Summarizing Satellites and Kepler’s Laws

Summary

• Kepler’s laws are stated for a small mass \(m\) orbiting a larger mass \(M\) in near-isolation. Kepler’s laws of planetary motion are then as follows:

  Kepler’s first law

  The orbit of each planet about the Sun is an ellipse with the Sun at one focus.

  Kepler’s second law

  Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.

  Kepler’s third law

  The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun:

  \(\cfrac{{T}_{1}^{ 2}}{{T}_{2}^{ 2}}=\cfrac{{r}_{1}^{ 3}}{{r}_{2}^{ 3}}\text{,}\)

  where \(T\) is the period (time for one orbit) and \(r\) is the average radius of the orbit.

• The period and radius of a satellite’s orbit about a larger body \(M\) are related by

  \({T}^{2}=\cfrac{{4\pi }^{2}}{\text{GM}}{r}^{3}\)

  or

  \(\cfrac{{r}^{3}}{{T}^{2}}=\cfrac{G}{{4\pi }^{2}}M\text{.}\)