When Johannes Met Tycho

Copernicus's work, containing a wealth of apparently irrefutable detail, put the Earth-centered universe to rest, and finally removed the constraints that had inhibited the quest to understand the solar system for over a millennium.

The next person to make progress on this quest was Johannes Kepler, who was born in Germany in 1571. As a theoretician of the first order, he brought his intellect to bear upon Copernicus's model of the solar system, and found it lacking. However, Kepler knew that precise observations of the planets' motions were required in order to expose the weaknesses of Copernicus's model and make further progress along the pathway. This need was satisfied by Kepler's chance association with Tycho Brahe, a Danish nobleman who spent much of his life and resources developing an astronomical observatory on an island off the coast of Denmark. This housed precision instruments, and Tycho compiled what was the most complete and accurate catalogue of planetary position measurements available at that time.

Johannes's brilliance as a theoretician and Tycho's observational genius complemented each other perfectly, to bring about the next revolution in understanding. However, their relationship was an uneasy one, and Tycho was reluctant to gift his life's work to a younger rival. Tycho did make his observations available to Kepler, but only in a frustratingly piecemeal manner. This impasse was finally resolved on the death of Tycho, after which Kepler was able to extract the full catalogue of measurements from Tycho's family.

Now that Kepler had accurate observations, he spent a number of years trying unsuccessfully to reconcile them with the notion that planetary orbits were circular. Looking at the orbit of Mars, he struggled for nearly a year to resolve a discrepancy between observation and theory of only 8 minutes of arc—a small angular measure of about one-quarter the diameter of the full moon. This in itself says a great deal about Kepler's integrity and honesty; clearly, it would have been easier to ignore such a small anomaly, or to regard it as an erroneous measurement. This struggle, however, led Kepler to the idea that was to be his core contribution to the understanding of the solar system—that planetary orbits were elliptical in shape. Making this step, he now found that Tycho's measurements fitted beautifully, and thereafter Kepler published his first two laws of planetary motion in 1609. His third law, to do with the relationship between the size of an orbit and its period, was also a tough one that took him a further 10 years to establish. Kepler's three laws of planetary motion are as follows (see also Figure 1.7):

Kepler 1 - The orbit of each planet is an ellipse, with the Sun at one focus. Kepler 2 - The line joining each planet to the Sun sweeps out equal areas in equal times.

Kepler 3 - The square of the period of a planet is proportional to the cube of its mean distance from the Sun.

It is worth dwelling a few moments on Kepler's laws, to explain the jargon, and to illustrate their meaning. The first law uses the word ellipse, which from high school geometry could be described as egg-shaped or a squashed

Figure 1.6: Drawing an ellipse. The ellipse's focal positions are where the pins penetrate the card.

planet planet

Figure 1.7: Illustrations of (a) Kepler's first law and (b) his second law.

circle. We know that to draw a circle, we use a compass. The resulting figure has one focus—the center where the point of the compass penetrates the paper. You may also have drawn an ellipse in school by pressing two pins into a piece of card, and placing a loose loop of string around the pins. Placing a pencil in the loop, and keeping the string tight, we can move the tip of the pencil over the paper to produce an ellipse, as shown in Figure 1.6. The two points where the pins penetrate are referred to as the focuses of the ellipse. From Kepler's first law (refer to Fig. 1.7a), we see that planets move along elliptical orbits, but also that the Sun is located at one of the focal positions.

Kepler's second law is a rather strange way of describing how fast a planet moves at different points on its orbit. Looking at Figure 1.7b, we can see that if a planet moves from point 1 to 2 in the same time that it takes to move from point 3 to 4, then Kepler's second law implies that the shaded areas— Area 1 and Area 2—must be equal to one another. This geometrical argument can be translated into a dynamical one, since it is easy to see that, for this to happen, the planet must move rapidly when close to the Sun, and more slowly when further away. Kepler's 17th century mind tended to think in terms of geometry, whereas a modern orbit analyst would tend to take the dynamical view.Kepler expressed his third law in terms of a word equation, and since we are trying to avoid the use of equations it is sufficient to say that planets in big orbits take longer to orbit the Sun than planets in small orbits—a fairly commonsensical notion.

Trying to summarize Kepler's activities in a few paragraphs, as we have done here, does no justice to the magnitude of his achievement in establishing the modern view of the way the planets move around the Sun. The time he took to do this does, perhaps, give a measure of the difficulty of the task. His achievement is emphasized by noting that his laws are still used today when engineers analyze the motion of spacecraft around the Sun, or indeed the orbit of a satellite around the Earth. It is also important to realize that Kepler developed his laws empirically, based purely on Tycho's catalogue of planetary measurements. He described how the planets moved around the Sun, but had no underlying theoretical foundation to explain why they moved in this way. This task was left to the intellectual giant that was Isaac Newton.

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