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

**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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**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