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 coefficients a and b correspond to the half-lengths of the longest (that is the major) and shortest (the minor) axis of the ellipse – both of which cross perpendicularly at the center point (x, y) = (0, 0). The two focal points contained within an ellipse are additionally, by definition, located at the points (x, y) = (± ae, 0). Importantly in terms of astronomical observations, an ellipse is uniquely determined by the measurement of any four points that lie upon its perimeter. This situation can actually be improved upon, and the mathematical techniques developed by such late 18th Century luminaries as Leonhard Euler, Johann Lambert, Pierre-Simon Laplace and Carl Friedrich Gauss enabled the computation of orbits from just three positional measurements4.
Just as there are numerous ways of describing an ellipse mathematically, so there are also many ways of drawing them mechanically. Such devices include the so-called Trammel of Archimedes, in which a ruler is attached via freely rotating pivots to two shuttles that are constrained (that is trammeled) to move along grooves that form a cross-shaped configuration – as the ruler rotates and the shuttles move along their respective grooves, so the free-end of the ruler sweeps out an elliptical curve. Other devices make use of moving linkage systems or follow the special hypertrochoid path swept out by the center of a small circle, of radius R, rolling upon the interior of a larger ring of radius 2R [3]. Of all the methods for drawing an ellipse, however, perhaps the simplest and in many ways the most versatile is that of the string compass. In this case the ends of a piece of string, of length 2a, are pinned down at the focal point locations of the ellipse to be drawn – the spacing of the two foci, recall, will be equal to a distance 2ae. Keeping the string taut at all times an ellipse can then be drawn-out with an appropriate pencil or marker (figure 1.11).
Figure 1.11. Compass and string method for constructing an ellipse. The string is adjusted to have a length equal to the required major-axis (AC = 2a), and its two ends are held in place at the focal points - separated by the distance F’F’ = 2ae. The image shows a variant of the compass ellipsograph patented by Johann Hardt in 1922.
The cometary orbit described
Comets move through space along orbits that have a 3-dimensional geometry (figure 1.12). The essential orbital path, as far as we shall be concerned, however, is that of a two-dimensional ellipse, with the Sun, in accordance with Kepler’s first law, being located at one of the focal points. The ellipse is defined by just two parameters: the semi-major axis, a, and the eccentricity e. Once these two terms are given then the entire shape and physical extent of the orbit are determined. Indeed, once these two quantities are specified it is possible to construct a scale diagram of the comet’s orbit. And, once the orbit has been compassed round, so predictions