Johannes Kepler's revolutionary insight

The great astromonical observer, Ticho Brahe, was concerned that he would be upstaged by his assistant, the mathematician Johannes Kepler, so gave him the task of analysing the orbit of Mars. Kepler was an adherent of Copernicus' heliocentric description of the heavens – i.e., in practice, what we now call the solar system – and analyzed Mars' orbit from that perspective. While the idea of a spinning Earth in a heliocentric model had showed up before, and was not opposed by The Church in Copernicus' life-time (he was even encouraged, by no less than an archbishop/cardinal, to publish), he has the distinction of being Kepler's and Galileo's source for this idea – making him one of the principal instigators of the scientific revolution.

The reason Brahe had chosen to give Kepler Mars was to slow him down: it was the hardest to understand – because Copernicus (while changing the centre) had retained orthodoxy's assumption that orbits are circles; while his heliocentric system made it possible to eliminate many of the epicycles of the prior orthodoxy, it did not eliminate them all. Mars required more epicyclic adjustments than others – and Kepler realised that a simpler model was available: the orbit of Mars is an ellipse. (Kepler tried various other ovoid shapes first, supposing the ellipse to be so simple that it must surely have been tried and found inadequate by earlier astronomers.) With this realisation, breaking free of the strait-jacket of circles, a far cleaner description of the motions of the planets presented itself.

Brahe (properly pronounced roughly as follows: drop the t off the end of the English pronunciation of brat; follow that by dropping the final h (if it's audible at all) off the end of the interrogative huh? – or get a Dane or Swede to pronounce it for you, and parrot whatever they say) was an advocate of a hybrid cellestial model, which had Sun and Moon orbiting Earth, with (the rest of) the planets orbiting The Sun (but, of course, advocates of both this system and the older orthodoxy didn't regard Earth as a planet, so would skip the rest of). This is formally equivalent to the Copernican system and can just as readilly be adapted to use elliptical orbits rather than circles and epicycles. Modern physics can happily accomodate such a description; it lets you chose your frame of reference (it is just as happy with Sherlock Holmes' preferred frame, in which the Earth's surface is at rest, as with a non-spinning inertial frame in which the cosmic microwave background has no over-all Doppler shift) and then tells you the physical laws you'll observe to be in force for your frame.

It is worth noting that, while Brahe made all of his famously accurate astronomical observations by eye, his student Kepler improved on the existing design of telescope (as used by Galileo, with one lens concave, the other convex) by combining two convex lenses to achieve higher magnifications (albeit with the image inverted). Kepler's design is the one generally used to this day (if only lenses are used – large telescopes generally use curved mirrors instead of lenses, introduced by Newton).

Kepler's three laws of planetary motion

Kepler's laws state:

From these it is possible to work out the (magnitude and direction of the) accelleration of a planet, using a little calculus. Fix, then, on spherical polar co-ordinates with The Sun as origin and axis perpendicular to the plane of the orbit. The latitude co-ordinate (θ or m) is thus constant (and, with my conventions, zero) so we can ignore it, reducing our co-ordinates to circular polar co-ordinates in the plane of orbit. Take the planet's perihelion as zero-point of longitude (φ or) n and measure longitude in the same direction as the planet's motion about the sun, so that its longitude is always increasing. Let the semi-latus rectum and eccentricity of the ellipse be L and e, respectively; the orbit is then described by

(this is the general form of a conic section, a family of curves of which the ellipse is the special case −1 < e < 1; and the above choice of co-ordinates ensures e ≥ 0). We are also given that the radius sweeps out area at a constant rate: this tells us that r.r.dn/dt/2 is constant, i.e. that dn/dt = J/r/r for some constant J.

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