The Earth-Moon tidal force function

The concept “the tidal force function of the Earth-Moon system” is introduced and its exact determination based on the Stokes constants (harmonic coefficients) in the external gravitational potential of both bodies is outlined. The exact determination of the torque due to the Moon exerted on the Earth may be performed in terms of the Stokes constants of both bodies and the mutual position of both ellipsoids of inertia.     

Tidal evolution of close binary asteroid systems

We provide a generalized discussion of tidal evolution to arbitrary order in the expansion of the gravitational potential between two spherical bodies of any mass ratio. To accurately reproduce the tidal evolution of a system at separations less than 5 times the radius of the larger primary component, the tidal potential due to the presence of a smaller secondary component is expanded in terms of Legendre polynomials to arbitrary order rather than truncated at leading order as is typically done in studies of well-separated system like the Earth and Moon. The equations of tidal evolution including tidal torques, the changes in spin rates of the components, and the change in semimajor axis (orbital separation) are then derived for binary asteroid systems with circular and equatorial mutual orbits. Accounting for higher-order terms in the tidal potential serves to speed up the tidal evolution of the system leading to underestimates in the time rates of change of the spin rates, semimajor axis, and mean motion in the mutual orbit if such corrections are ignored. Special attention is given to the effect of close orbits on the calculation of material properties of the components, in terms of the rigidity and tidal dissipation function, based on the tidal evolution of the system. It is found that accurate determinations of the physical parameters of the system, e.g., densities, sizes, and current separation, are typically more important than accounting for higher-order terms in the potential when calculating material properties. In the scope of the long-term tidal evolution of the semimajor axis and the component spin rates, correcting for close orbits is a small effect, but for an instantaneous rate of change in spin rate, semimajor axis, or mean motion, the close-orbit correction can be on the order of tens of percent. This work has possible implications for the determination of the Roche limit and for spin-state alteration during close flybys.     

Mutual gravitational potential,force, and torque of a homogeneous polyhedron and an extended body: an application to binary asteroids

Binary systems are quite common within the populations of near-Earth asteroids, main-belt asteroids, and Kuiper belt asteroids. The dynamics of binary systems, which can be modeled as the full two-body problem, is a fundamental problem for their evolution and the design of relevant space missions. This paper proposes a new shape-based model for the mutual gravitational potential of binary asteroids, differing from prior approaches such as inertia integrals, spherical harmonics, or symmetric trace-free tensors. One asteroid is modeled as a homogeneous polyhedron, while the other is modeled as an extended rigid body with arbitrary mass distribution. Since the potential of the polyhedron is precisely described in a closed form, the mutual gravitational potential can be formulated as a volume integral over the extended body. By using Taylor expansion, the mutual potential is then derived in terms of inertia integrals of the extended body, derivatives of the polyhedron’s potential, and the relative location and orientation between the two bodies. The gravitational forces and torques acting on the two bodies described in the body-fixed frame of the polyhedron are derived in the form of a second-order expansion. The gravitational model is then used to simulate the evolution of the binary asteroid (66391) 1999 KW4, and compared with previous results in the literature.     

Lie group variational integrators for the full body problem in orbital mechanics

Equations of motion, referred to as full body models, are developed to describe the dynamics of rigid bodies acting under their mutual gravitational potential. Continuous equations of motion and discrete equations of motion are derived using Hamilton’s principle. These equations are expressed in an inertial frame and in relative coordinates. The discrete equations of motion, referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method for simulating full body dynamics in orbital mechanics; they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long time periods. They are also efficient in only requiring a single evaluation of the gravity forces and moments per time step. The Lie group variational integrator also preserves the group structure without the use of local charts, reprojection, or constraints. Computational results are given for the dynamics of two rigid dumbbell bodies acting under their mutual gravity; these computational results demonstrate the superiority of the Lie group variational integrator compared with integrators that are not symplectic or do not preserve the Lie group structure.     

Mutual gravitational potential ofN solid bodies

The mutual gravitational potential ofN solid bodies is expanded without approximation in terms of harmonic coefficients of each body. As an application the Euler dynamical equations for the motion of the axis of figure of the rigid Earth are integrated analytically by the method of variation of parameters.     

Reduction,relative equilibria and potential in the two rigid bodies problem

In this paper the problem of two, and thus, after a generalization, of an arbitrary finite number, of rigid bodies is considered. We show that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system possesses natural symmetries. Using them we shown how to perform reduction of the number of degrees of freedom. We prove that on every stage of this process equations of motion are Hamiltonian and we give explicite form corresponding of non-canonical Poisson bracket. We also discuss practical consequences of the reduction. We prove the existence of 36 non-Lagrangean relative equilibria for two generic rigid bodies. Finally, we demonstrate that our approach allows to simplify the general form of the mutual potential of two rigid bodies.     

Lie-series for orbital elements: I. The planar case

Lie-integration is one of the most efficient algorithms for numerical integration of ordinary differential equations if high precision is needed for longer terms. The method is based on the computation of the Taylor coefficients of the solution as a set of recurrence relations. In this paper, we present these recurrence formulae for orbital elements and other integrals of motion for the planar $N$ -body problem. We show that if the reference frame is fixed to one of the bodies—for instance to the Sun in the case of the Solar System—the higher order coefficients for all orbital elements and integrals of motion depend only on the mutual terms corresponding to the orbiting bodies.     

Simulation of the full two rigid body problem using polyhedral mutual potential and potential derivatives approach

Herein we investigate the coupled orbital and rotational dynamics of two rigid bodies modelled as polyhedra, under the influence of their mutual gravitational potential. The bodies may possess any arbitrary shape and mass distribution. A method of calculating the mutual potential’s derivatives with respect to relative position and attitude is derived. Relative equations of motion for the two body system are presented and an implementation of the equations of motion with the potential gradients approach is described. Results obtained with this dynamic simulation software package are presented for multiple cases to validate the approach and illustrate its utility. This simulation capability is useful both for addressing questions in dynamical astronomy and for enabling spacecraft missions to binary asteroid systems.     

Resonances and bifurcations in axisymmetric scale-free potentials

We investigate an analytical treatment of bifurcations of families of resonant 'thin' tubes in axisymmetric galactic potentials. We verify that the most relevant bifurcations are due to the (1:1) resonance producing the 'inclined' orbits through two different mechanisms: from the disc orbit and from the 'thin' tube associated with the vertical oscillation. The closest resonances occurring after these are the (4:3) resonance in the oblate case and the (2:1) resonance in the prolate case. The (1:1) resonances are treated in a straightforward way using a second-order truncated normal form. The higher order resonances are instead cumbersome to investigate, because the normal form has to be truncated to a high degree and the number of terms grows very rapidly. We therefore adopt a further simplification giving analytical formulae for the values of the parameters at which bifurcations ensue and compare them with selected numerical results. Thanks to the asymptotic nature of the series involved, the predictions are reliable well beyond the convergence radius of the original series.     

On the Dynamics of a Deformable Satellite in the Gravitational Field of a Spherical Rigid Body

In this paper, a model is developed for the dynamics of a system of two bodies whose material points are under the influence of a central gravitational force. One of the bodies is assumed to be rigid and spherically symmetric, while the other is assumed to be deformable. To develop a tractable model for the system, the deformable body is modeled using Cohen and Muncaster's theory of a pseudo-rigid body. The resulting model of the system has several of the features, such as angular momentum conservation, exhibited by more restrictive models. We also show how the self-gravitation of the deformable body can be accommodated using appropriate constitutive equations for a force tensor. This enables our model to subsume many existing models of ellipsoidal figures of equilibrium. After the model and its conservations have been discussed, attention is restricted to steady motions of the system. Several results, which generalize recent works on rigid satellites, are established for these motions. For a specific choice of constitutive equations for the pseudo-rigid body, we determine the steady motions with the aid of a numerical continuation method. These results can also be considered as generalizations of earlier works on Roche's ellipsoids of equilibrium.     

On the motion of three rigid bodies: Central configurations

In the present paper, the motion of three rigid bodies is considered. With a set of new variables, and the 10 first integrals of the motion, the problem is reduced to a system of order 25 and one quadrature. The plane motions are characterized, and finally, an equation for the existence of central configurations (in particular, Lagrangian and Eulerian solutions) has been found. Besides, the case of three axisymmetric ellipsoids is studied.     

On the equilibrium solutions in the circular planar restricted three rigid bodies problem

In this paper, the restricted problem of three rigid bodies under central forces is considered, and the collinear and triangular equilibrium solutions are obtained. Finally, an application to the case of axisymmetric ellipsoids is made.     

The relative motion of two spheroidal rigid bodies

We consider two spheroidal rigid bodies of comparable size constituting the components of an isolated binary system. We assume that (1) the bodies are homogeneous oblate ellipsoids of revolution, and (2) the meridional eccentricities of both components are small parameters.We obtain seven nonlinear differential equations governing simultaneously the relative motion of the two centroids and the rotational motion of each set of body axes. We seek solutions to these equations in the form of infinite series in the two meridional eccentricities.In the zero-order approximation (i. e., when the meridional eccentricities are neglected), the equations of motion separate into two independent subsystems. In this instance, the relative motion of the centroids is taken as a Kepler elliptic orbit of small eccentricity, whereas for each set of body axes we choose a composite motion consisting of a regular precession about an inertial axis and a uniform rotation about a body axis.The first part of the paper deals with the representation of the total potential energy of the binary system as an infinite series of the meridional eccentricities. For this purpose, we had to (1) derive a formula for representing the directional derivative of a solid harmonic as a combination of lower order harmonics, and (2) obtain the general term of a biaxial harmonic as a polynomial in the angular variables.In the second part, we expound a recurrent procedure whereby the approximations of various orders can be determined in terms of lower-order approximations. The rotational motion gives rise to linear differential equations with constant coefficients. In dealing with the translational motion, differential equations of the Hill type are encountered and are solved by means of power series in the orbital eccentricity. We give explicit solutions for the first-order approximation alone and identify critical values of the parameters which cause the motion to become unstable.The generality of the approach is tantamount to studying the evolution and asymptotic stability of the motion.Research performed under NASA Contract NAS5-11123.     

Planetary theories with the aid of the expansions of elliptic functions

For coplanar circular orbits, the mutual perturbations between two bodies can be expressed in term of the argument of Jacobian elliptic functions instead of the difference of the mean longitudes. For a given pair of planets, such a change of time variable improves the convergence of the developments. At the first order of planetary masses an integration of Lagrange's equations for the osculating elements is performed. When compared to classical developments the results are reduced by an important factor. The method is then extended to the mutual perturbations of Jupiter and Saturn, at any order of planetary masses, either with Fourier series with two arguments, or with one argument solely, taking advantage of the close commensurability of the mean motions.     

Central configurations of four bodies with an axis of symmetry

A complete solution is given for a symmetric case of the problem of the planar central configurations of four bodies, when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry. The positions of the bodies on the axis of symmetry are described by angle coordinates with respect to the outside bodies. The solution is such, that giving the angle coordinates, the masses for which the given configuration is a central configuration, can be computed from simple analytical expressions of the angles. The central configurations can be described as one-parameter families, and these are discussed in detail in one convex and two concave cases. The derived formulae represent exact analytical solutions of the four-body problem.     

The general motion of three mutually-interacting,axisymmetric spherical rigid bodies

In this paper, the translational-rotational motions of an axisymmetric rigid body and two spherical rigid bodies under the influence of their mutual gravitational attraction are considered. The equations of motion in the canonical elements of Delaunay-Andoyer are obtained. The elements of motion in the zero and first approximations can be determined.     

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)     

The two rigid body interaction using angular momentum theory formulae

This work presents an elegant formalism to model the evolution of the full two rigid body problem. The equations of motion, given in a Cartesian coordinate system, are expressed in terms of spherical harmonics and Wigner D-matrices. The algorithm benefits from the numerous recurrence relations satisfied by these functions allowing a fast evaluation of the mutual potential. Moreover, forces and torques are straightforwardly obtained by application of ladder operators taken from the angular momentum theory and commonly used in quantum mechanics. A numerical implementation of this algorithm is made. Tests show that the present code is significantly faster than those currently available in literature.     

The motion of rotating spheroidal bodies in general relativity

The motion of two rotating spheroidal bodies, constituting the components of a binary system in a weak gravitational field, has been considered up to terms of the second order in the small parameterV/c, whereV denotes the velocity of the bodies andc is the velocity of light.The following simplifying assumptions, consistent with a problem of astronomical interest, have been made: (1) the dimensions of the bodies are small compared with their mutual distance; (2) the bodies consist of matter in the fluid state with internal hydrostatic pressure and their oblateness is due to their own rotation; (3) there exist axial symmetry about the axis of rotation and symmetry with respect to the equatorial plane, the same symmetry properties apply to mass densities and stress tensors.The Fock-Papapetrou method was used to ascertain those terms in the equations of motion which are due to the rotation and to the oblateness of each component. Approximate solutions to the Poisson and wave equations were obtained to express the potential and retarded potential at large distances from the bodies generating them. The explicit evaluation of certain integrals has necessitated the use of the Laplace-Clairaut theory for the equibrium configuration of rotating bodies. The final expressions require the knowledge of the mass density as a function of the mean radius of the equipotential surfaces.As an interpretation of the results, the Lagrangian perturbation equations were employed to evaluate the secular motion of the nodal line for the relative orbit of the two components. The results constitute a generalization of Fock's work and furnish the contribution of the mass distribution to the rotation effect of general relativity.     

On the stability parameters of periodic solutions

The stability parametersa, b, c, d of plane symmetric periodic solutions of non-integrable dynamical systems of two degrees of freedom are obtained in terms of their initial states of motion and elements of their variational matrics. Explicit formulae are given in the cases of the Störmer problem and the restricted problem of three bodies.