Non-integrability for Motion of Natural and Artificial Satellite in an Elliptic Orbit under the Influence of Aerodynamic Torque

This article deals with the non-linear oscillation of a satellite in an elliptic orbit around the Earth under the influence of aerodynamic and gravitational torque. It is assumed that the orbital plane coincides with the equatorial plane of the Earth. Using Euler's dynamical equation of motion, the equation of motion, Hamiltonian function and Melnikov's integral for the problem is derived. Taking aerodynamic torque of the order of eccentricity, it is observed that Hamiltonian function is dependent on aerodynamic torque. Using Melnikov's method it is observed that equations of motion are non-integrable. The non-integrability is observed graphically and numerically in Earth–Moon system and Earth–Resources at-1 system. It is concluded that for Earth–satellite system (Earth–Moon and Earth–Resourcesat-1) the Melnikov's function has simple zeros with the change in mass parameter and aerodynamic torque parameter, thus the non-linear system for equation of motion is non-integrable.