11th Hel.A.S Conference Athens, 9-12 September, 2013 CONTRIBUTED POSTER The global polytropic model for the solar and jovian systems revisited V.S. Geroyannis 1 , F.N. Valvi 2 , and T.G. Dallas 3 1 Department of Physics, University of Patras 2 Department of Mathematics, University of Patras 3 Department of History, Archaeology and Social Anthropology, University of Thessaly Abstract: The so-called “global polytropic model”is based on the assumption of hydrostatic equi- librium for the solar/jovian system, described by the Lane-Emden differential equation. A polytropic sphere of polytropic index n and radius R 1 represents the central component S 1 (Sun/Jupiter) of a polytropic configuration with further components the polytropic spherical shells S 2 , S 3 , ..., defined by the pairs of radii (R 1 ,R 2 ), (R 2 ,R 3 ), ..., respectively. R 1 ,R 2 ,R 3 ,... , are the roots of the real part Re(θ(R)) of the complex Lane-Emden function θ(R). Each polytropic shell is assumed to be an appropriate place for a planet/satellite to be “born” and “live”. This scenario has been studied numerically for the cases of the solar and the jovian systems. In the present paper, the Lane-Emden differential equation is solved numerically in the complex plane by using the Fortran code DCRKF54 (modified Runge-Kutta-Fehlberg code of fourth and fifth order for solving initial value problems in the complex plane along complex paths). We include in our numerical study some trans-Neptunian objects. 1 The Complex-plane Strategy 1.1 A review of the complex-plane strategy The so-called “complex-plane strategy” (CPS) proposes and applies numerical integration of “ordinary differential equations” (ODE, ODEs) in the complex plane, either along an interval I r ⊂ R when the independent variable r is real, or along a contour C ⊂ C when r is complex. Integrating in C is necessary when the “initial value problem” (IVP, IVPs) under consideration is defined on ODEs: (i) suffering from singularities and/or indeterminate forms in R, and/or (ii) involving terms that become undefined in R when the independent variable r exceeds a particular value. To the best of the knowledge of the authors, CPS is the first method that has used in certain problems of astrophysics the alternative of transforming real-valued functions of the real distance r ∈ R into complex-valued functions of the “complex distance” ([7], Sec. 3), r ∈ C, in order to avoid all pathologies. Thus the complex path C, along which integration proceeds, is a “complex distance detour”. CPS has been applied to astrophysical problems in which the polytropic “equation of state” (EOS, EOSs) is involved, or other EOSs with similar mathematical structure. Such problems are the poly- tropic models (see e.g. [3]); the solar system of planets and the jovian system of satellites ([4], [7]); white dwarf models obeying Chandrasekhar’s EOS (see e.g. [6]); and the general-relativistic polytropic models simulating neutron stars [8]. CPS extends numerical integration of the differential equations of an IVP well beyond the radius R of the nonrotating model instead of terminating integration just below R. Thus CPS knows the distortion caused by rotation over a sufficiently extended space surrounding the initially spherical configuration. So, to compute a particular rotating model, CPS does not extrapolate beyond the end of the function tables constructed by such extended numerical integrations. It is exactly the avoidance of any extrapolation which keeps the error in the computations appreciably small (note that, in numerical 1