mathrm{kg}}^2\right)$

  • $M$ = mass of the central body, in $\mathrm{kg}$

  • $m$ = mass of the orbiting satellite, in $\mathrm{kg}$

  • $v$ = orbital speed of the satellite, in $\mathrm{m}/\mathrm{s}$

  • $r$ = distance between the centers of the masses, in $\mathrm{m}$

• Therefore, the orbital speed of a satellite is:

$v=\sqrt{\frac{GM}{r}}$

• This means that all satellites, regardless of their mass, will travel at the same speed $v$ in a circular orbit of radius $r$

Angular momentum of a satellite

• When traveling in a circular path, the radius and the velocity of the satellite are always perpendicular to each other

• The angular momentum of the orbiting satellite is:

$L=mvr\sin 90°=mvr$

$L=mr\sqrt{\frac{GM}{r}}=m\sqrt{GMr}$

• This means that the angular momentum of a satellite traveling in a circular orbit of radius $r$ is constant

Total energy of a circular orbit

• The total mechanical energy of the system is the sum of kinetic and potential energies:

$E_{total}=K+U$

$E_{total}=\frac{1}{2}m{v}^2-\frac{GMm}{r}$

• Substituting the equation for orbital speed:

$E_{total}=\frac{1}{2}m\left(\frac{GM}{r}\right)-\frac{GMm}{r}$

$E_{total}=\frac{GMm}{2r}-\frac{GMm}{r}$

$E_{total}=-\frac{GMm}{2r}$

• Therefore, for a circular orbit, the following apply at all points:

  • $K=-\frac{1}{2}U$

  • $E_{total}=\frac{1}{2}U=-K$

Analyzing elliptical orbits

• In an elliptical orbit, the satellite's speed and radius are not constant, which means:

  • the system’s total mechanical energy and the satellite’s angular momentum are constant

  • the system’s gravitational potential energy and the satellite’s kinetic energy change throughout the satellite's motion

Satellite in an elliptical orbit

Angular momentum of a satellite

• When traveling in an elliptical path, the angle between the radius and the velocity of the satellite changes

• The angular momentum of the orbiting satellite is:

$L=mvr\sin \theta$

• Where:

  • $\theta$ = angle between the radius and the velocity of the satellite, in $°$

• As with a circular orbit, the gravitational attraction between the satellite and the central object is the only force that acts on the system

  • This force acts towards the focus of the ellipse where the central body is located

  • Since this force always acts in the same direction as the radius, it can never cause a torque $\left(\tau =Fr\sin 0=0\right)$

• Therefore, the angular momentum of the satellite is constant as no net external torque acts on the satellite-central-object system

Orbital speed of a satellite

• To conserve angular momentum, the satellite reaches maximum and minimum values of radius and velocity at the two extreme points

• At the point closest to the central object:

  • the radial distance is the shortest

  • the satellite has the highest speed

• At the point farthest from the central object:

  • the radial distance is the greatest

  • the satellite has the lowest speed

Total energy of an elliptical orbit

• To conserve mechanical energy, the satellite reaches maximum and minimum values of kinetic energy and gravitational potential energy at the two extreme points

• At the point closest to the central object:

  • the satellite has the highest kinetic energy

  • the system has the lowest (most negative) gravitational potential energy

• At the point farthest from the central object:

  • the satellite has the lowest kinetic energy

  • the system has the highest (least negative) gravitational potential energy