Ref; https://academic.oup.com/mnras/article/449/4/3522/1184971
[...]consequence of the disparity of several orders of magnitude between the orbit precession periods of the Earth and the Moon
[...]RESULTS AND DISCUSSION
These considerations have led us to investigate the role of lunar secular resonances in producing chaos and instability among the navigation satellites. We treat only the secular resonances of the lunar origin, since, despite recent implications to the contrary (Bordovitsyna et al. 2014), the solar counterparts will require much longer timespans than what interests us here for their destabilizing effects to manifest themselves (Ely 2002) – a consequence of the disparity of several orders of magnitude between the orbit precession periods of the Earth and the Moon. As a basis for our calculations, we have used a convenient trigonometric series development by Hughes (1980), corresponding largely to a harmonic analysis of the perturbations. The lunar disturbing function is developed as a Fourier series of complicated structure, whose arguments are combinations of the orbital phase and orientation angles of the satellite and the Moon, and whose coefficients depend on the size and shape (semimajor axes and eccentricities) of their orbits and the inclinations. Two considerable simplifications are possible, reducing this rather formidable expression in a marked degree. For satellites whose semimajor axis does not exceed one tenth of the Moon's distance from the Earth, we can truncate the series to second order in the ratio of semimajor axes, so that the lunar potential is approximated with sufficient accuracy by a quadrupole field (Musen 1961; Delhaise & Morbidelli 1993; Rosengren & Scheeres 2013; Tremaine & Yavetz 2014). To study the secular interactions, the short periodic terms of the disturbing function, depending on the mean anomalies of both the satellite and the Moon, can be averaged out (Morbidelli 2002).
[...] In the inner Solar system, overlapping secular resonances have been identified as the origin of chaos in the orbits of the terrestrial planets
[...] This approximation is satisfactory when compared with the rigorous theory. Neglect of the lunisolar perturbations on the frequencies of nodal and apsidal precession sets an upper limit to the radius of the orbit for which the theory is valid; on the other hand, the period of precession must be appreciably longer than a year for the double averaging procedure to be justifiable, which sets a LOWER LIMIT TO THE ORBITAL RADIUS. For these reasons, the analysis is most useful in the region of semimajor axes between 4 and 5 Earth radii.
[...] To depict such curves up to e = 1 is only theoretical, as the satellites will re-enter Earth's atmosphere when e > 1 − R/a. Nevertheless, we are mainly interested here in the early stages in the development of chaos.
http://academic.oup.com/mnras/article/449/4/3522/1184971
[...]consequence of the disparity of several orders of magnitude between the orbit precession periods of the Earth and the Moon
[...]RESULTS AND DISCUSSION
These considerations have led us to investigate the role of lunar secular resonances in producing chaos and instability among the navigation satellites. We treat only the secular resonances of the lunar origin, since, despite recent implications to the contrary (Bordovitsyna et al. 2014), the solar counterparts will require much longer timespans than what interests us here for their destabilizing effects to manifest themselves (Ely 2002) – a consequence of the disparity of several orders of magnitude between the orbit precession periods of the Earth and the Moon. As a basis for our calculations, we have used a convenient trigonometric series development by Hughes (1980), corresponding largely to a harmonic analysis of the perturbations. The lunar disturbing function is developed as a Fourier series of complicated structure, whose arguments are combinations of the orbital phase and orientation angles of the satellite and the Moon, and whose coefficients depend on the size and shape (semimajor axes and eccentricities) of their orbits and the inclinations. Two considerable simplifications are possible, reducing this rather formidable expression in a marked degree. For satellites whose semimajor axis does not exceed one tenth of the Moon's distance from the Earth, we can truncate the series to second order in the ratio of semimajor axes, so that the lunar potential is approximated with sufficient accuracy by a quadrupole field (Musen 1961; Delhaise & Morbidelli 1993; Rosengren & Scheeres 2013; Tremaine & Yavetz 2014). To study the secular interactions, the short periodic terms of the disturbing function, depending on the mean anomalies of both the satellite and the Moon, can be averaged out (Morbidelli 2002).
[...] In the inner Solar system, overlapping secular resonances have been identified as the origin of chaos in the orbits of the terrestrial planets
[...] This approximation is satisfactory when compared with the rigorous theory. Neglect of the lunisolar perturbations on the frequencies of nodal and apsidal precession sets an upper limit to the radius of the orbit for which the theory is valid; on the other hand, the period of precession must be appreciably longer than a year for the double averaging procedure to be justifiable, which sets a LOWER LIMIT TO THE ORBITAL RADIUS. For these reasons, the analysis is most useful in the region of semimajor axes between 4 and 5 Earth radii.
[...] To depict such curves up to e = 1 is only theoretical, as the satellites will re-enter Earth's atmosphere when e > 1 − R/a. Nevertheless, we are mainly interested here in the early stages in the development of chaos.
http://academic.oup.com/mnras/article/449/4/3522/1184971
