Kepler's Three Laws of Planetary Motion

Johannes Kepler 15711630

Johannes Kepler was born poor and sickly in what is now Germany. His father left home when Johannes
was five and never returned. It is believed he was killed in a war. While Johannes was pursuing higher
education his mother was tried as a witch. Johannes hired a legal team which was able to obtain her
release, mostly on legal technicalities.
Although he had an eventful life, Kepler is most remembered for "cracking the code" that
describes the orbits of the planets.
Prior to Kepler's discoveries, the predominate theory of
the solar system was an Earthcentered geometry as described by Ptolemy. A Suncentered theory
had been proposed by Copernicus, but its predictions were plagued with inaccuracies.
Working in Prague at the Royal Observatory of Denmark, Kepler succeeded by using the notes of his predecessor,
Tycho Brahe, which recorded the precise position of Mars relative to the Sun and Earth.
Kepler developed his laws empirically from observation, as opposed to deriving them
from some fundamental theoretical principles. About 30 years after Kepler died,
Isaac Newton was able to derive Kepler's Laws from basic laws of gravity.
Law 1. The orbits of the planets are ellipses, with the Sun at one focus.
Any ellipse has two geometrical points called the foci (focus for singular).
There is no physical significance of the focus without the Sun but it does have
mathematical significance. The total distance from a planet to each of the foci added together
is always the same regardless of where the planet is in its orbit.
The importance of this is that
by not assuming the orbits are perfect circles, the accuracy of predictions in the Suncentered theory was
(for the first time) greater than those of the Earthcentered theory.
Law 2. The line joining a planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.
In any given amount of time, 30 days for instance, the planet sweeps out the same amount of area
regardless of which 30 day period you choose. Therefore the planet moves faster when it is nearer the Sun
and slower when it is farther from the Sun.
A planet moves with constantly changing speed as it moves about its orbit.
The fastest a planet moves is at perihelion (closest) and the slowest
is at aphelion (farthest).
Law 3. The square of the total time period (T) of the orbit is proportional to the
cube of the average distance of the planet to the Sun (R).
This law is sometimes referred to as the law of harmonies.
It compares the orbital time period and radius of an orbit of any planet, to those of the other planets.
The discovery Kepler made is that the ratio of the squares of
the revolutionary time periods to the cubes of the average distances from the Sun, is the same for
every planet.
The Marvelous Lantern
Johannes Kepler found a marvelous way out of his dilemma, how to ascertain the real shape of
Earth’s orbit. Imagine a brightly shining lantern somewhere in the plane of the orbit.
Assume we know that this lantern remains permanently in its place and thus forms a kind of fixed
triangulation point for determining the Earth’s orbit, a point which the inhabitants of Earth can
take a sight on at any time of year. Let this lantern be further away from the Sun than the Earth.
With the help of such a lantern it is possible to determine the Earth’s orbit in the following way.
First of all, in every year there comes a moment when the Earth (E) lies exactly on the line joining
the Sun (S) and the Marvelous Lantern (M). If at this moment we look from the Earth (E) at the Lantern
(M) our line of sight will coincide with the line SunLantern (SM). Suppose the line to be marked
in the heavens. Now imagine the Earth in a different position and at a different time. Since the
Sun (S) and the Lantern (M) can both be seen from the Earth, the angle at E in the triangle SEM is
known. We might do this at frequent intervals during the year, each time we should get on our piece
of paper a position of the Earth with a date attached to it and a certain position in relation to the
permanently fixed base SM. The Earth’s orbit could thereby be determined. But, you will say, where did
Kepler get his lantern? His genius and nature gave it to him. There was the planet Mars, and the length
of the Martian year was known.

Albert Einstein on the occasion of the three hundredth anniversary of Kepler’s death – November 9, 1930.

