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
