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to support his argument is shown in Table 11.2.

Clearly, resonances with Jupiter are significant, but Kirkwood's explanation can be only part of the story since other resonances are marked by a clumping of asteroid orbits. This includes the 1 : 1 resonance, where the asteroids orbit with the same period as Jupiter at one of the Lagrange points for the Sun-Jupiter system (see p. 326), and the 3 : 2 resonance, in which there is a clustering of asteroids known as 'the Hildas'.

Resonances abound in the Solar System - we have discussed in Chapter 9 the approximate 5 : 2 ratio between the mean motions of Jupiter and Saturn

(p. 332) - and they have both dynamical and theoretical significance. Perhaps

More on the history of Kirkwood's analogy can be found in Numbers (1973). Adapted from that given in Kirkwood's The Asteroids between Mars and Jupiter published in the Annual Report of the Smithsonian Institution (1876) and reprinted in Shapley and Howarth (1929). Similar ideas were used by Kirkwood to explain gaps in Saturn's rings. Here it is commensurability with the moons of Saturn that is important. Kirkwood showed in 1867 that a body in the Cassini division - the largest of the gaps in the ring system of Saturn, discovered by G. D. Cassini in 1676 - would have a period in simple ratio with four of the eight moons of Saturn. Recently, the structure of the ring system of Saturn was used to predict the existence of an eighteenth satellite (Showalter (1991)).

Table 11.2. The distribution of the 169 asteroids the elements of which were known in 1876. The final column shows simple ratios between the period of Jupiter to that of an asteroid (given in parentheses). In the 2.75-2.85 AU range, eighteen of the nineteen asteroids are in the first half of the interval, thus demonstrating the lack of asteroids in a 5 : 2 resonance with Jupiter.

Table 11.2. The distribution of the 169 asteroids the elements of which were known in 1876. The final column shows simple ratios between the period of Jupiter to that of an asteroid (given in parentheses). In the 2.75-2.85 AU range, eighteen of the nineteen asteroids are in the first half of the interval, thus demonstrating the lack of asteroids in a 5 : 2 resonance with Jupiter.

Distance

No. of

Period at

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