Numerical stabilization of Keplerian motion

The time transformation dt/ds=r is studied in detail and numerically stablized differential equations are obtained for =1,2, and 3/2. The case =1 corresponds to Baumgarte's results.     

High order transfer maps for perturbed Keplerian motion

The paper presents a new semi-analytical technique for the propagation of near-Earth satellite motion. The approach uses differential algebra techniques to compute the high order expansion of the solution of the system’s ordinary differential equation for one orbital revolution, referred to as the transfer map. Once computed, a single high order transfer map (HOTM) can be reused to map an initial condition, or a set of initial conditions, forward in time for many revolutions. The only limiting factor is that the mapped objects must stay close to the reference orbit such that they remain within the region of validity of the HOTM. The performance of the method is assessed through a set of test cases in which both autonomous and non-autonomous perturbations are considered, including the case of continuously propelled trajectories.     

Numerical stabilization of the differential equations of Keplerian motion

A stabilization of the classical equations of two-body motion is offered. It is characterized by the use of the regularizing independent variable (eccentric anomaly) and by the addition of a control-term to the differential equations. This method is related to the KS-theory (Stiefel, 1970) which performed for the first time a stabilization of the Kepler motion. But in contrast to the KS-theory our method does not transform the coordinates of the particle. As far as the theory of stability and the numerical experiments are concerned we restrict ourselves to thepure Kepler motion. But, of course, the stabilizing devices will also improve the accuracy of the computation of perturbed orbits. We list, therefore, also the equations of the perturbed motion.     

On the Milankovitch orbital elements for perturbed Keplerian motion

We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace–Runge–Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution.     

Euler solution of a kinematic equation for nearly-parabolic Keplerian motion

In celestial mechanics the kinematic equation connecting the time and position in orbit is important. This equation is investigated in detail, but the case of nearly-parabolic motion remains little studied. The universal equations were derived by Euler, but he did not investigate then in detail. We present the solution in the form of series with respect to the small Euler parameter, with coefficients depending on time, and we solve the problem on determining the convergence domain of this series that occurs to be more complicated problem.     

Expansion of the principal functions of Keplerian motion using complex variables

A special system of canonical variables is considered. An algorithm for expanding the principal functions of Keplerian motion in new elements is presented. The advantage of the proposed system is a relatively small number of terms in the classical expansions of the unperturbed two-body problem. A method for expanding the time derivatives of the rectangular coordinates is proposed. Some estimates of the number of terms in the presented expansions have been obtained through numerical experiments.     

On the differential geometry of trajectory deviation with particular emphasis on Keplerian motion

The Pontryagin/Lawden scalar product of the deviation phase vector and the adjoint phase vector may be identified with Lagrange's reciprocal formula for two variant motions if the acceleration field is conservative. Hence for two slightly different trajectories with common end-points, the terminal velocity differences must have equal scalar product with Lawden's primer vectors. The final velocity difference is orthogonal to the constant time of flight locus for isoenergetic motions from a common initial point. A Keplerian trajectory behaves like a rigid curve as far as radial deviation is concerned in the case of a small change of direction of the initial velocity vector.     

Test of the extrapolation method for the numerical integration of the Keplerian motion

Two extrapolation methods (Gragg's method and GBS method) were tested for the numerical integration of the Keplerian motion. The tests cleared that the Gragg's method is better for the orbit computation and is 1) easy to control, 2) highly accurate, 3) tough, 4) economic, and 5) very flexible against the close encounter.     

Universal Keplerian state transition matrix

A completely general method for computing the Keplerian state transition matrix in terms of Goodyear's universal variables is presented. This includes a new scheme for solving Kepler's problem which is a necessary first step to computing the transition matrix. The Kepler problem is solved in terms of a new independent variable requiring the evaluation of only one transcendental function. Furthermore, this transcendental function may be conveniently evaluated by means of a Gaussian continued fraction.This work was supported at The Charles Stark Draper Laboratory, Inc., by the National Aeronautics and Space Administration under Contract NAS9-16023.     

Metric spaces of Keplerian orbits

Several metric spaces of Keplerian orbits and a set of their most important subspaces, as well as a factor space (not distinguishing orbits with the same longitudes of nodes and pericentres) are constructed. Topological and metric properties of them are established. Simple formulae to calculate the distance are deduced. Applications to a number of problems of Celestial Mechanics are discussed.     

Unstable modes of Keplerian discs

We derive the softened analogue of the Laplace–Lagrange secular theory of planetary motion, and use it to show that a small fraction of counter-rotating stars is all it takes for a hot Keplerian disc to grow unstable lopsided modes.     

Statistical mechanics of Keplerian orbits

Perturbations and gravitational encounters are incorporated into the statistical theory of Keplerian orbits. The birth of planets is discussed as an application of the theory. It turns out to be a consequence of the combined action of collisions, differential rotation and gravitational interaction.     

Statistical mechanics of Keplerian orbits

A statistical theory of Keplerian orbits is constructed for a system of particles, which are subject to partially elastic collisions. If the elasticity decreases with collisional velocity, the system shows an increased tendency to form condensations. Near the central body they are concentric rings, which are separated by gaps void of matter. At larger distances outside the Roche limit, the condensations probably form larger bodies. An application to Saturn's rings suggests that at least rings A and C would consist of separate ringlets.     

Ptolemaic Transformation in Keplerian Problem

We propose the Ptolemaic transformation: a canonical change of variables reducing the Keplerian motion to the form of a perturbed Hamiltonian problem. As a solution of the unperturbed case, the Ptolemaic variables define an intermediary orbit, accurate up to the first power of eccentricity, like in the kinematic model of Claudius Ptolemy. In order to normalize the perturbed Hamiltonian we modify the recurrent Lie series algorithm of HoriuuMersman. The modified algorithm accounts for the loss of a term's order during the evaluation of a Poisson bracket, and thus can be also applied in resonance problems. The normalized Hamiltonian consists of a single Keplerian term; the mean Ptolemaic variables occur to be trivial, linear functions of the Delaunay actions and angles. The generator of the transformation may serve to expand various functions in Poisson series of eccentricity and mean anomaly.     

Starn's rings and bimodality of Keplerian systems

The correction terms which are introduced by non-zero size of the particles into the mechanics of Keplerian systems can be replaced by relatively simple approximations which agree with computer simulations. The theory of finite particles confirms the bimodality of collisional systems which has previously been discussed in terms of the mass-point approximation. In Saturn's rings the ringlets correspond to the degenerate mode while the matter which fills the gaps is in the non-degenerate state. The predicted volume density of the ringlets (the fraction of space which is occupied by the particles), 0.2, is much higher than the conventional value which follows from the theory of mutual shadowing. Therefore, the opposition effect of Saturn's rings must originate in the particles themselves. The transition from one mode to the other which is needed to create a dense ring in a cloud of small particles follows from the growth of mass in the central body. This may be a recently-formed planet; but, more probably, the transition occurs in a loose pre-planetary disc.     

Statistical mechanics of Keplerian orbits

Theoretical predictions agree with computer simulations at least for those collisional systems in which the restitution coefficient is independent of impact velocity. An uncertainty principle for the orbits restricts the validity of the theory and its predictions. Discussion of the whole theory and of computer simulations shows that a velocity-dependent restitution coefficient provides the only astronomically interesting applications of the collisional processes. The Saturnian and Uranian ring systems correspond very well to theoretical expectations if the restitution coefficient is of this type.     

How to assemble a Keplerian processor

This article describes how a set of computer programs has been constructed to perform the literal series expansions of the two-body problem. The different steps of the approach are outlined, from the basic generation of fundamental Kepler functions with the aid of Bessel series to the construction of derived Kepler functions by elementary Poisson series operations. The different tests and checks which have been made are also described. The most extensive test application of the package of programs, the expansion of the lunar disturbing function, is included at the end of the article.This paper represents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under NASA Contract NAS 7-100     

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.     

Non-linear bending waves in Keplerian accretion discs

The non-linear dynamics of a warped accretion disc is investigated in the important case of a thin Keplerian disc with negligible viscosity and self-gravity. A one-dimensional evolutionary equation is formally derived that describes the primary non-linear and dispersive effects on propagating bending waves other than parametric instabilities. It has the form of a derivative non-linear Schrödinger (DNLS) equation with coefficients that are obtained explicitly for a particular model of a disc. The properties of this equation are analysed in some detail and illustrative numerical solutions are presented. The non-linear and dispersive effects both depend on the compressibility of the gas through its adiabatic index Γ. In the physically realistic case Γ < 3, non-linearity does not lead to the steepening of bending waves but instead enhances their linear dispersion. In the opposite case Γ > 3, non-linearity leads to wave steepening and solitary waves are supported. The effects of a small effective viscosity, which may suppress parametric instabilities, are also considered. This analysis may provide a useful point of comparison between theory and numerical simulations of warped accretion discs.