Relative equilibria of full dynamics of a rigid body with gravitational orbit-attitude coupling in a uniformly rotating second degree and order gravity field

The motion of a rigid body in a uniformly rotating second degree and order gravity field is a good model for the gravitationally coupled orbit-attitude motion of a spacecraft in the close proximity of an asteroid. The relative equilibria of this full dynamics model are investigated using geometric mechanics from a global point of view. Two types of relative equilibria are found based on the equilibrium conditions: one is the Lagrangian relative equilibria, at which the circular orbit of the rigid body is in the equatorial plane of the central body; the other is the non-Lagrangian relative equilibria, at which the circular orbit is parallel to but not in the equatorial plane of central body. The existences of the Lagrangian and non-Lagrangian relative equilibria are discussed numerically with respect to the parameters of the gravity field and the rigid body. The effect of the gravitational orbit-attitude coupling is especially assessed. The existence region of the Lagrangian relative equilibria is given on the plane of the system parameters. Numerical results suggest that the negative C₂₀ with a small absolute value and a negative C₂₂ with a large absolute value favor the existence of the non-Lagrangian relative equilibria. The effect of the gravitational orbit-attitude coupling of the rigid body on the existence of the non-Lagrangian relative equilibria can be positive or negative, which depends on the harmonics C₂₀ and C₂₂, and the angular velocity of the rotation of the gravity field.