Analytical research on the possibility of long orbital existence of submicron particles in the Earth’s plasmasphere by the methods of the KAM theory

Abstract

Particles which move in the magnetosphere’s plasma gain an electric charge which depends on the density and temperature of the plasma and the sunlight stream. If motion is slow enough, it is possible to consider that the micro-particle’s electric charge is in quasi-equilibrium. For certain conditions, the Hamilton function can be written for the problem with a variable electric charge and, hence, the methods of the analysis of systems of Hamilton equations can be applied for research of such micro-particle motion. Although these conditions are strong enough, they correspond to the statement of many real problems. The spatial distribution of plasma in the Earth’s plasmasphere is described by a model of a two-component plasma. In the present paper, the capability of propagation of results which were received earlier for a case of the motion of a quasi-equilibrium electric charge in the Earth plasmasphere has been shown. The received result shows that there is an opportunity for long orbital holding (not less than one month) of micro-particles of space dust in the Earth’s plasmasphere.

Analytical study of the motion of particles with variable electric charge

“…Using this way when the co-rotation field is not taken into account we showed that as well as for a case of a constant charge, the eternal affinity of such osculating elements as semimajor axis, eccentricity and an angle of inclination of an orbit to a plane of magnetic equator to initial values should take place (this fact for a case of a constant charge is proved in [13]). These conclusions are confirmed by results of the numerical simulation, some of them are given in [17].…”

“…We receive expansion coefficients ξ k by substitution of expression (8) in the equation 7. It has been shown in [17] the increase of polynomial degree essentially improves accuracy of approximation even for a greater interval of values L .…”

The restrictions on the assumption about conservation of parameters of orbit for submicron particles in the Earth’s plasmasphere in light of the corotational electric field

“…If the simplifying assumptions that were made in Yakovlev et al. (2017) are true, the Hamiltonian function for the motion of a MP with a variable electric charge in the superposition of the central gravitational field and the Lorentz force acting on the electric charge of the MP from the side of the dipole magnetic field of the Earth and the corotational electric field in a spherical coordinate system with the origin at the Earth’s center and the polar axis, which is passing through a southern magnetic pole, can be written as follows: where is the gravitational parameter of the Earth, is particle mass, , are the scalar and vector generalized charge-field potentials and are components of the generalized momentum vector (Kolesnikov et al. 2007; Yakovlev et al.…”

“…We obtain the expressions for generalized potential as were similarly made for in (Yakovlev et al. 2017). In dipolar coordinates , Here, is the parameter of the local magnetic shell, is the charge for , are coefficients of polynomial (see the expression (3.3) in Yakovlev et al.…”

“…In our paper, (Yakovlev, Kolesnikov & Chernov 2017), we offered an analytical model of a motion of particles with variable electric charge in superposition of the central gravitational field and a field of the central magnetic dipole and showed that the theory provides good agreement with the results of numerical simulation of the motion of MPs of space debris in the Earth’s plasmasphere in the time interval of 720 h (1 month). But, as our reviewers have pointed out, the question of the degree of influence of the forces that were not taken into account in our analytical model of dynamics of the specified microparticles was not sufficiently considered .…”