§7.2: Dance of the Planets

The Aristotelian view of the universe had the Earth at the center, with the stars and planets (and all celestial objects) attached to 55 concentric, crystalline spheres which rotated at different angular velocities.

By adjusting the velocities of these concentric spheres, many features of planetary motion could be explained. However, Greek astronomers also saw that the brightness of the planets changed through the course of a year, and that they appeared at times to move backwards against the backdrop of the other stars (retrograde motion).Since the spheres moved with constant angular velocity, and the objects attached to them were always the same distance from the earth, these phenomena of the planets were not explained.

This view of the universe was largely accepted through the Middle Ages and into the Renaissance, although it was enhanced by later astronomers. For example, Ptolemy dealt with the problem of retrograde motion by introducing the notion of epicycles.

In an epicycle, the primary sphere (known as the deferent) rotated normally with constant angular velocity. The planet, however, is not directly attached to the sphere any longer, but is attached to a circle called an epicycle. The epicycle is attached to the deferent, and also rotates with its own constant angular velocity through the deferent. The combined motion looks something like:

Further, the epicycle moved the planet to varying distances from the Earth, explaining the variation in brightness that was seen.

As time went by, more accurate astronomical information was collected. In some cases, even this use of epicycles and deferents was insufficient to accurately describe the motion of some planets. The accuracy of Aristotle's model for the universe could be improved by using epicycles on epicycles on deferents, and so on. . .

The first major change came with the proposed heliocentric model of Copernicus. Although he wasn't the first to propose placing the Sun at the center of the universe, the publication of his work had a lasting imprint that eventually took hold. Still, his approach continued using circular orbits. The results of his model were closer to what astronomers observed than that of Ptolemy, but not remarkably so.

However, his alteration of Aristotelian order was not well received by the Church and its adherents were charged with heresy, including Galileo Galilei, one of the founders of physics.

Tycho Brahe merged what he saw as the best points of the Ptolemaic and Copernican systems. He kept the Earth centered universe of Ptolemy and Aristotle, but placed the epicycles of the planets around the deferent of the Sun:

Tycho's universe was further improved by his assistant Johannes Kepler. Where Tycho Brahe excelled as a craftsman of precision instruments with the skill to use them to high levels of accuracy, Kepler's talents were stronger in mathematics. His introduction of elliptical orbits into the universe models, along with his elucidation of their properties, set the stage for a true understanding of celestial orbits and gravity.

The three laws he derived were based primarily on Tycho's measurements of the orbit of
the planet Mars that, because of its proximity to the earth and its relative proximity to the sun, completes a full circuit of our sky in roughly 2 years. (This gave Tycho the chance to accumulate many orbits' worth of data over several decades.)
  1. All planetary orbits are ellipses, with the sun at one focus of the ellipse. That is, orbits are never perfect circles, and the object orbited (in our case, the sun) is never at the center of the orbit.
  2. In its individual orbit, the closer a planet happens to be to the sun, the faster its orbital speed, and vice versa. Since all planetary orbits are ellipses, there are always times when the planet is closer to the sun, and times when it's farther away. When it's closest, its orbital speed at that moment is fastest of all, and when it's farthest away, its orbital speed at that moment is slowest of all. In fact, the greater the eccentricity of the orbit (the more highly elliptical it is), the greater the difference between the planet's maximum and minimum orbital speeds.
  3. Comparing all the planets, the greater a planet's average distance from the sun, the slower its average orbital speed. Here we are comparing one planet to another, and the actual shape of the orbit doesn't really matter. You average the distance a given planet is from the sun over its entire orbit, and then compare that to the average distance of another planet from the sun. The planet that is on average farther away has a slower average orbital speed. [Your instinct might tell you that the planet that is farthest away of course has the larger orbit, and will therefore take longer to get around. That's true, of course, but this law of Kepler says more than that. It's not only the size of the orbit that makes the full orbital period longer, but the fact that the speed of the planet with the larger orbit is considerably slower.]