Kepler’s Third Law Of Planetary Motion

      Edmund Halley was faced with a real puzzle when he began his quest to determine the distance of Earth from the sun.   One of the pieces to solving that puzzle came from the work of a German mathematician, astronomer and astrologer named Johannes Kepler.

      Early in the 17th Century, Kepler spent a lot of time observing the planets in our solar system as they orbited the sun.   He discovered that by taking note of the time it took for a planet to make one orbit around the sun, he could determine its relative distance from it.   He then compared his findings with other planets, noting the time it took for each to make this same journey.   His discovery would come to be known as Kepler’s Third Law of Planetary Motion.

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      But what exactly is meant by a planet’s “relative distance from the sun”?   In essence, it means that interplanetary distances, like just about everything else, are relative.   Put another way, heavenly bodies can be said to be a distance X relative to another heavenly body if you establish a value for X, whether it’s numerical or otherwise.

      In Kepler’s case, X would be the unknown value of a so-called astronomical unit (AU), where one AU is equal to the unknown distance from Earth to the sun.   Represented in equation form, this distance is:

rEarth-sun = 1 AU

      This relative marker of distance could then be used to show how far the other planets are from the sun, relative to Earth’s distance from the sun, the AU.   Kepler’s astronomical unit is simply a placeholder term for an unknown quantity, similar to any other unknown variable that might be used in an algebraic equation.

      For example, Kepler observed the orbits of Venus and Mars and determined their relative distances to the sun to be:

rVenus-sun = 0.72 × 1 AU = 0.72 AU

rMars-sun = 1.5 × 1 AU = 1.5 AU

     In other words, Venus’ distance from the sun is just under three quarters of Earth’s and Mars is one and a half times Earth’s distance from the sun.   In this way, Kepler was able to determine the relative distances from the sun in AU for all the observable planets in our solar system.

      Kepler felt sure that one day scientists would be able to accurately measure Earth’s distance from the sun, and when they accomplished this they could employ his astronomical unit system to determine distances between other planets in our solar system and the sun.

      Next time we’ll introduce the principle of parallax and see how Halley used this optical effect to devise a method for assigning a value to Kepler’s unknown AU.


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