Stability of Equilibrium Points and Behaviour of Nearby Trajectories

Abstract

In physics, an equilibrium point means the positions in body-fixed frame Oxyz where the gravity from the small body and the centrifugal force balance. For a real small body which has irregular overlook, the equilibrium points are usually isolated and of finite number. This chapter takes the equilibria as a start of surveying the system dynamic behaviours, not only because of its simplicity, but also that the equilibria may be a key to understand the global orbital behaviours. As shown by Eq. 3.1, an equilibrium point is also a critical point of the contour surfaces of the efficient potential V, which is correlated to the topological evolution of these surfaces (see Sect. 2.5 for the discussion on $\kappa$). In this chapter, we take a specific asteroid for example, and discuss the orbital motion within the neighbourhoods of equilibrium points using the polyhedral gravity model. Section 3.2 calculates the equilibrium points of the target body, and presents the geometry and topology of the contour surfaces of efficient potential V. Sections 3.3 and 3.4 use the linearized theory to determine the stability and type of an equilibrium point, and reveal the orbital motion on local invariant manifolds. Based on the analysis of local manifolds, Sect. 3.5 further discusses the general orbital patterns in the neighbourhoods of the equilibrium points.