Walker, Ian Walter
(1980)
Empirical stability criteria for hierarchical manybody dynamical systems in celestial mechanics.
PhD thesis, University of Glasgow.
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Abstract
The question of whether the Solar System is stable or not has been investigated by many researchers since the beginnings of celestial mechanics, with the formulation of the law of gravitation by Newton, up until the present day. As yet, even after some 300 years, celestial mechanics is unable to give a definite answer to this question. However throughout the studies which have been made there are strong indications that, among the major bodies i.e. the planets and (the majority of) satellites, the orbits are stable. Dynamical systems, such as those occurring in nature, may be broadly classified into two types: (i) those which contain large numbers of bodies, the gravitational attraction of the whole system dominating the movement of each of the bodies and (ii) those which contain relatively few bodies where the particular interactions of one body with each of the others is important. This thesis considers only the latter type: these are hierarchical systems where the orbits of the system are arranged so that close approaches of one body to another may be prevented e.g. the Solar System, triple stellar systems, etc. The concept of stability which is used in this thesis is that there should be no collisions between bodies within a hierarchical dynamical system, neither should any bodies escape the system. Having fulfilled these conditions a system may be called "stable". In other words we require a maintenance of the status quo with the orbits in the system being executed over many revolutions with only periodic changes in the semimajor axes, eccentricities and inclinations defining the osculating orbits; secular changes in these orbital elements should be absent. To investigate the stability of these systems empirical stability parameters are derived which are representative of the disturbances on the orbits of a hierarchical system due to the other bodies which are present. These parameters are obtained in the following manner. The equations of motion of the masses making up an nbody dynamical system are expressed in the Jacobian coordinate system. An expansion of the force function then gives rise naturally to a set (nl)(n2) dimensionless parameters, the epsilon parameters, representative of the size of the perturbations on the osculating Keplerian orbits of the various bodies in the system. In the case where there are only three bodies the relationship between these epsilon parameters and the analytical stability criterion employing the zerovelocity curves of the coplanar general threebody problem is examined. It is shown that the stability, in the sense of the zerovelocity curves of the problem being open or closed and thus whether an exchange between bodies is possible or not, is dependant, in a very simple fashion, on the magnitude of the epsilon parameters. The analytical stability criterion involving the use of the zerovelocity curves of the general threebody problem is then refined in order to take account of all the orbital parameters relevant to coplanar hierarchical threebody systems prior to a numerical investigation of the coplanar, corotational hierarchical threebody problem with initially circular orbits. By means of this numerical investigation it is demonstrated that the epsilon parameters can be used to both predict how stable or unstable a threebody system is, in the sense of the number of orbits it may be expected to execute before instability sets in, and also to predict the variations in semimajor axes and eccentricities of the system's constituent orbits. The effect commensurabilities in mean motion have on the stability of these systems is also demonstrated. It is shown how systems which are unstable can extend their predicted "lifetime" if they are close to a commensurable situation. Stable systems have their variations in semimajor axes and eccentricities greatly magnified by the presence of commensurabilities. This numerical study was continued into the fourbody problem. Although only a brief consideration was given to the problem it is apparent that results, similar to those obtained in the threebody case, may be derived. The applicability of the epsilon parameters to the planetary case of the hierarchical manybody problem is also considered by application of another expansion of the force function which takes into account the smallness of the planetary masses with respect to the central mass, namely the Sun. It is thus shown that the epsilon parameters are truly representative of the disturbances on the planetary orbits over a sufficiently wide range of the semimajor axes of the hierarchical planetary manybody problem. Another coordinate system is then developed which is applicable to all types of hierarchical system, including the type exemplified by the sextuple Castor system and similar multiple stellarsystems, Using this coordinate system a more general set of epsilon parameters are developed and the perturbations on the orbits of a system, such as that of Castor, are considered.
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