On sensitivity of the stability of equilibrium points with respect to the perturbations

Abstract

The present investigation considers the effect of small perturbations given in the Coriolis and centrifugal forces on the location and stability of the equilibrium points in the Robe’s circular restricted three-body problem with non-spherical primary bodies. The felicitous equations of motion of $m_3$ are obtained by taking into account the shapes of primaries $m_1$ and $m_2$, the full buoyancy force of the fluid which is filled inside $m_1$ of density $\rho _1$, the forces due to the gravitational attraction of the fluid and $m_2$. We assume that the massive body $m_1$ is an oblate spheroid and the $m_2$ a finite straight segment, and they move under a mutual gravitational attraction described by the Newton’s universal law of gravitation. In the present problem, $m_3$ is moving in the fluid and the rotating reference frame is used, its motion is bound to be affected by the perturbed Coriolis and centrifugal forces. In this attempt these effects along with the effects caused by the oblateness and length parameters A and l respectively, on the location and stability of the equilibrium points are observed. A pair of collinear equilibrium points $L_1$ and $L_2$ and infinite number of non-collinear equilibrium points are obtained. The stability of all the equilibrium points depends on the coefficients of their corresponding characteristic polynomials that are obtained with the help of linear variational equations.