Big Bang

2. the planets continuously vary their speed,

3. the Sun is not quite at the centre of these orbits.

When he knew he had the solution to the mystery of planetary orbits, Kepler shouted out: ‘O, Almighty God, I am thinking Thy thoughts after Thee.’

In fact, the second and third points in Kepler’s new model of the Solar System emerge out of the first, which states that planetary orbits are elliptical. A quick guide to ellipses and how they are constructed 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 sky. Thanks to this new invention, Galileo would discover the evidence that would prove that Aristarchus, Copernicus and Kepler were all correct.

Seeing Is Believing

Born in Pisa on 15 February 1564, Galileo Galilei has often been referred to as the father of science, and indeed his claim to that title is founded on a staggeringly impressive track record. He may not have been the first to develop a scientific theory, or the first to conduct an experiment, or the first to observe nature, or even the first to prove the power of invention, but he was probably the first to excel at all of these, being a brilliant theorist, a master experimentalist, a meticulous observer and a skilled inventor.

He demonstrated his multiple skills during his student years, when his mind wandered during a cathedral service and he noticed a swinging chandelier. He used his own pulse to measure the time of each swing and observed that the period for the back-and-forth cycle remained constant, even though the wide arc of the swing at the start of the service had faded to just a gentle sway by the end. Once home, he switched from observational to experimental mode and toyed with pendulums of different lengths and weights. He then used his experimental data to develop a theory that explained how the period of swing is independent of the angle of swing and of the weight of the bob, but depends only on the length of the pendulum. After pure research, Galileo switched into invention mode and collaborated on the development of the pulsilogia, a simple pendulum whose regular swinging allowed it to act as a timing device.

In particular, the device could be used to measure a patient’s pulse rate, thereby reversing the roles in his original observation when he used his pulse to measure the period of the swinging lamp. He was studying to be a doctor at the time, but this was his one and only contribution to medicine. Subsequently he persuaded his father to allow him to abandon medicine and pursue a career in science.

In addition to his undoubted intellect, Galileo’s success as a scientist would rely on his tremendous curiosity about the world and everything in it. He was well aware of his inquisitive nature and once exclaimed:‘When shall I cease from wondering?’

This curiosity was coupled with a rebellious streak. He had no respect for authority, inasmuch as he did not accept that anything was true just because it had been stated by teachers, theologians or the ancient Greeks. For example, Aristotle used philosophy to deduce that heavy objects fall faster than light objects, but Galileo conducted an experiment to prove that Aristotle was wrong. He was even courageous enough to say that Aristotle, then the most acclaimed intellect in history,‘wrote the opposite of truth’.

When Kepler first heard about Galileo’s use of the telescope to explore the heavens, he probably assumed that Galileo had invented the telescope. Indeed, many people today make the same assumption. In fact, it was Hans Lippershey, a Flemish spectacle-maker, who patented the telescope in October 1608. Within a few months of Lippershey’s breakthrough, Galileo noted that ‘a rumour came to our ears that a spyglass had been made by a certain Dutchman’, and he immediately set about building his own telescopes.

Galileo’s great accomplishment was to transform Lippershey’s rudimentary design into a truly remarkable instrument. In August 1609, Galileo presented the Doge of Venice with what was then the most powerful telescope in the world. Together they climbed St Mark’s bell-tower, set up the telescope and surveyed the lagoon. A week later, in a letter to his brother-in-law, Galileo was able to report that the telescope performed ‘to the infinite amazement of all’. Rival instruments had a magnification of about × 10, but Galileo had a better understanding of the optics of the telescope and was able to achieve a magnification of × 60. Not only did the telescope give the Venetians an advantage in warfare, because they could see the enemy before the enemy saw them, but it also enabled the shrewder merchants to spot a distant ship arriving with a new cargo of spices or cloth, which meant that they could sell off their current stock before market prices plummeted.

Galileo profited from his commercialisation of the telescope, but he realised that it also had a scientific value. When he pointed his telescope at the night sky, it enabled him to see farther, clearer and deeper into space than anyone ever before. When Herr Wackher told Kepler about Galileo’s telescope, the fellow astronomer immediately recognised its potential and wrote a eulogy: ‘O telescope, instrument of much knowledge, more precious than any sceptre! Is not he who holds thee in his hand made king and lord of the works of God?’ Galileo would become that king and lord.

First, Galileo studied the Moon and showed it to be ‘full of vast protuberances, deep chasms and sinuosities’, which was in direct contradiction to the Ptolemaic view that the heavenly bodies were flawless spheres. The imperfection of the heavens was later reinforced when Galileo pointed his telescope at the Sun and noticed blotches and blemishes, namely sunspots, which we now know to be cooler patches on the Sun’s surface up to 100,000 km across.

Figure 15 Galileo’s drawings of the Moon.

Then, during January 1610, Galileo made an even more momentous observation when he spotted what he initially thought were four stars loitering in the vicinity of Jupiter. Soon it became apparent that the objects were not stars, because they moved around Jupiter, which meant that they were Jovian moons. Never before had anybody seen a moon other than our own. Ptolemy had argued that the Earth was the centre of the universe, but here was indisputable evidence that not everything orbited the Earth.

Galileo, who was in correspondence with Kepler, was fully aware of the latest Keplerian version of the Copernican model, and he realised that his discovery of Jupiter’s moons was providing further support for the Sun-centred model of the universe. He had no doubt that Copernicus and Kepler were right, yet he continued to search for evidence in favour of this model in the hope of converting the establishment, which still clung to the traditional view of an Earth-centred universe. The only way to break the impasse would be to find a clear-cut prediction that differentiated between the two competing models. If such a prediction could be tested it would confirm one model and refute the other. Good science develops theories that are testable, and it is through testing that science progresses.

In fact, Copernicus had made just such a prediction, one which had been waiting to be tested as soon as the tools were available to make the appropriate observations. In De revolutionibus, he had stated that Mercury and Venus should exhibit a series of phases (e.g. full Venus, half Venus, crescent Venus) similar to the phases of the Moon, and the exact pattern of phases would depend on whether the Earth orbited the Sun, or vice versa. In the fifteenth century nobody could check the pattern of phases because the telescope had yet to be invented, but Copernicus was confident that it was just a matter of time before he would be proved correct: ‘If the sense of sight could ever be made sufficiently powerful, we could see phases in Mercury and Venus.’

Figure 16 Galileo’s sketches of the changing positions of Jupiter’s moons. The circles represent Jupiter, and the several dots either side show the changing positions of the moons. Each row represents one observation taken on a particular date and time, with one or more observations per night.

Leaving aside Mercury and concentrating on Venus, the significance of the phases is apparent in Figure 17. Venus always has one face illuminated by the Sun, but from our vantage point on the Earth this face is not always towards us, so we see Venus go through a series of phases. In Ptolemy’s Earth-centred model, the sequence of phases is determined by Venus’s path around the Earth, and its slavish obedience to its epicycle. However, in the Sun-centred model, the sequence of phases is different because it is determined by Venus’s path around the Sun without any epicycle. If somebody could identify the actual sequence of Venus’s waxing and waning, then it would prove beyond all reasonable doubt which model was correct.

In the autumn of 1610, Galileo became the first person ever to witness and chart the phases of Venus. As he expected, his observations perfectly fitted the predictions of the Sun-centred model, and provided further ammunition to support the Copernican revolution. He reported his results in a cryptic Latin note that read Haec immatura a me iam frustra leguntur oy (‘These are at present too young to be read by me’). He later revealed that this was a coded anagram that when unravelled read Cynthiæ figuras æmulatur Mater Amorum (‘Cynthia’s figures are imitated by the Mother of Love’). Cynthia was a reference to the Moon, whose phases were already familiar, and Mother of Love was an allusion to Venus, whose phases Galileo had discovered.

The case for a Sun-centred universe was becoming stronger with each new discovery. Table 2 (pp. 34—5) compared the Earth- and Sun-centred models based on pre-Copernican observations, showing why the Earth-centred model made more sense in the Middle Ages. Table 3 (overleaf) shows how Galileo’s observations made the Sun-centred model more compelling. The remaining weaknesses in the Sun-centred model would be removed later, once scientists had achieved a proper understanding of gravity and were able to appreciate why we do not sense the Earth’s motion around the Sun. And although the Sun-centred model did not chime with common sense, one of the criteria in the table, this was not really a weakness because common sense has little to do with science, as discussed earlier.

Figure 17 Galileo’s precise observations of the phases of Venus proved that Copernicus was right, and Ptolemy wrong. In the Sun-centred model of the universe, shown in diagram (a), both the Earth and Venus orbit the Sun. Although Venus is always half-lit by the Sun, from the Earth’s point of view it appears to go through a cycle of phases, turning from a crescent to a disc. The phase is shown next to each position of Venus.

In the Earth-centred model of the universe, both the Sun and Venus orbit the Earth, and in addition Venus moves round its own epicycle. The phases depend on where Venus is on its orbit and on its epicycle. In diagram (b), Venus’s orbit is such that it is roughly between the Earth and the Sun, which gives rise to the set of phases shown. By identifying the actual series of phases, Galileo could identify which model was correct.

At this point in history, every astronomer should have switched allegiance to the Sun-centred model, but no such major shift took place. Most astronomers had spent their entire lives convinced that the universe revolved around a static Earth, and they were unable to make the intellectual or emotional leap to a Sun-centred universe. When the astronomer Francesco Sizi heard about Galileo’s observation of Jupiter’s moons, which seemed to suggest that the Earth was not the hub of everything, he came up with a bizarre counter-argument: ‘The moons are invisible to the naked eye and therefore can have no influence on the Earth and therefore would be useless and therefore do not exist.’ The philosopher Giulio Libri took a similarly illogical stance and even refused to look through a telescope on a point of principle. When Libri died, Galileo suggested that he might at last see the sunspots, the moons of Jupiter and the phases of Venus on his way to heaven.

The Catholic Church was similarly unwilling to abandon its doctrine that the Earth was fixed at the centre of the universe, even when Jesuit mathematicians confirmed the superior accuracy of the new Sun-centred model. Thereafter, theologians conceded that the Sun-centred model was able to make excellent predictions of planetary orbits, but at the same time they still refused to accept that it was a valid representation of reality. In other words, the Vatican viewed the Sun-centred model in the same way that we regard this sentence: ‘How I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.’ This phrase is a mnemonic for the number π. By noting the number of letters in each word of the sentence, we obtain 3.141 592 653 589 79, which is the true value of π to fourteen decimal places. The sentence is indeed a highly accurate device for representing the value of π, but at the same time we know that π has nothing to do with alcohol. The Church maintained that the Sun-centred model of the universe had a similar status – accurate and useful, but not reality.

Table 3

This table lists ten important criteria against which the Earth-centred and Sun-centred models could be judged based on what was known in 1610, after Galileo’s observations. The ticks and crosses give crude indications of how well each model fared in relation to each criterion, and a question mark

indicates a lack of data. Compared to the assessment based on the evidence available before Copernicus (Table 2, pp. 34—5), the Sun-centred model now seems more convincing. This is partly down to new observations (points 8, 9 and 10) that were possible only with the advent of the telescope.

However, the Copernicans continued to argue that the Sun-centred model was good at predicting reality for the very reason that the Sun really was at the centre of the universe. Not surprisingly, this provoked a stern reaction from the Church. In February 1616, a committee of advisors to the Inquisition formally declared that holding the Sun-centred view of the universe was heretical. As a result of this edict, Copernicus’s De revolutionibus was banned in March 1616, sixty-three years after it had been published.

Galileo was unable to accept the Church’s condemnation of his scientific views. Although he was a devout Catholic he was also a fervent rationalist, and had been able to reconcile these two belief systems. He had come to the conclusion that scientists were best qualified to comment on the material world, whereas theologians were best qualified to comment on the spiritual ‘world and how one should live in the material world. Galileo argued: ‘Holy Writ was intended to teach men how to go to Heaven, not how the heavens go.’

Had the Church criticised the Sun-centred model by identifying weaknesses in the argument or poor data, then Galileo and his colleagues would have been willing to listen, but their criticisms were purely ideological. Galileo chose to ignore the views of the cardinals, and year after year he continued to press for a new vision of the universe. At last, in 1623, he saw an opportunity to overthrow the establishment when his friend Cardinal Maffeo Barberini was elected to the papal throne as Urban VIII.

Figure 18 Copernicus (top left),Tycho (top right), Kepler (bottom left) and Galileo were responsible for driving the shift from an Earth-centred to a Suncentred model of the universe. Together their achievements illustrate a key feature of scientific progress, namely how theories and models are developed and refined over time by several scientists building on each other’s work. Copernicus was prepared to make the theoretical leap that relegated the Earth to a mere satellite and promoted the Sun to the central role. Tycho Brahe, despite his brass nose, provided the observational evidence that would later help Johannes Kepler to identify the outstanding flaw in Copernicus’s model, namely that the planetary orbits are slightly elliptical, not perfectly circular. Finally, Galileo used a telescope to discover the key evidence that should have convinced doubters. He showed that the Earth is not at the centre of everything, because Jupiter has its own satellites. Also, he showed that the phases of Venus are only compatible with a Sun-centred universe.

Galileo and the new pope had known each other ever since they had attended the same university in Pisa, and soon after his election Urban VIII granted Galileo six lengthy audiences. During one audience, Galileo mentioned the idea of writing a book that compared the two rival views of the universe, and when he departed the Vatican he was left with the firm impression that he had received the Pope’s blessing. He returned to his study and made a start on what would turn out to be one of the most controversial books ever published in the history of science.

In his Dialogue Concerning the Two Chief World Systems, Galileo used three characters to explore the merits of the Sun-centred and Earth-centred world-views. Salviati presented Galileo’s preferred Sun-centred view and was clearly an intelligent, well-read and eloquent man. Simplicio, the buffoon, attempted to defend the Earth-centred position. And Sagredo acted as a mediator, guiding the conversation between the other two characters, although his bias sometimes emerged when he scolded and mocked Simplicio along the way. This was a scholarly text, but the device of using characters to explain the arguments and counter-arguments made it accessible to a wider readership. Also, it was written in Italian, not Latin, so clearly Galileo’s objective was to win widespread popular backing for a Sun-centred universe.

The Dialogue was eventually published in 1632, almost a decade after Galileo had apparently won the Pope’s approval. That huge delay between inception and publication turned out to have severe consequences, because the ongoing Thirty Years’ War had changed the political and religious landscape, and Pope Urban VIII was now ready to quash Galileo and his argument. The Thirty Years’ War had begun in 1618, when a group of Protestants marched into the Royal Palace in Prague and threw two of the town’s officials out of an upper window, an event known as the Defenestration of Prague. The local people had been angered because of the continual persecution of Protestants, and by taking this action they sparked a violent uprising by Protestant communities in Hungary, Transylvania, Bohemia and other parts of Europe.

By the time the Dialogue was published, the war had been raging for fourteen years, and the Catholic Church felt increasingly alarmed by the growing Protestant threat. The Pope had to be seen to be a strong champion of the Catholic faith, and he decided that part of his new hard-hitting populist strategy would be to make a deft U-turn and condemn the blasphemous writings of any heretical scientists who dared question the traditional Earth-centred view of the universe.