Kepler analyzed Tycho Brahe's observations and proposed that planets follow elliptical orbits around the Sun, which lies at one focus, overturning the traditional assumption of circular motion.

His laws of planetary motion connect orbital geometry, swept areas, and periods. Precise observations and mathematical analysis together supported this change.

History

Kepler's laws were developed based on a physical theory of planetary motion in which the Sun emitted magnetic fibrils which pulled the planets into orbits. The fibrils were somewhat elastic allowing non-circular motion driven by the inertia of the planets.

In Astronomia nova (1609), Kepler articulated his first law, showing that Mars' orbit is elliptical, having found them by analyzing the astronomical observations of Tycho Brahe. Kepler had believed in the Copernican model of the Solar System, which called for circular orbits, but he could not reconcile Brahe's highly precise observations with a circular fit to Mars' orbit – Mars coincidentally having the highest eccentricity of all planets except Mercury. His first law reflected this discovery.

In his Astronomia nova (1609), Kepler did not present his second law in its modern form. He did that only in his Epitome Astronomiae Copernicanae of 1621. Kepler had two versions of the second law, related in a qualitative sense: the first "distance law" and later the "area law". The distance form was only correct for orbits that were almost circular, but the area form was correct for all elliptical orbit. The "area law" is what became the second law in the set of three. This law had little impact on astronomy because calculations of planetary positions using the law were approximate and time consuming. The second law, in the "area law" form, was contested by Nicolaus Mercator in a book from 1664, but by 1670 his Philosophical Transactions were in its favour. As the century proceeded it became more widely accepted.

Kepler's third law was published in 1619 in his Harmonice Mundi. In 1621, Kepler noted that his third law applies to the four brightest moons of Jupiter. Godefroy Wendelin, the first well-known astronomer to adopt Kepler's laws, gave a detailed account of the third law in 1652.

Kepler's work had little initial impact. His work was a strong defense of Copernicanism which had fallen out of fashion in part because of opposition by Tycho Brahe. In 1627 Kepler published the Rudolphine Tables containing many accurate astronomical observations accumulated by Brahe. The breadth and accuracy of the tables allowed astronomers to compare Kepler's formula to good quality data. At first these difficult calculations were off putting, but once undertaken more astronomers became convinced of Kepler's approach.

The reception in Germany changed noticeably between 1688, shortly after Newton's Principia was published and was taken to be basically Copernican, and 1690, by which time work of Gottfried Leibniz on Kepler had been published. Newton understood that the second law is not special to the inverse-square law of gravitation, being a consequence just of the radial nature of that law, whereas the other laws do depend on the inverse-square form of the attraction. Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the phase space of planetary motion (the orthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.

History and proofs

Nevertheless, the result of the second law is exactly true, as it is logically equivalent to the conservation of angular momentum, which is true for any body experiencing a radially symmetric force. A correct proof can be shown through this. Since the cross product of two vectors gives the area of a parallelogram possessing sides of those vectors, the triangular area dA swept out in a short period of time is given by half the cross product of the r and dx vectors, for some short piece of the orbit, dx: for a small piece of the orbit dx and time to cover it dt. Thus Since the final expression is proportional to the total angular momentum, Kepler's equal area law will hold for any system that conserves angular momentum. Since any radial force will produce no torque on the planet's motion, angular momentum will be conserved.