Analysis of resonant curves and phase portrait in the earth–moon system by using its unperturbed solution and earth’s equatorial ellipticity

In this research article, we have investigated resonant curves due to the rate of change of earth’s equatorial ellipticity parameter $\left(\dot{\gamma }\right)$, steady-state value of the angular velocity of the moon $\left({\dot{\theta }}_{mo}\right)$, and angular velocity of barycenter $\left(\dot{{\alpha }_0}\right)$ in the Earth-Moon system. Equations of motion of the moon are determined in a spherical coordinate system with the help of the gravitational potential of the earth. By using the unperturbed solution, equations of motion of the moon reduced into the second-order differential equation. From the solution it is observed that resonance occurs due to the frequencies $\dot{\gamma }$, ${\dot{\theta }}_{m0}$, and $\dot{{\alpha }_0}$ at the resonant points ${\dot{\theta }}_{m0}=2\dot{\gamma }$, $3{\dot{\theta }}_{m0}=2\dot{\gamma }$, ${\dot{\theta }}_{m0}=\dot{\gamma }$, ${\dot{\theta }}_{m0}={\dot{\alpha }}_0$. Finally, we have analyzed the phase portrait and phase space by method of Poincaré section when the system is free from forces.