A study of the errors of the averaged models in the restricted three-body problem in a short time scale
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A study of the errors of the averaged models in the restricted three-body problem in a short time scale R. C. Domingos · A. F. B. De Almeida Prado · R. Vilhena De Moraes
Received: 31 December 2013 / Revised: 29 March 2014 / Accepted: 26 April 2014 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2014
Abstract The objective of the present research is to study the accuracy of the double- and single-averaged models that are usually considered to predict the motion of spacecrafts or celestial bodies that have their motion perturbed by a third-body. Those two models are compared with each other and then validated against the complete elliptic restricted threebody problem. Those models are developed to give a faster but general behaviour of the motion of the perturbed body in a medium or longer time scale. The researches performed here verify the accuracy of those methods for shorter time scales by showing the differences in terms of the values of the inclination and eccentricity of the perturbed body predicted by those models. Those differences are calculated both at every instant of time and as an integral over the time. The use of the integral along the time for the errors is a new form to study those differences and show a more complete comparison of the accuracy of those approximations, completing the instantaneous picture given by the usual approach of looking at the instantaneous measurement. If the value of the integral is divided by the time integration, the mean error is obtained. The results show that the single-averaged model is better in the short time scale and the difference among those models is smaller when predicting the eccentricity than the inclination. The effects of the time scale is verified by varying the study to values from 2 days up to 50 revolutions of the Moon (1,366 days). Another important point found in the present paper is the range of the eccentricities of the perturbing body that accelerates the dynamics.
Communicated by Elbert Macau and Cristiano Fiorilo. R. C. Domingos (B) Univ. Estadual Paulista (UNESP), São João da Boa Vista, SP, Brazil e-mail: [email protected] A. F. B. De Almeida Prado Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, SP, Brazil e-mail: [email protected] R. Vilhena De Moraes Universidade Federal de São Paulo (UNIFESP), São José dos Campos, SP, Brazil e-mail: [email protected]
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Keywords
Averaged models · Integral approach · Celestial mechanics
Mathematics Subject Classification
37N05 · 70F15 · 70F16
1 Introduction There are several studies in the literature considering the third-body perturbation in a spacecraft travelling around the Earth. One of the first is written by Kozai (1959) that shows the development of secular and long-period terms of the luni-solar perturbations. Another work on that line is the one made by Cook (1962), which applied the Lagrange’s planetary equations to study the motion of a spacecraft perturbed by the long-period terms of the disturbing potentia
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