On the spin-orbit phase-space of an artificial satellite

Artificial satellites are usually in a spin-orbit synchronous state, keeping the same face pointed towards the Earth. In this work, we study the spin-orbit dynamics of an elongated satellite under the gravitational attraction of the Earth, without any kind of altitude control. The shape of the satellite is characterized by the asphericity parameter ω=3(B-A)/C,$$\omega =\sqrt{3(B-A)/C},$$ where A<B<C$$A<B<C$$ are the principal moments of inertia. Using the Poincaré surface of section technique, we explore the spin-orbit phase space of the satellite for two representative values of ω.$$\omega .$$ The orbital eccentricity is a crucial parameter in such a study. Then, the dynamics is explored for a range of eccentricity values. The location and size of the main spin-orbit resonances are identified. We pay special attention to the 1:1 synchronous spin-orbit resonance. The periodic orbit associated to the 1:1 resonance bifurcates at a critical value of ω,$$\omega ,$$ and an analytical model is capable of reproducing its topological evolution. For practical purposes, there is a limiting amplitude of oscillation of the satellite that meets the mission requirements for synchronous behavior. To exemplify this, a case is assumed in which the satellite must remain pointing to the Earth’s surface. For a wide range of ω,$$\omega ,$$ the maximum eccentricity is estimated as a function of the satellite altitude, and an analysis of the parameters that would be compatible with such mission requirements is presented.