Cosmos - Carl Sagan [34]
The difference between a circular orbit and the true orbit could be distinguished only by precise measurement and a courageous acceptance of the facts: “The universe is stamped with the adornment of harmonic proportions, but harmonies must accommodate experience.” Kepler was shaken at being compelled to abandon a circular orbit and to question his faith in the Divine Geometer. Having cleared the stable of astronomy of circles and spirals, he was left, he said, with “only a single cartful of dung,” a stretched-out circle something like an oval.
Eventually, Kepler came to feel that his fascination with the circle had been a delusion. The Earth was a planet, as Copernicus had said, and it was entirely obvious to Kepler that the Earth, wracked by wars, pestilence, famine and unhappiness, fell short of perfection. Kepler was one of the first people since antiquity to propose that the planets were material objects made of imperfect stuff like the Earth. And if planets were “imperfect,” why not their orbits as well? He tried various oval-like curves, calculated away, made some arithmetical mistakes (which caused him at first to reject the correct answer) and months later in some desperation tried the formula for an ellipse, first codified in the Alexandrian Library by Apollonius of Perga. He found that it matched Tycho’s observations beautifully: “The truth of nature, which I had rejected and chased away, returned by stealth through the back door, disguising itself to be accepted … Ah, what a foolish bird I have been!”
Kepler had found that Mars moves about the Sun not in a circle, but in an ellipse. The other planets have orbits much less elliptical than that of Mars, and if Tycho had urged him to study the motion of, say, Venus, Kepler might never have discovered the true orbits of the planets. In such an orbit the Sun is not at the center but is offset, at the focus of the ellipse. When a given planet is at its nearest to the Sun, it speeds up. When it is at its farthest, it slows down. Such motion is why we describe the planets as forever falling toward, but never reaching, the Sun. Kepler’s first law of planetary motion is simply this: A planet moves in an ellipse with the Sun at one focus.
In uniform circular motion, an equal angle or fraction of the arc of a circle is covered in equal times. So, for example, it takes twice as long to go two-thirds of the way around a circle as it does to go one-third of the way around. Kepler found something different for elliptical orbits: As the planet moves along its orbit, it sweeps out a little wedge-shaped area within the ellipse. When it is close to the Sun, in a given period of time it traces out a large arc in its orbit, but the area represented by that arc is not very large because the planet is then near the Sun. When the planet is far from the Sun, it covers a much smaller arc in the same period of time, but that arc corresponds to a bigger area because the Sun is now more distant. Kepler found that these two areas were precisely the same no matter how elliptical the orbit: the long skinny area, corresponding to the planet far from the Sun, and the shorter, squatter area, when the planet is close to the Sun, are exactly equal. This was Kepler’s second law of planetary motion: Planets sweep out equal areas in equal times.
Kepler’s first law: A planet (P) moves in an ellipse with the Sun (S) at one of the two foci.
Kepler’s first two laws may seem a little remote and abstract: planets move in ellipses, and sweep out equal areas in equal times. Well, so what? Circular motion is easier to grasp. We might have a tendency to dismiss these laws as mere mathematical tinkering, something removed from everyday life. But these are the laws our planet obeys as we ourselves, glued by gravity to the surface of the Earth, hurtle through interplanetary space. We move in accord with laws of nature that Kepler first discovered. When we send spacecraft to