Kustaanheimo-Stiefel Research Articles

Long-Term Integration Error of Kustaanheimo-Stiefel Regularized Orbital Motion. II. Method of Variation of Parameters

We have discovered that application of the method of variation of parameters to the KustaanheimoStiefel (K-S) regularization drastically reduces the orbital integration errors of the perturbed two-body problem for arbitrary types of perturbations. This is because not only the errors of position, whose linear growth was determined previously (Paper I), but those of the physical time grow only linearly with respect to the —ctitious time even if using traditional integrators such as the Runge-Kutta, extrapolation,

Long-Term Integration Error of Kustaanheimo-Stiefel Regularized Orbital Motion

We confirm that the positional error of a perturbed two-body problem expressed in the Kustaanheimo-Stiefel (K-S) variable is proportional to the fictitious time s, which is the independent variable in the K-S transformation. This property does not depend on the type of perturbation, on the integrator used, or on the initial conditions, including the nominal eccentricity. The error growth of the physical time evolution and the Kepler energy is proportional to s if the perturbed harmonic oscillator part of the equation of motion is integrated by a time-symmetric integration formula, such as the leapfrog or the symmetric multistep method, and is proportional to s2 when using traditional integrators, such as the Runge-Kutta, Adams, Stormer, and extrapolation methods. Also, we discovered that the K-S regularization avoids the step size resonance/instability of the symmetric multistep method that appears in the unregularized cases. Therefore, the K-S regularized equation of motion is useful to investigate the long-term behavior of perturbed two-body problems, namely, those used for studying the dynamics of comets, minor planets, the Moon, and other natural and artificial satellites.

Zonal Harmonics of the Gravity Field in Def-Variables

To take advantage of thelinear and regularformulation and treatment of Celestial Mechanics problems (Kustaanheimo & Stiefel 1965; Stiefel & Scheifele 1971; Deprit, Elipe & Ferrer 1994), Sharaf & Saad (1997) have given an analytical expansion of the Earth’s gravitationalzonal potentialin terms of Kustaanheimo-Stiefel (KS) regular elements (Stiefel & Scheifele 1971, §19), with special emphasis on its application toelliptic-type two-body orbitsand, consequently, using a generalized (elliptic) eccentric anomaly as the independent variable.Motivated by these and other considerations based on the definition and use of KS elements, and following a treatment similar to that of Stiefel & Scheifele (1971, §19), we developelement equations corresponding to a DEF-formulationof the satellite problem under the effect of the zonal potential.

Time-Symmetrized Kustaanheimo-Stiefel Regularization

In this paper we describe a new algorithm for the long-term numerical integration of the two-body problem, in which two particles interact under a Newtonian gravitational potential. Although analytical solutions exist in the unperturbed and weakly perturbed cases, numerical integration is necessary in situations where the perturbation is relatively strong. Kustaanheimo--Stiefel (KS) regularization is widely used to remove the singularity in the equations of motion, making it possible to integrate orbits having very high eccentricity. However, even with KS regularization, long-term integration is difficult, simply because the required accuracy is usually very high. We present a new time-integration algorithm which has no secular error in either the binding energy or the eccentricity, while allowing variable stepsize. The basic approach is to take a time-symmetric algorithm, then apply an implicit criterion for the stepsize to ensure strict time reversibility. We describe the algorithm in detail and present the results of numerical tests involving long-term integration of binaries and hierarchical triples. In all cases studied, we found no systematic error in either the energy or the angular momentum. We also found that its calculation cost does not become higher than those of existing algorithms. By contrast, the stabilization technique, which has been widely used in the field of collisional stellar dynamics, conserves energy very well but does not conserve angular momentum.

Generalized Kustaanheimo-Stiefel transformations

A theory is given for the construction of generalized Kustaanheimo—Stiefel (KS) transformations for dimensionsq+1 (q=2 h ,h=0, 1, 2, ...) of the Kepler problem, and the following proposition is proved: A connection between the Kepler problem in a real space of dimensionq+1 and the problem of an isotropic harmonic oscillator in a real space of dimensionN exists and can be established by means of generalized KS transformations in the cases in whichN=2q andq=2 h (h=0, 1, 2, ...). A simple graphical prescription for constructing genralized KS transformations that realize this connection is proposed.

The KS-transformation revisited

We show that the Kustaanheimo-Stiefel (KS) regularizing transformation for the perturbed Kepler motion is so deeply rooted to the Keplerian orbital elements as to yield the position vector of a particle on the osculating orbit as the effect of a peculiar roto-dilatation in the physical Euclidean space. Adopting the conventional vector formulation, quaternions and spinors are also involved. A key role is played by (i): a simple hodographical approach to the integrals of the Kepler motion (angular momentum vector, Runge-Lenz vector); (ii) a polarized outlook on the attitude frame of the Kepler orbit; (iii) a simple kinematical expression for the orbital elements. The mechanical energy, the bilinear relation, the gauge transformation — fundamental in the KS-theory — are naturally arrived at, acquiring interesting kinematical interpretations.

An implementation ofN-body chain regularization

The chain regularization method (Mikkola and Aarseth 1990) for high accuracy computation of particle motions in smallN-body systems has been reformulated. We discuss the transformation formulae, equations of motion and selection of a chain of interparticle vectors such that the critical interactions requiring regularization are included in the chain. The Kustaanheimo-Stiefel (KS) coordinate transformation and a time transformation is used to regularize the dominant terms of the equations of motion. The method has been implemented for an arbitrary number of bodies, with the option of external perturbations. This formulation has been succesfully tested in a generalN-body program for strongly interacting subsystems. An easy to use computer program, written inFortran, is available on request.

Identités de Fierz et formes bilinéaires dans les espaces spinoriels

Following several authors on a recent geometrical interpretation of the Fierz identities, we give a general and geometrical demonstration of the Fierz identities, using the spinorial formalism of Chevalley, and without any matrix formalism. The Kustaanheimo-Stiefel (KS) transformation is presented as an elarged triality principle and a particular case of the Fierz identities.

On the connection among three classical mechanical problems via the hypercomplex KS-transformation

With reference to our previous notes [23–30] on the hypercomplex regularizations of the classical Kepler problem and on the hypercomplex descriptions of the classical rotational mechanics, we arrive at an intimate mathematical relationship among the three classical mechanical problems: the elliptic Kepler problem, the spherical Euler-Poinsot problem and the four-dimensional isotropic harmonic oscillator. A key role is played: (i) by the Kustaanheimo-Stiefel (KS) regularizing transformation viewed, in the light of Souriau's quantization procedure, as the projection map in the Hopf fibering of the contact 3-sphere; (ii) by a novel Lagrangian description of rotational kinematics expressed in terms of the Euler-Rodrigues parameters; (iii) by the unit vector a (characterizing both the attitude frame of the rotator and the direction of the major axis of the Kepler orbit) and shown to link ‘a la Cartan’ the KS-theory with the rotation theory. The Kepler-Poinsot-Oscillator connection is examined also in a quantistic Schroedinger approach and is related to the results of Ikeda and Miyachi.

A new derivation of the Kustaanheimo-Stiefel variables

First introducing the parabolic coordinates inR3, we can derive the Kustaanheimo-Stiefel (KS) variables quite simply and naturally. Through this derivation it becomes clearer where and how the additional dimension is introduced and what the bilinear relation means.

The development of the Poincar�-similar elements with eccentric anomaly as the independent variable

A new set of element differential equations for the perturbed two-body motion is derived. The elements are canonical and are similar to the classical canonical Poincare elements, which have time as the independent variable. The phase space is extended by introducing the total energy and time as canonically conjugated variables. The new independent variable is, to within an additive constant, the eccentric anomaly. These elements are compared to the Kustaanheimo-Stiefel (KS) element differential equations, which also have the eccentric anomaly as the independent variable. For several numerical examples, the accuracy and stability of the new set are equal to those of the KS solution. This comparable accuracy result can probably be attributed to the fact that both sets have the same time element and very similar energy elements. The new set has only 8 elements, compared to 10 elements for the KS set. Both sets are free from singularities due to vanishing eccentricity and inclination.

The uniform, regular differential equations of the KS transformed perturbed two-body problem

The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.

Concerning the derivation of the KS-transformation

The Kustaanheimo-Stiefel (KS) transformation is shown to follow naturally from the general solution of the two-body motion if ‘half-arguments’ are introduced. Application to collision orbits and to the exact triangular solutions of Lagrange (vide E. Stiefel and G. Scheifele: 1971,Linear and Regular Celestial Mechanics, Springer, Berlin-Heidelberg-New York, p. 23–35).