scholarly journals Low-Thrust Orbital Transfers in the Two-Body Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
A. A. Sukhanov ◽  
A. F. B. A. Prado
Keyword(s):  
Body Problem ◽  
Optimal Solution ◽  
Optimal Solutions ◽  
Low Thrust ◽  
Simple Method ◽  
Two Body Problem ◽  
Proper Number ◽  
Linearized Problem ◽  
Orbital Transfers ◽  
The Difference

Low-thrust transfers between given orbits within the two-body problem are considered; the thrust is assumed power limited. A simple method for obtaining the transfer trajectories based on the linearization of the motion near reference orbits is suggested. Required calculation accuracy can be reached by means of use of a proper number of the reference orbits. The method may be used in the case of a large number of the orbits around the attracting center; no averaging is necessary in this case. The suggested method also is applicable to the cases of partly given final orbit and if there are constraints on the thrust direction. The method gives an optimal solution to the linearized problem which is not optimal for the original nonlinear problem; the difference between the optimal solutions to the original and linearized problems is estimated using a numerical example. Also examples illustrating the method capacities are given.

Fast solution of minimum-time low-thrust transfer with eclipses

2018 ◽  
Vol 233 (7) ◽  
pp. 2699-2714
Author(s):  
Max Cerf
Keyword(s):  
Shooting Method ◽  
Optimal Solution ◽  
Search Space ◽  
Low Thrust ◽  
Indirect Methods ◽  
Transversality Conditions ◽  
Derivative Free ◽  
Fast Solution ◽  
Orbital Transfers ◽  
Zero Values

Optimizing low-thrust orbital transfers with eclipses by indirect methods raises several issues, namely the costate discontinuities at the eclipse entrance and exit, the initial costate guess sensitivity and the numerical accuracy required by the shooting method. The discontinuity issue is overcome by detecting the eclipse within the simulation and applying the costate jump derived analytically from the shadow constraint function. By fixing completely the targeted final position and velocity, the transversality conditions are removed and the shooting problem is recast as an unconstrained nonlinear programming problem. The numerical sensitivity issues are alleviated by using a derivative-free algorithm. The search space is reduced to four angles taking near zero values. This procedure yields a quasi-optimal solution from scratch in few minutes without requiring any specific user’s guess or tuning. The method is applicable whatever the thrust level and the eclipse configuration, as illustrated on transfers towards the geostationary orbit.

Comments on errors of “A simplified two-body problem in general relativity” by S Hod And Rectification of General Relativity

2016 ◽  
Vol 12 (1) ◽  
pp. 4188-4196 ◽  
Author(s):  
C. Y. Lo
Keyword(s):  
General Relativity ◽  
Gravitational Wave ◽  
Einstein Equation ◽  
Body Problem ◽  
Perturbation Approach ◽  
Common Error ◽  
Two Body Problem ◽  
Dynamic Solution ◽  
Dynamic Case ◽  
The Difference

Hod claimed to have a method to deal with a simplified two-body problem. The basic error of Hod and the previous researchers is that they failed to see that in general relativity there is no bounded dynamic solution for a two-body problem. A common error is that the linearized equation is considered as always providing a valid approximation in mathematics. However, validity of the linearization is proven only for the static and the stable cases when the gravitational wave is not involved. In a dynamic problem when gravitational wave is involved, since it is proven in 1995 that there is no bounded dynamic solution, the process of linearization is not valid in mathematics. This is the difference between Einstein and Gullstrand who suspected that a dynamic solution does not exist. In fact, for the dynamic case, the Einstein equation and the linearized equation are essentially independent equations, and the perturbation approach is not valid. Note that the linearized equation is a linearization of the Lorentz-Levi-Einstein equation, which has bounded dynamic solutions but not the Einstein equation, which has no bounded dynamic solution. Because of inadequacy in non-linear mathematics, many had made errors without knowing them.

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.

Planar High-Thrust and Low-Thrust Orbital Transfers from Earth to Europa

2004 ◽  
Vol 52 (4) ◽  
pp. 421-439
Author(s):  
Hans Seywald ◽  
Carlos M. Roithmayr ◽  
Daniel D. Mazanek ◽  
Frederic H. Stillwagen ◽  
Patrick A. Troutman ◽  
...  
Keyword(s):  
Low Thrust ◽  
Orbital Transfers ◽  
High Thrust

scholarly journals Temporal concatenation for Markov decision processes

2021 ◽  
pp. 1-28
Author(s):  
Ruiyang Song ◽  
Kuang Xu
Keyword(s):  
Markov Decision Processes ◽  
Large Scale ◽  
Optimal Solution ◽  
Upper Bounds ◽  
Black Box ◽  
Decision Processes ◽  
Optimal Solutions ◽  
Wide Range ◽  
Markov Decision ◽  
Speed Up

We propose and analyze a temporal concatenation heuristic for solving large-scale finite-horizon Markov decision processes (MDP), which divides the MDP into smaller sub-problems along the time horizon and generates an overall solution by simply concatenating the optimal solutions from these sub-problems. As a “black box” architecture, temporal concatenation works with a wide range of existing MDP algorithms. Our main results characterize the regret of temporal concatenation compared to the optimal solution. We provide upper bounds for general MDP instances, as well as a family of MDP instances in which the upper bounds are shown to be tight. Together, our results demonstrate temporal concatenation's potential of substantial speed-up at the expense of some performance degradation.

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