Jacobi Dynamics And The N-Body Problem With Variable Masses

An appropriate generalization of the Jacobi equation of motion for the polar moment of inertia I is considered in order to study the N-body problem with variable masses. Two coupled ordinary differential equations governing the evolution of I and the total energy E are obtained. A regularization scheme for this system of differential equations is provided. We compute some illustrative numerical examples, and discuss an average method for obtaining approximate analytical solutions to this pair of equations. For a particular law of mass loss we also obtain exact analytical solutions. The application of these ideas to other kind of perturbed gravitational N-body systems involving drag forces or a different type of mass variation is also considered. This revised version was published online in July 2006 with corrections to the Cover Date.     

Finiteness of spatial central configurations in the five-body problem

We strengthen a generic finiteness result due to Moeckel by showing that the number of spatial central configurations of the Newtonian five-body problem with positive masses is finite, apart from some explicitly given special cases of mass values.     

Symmetric planar central configurations of five bodies: Euler plus two

We study planar central configurations of the five-body problem where three of the bodies are collinear, forming an Euler central configuration of the three-body problem, and the two other bodies together with the collinear configuration are in the same plane. The problem considered here assumes certain symmetries. From the three bodies in the collinear configuration, the two bodies at the extremities have equal masses and the third one is at the middle point between the two. The fourth and fifth bodies are placed in a symmetric way: either with respect to the line containing the three bodies, or with respect to the middle body in the collinear configuration, or with respect to the perpendicular bisector of the segment containing the three bodies. The possible stacked five-body central configurations satisfying these types of symmetries are: a rhombus with four masses at the vertices and a fifth mass in the center, and a trapezoid with four masses at the vertices and a fifth mass at the midpoint of one of the parallel sides.     

Parabolic orbit determination. Comparison of the Olbers method and algebraic equations

In this paper, the Olbers method for the preliminary parabolic orbit determination (in the Lagrange–Subbotin modification) and the method based on systems of algebraic equations for two or three variables proposed by the author are compared. The maximum number of possible solutions is estimated. The problem of selection of the true solution from the set of solutions obtained both using additional equations and by the problem reduction to finding the objective function minimum is considered. The results of orbit determination of the comets 153P/Ikeya-Zhang and 2007 N3 Lulin are cited as examples.     

基于Broyden方法的不对称与对称周期解延拓算法与应用

考虑周期解的数值延拓问题并提出基于Broyden拟牛顿法来延拓周期解的一种有效算法,先后以布鲁塞尔振子、平面圆型限制性三体问题(Planar Circular Restricted Three-Body Problem, PCRTBP)的周期解为例进行了验证.这里的Broyden方法包含线性搜索、正交三角分解求线性方程组的步骤.对一般的周期解,周期性条件方程组中含有周期作为待延拓参数,可用周期来决定积分时长,将解代入周期性条件得到积分型的非线性方程组,利用Broyden方法迭代延拓直至初值收敛.根据两次垂直通过一个超平面的轨道是对称周期轨道的性质,可采用插值的方法求得再次抵达超平面的解分量,得到周期性条件方程组,再用Broyden方法求解.结合哈密顿系统的对称性和PCRTBP周期轨道的一些分类,对2/1、3/1的内共振周期解族进行了数值研究.最后,对算法和计算结果做了总结和讨论.     

Spatial restricted rhomboidal five-body problem and horizontal stability of its periodic solutions

The spatial restricted rhomboidal five-body problem, or shortly, SRRFBP, is a five body problem in which four positive masses, called the primaries, move two by two in coplanar circular motions with the center of mass fixed at the origin such that their configuration is always a rhombus, the fifth mass being negligible and not influencing the motion of the four primaries. The Hamiltonian function that governs the motion of the fifth mass is derived and has three degrees of freedom depending periodically on time. Using a synodical system of coordinates, we fix the primaries in order to eliminate the time dependence. With the help of the Hamiltonian structure, we characterize the regions of possible motion. The vertical $z$ axis is invariant and we study what we call the rhomboidal Sitnikov problem. Unlike the classical Sitnikov problem, no chaos exists and the behavior of the fifth mass is quite predictable, periodic solutions of arbitrary long periods are shown to exist and we study numerically their linear horizontal stability.     

Regularization of Motion Equations with L-Transformation and Numerical Integration of the Regular Equations

The sets of L-matrices of the second, fourth and eighth orders are constructed axiomatically. The defining relations are taken from the regularization of motion equations for Keplerian problem. In particular, the Levi-Civita matrix and KS-matrix are L-matrices of second and fourth order, respectively. A theorem on the ranks of L-transformations of different orders is proved. The notion of L-similarity transformation is introduced, certain sets of L-matrices are constructed, and their classification is given. An application of fourth order L-matrices for N-body problem regularization is given. A method of correction for regular coordinates in the Runge–Kutta–Fehlberg integration method for regular motion equations of a perturbed two-body problem is suggested. Comparison is given for the results of numerical integration in the problem of defining the orbit of a satellite, with and without the above correction method. The comparison is carried out with respect to the number of calls to the subroutine evaluating the perturbational accelerations vector. The results of integration using the correction turn out to be in a favorable position.     

On the stability of periodic solutions in the problem of the motion of a star inside an elliptical galaxy

The problem of the spatial motion of a star inside an inhomogeneous rotating elliptical galaxy with a homothetic density distribution is considered. Periodic solutions are constructed by the method of a small Poincaré parameter. Linear variational equations with periodic coefficients are used to analyze the Lyapunov stability of these solutions.     

The extended phase space formulation of the Vinti problem

The Vinti problem, motion about an oblate spheroid, is formulated using the extended phase space method. The new independent variable, similar to the true anomaly, decouples the radius and latitude equations into two perturbed harmonic oscillators whose solutions toO(J ₂ ⁴ ) are obtained using Lindstedt's method. From these solutions and the solution to the Hamilton-Jacobi equation suitable angle variables, their canonical conjugates and the new Hamiltonian are obtained. The new Hamiltonian, accurate toO(J ₂ ⁴ ) is function of only the momenta.     

A family of stacked central configurations in the planar five-body problem

We study planar central configurations of the five-body problem where three bodies, \(m_1, m_2\) and \(m_3\), are collinear and ordered from left to right, while the other two, \(m_4\) and \(m_5\), are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with \(m_1=m_3\), there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89–107, 2010), of stacked five-body central configuration where the segments \(m_4m_5\) and \(m_1m_3\) do not intersect.     

Conditionally periodic solutions near the libration points of the restricted circular three-body problem

Families of conditionally periodic solutions have been found by a slightly modified Lyapunov method of determining periodic solutions near the libration points of the restricted three-body problem. When the frequencies of free oscillations are commensurable, the solutions found are transformed into planar or spatial periodic solutions. The results are confirmed by numerically integrating the starting nonlinear differential equations of motion.     

Lie group for the Jordan and Brans-Dicke cosmology

A method based on the invariace under a continuous Lie group of transformations is worked out to reduce the problem of finding solutions to the cosmological equations of Jordan and Brans-Dicke theory of gravitation for the Robertson-Walker metrics and the cases of the dust universe and the vacuum universe. The reduction consists in a first-order differential equation and a quadrature for each case. Previously known cosmological solutions are re-obtained. In particular, it becomes apparent during the development of this scheme that the flat-space solutions are indeed the general solution.     

Numerical Solutions of Three-Dimensional Pressure-Bounded Magnetohydrostatic Flux Tubes

We present a three-dimensional technique for the solution of the magnetohydrostatic equations when we are modeling structures bounded by a current sheet that is free to move to satisfy pressure balance. The magnetic field is expressed in terms of Euler potentials and the equations are transformed to flux coordinates, greatly simplifying the problem of locating the free boundary. Multi-grid techniques are used to rapidly solve the resulting nonlinear elliptic partial differential equations. The method is tested against Low's (1982) exact solution of a bipolar plasma loop. It is shown that fast, accurate solutions can be found.     

利用3次方位观测天基初轨确定新方法

提出了一种适用于天基空间目标光学观测的初始轨道确定新方法. 通过对比地基和天基观测的几何构型, 分析了利用天基光学观测数据进行初轨确定时计算收敛到观测平台自身轨道的原因. 基于轨道半通径方程和改进Gauss方程, 推导出了斜距条件方程组的解析形式, 将天基光学观测的初轨确定问题转换为求解关于观测时刻斜距变量的非线性条件方程组的问题. 利用轨道能量约束减小了解的搜索区域, 消除了方程组的奇点. 最后利用天基实测数据验证并分析了非线性条件方程组根的性质, 利用低轨光学观测平台对低、中、高轨和大椭圆轨道空间目标的仿真观测数据验证了方法的有效性.     

A dipole field model for axisymmetric alfvén waves with finite ionosphere conductivities

An axisymmetric model for approximate solution of the magnetospheric Alfvén wave problem at latitudes above the plasmapause is proposed, in which a realistic dipole geometry is combined with finite anisotropic ionosphere conductivities, thus bringing together various ideas of previous authors. It is confirmed that the axisymmetric toroidal and poloidal modes interact via the ionospheric Hall effect, and an approximate method of solution is suggested using previously derived closed solutions of the uncoupled wave equations.A solution for zero Hall conductivity is obtained, which consists of sets of independent shell oscillations, regardless of the magnitude of the Pedersen conductivity. One set reduces to the classical solutions for infinite Pedersen conductivity, while another predicts a new set of harmonics of a quarter-wave fundamental, with longer eigenperiods than the classical solutions for a given L-shell.     

Numerical exploration of commensurable periodic solutions of the restricted problem of three bodies and their stability

The long period problem provides the initial conditions for numerical computation of close periodic solutions separated in three categories. For each type of commensurability a number of periodic solutions are computed and their stability is studied by computing the characteristic exponents of the matrizant. The Runge-Kutta method for the solution of differential equations of motion was used in all cases. The results obtained are presented for a four cases of commensurability.     

An elegant perturbation solution for the Lane–Emden equation of the second kind

Perturbation solutions are obtained for the Lane–Emden equation of the second kind which describe Bonnor–Ebert gas spheres. In particular, we employ the field-theoretic perturbative procedure due to Bender et al. to obtain analytical solutions to the nonlinear initial value problem. We find that the method allows one to construct perturbation solutions which converge rapidly to the true solutions in many cases, as it allows one to more accurately represent the influence of nonlinear terms in the linearized equations. The rapid convergence of the method results in qualitatively accurate solutions in relatively few iterations.     

A constructing method for a self-consistent three-dimensional solution of the magnetohydrostatic equations using full-disk magnetogram data

A technique is proposed for constructing self-consistent 3-D solutions satisfying the magnetohydrostatic (MHS) equations, and fitting observations along the line of sight of the magnetic field at the photosphere. The technique is a generalization of a potential-field extrapolation method (Rudenko, 2001) using full-disk magnetogram data. The solution of the problem under consideration is based on representing the magnetic field in terms of a scalar function, with its subsequent harmonic expansion in terms of the functional basic set of spherical functions that satisfies the specified boundary conditions. It is expected that a numerical realization of the proposed method will make possible a real-time modeling of the three-dimensional magnetic field, temperature, pressure and density distributions.     

Periodic solutions of the singly averaged hill problem

Based on the ideas of Lyapunov’s method, we construct a family of symmetric periodic solutions of the Hill problem averaged over the motion of a zero-mass point (a satellite). The low eccentricity of the satellite orbit and the sine of its inclination to the plane of motion of the perturbing body are parameters of the family. We compare the analytical solution with numerical solutions of the averaged evolutionary system and the rigorous (nonaveraged) equations of the restricted circular three-body problem.     

Elliptic Sitnikov five-body problem under radiation pressure

We consider the square configuration of photo-gravitational elliptic restricted five-body problem and study the Sitnikov motions. The four radiating primaries are of equal mass placed at the vertices of square and the fifth body having negligible mass performs oscillations along a straight line perpendicular to the orbital plane of the primaries. The motion of the fifth body is called vertical periodic motion and the main aim of this paper is to study the effect of radiation pressure on these periodic motions in the linear approximation. Moreover, the effects of radiation pressure on the motion of fifth body have been examined with the help of Poincare surfaces of section. By escalating the radiation pressure, surrounding periodic tubes and islands disappear and chaotic motion occurs near the hyperbolic points. Further, by escalating the radiation pressure, the main stochastic region joins the escaping one.