Wayward Comet: - Martin Beech

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the stars and attendant constellations, its constancy and very predictability is not absolute. To the human eye there are seven objects that behave differently to the stars - these are the Sun, the Moon and the planets Mercury, Venus, Mars, Jupiter and Saturn. These objects, vastly different in brightness, and vastly different in their daily motions, move across the backdrop of the stars - they do not move with them. The Sun, Moon and planets are obvious oddities when compared to the pinpoints of light that pockmark the celestial vault and move in lockstep unison, in mythical shapes and asterisms, rising in the east and setting in the west, but never changing their positions one relative to another. To the planets can be added the wayward motion of comets and the transitory flashes of meteors, although the fact that these latter objects even fell within the realm of astronomy is something of relatively recent genesis (see Appendix I).

      To the ancients, well versed in common sense, it seemed obvious that the Sun existed to illuminate the day and the Moon to periodically illuminate the night - but what of the planets? What was their purpose, and why did they move differently and with a brightness that varied from one month to the next? By considering the stars1 human insight moved in two, polar opposite, directions and laid the foundations for what later, much later in fact, became the stout and resolute body of science and its mercurial shape-shifting offspring astrology. Both areas of study are still with us today, but science, the practice of measured and rationale thought combined with prediction and experimentation, as applied to natural phenomena, is now in the ascendancy. For all their differences, however, each discipline, in its own way, has had much to say about the appearance and properties of comets.

      The Greek philosophers championed the early ideas, and Claudius Ptolemy, in the first century AD, gave us a summary of all that had been deduced. Ptolemy’s great work Syntaxis Mathematica, better known through its Arabic translations as the Almagest, was a tour de force - a superb encyclopedic work, full of brilliant, if not controversial ideas. For the ancients, the Earth stood fixed and un-moved – a spherical body at the center of the universe. Around the earth, in a series of concentric shells, were the assembled planetary realms, and about them all was the sphere of the stars. Ptolemy provided a set of detailed mathematical instructions to determine the motion of the planets, moon and Sun. Using an eccentric offset, epicyclic construction, Ptolemy was able to provide an excellent description of the motion of all the planets as they moved across the celestial sphere. He achieved this superb description, however, by introducing the idea of the equant point (figure 1.4). The equant point, while vital for making the Ptolemaic system work, was roundly criticized by subsequent commentators - it was a step too far since it gave importance to an otherwise empty point in space and it introduced the requirement of another point (the center of the epicycle) moving with a non-constant velocity into celestial calculations. The latter attribute went firmly against Plato’s original doctrine that all celestial motion should precede with a constant velocity and within a perfect circle. The circular motion component was contained in Ptolemy’s model, but to describe such observed characteristics as retrograde motion, and the periodic variations in the accumulated motion of a planet across the sky, in equal intervals of time, the equant was vital.

      While the Arabic and medieval astronomers did all they could to reduce the number of epicycles and to remove the equant from planetary theory: “epicycles correspond to nothing in nature” decried Henry of Langenstein in his 1373 Contra Astrologos. Ptolemy’s model worked well, indeed, it worked exceptionally well, but it offered no harmony of thought and no consistency of concept. At issue, ultimately, was the question of reality; how are the planets really distributed in space, and where exactly is the Earth located with respect to the other celestial objects. A purely mathematical description of planetary motion, such as that offered by Ptolemy, was all well and good, but did the mathematical model actually describe the reality of the heavens and God’s creation.

      Figure 1.4: Schematic diagram of Ptolemy’s planetary theory. The center of the deferent is located at O, while the Earth and the equant are located at E and Q. The center of the epicycle C moves around the deferent such that it sweeps out equal angles in equal intervals of time about the equant point. The planet P is positioned on the epicycle according to the requirement that CP is parallel to OS, where S indicates the location of the (fictitious) mean Sun which moves with a constant speed about the center O, completing one full rotation in the time interval of one year.

      Ptolemy’s Syntaxis was THE astronomy book for nearly one and a half thousand years - an incredibly run by any standards. His work provided a practical means for determining the positions of the Sun, moon and planets at any time, past, present and future, but as the centuries ticked by it was increasingly viewed as an esthetically unpleasing system. It was for these latter esthetic reasons, rather than because of any predictive deficiencies, that Nicolaus Copernicus set out to redefine and reshape the planetary realm in the mid-16th Century. He worked upon his ideas for nearly half of his life, but encouraged by his young disciple Georg Rheticus, eventually published his magnum opus, De Revolutionibus Orbium Coelestium, in 1543.

      Copernicus’s work was altogether something different, not because it was actually new, other philosophers, at various times, had suggested similar such ideas, but because Copernicus actually worked out the mathematical details. In his design, however, Copernicus looked backward to the postulates of Plato which required that planets must move with uniform speed along circular paths (or orbits as we now call them). This thinking was in some sense regressive, but by placing the Sun at the center of the universe (as Copernicus knew it) and making Earth the third planet out, then a reasonably good description of observed planetary motion could be obtained. The accuracy of the initial Copernican model was inferior to that of the Ptolemaic model, but as he had intended all along the model developed by Copernicus was simpler in concept and it provided a uniform description of the heavens. Each planet in the Copernican system had its own specific orbit and the spacing between the planets could be determined directly from the observations.

      It was the straightforward construction of the Copernican model that caught the imagination of other philosophers, but it took the mathematical genius of Johannes Kepler to make the model work. Building upon the highly accurate positional data obtained by Tycho Brahe, Kepler applied what in modern terminology would be called a reverse engineering study. Specifically, Kepler used the observational data to determine the shape of the orbit of Mars rather than assume, as Copernicus had, that it must be circular. Finding that the orbit of mars was elliptical, not circular, then automatically required that the speed with which a planet moved along its orbit must vary, being most rapid when close to the Sun and slower when further away. Not only this, the observational data indicated that the Sun must be located at one of the two focal point positions of a planet’s elliptical orbit (figure 1.5). Kepler explained the first two of his laws in Astronomia Nova, published in 1609, and later introduced a third law, in a somewhat obscure form, in his Harmonice Mundie, published in 1618. Kepler’s collected laws of planetary motion as we now known them are:

      K1: The orbit of every planet is an ellipse with the Sun located at one of its focal points

      K2: A line drawn between the Sun and a planet sweeps out equal areas in equal intervals of time

      K3: The square of the orbital period is proportional to the semi-major axis cubed

      It is K1 that describes the shape of planetary orbits and identifies the location of the Sun with respect to the orbit (see figure 1.5), while it is K2 that describes how the planet moves in its orbit and explains why the speed of the planet varies between perihelion and aphelion. When first announced in his Harmonice Mundie K3 was an absolute mystery - there was, at that time, no explanation as to why such a relationship between the orbital period and orbital size should exist. While Kepler tried to explain his laws in terms of magnetic planets interacting with a magnetic monopole Sun, it was Isaac Newton who eventually provided the first correct physical reasoning behind all three of the planetary laws. Indeed, Newton showed that any two objects interacting under a centrally acting force

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