Kepler's Laws of Planetary Motion

Kepler's Laws of Planetary Motion are simple, yet govern the mechanics of planets, solar systems and even galaxies.

By , 19th March 2008 in Astronomy

In 1609, a German mathematician by the name of Johannes Kepler discovered a simple relationship between the distance from the Sun and a planet's orbital period. This relationship became the foundation for Kepler's laws of planetary motion.
 

These laws were not anything new at the time, however, they did improve upon the heliocentric theory of Nicolaus Copernicus, and explained how the planets' speeds varied and had elliptical orbits rather than circular orbits with epicycles.

Kepler's Laws of Planetary Motion are simple and straightforward:

  1. The orbit of every planet is an ellipse with the Sun at one of the two foci.
  2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
  3. The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis (the "half-length" of the ellipse) of their orbits.

Law number 3, published by Kepler in 1619, is the most important as it describes the relationship between the distance of planets from the Sun and their orbital periods. This law can be represented mathematically as:

T^2=R^3
Equation 2 - Kelpers Law of Planetary Motion

Where T is the orbital period in years and R is the orbital distance in AU (1AU = the distance from the Sun to Earth, or 149,598,000 kilometres)

We can see that from this equation, Earth with an orbital period of one year has an orbital radius of 1AU.

Mars has an orbital radius of 1.524 AU, so its orbital period is given by:

T^2=1.524^3
T=sqrt{1.524^3}
T=1.88
Equation 3 - Kelpers Law of Planetary Motion (worked example)

This method can be used for all the planets in our solar system orbiting the Sun, as well as moons orbiting parent planets and even exoplanets orbiting other stars.

This law also works in reverse, for example, if you know the orbital period of a planet you can calculate its orbital distance. This is important for exoplanet discovery as most of the time we cannot directly observe an exoplanet's orbit.

The Moon's orbit around the Earth is an ellipse. Once side (perigee is closer to the Earth than the other (apogee).
The Moon's orbit around the Earth is an ellipse. Once side (perigee is closer to the Earth than the other (apogee).

Kepler's Laws of Planetary motion have since become part of the foundation of modern astronomy and physics.

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