The photogravitational Hill problem with oblateness: equilibrium points and Lyapunov families

Abstract

We introduce a new version of Hill's problem that incorporates the effects of radiation of the primary and oblateness of the secondary and study the basic dynamical features of this new model-problem. This formulation is more appropriate for some astronomical applications as an approximation to the corresponding restricted three-body problem. We use iterative methods for deriving approximate expressions of the equilibrium point locations and study their stability properties by using a linear stability analysis. All equilibrium points are unstable. We also employ singular perturbations methods for obtaining approximate expressions of the Lyapunov families emanating from equilibrium points, in both coplanar and spatial case, and numerical techniques for their continuation.

Collinear equilibrium points of Hill’s problem with radiation and oblateness and their fractal basins of attraction

“…Thus, the present work reveals findings that were not presented in our previous paper on this model (Markakis et al 2008), the existence of additional collinear equilibrium points being a result of potential importance in view of their stability. This makes a qualitative difference in the dynamics of the model.…”

“…This derivation is essentially contained in Markellos et al (2000Markellos et al ( , 2001, and Markakis et al (2008). In the final equations (7) the symbols α and Q are the oblateness and radiation coefficients of the present model problem derived from the respective coefficients of the restricted problem as scaled by (5).…”

“…When these conditions are satisfied, the two roots of the characteristic equation (for λ 2 ) are negative, giving four imaginary roots for λ. In Markakis et al (2008) it was found that when a > 0 this never happens for the outer collinear equilibria, which are therefore always unstable for an oblate secondary. For the inner collinear points we have computed the above quantities and the four roots of the characteristic equation at the node points of a dense grid in the (Q, a) plane, and have found that there is a substantial region in parameter space in which the inner equilibrium points are stable.…”

“…As mentioned above, this variant is more relevant for Solar System applications, and was considered in our previous paper (Markakis et al 2008) for positive oblateness coefficient.…”

Equilibrium points of the restricted three-body problem with equal prolate and radiating primaries, and their stability

“…In Markakis et al (2008) a variant of Hill's problem with radiation and oblateness was studied, and in Douskos (2010) new collinear equilibrium points of that model problem were found near a prolate secondary. In the present paper we complement that work by considering the restricted problem with equal prolate and radiating spheroids as primaries, with the aim to discuss systematically the existence, location and stability of its equilibrium points.…”