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THE MOST TREU PATH OF THE PLANER IS AN ELLIPSE

Before the 17th century, all astronomers were also astrologers. For many, including German astronomer Johannes Kepler, casting horoscopes was the main source of their income and influence. Knowing where the planets had been in the sky was important, but of greater significance for constructing astrological charts was the ability to predict where the planets would be over the next few decades.

To make predictions, astrologers assumed that the planets moved on specific paths around a central object. Before Copernicus, in the 16th century, this central body was thought by most to be Earth. Copernicus showed how the mathematics of planetary prediction could be simplified by assuming that the central body was the sun. However, Copernicus assumed that orbits were circular, and to provide any reasonable predictive accuracy, his system still required the planets to move around a small circle, the center of which moved around a larger circle. These circular velocities were always assumed to be constant.

Kepler supported the Copernican system, but the planetary tables it produced could still easily be out by a day or two. The planets, the sun, and the moon always appeared in a certain band of the sky, known as the ecliptic, but actual paths of individual planets around the sun were still a mystery, as was the mechanism that made them move.

“Kepler was never satisfied by a moderate agreement between theory and observation. The theory had to fit exactly otherwise some new possibility had to be tried.” Fred Hoyle

Kepler’s most productive years came in Prague under the patronage of Holy Roman Emperor Rudolf II (r.1576–1612). Rudolf was particularly interested in astrology and alchemy.

Finding the paths

To improve the predictive tables, Danish astronomer Tycho Brahe spent more than 20 years observing the planets. He next tried to ascertain a path of each planet through space that would fit the observational data. This is where the mathematical abilities of Kepler, Brahe’s assistant, came into play. He considered specific models for the solar system and the paths of the individual planets in turn, including circular and ovoid (egg-shaped) orbits. After many calculations, Kepler determined whether or not the model led to predictions of planetary positions that fit into Tycho’s precise observations. If there was not exact agreement, he would discard the idea and start the process again.

Abandoning circles

In 1608, after 10 years of work, Kepler found the solution, which involved abandoning both circles and constant velocity. The planets made an ellipse— a kind of stretched-out circle for which the amount of stretching is measured by a quantity called an eccentricity. Ellipses have two foci. The distance of a point on an ellipse from one focus plus the distance from the other focus is always constant. Kepler found that the sun was at one of these two foci.

These two facts made up his first law of planetary motion: the motion of the planets is an ellipse with the sun as one of the two foci. Kepler also noticed that the speed of a planet on its ellipse was always changing, and that this change followed a fixed law (his second): a line between the planet and the sun sweeps out equal areas in equal times. These two laws were published in his 1609 book Astronomia Nova.

Kepler had chosen to investigate Mars, which had strong astrological significance, thought to influence human desire and action. Mars took variable retrograde loops—periods during which it would reverse its direction of movement—and large variations in brightness. It also had an orbital period of only 1.88 Earth years, meaning that Mars went around the sun about 11 times in Tycho’s data set. Kepler was lucky to have chosen Mars, since its orbit has a high eccentricity, or stretch: 0.093 (where 0 is a circle and 1 is a parabola). This is 14 times the eccentricity of Venus. It took him another 12 years to show that the other planets also had elliptical orbits.

Studying Brahe’s observations, Kepler was also able to work out the planets’ orbital periods. Earth goes around the sun in one year, Mars in 1.88 Earth years, Jupiter in 11.86, and Saturn in 29.45. Kepler realized that the square of the orbital period was proportional to the cube of the planet’s average distance from the sun. This became his third law and he published it in 1619 in his book Harmonices Mundi, alongside lengthy tracts on astrology, planetary music, and platonic figures. The book had taken him 20 years to produce.

When just one body goes around a larger body undisturbed, the paths it can follow are known as Kepler orbits. These are a group of curves called conic sections, which include ellipses, parabolas, and hyperbolas. The shape of the orbit is defined by a property called eccentricity. An eccentricity of 0 is a circle (A). Eccentricity between 0 and 1 is an ellipse (B). Eccentricity equal to 1 produces a parabola (C), and greater than 1 a hyperbola (D).

Searching for meaning

Kepler was fascinated by patterns he found in the orbits of the planets. He noted that, once you accepted the Copernican system for the cosmos, the size of the orbits of the six planets—Mercury, Venus, Earth, Mars, Jupiter, and Saturn—appeared in the ratios 8 : 15 : 20 : 30 : 115 : 195.

Today, astronomers might look at a list of planetary orbital sizes and eccentricities and regard them as the result of the planetary formation process coupled with a few billion years of change. To Kepler, however, the numbers needed explanation. A deeply religious man, Kepler searched for a divine purpose within his scientific work. Since he saw six planets, he presumed that the number six must have a profound significance. He produced an ordered geometric model of the solar system in which the sun-centered spheres that contained each planetary orbit were inscribed and circumscribed by a specific regular “platonic” solid (the five possible solids whose faces and internal angles are all equal). The sphere containing the orbit of Mercury was placed inside an octahedron. The sphere that just touched the points of this regular solid contained the orbit of Venus. This in its turn was placed inside an icosahedron. Then followed the orbit of Earth, a dodecahedron, Mars, a tetrahedron, Jupiter, a cube, and finally Saturn. The system was beautifully ordered, but inaccurate.

Kepler’s great breakthrough was his calculation of the actual form of the planetary orbits, but the physics behind his three laws did not seem to concern him. Rather, he suggested that Mars was carried on its orbit by an angel in a chariot, or swept along by some magnetic influence emanating from the sun. The idea that the movements were due to a gravitational force only arrived with the ideas of Isaac Newton some 70 years later.

According to Kepler’s second law, the line joining a planet to the sun sweeps out equal areas in equal times. This is also known as the law of equal areas. It is represented by the equal areas of the three shaded areas ABS, CDS, and EFS. It takes as long to travel from A to B as from C to D and from E to F. A planet moves most rapidly when it is nearest the sun, at perihelion; a planet’s slowest motion occurs when it is farthest from the sun, at aphelion.

Wider contributions

Kepler also made important advances in the study of optics, and his 1604 book Astronomiae Pars Optica is regarded as the pioneer tome in the subject. Galileo’s telescope interested him greatly and he even suggested an improved design using convex lenses for both the objective and the magnifying eyepiece. He wrote, too, about the supernova that was first seen in October 1604, today commonly called Kepler’s supernova. Following Tycho, Kepler realized that the heavens could change, contradicting Aristotle’s idea of a “fixed cosmos.” A recent planetary conjunction coupled with this new star led him to speculate about the Biblical “Star of Bethlehem.” Kepler’s fervent imagination also produced the book Somnium, in which he discusses space travel to the moon and the lunar geography a visitor might expect on arrival. Many regard this as the first work of science fiction.

Kepler’s most influential publication, however, was a textbook on astronomy called Epitome Astronomiae Copernicanae, and this became the most widely used astronomical work between 1630 and 1650. He ensured that the Rudolphine Tables (named after Emperor Rudolf, his patron in Prague) were eventually published, and these tables of predicted planetary positions helped him greatly with the well-paid calendars that he published between 1617 and 1624. The accuracy of his tables, proven over a few decades, also did much to encourage the acceptance of both the Copernican sun-centered solar system and Kepler’s own three laws.

“Kepler was convinced that God created the world in accordance with the principle of perfect numbers, so that the underlying mathematical harmony … is the real and discoverable cause of the planetary motion. “ William Dampier

In Harmonices Mundi, Kepler experimented with regular shapes to unlock the secrets of the cosmos. He linked these shapes with harmonics to suggest a “music of the spheres.”

JOHANNES KEPLER

Born prematurely in 1571, Kepler spent his childhood in Leonberg, Swabia, in his grandfather’s inn. Smallpox affected his coordination and vision. A scholarship enabled him to attend the Lutheran University of Tübingen in 1589, where he was taught by Michael Maestlin, Germany’s top astronomer at the time. In 1600, Tycho Brahe invited Kepler to work with him at Castle Benátky near Prague. On Tycho’s death in 1601, Kepler succeeded him as Imperial Mathematician.

In 1611, Kepler’s wife died, and he became a teacher in Linz. He remarried and had seven more children, five of whom died young. His work was then disrupted between 1615 and 1621 while he defended his mother from charges of witchcraft. The Catholic Counter-Reformation in 1625 caused him further problems, and prevented his return to Tübingen. Kepler died of a fever in 1630.

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