A new regularization of the restricted three-body problem and an application

A new regularizing transformation for the three-dimensional restricted three-body problem is constructed. It is explicitly derived and is equivalent to a simple rational map. Geometrically it is equivalent to a rotation of the 3-sphere. Unlike the KS map it is dimension preserving and is valid inn dimensions. This regularizing map is applied to the restricted problem in order to prove the existence of a family of periodic orbits which continue from a family of collision orbits.     

Geometry of spinor regularization

The Kustaanheimo theory of spinor regularization is given a new formulation in terms of geometric algebra. The Kustaanheimo-Stiefel matrix and its subsidiary condition are put in a spinor form directly related to the geometry of the orbit in physical space. A physically significant alternative to the KS subsidiary condition is discussed. Derivations are carried out without using coordinates.     

Global regularization of the Magnetic-Binary problem

The author's aim is to achieve global regularization in the Magnetic-Binary problem by suitably transforming the state-time space of the system. The functions which perform the change of the physical time and the geometrical figures of the system, are connected by a special relation leaving the form of the equations of motion invariant. Additionally, a proposition for generalization of the process is discussed in an aspect as well, of how much such a regularization is profitable.     

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.     

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:

1. The equations of motion are regular for the two-body collisionsR ₁→0 orR ₂→0.
2. Provided thatR?R ₁ orR?R ₂, the equations of motion are numerically well behaved for close triple encounters.

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.     

Local regularization of the Magnetic-Binary problem

The singularities in a Magnetic-Binary system are regularized separately by changing both the coordinates and the time. It is shown why, in this problem it is more efficient to relate the geometric transformation to the rescaling of time.     

Algorithmic regularization of the few-body problem

Using a modified leapfrog method as a basic mapping, we produce a new numerical integrator for the stellar dynamical few-body problem. We do not use coordinate transformation and the differential equations are not regularized, but the leapfrog algorithm gives regular results even for collision orbits. For this reason, application of extrapolation methods gives high precision. We compare the new integrator with several others and find it promising. Especially interesting is its efficiency for some potentials that differ from the Newtonian one at small distances.     

嵌入式时间间隔计数器的研制

介绍了一种新研制的嵌入式时间间隔计数器，它可作为一种功能模块用于用户的设备或时统中，实现高精度时间测量。从基本原理、硬、软件和功能等不同角度对计数器进行了较为详细的讨论。在该计数器的研制过程中还开发了用来调试计数器和供计数器用户使用的计算机应用软件，对此也作了简要介绍。最后，给出了计数器的设计指标和一些实测数据。     

Time scaling as an infinitesimal canonical transformation

We show that time scaling transformations for Hamiltonian systems are infinitesimal canonical transformations in a suitable extended phase space constructed from geometrical considerations. We compute its infinitesimal generating function in some examples: regularization and blow up in celestial mechanics, classical mechanical systems with homogeneous potentials and Scheifele theory of satellite motion.Research partially supported by CONACYT (México), Grant PCCBBNA 022553 and CICYT (Spain).     

voboz: an almost-parameter-free halo-finding algorithm

We have developed an algorithm, called voboz (VOronoi BOund Zones), to find haloes in an N -body dark matter simulation; it has as little dependence on free parameters as we can manage. By using the Voronoi diagram, we achieve non-parametric, 'natural' measurements of each particle's density and set of neighbours. We then eliminate much of the ambiguity in merging sets of particles together by identifying every possible density peak, and measuring the probability that each does not arise from Poisson noise. The main halo in a cluster tends to have a high probability, while its subhaloes tend to have lower probabilities. The first parameter in voboz controls the subtlety of particle unbinding, and may be eliminated if one is cavalier with processor time; even if one is not, the results saturate to the parameter-free answer when the parameter is sufficiently small. The only parameter that remains, an outer density cut-off, does not influence whether or not haloes are identified, nor does it have any effect on subhaloes; it only affects the masses returned for supercluster haloes.     

A regularization of multiple encounters in gravitationalN-body problems

A method is introduced to regularize binary collisions between one of the bodies and any number of other bodies in the three-dimensional problem ofn-bodies. The coordinates are first transformed from an inertial system to a system relative to one of the bodies. The KS dependent variable transformation and a new independent variable transformation are introduced for the regularization.     

Time Evolution of the Altitude of an Observed Coronal Wave

The nature of coronal wave fronts is intensely debated. They are observed in several wavelength bands and are frequently interpreted as magnetosonic waves propagating in the lower solar atmosphere. However, they can also be attributed to the line-of-sight projection of the edges of coronal mass ejections. Therefore, estimating the altitude of these features is crucial for deciding in favor of one of these two interpretations. We took advantage of a set of observations obtained from two different view directions by the EUVI instrument onboard the STEREO mission on 7 December 2007 to derive the time evolution of the altitude of a coronal wave front. We developed a new technique to compute the altitude of the coronal wave and found that the altitude increased during the initial 5 min and then slightly decreased back to the low corona. We interpret the evolution of the altitude as follows: the increase in the altitude of the wave front is linked to the rise of a bubble-like structure depending on whether it is a magnetosonic wave front or a CME in the initial phase. During the second phase, the observed brightness of the wave front was mixed with the brightening of the underlying magnetic structures as the emission from the wave front faded because the plasma became diluted with altitude.     

Generalization of Levi-Civita regularization in the restricted three-body problem

A family of polynomial coupled function of n degree is proposed, in order to generalize the Levi-Civita regularization method, in the restricted three-body problem. Analytical relationship between polar radii in the physical plane and in the regularized plane are established; similar for polar angles. As a numerical application, trajectories of the test particle using polynomial functions of 2,3,…,8 degree are obtained. For the polynomial of second degree, the Levi-Civita regularization method is found.     

A chain regularization method for the few-body problem

A regularization method for integrating the equations of motion of small N-body systems is discussed. We select a chain of interparticle vectors in such a way that the critical interactions requiring regularization are included in the chain. The equations of motion for the chain vectors are subsequently regularized using the KS-variables and a time transformation. The method has been formulated for any number of bodies, but the most important application appears to be in the four-body problem which is therefore discussed in detail.     

A new regularization of the planar problem of three bodies

A new method of simultaneously regularizing the three types of binary collisions in the planar problem of three bodies is developed: The coordinates are transformed by means of certain fourth degree polynomials, and a new independent variable is introduced, too. The proposed transformation is in each binary collision locally equivalent to Levi-Civita's transformation, whereas the singularity corresponding to a triple collision is mapped into infinity. The transformed Hamiltonian is a polynomial of degree 12 in the regularized variables.Presented before the Division of Dynamical Astronomy at the 133rd meeting of the American Astronomical Society, Tampa, Florida, December 6–9, 1970.Department of Aerospace Engineering and Engineering Mechanics.