Analysis of Resonant Curve in a Synchronous Satellite under the Gravitational Effect of the Sun, the Moon, and the Earth Including Its Oblateness using Poincare Section

Abstract

In this paper, we have studied the effect of oblateness of the Earth on the resonant curve in synchronous satellite under the gravitational effect of the Sun, the Moon and the Earth. By following the procedure of Frick and Garber (1962), we obtained the equations of motion of the synchronous satellite in the form of a second order linear differential equation by using the perturbation technique. From the solution, we observed that oscillatory amplitude becomes infinitely large when the ratio of two frequencies $\dot {\alpha }$ (angular velocity of the bary-center system around the Sun) and ${{\dot {\theta }}_{{\text{E}}}}$ (spin rate of Earth) are commensurable. With the help of different graphs, we have shown the effect of $\alpha$ and ${{J}_{2}}$ (coefficient of Earth’s oblateness) on the resonant curves. It is observed that oscillatory amplitude increases when $\alpha$ (orbital angle of the bary-center system around the Sun) increases and the value of ${{J}_{2}}$ is very close to zero. We also analyzed the phase portrait and phase space by using the method of Poincare section when the system is free from all the forces. Finally, we have obtained the expression for the energy integral, motion of mean longitude near the synchronous satellite.