Celestial mechanics of planet shells
Abstract:The motion of a planet consisting of an external shell (mantle) and a core (rigid body), which are connected by a visco-elastic layer and mutually gravitationally interact with each other and with an external celestial body (considered as a material point), is studied (Barkin, 1999, 2002a,b; Vilke, 2004). Relative motions of the core and mantle are studied on the assumption that the centres of mass of the planet and external body move on unperturbed Keplerian orbits around the general centre of mass of the system. The core and mantle of the planet have axial symmetry and have different principal moments of inertia. The differential action of the external body on the core and mantle cause the periodic relative displacements of their centres of mass and their relative turns. An approximate solution of the problem was obtained on the basis of the linearization, averaging and small-parameter methods. The obtained analytical results are applied to the study of the possible relative displacements of the core and mantle of the Earth under the gravitational action of the Moon. For the suggested two-body Earth model and in the simple case of a circular (model) lunar orbit the new phenomenon of periodic translatory-rotary oscillations of the core with a fortnightly period the mantle was observed. The more remarkable phenomenon is the cyclic rotation with the same period (13.7 days) of the core relative to the mantle with a ‘large' amplitude of 152 m (at the core surface). The results obtained confirm the general concept described by Barkin (1999, 2002a,b) that induced relative shell oscillations can control and dictate the cyclic and secular processes of energization of the planets and satellites in definite rhythms and on different time scales. The results obtained mean that giant moments and forces produce energy which causes in particular deformations of the viscoelastic layer between planet shells. This process is realized with different intensities on different time scales. Here we have almost some machine of transformation of mechanical energy of translatory–rotary motions of the shells to elastic energy of deformation of the intermediate layer. Owing to the inelastic (dissipative) properties of this layer, part of elastic energy will become warm energy. This fundamental process has a cyclic character so the variations in the mechanical energy of translational and rotational motions of the shells are cyclic. The rhythms and types of relative wobble of the shells define periodic variations and transformations of mechanical, elastic and warm energies on different time scales. These fundamental positions maintain a constant value in the particular problem considered about the dynamics of the Earth's shell and core–mantle dynamics of resonant objects: the Moon and Mercury. The cyclic accumulation of elastic energy and warm energy of intermediate layer (between the core and mantle) in realized owing to the action of the inner moments and forces between shells. A considerable part of this energy transforms to the energy of numerous dynamic and physical processes on the planet. It is the mechanism of energization of the planet that defines its endogenous activity (Barkin, 2002a,b). In a number of studies (see for example Barkin (1999, 2002a,b) and Ferrandiz and Barkin (2003)), celestial bodies are studied as objects with a complex structure (elastic, liquid or gaseous core, with shells). The dynamics of such objects in a gravitational field are described by a system of integrodifferential equations in ordinary and partial derivatives (Vilke, 1997a,b) for which research is difficult. At the same time, the complex structure of planets can appear as one of the factors determining the course of dynamic processes (the rotation of a planet around the centre of mass, tidal phenomena, orbit evolution, and tectonic processes as a consequence of relative displacement of parts of a planet) (Barkin, 2002b). In this article the two-layer model of a planet as a system consisting of a core and a mantle, which are considered as gravitationally interacting rigid bodies and moving in a gravitational field of an external celestial body, is investigated. The problem is studied in a restricted way, namely an the assumption that the motions of the centres of mass of a planet and external body (considered as a material point) occur on unperturbed Keplerian orbits, and relative displacements of the core and mantle are subject to determination. The obtained results are illustrated with the example of the motion of the Earth–Moon system.
Document Type: Research Article
Affiliations: Department of Theoretical Mechanics, Leninskie Gory, Moscow State University, Moscow, 119899, Russia
Publication date: 2004-12-01