I'm particularly interested in how people like Kepler and Galileo, 400 years ago, were able to deduce that planets move in ellipses and work out laws like equal times means equal areas, among others.
Actually, Galileo didn't do work in quantitive description of orbits of celestial bodies. So it narrows down to Kepler.
Astronomical observations of position are inherently incomplete. The observations record angular position accurately, but distance can only be estimated.
For comparison: there is the classroom demonstration of letting a tethered object move around on an air table, to observe centripetal acceleration and angular momentum and so on.
You take pictures at short time intervals and then you record the position of the puck at each consecutive time interval. Imagine you are given only the angular positions of the puck, but not the distance values. Well, without the distance information you don't have enough to go on.
That is the problem that Kepler took upon himself to solve.
The way he achieved that is absolutely fascinating.
Kepler's efforts, including the many blind alleys he took, are recorded in his book 'New Astronomy'. I have come across an impressive resource for Kepler's New Astronomy.
Key passages are explained, and many of Kepler's diagrams are recreated as flash animations.
(The work has been done by a team that is part of the 'Larouche Youth Movement'. As far as I can tell this Lyndon Larouche isn't a scientist, so it's not clear where the interest in Kepler comes from, but whatever the motivation the team members have done a cracking job.)
The greek astronomers (culminating in the work of Ptolemy) used uniform circular motion as building blocks to contruct models for celestial motion. (Copernicus moved from geocentrism to Heliocentrism, but he continued the approach of using circles and epi-circles.)
At some point Kepler decided he had to move away from the motion along circles as basic building blocks. That meant that the only way for him to work on the problem was to try and guess.
To calculate orbit points you need:
- a rule for the shape of the orbit
- a rule for the velocity at each point in time.
Let's say you get the guess for the shape of the orbit right, but you've guessed wrong for the velocity change rule. Then the points that you calculate don't match the observations, and you don't know which of the two guesses was wrong (or both guesses may be wrong, of course.)
Before trying ellipse-shaped orbits (with the Sun at one focus) Kepler tried various egg-shaped orbits.
His first guess for the variable velocity was to make the velocity inversely proportional to the distance to the Sun. Right now I don't quite remember how Kepler came to try an area law.
The point is: for the numbers to come out right both guesses had to be right at the same time. There was no way for Kepler to first find the ellips-shape and then proceed to figure out the velocity change rule. The numbers come out right only when both guesses are right.
That makes Kepler's success an absolutely stunning achievement.
Kepler started with the planet Mars. In retrospect we know that the orbit of Mars is the only one with a large enough eccentricity for Kepler's first two laws to be discovered at all.
Earlier I claimed that in trying to figure out the orbits of the planets the observations don't give the distances, but I realized that is not quite true.
Kepler was aware he could assume that the orbits of the planets repeat themselves; they are cyclic. He had decades of observations to work with, spanning many orbital revolutions of Mars.
To work on the orbit of Mars Kepler converted the anguler positions relative to the Earth to positions relative to the Sun. I assume Kepler was able to infer the actual period of Mars' orbit from the observations, to a good accuracy. So he was able to take two observations of Mars exactly one Mars year apart, so that Mars was in the same position on both occasions (but Earth not in the same position), and then compare.
Comparisons like that gave Kepler enough data to know the shape of Mars' orbit roughly. In retrospect we can see that an ellipse-shaped orbit (with the Sun at one focus) is a strong candidate. Kepler was bound to try it sooner or later. As mentioned earlier, Kepler started trying various egg-shaped orbits.
Likewise, it was clear that Mars moves slower when far away from the Sun, and faster when close to the Sun.
Kepler had theories about how the Sun was guiding the planets in their orbits. For instance, he envisioned a role for magnetism. None of those theories had lasting value on its own merits, but they did prompt Kepler to include in his attempts a velocity change rule where equal areas are swept out in equal intervals of time.