Equilibrium points and stability in the restricted three-body problem with oblateness and variable masses

Abstract

The existence and stability of a test particle around the equilibrium points in the restricted three-body problem is generalized to include the effect of variations in oblateness of the first primary, small perturbations and given in the Coriolis and centrifugal forces α and β respectively, and radiation pressure of the second primary; in the case when the primaries vary their masses with time in accordance with the combined Meshcherskii law. For the autonomized system, we use a numerical evidence to compute the positions of the collinear points L 2κ , which exist for 0 < κ < ∞, where κ is a constant of a particular integral of the Gylden-Meshcherskii problem; oblateness of the first primary; radiation pressure of the second primary; the mass parameter ν and small perturbation in the centrifugal force. Real out of plane equilibrium points exist only for κ > 1, provided the abscissae ξ < ν(κ−1) β . In the case of the triangular points, it is seen that these points exist for < κ < ∞ and are affected by the oblateness term, radiation pressure and the mass parameter. The linear stability of these equilibrium points is examined. It is seen that the collinear points L 2κ are stable for very small κ and the involved parameters, while the out of plane equilibrium points are unstable. The conditional stability of the triangular points depends on all the system parameters. Further, it is seen in the case of the triangular points, that the stabilizing or destabilizing behavior of the oblateness coefficient is controlled by κ, while those of the small perturbations depends on κ and whether these perturbations are positive or negative. However, the destabilizing behavior of the radiation pressure remains unaltered but grows weak or strong with increase or decrease in κ. This study reveals that oblateness coefficient can exhibit a stabilizing tendency in a certain range of κ, as against the findings of the RTBP with constant masses. Interestingly, in the region of stable motion, these parameters are void for κ = 4 3 . The decrease, increase or non existence in the region of stability of the triangular points depends on κ, oblateness of the first primary, small perturbations and the radiation pressure of the second body, as it is seen that the increasing region of stability becomes decreasing, while the decreasing region becomes increasing due to the inclusion of oblateness of the first primary.

How does the oblateness coefficient influence the nature of orbits in the restricted three-body problem?

“…In fact, we may say that these two exits act as hoses connecting the interior region of the system where x(L 3 ) ≤ x ≤ x(L 2 ) with the "outside world" of the exterior region. The position of the Lagrangian points as well as the critical values of the energy are functions of the oblateness coefficient A 1 (e.g., Singh & Leke (2012)). In Table 1 we provide the location of the Lagrangian points and the critical values of the total orbital energy when A 1 = {0, 0.001, 0.01, 0.1}.…”

“…The oblateness or triaxiality of a celestial body can produce perturbation deviations from the two-body motion. The study of oblateness coefficient includes the series of works of Beevi & Sharma (2012); Markellos et al (1996Markellos et al ( , 2002; Kalantonis et al (2005Kalantonis et al ( , 2006Kalantonis et al ( , 2008; Kalvouridis & Gousidou-Koutita (2012); Perdiou et al (2012); Sharma & Subba Rao (1979, 1986; Subba Rao & Sharma (1988Sharma ( , 1997; Sharma (1981Sharma ( , 1987Sharma ( , 1989Sharma ( , 1990; Singh & Leke (2012 by considering the more massive primary as an oblate spheroid with its equatorial plane co-incident with the plane of motion of the primaries.…”

The Motion of Out-of-Plane Equilibrium Points in the Elliptic Restricted Three-Body Problem at <i>J₄</i>

“…Celestial bodies in the general restricted three-body problem are assumed to be spherical, but in nature, several celestial bodies have observed the significant effects of oblateness of their bodies [1,2,3,4,5], have observed the significant effects of oblateness of the bodies. The restricted three-body problem withoblateness of the primaries have received tremendous attention, specifically in the two and three-dimensional cases with respect to its five co-planar equilibrium points ( = 1, … ,5): The points 1 , 2 , 3 lying on the line joining the primaries are called collinear equilibrium points, while the points 4 , 5 forming the triangle with the line joining the primaries are called triangular points.…”

Convergence properties of equilibria in the restricted three‐body problem with prolate primaries

“…To incorporate the particular shape of the two main bodies into the equations explaining the motion of the test body (e.g., comet, asteroid, or spacecraft), the oblateness (or prolateness) parameter has been introduced and used initially in Sharma & Subba Rao (1975). From then, a large amount of research work has been devoted to the study of the influence of the oblateness (Kalantonis et al 2005, 2006, 2008; Markellos et al 1996, 2000; Perdiou et al 2012; Safiya Beevi & Sharma 2012; Sharma 1980, 1989; Sharma 2012; Singh & Leke 2012, 2013; Zotos 2015, 2018).…”