Second-order theory for the two-body problem with varying mass including periastron effect

2017 ◽  
Vol 88 (3) ◽  
pp. 1723-1732 ◽  
Author(s):  
M. I. El-Saftawy ◽  
F. A. Abd El-Salam
Keyword(s):  
Body Problem ◽  
Order Theory ◽  
Second Order ◽  
Two Body Problem ◽  
Varying Mass

Analytical treatment of the two-body problem with slowly varying mass

2009 ◽  
Vol 30 (3-4) ◽  
pp. 187-205 ◽  
Author(s):  
W. A. Rahoma ◽  
F. A. Abd El-Salam ◽  
M. K. Ahmed
Keyword(s):  
Body Problem ◽  
Analytical Treatment ◽  
Two Body Problem ◽  
Varying Mass

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.

scholarly journals Surface wavepackets subject to an abrupt depth change. Part 1. Second-order theory

2021 ◽  
Vol 915 ◽  
Author(s):  
Yan Li ◽  
Yaokun Zheng ◽  
Zhiliang Lin ◽  
Thomas A.A. Adcock ◽  
Ton S. van den Bremer
Keyword(s):  
Order Theory ◽  
Second Order

Abstract

Invariant relative orbits for satellite constellations: A second order theory

2006 ◽  
Vol 181 (1) ◽  
pp. 6-20 ◽  
Author(s):  
F.A. Abd El-Salam ◽  
I.A. El-Tohamy ◽  
M.K. Ahmed ◽  
W.A. Rahoma ◽  
M.A. Rassem
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Satellite Constellations
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