Kustaanheimo-Stiefel Regularization Research Articles

A Kustaanheimo–Stiefel regularization of the elliptic restricted three-body problem and the detection of close encounters with fast Lyapunov indicators

We present the Kustaanheimo–Stiefel (KS) regularization of the elliptic restricted three-body problem (ER3BP) at the secondary body P2, and discuss its use to study a category of transits through its Hill’s sphere (fast close encounters). Starting from the Hamiltonian representation of the problem using the synodic rotating–pulsating reference frame and the true anomaly of P2 as independent variable, we perform the regularization at the secondary body analogous to the circular case by applying the classical KS transformation and the iso-energetic reduction in an extended 10-dimensional phase-space; this translates into an efficient algorithm that can be readily implemented to numerically integrate the equations of motion. Using such a regularized Hamiltonian we recover a definition of fast close encounters in the ER3BP for small values of the mass parameter μ (while we do not require a smallness condition on the eccentricity of the primaries), and we show that for these encounters the solutions of the variational equations are characterized by an exponential growth during the fast transits through the Hill’s sphere. Thus, for small μ, we justify the effectiveness of the regularized fast Lyapunov indicators (RFLIs) to detect orbits with multiple fast close encounters. Finally, we provide numerical demonstrations and show the benefits of the regularization in terms of the computational cost.

Regularization of central forces with damping in two and three dimensions

Regularization of damped motion under central forces in two and three-dimensions are investigated and equivalent, undamped systems are obtained. The dynamics of a particle moving in $\frac{1}{r}$ potential and subjected to a damping force is shown to be regularized a la Levi-Civita. We then generalize this regularization mapping to the case of damped motion in the potential $r^{-\frac{2N}{N+1}}$. Further equation of motion of a damped Kepler motion in 3-dimensions is mapped to an oscillator with inverted sextic potential and couplings, in 4-dimensions using Kustaanheimo-Stiefel regularization method. It is shown that the strength of the sextic potential is given by the damping co-efficient of the Kepler motion. Using homogeneous Hamiltonian formalism, we establish the mapping between the Hamiltonian of these two models. Both in 2 and 3-dimensions, we show that the regularized equation is non-linear, in contrast to undamped cases. Mapping of a particle moving in a harmonic potential subjected to damping to an undamped system with shifted frequency is then derived using Bohlin-Sudman transformation.

Integrability of close encounters in the spatial restricted three-body problem

We extend to the spatial case a technique of integration of the close encounters formulated by Tullio Levi-Civita for the planar restricted three-body problem. We consider the Hamiltonian introduced in the Kustaanheimo–Stiefel regularization and construct a complete integral of the related Hamilton–Jacobi equation by means of a series convergent in a neighborhood of the collisions with the primary or secondary body.

Accurate orbit propagation in the presence of planetary close encounters

We present an efficient strategy for the numerical propagation of small Solar System objects undergoing close encounters with massive bodies. The trajectory is split in several phases, each of them being the solution of a perturbed two-body problem. Formulations regularized with respect to different primaries are employed in two subsequent phases. In particular, we consider the Kustaanheimo-Stiefel regularization and a novel set of non-singular orbital elements pertaining to the Dromo family. In order to test the proposed strategy, we perform ensemble propagations in the Earth-Sun CR3BP using a variable step-size and order multi-step integrator and an improved version of Everhart's RADAU solver of 15th order. By combining the trajectory splitting with regularized equations of motion in short-term propagations (one year), we gain up to six orders of magnitude in accuracy with respect to the classical Cowell's method for the same computational cost. Moreover, in the propagation of asteroid (99942) Apophis through its 2029 Earth encounter, the position error stays within 100 meters after 100 years. In general, as to improve the performance of regularized formulations, the trajectory must be split between 1.2 and 3 Hill radii from the Earth. We also devise a robust iterative algorithm to stop the integration of regularized equations of motion at a prescribed physical time. The results rigorously hold in the CR3BP, and similar considerations may apply when considering more complex models. The methods and algorithms are implemented in the NAPLES Fortran 2003 code, which is available online as a GitHub repository.

New Solutions for the Perturbed Lambert Problem Using Regularization and Picard Iteration

A new approach for solving two-point boundary value problems and initial value problems using the Kustaanheimo–Stiefel transformation and Modified Chebyshev–Picard iteration is presented. The first contribution is the development of an analytical solution to the elliptic Keplerian Lambert problem based on Kustaanheimo–Stiefel regularization. This transforms the nonlinear three-dimensional orbit equations of motion into four linear oscillators. The second contribution solves the elliptic Keplerian two-point boundary value problem and initial value problem using the Kustaanheimo–Stiefel transformation and Picard iteration. The Picard sequence of trajectories represents a contraction mapping that converges to a unique solution over a finite domain. Solving the Keplerian two-point boundary value problem in Kustaanheimo–Stiefel variables increases the Picard domain of convergence from about one-third of an orbit (Cartesian variables) to over 95% of an orbit (Kustaanheimo–Stiefel variables). These increases in the domain of Picard iteration convergence are independent of eccentricity. The third contribution solves the general spherical harmonic gravity perturbed elliptic two-point boundary value problem using the Kustaanheimo–Stiefel transformation and Picard iteration, and it does not require a Newton-like shooting method for fractional orbit transfers. For multiple revolution transfers, however, a shooting method can make use of the Modified Chebyshev–Picard iteration/Kustaanheimo–Stiefel/initial value problem and the Method of Particular Solutions to obtain solutions given a Keplerian Lambert solution as the starting iterative. The Kustaanheimo–Stiefel perturbed solution is illustrated using a (40,40) degree and order spherical harmonic gravity model. A general three-dimensional recipe is introduced for solving the perturbed Lambert Problem via Modified Chebyshev–Picard iteration without a Newton-like shooting method for the fractional orbit case. The increase in the domain of convergence of the Kustaanheimo–Stiefel transformed, perturbed Lambert problem via Modified Chebyshev–Picard iteration versus the Cartesian Modified Chebyshev–Picard iteration Lambert solution is analogous to the results for the Keplerian case. The three-dimensional two-impulse perturbed Lambert problem is efficiently convergent up to about 85% of the Keplerian orbit period with a (40,40) spherical harmonic gravity model. The efficiency of the current two-point boundary value problem solver is compared with MATLAB’s fsolve, where Runge–Kutta–Nystrom 12(10) and Gauss–Jackson are the integrators.

Kustaanheimo — Stiefel regularization and the quadrupolar conjugacy

In this note, we present the Kustaanheimo-Stiefel regularization in a symplectic and quaternionic fashion. The bilinear relation is associated with the moment map of the $S^{1}$- action of the Kustaanheimo-Stiefel transformation, which yields a concise proof of the symplecticity of the Kustaanheimo-Stiefel transformation symplectically reduced by this circle action. The relation between the Kustaanheimo-Stiefel regularization and the Levi-Civita regularization is established via the investigation of the Levi-Civita planes. A set of Darboux coordinates (which we call Chenciner-F\'ejoz coordinates) is generalized from the planar case to the spatial case. Finally, we obtain a conjugacy relation between the integrable approximating dynamics of the lunar spatial three-body problem and its regularized counterpart, similar to the conjugacy relation between the extended averaged system and the averaged regularized system in the planar case.

Efficient Orbit Integration by Orbital Longitude Methods with Sundmann Transformation on the Time Variable

We apply Sundmann's time transformation to two orbital longitude methods for numerical integration of perturbed two-body problems that we developed previously. The modified methods share many good features with the original orbital longitude methods. In addition, they are efficient enough in integrating highly eccentric orbits to be competitive with the quadruple scaling method with Kustaanheimo-Stiefel regularization, which we have found to have the best cost performance in integrating highly eccentric orbits. Moreover, the computational cost of the new methods is significantly less than that of the quadruple scaling method, in the sense that the number of components to be integrated for the perturbed two-body problem is reduced from 13 to seven. In the case of unperturbed orbits, the new orbital longitude methods reduce to the machine-epsilon level the errors in all the orbital elements except the mean longitude at epoch, which grows linearly with respect to the real time, independently of the precision of the numerical integration used. In the case of perturbed orbits, the orbital longitude methods with Sundmann transformation are significantly superior to the original ones when the eccentricity is large, and they show more robustness against oblateness perturbations than the quadruple scaling method. However, the quadruple scaling method remains the best when only the other types of perturbations are to be considered and the nominal eccentricity is extremely large.

A new conservative numerical integration algorithm for the three-dimensional Kepler motion based on the Kustaanheimo–Stiefel regularization theory

A new conservative numerical integration algorithm for the three-dimensional Kepler motion is presented which conserves all of the constants of motion including the Runge–Lenz vector. The Kustaanheimo–Stiefel regularization theory plays a central role in the discretization of the equations of motion.

Reversible adaptive regularization: perturbed Kepler motion and classical atomic trajectories

Reversible and adaptive integration methods based on Kustaanheimo–Stiefel regularization and modified Sundman transformations are applied to simulate general perturbed Kepler motion and to compute classical trajectories of atomic systems (e.g. Rydberg atoms). The new family of reversible adaptive regularization methods also conserve angular momentum and exhibit superior energy conservation and numerical stability in long–time integrations. The schemes are appropriate for scattering, for astronomical calculations of escape time and long–term stability, and for classical and semiclassical studies of atomic dynamics. The components of an algorithm for trajectory calculations are described. Numerical experiments illustrate the effectiveness of the reversible approach.

Reversible adaptive regularization methods for atomic N-body problems in applied fields

Reversible and adaptive integration methods based on Kustaanheimo-Stiefel regularization and modified Sundman transformations have recently been proposed for perturbed Kepler motion and certain types of N-body problems including atomic scattering and gravitation. These methods are explicit, and demonstrate superior conservation properties compared to standard explicit integrators applied with or without variable stepsize schemes. In this paper, reversible adaptive regularization is applied to treat (classical) atomic systems in applied electrical and/or magnetic fields.

Regularization of the three-dimensional gravitational potential

The ideas of geometrical algebra are used to investigate the physical content of the Kustaanheimo-Stiefel regularization procedure, which transforms the r 1 potential into the r 2 harmonic oscillator 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.

Orbital evolution of short-period comets treated as a Markov process

A fast new method is introduced to study the orbital evolution of active comets in the inner planetary region. It models the time evolution of the cometary orbit distribution as a Markov chain. An empirically derived distribution of perturbations of aphelion and perihelion distances is used. These perturbations have been computed in the nonplanar, elliptic restricted three-body problem (sun-Jupiter-comet) using the Kustaanheimo-Stiefel regularization for a sample of 60,000 fictitious comet orbits. The results obtained for the steady-state orbital distribution are in broad agreement with those obtained from the same perturbation sample by a different method: the Monte Carlo simulation (Froechle and Rickman, 1979). Because of its extreme efficiency the Markov method allows a systematic survey of models for observable lifetimes and initial orbits.

Kustaanheimo-Stiefel Regularization and nonclassical canonical transformations

This paper presents a method for constructing a transformation of coordinates in phase space yielding canonical equations. The general transformation, which does not necessarily yield a canonical system, has an arbitrary function in its right-hand members. The cases when the transformed equations may be made canonical by appropriate choice of this function are established. The method is illustrated by means of the Kustaanheimo-Stiefel Regularization and applied to the circular restricted three body problem.

On redundant variables in Lagrangian mechanics, with applications to perturbation theory and KS regularization

It is shown that it is possible to make a change of variables in a Lagrangian in such a way that the number of variables is increased. The Euler-Lagrange equations in the redundant variables are obtained in the standard way (without the use of Lagrange Multipliers!). These equations are not independent but they are all valid and consistent. In some cases they are simpler than if the minimum number of variables are used. The redundant variables are supposed to be related to each other by several constraints (not necessarily holonomic), but these constraints are not used in the derivation of the equations of motion. The method is illustrated with the well known Kustaanheimo-Stiefel Regularization. Some interesting applications to perturbation theory are also described.

A regularization of the three body problem

Letr 1,r 2,r 3 be arbitrary coordinates of the non-zero interacting mass-pointsm 1,m 2,m 3 and define the distancesR 1=|r 1−r 3|,R 2=|r 2−r 3|,R=|r 1−r 2|. An eight-dimensional regularization of the general three-body problem is given which is based on Kustaanheimo-Stiefel regularization of a single binary and possesses the properties: Although the requirementR≳ min (R 1,R 2) may involve occasional transformations to physical variables in order to re-label the particles, all integrations are performed in regularized variables. Numerical comparisons with the standard Kustaanheimo-Stiefel regularization show that the new method gives improved accuracy per integration step at no extra computing time for a variety of examples. In addition, time reversal tests indicate that critical triple encounters may now be studied with confidence. The Hamiltonian formulation has been generalized to include the case of perturbed three-body motions and it is anticipated that this procedure will lead to further improvements ofN-body calculations.

Direct Integration Methods of theN-Body Problem

AbstractA fourth-order polynomial method for the integration ofN-body systems is described in detail together with the computational algorithm. Most particles are treated efficiently by an individual time-step scheme but the calculation of close encounters and persistent binary orbits is rather time-consuming and is best performed by special techniques. A discussion is given of the Kustaanheimo-Stiefel regularization procedure which is used to integrate dominant two-body encounters as well as close binaries. Suitable decision-making parameters are introduced and a simple method is developed for regularizing an arbitrary number of simultaneous two-body encounters.

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.