Gravitational quantization of exoplanet orbits in HD 40307, $\mu$ Ara, Kepler-26, Kepler-62, and Kepler-275: Comparing predicted orbits

Abstract

According to the so-called "global polytropic model", hydrostatic equilibrium for a planetary system leads to the Lane-Emden differential equation. Solving this equation in the complex plane, we obtain polytropic spherical shells defined by succesive roots of the real part $\mathrm{Re}(\theta)$ of the Lane-Emden function $\theta$. Such shells provide hosting orbits for the members of a planetary system. Within the framework of the global polytropic model, we study the exoplanet systems HD 40307, $\mu$ Ara, Kepler-26, Kepler-62, and Kepler-275. We give emphasis on comparing our results with observations, and with orbit predictions derived from the "generalized Titius-Bode relation".