On the existence of the relative equilibria of a rigid body in the J ₂problem

The motion of a point mass in the J₂problem has been generalized to that of a rigid body in a J₂gravity field for new high-precision applications in the celestial mechanics and astrodynamics. Unlike the original J₂problem, the gravitational orbit-rotation coupling of the rigid body is considered in the generalized problem. The existence and properties of both the classical and non-classical relative equilibria of the rigid body are investigated in more details in the present paper based on our previous results. We nondimensionalize the system by the characteristic time and length to make the study more general. Through the study, it is found that the classical relative equilibria can always exist in the real physical situation. Numerical results suggest that the non-classical relative equilibria only can exist in the case of a negative J₂, i.e., the central body is elongated; they cannot exist in the case of a positive J₂when the central body is oblate. In the case of a negative J₂, the effect of the orbit-rotation coupling of the rigid body on the existence of the non-classical relative equilibria can be positive or negative, which depends on the values of J₂and the angular velocity Ω_e. The bifurcation from the classical relative equilibria, at which the non-classical relative equilibria appear, has been shown with different parameters of the system. Our results here have given more details of the relative equilibria than our previous paper, in which the existence conditions of the relative equilibria are derived and primarily studied. Our results have also extended the previous results on the relative equilibria of a rigid body in a central gravity field by taking into account the oblateness of the central body.