Recurrent Power Series Research Articles

On the integration of an axially symmetric galaxy model

A numerical exploration of the escape of a particle from a potential well modeling an axially symmetric galaxy requires the integration of millions of orbits over very long spans of time. In this paper, we propose an integration of the equations of motion of the system by recurrent power series. We prove the convergence of the method, compare the computational time of the recurrent series method against the Dormand-Prince method, and study the propagation of the truncation errors in the numerical solutions.

On the long term numerical integration of planetary rings

The N -body ring problem is a classical problem that describes the motion of particle attracted by the gravitational field of N + 1 primary bodies. These bodies are distributed in a planar ring configuration, that is, a central primary and N primaries of equal mass located at the vertices of a regular polygon that is rotating on its own plane about the center with a constant angular velocity. If N is large enough, this system models the motion of a small particle close to a planetary ring. The study of the escape of a particle from the potential well of this system requires the use of very accurate numerical integration methods, as we are interested in the integration of the problem over very long spans of time, and large values of N . In this paper, we analyze the integration of the equations of motion of this problem by recurrent power series, and compare the resulting numerical solutions against some other methods.

Numerical integration of the N-body ring problem by recurrent power series

The aim of this article is to present a method for the integration of the equations of motion of the N-body ring problem by means of recurrent power series. We prove that the solution is convergent for any set of initial conditions, excluding those corresponding to binary collisions.

A new algorithm of the recurrent power series method for the non-gravitational motion of comets

In this paper, an automatic symbolic version of the recurrent power series method has been established for integrating the equations of motion of comets taking into account the perturbations of the giant planets and the non-gravitational effects terms. This new version has the advantage of selecting the coefficients of the power series solutions from the equations of motion of the respective body directly. In this case, it is no need to make derivations of many recurrence relations for the power expansions both of cometary and planetary data. The program is also provided by optimum step size integrations and elaborated using Mathematica software packages. Application to the orbital motion of short-period comets is given.

Symbolic analytical solutions for the abundances differential equations of the Helium burning phase

AbstractIn this paper, a literal analytical solution is developed for the abundances differential equations of the helium burning phase in hot massive stars. The abundance for each of the basic elements 4He,12C,16O and 20Ne is obtained as a recurrent power series in time, which facilitates its symbolic and numerical evaluations. Numerical comparison between the present solution and the numerical integration of the differential equations for the abundances show good agreement.

Numerical integration of the variational equations of satellite orbits

A recurrent power series (RPS) method is constructed for the numerical integration of the equations of motion together with the variational equations of N point masses orbiting around an oblate spheroid. By the term “variational equations” we mean the equations of the partial derivatives of the bodies’ position and velocity components with respect to the initial conditions, the relative masses and the spheroid's oblateness coefficients J 2 and J 4. The construction of recursive relations for the partial derivatives involved in the variational equations is based on partial differentiation of the corresponding recursive relations for the integration of the equations of motion. Since the number of the auxiliary variables needed for this complex system becomes tremendously large when N>1, special care must be taken during computer implementation, so as to minimize the amount of computer memory needed as well as the cost in CPU time. The RPS method constructed in this way is tested for N=1,…,4 using real initial conditions of the Saturnian satellite system. For various sets of satellites, we monitor the behaviour of all the corresponding partial derivatives. The results show a prominent difference in the behaviour of the partial derivatives between resonant and non-resonant orbital systems.

Automatic programming of recurrent power series

The integration of differential equations by recurrent power series is a classical method in ODE. This method is valid on very long spans of integration and unusually large step-sizes. However, this method is rarely used, mainly since each problem requires a specific formulation. In this paper, we take advantage of the facilities of the modern general purpose algebraic manipulators to elaborate a general program that provides as output the Fortran code with the recurrent formulas needed for integrating any first order differential system. We present two applications, the Hénon–Heiles problem, and the van der Pol's oscillator. For the last one, we make numerical comparisons between the recurrent power series method and a well tested Runge–Kutta one.

Comparison of Numerical Methods for the Integration of Natural Satellite Systems

We present a new implementation of the recurrent power series (RPS) method which we have developed for the integration of the system of N satellites orbiting a point-mass planet. This implementation is proved to be more efficient than previously developed implementations of the same method. Furthermore, its comparison with two of the most popular numerical integration methods: the 10th-order Gauss–Jackson backward difference method and the Runge–Kutta–NystrRKN12(10)17M shows that the RPS method is more than one order of magnitude better in accuracy than the other two. Various test problems with one up to four satellites are used, with initial conditions obtained from ephemerides of the saturnian satellite system. For each of the three methods we find the values of the user-specified parameters (such as the method's step-size (h or tolerance (TOL)) that minimize the global error in the satellites' coordinates while keeping the computer time within reasonable limits. While the optimal values of the step-sizes for the methods GJ and RKN are all very small (less than T/100, the ones that are suitable for the RPS method are within the range: T/13<h<T/6 (T being the period of the innermost satellite of the problem). Comparing the results obtained by the three methods for these step-sizes and for the various test problems we observe the superiority of the RPS method over GJ in terms of accuracy and over RKN both in accuracy and in speed.

Linkage of all the apparitions of comet P/Wolf

We collected 602 astrometric observations from 14 apparitions of comet P/Wolf during the interval 1884–1993. Equations of motion of the comet have been integrated using recurrent power series including all the planetary perturbations. We also took into account nongravitational effects in the comet's motion using Marsden's method, but with the four parameters A, η, I, φ connected with the rotating nucleus of the comet; a precessional motion of the spin axis of the comet's nucleus was taken into account by determining the constant daily change of the angle φ. Thus, it was possible to link all the observations of the comet with the RMS residual equal to 01.65”.

Numerical Simulations of Planetary Systems of the Jupiter-Saturn Type

We made a numerical study of the General Three-Body Problem in two dimensions, with the intention to obtain some statistical estimates of the outcome of the system after a long time. Two different sets of masses were used. In the first series of experiments we use masses in the ratio of 0.95, 0.04 and 0.01. In the second series, we use masses that are exactly in the Sun-Jupiter-Saturn ratio. To facilitate the discussion, we use the names Sun, Jupiter and Saturn for the three masses, in both cases. In all our experiments, the orbit of Jupiter starts with zero eccentricity and with a unit radius. However, the orbit of Saturn varies in two ways: the initial value of the semi-major axis varies from 1.1 to 3.5 and the eccentricity from 0.0 to 0.75. In total about 4000 cases were run for the two series of masses. All the numerical integrations were done with the method of recurrent power series of order 14, in a heliocentric frame of reference, integrating thus eight simultaneous first-order differential equations. All integrations were performed for a maximum of 12,500 canonical units of time, corresponding to about 2000 revolutions of Jupiter. The cause of termination or type of catastrophe for the system has been determined in all cases. In most cases, this is a close approach of Saturn with Jupiter, followed by ejection of Saturn from the system.

Motion of artificial satellites in the set of Eulerian redundant parameters, II

In this paper the full recurrent power series solution is established for J2-gravity perturbed motion in terms of the Eulerian redundant parameters. Applications of the method for the problem of the final state prediction are illustrated by numerical examples of some typical ballistic missiles, a final state of very high accuracy is obtained for each case study.

Dynamical Problems in the Preparation of the Catalogue of One-Apparition Comets

Subtle effects influencing the results of orbit determination have to be taken into consideration when recomputing the cometary orbits for the Catalogue. It is very important to apply the numerical integration method, taking into account the specific conditions of the comet's motion, which almost eliminates a cumulation of numerical errors of integration. In this point of view the recurrent power series integration is adapted for cometary orbital computations. The optimal value of step-size of integration according to the required accuracy is calculated at every integration step. The method gives excellent results for “normal” comets as well as in the cases of close approaches of a comet to the Sun and to the planets.

An optimal (m+3)[m+4] Runge Kutta algorithm

Recurrent power series methods are particularly applicable to problems in celestial mechanics since the Taylor coefficients may be expressed by recurrence relations. However, as the number of Taylor coefficients increases as is often necessary because of accuracy requirements, the computing time grows prohibitively large. In order to avoid this unfavorable situation, Dr E. Fehlberg introduced in 1960 Runge-Kutta methods that use the firstm Taylor coefficients obtained by recursive relations, or some other technique. Optimalm-fold Runge-Kutta methods are introduced. Embedded methods of order (m+3)[m+4] and (m+4)[m+5] are presented which have coefficients that produce minimum local truncation errors for the higher order pair of solutions of the method, as well as providing a near maximum absolute stability region. It is emphasized that the methods are formulated such that the higher order pair of solutions is to be utilized. These optimal methods are compared to the existingm-fold methods for several test problems. The numerical comparisons show that the optimal methods are more efficient. It is stressed that these optimal methods are particularly efficient whenm is small.

Solution of theN-body problem with recurrent power series

A Recurrent Power Series solution is given for the classicalN-body problem. The application to numerical integration is also pointed out.

Stability of periodic orbits in the elliptic, restricted three-body problem.

A systematic study has been made of periodic orbits in the two-dimensional, elliptic, restricted three-body problem. All ranges of eccentricities, from 0 to 1, and mass-ratios, from 0 to J, have been investigated. Eleven hundred periodic orbits have been obtained. It is concluded that the elliptic problem behaves in a way which is completely different from the circular problem. The main difference is in the stability properties of the periodic orbits. Because of the nonexistence of the Jacobi integral (the elliptic problem is not conservative), the characteristic equation of the monodromy matrix does not have a pair of unit roots, in general. The stability is denned by two real numbers (stability indices) rather than one. Because of that, there are seven general classes of periodic orbits, according to their stability properties. The stability of the periodic orbits has been determined by numerically integrating the variational equations with a recurrent power series method. The results are in contrast with the circular problem, where there are only three classes of orbits (stability, even instability, and odd instability): in the elliptic problem there are one stable class and six unstable classes. The elliptic, restricted three-body problem can be considered as the prototype of all nonintegrable, nonconservative Hamiltonian systems, and in this paper, probably for the first time, a classification of the multipliers is given for these systems. I. Introduction

Numerical integration of an orbit and its concomitant variations by recurrent power series

Lorsqu'il s'agit de calculer tout en une fois une orbite et la resolvante de ses equations aux variations dans le probleme restreint des trois corps, l'integration par series de puissances recurrentes s'avere nettement plus avantageuse que d'autres procedes numeriques. A precision egale, les developpements en series ne prennent que 30% du temps mis par les methodes a pas multiples, et seulement 12 a 15% du temps employe par la methode deRunge-Kutta-Nystrom.

The computation of characteristic exponents in the planar restricted problem of three bodies

view Abstract Citations (34) References (8) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The computation of characteristic exponents in the planar restricted problem of three bodies Deprit, Andre ; Price, J. F. Abstract The canonical equations of motion in barycentric synodical Cartesian coordinates and momenta are integrable by means of recurrent power series; these series are proved to be convergent for initial conditions anywhere in the phase space except in the two phase planes of binary collisions. The integration by recurrent power series is extended to the variation equations. It is used to compute the monodromy matrix associated to the fundamental period of a periodic orbit. A simple formula is de- rived, which relates the trace of the monodromy matrix and the characteristic exponents. These numerical methods are applied to evaluate the characteristic exponents of Rabe's Trojan orbits; they are found to be of the stable type for the ovals, and of the unstable type for the horseshoe shaped orbit. When the periodic orbit is symmetric with respect to the axis of syzygies, four independent variational solutions computed only over half the period are shown to be sufficient for evaluating the characteristic exponents. Publication: The Astronomical Journal Pub Date: December 1965 DOI: 10.1086/109823 Bibcode: 1965AJ.....70..836D full text sources ADS |