Big Bang. Simon Singh
reveals why this is so. One way to draw an ellipse is to pin a length of string to a board, as shown in Figure 13, and then use a pencil to extend the string. If the pencil is moved around the board, keeping the string taut, it will trace out half an ellipse. Switch to the other side of the string, and make it taut again, and the other half of the ellipse can be traced out. The length of the string is constant and the pins are fixed, so a possible definition of the ellipse is the set of points whose combined distance to the two pins has a specific value.
Figure 13 A simple way to draw an ellipse is to use a piece of string attached to two pins, as shown in diagram (a). If the pins are 8 cm apart and the string is 10 cm long, then each point on the ellipse has a combined distance of 10 cm from the two pins. For example, in diagram (b), the 10 cm of string forms two sides of a triangle, both 5 cm long. From Pythagoras’ theorem, the distance from the centre of the ellipse to the top must be 3 cm. This means that the total height (or minor axis) of the ellipse is 6 cm. In diagram (c), the 10 cm of string is pulled to one side. This indicates that the total width (or major axis) of the ellipse is 10 cm, because it is 8 cm from pin to pin plus 1 cm at both ends.
The ellipse is quite squashed, because the minor axis is 6 cm compared with the major axis of 10 cm. As the two pins are brought closer together, the major and minor axes of the ellipse become more equal and the ellipse becomes less squashed. If the pins merge into a single point, then the string would form a constant radius of 5 cm and the resulting shape would be a circle.
The positions of the pins are called the foci of the ellipse. The elliptical paths followed by the planets are such that the Sun sits at one of the foci, and not at the centre of the planetary orbits. Therefore there will be times when a planet will be closer to the Sun than at other times, as if the planet has fallen towards the Sun. This process of falling would cause the planet to speed up and, conversely, the planet would slow down as it moved away from the Sun.
Kepler showed that, as a planet follows its elliptical path around the Sun, speeding up and slowing down along the way, an imaginary line joining the planet to the Sun will sweep out equal areas in equal times. This somewhat abstract statement is illustrated in Figure 14, and it is important because it precisely defines how a planet’s speed changes over the course of its orbit, contrary to Copernicus’s belief in constant planetary speeds.
The geometry of the ellipse had been studied since ancient Greek times, so why had nobody ever before suggested ellipses as the shape of the planetary orbits? One reason, as we have seen, was the enduring belief in the sacred perfection of circles, which seemed to blinker astronomers to all other possibilities. But another reason was that most of the planetary ellipses are only very slightly elliptical, so under all but the closest scrutiny they appear to be circular. For example, the length of the minor axis divided by the length of the major axis (see Figure 13) is a good indication of how close an ellipse is to a circle. The ratio equals 1.0 for a circle, but the Earth’s orbit has a ratio of 0.99986. Mars, the planet that had given Rheticus nightmares, was so problematic because its orbit is more squashed, but the ratio of the two axes is still very close to 1, at 0.99566. In short, the Martian orbit was only slightly elliptical, so it duped astronomers into thinking it was circular, but the orbit was elliptical enough to cause real problems for anybody who tried to model it in terms of circles.
Figure 14 The diagram shows a highly exaggerated planetary orbit. The height of the ellipse is roughly 75% of its width, whereas for most planetary orbits in the Solar System this proportion is typically between 99% and 100%. Similarly, the focus occupied by the Sun is far off-centre, whereas it is only slightly off-centre for actual planetary orbits. The diagram demonstrates Kepler’s second law of planetary motion. He explained that the imaginary line joining a planet to the Sun (the radius vector) sweeps out equal areas in equal times, which is a consequence of a planet’s increase in speed as it approaches the Sun. The three shaded sectors all have equal areas. When the planet is closer to the Sun the radius vector is short, but this is compensated by its greater speed, which means that it covers more of the ellipse’s circumference in a fixed time. When the planet is far from the Sun the radius vector is much longer, but it has a slower speed so it covers a smaller section of the circumference in the same time.
Kepler’s ellipses provided a complete and accurate vision of our Solar System. His conclusions were a triumph for science and the scientific method, the result of combining observation, theory and mathematics. He first published his breakthrough in 1609 in a huge treatise entitled Astronomia nova, which detailed eight years of meticulous work, including numerous lines of investigation that led only to dead ends. He asked the reader to bear with him: ‘If thou art bored with this wearisome method of calculation, take pity on me who had to go through with at least seventy repetitions of it, at a very great loss of time.’
Kepler’s model of the Solar System was simple, elegant and undoubtedly accurate in terms of predicting the paths of the planets, yet almost nobody believed that it represented reality. The vast majority of philosophers, astronomers and Church leaders accepted that it was a good model for making calculations, but they were adamant that the Earth remained at the centre of the universe. Their preference for an Earth-centred universe was based largely on Kepler’s failure to address some of the issues in Table 2 (pp. 34—5), such as gravity – how can the Earth and the other planets be held in orbit around the Sun, when everything that we see around us is attracted to the Earth?
Also, Kepler’s reliance on ellipses, which was contrary to the doctrine of circles, was considered laughable. The Dutch clergyman and astronomer David Fabricius had this to say in a letter to Kepler: ‘With your ellipse you abolish the circularity and uniformity of the motions, which appears to me increasingly absurd the more profoundly I think about it… If you could only preserve the perfect circular orbit, and justify your elliptic orbit by another little epicycle, it would be much better.’ But an ellipse cannot be built from circles and epicycles, so a compromise was impossible.
Disappointed by the poor reception given to Astronomia nova, Kepler moved on and began to apply his skills elsewhere. He was forever curious about the world around him, and justified his relentless scientific explorations when he wrote: ‘We do not ask for what useful purpose the birds do sing, for song is their pleasure since they were created for singing. Similarly, we ought not to ask why the human mind troubles to fathom the secrets of the heavens… The diversity of the phenomena of Nature is so great, and the treasures hidden in the heavens so rich, precisely in order that the human mind shall never be lacking in fresh nourishment.’
Beyond his research into elliptical planetary orbits, Kepler indulged in work of varying quality. He misguidedly revived the Pythagorean theory that the planets resonated with a ‘music of the spheres’. According to Kepler, the speed of each planet generated particular notes (e.g. doh, ray, me, fah, soh, lah and te). The Earth emitted the notes fah and me, which gave the Latin word fames, meaning ‘famine’, apparently indicating the true nature of our planet. A better use of his time was his authorship of Somnium, one of the precursors of the science fiction genre, recounting how a team of adventurers journey to the Moon. And a couple of years after Astronomia nova, Kepler wrote one of his most original research papers, ‘On the Six-Cornered Snowflake’, in which he pondered the symmetry of snowflakes and put forward an atomistic view of matter.
‘On the Six-Cornered Snowflake’ was dedicated to Kepler’s patron, Johannes Matthaeus Wackher von Wackenfels, who was also responsible for delivering to Kepler the most exciting news that he would ever receive: an account of a technological breakthrough that would transform astronomy in general and the status of the Sun-centred model in particular. The news was so astonishing that Kepler made a special note of Herr Wackher’s visit in March 1610: ‘I experienced a wonderful emotion while I listened to this curious tale. I felt moved in my deepest being.’
Kepler had just heard for the first time about the telescope, which was being used by Galileo to explore the heavens and reveal completely new features of the night