Regularization of the circular restricted three-body problem using ‘similar’ coordinate systems

The regularization of a new problem, namely the three-body problem, using ‘similar’ coordinate system is proposed. For this purpose we use the relation of ‘similarity’, which has been introduced as an equivalence relation in a previous paper (see Roman in Astrophys. Space Sci. doi:, 2011). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function, we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to regularize the equations of motion. Explicit formulas are given for the regularization in the coordinate systems centered in the more massive and the less massive star of the binary system. The ‘similar’ polar angle’s definition is introduced, in order to analyze the regularization’s geometrical transformation. The effect of Levi-Civita’s transformation is described in a geometrical manner. Using the resulted regularized equations, we analyze and compare these canonical equations numerically, for the Earth-Moon binary system.     

Stabilized Kepler motion connected with analytic step adaptation

In the publication Baumgarte and Stiefel (1974a) a stabilization of the Keplerian motion was offered by making use of a manipulation of the Hamiltonian. By this stabilization technique the given HamiltonianH(p _i,q _i) is replaced by a new HamiltonianH ^*, which leads to Lyapunov-stable differential equations of motion.Whereas, in the quoted publication, the physical timet was used as the independent variable we now develop a generalization which allows to combine the stabilization with the introduction of a new independent variables. Such a fictitious times is popular for achieving an analytic step-size adaptation (Baumgarte and Stiefel, 1974c). Perturbations of Kepler motion are discussed.     

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.     

Time-varying transformations for Hill–Clohessy–Wiltshire solutions in elliptic orbits

The relative motion of chief and deputy satellites in close proximity with orbits of arbitrary eccentricity can be approximated by linearized time-periodic equations of motion. The linear time-invariant Hill–Clohessy–Wiltshire equations are typically derived from these equations by assuming the chief satellite is in a circular orbit. Two Lyapunov–Floquet transformations and an integral-preserving transformation are here presented which relate the linearized time-varying equations of relative motion to the Hill–Clohessy–Wiltshire equations in a one-to-one manner through time-varying coordinate transformations. These transformations allow the Hill–Clohessy–Wiltshire equations to describe the linearized relative motion for elliptic chief satellites.     

Accurate computation of highly eccentric satellite orbits

For computing highly eccentric (e0.9) Earth satellite orbits with special perturbation methods, a comparison is made between different schemes, namely the direct integration of the equations of motion in Cartesian coordinates, changes of the independent variable, use of a time element, stabilization and use of regular elements. A one-step and a multi-step integration are also compared.It is shown that stabilization and regularization procedures are very helpful for non or smoothly perturbed orbits. In practical cases for space research where all perturbations are considered, these procedures are no longer so efficient. The recommended method in these cases is a multi-step integration of the Cartesian coordinates with a change of the independent variable defining an analytical step size regulation. However, the use of a time element and a stabilization procedure for the equations of motion improves the accuracy, except when a small step size is chosen.     

Stabilization by modification of the Lagrangian

In order to reduce the error growth during a numerical integration, a method of stabilization, of the differential equations of the Keplerian motion is offered. It is characterized by the use of the eccentric anomaly as independent variable in such a way that the time transformation is given by a generalized Lagrange formalism. The control terms in the equations of motion obtained by this modified Lagrangian give immediately a completely Lyapunov-stable set of differential equations. In contrast to other publications, here the equation of time integration is modified by a control term which leads to an integral which defined the time element for the perturbed Keplerian motion.This paper was supported by the National Research Council and the National Aeronautics and Space Administration and also by the Deutsche Forschungsgemeinschaft. It was presented at the Flight Mechanics/Estimation Theory Symposium, Goddard Space Flight Center, Greenbelt, Md., April 15–16, 1975.     

On the motion of a mass point inside a rotating inhomogeneous ellipsoidal body

The plane motion of a mass point inside an inhomogeneous rotating ellipsoidal body with a homothetic density distribution is considered. The force function of the problem is expanded in terms of the ellipsoid's second eccentricities up to the fourth order, which are taken as small parameters. We derive an expression for the perturbing function and solve the equations of perturbed motion in orbital elements.     

Motion of artificial satellites in the set of Eulerian redundant parameters (III)

In this paper, the classical and generalized Sundman time transformations are used to establish new generating set of differential equations of motion in terms of the Eulerian redundant parameters. The implementation of this set on digital computers for the commonly used independent variables is developed once and for all. Motion prediction algorithms based on these equations are developed in a recursive manner for the motions in the Earth's gravitational field with axial symmetry whatever the number of the zonal harmonic terms may be. Applications for the two types of short and long term predictions are considered for the perturbed motion in the Earth's gravitational field with axial symmetry with zonal harmonic terms up to J ₃₆ . Numerical results proved the very high efficiency and flexibility of the developed equations.     

Theory of the motion of an artificial Earth satellite

An improved analytical solution is obtained for the motion of an artificial Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational model includes zonal harmonics throughJ ₄, and the atmospheric model assumes a nonrotating spherical power density function. The differential equations are developed through second order under the assumption that the second zonal harmonic and the drag coefficient are both first-order terms, while the remaining zonal harmonics are of second order.Canonical transformations and the method of averaging are used to obtain transformations of variables which significantly simplify the transformed differential equations. A solution for these transformed equations is found; and this solution, in conjunction with the transformations cited above, gives equations for computing the six osculating orbital elements which describe the orbital motion of the satellite. The solution is valid for all eccentricities greater than 0 and less than 0.1 and all inclinations not near 0^o or the critical inclination. Approximately ninety percent of the satellites currently in orbit satisfy all these restrictions.     

A Note on the Regularization of the Restricted Three-body Problem

The requirement that near a singular point of the equations of motion the power series expansions of the old variables in terms of the new ones start with second order terms leads to the transformation z = sin²1/2w related to that of THIELE -BURRAU . Using this new transformation, a derivation of the regularized equations of motion is given. The original as well as the regularized equations of motion are of interest, for example, for calculating the initial values of the orbital elements for SCHWARZSCHILD's periodic solutions (LEIMANIS and OLUND 1972).     

Stablization and time transformation

We suggest a simple stabilization technique to reduce the along-track error in the numerical Integration of the Lagrange's equations of motion. We also investigate the equations of motion of the two-body problem after applying the Sundman transformation dt = r^αds, both with and without introducing the energy integral. In both cases, we show how the stability of the equations varies with α and in the case with the energy integral, we show that every solution is a quasi-periodic function of s with two frequencies.     

A Simplified Kinetic Element Formulation for the Rotation of a Perturbed Mass-Asymmetric Rigid Body

Euler's equations, describing the rotation of an arbitrarily torqued mass asymmetric rigid body, are scaled using linear transformations that lead to a simplified set of first order ordinary differential equations without the explicit appearance of the principal moments of inertia. These scaled differential equations provide trivial access to an analytical solution and two constants of integration for the case of torque-free motion. Two additional representations for the third constant of integration are chosen to complete two new kinetic element sets that describe an osculating solution using the variation of parameters. The elements' physical representations are amplitudes and either angular displacement or initial time constant in the torque-free solution. These new kinetic elements lead to a considerably simplified variation of parameters solution to Euler's equations. The resulting variational equations are quite compact. To investigate error propagation behaviour of these new variational formulations in computer simulations, they are compared to the unmodified equations without kinematic coupling but under the influence of simulated gravity-gradient torques.     

General relativity and satellite orbits: The motion of a test particle in the Schwarzschild metric

The motion of a satellite with negligible mass in the Schwarzschild metric is treated as a problem in Newtonian physics. The relativistic equations of motion are formally identical with those of the Newtonian case of a particle moving in the ordinary inverse-square law field acted upon by a disturbing function which varies asr ^?3. Accordingly, the relativistic motion is treated with the methods of celestial mechanics. The disturbing function is expressed in terms of the Keplerian elements of the orbit and substituted into Lagrange's planetary equations. Integration of the equations shows that a typical Earth satellite with small orbital eccentricity is displaced by about 17 cm from its unperturbed position after a single orbit, while the periodic displacement over the orbit reaches a maximum of about 3 cm. Application of the equations to the planet Mercury gives the advance of the perihelion and a total displacement of about 85 km after one orbit, with a maximum periodic displacement of about 13 km.     

On Pfaff's equations of motion in dynamics; Applications to satellite theory

In this article we study a form of equations of motion which is different from Lagrange's and Hamilton's equations: Pfaff's equations of motion. Pfaff's equations of motion were published in 1815 and are remarkably elegant as well as general, but still they are much less well known. Pfaff's equations can also be considered as the Euler-Lagrange equations derived from the linear Lagrangian rather than the usual Lagrangian which is quadratic in the velocity components. The article first treats the theory of changes of variables in Pfaff's equations and the connections with canonical equations as well as canonical transformations. Then the applications to the perturbed two-body problem are treated in detail. Finally, the Pfaffians are given in Hill variables and Scheifele variables. With these two sets of variables, the use of the true anomaly as independent variable is also considered.     

Green's functions of the induction equation on regions with boundary. I. General properties and a construction principle

The evolution of a magnetic field is considered which pervades an electrically conducting fluid and its non-conducting surroundings under the influence of electromotive forces due to internal motion and other causes. The governing equations -- among which the induction equation of magnetohydrodynamics is the most prominent -- pose an initial value problem for the magnetic flux density. Properties of this initial value problem and of the solving Green's function are reviewed and a general construction principle for the Green's function is given. Detailed treatment of cases where the fluid occupies a sphere or an evenly bounded half-space are presented in two subsequent papers     

Stabilization by manipulation of the Hamiltonian

Numerical integration of unstable differential equations should be avoided since a numerical error during thenth step produces erroneous initial values for the next step and thus deteriorates the subsequent integration in an unstable manner. A method is offered to stabilize the equations of motion corresponding to a given HamiltonianH by transformingH into a new HamiltonianH ^* which is equivalent to the Hamiltonian of a harmonic oscillator. In contrast to other methods of stabilization the realm of canonical mechanics is thus not abandoned. Perturbations are discussed and as examples the Keplerian motion and the motion of a gyroscope are presented.     

Regularization and the computation of optimal trajectories

A completely regular form for the differential equations governing the three-dimensional motion of a continuously thrusting space vehicle is obtained by using the Kustaanheimo-Stiefel regularization. The differential equations for the thrusting rocket are transformed using the K-S transformation and an optimal trajectory problem is posed in the transformed space. The canonical equations for the optimal motion in the transformed space are regularized by a suitable change of the independent variable. The transformed equations are regular in the sense that the differential equations do not possess terms with zero divisors when the motion encounters a gravitational force center. The resulting equations possess symmetry in form and the coefficients of the dependent variables are slowly varying quantities for a low-thrust space vehicle.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

On the determination of potential function

Szebehely's renowned equation given in 1974, allowing for potential determination from a given orbit or family of orbits, is proved to be equivalent with an equation deduced in 1963 by Drǎmbǎ. This basic equation in the inverse problem of dynamics, for which the denomination of Drǎmbǎ –Szebehely equation is proposed, is generalized for the motion in the n-dimensional Euclidean space. A method for the determination of the potential function from motion equations is extended to this space.