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 three-body 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.
Similar content being viewed by others
References
Kozai Y (1959) On the effects of the Sun and Moon upon the motion of a close Earth satellite, Smithsonian Inst. Astrophysical Observatory, Special Report 22
Cook GE (1962) Luni-solar perturbations of the orbit of an earth satellite. Geophys J R Astron Soc 6(3):271
Kozai Y (1962) Secular perturbations of asteroids with high inclination and eccentricity. Astron J 67:591
Kaula WM (1962) Development of the lunar and solar disturbing functions for a close satellite. Astron J 67:300
Giacaglia GEO (1973) Lunar perturbations on artificial satellites of the earth, Smithsonian Astrophysical Observatory, Special Report 352, 1
Kozai Y (1973) A new method to compute lunisolar perturbations in satellite motions, Smithsonian Astrophysical Observatory. Special Report 349, 27
Collins SK, Cefola PJ (1979) Double averaged third body model for prediction of super-synchronous orbits over long time spans. AIAA 64
Broucke RA (2003) J Guid Control Dyn 26(1):27
Prado AFBA (2003) J Guid Control Dyn 26(1):33
Carvalho JPS, Elipe A, Vilhena de Moraes R, Prado AFBA (2010) Some orbital characteristics of lunar artificial satellites. Celest Mech Dyn Astron 108:371
Carvalho JPS, Elipe A, Vilhena de Moraes R, Prado AFBA (2011) Planetary satellite orbiters: applications for the moon. Math Probl Eng 2011:1
Lara M (2011) Design of long-lifetime lunar orbits: a hybrid approach. Acta Astronaut 69(3–4):186–199
Paskowitz ME, Scheeres DJ (2006a) J Guid Control Dyn 29(5):1147
Paskowitz ME, Scheeres DJ (2006b) J Guid Control Dyn 29(2):342
Domingos RC, Vilhena de Moraes R, Prado AFBA (2008) Third-body perturbation in the case of elliptic orbits for the disturbing body. Math Probl Eng. doi:10.1155/2008/763654
Xiaodong L, Hexi B, Xingrui M (2012) Astrophys Space Sci 339:295
Domingos RC, Prado AFBA, Vilhena de Moraes R (2013a) Studying the behaviour of averaged models in the third body perturbation problem. Math Probl Eng. doi:10.1155/2013/260830
Domingos RC, Prado AFBA, Vilhena de Moraes R (2013b) A study of single- and double-averaged second-order models to evaluate third-body perturbation considering elliptic orbits for the perturbing body. J Phys Conf Ser. doi:10.1088/1742-6596/465/1/012017
Acknowledgments
The authors wish to express their appreciation for the support provided by grants # 473387/2012-3, 150195/2012-5 and 304700/2009-6, from the National Council for Scientific and Technological Development (CNPq); grants # 2011/09310-7, 2012/21023-6 and 2011/08171-3, from São Paulo Research Foundation (FAPESP) and the financial support from the National Council for the Improvement of Higher Education (CAPES).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Elbert Macau and Cristiano Fiorilo.
Rights and permissions
About this article
Cite this article
Domingos, R.C., De Almeida Prado, A.F.B. & Vilhena De Moraes, R. A study of the errors of the averaged models in the restricted three-body problem in a short time scale. Comp. Appl. Math. 34, 507–520 (2015). https://doi.org/10.1007/s40314-014-0148-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-014-0148-5