Ptolemaic Transformation in Keplerian Problem

We propose the Ptolemaic transformation: a canonical change of variables reducing the Keplerian motion to the form of a perturbed Hamiltonian problem. As a solution of the unperturbed case, the Ptolemaic variables define an intermediary orbit, accurate up to the first power of eccentricity, like in the kinematic model of Claudius Ptolemy. In order to normalize the perturbed Hamiltonian we modify the recurrent Lie series algorithm of HoriuuMersman. The modified algorithm accounts for the loss of a term's order during the evaluation of a Poisson bracket, and thus can be also applied in resonance problems. The normalized Hamiltonian consists of a single Keplerian term; the mean Ptolemaic variables occur to be trivial, linear functions of the Delaunay actions and angles. The generator of the transformation may serve to expand various functions in Poisson series of eccentricity and mean anomaly.     

On the equinoctial orbit elements

This paper investigates the equinoctial orbit elements for the two-body problem, showing that the associated matrices are free from singularities for zero eccentricities and zero and ninety degree inclinations. The matrix of the partial derivatives of the position and velocity vectors with respect to the orbit elements is given explicitly, together with the matrix of inverse partial derivatives, in order to facilitate construction of the matrizant (state transition matrix) corresponding to these elements. The Lagrange and Poisson bracket matrices are also given. The application of the equinoctial orbit elements to general and special perturbations is discussed.This work was initiated while the second author was a Postdoctoral Scholar in the School of Engineering and Applied Science, University of California, Los Angeles.     

Expansion of the Hamiltonian of the Two-Planetary Problem into the Poisson Series in All Elements: Application of the Poisson Series Processor

This paper is the third in a series of articles devoted to one of the basic problems of celestial mechanics: the study of the evolution of solar-type planetary systems. In the previous papers a brief review of the history and current state of the problem was given; the plan of the study was outlined; the Jacobi coordinates and the related osculating elements were introduced; the form of the Poisson expansion of the Hamiltonian in all elements was given; and the expansion coefficients for the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem were obtained (though with impure accuracy) by a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. In the present paper the expansion of the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem into the Poisson series in all elements is constructed with the help of the PSP Poisson series processor, which is capable of required accuracy.     

The Expansion of the Hamiltonian of the Planetary Problem into the Poisson Series in All Keplerian Elements (Theory)

The study of the evolution of planetary systems, primarily of the Solar System, is one of the basic problems of celestial mechanics. The stability of motion of giant planets on cosmogonic time scales was established by numerical and analytical methods, but the question about the evolution of orbits of terrestrial planets and arbitrary solar-type planetary systems remained open. This work initiates a series of papers allowing one to advance in solving the problem of the evolution of the solar-type planetary systems on cosmogonic time scales by using powerful analytical tools. In the first paper of this series, we choose the optimum reference system and obtain the Poisson series expansion of the Hamiltonian of the problem in all Keplerian elements. We propose to use the integral representation of the corresponding coefficients or the Poisson processor means instead of conventionally addressing any possible special functions. This approach extremely simplifies the algorithm. The next paper of this series deals with the calculation of the expansion coefficients.     

The elimination of short and intermediate period terms from the problem of a high altitude earth satellite

The Delaunay-Similar elements of Scheifele are applied to the problem of an Earth satellite that is perturbed by the Sun, Moon andJ ₂. All three effects are assumed to be the same order of magnitude. Since the external body terms depend explicitly on time, the time element appears as an additional angle variable. Also, the eccentric anomaly is used as a noncanonical auxiliary variable. A solution to the first Von Zeipel equation allows simultaneous elimination of short and intermediate period terms. The canonical transformation to mean elements is defined by a generating function that is a series involving Bessel coefficients.     

Perturbations in close binary systems produced by the tides lagging in latitude

The aim of this investigation is to present the periodic and secular perturbations of the orbital elements of close binary systems due to tidal lag in latitude. The variational equations of the problem of plane motion will be set up in terms of the rectengular componentsR, S, andW of the disturbing accelerations. These equations are highly nonlinear with respect to the orbital elements and we present analytic approximations to the effects produced by the perturbing acceleration due to dynamical tides lagging in latitude. The perturbed elements of the orbit have been expressed by means of Hansen coefficients in the compact form of summations.     

受摄限制性三体问题平动点线性稳定性的判断条件及在Robe问题的应用

给出了受摄限制性三体问题平动点线性稳定性的一个判断条件．条件只与平动点切映像的特征方程系数有关，使用方便．用判断条件，讨论了Robe问题平动点在阻力摄动下的线性稳定性，得到了Hallan等给出的Robe问题平动点在阻力摄动下的线性稳定范围．并改进了Giordanoc等的结果．     

Lie groups of motor integrals of generalized Kepler motion

The singularity of the Kepler motion can be eliminated by means of the spinor regularization. The extensive integrals of the Kepler motion form a Lie algebra with respect to the Poisson bracket operation. Mayer-Gürr has shown that in the caseH>0 the corresponding Lie group is the multiplicative group of all real 4×4 unimodular matrices SL(4,R). Kustaanheimo has posed the problem of the identification of the corresponding Lie groups in the elliptic and parabolic cases. We solve this problem here and use the opportunity to introduce the concept of the Clifford algebra which is needed in our solution.     

A third-order solution of Vinti's problem with explicit expressions for the poisson brackets

We have derived an explicit third-order solution of Vinti's problem including J₃. Poisson brackets for the elements a, e, s, M_s, ψ_s and $\text{Ω}{}_{\mathrm{s}}$ are given. These may be used in the construction of a third-order theory of artificial satellites.     

Perturbation theory in rectangular coordinates

For treating the perturbed two-body problem in rectangular coordinates a new method is developed. The method is based on the reduction of the variational equations of the two-body problem with arbitrary elements to the Jordan system. The equations of perturbed motion rewritten in the quasi-Jordan form are subjected to a transformation excluding fast variables and leading to a system governing the long term evolution of motion. The method may be easily extended to the problem of the heliocentric motion of the major planets. For performing this method on computer it is suitable to use facilities of Poissonian and Keplerian processors.     

The Expansion of the Hamiltonian of the Two-Planetary Problem into a Poisson Series in All Elements: Estimation and Direct Calculation of Coefficients

This is the second paper in a series of articles devoted to one of the basic problems of celestial mechanics: the evolution of solar-type planetary systems. In the first paper (Kholshevnikov et al., 2001), we reviewed the history and the current state of the issue, outlined the scheme of the study, introduced Jacobi coordinates and related osculating elements, and indicated the form of the Hamiltonian expansion into a Poisson series in all elements. In this paper, the expansion coefficients are found according to a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. At the first stage, we restricted our analysis to the two-planetary problem (Sun–Jupiter–Saturn). The general case will be investigated in a forthcoming paper.     

The Hamiltonian Dynamics of the Two Gyrostats Problem

The problem of two gyrostats in a central force field is considered. We prove that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system posseses symmetries. Using them we perform the reduction of the number of degrees of freedom. We show that at every stage of the reduction process, equations of motion are Hamiltonian and give explicit forms corresponding to non-canonical Poisson brackets. Finally, we study the case where one of the gyrostats has null gyrostatic momentum and we study the zero and the second order approximation, showing that all equilibria are unstable in the zero order approximation. This revised version was published online in July 2006 with corrections to the Cover Date.     

A criterion for the linear stability of theequilibrium points in the perturbed restricted three-body problem and its application in Robe''s problem

A criterion for the linear stability of the equilibrium points in the perturbed restricted three-body problem is given. This criterion is related only to the coefficients of the characteristic equation of the tangent map of an equilibrium point, and this is convenient to use. With this criterion, we have discussed the linear stability of the equilibrium points in the Robe problem under the perturbation of a drag force, derived the linearly stable region of the equilibrium point in the perturbed Robe's problem with the drag given by Hallen et al., and improved as well the results obtained by Giordano et al.     

有摄圆型限制性三体问题平动点稳定的充要条件及应用

摘要给出了一个判断有摄圆型限制性三体问题平动点稳定性的充要条件．该条件只依赖于平动点变分方程的特征方程系数的一个简单关系，使用很方便．用所得到的条件，讨论了任意外力摄动对经典圆型限制性三体问题三角平动点稳定性的影响和惯性阻力摄动对Robe圆型限制性三体问题主要平动点的稳定性的影响．     

Improved semi-analytical computation of center manifolds near collinear libration points

In the framework of the circular restricted three-body problem, the center manifolds associated with collinear libration points contain all the bounded orbits moving around these points. Semianalytical computation of the center manifolds and the associated canonical transformation are valuable tools for exploring the design space of libration point missions. This paper deals with the refinement of reduction to the center manifold procedure. In order to reduce the amount of calculation needed and avoid repetitive computation of the Poisson bracket, a modified method is presented. By using a polynomial optimization technique, the coordinate transformation is conducted more efficiently. In addition, an alternative way to do the canonical coordinate transformation is discussed, which complements the classical approach. Numerical simulation confirms that more accurate and efficient numerical exploration of the center manifold is made possible by using the refined method.     

Equivalence of the perturbation theories of Hori and Deprit

A comparison is made between the perturbation theories of Hori and Deprit which are based on the use of Poisson brackets. General recurrence formulae are presented for Hori's theory which are analogous to those in Deprit's theory. Explicit relations between the determining functions for the two theories are indicated through the sixth order, these results having been obtained by a novel computer program. A general argument for the equivalence of the theories to all orders is given.     

On the motion of a mass point inside a rotating inhomogeneous ellipsoidal body

The plane motion of a mass point inside an inhomogeneous rotating ellipsoidal body with a homothetic density distribution is considered. The force function of the problem is expanded in terms of the ellipsoid's second eccentricities up to the fourth order, which are taken as small parameters. We derive an expression for the perturbing function and solve the equations of perturbed motion in orbital elements.     

Conditions for the linear stability of libration points and effects of drag on the linear stability of triangular libration points in the restricted three-body problem

A number of criteria for linear stability of libration points in the perturbed restricted three-body problem are presented. The criteria involve only the coefficients of the characteristic equation of the tangent map of the libration points and can be easily applied. With these criteria the effect of drag on the linear stability of the triangular libration points in the classical restricted three-body problem is investigated. Some of Murray et al.'s results are improved.     

Application of the stroboscopic method to the stellar three-body problem

The stellar three-body problem has been approached by directly integrating the equations of motion in the orbital elements. The problem is set up in a barycentric chain with the orbits as perturbed ellipses.The integration was performed using the semi-analytical stroboscopic method, which is particularly useful for solving differential systems that depend on several slow variables and one fast, angular variable. Perturbation theory is applied and the solution for a particular order is obtained by way of successive approximations on the fundamental period of the fast variable.     

Asymptotic solution for the two-body problem with constant tangential thrust acceleration

An analytical solution of the two body problem perturbed by a constant tangential acceleration is derived with the aid of perturbation theory. The solution, which is valid for circular and elliptic orbits with generic eccentricity, describes the instantaneous time variation of all orbital elements. A comparison with high-accuracy numerical results shows that the analytical method can be effectively applied to multiple-revolution low-thrust orbit transfer around planets and in interplanetary space with negligible error.