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 progress across the sky, keeping track of its tail and coma variations. Most importantly for the history of astronomy, however, Brahe was able to combine observations of the comets motion made from different locations, and this enabled a parallax distance estimate to be made. Incredibly, the comet was located at least 230 Earth radii away, and its coma was nearly ¼ the size of the Earth. At the deduced distance, the comet was situated well beyond the orbit of the Moon, which sits at a distance of about 65 Earth radii away. Having placed the comet in the celestial realm, however, Brahe argued that it was nonetheless a temporary object that would eventually fade from all view. Brahe additionally suggested a mechanism for the appearance of the comet’s tail – it being the result of sunlight passing trough the rarified and porous matter that constituted the comet’s head.
Figure 1.14. The Comet of 1577 placed (and labeled X) in its own circular orbit (STVX) about the Sun (C) – as deduced by Tycho Brahe in his De Mundi, published in 1588.
Brahe’s master work that included his analysis of the comet of 1577, De Mundi Aetherei Recentiorbus, was published in 1588, and here he placed the comet upon a circular orbit about the Sun just beyond the orbit of Venus (figure 1.14). Brahe’s planetary system, although being post-Copernican in date, was (seemingly) unusual for having the planets Mercury through to Saturn orbiting the Sun, with the Sun and Moon orbiting a stationary Earth [5]. The comet is drawn as moving along a circular orbit, but Brahe additionally suggested the path might be more oval in shape. Since Brahe’s planetary model was never widely accepted the question concerning the shape of cometary paths was not considered settled. Indeed, later on, in the 1630s, Jeremiah Horrocks suggested that the comet of 1577 might have been a temporary ejection of material from the Sun, moving initially along a straight line (rectilinear) path, but then, being strongly affected by the Sun’s magnetic field, it eventually fell back to the place of its origin. The idea that comet’s might move along rectilinear paths had actually been developed earlier by Johannes Kepler in his De Cometis Libelli Tres, published in 1619, although Kepler based his arguments upon the comets seen in 1607 and 1618.
“When something unusual arises in the heavens, whether from strong constellations or from new hairy stars [comets], then the whole of nature, and all living forces of all natural things feel it and are horror stricken”. So wrote Johannes Kepler on comets. Recipient of Brahe’s observational data and Royal patronage, upon the latter’s death in 1601, Kepler, as we have seen, played the vital role of both confirming and correcting the Copernican hypothesis. Planets, as Kepler showed, moved along elliptical paths about the Sun, but comets, Kepler believed, were ephemeral, formed spontaneously from various rising vapors and seen only the once. On this basis Kepler assigned the comets not an elliptical path about the Sun, but a straight-line, rectilinear motion through the celestial realm (figure 1.15). Kepler’s magnum opus De Cometis Libelli, was published in 1619 – the same year as his (perhaps) more famous text Harmonius Mundi in which his third law of planetary motion