The four longest period Kuiper belt objects have orbital periods close to integer ratios with each other. A hypothetical planet with orbital period ∼ 17,117 years, semimajor axis ∼ 665 AU, would have N/1 and N/2 period ratios with these four objects. The orbital geometries and dynamics of resonant orbits constrain the orbital plane, the orbital eccentricity and the mass of such a planet, as well as its current location in its orbital path.

Abstract. The equilibrium points of the gravitational potential field of minor celestial bodies, including asteroids, comets, and irregular satellites of planets, are studied. In order to understand better the orbital dynamics of massless particles moving near celestial minor bodies and their internal structure, both internal and external equilibrium points of the potential field of the body are analyzed. In this paper, the location and stability of the equilibrium points of 23 minor celestial bodies are presented. In addition, the contour plots of the gravitational effective potential of these minor bodies are used to point out the differences between them. Furthermore, stability and topological classifications of equilibrium points are discussed, which clearly illustrate the topological structure near the equilibrium points and help to have an insight into the orbital dynamics around the irregular-shaped minor celestial bodies.The results obtained here show that there is at least one equilibrium point in the potential field of a minor celestial body, and the number of equilibrium points could be one, five, seven, and nine, which are all odd integers. It is found that for some irregular-shaped celestial bodies, there are more than four equilibrium points outside the bodies while for some others there are no external equilibrium points. If a celestial 2 body has one equilibrium point inside the body, this one is more likely linearly stable.

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.

The stability and topological structure of equilibrium points in the potential field of the asteroid 101955 Bennu have been investigated with a variable density and rotation period. A dimensionless quantity is introduced for the nondimensionalization of the equations of motion, and this quantity can indicate the effect of both the rotation period and bulk density of the asteroid. Using the polyhedral model of the asteroid Bennu, the number and position of the equilibrium points are calculated and illustrated by a contour plot of the gravitational effective potential field. The topological structure and the stability of the equilibrium points are also investigated using the linearized method. The results show that there are nine equilibrium points in the potential field of the asteroid Bennu, eight in the exterior of the body and one in the interior of the body. Moreover, bifurcation will occur with a variation of the density and rotation period. Different equilibrium points will encounter each other and mix together. Thus, the number of equilibrium points will change. The stability and topological structure of the equilibrium points will also change because of the variation of the density and rotation period of the asteroid. When considering the error of the density of Bennu, the range of the dimensionless quantity covers the critical values that will lead to bifurcation. This means that the stability of the equilibrium points is uncertain, making the dynamical environment of Bennu much more complicated. These bifurcations can help better understand the dynamic environment of an irregular-shaped asteroid.

The observationally complete sample of the main belt asteroids now spans more than two orders of magnitude in size and numbers more than 64,000 (excluding collisional family members). We undertook an analysis of asteroids' eccentricities and their interpretation with simple physical models. We find that Plummer's (1916) conclusion that the asteroids' eccentricities follow a Rayleigh distribution holds for the osculating eccentricities of large asteroids, but the proper eccentricities deviate from a Rayleigh distribution: there is a deficit of eccentricities smaller than ∼ 0.1 and an excess of larger eccentricities. We further find that the proper eccentricities do not depend significantly on asteroid size but have strong dependence on heliocentric distance: the outer asteroid belt follows a Rayleigh distribution, but the inner belt is strikingly different. Eccentricities in the inner belt can be modeled as a vector sum of a primordial eccentricity vector of random orientation and magnitude drawn from a Rayleigh distribution of parameter ∼ 0.06, and an excitation of random phase and magnitude ∼ 0.13. These results imply that a late dynamical excitation of the asteroids occurred, it was independent of asteroid size, it was stronger in the inner belt than in the outer belt. We discuss implications for the primordial asteroid belt and suggest that the observationally complete sample size of main belt asteroids is large enough that more sophisticated model-fitting of the eccentricities is warranted and could serve to test alternative theoretical models of the dynamical excitation history of asteroids and its links to the migration history of the giant planets.