Hyper-periods and kirkwood gaps

Abstract

Section 1 shows that the time of perihelion passage in a Keplerian orbit is unstable in the same sense as a rod standing on end and, moreover, both can be stabilized by a suitable periodic force. Stability is associated with the existence of a hyper-period, corresponding to one type of solution of a Hill's equation. Formulae for calculating the hyper-period are given in order to refute Öpik's (1972) suggestion that the hyper-period is merely a numerical effect in computers. Section 2 refers to my recent work, briefly reported in 1978, in which the problem of the Kirkwood Gaps is attacked in a novel way by means of a Hill's equation. A comparison in methodology between three applications of the Hill's equation is given, partly to clarify my general stand and partly in answer to a criticism by Aoki (1978). Section 3 describes the previous attempts at explaining the Gaps and gives my reason for not subscribing to the collisional hypothesis. Section 4 presents, for the general reader, the well-known sequence of canonical systems in the planar circular model of three bodies that culminates in Poincaré's resonance variables. A shorthand notation for canonical systems is suggested. Section 5 discusses Schubart's (1964) model of one degree of freedom (the Sσ- model) and points out that while this model is useful in helping to define our goal as the writing down of a Hill's equation for the variation in the completely known solution, the approximate character of the model means that some additional assumption is both necessary and justified for this goal to be attained. The equation is derived in Section 6 under the assumption that the partial derivatives of the Hamiltonian of the Sσ-model be regarded as strictly time-independent. The results of applying the present method to a selection of hypothetical asteroids around the 2 1 and 3 2 resonances are given in Section 7. With the removal of a technical error in the first calculations, the observed contrast between the two resonances can now be accounted for even more naturally.