Noncanonical Parametrization of Poisson Brackets in Celestial Mechanics

The formulas for the Poisson bracket of a perturbed two-body problem and a perturbed planetary problem are found in different systems of Keplerian elements. As with canonical parametrization, the Poisson bracket is equal to a linear combination of partial brackets, but it contains coefficients depending on semimajor axis, eccentricity, and inclination. A simple relation between the Poisson brackets and matrices of coefficients of Lagrange-type equations determining the variations of osculating elements is derived. The Poisson bracket of D'Alembertian functions is proved to be a D'Alembertian one by itself.     

Exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral harmonics

A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations.     

Coordinates for perturbed keplerian systems with axial symmetry

A coordinate system is defined on the phase space of a perturbed Keplerian system after the mean anomaly has been averaged out, for the purpose of explaining how eliminating the longitude of the ascending node reduces the orbital space to a two-dimensional sphere in case the system admits an axial symmetry. Concomitantly, on the submanifold of direct osculating ellipses, the CDM variables are replaced by functions which form the basis of a Poisson algebra isomorphic to the Lie algebra so(3) of the rotation group SO(3); furthermore, in these variables, the doubly reduced phase flow appears like a rotation of the reduced phase space.     

The Expansion of the Hamiltonian of the Planetary Problem into the Poisson Series in All Keplerian Elements (Theory)

The study of the evolution of planetary systems, primarily of the Solar System, is one of the basic problems of celestial mechanics. The stability of motion of giant planets on cosmogonic time scales was established by numerical and analytical methods, but the question about the evolution of orbits of terrestrial planets and arbitrary solar-type planetary systems remained open. This work initiates a series of papers allowing one to advance in solving the problem of the evolution of the solar-type planetary systems on cosmogonic time scales by using powerful analytical tools. In the first paper of this series, we choose the optimum reference system and obtain the Poisson series expansion of the Hamiltonian of the problem in all Keplerian elements. We propose to use the integral representation of the corresponding coefficients or the Poisson processor means instead of conventionally addressing any possible special functions. This approach extremely simplifies the algorithm. The next paper of this series deals with the calculation of the expansion coefficients.     

Normal forms for the epicyclic approximations of the perturbed Kepler problem

We compute the normal forms for the Hamiltonian leading to the epicyclic approximations of the (perturbed) Kepler problem in the plane. The Hamiltonian setting corresponds to the dynamics in the Hill synodic system where, by means of the tidal expansion of the potential, the equations of motion take the form of perturbed harmonic oscillators in a rotating frame. In the unperturbed, purely Keplerian case, the post-epicyclic solutions produced with the normal form coincide with those obtained from the expansion of the solution of the Kepler equation. In all cases where the perturbed problem can be cast in autonomous form, the solution is easily obtained as a perturbation series. The generalization to the spatial problem and/or the non-autonomous case is straightforward.     

Secular orbit-orbit resonance between two satellites with non-zero masses

After the mean anomaly has been removed from the perturbations, the reduced Hamiltonian becomes a function over the Lie algebra determined by the infinitesimal generators associated with the dynamical symmetries of an unperturbed Keplerian system. The phase space being now the group SO(3), average motions consist of rotations, and the normalized Hamiltonian serves as a Morse function whose critical points determine the intrinsic topology of the perturbed system.     

Specialized celestial mechanics systems for symbolic manipulation

Construction and application of the current high accuracy analytical theories of motion of celestial bodies necessitates the development of specialized software for the implementation of analytical algorithms of celestial mechanics. This paper describes a typical software package of this kind. This package includes a universal Poisson processor for the rational functions of many variables, a tensorial processor for purposes of relativistic celestial mechanics, a Keplerian processor valid for the solutions of the two body problem in the form of a Poisson series, Taylor expansions in powers of time and closed expressions, and an analytical generator of celestial mechanics functions, facilitating the immediate implementation of the present analytical methods of celestial mechanics. The package is completed with a numerical-analytical interface designed, in particular, for the fast evaluation of the long Poisson series.     

About averaging procedures in the problem of asteroid motion

A new mathematically correct approach to construct an averaging procedure for the motion of a massless body around the central body perturbed by fully interacting planets is developed and the errors of the standard solution are discussed. The new technique allows to combine the advantages of the Hamiltonian representation with the usage of standard osculating elements in combination with all the standard expansions of the perturbing functions. The main idea is to introduce new additional variables conjugate to all the standard elements and to work in a corresponding super phase space. In this way, the number of variables is doubled at first, but one has to deal with only one Hamiltonian. The artificially introduced variables disappear from the final averaged equations as well as from the transformation formulae connecting the osculating and the mean elements.     

Explicit Symplectic Integrator for Highly Eccentric Orbits

An explicit symplectic integrator is constructed for perturbed elliptic orbits of an arbitrary eccentricity. The perturbation should be Hamiltonian, but it may depend on time explicitly. The main feature of the integrator is the use of KS variables in the ten-dimensional extended phase space. As an example of its application the motion of an Earth satellite under the action of the planet's oblateness and of lunar perturbations is studied. The results confirm the superiority of the method over a classical Wisdom–Holman algorithm in both accuracy and computation time. This revised version was published online in July 2006 with corrections to the Cover Date.     

Perturbations of the Kepler Problem in Global Coordinates

The usual action-angle variables for the Kepler Problem (the Delaunay variables) are not globally defined, leaving out some orbits (circular orbits or those lying on the xy-plane). Moreover they are trascendental functions of the physical variables, making it quite difficult to write the perturbed Hamiltonian. The way-out proposed here is to pass to a 8-dimensional rank-6 Poisson manifold, that is, to parametrize the state of the Kepler Problem with two 4-dimensional vectors mutually orthogonal and of equal norm. This revised version was published online in July 2006 with corrections to the Cover Date.     

Perturbation theory in rectangular coordinates

For treating the perturbed two-body problem in rectangular coordinates a new method is developed. The method is based on the reduction of the variational equations of the two-body problem with arbitrary elements to the Jordan system. The equations of perturbed motion rewritten in the quasi-Jordan form are subjected to a transformation excluding fast variables and leading to a system governing the long term evolution of motion. The method may be easily extended to the problem of the heliocentric motion of the major planets. For performing this method on computer it is suitable to use facilities of Poissonian and Keplerian processors.     

Gyldén systems: Rotation of pericenters

A canonical transformation in phase space and a rescaling of time are proposed to reduce a Keplerian system with a time-dependent Gaussian parameter to a perturbed Keplerian system with a constant Gaussian parameter. When the time variation is slow, the perturbation through second order in the reduced problem is conservative, and, as a result, the orbital space of the averaged system is a sphere on which the phase flow causes a differential rotation representing a circulation of the line of apsides. The flow presents two isolated singularities corresponding to circular orbits travelled respectively in the direct and in the retrograde sense, and a degenerate manifold of fixed points corresponding to the collision orbits. Normalization beyond order two does not break the degeneracy. Adiabatic invariants, which are conservative functions, may be computed from the normalized Hamiltonian evaluated here to the fourth order. Nonetheless so high an approximation gives little information because the normalizing Lie transformation depends explicitly on the time through mixed secular-periodic terms. As an application, an estimate is offered for the apsidal rotation that a second order time derivative in the mass of the sun would induce on planetary orbits. This suggests an observational method for determining the latter parameter for the solar wind, but the predicted motions are too slow for the current level of observational precision.     

Expansion of the Hamiltonian of the Two-Planetary Problem into the Poisson Series in All Elements: Application of the Poisson Series Processor

This paper is the third in a series of articles devoted to one of the basic problems of celestial mechanics: the study of the evolution of solar-type planetary systems. In the previous papers a brief review of the history and current state of the problem was given; the plan of the study was outlined; the Jacobi coordinates and the related osculating elements were introduced; the form of the Poisson expansion of the Hamiltonian in all elements was given; and the expansion coefficients for the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem were obtained (though with impure accuracy) by a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. In the present paper the expansion of the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem into the Poisson series in all elements is constructed with the help of the PSP Poisson series processor, which is capable of required accuracy.     

Ideal frames for perturbed Keplerian motions

Cartan's exterior calculus is used to refer a perturbed Keplerian motion to an ideal frame by means of either the Eulerian parameters or the Eulerian angles, in which case the equations are given a Hamiltonian form. The results are compared with the corresponding systems in the orbital and nodal frames.     

Generation of a secular system in the analytical theory for the motion of the moon

In order to generate an analytical theory of the motion of the Moon by considering planetary perturbations, a procedure of general planetary theory (GPT) is used. In this case, the Moon is considered as an addition planet to the eight principal planets. Therefore, according to the GPT procedure, the theory of the Moon’s orbital motion can be presented in the form of series with respect to the evolution of eccentric and oblique variables with quasi-periodic coefficients, which are the functions of mean longitudes for principal planets and the Moon. The relationship between evolution variables and the time is determined by a trigonometric solution for the independent secular system that describes the secular motion of a perigee and the Moon node by considering secular planetary inequalities. Principal planetary coordinates required for generating the theory of the motion of the Moon includes only Keplerian terms, the intermediate orbit, and the linear theory with respect to eccentricities and inclinations in the first order relative to the masses. All analytical calculations are performed by means of the specialized echeloned Poisson Series Processor EPSP.     

Explicit Symplectic Integrator for Rotating Satellites

An explicit symplectic integrator is constructed for the problem of a rotating planetary satellite on a Keplerian orbit. The spin vector is fixed perpendicularly to the orbital plane. The integrator is constructed according to the Wisdom-Holman approach: the Hamiltonian is separated in two parts so that one of them is multiplied by a small parameter. The parameter depends on the satellite’s shape or the eccentricity of its orbit. The leading part of the Hamiltonian for small eccentricity orbits is similar to the simple pendulum and hence integrable; the perturbation does not depend on angular momentum which implies a trivial ‘kick’ solution. In spite of the necessity to evaluate elliptic function at each step, the explicit symplectic integrator proves to be quite efficient. This revised version was published online in July 2006 with corrections to the Cover Date.     

Computing secular motion under slowly rotating quadratic perturbation

We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potential that can be approximated to sufficient accuracy by expanding it to second order in the coordinates. After averaging over time to obtain the secular Hamiltonian, we use angular momentum and eccentricity vectors as elements. The method of variation of constants then leads to a set of equations of motion that are simple and regular, thus allowing efficient numerical integration. Some possible applications are briefly described.     

Analytical treatment of the two-body problem with slowly varying mass

The present work is concerned with the two-body problem with varying mass in case of isotropic mass loss from both components of the binary systems. The law of mass variation used gives rise to a perturbed Keplerian problem depending on two small parameters. The problem is treated analytically in the Hamiltonian frame-work and the equations of motion are integrated using the Lie series developed and applied, separately by Delva (1984) and Hanslmeier (1984). A second order theory of the two bodies eject mass is constructed, returning the terms of the rate of change of mass up to second order in the small parameters of the problem.