The subject of the 1856 University of Cambridge Adams Prize was ldquoThe Motion of the Rings of Saturnrdquo Different hypotheses were given regarding the rings namely that the rings were (i) rigid (ii) fluid or partly gaseous or (iii) composed of isolated masses

Maxwell and the other candidates were asked which of these hypotheses would be mechanically stable

Saturnrsquos Rings

Maxwell developed equations to model Saturn and its rings He showed that on hypothesis (i) a uniform or non-uniform solid ring could not be stable except for a bizarre case where a uniform solid ring had a large point mass attached to it Observations however indicated that no large mass was part of the ring

On hypothesis (ii) Maxwell showed that a fluid ring could not be stable and would break up into small portions

Maxwell then examined a single ring composed of unconnected satellites and the propagation of waves round this ring He extended this analysis to satellites of unequal mass multiple rings of satellites and other issues

Maxwell concluded that the rings were made of lsquounconnected particlesrsquo - the isolated masses of

hypothesis (iii) In order to appear to the eye as a solid ring each ring would have to be made up of a very large number of independent particles

Maxwellrsquos essay won the 1856 Adams Prize - demonstrating his powerful ability to analyse a difficult problem mathematically Indeed such was the difficulty of the problem that it appears that Maxwellrsquos essay was the only entry

In 1858 Maxwell built a mechanical model shown below to demonstrate the wave motion of individual satellites forming a ring of Saturn

Maxwell is rightly commemorated by having a feature of the Rings of Saturn named after him ndash the lsquoMaxwell Gaprsquo - within the C ring

In 2004 the NASA Cassini probe to Saturn showed that Maxwellrsquos conclusion was right

13 Reproduction13 of13 Maxwellrsquos13 mechanical13 model13 of13 the13 motion13 of13 the13 individual13 particles13 which13 form13 the13 Rings13 of13 Saturn13 courtesy13 Cavendish13 Laboratory13 Cambridge13

13 (c) James Clerk Maxwell Foundation 2015

13 13 This13 panel13 is13 sponsored13 by13 the13 School13 of13 Physics13 and13 Astronomy13 at13 The13 University13 of13 Edinburgh

13

13

Maxwellrsquos approach

For the system represented by Saturn and a solid ring Maxwell derived the three differential equations of motion To investigate a solid ring of variable breadth he used Fourier analysis By omitting higher order terms he linearised the three equations for three unknowns By setting the determinant to zero he derived a biquadratic equation and obtained the conditions that would give rise to stability As Maxwell determined that a solid ring would never satisfy these conditions he concluded that except for the bizarre case a solid ring could not be stable

Maxwell then investigated the stability of a circle of unconnected satellites He established that such an arrangement could be stable and that four types of wave could be propagated round the ring provided Saturn was massive enough (S gt 04352micro2R where S is the mass of Saturn R is the mass of the ring and micro is the number of satellites)

As Maxwell said in his essay

ldquoI have shown that such a destructive tendency (for the individual satellites to gather together under self-gravity and form one large satellite) actually exists but that by the revolution of the ring it is converted into the condition of dynamical stability that the only system of rings which can exist is one composed of an infinite number of unconnected particles revolving round the planet with different velocities according to their respective distancesrdquo

Reproduction of Maxwellrsquos mechanical model of the motion of the individual particles which form the rings of Saturn courtesy Cavendish Laboratory Cambridge

Saturnrsquos13 Rings13

The13 subject13 of13 the13 185613 (University13 of13 Cambridge)13 Adams13 Prize13 was13 ldquoThe13 Motion13 of13 the13 Rings13 of13 Saturnrdquo13 13 13

Different13 hypotheses13 were13 given13 regarding13 the13 rings13 namely13 that13 the13 rings13 were13 13

(i) rigid13 13 (ii) fluid13 or13 partly13 gaseous13 or13 13 (iii) comprised13 of13 isolated13 masses13 13

Candidates13 were13 asked13 which13 of13 these13 hypotheses13 would13 be13 mechanically13 stable13 13

13 13 The13 Rings13 of13 Saturn13 courtesy13 NASA13

Maxwell13 developed13 equations13 to13 model13 Saturn13 and13 its13 rings13 13 He13 was13 able13 to13 show13 that13 on13 hypothesis13 (i)13 a13 uniform13 or13 non-shy‐uniform13 solid13 ring13 could13 not13 be13 stable13 except13 for13 a13 bizarre13 case13 where13 a13 uniform13 solid13 ring13 had13 a13 large13 point13 mass13 attached13 to13 it13 Observations13 however13 indicated13 that13 no13 large13 mass13 was13 part13 of13 the13 ring13 13

James Clerk Maxwell Foundation 2015

James Clerk Maxwell Foundation

This panel is sponsored by The School of Physics and Astronomy at The University of Edinburgh

Image courtesy of NASA