Kepler’s Laws

How do the planets move around the sun?

The problem is made much more difficult because we observe it from a moving platform.

horocycles

Astronomy was considered a branch of Mathematics, not part of what we would call Physics.

Early astronomers looked for mathematical explanations of the paths of the planets. They were severely hampered by the assumption that the earth was at the centre, and everything else moved around it.

Tycho BraheTycho Brahe (1546-1601) was a Danish astronomer who made detailed and meticulous measurements of the positions of the planets over a long period of time.

His measurements were used by others to try to explain the motion of the planets.

Tycho Brahe lost part of his nose in a duel.

He had a pet moose who died after drinking too much beer and falling down a flight of stairs.

He died of mercury poisoning, quite possibly murdered. One possibility is that the Danish king had him killed because he believed Tycho had an affair with the king’s mother.

Johannes Kepler

Johannes Kepler (1571-1630) was a German astronomer interested in planetary motion.

He worked as an assistant to Tycho Brahe in Prague.

There is a theory that he murdered Tycho to get his data.

Johannes Kepler

Kepler tried various schemes to explain the mystery of the planets.

At one point he tried to explain the sizes of the orbits of the planets in terms of the sizes of the “platonic solids”:

tetrahedron cube octahedron icosahedron dodecahedron

Eventually, he hit upon Kepler’s Laws. The first says that planets follow elliptical orbits.

The second is a precise formula relating the size of the orbit to the period (the time it takes the planet to go around the orbit).

The third is a precise formula involving the speed of the planet at each point in the orbit; it moves faster when it is closer to the sun.

Isaac Newton (1643 – 1727) proved Kepler’s Laws using calculus.

He invented calculus, but did not publish it for about twenty years.

He used it to prove Kepler’s Laws. The elegant proof is one of the great achievements in the history of science. It also guaranteed the importance of calculus.

Gottfried Wilhelm Leibniz1646 — 1716

After Newton invented calculus, but before he published it, Gottfried Leibniz invented it independently.

Leibniz was busy with so many projects that he did not get around to publishing his work on calculus for many years.

Eventually, he did publish.

As President, Newton appointed a committee to hear the case.

But, in fact, it was Newton himself who wrote the report.

He subsequently hounded Leibniz.

In his later life, Newton became Master of the Royal Mint. He spent his declining years hunting counterfeiters, and any unlucky enough to be caught he tortured and executed with unseemly enthusiasm.

Leibniz made a strategic blunder by applying to the Royal Society in London to adjudicate the dispute over priority. He may not have realized that Newton was President of the Royal Society.

Newton was furious, and he set out to destroy Leibniz.

An important element in Newton’s proof of Kepler’s Laws was the realization that there had to be a force (gravity) that held the planets in their orbits.

He also realized that such a force would be the same in all directions, which means that the mathematical model of the solar system has a built-in symmetry.

He exploited this symmetry to simplify the problem.

His proof of Kepler’s Laws is one of the great mathematical achievements of all time, but ironically, relating planetary motion to the force of gravity moved the subject of Astronomy from Mathematics to Physics, where it has resided ever since.

Applications of Kepler’s Laws

In 1945, Arthur C. Clarke, a writer of science fiction,invented the idea of the “geosynchronous satellite”.

His observation was that the higher up a satellite is, the longer it takes to go around the earth.

Relatively low-flying artificial satellites orbit the earth in about an hour and a half. The moon takes twenty-eight days. Somewhere in between, there should be an orbit which would take exactly twenty-four hours.

Using Kepler’s Laws, it is easy to calculate that a circular orbit with a radius of about 42,000 km will have a period of 24 hours.

A satellite in such an orbit will travel around exactly once each time the earth revolves on its axis.

From the vantage point of somebody on earth’s surface, such a satellite would appear to be stationary. Actually, it would be stationary if it lies over the equator; elsewhere it would appear to move north and south each day, but always in the same path.

A stationary satellite is ideal for communications purposes, because it can use a stationary ground antenna, as opposed to a very expensive and delicate antenna that can follow it across the sky.

This was the idea Arthur C. Clarke had in 1945.

The first geosynchronous satellite was launched in 1963, and now there are many.

Another use of Kepler’s Laws involves planets around distant stars.

If two stars form a double star, orbiting around each other, they both move.

A planet orbiting a star is much lighter than the star, so it moves much further.

But the star does still move.

From earth, we can detect the slight changes in the star’s light resulting from this motion, using the Doppler effect.

From the results, we can infer the period of the satellite, and, if we can estimate the size of the sun, we can use Kepler’s Laws to find the size of the planet’s orbit.

Hundreds of “extrasolar” planets have been discovered, most by this method or a variant of it.