Application of Topological Degree Theory to the Equilibrium Point in Irregular Gravitational Field

The equilibrium point in an irregular gravitational field has been a focus of recent research. The numerical results indicated the number of equilibrium points in an irregular gravitational field is always odd, regardless of the shape or density of the celestial body. This article applies the topological degree theory and demonstrates the mathematical theory and physical mechanism of this phenomenon. We study the equilibrium points of the force field in the fixed coordinate system of a rotating homogeneous ball and generalize the results to the irregular gravitational field by the homotopy method. We demonstrate intrinsic properties of equilibrium points in the irregular gravitational field, which are independent of the shape, angular velocity, or density distribution of the celestial body under certain assumptions. The conclusions are also extended to the multiple celestial bodies system. In the end, real asteroids are used as numerical examples to verify the conclusions, and the results agree well with our analysis.