Big Bang. Simon Singh

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      Kepler was born into a lowly family that struggled to survive the upheavals caused by war, religious strife, a wayward criminal father and a mother who had been exiled after accusations of witchcraft. Not surprisingly, he grew up as an insecure hypochondriac with little self-esteem. In his own self-deprecating horoscope, written in the third person, he described himself as a little dog:

      He likes gnawing bones and dry crusts of bread, and is so greedy that whatever his eyes chance on he grabs; yet, like a dog, he drinks little and is content with the simplest food… He continually seeks the goodwill of others, is dependent on others for everything, ministers to their wishes, never gets angry when they berate him and is anxious to get back into their favour… He has a dog-like horror of baths, tinctures and lotions. His recklessness knows no limits, which is surely due to Mars in quadrature with Mercury and in trine with the Moon.

      His passion for astronomy seems to have been his only respite from self-loathing. At the age of twenty-five he wrote Mysterium cosmographicum, the first book to defend Copernicus’s De revolutionibus. Thereafter, convinced of the veracity of the Sun-centred model, he dedicated himself to identifying just what it was that made it inaccurate. The greatest error was in predicting the exact path of Mars, a problem that had plagued Copernicus’s assistant, Rheticus. According to Kepler, Rheticus had been so frustrated with his failure to solve the Mars problem that ‘he appealed as a last resort to his guardian angel as an Oracle. The ungracious spirit thereupon seized Rheticus by the hair and alternately banged his head against the ceiling, then let his body down and crashed it against the floor.’

      With access at last to Tycho’s observations, Kepler was confident that he could solve the problem of Mars and remove the inaccuracies in the Sun-centred model within eight days; in fact, it took him eight years. It is worth stressing the amount of time that Kepler spent perfecting the Sun-centred model– eight years!– because the brief summary that follows could easily underplay his immense achievement. Kepler’s eventual solution was the result of arduous and tortuous calculations that filled nine hundred folio pages.

      Kepler made his great breakthrough by jettisoning one of the ancient tenets, namely that the planets all move in paths that are circles or combinations of circles. Even Copernicus had clung loyally to this circular dogma, and Kepler pointed out that this was just one of Copernicus’s flawed assumptions. In fact, Kepler claimed that his predecessor had wrongly assumed the following three points:

      1. the planets move in perfect circles,

      2. the planets move at constant speeds,

      3. the Sun is at the centre of these orbits.

      Although Copernicus was right in stating that the planets orbit the Sun and not the Earth, his belief in these three false assumptions sabotaged his hopes of ever predicting the movements of Mars and the other planets with a high degree of accuracy. However, Kepler would succeed where Copernicus had failed because he discarded these assumptions, believing that the truth emerges only when all ideology, prejudice and dogma are set aside. He opened his eyes and mind, took Tycho’s observations as his rock and built his model upon Tycho’s data. Gradually an unbiased model of the universe began to emerge. Sure enough, Kepler’s new equations for the orbits matched the observations, and the Solar System took shape at last. Kepler exposed Copernicus’s errors, and showed that:

      1. the planets move in ellipses, not perfect circles,

      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.

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