Fully analytical solution of the electromagnetic perturbations on the motion of the charged satellites in earth’s magnetic field

In this work, the perturbations in all orbital elements due to the motion of the charged satellites in the Earth’s magnetic field using Lagrange’s planetary equations are solved exactly. The geomagnetic and electric force components in terms of orbital elements in full analytic expressions are derived. Lagrange’s planetary equations after substitution with the considered perturbations in Keplerian orbital elements are obtained. To solve the problem in full analytical way, some integrals by Ahmed (Astron. J. 107, 1900 (1994)) are utilized. The time rate of change of all orbital elements of the spacecraft are integrated and represented in matrix notations.