(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 of where the comet will be as time passes by can be made – the wayward path of the comet is accordingly corralled by the ellipse.
Figure 1.12. The cometary orbit in space. While the semi-major axis and eccentricity describe the shape of the orbit, the orbit’s orientation to the ecliptic is described by its inclination (i), its argument of perihelion (ω), and the longitude of the ascending node (Ω). The position of a comet, at any time t, in its orbit is fully determined once the parameters (a, e, i, ω, Ω and T) are known, where T is the time of perihelion passage. Table 1.2 provides the orbital data relating to Halley’s Comet during its 1986 return to perihelion – see Chapter 3 for further details on Kepler’s Problem.
| Parameter | Value |
| Semi-major axis a | 17.9411044 AU |
| Eccentricity e | 0.9672760 |
| Inclination i | 162.23928 deg. |
| Argumetn of perihelion ω | 111.84809 deg. |
| Longitude of the ascending node Ω | 58.14536 deg. |
| Time of perihelion T | 1986 February 9.45175 |
Table 1.2. The orbital data set for Halley’s Comet during its 1986 return to perihelion. Data from D. W. Hughes (Journal of the Britisah Astronomical Association, 95, 162-163, 1985).
Rather than describing the planar elliptical orbit of a comet in terms of the (x, y) coordinates of equation (1.2) it is common practice to consider the variation in heliocentric distance. If the angle subtended between the comet and perihelion point is ν (the so-called true anomally) then the heliocentric distance is
| r=a(1−e2)1+ecosv | (1.3) |
From equation (1.3) it is easily found that the perihelion distance, when the comet is at its closest point to the Sun and ν = 0°, is given by q = a(1 – e). Likewise the aphelion point where the comet is at its greatest distance from the Sun, and ν = 180°, is Q = a(1 + e).
Having deduced that the orbit of a periodic comet has the shape of an ellipse, with the Sun located at one of the focal points, we can proceed to illustrate the observable consequences of Kepler’s second law, which requires that the line drawn between the comet and the Sun sweeps out equal areas in equal intervals of time. Indeed, inspired by the return of Halley’s in 1986, David W. Hughes (Sheffield University) has described a Year Post model to describe its orbit. Using 76 posts, with the separation between each post corresponding to a time inteval of one year, Hughes provides tabulated data so that the motion of the comet might be visualized, “in a school playground or an astronomy park”. Figure 1.13 shows the relative positioning of the 76 posts for Halley’s Comet as tabulated by Hughes. It is immediately clear from the diagram that as the comet rounds perihelion its orbital displacement (motion from one post to the next) is much greater than that when it rounds aphelion. Since by construction the time to travel from one post to the next is constant (1 year), so the change in spacing indicates a concomittent change in the velocity – the comet is traveling at a much greater speed when close to perihelion than at aphelion. It is this change in speed of the comet that provides the most visual consequence of Kepler’s second law, and it is this very phenomenon that the cometarium (recall figure 1.2) is designed to illustrate.
Figure 1.13. The location of Halley’s Comet around its orbit at equal intervals of time (Δt = 1 year). The area swept-out between each successive set of points and the Sun is constant in accordance with Kepler’s second law. Orbital data from Table 1.2.
Extending the Solar System
Throughout human history innumerable pictures have been produced of comets within the sky, but it was only with the work of Tycho Brahe, and the appearance of a particularly conspicuous comet in 1577 (C/1577 V1), that they were placed within the realm of the planets (figure 1.14). First observed by Japanese court astronomers on the 8 November of 1577, the comet appeared “like a man standing with legs opened and arms stretched, both sideways” [4]. Having rounded perihelion on 27 October, 1577, the comet was first seen by Brahe on November 13th – he immediately recognized it as something new in the heavens. From the first, Brahe followed the comets
