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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.

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Kepler's Laws of Planetary Motion
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.

Keplers First Law: The Law of Ellipses

Kepler's first law explains that planets are orbiting the Sun in a path described as an ellipse. An ellipse has two points of focus, unlike a circle which has one. In an ellipse, the sum of the distances from every point on the curve to two other points is a constant.

Elliptical Orbits - Aphelion, Perihelion, Perigee, Apogee
Elliptical Orbits - Aphelion, Perihelion, Perigee, Apogee

Keplers Second Law: The Law of Equal Areas

Kepler's second law describes the speed at which any given planet will move while orbiting the Sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. If an imaginary line were drawn from the centre of the planet to the centre of the Sun, that line would sweep out the same area in equal periods of time.

An imaginary line drawn from the Sun to any point on the ecliptic orbit sweeps out an equal area for equal amounts of time
An imaginary line drawn from the Sun to any point on the ecliptic orbit sweeps out an equal area for equal amounts of time

Keplers Third Law: The Law of Harmonies

Kepler's third law - sometimes referred to as the law of harmonies - compares the orbital period and radius of the orbit of a planet to those of other planets. Unlike Kepler's first and second laws that describe the motion characteristics of a single planet, the third law makes a comparison between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets.

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.

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

Last updated: 2017-07-19

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