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In the so-called "global polytropic model", we assume planetary systems inhydrostatic equilibrium and solve the Lane--Emden equation in the complexplane. We thus find polytropic spherical shells providing hosting orbits toplanets. On the basis of this model, we develop a numerical method which hasthree versions. In its three-dimensional version, the method is effective forsystems with substantial uncertainties in the observed host star radius, and inthe orbit of a particular planet (compared to the uncertainties in the orbitsof the other planets); the method uses as fixed entry values the observedorbits of the remaining planets. In its two-dimensional version, the method iseffective for systems with substantial uncertainty in the host star radius; inthis case, the method uses as fixed entry values the observed orbits of theplanets. The one-dimensional version was previously developed and applied toseveral systems; in this version, the observed values of the host star radiusand of the planetary orbits are taken as fixed entry values. Our method cancompute optimum values for the polytropic index of the global polytropic modelwhich simulates the exoplanetary system, for the orbits of the planets, and(excluding the one-dimensional version) for the host star radius.

international journal of modern physics c

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