Abstract. We study the orbits and manifolds near the equilibrium points of a rotating asteroid. The linearised equations of motion relative to the equilibrium points in the gravitational field of a rotating asteroid, the characteristic equation and the stable conditions of the equilibrium points are derived and discussed. First, a new metric is presented to link the orbit and the geodesic of the smooth manifold. Then, using the eigenvalues of the characteristic equation, the equilibrium points are classified into 8 cases. A theorem is presented and proved to describe the structure of the submanifold as well as the stable and unstable behaviours of a massless particle near the equilibrium points. The linearly stable, the non-resonant unstable, and the resonant equilibrium points are discussed. There are three families of periodic orbits and four families of quasi-periodic orbits near the linearly stable equilibrium point. For the non-resonant unstable equilibrium points, there are four cases; for the periodic orbit and the quasi-periodic orbit, the structures of the submanifold and the subspace near the equilibrium points are studied for each case. For the resonant equilibrium points, the dimension of the resonant manifold is greater than four, and we find at least 1 family of periodic orbits near the resonant equilibrium points. Besides, this theory is 2 applied to asteroids 216 Kleopatra,

We are interested in the periodic motion and bifurcations near the surface of an asteroid. The gravity field of an irregular asteroid and the equation of motion of a particle near the surface of an asteroid are studied. The periodic motions around the major body of triple asteroid 216 Kleopatra and the OSIRIS-REx mission target-asteroid 101955 Bennu are discussed. We find that motion near the surface of an irregular asteroid is quite different from the motion near the surface of a homoplastically spheroidal celestial body. The periodic motions around the asteroid 101955 Bennu and 216 Kleopatra indicate that the geometrical shapes of the orbits are probably very sophisticated. There exist both stable periodic motions and unstable periodic motions near the surface of the same irregular asteroid. This periodic motion which is unstable can be resonant or non-resonant. The period-doubling bifurcation and pseudo period-doubling bifurcation of periodic orbits coexist in the same gravity field of the primary of the triple asteroid 216 Kleopatra. It is found that both of the period-doubling bifurcations of periodic orbits and pseudo period-doubling bifurcation of periodic orbits have four different paths. The pseudo period-doubling bifurcation found in the potential field of primary of triple asteroid 216 Kleopatra shows that there exist stable periodic orbits near the primary's equatorial plane, which gives an explanation for the motion stability of the triple asteroid 216 Kleopatra's two moonlets, Alexhelios and Cleoselene. spacecraft or moonlet which is orbiting an irregular asteroid can be modeled by a massless particle (Scheeres 2012;Takahashi et al. 2013;Chanut et al. 2014); if the particle is sufficiently far from the irregular asteroid, the asteroid can be approximately modelled as a sphere and the solar gravitation to the particle should be considered; conversely, if the particle is sufficiently close to the irregular asteroid, the irregular asteroidal gravity field provides the decisive effect on the motion of the particle, and the irregular asteroidal gravity field should be considered and the solar gravitation can be neglected (Borisov and Zakharov 2014;Wang et al. 2014).Recently, a few irregular asteroids were selected to analyze motions around them, including asteroids 4 Vesta (Mondelo et al.

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Abstract. In this work, we investigate the bifurcations of relative equilibria in the gravitational potential of asteroids. A theorem concerning a conserved quantity, which is about the eigenvalues and number of relative equilibria, is presented and proved. The conserved quantity can restrict the number of non-degenerate equilibria in the gravitational potential of an asteroid. It is concluded that the number of non-degenerate equilibria in the gravitational field of an asteroid varies in pairs and is an odd number. In addition, the conserved quantity can also restrict the kinds of bifurcations of relative equilibria in the gravitational potential of an asteroid when the parameter varies. Furthermore, studies have shown that there exist transcritical bifurcations, quasi-transcritical bifurcations, saddle-node bifurcations, saddle-saddle bifurcations, binary saddle-node bifurcations, supercritical pitchfork bifurcations, and subcritical pitchfork bifurcations for the relative equilibria in the gravitational potential of asteroids. It is found that for the asteroid 216 Kleopatra, when the rotation period varies as a parameter, the number of relative equilibria changes from 7 to 5 to 3 to 1, and the bifurcations for the relative equilibria are saddle-node bifurcations and saddle-saddle bifurcations.

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With the rapid development of satellite technology and the need to satisfy the increasing demand for location-based services, in challenging environments such as indoors, forests, and canyons, there is an urgent need to improve the position accuracy in these environments. However, traditional algorithms obtain the position solution through time redundancy in exchange for spatial redundancy, and they require continuous observations that cannot satisfy the real-time location services. In addition, they must also consider the clock bias between the satellite and receiver. Therefore, in this paper, we provide a single-satellite integrated navigation algorithm based on the elimination of clock bias for broadband low earth orbit (LEO) satellite communication links. First, we derive the principle of LEO satellite communication link clock bias elimination; then, we give the principle and process of the algorithm. Next, we model and analyze the error of the system. Subsequently, based on the unscented Kalman filter (UKF), we model the state vector and observation vector of our algorithm and give the state and observation equations. Finally, for different scenarios, we conduct qualitative and quantitative analysis through simulations, and the results show that, whether in an altimeter scenario or non-altimeter scenario, the performance indicators of our algorithm are significantly better than the inertial navigation system (INS), which can effectively overcome the divergence problem of INS; compared with the medium earth orbit (MEO) constellation, the navigation trajectory under the LEO constellation is closer to the real trajectory of the aircraft; and compared with the traditional algorithm, the accuracy of each item is improved by more than 95%. These results show that our algorithm not only significantly improves the position error, but also effectively suppresses the divergence of INS. The algorithm is more robust and can satisfy the requirements of cm-level real-time location services in challenging environments.

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Abstract. The order and chaos of the motion near equilibrium points in the potential of a rotating highly irregular-shaped celestial body are investigated from point of view of the dynamical system theory. The positions of the non-degenerate equilibrium points vary continuously when the parameter changes. The topological structures in the vicinity of equilibrium points are classified into several different cases.Bifurcations at equilibrium points and the topological transfers between different cases for equilibrium points are also discussed. The conclusions can be applied to all kinds of rotating celestial bodies, simple-shaped or highly irregular-shaped, including asteroids, comets, planets and satellites of planets to help one to understand the dynamical behaviors around them. Applications to asteroids 216 Kleopatra, 2063 Bacchus, and 25143 Itokawa are significant and interesting: eigenvalues affiliated to the equilibrium points for the asteroid 216 Kleopatra move and always belong to the same topological cases; while eigenvalues affiliated to two different equilibrium points for the asteroid 2063 Bacchus and 25143 Itokawa move through the resonant cases of equilibrium points, and the collision of eigenvalues in the complex plane occurs. Poincaré sections in the potential of the asteroid 216 Kleopatra show the chaos behaviors of the orbits in large scale.