Existence and Stability of Equilibrium Points in the Robe’s Restricted Three-Body Problem with Variable Masses

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 1   ;  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 ; 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.

Effect of variation of charge in the circular restricted three-body problem with variable masses

“…To solve Equation (11), let us set a = αe λτ , b = βe λτ , c = γ e λτ where α, β and γ are some parameters.…”

“…Simultaneously, Lukyanov [7] elaborates a study on the particular solution of the restricted problem of three bodies with variable mass by using Meshcherskii transformation. On the other hand, Singh et al in a series of papers, [8][9][10][11] explored the restricted three-body problem with different perturbations. Zhang et al [12] investigated the photogravitational restricted three-body problem, with the assumption that the infinitesimal body has a variable mass according to Jeans' law.…”

The motion of infinitesimal body in CR4BP with variable masses and Albedo effect

“…Many mathematicians studied CR3BP and CR4BP when the masses vary with time as Jeans [13], Meshcherskii [14], Shrivastava et al [15], Lichtenegger [16], Singh et al [17][18][19][20][21], Lukyanov [22], Zhang et al [23], Abouelmagd et al [24], Ansari et al [25][26][27][28][29][30], Alhussain [31,32].…”

Effects of Poynting–Robertson drag on the circular restricted Three-Body Problem with variable masses

“…Ishwar [16] studied in his thesis about the restricted problem of three bodies with a variable mass. Singh [17][18][19][20][21] investigated the effect of perturbations on the locations and stability of triangular equilibrium points in the restricted Three-body problem with variable mass. Zhang [22] investigated the stability of the triangular equilibrium points in the restricted Threebody problem with a variable mass by using spacetime transformation of Meshcherskii.…”