Constraints on bounded motion and mutual escape for the full 3-body problem

When gravitational aggregates are spun to fission they can undergo complex dynamical evolution, including escape and reconfiguration. Previous work has shown that a simple analysis of the full 2-body problem provides physically relevant insights for whether a fissioned system can lead to escape of the components and the creation of asteroid pairs. In this paper we extend the analysis to the full 3-body problem, utilizing recent advances in the understanding of fission mechanics of these systems. Specifically, we find that the full 3-body problem can eject a body with as much as 0.31 of the total system mass, significantly larger than the 0.17 mass limit previously calculated for the full 2-body problem. This paper derives rigorous limits on a fissioned 3-body system with regards to whether fissioned system components can physically escape from each other and what other stable relative equilibria they could settle in. We explore this question with a narrow focus on the Spherical Full Three Body Problem studied in detail earlier.     

Linear Stability of the Lagrangian Triangle Solutions for Quasihomogeneous Potentials

In this paper, we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. First, in the case the potential is a homogeneous function of degree −a, we find that any relative equilibrium of the n-body problem with a>2 is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on the size of the configuration.     

Energy and stability in the Full Two Body Problem

The conditions for relative equilibria and their stability in the Full Two Body Problem are derived for an ellipsoid–sphere system. Under constant angular momentum it is found that at most two solutions exist for the long-axis solutions with the closer solution being unstable while the other one is stable. As the non-equilibrium problem is more common in nature, we look at periodic orbits in the F2BP close to the relative equilibrium conditions. Families of periodic orbits can be computed where the minimum energy state of one family is the relative equilibrium state. We give results on the relative equilibria, periodic orbits and dynamics that may allow transition from the unstable configuration to a stable one via energy dissipation.      

The effect of radiation pressure on the particle dynamics in ring-type N-body configurations

The paper deals with a simple photo-gravitational model of N+1 bodies. The motion of a small particle which subjects both the gravitational attraction and the radiation pressure is studied in a regular polygon configuration of N bodies. The dynamical features of this model are investigated for a wide range of values of the radiation parameters by mapping its equilibrium points and periodic orbits. The results show that for these values, radiation merely affects quantitatively the characteristics of the system, while it leaves unaffected the stability of the particle periodic motions and equilibria. This revised version was published online in July 2006 with corrections to the Cover Date.     

Equilibrium Points and Central Configurations for the Lennard-Jones 2- and 3-Body Problems

In this paper we study the relative equilibria and their stability for a system of three point particles moving under the action of a Lennard-Jones potential. A central configuration is a special position of the particles where the position and acceleration vectors of each particle are proportional, and the constant of proportionality is the same for all particles. Since the Lennard-Jones potential depends only on the mutual distances among the particles, it is invariant under rotations. In a rotating frame the orbits coming from central configurations become equilibrium points, the relative equilibria. Due to the form of the potential, the relative equilibria depend on the size of the system, that is, depend strongly of the momentum of inertia I. In this work we characterize the relative equilibria, we find the bifurcation values of I for which the number of relative equilibria is changing, we also analyze the stability of the relative equilibria. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Equilibrium figures of spinning bodies with self-gravity

The study of the equilibrium and stability of spinning ellipsoidal fluid bodies with gravity began with Newton in 1687, and continues to the present day. However, no smaller bodies of the Solar System are fluid. Here I model those bodies as elastic-plastic solids using a cohesionless Mohr-Coulomb yield envelope characterized by an angle of friction. This study began in Holsapple 2001. Here new closed-form algebraic formulas for the spin limits of ellipsoidal shapes are derived using an energy method. The fluid results of Maclaurin and Jacobi are again recovered as special cases. I then consider the stability of those equilibrium states. For elastic-plastic solids the common methods cannot be used, because the constitutive equations lack sufficient smoothness at the limiting plastic states. Therefore, I propose and study a new measure of the stability of dynamic processes in general bodies. An energy-based approach is introduced which is shown to include stability approaches used in the statics of nonlinear elastic and elastic-plastic bodies, spectral definitions and the Liapunov methods used for finite-dimensional dynamical systems. The method is applied to spinning, solid, strained bodies. In contrast to the special fluid case, it is found that the strain energy term of solid materials generally induces stability of all equilibrium shapes, except for two possible cases. First, strain softening in the elastic-plastic law can result in instability at the plastic limit spin. Second, a loss of shear stiffness can give unstable states at specific spins less than the limit equilibrium spins. In the latter case, a solid spinning ellipsoidal body without elastic shear stiffness can spin no faster than with a period of about 3.7 hr, else it will fail by shearing deformations. That is distinctly slower than the oft-quoted limit of 2.1 hr at which material would be flung off the equator by tensile forces. However, the final conclusion is that neither cohesion nor tensile strength is required for the shapes and spins of almost all of the larger observed asteroids: we cannot rule out rubble-pile structures.     

Motion in a modified Chermnykh’s restricted three-body problem with oblateness

In this paper, the restricted problem of three bodies is generalized to include a case when the passively gravitating test particle is an oblate spheroid under effect of small perturbations in the Coriolis and centrifugal forces when the first primary is a source of radiation and the second one an oblate spheroid, coupled with the influence of the gravitational potential from the belt. The equilibrium points are found and it is seen that, in addition to the usual three collinear equilibrium points, there appear two new ones due to the potential from the belt and the mass ratio. Two triangular equilibrium points exist. These equilibria are affected by radiation of the first primary, small perturbation in the centrifugal force, oblateness of both the test particle and second primary and the effect arising from the mass of the belt. The linear stability of the equilibrium points is explored and the stability outcome of the collinear equilibrium points remains unstable. In the case of the triangular points, motion is stable with respect to some conditions which depend on the critical mass parameter; influenced by the small perturbations, radiating effect of the first primary, oblateness of the test body and second primary and the gravitational potential from the belt. The effects of each of the imposed free parameters are analyzed. The potential from the belt and small perturbation in the Coriolis force are stabilizing parameters while radiation, small perturbation in the centrifugal force and oblateness reduce the stable regions. The overall effect is that the region of stable motion increases under the combine action of these parameters. We have also found the frequencies of the long and short periodic motion around stable triangular points. Illustrative numerical exploration is rendered in the Sun–Jupiter and Sun–Earth systems where we show that in reality, for some values of the system parameters, the additional equilibrium points do not in general exist even when there is a belt to interact with.     

Reduction and relative equilibria for the two-body problem on spaces of constant curvature

We consider the two-body problem on surfaces of constant nonzero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each \(q>0\) we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except \(\pi /2\). When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal, there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (‘isosceles RE’) and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At \(\pi /2\), the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a five-dimensional phase space and possess one Casimir function.     

Capture into first-order resonances and long-term stability of pairs of equal-mass planets

Massive planets form within the lifetime of protoplanetary disks, and therefore, they are subject to orbital migration due to planet–disk interactions. When the first planet reaches the inner edge of the disk, its migration stops and consequently the second planet ends up locked in resonance with the first one. We detail how the resonant trapping works comparing semi-analytical formulae and numerical simulations. We restrict to the case of two equal-mass coplanar planets trapped in first-order resonances, but the method can be easily generalized. We first describe the family of resonant stable equilibrium points (zero-amplitude libration orbits) using series expansions up to different orders in eccentricity as well as a non-expanded Hamiltonian. Then we show that during convergent migration the planets evolve along these families of equilibrium points. Eccentricity damping from the disk leads to a final equilibrium configuration that we predict precisely analytically. The fact that observed multi-exoplanetary systems are rarely seen in resonances suggests that in most cases the resonant configurations achieved by migration become unstable after the removal of the protoplanetary disk. Here we probe the stability of the resonances as a function of planetary mass. For this purpose, we fictitiously increase the masses of resonant planets, adiabatically maintaining the low-amplitude libration regime until instability occurs. We discuss two hypotheses for the instability, that of a low-order secondary resonance of the libration frequency with a fast synodic frequency of the system, and that of minimal approach distance between planets. We show that secondary resonances do not seem to impact resonant systems at low amplitude of libration. Resonant systems are more stable than non-resonant ones for a given minimal distance at close encounters, but we show that the latter nevertheless play the decisive role in the destabilization of resonant pairs. We show evidence that as the planetary mass increases and the minimal distance between planets gets smaller in terms of mutual Hill radius, the region of stability around the resonance center shrinks, until the equilibrium point itself becomes unstable.     

Dynamical evolution of two associated galactic bars

We study the dynamical interactions of mass systems in equilibrium under their own gravity that mutually exert and ex‐perience gravitational forces. The method we employ is to model the dynamical evolution of two isolated bars, hosted within the same galactic system, under their mutual gravitational interaction. In this study, we present an analytical treatment of the secular evolution of two bars that oscillate with respect to one another. Two cases of interaction, with and without geometrical deformation, are discussed. In the latter case, the bars are described as modified Jacobi ellipsoids. These triaxial systems are formed by a rotating fluid mass in gravitational equilibrium with its own rotational velocity and the gravitational field of the other bar. The governing equation for the variation of their relative angular separation is then numerically integrated, which also provides the time evolution of the geometrical parameters of the bodies. The case of rigid, non‐deformable, bars produces in some cases an oscillatory motion in the bodies similar to that of a harmonic oscillator. For the other case, a deformable rotating body that can be represented by a modified Jacobi ellipsoid under the influence of an exterior massive body will change its rotational velocity to escape from the attracting body, just as if the gravitational torque exerted by the exterior body were of opposite sign. Instead, the exchange of angular momentum will cause the Jacobian body to modify its geometry by enlarging its long axis, located in the plane of rotation, thus decreasing its axial ratios. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parametric evolution of periodic orbits in the restricted four-body problem with radiation pressure

The backbone of the analysis in most dynamical systems is the study of periodic motions, since they greatly assist us to understand the structure of all possible motions. In this paper, we deal with the photogravitational version of the rectilinear restricted four-body problem and we investigate the dynamical behaviour of a small particle that is subjected to both the gravitational attraction and the radiation pressure of three bodies much bigger than the particle, the primaries. These bodies are always in syzygy and two of them have equal masses and are located at equal distances from the third primary. We study the effect of radiation on the distribution of the periodic orbits, their stability, as well as the evolution of the families and their main features.     

Robe’s circular restricted three-body problem under oblate primaries with perturbations in Coriolis and centrifugal forces

This paper examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid, and the shape of the second primary is also an oblate spheroid. The problem is perturbed in the sense that small perturbations given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. The linear stability of this configuration is examined; it is observed that the first axial point is unstable while the second one is conditionally stable and the circular points are unstable.     

Dynamics of a triaxial gyrostat at a Lagrangian equilibrium of a binary asteroid

The non-canonical Hamiltonian dynamics of a triaxial gyrostat in Newtonian interaction with two punctual masses is considered. This serves as a model for the study of the attitude dynamics of a spacecraft located at a Lagrangian equilibrium point of the system formed by a binary asteroid and a spacecraft. Using geometric-mechanics methods, the approximated dynamics that arises when developing the potential in series of Legendre functions and truncating the series to the second harmonics is studied. Working in the reduced problem, the existence of equilibria in Lagrangian form are studied, in analogy with classic results on the topic. In this way, the classical results on equilibria of the three-body problem, as well as other results by different authors that use more conventional techniques for the case of rigid bodies, are generalized. The rotational Poisson dynamics of a spacecraft located at a Lagrangian equilibrium and the study of the nonlinear stability of some important equilibria are considered. The analysis is done in vectorial form avoiding the use of canonical variables and the tedious expressions associated with them.     

Second Fundamental Model of Resonance with Asymmetric Equilibria

We study the position and the stability of the equilibria for a generic Hamiltonian function developed up to the second harmonic and depending on two parameters; we describe the topology of the phase space for fixed values of these parameters. We show that for some values of the parameters asymmetric equilibria (unstable or stable) may appear. We deduce the conditions of capture into first order resonances for slowly drifting systems. We apply this model to the restricted three-body problem.     

A symplectic mapping for the synchronous spin-orbit problem

We derive a symplectic mapping model based on Hadjidemetriou’s method for the synchronous spin-orbit problem with and without the additional precession of the nodes. The mapping is derived from the averaged potential of the spin-orbit dynamical model and includes the main spin-orbit interactions, i.e. the non-zero obliquity and wobble motion of the rotating body. In addition the orbit of the perturbing body allows non-zero inclination and eccentricity. To obtain the equilibrium configuration we calculate the position and stability of the fixed points in the 1:1 spin-orbit resonance and relate them to the equilibria of the continuous system. We use the mapping equations to investigate the long-term stability close to the fixed point solutions of the mapping. We also apply the mapping method to the case of the moon Titan and validate the mapping approach by means of numerical integrations. The mapping model reproduces all the characteristics of Deprit’s model of free rotation as well as the dynamical features of Henrard’s averaged model of spin-orbit interaction with great precision.     

Low-thrust propulsion in a coplanar circular restricted four body problem

This paper formulates a circular restricted four body problem (CRFBP), where the three primaries are set in the stable Lagrangian equilateral triangle configuration and the fourth body is massless. The analysis of this autonomous coplanar CRFBP is undertaken, which identifies eight natural equilibria; four of which are close to the smaller body, two stable and two unstable, when considering the primaries to be the Sun and two smaller bodies of the Solar System. Following this, the model incorporates ‘near term’ low-thrust propulsion capabilities to generate surfaces of artificial equilibrium points close to the smaller primary, both in and out of the plane containing the celestial bodies. A stability analysis of these points is carried out and a stable subset of them is identified. Throughout the analysis the Sun-Jupiter-asteroid-spacecraft system is used, for conceivable masses of a hypothetical asteroid set at the libration point L ₄. It is shown that eight bounded orbits exist, which can be maintained with a constant thrust less than 1.5 × 10⁻⁴ N for a 1000 kg spacecraft. This illustrates that, by exploiting low-thrust technologies, it would be possible to maintain an observation point more than 66% closer to the asteroid than that of a stable natural equilibrium point. The analysis then focusses on a major Jupiter Trojan: the (624) Hektor asteroid. The thrust required to enable close asteroid observation is determined in the simplified CRFBP model. Finally, a numerical simulation of the real Sun-Jupiter-(624) Hektor-spacecraft is undertaken, which tests the validity of the stability analysis of the simplified model.     

Magnetohydrodynamic equilibrium and stability of pre-flare loops

The equilibrium and stability of a loop in which energy storage occurs prior to a solar flare is discussed. Working on the hypothesis, that the onset of the flare begins only after sufficient magnetic energy has been stored in the loop typical values of parameters which describe the equilibrium are found for a magnetic field with a constant twist. The stability of this configuration is examined next and it is shown that for the force-free case, the structure is always unstable to kinking for any degree of twist. However, a slight deviation from the force-free configuration, through the presence of a small positive transverse pressure gradient, can stabilize the loops for moderate degrees of twist. The range of wave-numbers for which instability occurs and the maximum growth rates are also presented. Lastly, it is shown that the pressure gradients required to stabilize a pre-flare loop do not lead to conflict with observations.