scholarly journals CMMSE: Study of a new symmetric anomaly in the elliptic, hyperbolic and parabolic Keplerian motion

2021 ◽  
Author(s):  
jose antonio lópez ortí ◽  
Vicemte Agost Gómez ◽  
Miguel Barreda rochera
Keyword(s):  
Body Problem ◽  
True Anomaly ◽  
Keplerian Motion ◽  
Hyperbolic Motion ◽  
Two Body Problem ◽  
Movement Type ◽  
Numerical Integration Methods ◽  
The Mean ◽  
Temporal Variable ◽  
Integration Errors

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.

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
Keyword(s):  
Body Problem ◽  
Numerical Solutions ◽  
Phase Error ◽  
Integral Invariant ◽  
True Anomaly ◽  
Integration Time ◽  
Angular Momentum Vector ◽  
Orbital Elements ◽  
Eccentric Anomaly ◽  
Two Body Problem

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.

Dynamics of a stochastically perturbed two-body problem

2007 ◽  
Vol 463 (2080) ◽  
pp. 979-1003 ◽  
Author(s):  
Shambhu N Sharma ◽  
H Parthasarathy
Keyword(s):  
Body Problem ◽  
Random Force ◽  
Second Order ◽  
Differential Model ◽  
Two Body Problem ◽  
System Nonlinearity ◽  
The Mean ◽  
Perturbed Model ◽  
Perturbation Effect ◽  
Stochastically Perturbed

In classical mechanics, the two-body problem has been well studied. The governing equations form a system of two-coupled second-order nonlinear differential equations for the radial and angular coordinates. The perturbation induced by the astronomical disturbance like ‘dust’ is normally not considered in the orbit dynamics. Distributed dust produces an additional random force on the orbiting particle, which can be modelled as a random force having ‘Gaussian statistics’. The estimation of accurate positioning of the orbiting particle is not possible without accounting for the stochastic perturbation felt by the orbiting particle. The objective of this paper is to use the stochastic differential equation (SDE) formalism to study the effect of such disturbances on the orbiting body. Specifically, in this paper, we linearize SDEs about the mean of the state vector. The linearization operation performed above, transforms the system of SDEs into another system of SDEs that resembles a bilinear system, as described in signal processing and control literature. However, the mean trajectory of the resulting bilinear stochastic differential model does not preserve the perturbation effect felt by the orbiting particle; only the variance trajectory includes the perturbation effect. For this reason, the effectiveness of the dust-perturbed model is examined on the basis of the bilinear and second-order approximations of the system nonlinearity . The bilinear and second-order approximations of the system nonlinearity allow substantial simplifications for the numerical implementation and preserve some of the properties of the original stochastically perturbed model. Most notably, this paper reveals that the Brownian motion process is accurate to model and study the effect of dust perturbation on the orbiting particle. In addition, analytical findings are supported with finite difference method-based numerical simulations.

The post-Newtonian approximation

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Body Problem ◽  
Equations Of Motion ◽  
Two Body Problem ◽  
Newtonian Approximation ◽  
Actual Motion ◽  
Law Of Attraction ◽  
Universal Law ◽  
Two Bodies

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.

The two-body problem: an effective-one-body approach

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Body Problem ◽  
Equations Of Motion ◽  
Reaction Force ◽  
Kerr Black Hole ◽  
Radiation Reaction ◽  
Spherically Symmetric ◽  
Geodesic Motion ◽  
Two Body Problem ◽  
Symmetric Spacetime ◽  
Radiation Reaction Force

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.

A Fitted Runge-Kutta-Nyström Method with Six Stages for the Integration of the Two-Body Problem

2011 ◽  
Author(s):  
A. A. Kosti ◽  
Z. A. Anastassi ◽  
T. E. Simos ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
...  
Keyword(s):  
Body Problem ◽  
Two Body Problem ◽  
Runge Kutta
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