Gravitational quantization of exoplanet orbits in 55 Cnc, $\upsilon$ And, Kepler-11, Kepler-20, and Kepler-90

In the framework of the so-called “global polytropic model”, we assume hydrostatic equilibrium for a planetary system, and solve the resulting Lane–Emden differential equation in the complex plane. We thus obtain 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 provide hosting orbits for the planets of the system(s) under consideration. In the present investigation, we study within this framework the exoplanet systems 55 Cnc, $\upsilon$ And, Kepler-11, Kepler-20, and Kepler-90.