Kepler’s Laws of Planetary Motion

Written by on Wednesday 19th March 2008 under Celestial Mechanics,Exoplanets

Kepler's Laws of Planetary Motion describe the motion of a planet orbiting around a star using the planets orbital period and orbital distance.


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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.

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 exoplanets 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).

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