Hill stability in the many-body problem

We established a criterion for the Hill stability of motions in the problem of many spherical bodies with a spherical density distribution. The region of Hill stability was determined. The sizes of this region are comparable to the total volume of all of the bodies in the system, which sharply increases the probability of mutual collisions. This result may be considered as a confirmation that a supermassive core can be formed at the center of a globular star cluster. The motions in the n-body problem are shown to be unstable according to Hill.     

On the Hill stability in the general problem of three finite bodies

We construct zero-kinetic-energy surfaces and determine the regions where motion is possible. We show that for bodies with finite sizes, there are bounded regions of space within which a three-body system never breaks up. The Hill stability criterion is established.     

Sundman surfaces and Hill stability in the three-body problem

Using the famous Sundman inequality, we have constructed for the first time the surfaces for the general three-body problem that we suggest calling Sundman surfaces. These surfaces are a generalization of the widely known Hill surfaces in the restricted circular three-body problem. The Sundman surfaces are constructed in a rectangular coordinate system that uses the mutual distances between the bodies as the Cartesian rectangular coordinates. The singular points of the family of these surfaces have been determined. The possible and impossible regions of motion of the bodies have been constructed in the space of mutual distances. We have shown the existence of Hill stable motions and established sufficient criteria for Hill stability of motions. Some of the astronomical applications are considered.     

The Photogravitational Hill Problem: Numerical Exploration

We consider the recently introduced version of the classical Lunar Hill problem, the photogravitational Hill problem, and study it's equilibrium points and zero-velocity curves. The full network of families of periodic orbits is numerically explored, their stability is computed and critical orbits are determined. Non-periodic orbits are also computed as points on a surface of section, providing an outlook of the stability regions, chaotic motions and escape.     

Connection between Hill stability and weak stability in the elliptic restricted three-body problem

This paper provides a study on the connection between Hill stability and weak stability in the framework of the spatial elliptic restricted three-body problem. We determine a necessary condition for weak stability by giving an upper and a lower bound of qualitative measure of the Hill stability. The sufficient condition for weak stability and the symmetry of weak stable regions around the planets of the Solar System is also investigated.     

Stability of binary asteroids

The restricted problem of 2+2 bodies is applied to the study of the stability and dynamics of binary asteroids in the solar system. Numerical investigation of the behavior of the orbital elements and the maximal Lyapunov characteristic number of binary asteroids reveal extensive regions where bounded quasiperiodic motion is possible. These regions are compared to the bounded regions which are predicted by the classical restricted problem of three bodies. Regions of bounded chaotic solutions are also found.     

Surfaces of zero kinetic energy in a general three-body problem

We generalize the concept of zero-velocity surface and construct zero-kinetic-energy surfaces. In the space of three mutual distances, we determine the regions where motion is possible; these regions are in the shape of an infinitely long tripod. Motions in the three-body problem are shown to be unstable according to Hill.     

Mission design through averaging of perturbed Keplerian systems: the paradigm of an Enceladus orbiter

Preliminary mission design for planetary satellite orbiters requires a deep knowledge of the long term dynamics that is typically obtained through averaging techniques. The problem is usually formulated in the Hamiltonian setting as a sum of the principal part, which is given through the Kepler problem, plus a small perturbation that depends on the specific features of the mission. It is usually derived from a scaling procedure of the restricted three body problem, since the two main bodies are the Sun and the planet whereas the satellite is considered as a massless particle. Sometimes, instead of the restricted three body problem, the spatial Hill problem is used. In some cases the validity of the averaging is limited to prohibitively small regions, thus, depriving the analysis of significance. We find this paradigm at Enceladus, where the validity of a first order averaging based on the Hill problem lies inside the body. However, this fact does not invalidate the technique as perturbation methods are used to reach higher orders in the averaging process. Proceeding this way, we average the Hill problem up to the sixth order obtaining valuable information on the dynamics close to Enceladus. The averaging is performed through Lie transformations and two different transformations are applied. Firstly, the mean motion is normalized whereas the goal of the second transformation is to remove the appearance of the argument of the node. The resulting Hamiltonian defines a system of one degree of freedom whose dynamics is analyzed.     

Stability in the Full Two-Body Problem

Stability conditions are established in the problem of two gravitationally interacting rigid bodies, designated here as the full two-body problem. The stability conditions are derived using basic principles from the N-body problem which can be carried over to the full two-body problem. Sufficient conditions for Hill stability and instability, and for stability against impact are derived. The analysis is applicable to binary small-body systems such as have been found recently for asteroids and Kuiper belt objects.     

The restricted three-body problem with logarithm potential

We present a study of the restricted three body problem with logarithm potential. We discuss equilibria, stability, Hill’s regions of motion and the families of periodic orbits near equilibria. Moreover, we show that equilibria and some periodic orbits continue in the logarithm three body problem.     

On the stability of particular solutions of the singly averaged Hill problem

We consider the particular solutions of the evolutionary system of equations in elements that correspond to planar and spatial circular orbits of the singly averaged Hill problem. We analyze the stability of planar and spatial circular orbits to inclination and eccentricity, respectively. We construct the instability regions of both particular solutions in the plane of parameters of the problem.     

Equilibrium points and Lyapunov families in the circular restricted three-body problem with an oblate primary and a synchronous rotating dipole secondary: Application to Luhman-16 binary system

We numerically study a version of the synchronous circular restricted three-body problem, where an infinitesimal mass body is moving under the Newtonian gravitational forces of two massive bodies. The primary body is an oblate spheroid while the secondary is an elongated asteroid of a combination of two equal masses forming a rotating dipole which is synchronous to the rotation of the primaries of the classic circular restricted three-body problem. In this paper, we systematically examine the existence, positions, and linear stability of the equilibrium points for various combinations of the model's parameters. We observe that the perturbing forces have significant effects on the positions and stability of the equilibrium points as well as the regions where the motion of the particle is allowed. The allowed regions of motion as determined by the zero-velocity surface and the corresponding isoenergetic curves as well as the positions of the equilibrium points are given. Finally, we numerically study the binary system Luhman-16 by computing the positions of the equilibria and their stability as well as the allowed regions of motion of the particle. The corresponding families of periodic orbits emanating from the collinear equilibrium points are computed along with their stability properties.     

Hill stability of the general problem of three bodies

We discuss Hill stability in the general three-body problem. The Hill curves in the general problem are the same as in the planar problem. We show that the bifurcation points correspond to the five equilibrium solutions, and derive the criterion for stability in the general case. Application of this criterion to 19 natural satellites of the Solar system leads to the result that, apart from Neptune 1, all the other 18 satellites are unstable in the sense of Hill. The dominant factor in producing this result is the finite eccentricity of the planetary orbits around the Sun.     

Instability of the 2+2 body problem

The equations of motion of the 2+2 body problem (two interacting particles in the gravitational field of two much more massive primaries m₁ and m₂ in circular keplerian orbit) have an integral analogous to the Jacobi integral of the circular 2+1 body problem. We show here that with 2+2 bodies this integral does not give rise to Hill stability, i.e. to confinement for all time in a portion of the configuration space not allowing for some close approaches to occur. This is because all the level manifolds are connected and all exchanges of bodies between the regions surroundingm ₁,m ₂ and infinity do not contradict the conservation of the integral. However, it is worth stressing that some of these exchanges are physically meaningless, because they involve either unlimited extraction of potential energy from the binary formed by the small bodies (without taking into account their physical size) or significant mutual perturbations between the small masses without close approach, a process requiring, for the Sun-Jupiter-two asteroids system, timescales longer than the age of the Solar System.     

The Hill stability of a binary or planetary system during encounters with a third inclined body moving on a hyperbolic orbit

The dynamical interaction of a binary or planetary system and a third body moving on a hyperbolic orbit inclined to the system is discussed in terms of Hill stability for the full three-body problem. The situation arises in binary star disruption and exchange and planetary system exchange or capture. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the third body also produces similar effects. These type of changes make exchange or disruption of the component masses more likely as also does increasing the eccentricity of the binary.

The critical distances and Hill stability ranges associated with the possible formation of roughly equal mass trans-Neptunian binaries from three-body interactions are determined for a range of secondary component masses.     

The Hill stability of inclined small mass binary systems in three-body systems with special application to triple star systems, extrasolar planetary systems and Binary Kuiper Belt systems

The dynamical stability of a bound triple system composed of a small binary or minor planetary system moving on a orbit inclined to a central third body is discussed in terms of Hill stability for the full three-body problem. The situation arises in the determination of stability of triple star systems against disruption and component exchange and the determination of stability of extrasolar planetary systems and minor planetary systems against disruption, component exchange or capture. The Hill stability criterion is applied to triple star systems and extrasolar planetary systems, the Sun-Earth-Moon system and Kuiper Belt binary systems to determine the critical distances for stable orbits. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the binary also produces similar effects.These type of changes make exchange or disruption of the component masses more likely. Increasing the eccentricity of the binary orbit relative to the third body substantially decreases stability regions as the eccentricity reaches higher values. The Kuiper Belt binaries were found to be stable if they move on circular orbits. Taking into account the eccentricity, it is less clear that all the systems are stable.     

The Hill stability of a binary or planetary system during encounters with a third inclined body

The dynamical interaction of a binary or planetary system and a third body moving on a parabolic orbit inclined to the system is discussed in terms of Hill stability for the full three-body problem. The situation arises in binary star disruption and exchange, in extrasolar planetary system disruption, exchange and capture. It is found that increasing the inclination of the third body decreases the Hill regions of stability. This makes exchange or disruption of the component masses more likely as does increasing the eccentricity of the binary.
The stability criteria are applied to determine possible disruption and capture distances for currently known extrasolar planetary systems.     

Restricted four-body problem. The case of a central configuration: Libration points and their stability

We consider the problem of the motion of a zero-mass body in the vicinity of a system of three gravitating bodies forming a central configuration.We study the case where two gravitating bodies of equal mass lie on the same straight line and rotate around the central body with the same angular velocity. Equations for calculating the equilibrium positions in this system have been derived. The stability of the equilibrium points for a system of three gravitating bodies is investigated. We show that, as in the case of libration points for two bodies, the collinear points are unstable; for the triangular points, there exists a ratio of the mass of the central body to the masses of the extreme bodies, 11.720349, at which stability is observed.     

A Review on Co-orbital Motion in Restricted and Planetary Three-body Problems

The 1:1 mean motion resonance may be referred to as the lowest order mean motion resonance in restricted or planetary three-body problems. The five well-known libration points of the circular restricted three-body problem are five equilibriums of the 1:1 resonance. Coorbital motion may take different shapes of trajectory. In case of small orbital eccentricities and inclinations, tadpole-shape and horseshoe-shape orbits are well-known. Other 1:1 libration modes different from the elementary ones can exist at moderate or large eccentricities and inclinations. Coorbital objects are not rare in our solar system, for example the Trojans asteroids and the coorbital satellite systems of Saturn. Recently, dozens of coorbital bodies have been identified among the near-Earth asteroids. These coorbital asteroids are believed to transit recurrently between different 1:1 libration modes mainly due to orbital precessions, planetary perturbations, and other possible effects. The Hamiltonian system and the Hill’s three-body problem are two effective approaches to study coorbital motions. To apply the perturbation theory to the Hamiltonian system, standard procedures involve the development of the disturbing function, averaging and normalization, theory of ideal resonance model, secular perturbation theory, etc. Global dynamics of coorbital motion can be revealed by the Hamiltonian approach with a suitable expansion. The Hill’s problem is particularly suitable for the studies on the relative motion of two coorbital bodies during their close encounter. The Hill’s equation derived from the circular restricted three-body problem is well known. However, the general Hill’s problem whose equation of motion takes exactly the same form applies to the non-restricted case where the mass of each body is non-negligible, namely the planetary case. The Hill’s problem can be transformed into a “canonical shape” so that the averaging principle can be applied to construct a secular perturbation theory. Besides the two analytical theories, numerical methods may be consulted, for example the approach of periodic orbit, the surface of section, and the computation of invariant manifolds carried by equilibriums or periodic orbits.