Abstract
The positions and linear stability of the equilibrium points of the Robe’s circular restricted three-body problem, are generalized to include the effect of mass variations of the primaries in accordance with the unified Meshcherskii law, when the motion of the primaries is determined by the Gylden-Meshcherskii problem. The autonomized dynamical system with constant coefficients here is possible, only when the shell is empty or when the densities of the medium and the infinitesimal body are equal. We found that the center of the shell is an equilibrium point. Further, when k﹥1; k being the constant of a particular integral of the Gylden-Meshcherskii problem; a pair of equilibrium point, lying in the -plane with each forming triangles with the center of the shell and the second primary exist. Several of the points exist depending on k; hence every point inside the shell is an equilibrium point. The linear stability of the equilibrium points is examined and it is seen that the point at the center of the shell of the autonomized system is conditionally stable; while that of the non-autonomized system is unstable. The triangular equilibrium points on the -plane of both systems are unstable.
Highlights
The restricted three-body problem (R3BP) describes the motion of an infinitesimal mass moving under the gravitational effects of the two finite masses, called primaries, which move in circular orbits around their common center of mass on account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries
In this paper, we investigate the motion of a test particle of infinitesimal mass under the set up of the Robe [9] model given that the masses of both primaries vary in proportion to each other according to the unified Meshcherskii [4] law and their motion determined by the Gylden-Meshcherskii problem (Gylden [2]; Meshcherskii [3])
We have derived the equations of motion of the infinitesimal mass under the effects of the buoyancy force exerted by the fluid, the gravitational attraction of the fluid and the attraction of the second primary; when the masses of the primaries vary with respect to time in the absence of reactive forces
Summary
The restricted three-body problem (R3BP) describes the motion of an infinitesimal mass moving under the gravitational effects of the two finite masses, called primaries, which move in circular orbits around their common center of mass on account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries. A new kind of the restricted three-body problem was formulated by Robe [9], in which one of the primaries of mass m1 , is a rigid spherical shell, filled with homogenous, incompressible fluid of density 1 , with the second mass point m2 outside the shell and moving around the first primary in a Keplerian orbit; and the infinitesimal mass m3 as a small solid sphere of an infinitesimal radius, and of density 3 , moving inside the shell and is subject to the attraction of m2 and the buoyancy force due to the fluid He discussed the linear stability of an equilibrium point obtained in two cases; the first being the case when the orbit of m2 around m1 is circular and in the second case, when it is elliptic, but the shell is empty (there is no fluid inside it) or densities of m1 and m3 are equal.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: International Journal of Astronomy and Astrophysics
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
