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# Invariant relative orbits for satellite constellations: A second order theory

Faculty of Science, Cairo University, Al Qāhirah, Muḩāfaz̧at al Qāhirah, Egypt
(Impact Factor: 1.55). 10/2006; 181(1):6-20. DOI: 10.1016/j.amc.2006.01.004
Source: dx.doi.org

ABSTRACT

Working with the mean orbit elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits. By having both orbits drift at equal angular rates on the average, they will not separate over time due to the influence of the perturbative effects of the asphericity of the Earth, as is considered in this work. The problem is stated. The expressions for the time rate of change of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly in terms of the Delaunay canonical elements are obtained. The expressions for the second order conditions that guarantee that the drift rates of two neighboring orbits are equal on the average are derived.

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• "Now we need to eliminate the short as well as the long periodic terms of the satellite motion in addition to the short periodic terms of the distance perturbing body. Using the perturbation technique based on Lie series and Lie transform, Kamel [9], the transformed Hamiltonian, for different orders 0, 1, 2 can be written as, Abd El- Salam et al. [7] and Domingos et al. [8] "
##### Article: Invariant Relative Orbits Taking into Account Third-Body Perturbation
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ABSTRACT: For a satellite in an orbit of more than 1600 km in altitude, the effects of Sun and Moon on the orbit can’t be negligible. Working with mean orbital elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits to negate the separation over time due to the potential of the Earth and the third body effect. The expressions for the second order conditions that guarantee that the drift rates of two neighboring orbits are equal on the average are derived. To this end, the Hamiltonian was developed. The expressions for the non-vanishing time rate of change of canonical elements are obtained.
Applied Mathematics 02/2012; 3(02):113-120. DOI:10.4236/am.2012.32018 · 0.19 Impact Factor
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##### Article: Second order constraints in the theory of invariant relative orbits including relativistic and direct solar radiation pressure effects
[Hide abstract]
ABSTRACT: Working with the mean orbit elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits. By having both orbits drift at equal angular rates on the average, they will not separate over time due to the influence of the perturbative effects of the asphericity of the Earth. The relativistic corrections and the direct solar radiation pressure to the equations of motion, as is considered in this work. The problem is stated. The expressions for the time rate of change of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly in terms of the augmented Delaunay canonical elements are obtained. The expressions for the second order conditions that guarantee that the drift rates of two neighboring orbits are equal on the average are derived.
Indian Journal of Science and Technology 09/2012; 5(9):1-14. · 1.05 Impact Factor
• ##### Article: Lunisolar invariant relative orbits
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ABSTRACT: The present study deal with constructing an analytical model within Hamiltonian formulation to design invariant relative orbits due to the perturbation of J2 and the lunisolar attraction. To fade the secular drift separation over the time between two neighboring orbits, two second order conditions that guarantee that drift are derived and enforced to be equal.
American Journal of Applied Sciences 04/2013; 10(4):307-312. DOI:10.3844/ajassp.2013.307.312