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

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 hosting orbits to planets. On the basis of this model, we develop a numerical method which has three versions. In its three-dimensional version, the method is effective for systems with substantial uncertainties in the observed host star radius, and in the orbit of a particular planet (compared to the uncertainties in the orbits of the other planets); the method uses as fixed entry values the observed orbits of the remaining planets. In its two-dimensional version, the method is effective for systems with substantial uncertainty in the host star radius; in this case, the method uses as fixed entry values the observed orbits of the planets. The one-dimensional version was previously developed and applied to several systems; in this version, the observed values of the host star radius and of the planetary orbits are taken as fixed entry values. Our method can compute optimum values for the polytropic index of the global polytropic model which simulates the exoplanetary system, for the orbits of the planets, and (excluding the one-dimensional version) for the host star radius.