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The present work deals with constructing a conditionally periodic solution for the motion of an Earth artificial satellite taking into account the oblateness of the Earth and the Luni-Solar attractions. The oblateness of the Earth is truncated beyond the second zonal harmonic 2 J . The resonances resulting from the commensurability between the mean...
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Citations
... This can be accomplished via the averaging principle, authors are usually focusing on the specific resonant term and eliminating other periodic terms. The long-term behavior of the system is now governed by the reduced Hamiltonian that represents a system of one degree of freedom with the resonance angle as the action variable, see (Henrard & Lemaitre [10]; Feng et al. [11]; Abd El-Salam et al. [12] see Tan et al. [13]; Daquin et al. [14]; and Sukhov [15]. The second approach is studying the periodic orbits. ...
In this research work, the effects of perturbations due to resonant geopotential harmonics on the commensurate semi-major axis \(a_{c}^{{}}\)of the GPS orbits are being studied and analyzed. A second order approximation of the commensurate semi-major axis due to the higher zonal perturbations (up to \({J_6}\)) is obtained iteratively, starting with the unperturbed case as the zeroth-order. In the resonance cases a certain number of tesseral harmonics can produce effects of large amplitude and very long period, these important tesseral harmonics can be computed by Kaula’s resonant perturbation theory [1]. Using Lagrange’s planetary equations, the rate of change of the semi-major axis is computed. The drift rate as a function of longitudinal position is derived and plotted, revealing the existence of four roots which can be classified as stable positions at \({25.94^ \circ }\)E, \({205.9^ \circ }\)E and metastable positions at \({118.7^ \circ }\)E, \({298.7^ \circ }\)E. Motion about these stable and metastable points is investigated using the Poincare section method. The analysis reflects the existence of periodic, quasi-periodic, and even chaotic orbits in the vicinity of these points.
We have investigated the resonances due to the perturbations of a geo-centric synchronous satellite under the gravitational forces of the Sun, the Moon and the Earth including it’s equatorial ellipticity. The resonances at the points resulting from (i) the commensurability between \(\dot{\theta}_{0}\) (steady-state orbital angular rate of the satellite) and \(\dot{\theta}_{m}\) (angular velocity of the moon around the earth) and (ii) the commensurability between \(\dot{\theta}_{0}\) and \(\dot{\psi}_{0}\) (steady-state regression rate of the synchronous satellite) are analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure as given in Brown and Shook (Planetary Theory, Cambridge University Press, Cambridge, 1933). We have observed that as θ
m
(0∘≤θ
m
≤45∘) and ψ (0∘≤ψ≤135∘) increase, the amplitude decreases and the time period also decreases. We have also shown the effect of ψ on amplitude and time period for 0∘≤Γ≤45∘, where Γ is the angle measured from the minor axis of the earth’s equatorial ellipse to the projection of the satellite on the plane of the equator.
We have investigated the resonances in the earth-moon system around the sun including earth’s equatorial ellipticity. The resonance resulting from the commensurability between the mean motion of the moon and Γ (angle measured from the minor axis of the earth’s equatorial ellipse to the projection of the moon on the plane of the equator) is analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure of Brown and Shook. We have shown the effects of Γ on the amplitude and the time period of the resonance oscillation using the data of the moon. It is observed that the amplitude decreases and the time period also decreases as Γ increases from 0∘ to 45∘.
