The present paper addresses the existence of J 2 invariant relative orbits with arbitrary relative magnitude over the infinite time using the Routh reduction and Poincaré techniques in the J 2 Hamiltonian problem. The current research also proposes a novel numerical searching approach for J 2 invariant relative orbits from the dynamical system point of view. A new type of Poincaré mapping is defined from different central manifolds of the pseudocircular orbits (parameterized by the Jacobi energy E, the polar component of momentum H z and the measure of distance r between the fixed point and its central manifolds) to the nodal periods T d and the drifts of longitude of the ascending node during one period ( ), which differs from Koon et al.'s (AIAA 2001) definition on central manifolds parameterized by the same fixed point. The Poincaré mapping is surjective because it compresses the three-dimensional variables into two-dimensional images, and the mapping degenerates into a bijective mapping in consideration of the fixed points. An iteration algorithm to the degenerated bijective mapping is proposed from the continuation procedure to perform the ergodic representation of E-and H z -contour maps on the space of T d -. For the surjective mapping with r = 0, different pseudo-circular or elliptical orbits may share the same images. Hence, the inverse surjective mapping may achieve non-unique variables from a single image, which makes the generation of J 2 invariant relative orbits possible. The pseudo-circular or elliptical orbits generated from the surjective mapping will be defined in different meridian planes. Hence, the critical contribution of the present paper is the assignment of J 2 invariant relative orbits to different invariant parameters E and H z depending on the E-and H z -contour map, which will hold J 2 invariant relative orbits for extended durations. To investigate the high-order nonlinearity neglected by previous studies, a formation configuration with a 123 428 M. Xu et al.large magnitude of 500 km is successfully generated from the theory developed in the present work, which is beyond the scope of the linear conditions of J 2 invariant relative orbits. Therefore, the existence of J 2 invariant relative orbit with an arbitrary relative magnitude over the infinite time is achieved from the dynamical system point of view.
The motion of a point mass in the J 2 problem is generalized to that of a rigid body in a J 2 gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal (S. Xu). 2 harmonic J 2 and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively.The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system.
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