Recent developments of integrating the gravitational problem ofN-bodies

This paper discusses the formulation and the numerical integration of large systems of differential equations occurring in the gravitational problem ofn-bodies.Different forms of the pertinent differential equations of motion are presented, and various regularizing and smoothing transformations are compared. Details regarding the effectiveness and the efficiency of the Kustaanheimo-Stiefel and of other methods are discussed. In particular, a method is described in which some of the phase variables are treated in the regularized system and others in the ordinary system. This mixed method of numerical regularization offers some advantages.Several numerical integration techniques are compared. A high order Runge-Kutta method, Steffensen's method, and a finite difference method are investigated, especially with regard to their adaptability to regularization.The role of integrals and integral invariants is displayed in controlling the accuracy of the numerical integration.Numerical results are described with 5, 25 and 500 bodies participating. These examples compare the various integration techniques, several regularization methods and different logics in treating binaries.     

Regularization and the artificial Earth satellite problem

Some of the properties of singularities of a system of differential equations, which includes several formulations of the artificial Earth satellite problem, are derived. Using them, it is shown that this problem cannot be regularized by using the current commonly used ideas and definitions of regularization.     

Stabilization of Kepler's problem

A regularization of Kepler's problem due to Moser is used to ‘stabilize’ the equations of motion, that is, imbed a particular solution of Kepler's problem in a Lyapounov stable system.     

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.     

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.     

Non-parametric methods in density profile estimation of stellar systems

In this paper a specific inverse astronomical problem is treated. We study the problem of inferring the spatial density profile of a spherically symmetric stellar system from values of its surface density or from discrete data as the projected positions of its stars on the plane of sky. Non parametric techniques were used based on the least-squares method or on the maximum likelihood estimation. The smoothness of the resulting profiles is increased by the method of regularization.     

A method of regularizing the equations of motion in the central force-field

This paper deals with a method of regularization and linearization of the equations of motion in the central force-field, when the potential is given.This method of regularization of the equations of motion is known (Sundman, 1913), and is based on the transformation of time by means of introducing a new independent variable.In this article a condition has been obtained for the regularizing function when the potential is given.Some examples of the perturbed Keplerian motions are discussed.     

A Regularized Image Restoration Algorithm for Lossy Compression in Astronomy

Astronomical images currently provide large amounts of data. Lossy compression algorithms have recently been developed for high compression ratios. These compression technique introduce distortion in the compressed images and for high compression ratios, a blocking effect appears. We propose a modified compression algorithm based on the hcompress scheme, and we introduce a new decompression method based on the regularization theory The image is restored scale by scale in a multiresolution scheme and the information lost during the compression is recovered by applying a Tikhonov regularization constraint. The experimental results show that the blocking effect is reduced and some measurements made on a simulated image show that the astrometric and the photometric properties of the restored images are improved.     

The deconvolution of brightness profiles of galaxies from seeing: Application to M32

The integral equation relating to the brightness distribution in a galaxy and in its image formed by an optical system characterized by a Gaussian (or a sum of Gaussians) point-spread function (PSF) is derived. Since the solution of this equation, attainable by any classical method, is numerically unstable, according to the ill-posed nature of the problem, an approximate and stable solution is obtainable by a first-order regularization in Tikhonov's sense. For the bright spike the application to M32 gives a radius of 2.1 arc sec a central surface brightness of 13.10 V mag arc sec^–2 and a 12 V integrated magnitude.     

Regularized Energy-Dependent Solar Flare Hard X-Ray Spectral Index

The deduction from solar flare X-ray photon spectroscopic data of the energy-dependent model-independent spectral index is considered as an inverse problem. Using the well-developed regularization approach we analyze the energy dependency of spectral index for a high-resolution energy spectrum provided by Ramaty High Energy Solar Spectroscopic Imager (RHESSI). The regularization technique produces much smoother derivatives while avoiding additional errors typical of finite differences. It is shown that observations imply a spectral index varying significantly with energy, in a way that also varies with time as the flare progresses. The implications of these findings are discussed in the solar flare context.     

Quaternions for regularizing Celestial Mechanics: the right way

Quaternions have been found to be the ideal tool for describing and developing the theory of spatial regularization in Celestial Mechanics. This article corroborates the above statement. Beginning with a summary of quaternion algebra, we will describe the regularization procedure and its consequences in an elegant way. Also, an alternative derivation of the theory of Kepler motion based on regularization will be given. Furthermore, we will consider the regularization of the spatial restricted three-body problem, i.e. the spatial generalization of the Birkhoff transformation. Finally, the perturbed Kepler motion will be described in terms of regularized variables.     

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.     

Numerical Treatment of Small Stellar Systems with Binaries

The use of regularization methods based on the Kustaanheimo-Stiefel transformation (KS) is reviewed. The history of the development of such methods is summarized and anecdotal information about the related events is told. Details of the multi-particle regularization methods are given, including the most recent developments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Treatment of close approaches in the numerical integration of the gravitational problem ofN bodies

One of the main difficulties encountered in the numerical integration of the gravitationaln-body problem is associated with close approaches. The singularities of the differential equations of motion result in losses of accuracy and in considerable increase in computer time when any of the distances between the participating bodies decreases below a certain value. This value is larger than the distance when tidal effects become important, consequently,numerical problems are encounteredbefore the physical picture is changed. Elimination of these singularities by transformations is known as the process of regularization. This paper discusses such transformations and describes in considerable detail the numerical approaches to more accurate and faster integration. The basic ideas of smoothing and regularization are explained and applications are given.     

Maximum Entropy Regularization Method of Ionospheric Occultation

With the increasing of applications of Global Positioning System (GPS), the research on the factors affecting the radio signals is becoming more and more important. One of the most significant effects on ionosphere monitoring is the ionospheric radio occultation (IRO), and the IRO data is now serving as one of the most important monitoring sources for the ionospheric measurements. Meanwhile the inversion of the occultation data is a popular topic. The traditional Abel inversion process through compensated total electron content (TEC) is a linear inversion method, thus it would transfer the measurement errors directly to the inversion results. In order to improve the occultation results, we introduce in this paper two nonlinear methods, namely the regularization method and the maximum entropy regularization method. Through designed simulative experiments, we verify and compare these three methods, and conclude that the maximum entropy regularization method can reduce significantly the influence of measurement errors.     

电离层掩星反演的正则最大熵法

电离层掩星数据现已成为电离层观测数据的重要来源,对掩星数据的反演研究一直是掩星研究的热点.传统采用的改正TEC(1btal Electron Content)的Abel变换反演法为线性反演法,它会把测量误差带入反演结果中.为改善反演效果受测量误差的影响,引入两种非线性的反演方法一正则化和正则最大熵反演法.随后设计模拟试验,对3种方法进行验证、比较,得到正则最大熵反演法可以很好地减小测量误差的影响.     

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.     

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 study of the global magnetic activity of the star Kepler-17 using light curve inversion with bipartite regularization

A new method of light curve inversion with bipartite regularization(LIBR),which is complementary to the previous treatments by Bonomo and Lanza and Estrela and Valio,is used to reconstruct the physical properties of star spots on the solar-type star Kepler-17 by using the full Q1-Q17 data set.The Markov Chain Monte Carlo(MCMC) method was applied to find the best profile of the reconstructed surface.The known value of the rotation inclination of Kepler-17 allows the generation of a star spot model in a sequence of stellar rotation with a period of 12.26 d.Because of the nature of the light curve inversion,the spot model is limited to the equatorial region.We also investigated the starspot lifetimes of Kepler-17 utilizing the MCMC method.Combined with the LIBR inversion results,it was found that the star spots typically last from one to several stellar rotations.From the time evolution of the spot size,a magnetic cycle period of 437 d can be derived.This value is comparatively shorter than the solar cycle which might be a consequence of the younger age(～1.78 Gyr) of Kepler-17.The light curve of Kepler-17 is characterized by the presence of large-amplitude variation caused by star spots but no superflare activity.An interesting possibility is that the magnetic energy stored in the star spot regions could have been constantly dissipated by electrodynamic interaction between the central star and the hot Jupiter,Kepler-17 b,via a lower-level energy release process.