Wayward Comet: - Martin Beech

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rel="nofollow" href="#litres_trial_promo">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.

      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].

      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, purely by chance emerging from behind the Sun on December 20th after Kirch’s had disappeared from view. Other observer’s, Britain’s Astronomer Royal John Flamsteed foremost among them, argued that just one comet had been seen, and that remarkably it has been repulsed by the Sun and set upon a parabolic path back into the outer realms of the solar system. Much debate followed, but it appears that it was not until at least mid-1684 that Newton came around to the viewpoint that comets might move along elliptical orbits and that the comet of 1680 had, “fetched a compass about the Sun”. Applying his formidable intellect to the problem, Newton developed the mathematical techniques needed to deduce, from a set of three evenly spaced observations, the parameters that describe the shape of a comet’s orbit (to be discussed below). The fruit of Newton’s labors were eventually revealed in Book III of his 1687 Principia (figure 1.7), where he demonstrated that the Great Comet of 1680 had traveled along a parabolic path with, importantly, the Sun located at the focal point. Newton had not only shown how to deduce the path of a comet from the observations, however, he had also placed them within the realm of his gravitational theory, and this, of course, was his greatest triumph.

      Figure 1.7: Path of the Great Comet of 1680 as revealed in Newton’s 1687 Principia – the original diagram was an impressive 2-page foldout from the text. In this diagram HG indicates Earth’s orbit, while the comet’s path is the parabola ABC. The Sun is located at the focal point D. The line DF is the comet’s line of nodes, and B indicates the perihelion point (which the comet reached on 18 December 1860). See table 1.1 for additional details.

      Newton both observed the comet of 1680 directly and he collected data upon its appearance from across Europe (Table 1.1). In the Principia Newton indicates that he used a 7-foot telescope to observe the comet from Cambridge. He uses the combined observational reports, however, to determine the tail-length evolution of the comet. Newton’s own observations reveal a tail length of less then ½ a degree on November 11 (Gregorian calendar date), increasing to 40 degrees on the sky on January 5; falling to 2 degrees and zero tail on February 10 and 25th respectively. The comet was last observed by Newton on March 9th (March 19th in the Julian calendar). Newton’s diagram (our figure 1.7) is not only revolutionary for showing a parabolic orbit for the comet, but for also showing the time evolution of the comet’s tail, before and after perihelion passage.

      Upon

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