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ОглавлениеCHAPTER 1
THE COMET CONSTRAINED
The education of Euphrosyne
Obscure in the modern-era, the heyday of the cometarium was set in the first half of the 19th Century. It is a device that was built upon the intellectual certainty of Newtonianism and the observational triumphs of 18th Century astronomy. Before exploring the details of all these multifaceted connections, however, let us first see how famed instrument maker and jubilant author Benjamin Martin (1714 - 1782) introduced the cometarium in his imagined dialogue between a young natural philosopher, Cleonicus, and his eager and able pupil, Euphrosyne (figure 1.1). The year of the work is 1772, and Halley’s Comet had but 13 years earlier appeared in the night sky to vindicate Edmund Halley’s 1705 prediction and thereby sealing the hegemony of Newton’s theory of gravitational attraction as applied to celestial bodies. The dialogue proceeds thus:
Figure 1.1: The Young Gentleman and Lady’s Philosophy – Dialogue XVL by Benjamin Martin (1772)
Euphrosyne: Since you gave me the lecture on comets, you have filled my head with such odd kinds of ideas, and I scarcely known whether I hope or fear most to see a comet; but dear Cleonicus, since that is shortly to be the case, and a comet we must behold, if your astronomical prediction is to be regarded, I think I may as well take courage, and resolve to attend the important event undauntedly.
Cleonicus: Fortitude, my Euphrosyne, is an excellent virtue; and hence I must admonish you to speak with more reverence of astronomical predictions.
Euphrosyne: If I remember right, you once told me, that you could make the manner of the comet’s motion intelligible by a proper instrument, as well as those of the planets.
Cleonicus: I did so; the instrument I mean is called the cometarium, and which I shall now spend one quarter of an hour in explaining to you – Here is the machine.
Euphrosyne: And a beautiful one it is; I can almost tell the use of it by its very appearance.
Cleonicus: Observer, when I turn the winch, the brazen comet moves, and with a very unequal pace in its elliptical orbit, about the focal Sun.
This wonderful dialogue, in just a few short refrains, sets the scene for the appearance and application of the cometarium, invented, in fact, some forty-years before Martin was writing, and it also brings out the new order that had been imposed – or more correctly revealed – in relation to cometary orbits. Indeed, the final stanza of the discussion presented points towards serious astronomy and the visual illustration of the first two of Kepler’s laws of planetary motion. In contrast, however, the first stanza reveals an initial sense of doubt on behalf of Euphrosyne. The fact that the prediction is to be regarded as correct, as emphasized by Cleonicus rests upon Newton’s laws and the (mostly) correct predictions by Edmund Halley in 1705 (to be described below).
But what was this device that Cleonicus showed to Euphrosyne? The answer to this lies partly within an earlier work by Martin. Businessman that he was, Martin knew that the predicted return of Halley’s Comet in 1758 was bound to cause a ripple of public interest in matters astronomical. To this end he published in 1757 a small pamphlet entitled The Theory of Comets Illustrated, and to go with this work he re-invented and re-named a device previously called the equal-area machine; a demonstration device made known through the popular public lectures of Scottish astronomer James Ferguson. It was Martin, in 1757, who first coined the name cometarium, and for the tidy sum of just 5-Guineas such a device could be purchased to further enhance ones viewing of the spectacle soon to be portrayed in the heavens above.
From the very outset the appearance of the cometarium is somewhat odd and perhaps even intimidating (figure 1.2). Its working face displays a number of graduated dials and pointers, and set within a large ring is an elliptical track-way. Certainly it has the appearance of a serious demonstration device - there are no frills or extraneous details. But what exactly is the device trying to tell the user?
Figure 1.2: The cometarium as improved by Benjamin Martin. The text around the elliptical track reads: “the come of the year 1682”.
The cometarium is not exactly like the planetarium and orrery, whose function is essentially evident at first glance; they are devises to show the relative motion of the planets around a central Sun (see Chapter 2). This being said, some features of the cometarium, as Euphrosyne so eloquently observed, are readily understandable. Having already been told that comets move along elliptical orbits about the Sun, it is clear that the central elliptical track must represent the path of the comet. Indeed, inscribed around the edge of the inner elliptical plate are the words “the Comet of the year 1682”. This track, therefore, represents the orbit of Halley’s Comet, and indeed, the year corresponds to the very time at which Halley observed the comet and began to wonder about its origins and past history (see Appendix 1).
Given that the elliptical track corresponds to the orbit of the comet the rotation point of the comet-driving radial arm must correspond to the location of the Sun - and, indeed this rotation point is distinguished by a spiked coronal motif. From this offset position of the Sun, it is immediately seen that there are two points in any comets orbit where it is at its closest and most distant locations. These are the perihelion and aphelion points respectively, and it takes the comet exactly half of its orbital period to move from one to the other. It is the rotation of the radial arm extending form the Sun focus point that drives the comet ball around the elliptical track, and the large circular ring, centered on the Sun, corresponds to the great circle of the comet path projected onto the celestial sphere. The lower circular dial of the cometarium indicates the time elapsed since perihelion passage, and its scale is divided into 75 divisions, corresponding to an orbital period of 75 years.
It is now well known that the period of Halley’s Comet is not exactly 75 years, and that from one perihelion passage to the next its orbital eccentricity and orientation can change. The cometarium is not intended to represent, therefore, any specific orbit as it might have been observed at some specific return of Halley’s Comet, nor does it act as a predictive instrument to tell the user where the comet is going to be in the sky. Indeed, although Martin labeled his cometarium as corresponding to that of the comet for 1682, this designation is entirely irrelevant, and in reality any orbital period can be modeled. The only relevant point is that one 360-degree rotation of the time dial corresponds to the orbital period of the comet. The only way that different orbital eccentricities can be accommodated is by literally changing the eccentricity of the track in which the comet ball is constrained to move. This being said, Halley’s Comet was the very first verified periodic comet, and in 1755 it was still the only comet known to have returned more than once to the inner solar system.
Figure 1.3: The interior wheel-work to Benjamin Martin’s cometarium.
To make the cometarium work the demonstrator turns the small crank K (as seen in the lower left hand corner of figure 1.3). Connected to a continuous worm gear, the crank directly engages with the first circular gear c. The center of gear c is connected to the lower time dial located on the front of the cometarium. The crank K, therefore, is the time input of the machine; the more turns being given to crank K, so the greater the time interval being recreated along the comet’s orbital path. Gear C is directly meshed to a second circular gear Q whose center is located at one of the focal points of the elliptical track. The spindle at Q is further fixed to a rigid elliptical former LM. If we now imagine that the demonstrator turns the crank K at a constant rate, the gearing - so far described - is such that the elliptical former will be driven about its focal point at a constant rate. This constant rate of rotation is transformed into a non-constant rate of rotation by letting the elliptical former LM drive a second elliptical former NO about its focal point P. This drive is achieved via a figure of eight catgut string constrained to move along a v-grove cut into the edges of the elliptical formers. A spindle attached at P to the elliptical former NO will now rotate at a non-constant rate. By attaching a drive arm to the spindle at P on the front face of the cometarium, the comet ball will be driven around its track with varying speed. Spindle P represents the Sun-occupied focal point of the comet’s orbit, and the drive rate of the comet ball will be at its fastest when near to the Sun and at its slowest when far away from the Sun. Further details of the motion of the comet ball will be presented later in Chapter 2 and in Appendix II. In the mean time, we now regain contact with the dialogue of Cleonicus and begin to understand his final quoted words which tell us that the “brazen comet moves, and with a very unequal pace in its elliptical orbit about the focal Sun”. The cometarium being described by Cleonicus is clearly, given the focal point location of the Sun, a product of the Copernican hypothesis and it is the development of this idea that we shall briefly consider next.
The workings of the solar system: a brief history
Although the night sky is animated by the stately movement of the stars and attendant constellations, its constancy and very predictability is not absolute. To the human eye there are seven objects that behave differently to the stars - these are the Sun, the Moon and the planets Mercury, Venus, Mars, Jupiter and Saturn. These objects, vastly different in brightness, and vastly different in their daily motions, move across the backdrop of the stars - they do not move with them. The Sun, Moon and planets are obvious oddities when compared to the pinpoints of light that pockmark the celestial vault and move in lockstep unison, in mythical shapes and asterisms, rising in the east and setting in the west, but never changing their positions one relative to another. To the planets can be added the wayward motion of comets and the transitory flashes of meteors, although the fact that these latter objects even fell within the realm of astronomy is something of relatively recent genesis (see Appendix I).
To the ancients, well versed in common sense, it seemed obvious that the Sun existed to illuminate the day and the Moon to periodically illuminate the night - but what of the planets? What was their purpose, and why did they move differently and with a brightness that varied from one month to the next? By considering the stars1 human insight moved in two, polar opposite, directions and laid the foundations for what later, much later in fact, became the stout and resolute body of science and its mercurial shape-shifting offspring astrology. Both areas of study are still with us today, but science, the practice of measured and rationale thought combined with prediction and experimentation, as applied to natural phenomena, is now in the ascendancy. For all their differences, however, each discipline, in its own way, has had much to say about the appearance and properties of comets.
The Greek philosophers championed the early ideas, and Claudius Ptolemy, in the first century AD, gave us a summary of all that had been deduced. Ptolemy’s great work Syntaxis Mathematica, better known through its Arabic translations as the Almagest, was a tour de force - a superb encyclopedic work, full of brilliant, if not controversial ideas. For the ancients, the Earth stood fixed and un-moved – a spherical body at the center of the universe. Around the earth, in a series of concentric shells, were the assembled planetary realms, and about them all was the sphere of the stars. Ptolemy provided a set of detailed mathematical instructions to determine the motion of the planets, moon and Sun. Using an eccentric offset, epicyclic construction, Ptolemy was able to provide an excellent description of the motion of all the planets as they moved across the celestial sphere. He achieved this superb description, however, by introducing the idea of the equant point (figure 1.4). The equant point, while vital for making the Ptolemaic system work, was roundly criticized by subsequent commentators - it was a step too far since it gave importance to an otherwise empty point in space and it introduced the requirement of another point (the center of the epicycle) moving with a non-constant velocity into celestial calculations. The latter attribute went firmly against Plato’s original doctrine that all celestial motion should precede with a constant velocity and within a perfect circle. The circular motion component was contained in Ptolemy’s model, but to describe such observed characteristics as retrograde motion, and the periodic variations in the accumulated motion of a planet across the sky, in equal intervals of time, the equant was vital.
While the Arabic and medieval astronomers did all they could to reduce the number of epicycles and to remove the equant from planetary theory: “epicycles correspond to nothing in nature” decried Henry of Langenstein in his 1373 Contra Astrologos. Ptolemy’s model worked well, indeed, it worked exceptionally well, but it offered no harmony of thought and no consistency of concept. At issue, ultimately, was the question of reality; how are the planets really distributed in space, and where exactly is the Earth located with respect to the other celestial objects. A purely mathematical description of planetary motion, such as that offered by Ptolemy, was all well and good, but did the mathematical model actually describe the reality of the heavens and God’s creation.
Figure 1.4: Schematic diagram of Ptolemy’s planetary theory. The center of the deferent is located at O, while the Earth and the equant are located at E and Q. The center of the epicycle C moves around the deferent such that it sweeps out equal angles in equal intervals of time about the equant point. The planet P is positioned on the epicycle according to the requirement that CP is parallel to OS, where S indicates the location of the (fictitious) mean Sun which moves with a constant speed about the center O, completing one full rotation in the time interval of one year.
Ptolemy’s Syntaxis was THE astronomy book for nearly one and a half thousand years - an incredibly run by any standards. His work provided a practical means for determining the positions of the Sun, moon and planets at any time, past, present and future, but as the centuries ticked by it was increasingly viewed as an esthetically unpleasing system. It was for these latter esthetic reasons, rather than because of any predictive deficiencies, that Nicolaus Copernicus set out to redefine and reshape the planetary realm in the mid-16th Century. He worked upon his ideas for nearly half of his life, but encouraged by his young disciple Georg Rheticus, eventually published his magnum opus, De Revolutionibus Orbium Coelestium, in 1543.
Copernicus’s work was altogether something different, not because it was actually new, other philosophers, at various times, had suggested similar such ideas, but because Copernicus actually worked out the mathematical details. In his design, however, Copernicus looked backward to the postulates of Plato which required that planets must move with uniform speed along circular paths (or orbits as we now call them). This thinking was in some sense regressive, but by placing the Sun at the center of the universe (as Copernicus knew it) and making Earth the third planet out, then a reasonably good description of observed planetary motion could be obtained. The accuracy of the initial Copernican model was inferior to that of the Ptolemaic model, but as he had intended all along the model developed by Copernicus was simpler in concept and it provided a uniform description of the heavens. Each planet in the Copernican system had its own specific orbit and the spacing between the planets could be determined directly from the observations.
It was the straightforward construction of the Copernican model that caught the imagination of other philosophers, but it took the mathematical genius of Johannes Kepler to make the model work. Building upon the highly accurate positional data obtained by Tycho Brahe, Kepler applied what in modern terminology would be called a reverse engineering study. Specifically, Kepler used the observational data to determine the shape of the orbit of Mars rather than assume, as Copernicus had, that it must be circular. Finding that the orbit of mars was elliptical, not circular, then automatically required that the speed with which a planet moved along its orbit must vary, being most rapid when close to the Sun and slower when further away. Not only this, the observational data indicated that the Sun must be located at one of the two focal point positions of a planet’s elliptical orbit (figure 1.5). Kepler explained the first two of his laws in Astronomia Nova, published in 1609, and later introduced a third law, in a somewhat obscure form, in his Harmonice Mundie, published in 1618. Kepler’s collected laws of planetary motion as we now known them are:
K1: The orbit of every planet is an ellipse with the Sun located at one of its focal points
K2: A line drawn between the Sun and a planet sweeps out equal areas in equal intervals of time
K3: The square of the orbital period is proportional to the semi-major axis cubed
It is K1 that describes the shape of planetary orbits and identifies the location of the Sun with respect to the orbit (see figure 1.5), while it is K2 that describes how the planet moves in its orbit and explains why the speed of the planet varies between perihelion and aphelion. When first announced in his Harmonice Mundie K3 was an absolute mystery - there was, at that time, no explanation as to why such a relationship between the orbital period and orbital size should exist. While Kepler tried to explain his laws in terms of magnetic planets interacting with a magnetic monopole Sun, it was Isaac Newton who eventually provided the first correct physical reasoning behind all three of the planetary laws. Indeed, Newton showed that any two objects interacting under a centrally acting force that varied as the inverse square of distance must automatically obey Kepler’s laws (figure 1.6). The force that Newton introduced, of course, was that of gravity, and this combined with the laws pertaining to the conservation of energy and angular momentum provided a full description of celestial dynamics. These remarkable results were first articulated in Newton’s masterpiece, Philosophiæ Naturalis Principia Mathematica, published by the Royal Society of London, with financing by Edmund Halley, in 1687. Not only was Halley the paymaster and editor of Newton’s remarkable text, he was also the inspired astronomer who wrote an ebullient dedication to the tome’s esteemed author. Indeed, Halley eulogized that, “now we know the sharply veering ways of comets, once a source of fear and dread, no longer do we quail beneath appearances of bearded stars”. In the same way that the pen is mightier than the sword, so Newton’s mathematical analysis and physical insight were mightier than two thousand years, and more, of superstition and fear (see Appendix I).
Figure 1.5: The key characteristics of Kepler’s 1st law are that the orbital path is elliptical in shape and that the Sun is located at one of the two focal points (F1 and F2) of the ellipse. The two focal points are situated at equal distances away from the center (O) of the ellipse. A line drawn between the planet (or comet) and the non-Sun, or empty focus, has dynamical characteristics similar to that of Ptolemy’s equant point.
Coda: Newton’s proof of K2
With reference to figure 1.6, imagine an object at point B subject to the gravitational pull of an object located a point S. The arc ABCDEF indicates the successive positions of the object at equal time intervals t. In the first time interval, the objects moves from A to B and sweeps out the area SAB – at this stage the motion is rectilinear and the object moves along the straight line path AB. Once at point B, however, the gravitational force begins to pull the object away from its straight line path with the result that it moves along the path BC rather than Bc. The proof of Kepler’s 2nd law now proceeds by demonstration that in the second time interval the area swept out by the object SBC, as it moves from B to C, is exactly equal to area SAB. Firstly, Newton noted that the area of triangles SAB and SBc are equal – this follows since they have the same altitude (the perpendicular line dropped from S to the line extending through ABc) and they have the same base lengths: AB = Bc. The next step is to show that the area of the triangle SBc is equal to SBC. This is accomplished by noting that the displacement cC (due to the gravitational force acting at S) runs parallel to the line SB. With this condition in place Newton had his proof of equal areas since the two triangles of interest have the same base length SB and identical altitudes (the line dropped from c – and C – to the line extending along SB). Having shown that triangles SAB and SBC have equal areas, the final part of the proof is just a generalization – the same result, as just proven, must apply to the motion of the object from point C, with the triangles SBC and SCD also having equal areas.
Figure 1.6: Newton’s diagram explaining the motion of an object under a central force and his proof of Kepler’s second law. Newton’s proof of K2 proceeds geometrically and requires a demonstration that the area of triangle SAB is the same as triangle SBC, which is the same as SCD and so on.
Newton’s proof of Kepler’s second law is a remarkable geometric construction – he has employed nothing more than straight lines and triangles, and the result is independent of the actual value of the time step t. Likewise the proof is independent of the magnitude of the centrally acting force at S [1].
Comet C/1680 V1 - the game changer
It took over 70 years to come about, but eventually, on 14 November 1680 German astronomer Gottfried Kirch became the first person to discover a comet with the aid of a telescope. Working in the early morning hours, and using a 2-foot focal length refractor Kirch was observing the Moon and Mars when by chance, in the constellation of Leo, he sighted “a sort of nebulous spot, of an uncommon appearance….. a nebulous star, resembling that in the girdle of Andromeda2”. Kirch followed his nebulous star over ensuing nights, and on November 21st using a 10-foot focal length telescope confirmed the appearance of a small but distinct tail. The comet was increasing in brightness and heading towards the Sun. By the close of November the new comet was visible to the naked-eye and its tail was reckoned to be over 15o long.
As December proceeded the comet grew ever brighter but by mid-month it was lost to view within the Sun’s glare. Perihelion occurred on December 18th. Rapidly rounding the Sun the comet re-emerged to view sporting an extraordinary long, near 90o tail on December 20th. Many years later Augustan missionary Casimo Diaz recalled his sighting of the comet from Manila in the Philippines: “the frightful comet [was] so large it extended, like a wide belt, from one horizon to the other… causing in the darkness of the night nearly as much light as the Moon in her quadrature”.
Having caught the eye of the world’s populace the Great Comet of 1680 now required an explanation from the astronomers. The first question that needed to be settled was whether one or two comets had actually been seen. Some observer’s, Isaac Newton among them, initially argued that two comets has been seen: the first being Kirch’s comet heading inwards towards the Sun, with a second comet, purely by chance emerging from behind the Sun on December 20th after Kirch’s had disappeared from view. Other observer’s, Britain’s Astronomer Royal John Flamsteed foremost among them, argued that just one comet had been seen, and that remarkably it has been repulsed by the Sun and set upon a parabolic path back into the outer realms of the solar system. Much debate followed, but it appears that it was not until at least mid-1684 that Newton came around to the viewpoint that comets might move along elliptical orbits and that the comet of 1680 had, “fetched a compass about the Sun”. Applying his formidable intellect to the problem, Newton developed the mathematical techniques needed to deduce, from a set of three evenly spaced observations, the parameters that describe the shape of a comet’s orbit (to be discussed below). The fruit of Newton’s labors were eventually revealed in Book III of his 1687 Principia (figure 1.7), where he demonstrated that the Great Comet of 1680 had traveled along a parabolic path with, importantly, the Sun located at the focal point. Newton had not only shown how to deduce the path of a comet from the observations, however, he had also placed them within the realm of his gravitational theory, and this, of course, was his greatest triumph.
Figure 1.7: Path of the Great Comet of 1680 as revealed in Newton’s 1687 Principia – the original diagram was an impressive 2-page foldout from the text. In this diagram HG indicates Earth’s orbit, while the comet’s path is the parabola ABC. The Sun is located at the focal point D. The line DF is the comet’s line of nodes, and B indicates the perihelion point (which the comet reached on 18 December 1860). See table 1.1 for additional details.
Newton both observed the comet of 1680 directly and he collected data upon its appearance from across Europe (Table 1.1). In the Principia Newton indicates that he used a 7-foot telescope to observe the comet from Cambridge. He uses the combined observational reports, however, to determine the tail-length evolution of the comet. Newton’s own observations reveal a tail length of less then ½ a degree on November 11 (Gregorian calendar date), increasing to 40 degrees on the sky on January 5; falling to 2 degrees and zero tail on February 10 and 25th respectively. The comet was last observed by Newton on March 9th (March 19th in the Julian calendar). Newton’s diagram (our figure 1.7) is not only revolutionary for showing a parabolic orbit for the comet, but for also showing the time evolution of the comet’s tail, before and after perihelion passage.
Halley’s bold predictions
Upon seeing Newton’s orbital solution 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 (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 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 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 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 was first articulated. In De Cometis Libelli, Kepler presented his observations and thoughts relating to the comet of 1607 (C/1607 S1), and three bright comets observed in the fall of 1618. Kepler first saw the comet of 1607 on September 26 – after a firework display he had been enjoying while in Prague. The comet was widely observed from across Europe and Kepler, forcing the path to be rectilinear, attempted to map out its path through the heavens (figure 1.15). While Kepler acknowledge that he had problems in forcing the comet’s path to be a straight line, the ultimate irony of this particular story is that the comet of 1607, was in fact, Halley’s Comet. In 1618 Kepler became the first person, as far as is recorded, to see a comet through a telescope. Observing comet C/1618 Q1 on September 6th, Kepler described it as being “large and resembling a cloud”. Perhaps surprisingly Galileo, the great self-promoter, did not bother to observe any of the three comets seen towards the close of 1618. True, Galileo was apparently ill and confined to bed at the time of interest, but it does not seem unreasonable to suppose that if he had felt that comet’s were worthy of his study, he would have done so. True to form, however, while Galileo made no observations of the comet himself he produce a vitriolic attack upon the observations and ideas relating to the comet’s appearance published by Jesuit astronomer Horatio Grassi. Being generous, it could be said, Galileo played the role of Devil’s Advocate, but one rather suspects that he just liked nothing better than a good old public scrap. While he denounced Grassi’s observations, and even brought into question Brahe’s parallax observations relating to the comet of 1577, Galileo offered no particularly new or useful insights to the study of cometary phenomena.
Figure 1.15. The rectilinear path for the comet of 1607 [which was actually Halley’s Comet] as derived by Johannes Kepler. The comet’s path takes it through the orbits of the Earth, and Venus. Image from Kepler’s De Cometis Libelli (1619).
The great French philosopher René Descartes outlined a detailed vision of the stellar realm in his highly influential text Le Monde (published in 1633). Espousing a mechanical philosophy, Descartes brought order to the primordial chaos by invoking the formation of a swirling broth of agitated particles – a space-filling, particulate ocean of ever moving circular eddies and tourbillions (vortices). At the center of each cosmic vortex was a star at rest, with any accompanying planets, trapped like so much flotsam in the ocean swell, being carried around in their orbits by the outer circular flow (figure 1.16). “Picture the bend in a river, where the water coils itself, tuning in circles, some major, some minor,” writes Descartes in his Principles of Philosophy (first published in 1644), “Things floating in this current, we notice, are carried along by it and turned around and around, even heavy objects, some of which revolve about their own centers. Those objects nearer the center of the eddy containing them complete their revolution sooner than do those farther from the center. Finally, although the eddies always turn in circles they hardly ever describe a perfect circle …. Let us likewise imagine the same things happening to the planets. And this is all we need in order to explain all of their phenomena”. Descartes analogy catches the minds-eye, his arguments sound reasonable, the dynamics can be envisioned, and the picture appears to encapsulate the observed properties of planetary orbits. Unfortunately for Descartes, however, as Isaac Newton was eventually to show, Nature is not so easily explained – more than just a good picture is required of a philosophy, and planetary behaviors in particular are more directly explained by mathematics and physical principles.
While falling short on physical realism and lacking mathematical detail, Descartes cosmos was, none-the-less, far from static, and he argued that adjacent vortices would compete for space, expanding and pushing against each other in a tumultuous battle for dominance. As with all battles there would be winners and losers, and in the cosmic battle a star that lost control of its vortex was destined to become a wandering comet. Such star-comets would drift and curve through space skimming around the outer edges of the various vortices that it chanced to encounter. In Descartes’ philosophy, comets followed a trail similar to that of a meandering stream, randomly turning this way and that; sometimes passing close to another vortex-centered star and at other times drifting haplessly through the depths of interstellar space.
It was the appearance of two bright comets in 1664 and 1665 that caught the imagination and eye of Polish astronomer Johannes Hevelius. Turning both his telescope and his positional instruments to these comets, he traced their path across the heavens, and wrote of their behavior in a short text, simply titled, Prodomus Cometicus - Historia Cometae anno 1664 et 1665 (Published by S. Reininger, Gedani, 1665). What sets this particular text apart from the many others written at the same time is that on 4 February 1665, Hevelius set out to estimate the horizontal parallax of the comet C/1664 W1. Remarkably, he determined a “sensible parallax” of 41 arc seconds, and this indicated that the comet was at least 5000 Earth radii away. This distance, much larger than the 230 Earth radii deduced for the comet of 1577 by Tycho Brahe, clearly placed comets within the planetary realm, and the question as to their origins and true paths became paramount.
Figure 1.16. Descartes vortices, with a star located at each center (Y, f, F, S, L, D). The meandering path of a comet (starting at point N in the lower center and continuing to the upper right) works its way from one outer vortex boundary region to another.
Hevelius continued to study comets and eventually published his magnum opus on the subject Cometographia in 1668. Within this tome he attempted to describe and classify cometary tails, and posited that comets move along paths corresponding to conic sections (recall figure 1.10) with the Sun located at a focal point (although he did vacillate somewhat on this latter condition). Since comets were still believed to be ephemeral, one-off displays, Hevelius suggested they were some kind of spurious planet, formed from material exhalations derived from either one of the outermost planets Jupiter or Saturn – the planet of origin being determined by the comet’s color. The frontispiece to Cometographia reveals the Hevelius viewpoint on comet orbits and origins (see figure 1.17). The image shows Hevelius flanked by the ghost of two philosophers past: Aristotle and Kepler. To his right is Aristotle, ignored and pointing somewhat vaguely to a diagram showing comets as atmospheric phenomena constrained to the sublunar region, their tails pointing in any and all directions. To the left of Hevelius, in more earnest stance is Johannes Kepler, who holds his diagram revealing the rectilinear motion of cometary paths; their tails streaming away from the Sun. Hevelius, however, cannot hold the eye of Kepler, and with a gentle, staying wave of his left hand points emphatically with his right to the new interpretation. Comets move along gently curving paths (parabolic or hyperbolic – but not elliptical). Hevelius essentially truncated the long and gently curving (interstellar) pathways depicted by Descartes (figure 1.16); downgrading comets thereby from vortex skimming stars of unknown origin, to locally birthed planet-born progeny.
While justly famed for his lunar and star mapping work, Hevelius, in later life, was overwhelmed by the scientific and instrumental advancements that had developed around him. The beginning of the end for Hevelius began, in fact, with his observations relating to the comet of 1664, for which he presented sky positions that were at odds with all other observers. Then in 1674 Robert Hooke, in his Animadversions, launched into a long critique of Helvelius’s Machina Coelestis (published in 1673) – chastising Hevelius for relying on the eye to make positional observations when telescopic sights offered a much more precise means of measurement. Further misfortune followed in September of 1679 when his observatory in Danzig, Stellaburgum, burnt to the ground. And finally, with the appearance of the comet of 1680, and Newton’s 1687 triumphant description of its orbit as a parabola, with the Sun at the focal point (recall figure 1.5), Helvelius’s notion of comets moving along gentle curving paths was laid to rest. Hevelius died in 1687, the last of the great Tychonic School of naked-eye observers, being spared the news of Newton’s work and the 1705 publication of A Synopsis of the Astronomy of Comets by Edmund Halley in which the periodic nature of the comet of 1682 was revealed.
Figure 1.17. The title page from Hevelius’s Cometographia (1668). To the left in the image is Aristotle, centrally seated is Hevelius and to the right is Kepler. In the center background a crowd of onlookers, some with arms raised to the sky, contemplate the approaching comet. Meanwhile, lofted above the superstitious plebeians, are seen the “astronomers” at work. Three large sextants (characteristic of the design promoted by Tycho Brahe) can be seen in the image, along with an observer viewing the comet through a telescope. The dividers placed on the table next to Helvelius’s right hand are a sign of authority and exactness.
Kepler, Galileo, and Helvelius (and indeed all astronomers prior to their time) accepted that comets were large ephemeral objects, quasi-planets perhaps, but objects with a limited lifetime. This notion began to change, however, with the realization, through Edmund Halley’s investigations, that at least some comets moved along elliptical paths and were in principle at least long-lived and subject to repeated returns to the inner solar system. Newton’s thoughts on the topic were discussed in his Principia (1687) and in his second magnum opus, Opticks (first published in 1704). In these works Newton suggested that while comets had a solid and durable nucleus, they were additionally surrounded by a tenuous atmosphere. Newton additionally suggested an alchemical link with respect to the utility of the vapors emanating from cometary tails, arguing that planets needed to accrete such matter in order to replenish the fluids vital to the process of vegetation and putrefaction [6]. Indeed, in his Opticks Newton asked, “to what end are comets?” - answering in effect that it was only by the accretion of cometary material that sustained life on Earth was possible. With Newton, the swirling vortices of Descartes were laid to rest. “The motions of the Comets”, Newton writes in his Principia, “are exceedingly regular, are governed by the same laws with the motions of the planets, and can by no means be accounted for by the hypothesis of vortices. For comets are carried with very eccentric motion through all parts of the heavens indifferently, with a freedom that is incompatible with the notion of a vortex”. While Newton freed comets from the random buffetings supplied by Descartes swirling tourbillions, he none-the-less entrapped them within the ever searching web of gravity. By this act, Newton placed comets within the realm of direct calculation; their paths, at least in principle, being described by mathematical formula and direct computation.
With the mathematical techniques put in place by Isaac Newton, the 18th Century saw ever more calculations of cometary orbits being made. With, in principle, their orbits now being understood and numerated, the issue as to their origins and purpose became a matter of greater interest. It was now clear that comets, for at least some of the time, moved in a realm beyond the planets. Indeed, Halley’s Comet, at aphelion, is located three times further from the Sun than Saturn, the outermost of the known planets until the discovery of Uranus, by William Herschel, in 1781. One of the first diagrams to clearly illustrate cometary orbits and contrast them against those of the planets was that constructed by Thomas Wright of Durham. Writing in his text Clavis Cœlestis: being the explanation of a diagram entitled a Synopsis of the Universe (published 1742). Wright notes that, “Besides the Planets, which all move round the Sun from West to East near the Plain of the Ecliptic; there are other surprising Bodies in the System call’d Comets, whose Motions are perform’d in very different Plains, and in all manner of Directions, both direct and retrograde”. Lavishly illustrated, and massive in scale, Wright shows in the supplementary diagrams that accompanied Clavis Cœlestis the paths deduced for Halley’s three ‘periodic’ comets near to the Sun (figure 1.18), but leaves as a mystery their more distant peregrinations. Other diagrams of the time simple stop once the cometary distance extends beyond that of the orbit of Saturn (figure 1.19) – in many ways, rather than invoking the terrestrial map maker’s “here be dragons”, the astronomers responded with the notion that in the realm beyond Saturn, “there be comets”.
Figure 1.18. A small section of Thomas Wright of Durham’s A Synopsis of the Universe, published in 1742, showing the orbits deduced for the comets of 1661, 1680 and 1682 interior to the orbit of Mars. These three comets were highlighted by Edmund Halley as being periodic – that of 1661 being, in fact, Halley’s Comet.
At the time of publishing Clavis Cœlestis Wright was actively engaged in the study of comet C/1742 C1, describing his observations in the February, March, and April issues of The Gentleman’s Magazine. Wright ultimately submitted his derived orbit for the comet to the Royal Society. Employing these same mathematical skills a few years later, Wright speculated upon a possible extension of Kepler’s laws of planetary motion with respect to the comets. Writing in his 1750 work, An Original Theory or New Hypothesis of the Universe, Wright noted (in letter III) that, “I am strongly of the opinion that the comets in general, throughout all their respective orbits, describe one common area”. This idea, that all cometary orbits should encompass the same area, moves beyond Kepler’s laws, and there is no physical reason, other than the direct influence of an omnipotent maker, that such a constant condition should apply to cometary orbits. The area of an ellipse is given by A = π a b, where a and b are the semi-major and semi-minor axes. The major and minor axes are further related through the eccentricity, e, with b = a(1 - e2)1/2. Accordingly, Wright was suggesting that with A being constant, so the semi-major axis and the eccentricity for any comet’s orbit were related as: a2 = (A/π)/(1 - e2)1/2. Substituting modern-day numbers for cometary a and e values, however, reveals that Wright’s idea is entirely fallacious: for 1P/Halley, 2P/Encke and 3D/Biela, we find areas of 253.6, 8.1 and 25.6 astronomical units squared respectively. Even though Wright was wrong in his supposition, and indeed there is no dynamical requirement for the semi-major axis and eccentricity to be linked according to a constant area, the idea is interesting since it builds upon the notion that comets are somehow ‘different’ from the planets; their orbits being constrained not only by Kepler’s three laws of planetary motion but by an additional, dare one say divinely engineered, constant orbital area rule.
Figure 1.19. A scheme of the solar system, by William Whiston, published circa 1720, and engraved by John Senex. The cometary orbits are those deduced by Edmund Halley in his 1705 Synopsis of the Astronomy of Comets (recall figure 1.9). The map was first composed circa 1712 and Whiston writes in the (densely packed) surrounding text that comets, “are the more numerous bodies of the entire Solar system”. The large number of comets that cut across the circular planetary orbits have become a dominant feature of the map, and the sence that collsions might occasionally occur between planets and comets is apparent, albeit geometrically enhanced by the two-dimensional restriction of the page.
Beyond the 2-dimensional page
Human beings see the world in three-dimensions and this trait can be exploited to provide a better minds-eye view of a comet’s path through space. The ecliptic provides a natural plane for orienting the cometary orbit (recall figure 1.13), and the circle of the Earth’s orbit, centered upon the Sun, provides a graphic representation for the advancement of time when divided into appropriate date markers. Cometaria constructed in this manner have no moving parts as such, but when constructed to scale, sky locations, phase angles5 and distances relative to the Earth and/or Sun can be read-off with a straightedge or piece of string.
The 2-dimensional page can be transformed, and literally lifted into the third dimension, by the ingenious use of folded inserts and origami-style constructions. In this way, by turning the page, a 3-dimensional vista can be made to unfold – the scene animating itself before the reader’s eye. The history of the pop-up book stretches back many centuries, and a number of early astronomy texts are known to have used combinations of fold-out pages, fold-up diagrams and diagrams composed of multiple moving layers and/or moving parts – books containing the latter kind of diagrams generally being known as volvelle. Indeed, the addition of rotating discs within books dates back to at least the 13th Century, when, for example, they were used to enable the calculation of holy days within the Christian calendar, or to demonstrated the many uses of the astrolabe6. Volvelle, or wheel charts, are typically designed to act as some form of analogy computer – indeed; they are distant relatives of the circular slide rule. The modern day descendent of the astrolabe is the planisphere, which in turn is a two component volvelle-like device. Made-up of two circular disks that rotate about the same center (corresponding to the celestial pole on the sky) the planisphere is a 2-dimensional model of the celestial sphere. The lower disk contains a projection of the stars and constellations that will be visible from a given location (specified by latitude) on Earth’s surface during the course of one year. The front disk is opaque except for a cutout window that enables the user to see the star disk situated below. By rotating the two disks so that they are aligned at a specified hour and day of the month, the window in the upper disk will reveal the stars that will be visible above the observer’s horizon. The planisphere is a wonderfully simple, but incredibly useful device, enabling the determination of approximate rise and set times for stars and constellations, and, of course, if the predicted path of a comet through the sky is printed on the lower dial, then the rise and set times, as well as sky location of the comet on a specific day can be determined. Planispheres showing the path of Halley’s Comet were certainly sold during its 1986 return (figure 1.20), and at other times planispheres showing the progress of other comets across the sky have been manufactured and sold to the public (see later with respect to the 1832 return of comet Biela).
Figure 1.20. Planisphere window view for the path of Halley’s Comet across the sky from December 1985 to March 1986. Note the change in direction of the tail on the sky as the comet approaches perihelion on 9 February 1986.
In celebration of its imminent return in 1986, renowned astronomy writers Patrick Moore and Heather Couper (with illustrations by Paul Doherty) produced a pop-up book on Halley’s Comet. The text featured pull-out tabs to show the expected appearance of the comet’s tail, in specific months, as it approached and rounded perihelion; it also contained a 3D-pop-up representation of the comet’s orbit, showing the location of the comet at specific times between 1948 and 2024 (the latter two dates being times when the comet was and will next be at aphelion). While the book by Moore and Couper was written for a juvenile audience its interactive and explorative nature makes it a delightful read – the story and the science jump right out of the page.
The first print to reveal the 3-dimensional geometry of cometary orbits was published by Dutch polymath Nicolaas Struyck in volume 2 (1753) of his Vervolg van de beschryving der staartsterren. The image (figure 1.21) is a 2-dimensional hybrid of a pop-up book – indeed, one can easily imagine opening-out a plush, semi-circular binder with the orbits rising up, hydra-like, from the inside of the cover pages. The cometarium by Stuyck is highly detailed and shows near-to-perihelion, and above the ecliptic, portions of the orbits to some 14 comets (table 1.3) as recorded between 1533 and 1748. No specific scale for the cometarium is indicated, but its design incorporates the Sun at the center of a circular base plate, with a circle band representing the Earth’s orbit. Although not clear from the diagram, the model Earth, which is located at the center of two circular protractors, is probably moveable and can be set to a particular day and month of the year location. A string is attached to the Earth model and this, stretched to any one of the cometary tracks, would enable an estimate of its ecliptic coordinates to be made. Furthermore, by stretching the string from the Earth model, to the comet location and then onto the Sun, the comets phase angle and solar elongation could be determined. The orbit tracks are clearly marked with one-day interval striations and they are orientated correctly with respect to inclination, argument of perihelion and longitude of the ascending node – the (i, ω, Ω) co-ordinates in figure 1.13. The location of the comet is set by sliding a small ball, complete with cometary tail, to the appropriate data as indicated along the orbital track. While the image of Struyck’s cometarium is highly detailed, and could certainly have functioned as a demonstration device, there is no specific evidence to indicate that the cometarium was ever physically made and/or used for lecturing purposes.
Figure 1.21. The cometarium of Nicolaas Struyck. The diagram is from volume 2 of the Vervolg van de beschryving der staartsterren (1753). See table 1.3 for details.
| label | Comet | Designation | P(yr) | e | i (°) | notes |
| A | 1748 | C/1748 H1 | --- | 1.00 | 94.54 | |
| B | 1742 | C/1742 C1 | --- | 1.00 | 112.95 | |
| C | 1744 | C/1743 X1 | --- | 1.00 | 47.14 | (1) |
| D | 1533 | C/1533 X1 | --- | 1.00 | 149.59 | |
| E | 1748 | C/1748 K1 | --- | 1.00 | 67.08 | |
| F | 1706 | C/1706 F1 | --- | 1.00 | 55.27 | |
| G | 1678 | 6P/d’Arrest | 6.62 | 0.67 | 2.81 | (2) |
| H | 1707 | C/1707 W1 | --- | 1.00 | 88.65 | |
| I | 1743 | C/1743 C1 | --- | 1.00 | 2.28 | |
| K | 1702 | C/1702 H1 | --- | 1.00 | 3.38 | |
| L | 1699 | C/1699 D1 | --- | 1.00 | 109.42 | |
| M | 1739 | C/1739 K1 | --- | 1.00 | 124.26 | (3) |
| N | 1743 | C/1743 Q1 | --- | 1.00 | 134.42 | |
| O | 1718 | C/1718 B1 | --- | 1.00 | 148.84 |
Table 1.3. Details of the comets studied by Nicolaas Struyck. Column 1 is the letter labeled as used in his illustration (our figure 1.21). Column 2 is the year of the comet as given by Struyck, while column 3 provides the present-day cometary identification. Columns 4, 5 and 6 indicate the orbital period (technically infinite if the orbit is parabolic: e = 1.00), the eccentricity and orbital inclination. Notes: (1) Often identified as Chéseaux’s comet, this comet ranks as one of the greatest in all of history. It was remarkable for being both very bright and for showing multiple tails. (2) The linkage between comet C/1678 R1 and periodic comet 6P/d’Arrest was made by Carusi et al., in 1991. In his 1753 study Struyck deduced a parabolic orbit. Heinrich Louis d’Arrest discovered his comet in June of 1951, indicating a remarkable 41 returns to perihelion since the 1678 sighting. (3) This is the parent comet to the October Leonis Minorid meteor shower.
With Struyck we see the first 3-dimensional, albeit on a 2-dimensional page, rendition of cometary orbits. And accordingly the Solar system takes-on a more dynamical feel: the sedate and circular motions of the planets being enveloped in a Gordian knot of cometary paths. The image also hints at the distinct possibility that collisions might occasionally occur between planets and comets – this being more apparent, in some strange optical sense, than in the case of the standard 2-dimensional diagrams (e.g., figure 1.19), where the orbits must, of necessity, cut across each other. It is as if by adding the third dimension, the real threat of cometary collisions is lifted, literally, right from the page – it is the real solar system that is now being portrayed, raw and primordial. No longer the neatly drawn and ordered orbital diagram of the earlier philosophers, the 3-dimensional cometarium of Struyck reveals the inner solar system to be dynamic and perhaps, alarmingly, somewhat overcrowded. Isaac Newton had already hinted at the possibility of collisions between comets and planets in his Opticks (1704), and William Whiston (recall figure 1.19) had additionally suggested that a comet passing close by the Earth might have precipitated the biblical deluge. Furthermore, Whiston argued that comets, “seem fit to cause vast mutations in the planets, particularly in bringing on them Deluge and Conflagration…. [they are] instruments of Devine vengeance upon the wicked inhabitants of any of these worlds” (here Whiston is invoking the commonly held idea at that time that all of the planets in the solar system were inhabited). Indeed, Newton writes, “whence is it that Nature doth nothing in vain; and whence arises all that Order and Beauty which we see in the World? To what end are comets, and whence is it that Planets move all one and the same way in orbs concentrick, while Comets move all manner of ways in Orbs very eccentrick”. The general idea that comets might on occasion collide with a planet, and specifically Earth, however, was not a strongly supported idea during either the 18th or 19th centuries. Indeed, a form of group denial, in effect, set-in during these centuries, with philosophers choosing to believe that the solar system was so well ordered and so well constructed that collisions were either impossible, negligibly rare, or, if they occurred at all, were for very specific God-ordained purposes. And indeed, as a mirror to the heavens, so it was on Earth too – the idea of slow, gradual change, rather than catastrophic punctuations, was dominant amongst natural philosophers and the proof of concept was provided for by Charles Lyell in his highly influential, multi-volume, Principles of Geology (published between 1830 and 1833).
Not all predictions concerning comets come true, and such was the situation with the (non)comet of 1789. Highlighted as one of Edmund Halley’s originals, it had been reasoned that the comets sighted in 1532 and 1661 were one and the same, and with an apparent period of some 129 years its return some time circa 1789 was expected. Indeed, Bartholomew Burges in his work, A short account of the solar system and comets in general: together with a particular account of the comet that will appear in 1789 (published in 1789), reasoned that, “we may reasonably expect the comet in question, to be visible towards the latter end of 1788 or the beginning of the year 1789, and certainly some time before the 27th of April 1789”. For all of the great confidence that Burges invested in his prediction, no comet appeared. In spite of pinning his hopes upon the wrong cometary return, Burges’s text remains interesting because of two attached fold-out charts. The first chart (figure 1.22) shows to scale the predicted path for the comet of 1789 as it moves interior to the orbit of Uranus and in towards the inner solar system. While the first chart is not specifically innovative in its construction (other than showing Uranus as the new outermost planet), the second chart is altogether something different. Burges writes, “I have annexed to this work, a diagram, to which I have fixed threads, which when extended shows the perihelion distance of the comet – show the angle its orbit makes with the ecliptic, and the distance the Earth will be from it at that time – to be measured from the scale of the diagram”. Here we have a three-dimensional, predictive, scale model for the comet. While not the first pop-up book to be manufactured, the text by Burges may well be the first that was specifically produced to reveal the true spatial and temporal motion of a comet – albeit a comet that didn’t actually appear. The path of the comet, above and below the ecliptic plane, has been drawn on a semi-circular piece of paper that folds-up above and below the base page representing the ecliptic. The Earth’s orbit is drawn to scale along with date and position markers, and the attached threads can be set and used to determine the Earth and heliocentric distances to the comet on specific dates. For the price of the pamphlet, one could literally hold the orbit of the comet in the palm of ones hand.
Figure 1.22. Foldout map of the solar system by Bartholomew Burges (1789). The orbits are shown out to “planet Herschel” (Uranus), which was first identified by William Herschel in March of 1781. Along the cometary path can be read the following, “The path of the comet that appeared in the year 1532 and 1661 with period 128 yrs. 89 dys. 24 m. & will consequently be visible again on or before the 27 of April 1789”.
In addition to producing his pamphlet on the solar system and hyping the supposed appearance of the comet in 1789, Burges also set out to deliver a series of lectures on the topic in the Boston area. An advertisement [7] in the Boston Gazette for 16 February, 1789 indicates that these lectures will feature a display of astronomical apparatus. Indeed, we are told that the display constitutes, “two large Planispheres and Gilded Projectiles, displaying the Celestial Bodies, suspending in their due Proportion, and their orbits; and their relative and mean Distances from the Sun, with their moons and Satellites”. The exact layout of the system is not clear from the advertisement, but it would appear to be some kind of ceiling-mounted planetarium display that was being shown (figure 1.23
