The Origin and Evolution of the Solar System

18 The structure of the Solar Systemin colour, probably due to an undisturbed layer of dust from meteorites. There arealso younger and brighter regions characterized by bundles of parallel grooves.The final Galilean satellite, Callisto, has a thick icy crust that is dark andshows a large number of impact features. There is a very large ‘bulls-eye’ feature,Valhalla, in the form of a series of concentric rings. This is similar to the Orientalefeature on the Moon and is certainly due to a very large impact.Before Voyager I reached Jupiter in 1979 the ring system of Uranus had beendetected from Earth observation and, with two known ring systems, there wasinterest in seeing if Jupiter also had a ring. A single thin ring was discovered—which then raised the possibility that rings were a universal feature of major planetsand that Neptune too would have rings.1.4.3 The Saturnian systemWith 18 members identified so far Saturn has the most heavily populated satellitesystem (table 1.6). Only Titan, slightly larger than Mercury, matches the Galileansbut four others—Tethys, Dione, Rhea and Iapetus—have diameters greater than1000 km. The system has a number of striking commensurabilities (Roy 1977)with both Enceladus–Dione and Mimas–Tethys having mean motions in the ratio2:1. In addition the 4:3 ratio for Titan–Hyperion ensures that this pair have conjunctionsnear the aposaturnium (furthest orbital point from Saturn) of Hyperion.The smallest separation of these two bodies is thus about 400 000 km rather thanthe 100 000 km implied by a simple consideration of the sizes of the two orbitsSpacecraft discoveries of smaller satellites show a number of 1:1 commensurabilitieswhich are really examples of special solutions in the restricted threebodyproblem. It is well known that general solutions of the gravitational problemof Ò (¿) bodies do not exist. However, Lagrange (1736–1813) showed thatspecial configurations of three bodies do satisfy the equations of motion. Theseinvolve collinear and equilateral triangular arrangements of the bodies, as illustratedin figure 1.9, in which two of the bodies are placed at the points A and Band the third (C) can occupy one of the five points, L ½ to L , known as the Lagrangepoints. The whole system must rotate about the centre of mass. Generallythese solutions are unstable and any small displacement will rapidly destroy thesymmetry. Since no three-body system can properly be isolated from the perturbingeffects of other bodies, this suggests that the Lagrange solutions cannot beachieved in practice. In certain restricted conditions the triangular solutions arestable; they require the third body, C, to be of negligible mass and for the ratioof the masses of A and B to exceed 25. In such cases small displacements ofbody C from L and L do not become unbounded and, of course, A and B executetwo-body motion. The conditions are satisfied by Saturn–Tethys–Calypso,Saturn–Tethys–Telesto and also by Saturn–Dione–Dione B. Effectively Calypsoand Telesto move in the same orbit as Tethys (hence the 1:1 commensurability)but maintain a position on average ¼ Æ in front and ¼ Æ behind Tethys in its orbit.Saturn has a single very large satellite, Titan, which is very little below