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.

- It is very rare that two models are predictively equivalent, although it is common that
two models are predictively equivalent
*with respect to*a restricted set of quantities. E. g., for every Copernican model (without equants) there is a Ptolemaic model that is predictively equivalent*with respect to*the angular positions of the planets. - In planetary astronomy, predictions are made from a model of the solar system
*as a whole*. This important because the Ptolemaic and Copernican models of the whole solar system makes different predictions about the phases of Venus, for example, even if they made the same predictions about the angular positions of the planets. Therefore, Ptolemaic and Copernican models are not predictively equivalent in any unrestricted sense.

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

- Ptolemaic and Copernican models of a single planets make different assertions about absolute motions. Absolute motions are not observed (they are inferred with the aid of background theory), but this difference is sufficient to ensure that they are not logically equivalent even if they are predictively equivalent.

Galileo

New observational data:

- The discovery of mountains on our moon, Earthlight on the moon, sunspots, and supernova were important in the way they undermined the authority of Aristotle.
- The moons of Jupiter provided an argument for the idea that our moon could follow the Earth even though the Earth is moving.
- It was only by using the telescope that Galileo was able to check some of the predictions that the Copernican model made that Ptolemaic models did not, like the phases of Venus. So, this example is especially important.

**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.

- Galileo’s rebuttal of Aristotelian arguments against the second Copernican thesis (b).
- 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.

- If the Earth is not moving, then the stone will land at its base.
- If the Earth is moving, then the stone will land away from the base.

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

- Experience tells us that stones dropped from towers land at their base.
- By the Aristotelian test: if the Earth moves, then stones dropped from towers will not land at their base.
- 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.

- Galileo analyzed projectile motion has being composed of two
*independent*components: (a) the horizontal "inertial" component, (b) the downward accelerative component.

(Multimedia animation.)

Why is Galileo's analysis of projectile motion important?

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

Kepler:

**Kepler’s First Law**(Elliptical path law): Each planet follows an elliptical path with the sun at one focus.**Kepler’s Second Law**(Area law): The line drawn from the sun to the planet sweeps out equal areas in equal times.

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.

**Kepler’s Third Law**(Harmonic law): The ratio*D*^{3}/*T*^{2 }is the same for all planets. The values of*D*and*T*for Mercury are different from the values for Venus, but the ratio*D*^{3}/*T*^{2 }for Mercury is the same as the ratio for Venus. Similarly for all the planets.

**Advantages of Kepler over Copernicus**:

- Copernicus can account for the fact that the periods of the outer planets are longer, but only ad hoc. But given Kepler’s model, this fact is necessary.
- The constant ratio in Kepler’s third law is proportional to the mass of the sun. Thus, Kepler’s model explains many effects in terms of a common cause, namely the mass of the sun. (Newton made this argument on Kepler’s behalf.)

**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.