Wayward Comet:. Martin Beech

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СКАЧАТЬ 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 that varied as the inverse square of distance must automatically obey Kepler’s laws (figure 1.6). The force that Newton introduced, of course, was that of gravity, and this combined with the laws pertaining to the conservation of energy and angular momentum provided a full description of celestial dynamics. These remarkable results were first articulated in Newton’s masterpiece, Philosophiæ Naturalis Principia Mathematica, published by the Royal Society of London, with financing by Edmund Halley, in 1687. Not only was Halley the paymaster and editor of Newton’s remarkable text, he was also the inspired astronomer who wrote an ebullient dedication to the tome’s esteemed author. Indeed, Halley eulogized that, “now we know the sharply veering ways of comets, once a source of fear and dread, no longer do we quail beneath appearances of bearded stars”. In the same way that the pen is mightier than the sword, so Newton’s mathematical analysis and physical insight were mightier than two thousand years, and more, of superstition and fear (see Appendix I).

      Figure 1.5: The key characteristics of Kepler’s 1st law are that the orbital path is elliptical in shape and that the Sun is located at one of the two focal points (F1 and F2) of the ellipse. The two focal points are situated at equal distances away from the center (O) of the ellipse. A line drawn between the planet (or comet) and the non-Sun, or empty focus, has dynamical characteristics similar to that of Ptolemy’s equant point.

Coda: Newton’s proof of K2

      With reference to figure 1.6, imagine an object at point B subject to the gravitational pull of an object located a point S. The arc ABCDEF indicates the successive positions of the object at equal time intervals t. In the first time interval, the objects moves from A to B and sweeps out the area SAB – at this stage the motion is rectilinear and the object moves along the straight line path AB. Once at point B, however, the gravitational force begins to pull the object away from its straight line path with the result that it moves along the path BC rather than Bc. The proof of Kepler’s 2nd law now proceeds by demonstration that in the second time interval the area swept out by the object SBC, as it moves from B to C, is exactly equal to area SAB. Firstly, Newton noted that the area of triangles SAB and SBc are equal – this follows since they have the same altitude (the perpendicular line dropped from S to the line extending through ABc) and they have the same base lengths: AB = Bc. The next step is to show that the area of the triangle SBc is equal to SBC. This is accomplished by noting that the displacement cC (due to the gravitational force acting at S) runs parallel to the line SB. With this condition in place Newton had his proof of equal areas since the two triangles of interest have the same base length SB and identical altitudes (the line dropped from c – and C – to the line extending along SB). Having shown that triangles SAB and SBC have equal areas, the final part of the proof is just a generalization – the same result, as just proven, must apply to the motion of the object from point C, with the triangles SBC and SCD also having equal areas.

      Figure 1.6: Newton’s diagram explaining the motion of an object under a central force and his proof of Kepler’s second law. Newton’s proof of K2 proceeds geometrically and requires a demonstration that the area of triangle SAB is the same as triangle SBC, which is the same as SCD and so on.

      Newton’s proof of Kepler’s second law is a remarkable geometric construction – he has employed nothing more than straight lines and triangles, and the result is independent of the actual value of the time step t. Likewise the proof is independent of the magnitude of the centrally acting force at S [1].

Comet C/1680 V1 - the game changer

      It took over 70 years to come about, but eventually, on 14 November 1680 German astronomer Gottfried Kirch became the first person to discover a comet with the aid of a telescope. Working in the early morning hours, and using a 2-foot focal length refractor Kirch was observing the Moon and Mars when by chance, in the constellation of Leo, he sighted “a sort of nebulous spot, of an uncommon appearance….. a nebulous star, resembling that in the girdle of Andromeda2”. Kirch followed his nebulous star over ensuing nights, and on November 21st using a 10-foot focal length telescope confirmed the appearance of a small but distinct tail. The comet was increasing in brightness and heading towards the Sun. By the close of November the new comet was visible to the naked-eye and its tail was reckoned to be over 15o long.

      As December proceeded the comet grew ever brighter but by mid-month it was lost to view within the Sun’s glare. Perihelion occurred on December 18th. Rapidly rounding the Sun the comet re-emerged to view sporting an extraordinary long, near 90o tail on December 20th. Many years later Augustan missionary Casimo Diaz recalled his sighting of the comet from Manila in the Philippines: “the frightful comet [was] so large it extended, like a wide belt, from one horizon to the other… causing in the darkness of the night nearly as much light as the Moon in her quadrature”.

      Having caught the eye of the world’s populace the Great Comet of 1680 now required an explanation from the astronomers. The first question that needed to be settled was whether one or two comets had actually been seen. Some observer’s, Isaac Newton among them, initially argued that two comets has been seen: the first being Kirch’s comet heading inwards towards the Sun, with a second comet, СКАЧАТЬ