# Johannes Kepler and the Proof of His First Planetary Law

*Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.*

## Introduction

Johannes Kepler lived in a time of great astronomical and mathematical discovery. Telescopes were invented, asteroids were being discovered, observations of the heavens improved, and the precursors to calculus were in the works during his lifetime, leading to a deeper development of celestial mechanics. But Kepler himself made numerous contributions not only to astronomy but also in mathematics as well as philosophy. It is, however, his Three Planetary Laws that he is most remembered for and whose practicality has not been lost to this day.

## Early Life

Kepler was born on December 27, 1571 in Weil der Stadt, Wurttemberg, what is now Germany. As a child, he assisted his grandfather at his inn, where his mathematical skills were honed and noticed by the patrons. As Kepler grew older, he developed deep religious views, in particular that God made us in His image and thus gave His creations a way to understand His universe, which in Kepler’s eyes was mathematical. When he went to school, he was taught the Geocentric Model of the universe, in which the Earth was the center of the cosmos and everything revolved around it. After his instructors realized his talents when he nearly aced all his classes, he was taught the (at the time) controversial model of the Copernican System in which the universe still revolves around a central point but it is the Sun and not the Earth (Heliocentric). However, something struck Kepler as odd: why were the orbits assumed to be circular? (Fields)

## Mystery of the Cosmos

After leaving school, Kepler gave his orbit problem some thought and arrived at a mathematically beautiful, though incorrect, model. In his book *Mystery of the Cosmos*, he postulated that if you treat the moon as a satellite, a total of six planets remain. If the orbit of Saturn is the circumference of a sphere, he inscribed a cube inside the sphere and inside that cube inscribed a new sphere, whose circumference was treated as the orbit of Jupiter, seen on the upper right. Using this pattern with the remaining four regular solids that Euclid proofed in his *Elements*, Kepler had a tetrahedron between Jupiter and Mars, a dodecahedron between Mars and Earth, an icosahedron between Earth and Venus, and an octahedron between Venus and Mercury as seen to the lower right. This made perfect sense to Kepler since God designed the Universe and geometry was an extension of His work, but the model contained a small error in the orbits still, something not fully explained in *Mystery* (Fields).

## Mars and the Mysterious Orbit

This model, one of the first defenses of the Copernican theory, was so impressive to Tycho Brahe that it got Kepler a job at his observatory. At the time, Tycho was working on the mathematical properties of the orbit of Mars, making tables upon tables of observations in hopes of uncovering its orbital mysteries (Fields). Mars was chosen for study because of (1) how fast it moves through its orbit, (2) how it is viewable without being near the Sun, and (3) its non-circular orbit being the most prominent of the known planets at the time (Davis). Once Tycho passed away, Kepler took over and eventually discovered that the orbit of Mars was not just non-circular but elliptical (his 1^{st} Planetary Law) and that the area covered from the planet to the Sun in a certain timeframe was consistent no matter what that area might be (his 2^{nd} Planetary Law). He eventually was able to extend these laws to the other planets and published it in *Astronomia Nova* in 1609 (Fields, Jaki 20).

## 1st Attempt at the Proof

Kepler did prove that his three laws are true, but Laws 2 and 3 are shown to be true by using observations and not with much proof techniques as we would call them today. Law 1, however, is a combination of physics as well as some mathematical proof. He noticed that at certain points of Mar’s orbit it was moving slower than expected and at other points it was moving faster than expected. To compensate for this, he began to draw the orbit as an oval shape, seen right, and approximated its orbit using an ellipse he found that, with a radius of 1, that the distance AR, from the circle to the minor axis of the ellipse, was 0.00429, which was equal to e^{2}/2 where e is CS, the distance from between the center of the circle and one of the foci of the ellipse, the Sun. Using the ratio CA/CR = [1-(e^{2}/2)]^{-1 }where CA is the radius of the circle and CR is the minor axis of the ellipse, was approximately equal to 1+(e^{2}/2). Kepler realized that this was equal to the secant of 5° 18’, or ϕ, the angle made by AC and AS. With this he realized that at any beta, the angle made by CQ and CP, the ratio of the distance SP to PT was also the ratio of VS to VT. He then assumed that the distance to Mars was PT, which equals PC +CT = 1 + e*cos(beta). He tried this out using SV = PT, but this produced the wrong curve (Katz 451)

## The Proof Is Corrected

Kepler corrected this by making the distance 1 +e*cos (beta), labeled p, the distance from a line perpendicular to CQ ending at W as seen to the right. This curve accurately predicted the orbit. To give a final proof, he assumed that an ellipse was centered at C with a major axis of a=1 and a minor axis of b = 1-(e^{2}/2), just like before, where e = CS. This can also be a circle of radius 1 by reducing terms perpendicular to QS by b since QS lies on the major axis and perpendicular to that would be the minor axis. Let v be the the angle of the arc RQ at S. Thus, p * cos(v) = e + cos (beta) and p * sin(v) = b * sin^{2}(beta). Squaring both of them and adding will result in

p^{2}[cos^{2 }(v) + sin^{2}(v)] = e^{2 }+ 2e * cos (beta) + cos^{2} (beta) + b^{2} * sin^{2} (beta)

which reduces to

p^{2}= e^{2 }+ 2e * cos (beta) + cos^{2} (beta) + [1-(e^{2} /2)] ^{2} * sin^{2} (beta)

which reduces further down to

p^{2}= e^{2 }+ 2e * cos (beta) + 1 - e^{2} * sin^{2} (beta) + (e^{4}/4)*sin (beta)

Kepler now ignores the e^{4} term, giving us:

p^{2}= e^{2 }+ 2e * cos (beta) + 1 - e^{2} * sin^{2} (beta)

= e^{2 }+ 2e * cos (beta) + e^{2} * cos^{2} (beta)

=[1 + e * cos (beta)]^{2}

p = 1 + e * cos (beta)

The same equation that he found empirically (Katz 452).

## Kepler Explores

After Kepler solved the Mars orbit problem, he began to focus on other areas of science. He did work on optics while he was waiting for *Atronomica Nova* to be published and created the standard telescope using two convex lenses, otherwise known as the refracting telescope. While at the wedding reception of his second wedding, he noticed that the volumes of the wine barrels were calculated by inserting a rob into the barrel and seeing how much of the rod was wet. Using Archemedian techniques, he uses indivisibles, a precursor to calculus, to solve the problem of their volumes and publishes his results in *Nova Stereometria Doliorum* (Fields).

## Kepler Returns to Astronomy

Eventually though, Kepler found his way back to the Copernican system. In 1619, he publishes *Harmony of the World*, which expands upon *Mystery of the Cosmos. *He proofs that there are only thirteen regular convex polyhedral and also states his 3^{rd} planetary law, P^{2 }= a^{3} ,where P is the period of the planet and a is the mean distance from the planet to the Sun. He also attempts to further demonstrate the musical properties of the ratios of the planetary orbits. In 1628, his astronomical tables are added to the *Rudolphine Tables*, as well as his demonstration of logarithms (usind Euclids *Elements*) that proved so accurate in their use for astronomy that they were the standard for years to come (Fields). It was through his use of logarithms that he most likely derived his third law, for if log(P) is plotted against log(a), the relation is clear (Dr. Stern).

## Conclusion

Kepler passes away November 15. 1630 in Regensburg (now Germany). He was buried at the local church, but as the Thirty Years War progressed, the church was destroyed and nothing remains of it or Kepler. However, Kepler and his contributions to science are his enduring legacy even if he has no tangible remains left on Earth. Through him, the Copernican system was given a proper defense and the mystery of planetary orbit shapes was solved.

## Works Cited

Davis, A E L. Kepler's Planetary Laws. October 2006. 9 March 2011 <http://www-history.mcs.st-and.ac.uk/HistTopics/Keplers_laws.html>.

Dr. Stern, David P. Kepler and His Laws. 21 June 2010. 9 March 2011 http://www.phy6.org/stargaze/Skeplaws.htm.

Fields, J.V. Kepler Biography. April 1999. 9 March 2011 http://www-history.mcs.st-and.ac.uk/Biographies/Kepler.html.

Jaki, Stanley L. Planets and Planetarians: A History of Theories of the Origin of the Planetary Systems. John Wiley & Sons, Halstead Press: 1979. Print. 20.

Katz, Victor. A History of Mathematics: An Introduction. Addison-Wesley: 2009. Print. 446-452.

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**© 2011 Leonard Kelley**