## Wednesday, April 13, 2011

Kepler

I am continuing my posting as an amateur historian of science. This post will be alll from memory -- no links and certainly no references to actual historical evidence.

He had 4 claims the fourth of which is, by far, the most astonishing.

1. Planets (including the Earth) move on ellipses with the Sun at the focus. The Moon moves on an ellipse with the Earth at the focus. This is a very radical claim. All previous Western astronomers worked only with circles. The path of the planets is close to a circle centered on a point near the sun (a not very eccentric ellipse is very close to a circle centered on the point midway between the foci). Only much more measurements (made by/under the supervision of Tycho Brahe) made it possible to show that the path wasn't an off center circle.

It is possible to approximate and ellipse very well with a big circle and an epicycle. This is what Copernicus did. His epicycles were tiny, because the orbits of the planets happen to be ellipses with low eccentricity -- that is very close to circles.

2. Angular momentum is conserved. Take the vector from the sun to a planet. The cross product of that ray and the velocity of the planet is constant. To put it in terms Kepler would understand, the product of the distance to the sun and the planet's motion in the directino perpendicular to the line segment from the sun to the planet is constant. Kepler really only noted this (in a diagram which I remember seeing a photo of it somewhere) for positions near the perihelion and near the apehelion (sp ?). It means that if you make a triangle of the Sun, the planet at time t and the planet at time t+x then the area depends on x but not on t.

This means that the angle swept out per unit of time is not at all constant when viewed from the Sun. It is proportional to the inverse of the square of the distance from the Sun to the Planet. Taking first order approximations (which are very close to accurate since the ellipses have low eccentricity) the angle swept out in a unit of time is almost exactly constant when viewed from the *other* focus of the ellipse. This means that planetary orbits are very nearly circles centered on a point other than the sun (the point midway between the foci) and that angle swept out is very nearly constant when viewed from the point twice as far from the Sun (in the same directin). This means that Ptolemy's system of eccentrics and equants is almost exact. Copernicus's conviction that motion must be constant when viewed from the center (of the circle) has nothing to do with reality.

So far I score it Copernicus 1 Ptolemy 2. On one point of disagreement (does the Earth stand still) Copernicus was right. On two points eccentrics and equants, Ptolemy was ahead of Copernicus. On the point of agreement -- perpetual motion must be circular, they were both wrong.

I am being generous to Copernicus, because I have skipped Brahe. Brahe had a compromise model in which the Sun and the Moon orbit the Earth and the other planets orbit the Sun. Brahe's model can't be distinguished from Copernicus's model by astronomical observations. If he had been a loyal employee/student Kepler could have Braheed his model with the Sun oribiting on an ellipse with the Earth at its center and the other planets oribiting on ellipses with the Sun at the center.

The only reason we are sure such a model is wrong is that it is inconsistent with Newton's theory of gravity which -- well that was a megagigantic huge very large scientific advance and just fits so many facts that the only way to remember them is to remember that Newton's model worked perfectly except for the precession of the parihelion of Mercury.

3. I have to redo the math each time working back from Newton. OK the square of the period of the planet is proportional to the cube of the average distance from the sun to the planet.

Stylized facts 1, 2 and 3 are all implied by Newton's theory of motion and of gravity.

So what is the really amazing fact 4 which is much more impressive than 1-3 and such that it just strains credulity to argue that it is a coincidence ?

There are only five regular solids: tetrahedron, cube, octahedron, dodecahedron and icosahedron (sp?). This is an easily proven yet still amazing fact in geometry.

There are six planets visible to the naked eye mercury, Venus, Earth (ha ha you looked) Mars, Jupiter and Saturn. If you make spheres including the circles which approximate their orbits (Copernicus's model without the epicycles) then the following amazing fact is true.

The corners of the smallest tetrahedron which includes the sphere of Mercury just touch the sphere of Venus.
The corners of the smallest cube which contains the sphere of Venus just touch the sphere of the Earth.
The corners of the smallest octahedron which constains the sphere of Earth just touch the sphere of Mars.

you get the picture.

This hints at an explanation of why there are 6 planets, since there are 5 regular solids to be between the orbits of successive planets. It also fits 5 numbers with no free parameters at all.

Any sensible rule of statistics or science or common sense must imply that one has to be a total idiot, insane or very very stubborn to assert that this is just a coincidence which tells us nothing.

The consensus view of basically all living people is that this is just a coincidence. Almost nobody knows the fact. It clearly can't tell us about physics, because it depends on the word "visible." There are other planets. If we had better eyes, then Kepler would never have noted the patter. If we had worse eyes and couldn't see Saturn (or mercury or something) then Kepler would never have noted the pattern

It is an amazing astonishing hard to believe it is a coincidence statement simultaneously about the Solar system and the sensitivity of the human eye.

Maybe it was a weird coincidence.

Another possibility is suggested by the Church of the God with a Sick and Twisted sense of Humor. This weird fact could be explained if He was Messing with Kepler's mind.

To recap

Peolemy: Ptolemy wrote down a model which was used to forecast 1350 years out of sample. For some reason, this gigantic scientific triumph is used as an example of bad science. The problem, as everyone thinks they know, is that clinging to the core belief that orbits were made of circles, Ptolemaic astronomers added epicycle after epicycle, pleased that they had reconciled the data with the claim that orbits could be approximated with enough circles and not noticing that any curve can be so approximated. This, first of all, sounds a whole lot like economic reasearch to me and second of all would clearly be bad science. The problem is that there is absolutely no historical evidence that Ptolemy's followers added an additional epicycle to his model.

I learned something recently. I thought they added eccentrics and equants. Now I know that Ptolemy had them in his model, and that they were used long before Ptolemy.

Copernicus: Copernicus was a pro-circle hard liner. His stated aim was to make a model without equants. OK I have to define. In Ptolemy's model something goes around a circle, but the rate at which it goes around is not constant when viewed from the center of the circle but rather from another point called an equant. This was not acceptable to Copernicus. I don't think we can understand why not (we can't read his mind).

The Copernican model had epicycles. It also suggested that they might not be necessary as Copernicus's epicycles were tiny.

What's this about 50 years ? Well for around 50 years after Copernicus published, no one was convinced. Then two things happened. One is that Galileo observed the phases of Venus with a telescope. That mean't he could estimate the angle between the line from the Sun to Venus and from Venus to the Earth. That angle corresponded very closely to the angle predicted by Copernicus and not at all to the angle predicted by Ptolemy. After 1450 data were observed which Ptolemy's model couldn't fit (to the stated precision of around a degree or so).

Second, Kepler published. Kepler's model is heliocentric and it gives predictions which are almost exact. Measurements precise enough to reject Kepler's model weren't available for centuries. Both Ptolemy and Copernicus admitted that their model didn't give exact predictions (which means forecasts errors can be detected with the naked eye).

Copernicus's model had very limited appeal and this makes sense. It gave predictions about as good as Ptolemy's and was about as complicated as Ptolemy's. New data showed it corresponded more closely to reality. Also a new model which we perceive to be similar to Copernicus's model was far superior to the models of Copernicus and of Ptolemy.

OK finally what did Kepler say ?

I've moved it up to before recap of Ptolemy and Copernicus so now it is at the top of the post and probably incomprehensible.

cactus said...