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 (figure 1.10). According to the angle at which the plane intersects the cone the conic section produced will be a circle, an ellipse, a parabola or a hyperbola – these names being first introduced by Apollonius of Perga circa 230 B.C. The ellipse and circle are closed curves, while the parabola and hyperbola are open curves. The properties of the various conic sections are typically described in terms of their eccentricity e, with a circle having e = 0, and the class of all ellipses satisfying 0 ≤ e < 1. A parabola has an eccentricity of e = 1, while hyperbola have e > 1. In this methodology, the conic section is defined geometrically as the locus of a point P moving in such a manner that its distance (FP) from a fixed point (the focus), is proportional to its distance (DP) from a fixed line (the directrix). It is the eccentricity that defines the fixed proportionality, with e = FP / DP. In terms of actual cometary orbits it is not uncommon for a parabolic path to be derived, but no strongly hyperbolic trajectory has ever been observed (see Appendix I). Periodic comets, by their very nature, must have closed elliptical orbits, and when constructing a cometarium it is tacitly assumed that the eccentricity being modeled is less than unity.
Figure 1.10. The conic sections produced when a right circular cone is cut by a plane. If the cut is parallel to the base of the cone then a circle is produced; if the cut is at an angle smaller than that of the cone-angle then an ellipse results. For cuts equal to and larger than the cone-angle, parabola and hyperbola are produced respectively.
While Pappus of Alexandria introduced the description of conic sections in terms of their eccentricity circa A.D. 320, interest in such curves waned with the close of the classic Greek period; only to be seriously revisited in the 17th Century by Johannes Kepler, Gerard Desargues (1593 – 1662) and René Descartes (1596 – 1650). Kepler’s initial foray into such matters took place in 1603 when he was working on his Astronomiae Pars Optica (published in 1604). In this remarkably work Kepler describes the inverse-square law of light intensity and he explores the way in which light is reflected from flat and curved surface – he also studied the properties of the eye, and tried (in vain) to understand the way in which images were realized by the brain. Along with these practical issues of optics relating to astronomy, Kepler also investigated the properties of conic sections, introducing, in fact, the term focus to describe points of convergence. He additionally realized that the properties of conic sections could be described according to the separation of their focal points (this latter condition eventually becoming an integral part of his laws of planetary motion). Working from the definition of an ellipse as the locus of all those points for which the distances r1 and r2, measured from two fixed focal points F1 and F2, are constant (that is: r1 + r2 = constant), so, a circle results if F1 and F2 are coincident; if, in contrast, F2 is moved further and further away from F1 so at infinity the ellipse morphs into a parabola. In the modern era this distinction is usually expressed through the eccentricity term e, which is defined as the distance of either one of the focal points from the center3 of the ellipse OF divided by the ellipses semi-major axis a – accordingly: e = OF / a. For a circle the two foci are coincident and located at the center, dictating that OF ≡ 0, and so e = 0. This latter condition essential tells us that a circle is a special limiting case of the ellipse. As the distance of the focal points from the center approach the perimeter of an ellipse so OF → a, and e → 1. The infinity condition described by Kepler, therefore corresponds to the coincidence of OF and a, for which e = 1. We now begin to reveal an optical synergy between parabolic mirrors, which bring all parallel light rays from infinity to a single focal point, and the elliptical curved mirror. The latter, in fact, has the special property that any light ray passing through one of the focal points must always pass through the other focal point after one reflection at the mirrors perimeter. For cometary orbits the ellipse-to-parabola divide where 0 ≤ e < 1 becomes e = 1 corresponds to an infinite divide. For a cometary orbit the transition to e = 1 literally divides the cosmos into bounded and unbounded space. A comet moving along an orbit with 0 ≤ e < 1, mutatis mutantis must always be periodic and it will eventually, sooner or later, return to any given point on its perimeter – this result being encoded within Kepler’s 3rd law of planetary motion. A comet moving along an orbit having e = 1 (also e > 1, the hyperbolic case) will be seen just the once in the inner solar system and nevermore thereafter.
It was René Descartes who first discussed the analytical properties of the various conic sections. Realizing that all such curves can be described in terms of an equation of the second degree in two variables x and y, a general description of any conic section can be cast in the form
| ax2+by2+2cxy+dx+fy+g=0 | (1.1) |
where the coefficients in (1.1) are constant (with a, b, and c not all being zero at the same time) and vary from one conic section to another. Equation (1.1) further reveals that a conic section is fully determined if five points upon its curve are specified. For an ellipse it turns out that a and b are connected via the eccentricity e so that b2 = a2 (1 – e2), and the general Cartesian equation for an ellipse becomes
| x2a2+y2b2=1 | (1.2) |
the
