A numerical method for computing optimum radii of host stars and orbits of planets, with application to Kepler-11, Kepler-90, Kepler-215, HD 10180, HD 34445 and TRAPPIST-1

Abstract

In the so-called “global polytropic model”, we assume planetary systems in hydrostatic equilibrium and solve the Lane–Emden equation in the complex plane. We thus find polytropic spherical shells providing accommodation to planetary orbits. On the basis of this model, we develop a numerical method which can compute optimum values for the polytropic index of the global polytropic model that simulates the planetary system, for the orbits of the planets, and for the host star radius. We apply our method to the exoplanetary systems Kepler-11, Kepler-90, Kepler-215, HD 10180, HD 34445 and TRAPPIST-1.