Filling the Gap

KEPLER: MIND THE GAP!

The minor planets of the Kuiper Belt extended the Solar System outwards into space. But there had been an inner space in the Solar System, filled earlier. The first asteroids to be discovered, Ceres and Pallas, occupied a gap in the Solar System that had emerged when its structure first became clear, with the Sun in the center.

In 1543, in the very last days of his life, the Polish cleric Nicolaus Copernicus put forward his idea publicly in a book that the planets orbited the Sun, not Earth. It was an historic change in the way that we look at the universe, and like all great ideas it formed the basis of advances of knowledge on many fronts. It was particularly important for astronomers to have discovered the realistic structure of the Solar System. It became possible to map planetary orbits, to find out their exact shape and determine the distances of the planets from the Sun. As the detail of this map emerged Isaac Newton was able to develop the reason why the planets moved around the Sun, namely through the force of gravity.

The breakthrough in mapping the structure of the Solar System was made by the astronomer Johannes Kepler, (1134) Kepler, born in Tubingen in 1571 what is now Germany. Kepler was a curious mixture of a modern scientist and a medieval mystic. He learned astronomy from an early age, recalling in later life that he had been taken outside at the age of six to view the Great Comet of 1577. He studied theology at Tubinger Stift, a Lutheran seminary. As a student he had for taken a range of arts courses, and then philosophy and theology. He was inspired by one of his teachers, Michael Maestlin, (11771) Maestlin, to take up the sciences, astronomy, geometry and mathematics. But he was also attracted by the pseudo-science of astrology and cast horoscopes for credulous fellow students. He kept up this skill in later life, turning it to good advantage when he became the court astronomer in Prague, when, especially in times when his patron fell behind in paying his stipend, he cast horoscopes for the rich members of the court, including the Holy Roman Emperor himself. It was Maestlin who privately, outside his public lectures, introduced Kepler to the works of Copernicus, at a time when this was seen to be unsound theology. Kepler became a Copernican in his early 20s, committed to the view that it was the Sun at the center of the Solar System, not Earth.

Even before he had finished his studies in 1594, Kepler was invited to become a teacher of astronomy at a Protestant school in Graz in Austria. A great mathematician, Kepler was a poor lecturer, and the subject that he taught was not a popular one. He expected too much of his students, gabbled too much, and ventured down too many side streets away from the simple main road of the subject. His teaching was confused. Thus he attracted few students to his classes. But ideas incessantly bubbled up in his mind, and one summer’s day in 1595 he had a brainwave about the © Springer International Publishing Switzerland 2016

P. Murdin, Rock Legends, Springer Praxis Books, DOI 10.1007/978-3-319-31836-3_8

orbits of the planets. “There were three things in particular about which I persistently sought the reasons why they were such and not otherwise: the number, the size, and the motions of the orbits,” he wrote in the Preface to his book Mysterium Cosmographicum (The CosmographicMystery), first published in 1596 with a second edition in 1621. In the book, he explained his brainwave, according to the title page of the book. It was “the Secret of the Universe: the Marvellous Proportion of the Celestial Spheres, and the True and Particular Causes of the Number, Magnitude, and Periodic Motions of the Heavens; Established by Means of the Five Regular Geometric Solids.”

Thus, Kepler quite consciously set out to discover the plan of creation. On July 19, 1595, Kepler was preparing to teach a class in geometry. He drew a diagram on a blackboard of a large number of equilateral triangles within a circumscribed circle, which joined their corners. There was within all these triangles another circle inscribed within, touching the triangles’ sides. He realized that the ratio of size of the two circles was the same as the ratio of the size of the orbits of Jupiter and Saturn. He wondered whether he could fit the orbits of all the planets with geometric figures—a triangle, a square, a pentagon, and so on. This did not work out. He tried the same thing with three-dimensional geometric solids, better representatives (he thought) of the planets as corporeal bodies. This idea proved to be more of a success.

Six planets were known to Kepler—Mercury, Venus, Earth, Mars, Jupiter and Saturn, and Kepler wondered why there were this number, rather than the seven planets that the ancients considered—Mercury, Venus, Moon, Sun, Mars, Jupiter and Saturn. Seven is a renowned mystical number; there are seven stars in the Plough or Big Dipper, seven stars in the Pleiades, seven days in Creation, seven deadly sins, seven wonders of the ancient world, seven colors in the spectrum, and so on. Astrologers and alchemists considered that the number seven was split into the spiritual three and the material four; the seven liberal arts were divided in the same way into the trivium and quadrivium of subjects, as taught in medieval universities (grammar, logic, and rhetoric and arithmetic, geometry, music and astronomy, respectively). Seven had a special place in the studies that Kepler would have made, but what was sacred about the number six?

Kepler found that he could fit simple, regular, geometric solids into the orbits of the planets, starting on the outside with a sixth solid sphere that represented the orbit of Saturn. He fitted a cube into the sphere, and within that fitted a second sphere that represented the orbit of Jupiter. Inside that sphere he fitted a tetrahedron, within which a sphere represented the orbit of Mars. Inside that sphere he fitted a dodecahedron (Earth), followed by an icosahedron (Venus), and, finally, the innermost sphere that represented the orbit of Mercury was fitted inside an octahedron.

These solids are known as the Platonic solids—regular solids whose faces are plane figures with equal sides: squares, equilateral triangle, etc. Mathematically, there are only five of them. Add a sphere and that makes

Table 8.1 Distances of the planets from the Sun

Planet

Distance from Sun (in terms where the distance of Earth from the Sun is 1.0)

Kepler’s estimate from his geometric construction

Mercury

0.39

0.56

Venus

0.72

0.79

Earth

1.0

1.00

Mars

1.52

1.26

Jupiter

5.20

3.77

Saturn

9.54

6.54

six solids, which correspond to the six planets. The Platonic solids are called after the Greek philosopher, Plato, who wrote about them in his work of philosophy, Timaeus, but they were known long before him. They were especially important to Greek science because they became associated with what were thought to be the five elements from which everything was made: air, water, earth, fire and the ether. If the orbits of the planets were associated with the Platonic solids, it followed that celestial motions were directly connected with the elements.

With this geometric construction, Kepler estimated the distances of the planets from the Sun and found what he thought at first was an impressive fit. The planets orbited at distances from the Sun as shown in Table 8.1.

The goodness of the fit brought Kepler to tears. What had started out as an intellectual speculation had ended up at what seemed to hint at a profound truth. Planetary orbits, mathematical solids, universal elements— Kepler thought that he had discovered some fundamental, divinely inspired connection between mathematics, astronomy and the nature of the universe. He had glimpsed God’s profound glory.

However, the geometric construction was not perfect. The agreement between the calculated and actual distances of the planets was not perfect. In fact, like many eureka moments that are viewed in the cold dawn after an evening’s calculations, it was rather weak. It went a long way off track at the outer planets. Kepler was also struck by the gap between Jupiter and Mars, as well as the less striking gap between Venus and Mercury. Broadly speaking, the distances of each planet from the Sun were doubling up from one planet to the next, except that the distance to Jupiter was nearly a factor of four times that to Mars. Kepler wondered whether there were undiscovered planets there:

Between Jupiter and Mars I placed a new planet, and also another between Venus and Mercury, which were to be invisible on account of their tiny size, and I assigned periodic times to them. For I thought that in this way I should produce some agreement between the ratios, as the ratios between the pairs would be respectively reduced in the direction of the Sun and increased in the direction of the fixed stars... Yet the interposition of a single planet was not sufficient for the huge gap between Jupiter and Mars; for the ratio of Jupiter to the new planet remained greater than is the ratio of Saturn to Jupiter.

Kepler had to reconcile the urge to populate the gap between Mars and Jupiter with further planets. Adding the sphere to the five Platonic solids brought the number to six, exactly the same as the sacred number of planets. To preserve this connection, Kepler thought that any extra planet had to be a lesser sort from the six major planets. This is why he suggested that the hypothetical new planet was small.

But Kepler rejected the thought that there were extra planets. If one extra, why not two? Or three? Or any other number? Kepler favored keeping his geometric construction, limiting the number of planets to 6, and the gap continued to pose a puzzle.

Kepler was also not satisfied with the completeness of the fit of the geometric construction shown in Table 8.1, and continued to seek further explanations, or laws, about planetary distances. Even in the book in which he proposed his geometrical construction he was thinking whether there was a relationship between the distances of the planets from the Sun and their orbital periods. In 1599 religious conflict was growing in Graz between Lutherans and Roman Catholics, and, a Lutheran, Kepler was preparing to leave Graz. He was invited to move to Prague by the Danish astronomer Tycho Brahe, and seized the opportunity, using Brahe’s observations to show three foundational “laws” of planetary motion.

Brahe was a rich Danish nobleman who had the means to indulge in his eccentric interests. He kept a pet moose, which tragically died when, drunk, it fell down a flight of stairs. As a student he had lost most of his nose in a duel, and habitually wore a prosthetic one made of gold. More significantly, he devoted much of his energy to establishing his astronomical observatories, Uraniborg and Stjerneborg (both names meaning “Star City”) on the Danish island of Hven. The enterprise was supported by the then king of Denmark, Frederick II, but when his successor Christian IV came to the throne in 1588, royal support began to dry up as Christian imposed an age of austerity in the national budget to compensate for Frederick’s profligacy. Brahe’s observing program began to run down and he looked for opportunity elsewhere. In 1597 Brahe moved to Prague to benefit from the patronage of Emperor Rudolf of the Holy Roman Empire, where he became imperial astronomer. Soon after Kepler had arrived, Brahe died in 1601 of retention of urine, having been too embarrassed to leave the table at a formal banquet and empty his bladder. Kepler inherited Brahe’s papers and his measurements. Kepler pored over them, trying to understand better why God had made the Solar System as he had.

Isaac Newton addressed the same issues in letters exchanged with a classicist, Richard Bentley. In 1692 Bentley was appointed as the first Boyle Lecturer, whose duties were to give eight sermons about the relationship between Christianity and science. (This lecture series has continued, with gaps, to the present day.) In preparation for his lectures, which he entitled A Confutation of Atheism, Bentley studied Newton’s view of the universe expressed through his physics, and asked him some hard questions. In reply Newton offered his explanation for the gap in the Solar System beyond Mars that Kepler had identified. In order to take care of his human creation, Newton said, God had separated Jupiter from the rest of the planets so that it would not disturb the motion of the Earth. Newton’s letter survives in the library of Trinity College, Cambridge:

...the Planets of Iupiter & Saturn as they are rarer then the rest so they are vastly greater & contein a far greater quantity of matter & have many Satellites about them: which qualifications surely arose not from their being placed at so great a distance from the Sun but were rather the cause why the creater placed them at that great distance. ffor by their gravitating powers they disturb one anothers motions very sensibly as I find by some late Observations of Mr Flamsteed, & had they been placed much nearer to the Sun & to one another they would by the same powers have caused a considerable disturbance in the whole Systeme.

The letter expresses, in Newton’s own handwriting and spelling, his belief, held by few people today, that the universe has been constructed for human benefit.

As an astrologer as well as an astronomer, Kepler was inclined to numerological explanations within a religious framework; for example, he was convinced that the motions of the planets were connected to musical notes, and that the angels could hear “the music of the spheres.” He tried a large number of calculations to find what was underlying the distances of the six planets. In 1618, Kepler had a flash of inspiration, which related the distance of each planet to the period of its orbit around the Sun. In what became known as Kepler’s Third Law of Planetary Motion, he explained that “The square of the periodic times of each planet in orbit round the Sun are to each other as the cubes of the mean distances.” Table 8.2 lists the periods and the distances; if Kepler’s Law is exact then the ratio of the period-squared divided by distance-cubed should be the same for each planet—and it is.

Around the 1680s a number of people were able to explain the origin of Kepler’s Third Law by supposing that the planets are attracted to the Sun by the force of gravity, provided that the force follows an inverse square law. Isaac Newton published the most comprehensive explanation in a book, known as the Principia, in which he set out his theory of gravity and dynamics. He successfully knitted together a large range of facts and laws that had seemed to that point arbitrary. The motion of the planets had inspired the discovery that everything in science was rational. The underlying idea was powerful. It was for example taken as one of the foundation stones of the French Enlightenment. We see the idea persisting in modern politics in the concept of “evidence-based policy.”

Table 8.2 Kepler’s third law

Planet

Period (P, year)

Average distance (R, AU)

P2/R3

Mercury

0.241

0.39

0.98

Venus

0.615

0.72

1.01

Earth

1.00

1.00

1.00

Mars

1.88

1.52

1.01

Jupiter

11.8

5.20

0.99

Saturn

29.5

9.54

1.00

 
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