Planetary MotionPlanetary Motion

Kepler devised three laws of planetary motion:

  1. The orbit of a planet is an ellipse with the Sun at one focus
  2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time (planets move faster closer to the Sun
  3. T^{2}propto r^{2}

Most planets in the Solar system have almost circular orbits. This means that we can use the mathematics of circular motion along with Newton’s law of gravitation to derive Kepler’s third law. 

ΣF = ma

GmM/r2 = mv2/r

Cancelling the m’s and a factor of r gives:

GM/r = v2

We can bring in the period using:

v = 2πr / T

This gives GM/r = 4π2r2/T2

Re-arranging gives:

Kepler’s Third Law: T2 = (4π2/GM) r3

The centripetal force on a planet is provided by the gravitational force between it and the Sun. Another useful equation when dealing with orbital motion of planets is:  v=sqrt{frac{GM}{r}}

A geostationary orbit is one whose orbital period is the same as the period of Earth’s rotation. It stays above the same point of the equator of the Earth. Satellites in geostationary orbits are principally used for telecommunications.