A new simple method for the analytical solution of Kepler's equation

A new simple method for the closed-form solution of nonlinear algebraic and transcendental equations through integral formulae is proposed. This method is applied to the solution of the famous Kepler equation in the two-body problem for elliptic orbits. The resulting formulae are quite elementary and, beyond their analytical interest, they can also provide quite accurate numerical results by using Gausstype quadrature rules.     

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.     

Increased accuracy of computations in the main satellite problem through linearization methods

The set of canonical redundant variables previously introduced by the first author is derived from Cartesian coordinates in a simplified form which allows the reduction of the Kepler problem to four harmonic oscillators with unit frequency. The coordinates are defined to be the direction cosines of the position of the particle along with the inverse of its distance. True anomaly is the new independent variable. The behavior of this new transformation is studied when applied to the numerical integrations of the main problem in satellite theory. In particular, computation time and accuracy of orbits in the new variables are compared with those in K-S and Cartesian variables. It is noteworthy that for high eccentricities the new variables require the least computation time for comparable accuracy, regardless of the integration scheme.     

Solution algorithm of a quasi-Lambert’s problem with fixed flight-direction angle constraint

A two-point boundary value problem of the Kepler orbit similar to Lambert’s problem is proposed. The problem is to find a Kepler orbit that will travel through the initial and final points in a specified flight time given the radial distances of the two points and the flight-direction angle at the initial point. The Kepler orbits that meet the geometric constraints are parameterized via the universal variable z introduced by Bate. The formula for flight time of the orbits is derived. The admissible interval of the universal variable and the variation pattern of the flight time are explored intensively. A numerical iteration algorithm based on the analytical results is presented to solve the problem. A large number of randomly generated examples are used to test the reliability and efficiency of the algorithm.     

The constrained normal form algorithm

This paper describes an algorithm which brings a regularizable polynomial perturbation of a three degree of freedom Kepler problem into a normal form which Poisson commutes with the Kepler Hamiltonian. We illustrate the alogrithm with an example: the quadratic Zeeman effect. For other applications of this algorithm see [1],[4], and [5]. The authors have written a program in MAPLE which implements the constrained normal form.     

Some investigations into the atmospheric drag problem

This paper calls into question the validity of the well-known formulae for the perturbations in the Keplerian elements, over one revolution of an orbit, for the motion of a drag-perturbed artificial satellite. These formulae are derived from Gauss's form of the planetary equations, by averaging over a single revolution of the orbit, and using the eccentric anomaly as the independent variable.It is shown that for light balloon-type satellites in near-circular orbits neither the eccentric anomaly nor the true longitude is a suitable choice of independent variable for the averaging procedure. Under these circumstances, it would seem that simple formulae for the variations in the elements cannot be derived from Gauss's equations.     

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.     

Radial,transverse and normal satellite position perturbations due to the geopotential

Perturbations in the position of a satellite due to the Earth's gravitational effects are presented. The perturbations are given in the radial, transverse (or alongtrack) and normal (or cross-track) components. The solution is obtained by projecting the Kepler element perturbations obtained by Kaula [Kaula, 1966] into each of the three components. The resulting perturbations are presented in a form analogous to the form of Kaula's solution which facilitates implementation and interpretation.     

The general Kepler equation and its solutions

My father K. Stumpff (1947, 1949, 1951, 1959, 1962) developed a transcendental equation which replaces the original Kepler equation but is valid for all types of orbits. Other advantages over the classical methods are: a) the independent arguments of the equation follow from the vectors of position and velocity at any instant T_o, where T_o is not necessarily the perihelion time; b) an explicit knowledge of the classical orbital elements is not required; c) transformations of coordinate systems are avoided. The present paper discusses the properties of the general Kepler equation in a wide range of its independent arguments, and it is shown that analytic solutions, existing in special cases, can be used for the numerical solution of general cases. The theory is generalized insofar as it now can handle not only attracting forces but also repulsive ones. As a result of this investigation, FORTRAN subroutines were developed which can be used in connection with any two-body problem for the computation of position and velocity as function of time along any unperturbed orbit.     

Lie groups of motor integrals of generalized Kepler motion

The singularity of the Kepler motion can be eliminated by means of the spinor regularization. The extensive integrals of the Kepler motion form a Lie algebra with respect to the Poisson bracket operation. Mayer-Gürr has shown that in the caseH>0 the corresponding Lie group is the multiplicative group of all real 4×4 unimodular matrices SL(4,R). Kustaanheimo has posed the problem of the identification of the corresponding Lie groups in the elliptic and parabolic cases. We solve this problem here and use the opportunity to introduce the concept of the Clifford algebra which is needed in our solution.     

The extended Lissajous–Levi-Civita transformation

Action-angle variables for the Levi-Civita regularized planar Kepler problem were introduced independently first by Chenciner and then by Deprit and Williams. The latter used explicitly the so-called Lissajous variables. When applied to the transformed Keplerian Hamiltonian, the Lissajous transformation encounters the difficulty of being defined in terms of the constant frequency parameter, whereas the Kepler problem transformed into a harmonic oscillator involves the frequency as a function of an energy-related canonical variable. A simple canonical transformation is proposed as a remedy for this inconvenience. The problem is circumvented by adding to the physical time a correcting term, which occurs to be a generalized Kepler’s equation. Unlike previous versions, the transformation is symplectic in the extended phase space and allows the treatment of time-dependent perturbations. The relation of the extended Lissajous–Levi-Civita variables to the classical Delaunay angles and actions is given, and it turns out to be a straightforward generalization of the results published by Deprit and Williams.     

Exponential instability of collision orbit in the anisotropic Kepler problem

The straight-line collision solution in the anisotropic Kepler problem is extended to a periodic solution by means of Sundman's analytic continuation. It is shown that this collision periodic solution is always exponentially unstable.     

Periodic Solutions for any Planar Symmetric Perturbation of the Kepler Problem

We consider perturbations of the Kepler problem that are symmetric with respect to the origin and admit a first integral of motion which is also symmetric with respect to the origin. It has been proved that each circular solution of the unperturbed problem gives rise to a periodic solution of the perturbed system.     

Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury

In this paper, we consider the elliptic collinear solutions of the classical n-body problem, where the n bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler–Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler–Moulton collinear solution of n-bodies splits into \((n-1)\) independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler 2-body problem at Kepler elliptic orbit, and each of the other \((n-2)\) systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3-body problem whose mass parameter is modified. Then the linear stability of such a solution in the n-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-body problems, which for example then can be further understood using numerical results of Martínez et al. on 3-body Euler solutions in 2004–2006. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler–Moulton solution of the 4-body problem with two small masses in the middle.     

The Kepler problem and geodesic flows in spaces of constant curvature

The main result of this paper is a theorem on the trajectory equivalence of phase flows on isoenergetic surfaces with a positive energy level in the Kepler problem and perturbed kepler problem. The following two facts are crucial for proving it: firstly, an isomorphism of the phase flow on an isoenergetic surface in the Kepler problem and the geodesic flow in a constant curvature space. The isomorphism is studied in detail. In particular, all the integrals of the Kepler problem are obtained proceeding from the group-theory considerations. The second fact is a generalization of the theorem on structural stability of Anosov flows onto non-compact manifolds.     

Integration of the elliptic restricted three-body problem with Lie series

The method of Lie series is used to construct a solution for the elliptic restricted three body problem. In a synodic pulsating coordinate system, the Lie operator for the motion of the third infinitesimal body is derived as function of coordinates, velocities and true anomaly of the primaries. The terms of the Lie series for the solution are then calculated with recurrence formulae which enable a rapid successive calculation of any desired number of terms. This procedure gives a very useful analytical form for the series and allows a quick calculation of the orbit.The project is supported by the Austrian Fonds zur Förderung der wissénschaftlichen Forschung under Project No. 4471.     

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.     

The two-body problem with drag and radiation pressure

The Kepler problem including radiation pressure and drag is treated. The equation of the orbit is derived and the scalar and vector integrals of motion are obtained by direct operation on the vector form of the equation of motion.     

Adiabatic invariants for the nonconservative Kepler's problem

This paper considers adiabatic invariants for the classical Kepler problem with resisting forces. The analysis is based on the theory of integrating factors and theory of adiabatic invariants in the Krylov-Bogoliubov-Mitropolski variables. The adiabatic invariants are series with respect to a small parameter. Also, for every particular case of nonconservative forces, it is shown that, with a complete set of adiabatic invariants, an approximate solution of the problem can be obtained. Four problems are analyzed in detail where approximate solutions are compared with numerical.