I don't do this any more, but in the past I did what many astronomy professors do when teaching introductory astronomy: tell the tale of Tycho, Kepler, and Newton, as a way of introducing and describing planetary orbits. It's such a great story, as it shows the concrete struggle we as a race went through to fully codify and understand the heliocentric, Copernican picture of the Solar System. It also highlights the contributions of three very different sorts of scientists.
We have Tycho, the observer. We have Kepler, the phenomenologist. And, we have Newton, the theorist. Each played a crucial role, without which the contributions of each of the others would have been greatly lessened. The result was a revolution in our way of thinking about the Solar System, and the Universe at large.
Tycho Brahe was an observer. He compulsively took very precise data (for the time)— reams of data, huge amounts of data, on the positions of the planets as observed in the sky. Ultimately, this data would be used to once and for all show that the Copernican, Sun-centered picture of the Solar System was superior to the Earth-centered picture of the Solar System. Tycho, however, didn't believe it; he believed in a geocentric model. Like a good scientist, what he did was take lots of data, lots of very careful data, to test his theory. Perhaps he was doing it to prove the theory he favored— that is not the motivation that scientists are supposed to have, but as long as they approach it honestly and carefully, and don't fudge their data, it may be mere semantics whether they are trying to test their favorite theory, or whether they are trying to prove their favorite theory.
Ultimately, Tycho didn't really do either. He is the consummate observer or experimentalist. He excelled in the acquisition of first-rate data. He was the principal investigator who led and executed the project that produced the "planetary orbit survey" data of his time, a data set that later scientists would use for important and fundamental understanding about the nature of our world.
Kepler was a student and assistant of Tycho. Not the observer that Tycho was, Kepler's contribution came in the analysis of Tycho's data. And, yes, this is academia, so there are tales of acrimony and distrust on all sides, but we shall leave that aside for now.
Kepler took Tycho's data, and showed once and for all that you make a much simpler, much more straightforward, and much more powerful model of the Solar System if you place the Sun at the center and put the planets in orbit around it. This wasn't a new idea; Copernicus is generally credited, at least popularly, with the notion of the heliocentric model. The dominant model of the day, however, was Ptolemy's geocentric model, where the Earth was (sort of) at the center, and the planets (including, in the nomenclature of the time, the Sun and the Moon) orbited the Earth in a particularly complicated way. I suspect few really understood the model— heaven knows that I don't know all the details!— for the Earth wasn't exactly at the center. It was the thing closest to the center, but there were all sorts of ad-hoc offsets needed to make everything work right. Still, it wasn't bad; Ptolemy's model was able to predict the positions of the planets reasonably well, and as such was useful, even if cumbersome.
Kepler sat down with Tycho's data and, through a lot of work, was able to match all of the planet orbit data with three simple empirical laws:
The planets orbit the sun in ellpises. Copernicus' original idea was that it was circles. At the time, there was still this notion that "the heavens" was a place of geometrical perfection, in contrast to the Earth. Circles are perfect geometrical figures, so there was a philosophical bias in favor of them. Kepler showed that circles didn't quite do it, but ellipses (which are slightly squashed circles, the amount of squashing being parameterized by the "eccentricity") did. The sun is not at the center of of the ellipse, but off center at the focus (a geometricall well-defined position). Most of the planets orbit in ellipses of very low eccentricity, so that they are almost circles. Mars' orbit is still very close to circular, but was eccentric enough that the difference between a circular and an elliptical orbit was easily discerned in Tycho's careful data.
In their elliptical orbits, the planets move faster when closer to the sun than when farther from the Sun. (The actual law precisely specifies how this works.)
The orbital period of a planet's orbit (how long its "year" is) is related to the "semi-major axis" of the orbit's ellipse (something like the average distance from the Sun), with larger orbits having longer periods according to the famous "P^2=A^3" law (P is period in Earth-years, A is the semi-major axis in astronomical units, or in units of the distance between the Earth and the Sun).
What Kepler had done was take Tycho's data, and through tremendous effort, distilled it down to a very small number of straightforward empirical laws. With those laws and a few free parameters— a semi-major axis, eccentricity, and position in the orbit on any one specified day— you could use Kepler's laws to predict where a planet would be any time in the future.
Why do I call these "empirical" laws? Because there is no underlying physics that says why these laws should be so. They are descriptions of the data. Elegant, extremely useful, beautiful, and compact descriptions of the data, yes. This is why I call Kepler a phenomenologist. He wasn't after the underlying fundamental theory, necessarily. Rather, he gave simple laws that described how all of the data worked.
Without Tycho, Kepler could never have discovered his three laws. Without Kepler, or another doing the work he did, Tycho's data would not have been nearly as important to the progress of human science as it turned out to be.
The last person in the story is Newton. Newton gave us his three laws of mechanics— the laws that include inertial (objects in motion stay in motion), reaction force (every action has an equal and opposite reaction), and the law of motion (the acceleration of an object is proportional to the force on that object; the constant of proportionality is the object's mass).
Newton also gave us his Universal Law of Gravitation. This was a single, simple equation that said that the gravitational force between two objects (which is the same on either object, thanks to the action/reaction law) is proportional to the masses of the two objects, and goes down as the square of the distance between the two objects. If you start from this law, you can derive that when a small mass is orbiting a large mass, the orbits are in ellipses, the small mass moves faster when closer to the large mass, and the period of the orbit goes up with the semi-major axis of the ellipse with P^2 proportional to A^3.
Notice what has happened. With universal laws of mechanics— not specific to planetary orbits at all— and with a single equation describing gravity, Newton is able to derive Kepler's three laws. What's more, there is now a physical theory that explains why Kepler's Laws are the way they are: it's just gravity.
There are two important, major contributions here. First is the unification of the heavens and the earth. No longer are the planets in a "celestial" realm that obeys "more perfect" laws of its own. The force that explains why planets orbit the way they do is exactly the same as the force that explains why apples fall to the ground. The force is described by exactly the same equation. Second, we have the conversion of Kepler's purely empirical laws into something that can be derived from a fundamental physical theory. Whereas Tycho was the observer and Kepler was the phenomenologist in this story, Newton is the fundamental theorist.
Could Kepler have been skipped? Perhaps. Perhaps Newton could have come up with his law of Universal gravitation, and then showed through long and arduous calculation that it is able to predict and explain Tycho's data. However, by distilling Tycho's data, Kepler gave a much more tractable and a much more meaningful set of observations which Newton's theory needed to describe. Newton's theory didn't need to be compared with each and every data point produced by Tycho, because Kepler had already distilled Tycho's data down to its core. At that point, if Newton's theory is shown to be consistent with Kepler's empirical laws, we know that Newton's theory is consistent with Tycho's data— because Kepler's laws are consistent with Tycho's data! Beyond that, the form of Kepler's laws can help guide the process of discovery. Just a lot of planetary positions themselves may not give much physical insight. The notion of planets orbiting in ellipses with the Sun at the focus, however, can. Those simple laws suggest that a simple underlying physical theory may be sufficient, and may even help guide the scientist in understanding the form of that theory.
The Copernican Revolution was complete. We had finally shown that not only is the heliocentric model much simpler and more tractable (according to Kepler), but we had a fundamental theory of gravity that showed that in a real way the Sun really is at the center of the Solar System— for the Sun has most of the mass of the Solar System, and is located very close to the "center of mass" of the Solar System. What's more, we had gone through the steps of understanding just the way science is supposed to. We had started with a lot of careful data. We had gone through the arduous process of trying to understand, manage, and simplify that data. And, finally, we had taken the understanding of that data we'd obtained and turned it into confirmation of a basic physical theory.
Tycho, Kepler, and Newton: observer, phenomenologist, and theorist. Respect them all, for all of their contributions to science are essential. Without any one, the contributions of the other two would have been less, or even impossible.