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Maxwell and the Rings of Saturn

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Maxwell and the Rings of Saturn newsletter OF THE James Clerk Maxwell Foundation Issue No.5 Spring 2 015 Maxwell a nd t he Rings of Saturn by Professor Andrew Whitaker, Emeritus Professor of Physics, Queen’s University, Belfast Le Verrier, or to be worthy of the credit bestowed on him by Britain and, in particular, Cambridge. The funds for the Adams Prize were A view of the rings of Saturn (from Gribbin, John and provided by alumni of Adams’ Cambridge Goodwin, Simon (1998). Empire of the Sun: Planets college, St. John’s, and the prize was in and Moons of the Solar System. Constable, London.) the gift of the University. Airy and Challis were heavily involved in choosing topics Maxwell considered the rings of Saturn Against this background, why was in the first few years of the prize, and in to be ‘ the most remarkable bodies in the Maxwell prepared to put in so much work Maxwell’s year William Thomson was heavens ’, second only, in his opinion, to over several years on the rings of Saturn? also an examiner. Thomson, later Lord the spiral nebulae. Yet it seems extremely Admittedly the prize was substantial – Kelvin, had, of course, been Professor of unlikely that he would have devoted £130, or perhaps £13,000 in today’s terms, Natural Philosophy at Glasgow University a substantial amount of time and and this was rather more than a third of since 1846, but retained very strong links effort to determining their structure his full year’s salary at Aberdeen. But it is with Cambridge throughout his career. mathematically had the topic not been more likely that the reasons for taking Thus the institution of the Adams Prize chosen in 1855 as the subject of the on the challenge were, firstly, personal and the quality of the prize-winning Adams Prize of 1856. prestige, but, secondly, and even more essays may be regarded as an element of importantly, the wish to maintain or The deadline for submission was academic rivalry with France, and it is increase the scientific stature of Britain quite likely that Maxwell saw his heavy December 1856, and Maxwell submitted and in particular Cambridge University. his attempt – the only one – just before labours in this way. the deadline. He was awarded the prize The Adams Prize was instituted in As early as 1610, Galileo had observed in June 1857, but continued to work on recognition for the contribution of the structure about Saturn, though it was the topic for a further year, finally young Cambridge mathematician John Christian Huygens in 1656 who was able publishing his results as a book in 1859. Couch Adams, made only eight years to identify its ring nature and Jean Cassini, earlier, to the discovery of the new a century later, who discovered that This was at a time when Maxwell had a planet Neptune. His contribution, there were, in fact, two concentric rings great range of other interests and duties. though, was disputed, particularly in separated by a dark band. Shortly before From March 1855, he must certainly have France. Adams had told the senior the setting of the Adams Prize topic, been absorbed by his first studies on Cambridge observational astronomers, several astronomers identified an inner electromagnetism. In April 1856 his George Biddell Airy and James Challis, dark ring, which was partially transparent. father died, and even apart from the time of his prediction of the position of the There were, of course, many ideas about required for grieving, he spent much planet as early as September 1844, but he the structure of the rings, in particular effort taking over the charge of the family had not published his work, and no whether each ring was solid or consisted estate. Also, in February 1856, just before search had been made for the planet of a large number of rocks. However there his father died, he had applied for the before the French mathematician Urbain was very little rigorous mathematical vacant Chair of Natural Philosophy at Le Verrier published a similar prediction, analysis, and, as Airy pointed out, even the Aberdeen’s Marischal University, and was also able to persuade the conclusions of the great mathematician requiring him to take time requesting German astronomer Johann Galle to Pierre-Simon Laplace were somewhat testimonials. He was duly appointed and search for the planet. Galle speedily met flawed, though many of his arguments took up his post in November of that year, with complete success in September were valid –he showed that the rings and this, of course, required him to 1847. To put it mildly, French scientists could not be a uniform solid but claimed prepare teaching and also to write and were not convinced that Adams deserved they therefore must be an irregular solid. present an inaugural lecture. to share the honour of the discovery with newslette r OF THE James Clerk Maxwell Foundation So we may consider Maxwell’s work as neighbouring regions, the system moving nearer one of its neighbours, it would seem detailed rigorous and comprehensive towards equilibrium. Yet Maxwell’s that gravitation would act to decrease mathematics on an essentially blank canvas. mathematics shows the opposite effect further the distance between the two Maxwell’s methods were not actually very occurring. satellites. But the reverse occurs, and for advanced, consisting of potential theory, Suppose a portion of fluid needs to increase the same reason discussed in the fluid case. mainly using linear differential equations, its speed around the planet in order that the When a satellite speeds up, it swings out Taylor series and Fourier analysis, but the system should move towards equilibrium. from its orbit, and so essentially its angular analysis was sophisticated and exceptionally As it increases its speed, the balance speed decreases. The situation is indeed sure-footed. In general, he searched for between gravitational attraction and stable. equilibrium configurations of planet and centrifugal force is broken, and this portion It must be remembered that there is more ring, and then used Taylor series to study of fluid moves outwards. Thus its path has than one ring, and Maxwell proceeded to whether any such configuration might be a larger circumference and so its angular investigate a system of two rings, each of the stable to small disturbances. speed decreases. So compared to adjacent same form as the single ring just discussed. Maxwell was first able to show that the cases portions of the ring, it is going backwards, Complicated phenomena may occur; if, for of a uniform ring or a non-uniform ring did and the expected catching up is totally example, a wave in one ring is in resonance not lead to stable solutions. He did, though, stymied. Thus the disturbance does not with a different type of wave in the other find a highly bizarre case of stability – a result in a restoring effect but in an increasing ring, a combined wave may increase uniform ring loaded at a particular point divergence from equilibrium. In other words without limit leading to collapse of the ring with a very heavy weight. The conditions a fluid ring system is intrinsically unstable. system. Maxwell argued, though, that the for stability of this, somewhat odd, type Maxwell was to write that: “ Thiseffectisvery ring system could exist for a very long time of ring were stringent; the ratio of the remarkableasshowingthedestructiveeffectof without such destruction. displacement of the centre of gravity of the anapparentlyconservativeforce. ” Maxwell knew, of course, that his regular ring to its radius must be between 0.8159 As Maxwell pointed out, the fact that Saturn rings were an over-simplification, and and 0.8279, and the weight at the special itself could be seen through the inner dark attempted to generalize to an irregular point must be between 4.43 and 4.81 times ring in its true position indicated the distribution of satellites; it seems likely that that of the rest of the ring. Amusing as correctness of this suggestion. The light these thoughts led to his important work this solution was, it was clear from direct must be passing through gaps in the ring; on kinetic theory. observation that it bore no relation to the if the ring were fluid the image would be Maxwell’s explanation has broadly been actual ring system, so Maxwell’s overall refracted. Thus the only possibility seems found to be correct by the Voyager mission conclusion was that each ring was not a to be that each ring consists of a very of the 1980s and the Cassini probe of 2009. solid system. large number of unconnected satellites. Each ring consists of an immense number Maxwell now considered the possibility that Now the challenge was for Maxwell to of particles – mostly ice but some rocks – the ring system was fluid, which initially show the stability of such a system. each rotating at the appropriate speed for attracted him. Naturally in order to balance Maxwell modelled the situation with a its distance from Saturn. Each ring, though, centrifugal effects, the ring must rotate. His system of equivalent rocks or satellites consists of much variation of brightness – method was to analyze wave motion in the spaced equally round the ring, and maxima and minima with total gaps and ring and to consider whether such waves calculated whether any disturbance would ringlets of very high density. It is nice to note will be stable. Let us imagine, for example, lead to a wave travelling round the ring or a that there is a Maxwellgapand a Maxwell a place where, as part of the wave motion total disruption of the system.
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