scholarly journals The use of Kepler solver in numerical integrations of quasi-Keplerian orbits

2020 ◽  
Vol 496 (3) ◽  
pp. 2946-2961
Author(s):  
Chen Deng ◽  
Xin Wu ◽  
Enwei Liang

ABSTRACT A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the eccentric anomaly relies on the true anomaly that is associated with a unit radial vector calculated by an integrator. This scheme rigorously conserves all integrals and orbital elements except the mean longitude. However, the Kepler energy, angular momentum vector, and Laplace–Runge–Lenz vector for perturbed Kepler problems are slowly varying quantities. However, their integral invariant relations give the quantities high-precision values that directly govern five slowly varying orbital elements. These elements combined with the eccentric anomaly determine the desired numerical solutions. The newly proposed method can considerably reduce various errors for a post-Newtonian two-body problem compared with an uncorrected integrator, making it suitable for a dissipative two-body problem. Spurious secular changes of some elements or quasi-integrals in the outer Solar system may be caused by short integration times of the fourth-order Runge–Kutta algorithm. However, they can be eliminated in a long integration time of 108 yr by the proposed method, similar to Wisdom–Holman second-order symplectic integrator. The proposed method has an advantage over the symplectic algorithm in the accuracy but gives a larger slope to the phase error growth.

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
José Antonio López Ortí ◽  
Francisco José Marco Castillo ◽  
María José Martínez Usó

This paper aimed to address the study of a new family of anomalies, called natural anomalies, defined as a one-parameter convex linear combination of the true and secondary anomalies, measured from the primary and the secondary focus of the ellipse, and its use in the study of analytical and numerical solutions of perturbed two-body problem. We take two approaches: first, the study of the analytical development of the basic quantities of the two-body problem to be used in the analytical theories of the planetary motion and second, the study of the minimization of the errors in the numerical integration by an appropriate choice of parameters in our family for each value of the eccentricity. The use of an appropriate value of the parameter can improve the length of the developments in the analytical theories and reduce the errors in the case of the numerical integration.


2020 ◽  
Vol 160 (5) ◽  
pp. 203
Author(s):  
Alberto Abad ◽  
Manuel Calvo ◽  
José A. Docobo ◽  
Antonio Elipe

Author(s):  
jose antonio lópez ortí ◽  
Vicemte Agost Gómez ◽  
Miguel Barreda rochera

In the present work, we define a new anomaly, $\Psi$, termed semifocal anomaly. It is determined by the mean between the true anomaly, $f$, and the antifocal anomaly, $f^{\prime}$; Fukushima defined $f^{\prime}$ as the angle between the periapsis and the secondary around the empty focus. In this first part of the paper, we take an approach to the study of the semifocal anomaly in the hyperbolic motion and in the limit case correspoding to the parabolic movement. From here we find a relation beetween the semifocal anomaly and the true anomaly that holds independently of the movement type. We focus on the study of the two-body problem when this new anomaly is used as the temporal variable.\\ In the second part, we show the use of this anomaly —combined with numerical integration methods— to improve integration errors in one revolution. Finally, we analyze the errors committed in the integration process —depending on several values of the eccentricity— for the elliptic, parabolic and hyperbolic cases in the apsidal region.


2015 ◽  
Vol 454 (3) ◽  
pp. 2890-2908 ◽  
Author(s):  
Giulio Baù ◽  
Claudio Bombardelli ◽  
Jesús Peláez ◽  
Enrico Lorenzini

Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.


Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan

This chapter presents the basics of the ‘effective-one-body’ approach to the two-body problem in general relativity. It also shows that the 2PN equations of motion can be mapped. This can be done by means of an appropriate canonical transformation, to a geodesic motion in a static, spherically symmetric spacetime, thus considerably simplifying the dynamics. Then, including the 2.5PN radiation reaction force in the (resummed) equations of motion, this chapter provides the waveform during the inspiral, merger, and ringdown phases of the coalescence of two non-spinning black holes into a final Kerr black hole. The chapter also comments on the current developments of this approach, which is instrumental in building the libraries of waveform templates that are needed to analyze the data collected by the current gravitational wave detectors.


2011 ◽  
Author(s):  
A. A. Kosti ◽  
Z. A. Anastassi ◽  
T. E. Simos ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
...  

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