Gravitational quantization of satellite orbits in the giant planets

A fundamental assumption in the so-called “global polytropic model” is hydrostatic equilibrium for a system of planets or statellites. By solving the Lane-Emden differential equation for such a system in the complex plane, we find polytropic spherical shells defined by succesive roots of the real part $\mathrm{R}\mathrm{e}(\theta )$ of the Lane-Emden function $\theta$. These shells seem to be appropriate places for accomodating planets or satellites. In the present study, we apply the global polytropic model to the systems of satellites of the giant planets: Jupiter, Saturn, Uranus, and Neptune.