Wayward Comet:. Martin Beech

for the Great Comet of 1680, Halley immediately suggested that the new mathematical techniques might be applied to other comets. Newton was non-committal, and Halley, eventually some eight years later, set about the work himself. Writing to Newton on 7 September 1695 Halley noted that the orbital solutions for the comets observed in 1607 and 1682 were nearly identical. In a second letter dated for September 1695, Halley further noted that a re-analysis of the data for the Great Comet of 1680 indicated that its orbit might be better described by an ellipse rather than a parabola. Over the following months more letters were exchanged between Halley and Newton, and it was on 3 June 1696 that Halley first explained to the assembled Fellows of the Royal Society of London, that the comets of 1607 and 1682 were one and the same object – a month later Halley announced to the again assembled Fellows that the comet of 1618 had followed a parabolic path around the Sun and that at its perihelion point was located interior to the orbit of Mercury.

Letter	Date	Tail (deg.)	Letter	Date	Tail (deg.)
I	Nov. 04	No tail	P	Jan. 05	40 *
K	Nov. 11	< ½ *	Q	Jan. 25	~ 6
L	Nov. 14	~ 15	R	Feb. 05	No tail #
M	Dec. 12	~ 40 #	S	Feb. 25	No tail *
N	Dec. 21	~ 90 +, #	T	Mar. 05	----
O	Dec. 29	~ 50	V	Mar. 09	----

Table 1.1. Key to Newton’s diagram showing the Great Comet of 1680. Note: the dates given by Newton correspond to the Gregorian calendar, then still in use within England - to convert to the common Julian calendar, add 10 days to each entry. See figure 1.7 for the letter sequence in relation to the orbit. Symbol key: * indicates measurements made by Newton at Cambridge; # telescopic observations by Astronomer Royal, John Flamsteed; + observations by Robert Hooke.

      The years bracketing the beginning of the 18th Century saw Halley, as a Captain in the Royal Navy, pursuing a number of expeditions related to coastal mapping and the measurement of magnetic declination variations. By 1702, however, Halley had the basics of his great cometary treatise prepared, although it was not to see publication until 1705. This relatively short work, A Synopsis of the Astronomy of Comets, contained the derived orbital data (figure 1.8) for 24 comets which had been witnessed between 1337 and 1698. “Having collated all the observations of comets I could”, writes Halley, “I fram’d this Table [figure 1.8], the Result of a prodigious deal of Calculation”. Halley initially assumes that all of the orbits are “exactly parabolic”, noting that, “upon which supposition it would follow, that comets being impell’d towards the Sun by a Centripetal Force [i.e. gravity], descend as from spaces infinitely distant”. Such a situation might reasonably arise if Descartes vortex theory were true, but since Newton had roundly dismissed such vortices, Halley continued, “But, since they appear frequently enough, and since none of them can be found to move with Hyperbolick Motion [that is, with eccentricity e > 1], or a motion swifter that a Comet might acquire by its Gravity to the Sun, ‘tis highly probable they move in very exentrick Orbits and make their return after long Periods of Time”. This is an interesting set of arguments, with Halley in the first case essentially bulking at the idea that there might be an infinite, or at least a vast number, of comets swirling around in the heavens, and secondly he applies an observational constraint that no truly hyperbolic orbit with e > 1 had every been recorded for a comet approaching the Sun [the issue of hyperbolic comets is discussed further in Appendix I]. With these observations in place, however, Halley sets out to see if any of his comets follow elliptical paths. Importantly, Halley noted that the comets of 1531, 1607 and 1682 had nearly identical orbital parameters, and it was upon this basis that he boldly predicted the comet’s return for late 1758. History now tells us that Halley’s Comet, now so called, was dually swept-up in the telescope of German astronomer Johann Palitzsch on 25 December 1758. With the conformation of Halley’s prediction the size of the known solar system grew by a factor of nearly times four, pushing its outer limits to just beyond 35 AU. Not only this the return of the comet confirmed, if indeed there was any doubt left by 1758, that the dynamics of the solar system was understandable, and more importantly predictable, in terms of Newtonian dynamics, and, of course, it also confirmed that comets obey the laws outlined by Kepler.

      Figure 1.8. Cometary orbits, as produced by “a prodigious deal of calculation”, from Halley’s Synopsis of the Astronomy of Comets.

      In his 1705 Synopsis Halley sagely writes that, “astronomers have a large field to exercise themselves in for many ages, before they will be able to know the number of these many great bodies [comets] revolving about the common center of the Sun; and reduce their motions to certain rules”. Indeed, the process of observing, recording and reducing orbits continues to this very day [2], although as of the end of 2012 just 272 comets are known to be periodic – that is observed at least twice (figure 1.9). Having decided that the comets of 1531, 1607 and 1682 were one and the same object, Halley goes on to argue that should it return in 1758 then, “we shall have no reason to doubt but the rest must return too”. Here Halley somewhat overstepped the mark and we find that of the 24 comets discussed by Halley, only two are actually periodic (accounting for 5 of the appearances in his table), with the remainder, some 19 comets, being single-encounter long-period bodies derived from the Oort cloud (see Appendix 1). Halley’s Comet was the only periodic comet known for well over 100 years; the orbit and past activity of the second periodic comet, comet 2P/Encke, being described by Johann Encke in 1819 (figure 1.9).

      Figure 1.9: Cumulative number of known periodic comets (lower line) and comets observed (upper line) plotted against time: 1650 to 1950. While sightings of Halley’s Comet (indicated by large dots) can be traced back to 240 BC, we use the 1682 return as being its discovery year. Over the time interval considered in this data display, six of the periodic comets are now listed as being ‘lost’, and have either become totally dormant or have been destroyed through catastrophic fragmentation (see Appendix I).

An aside on Conic Sections – especially ellipses

      Not only did Newton describe the parabolic path of the Great Comet in his Principia, he additionally showed that the orbit of any object, be it a comet or a planet, moving under a centrally acting force must follow the path described by a conic section. Such curves were studied in antiquity by Menaechmus of Greece circa 350 B.C. and they conform to the boundary curve produced when a 2-dimensional plane slices through a right circular cone (