Galileo and Kepler

Introduction

predictively equivalent models: Also, observationally equivalent models. Two models that, when combined with the same observed data, make the same predictions.

logically equivalent models: Also, equivalent models. Two models that make all the same assertions, including theoretical assertions.

Galileo

New observational data:

Theoretical contributions:

Galileo defended two principal theses of Copernican astronomy; (a) the thesis that the planets revolve around the sun, and (b) the thesis that the Earth moves.

  1. Galileo’s rebuttal of Aristotelian arguments against the second Copernican thesis (b).
  2. Galileo's decomposition of motion, and the law of projectile motion (Galileo's law).

The old Aristotelian argument against a moving Earth was quite simple. It is based on what I will call the Aristotelian test. To test whether the Earth is moving, drop a stone from the top of a tower.

The first clause is uncontroversial (even Galileo accepts that). The Aristotelian argument for the second clause is as follows:

  1. Experience tells us that stones dropped from towers land at their base.
  2. By the Aristotelian test: if the Earth moves, then stones dropped from towers will not land at their base.
  3. Therefore, the Earth does not move.

Galileo undermined Premise 2 by showing the assertion that "if a cannonball is dropped from the mast of a moving ship then it will not land at its base" is false.

(Multimedia animation.)

Why is Galileo's analysis of projectile motion important?

  1. As a part of Galileo's against the prevailing Aristotelian dogma.
  2. As a way of unifying disparate phenomena.

Kepler:

  1. Kepler’s First Law (Elliptical path law): Each planet follows an elliptical path with the sun at one focus.
  2. Kepler’s Second Law (Area law): The line drawn from the sun to the planet sweeps out equal areas in equal times.
  3. The second law determines the comparative speed of the planet in different parts of its elliptical orbit. For example, the area law implies that the planet will move faster at the end of the ellipse closer to the sun (called the perihelion) than at the point farthest from the sun (the aphelion) because the shorter radius has to move faster to sweep out the same area in the same time.

However, Kepler’s first two laws do not require that the outer planets move faster around their orbits than the inner planets. This interplanetary relationship is given a precise quantified formulation in Kepler’s third law. Let T be the period of a planet’s circumsolar navigation, and D be its mean radius. The mean radius is just the radius of the deferent circle in the Copernican system.

  1. Kepler’s Third Law (Harmonic law): The ratio D3/T2 is the same for all planets. The values of D and T for Mercury are different from the values for Venus, but the ratio D3/T2 for Mercury is the same as the ratio for Venus. Similarly for all the planets.

Advantages of Kepler over Copernicus:

Note: Kepler’s laws are not conjoined with auxiliary assumptions in order to derive a model. The laws already define a model, and there is no sense in which the model may be changed. Kepler allowed no addition of epicycles, for instance. However, Kepler’s model was a very serious contender. According to the two methods of comparing models described earlier, Kepler’s model is predictively more accurate than the best Copernican model because it is much simpler than the Copernican models that achieve comparable fit. It is not better because it fits than any Copernican model. We know that this is not the case by Fourier’s theorem.