Effect of Earth’s Equatorial Ellipticity on the Resonant Curve and Phase Portrait of Geo-centric Satellite Under the Gravitational Effect of the Earth–Moon–Sun System by Using Unperturbed Solution

Abstract

In this article, we investigated the resonant curves of geo-centric satellite due to the rate of change in earth’s equatorial ellipticity parameter $(\dot{\gamma })$, moon’s angular velocity around the earth $(\dot{\vartheta })$, and angular velocity of bary-center $(\dot{\Theta })$ around the sun. We expressed the equations of the motion of the satellite in a spherical coordinate system using the earth’s potential. The system of equations are reduced into a second-order ordinary differential equation and determined its solution. From the solution, it is noted that resonance appears due to the frequencies $\dot{\delta _{s0 }}$, $\dot{\gamma }$, $\dot{\Theta _0}$ and $\dot{\vartheta _0}$ at ten resonant points ${\dot{\delta }_{s0}}={2\dot{\gamma }}$, ${\dot{\delta }_{s0}}={\dot{\gamma }}$, ${\dot{\delta }_{s0}}={2\dot{\Theta }_0}$, ${\dot{\delta }_{s0}}={\dot{\Theta }_0}$, ${\dot{\delta }_{s0}}=\frac{{2 \dot{\Theta }_0}}{3}$, ${\dot{\delta }_{s0}}=\frac{\dot{\Theta }_0}{2}$, ${\dot{\delta }_{s0}={2\dot{\vartheta _0}}}$, ${\dot{\delta }_{s0}={\dot{\vartheta _0}}}$, ${\dot{\delta }_{s0}=\frac{{2\dot{\vartheta _0}}}{3}}$, and ${\dot{\delta }_{s0}=\frac{{dot{\vartheta _0}}}{2}}$, where ${\dot{\delta }_{s0}}$, ${\dot{\Theta }_0}$ and ${\dot{\vartheta _0}}$ are the steady-state value of ${\dot{\delta }_{s}}$, ${\dot{\Theta }}$ and ${\dot{\vartheta }}$ respectively. We have drawn the resonant curves corresponding to the resonant points. From the graph, it is observed that the effect of earth’s equatorial ellipticity parameter on the resonant curves are very small. Also, we have shown the effect of eccentricity (e) and semi-major axes (a) on the oscillatory amplitudes $(A_i)$, where $i=1,2,3\ldots 21$. Finally, we have analyzed the phase portrait by using Poincaré section method when the system is free from forces.