The so-called “global polytropic model” is based on the assumption of hydrostatic equilibrium for the solar system, or for a planet’s system of statellites (like the Jovian system), described by the Lane-Emden differential equation. A polytropic sphere of polytropic index n and radius R₁ represents the central component S₁ (Sun or planet) of a polytropic configuration with further components the polytropic spherical shells S₂, S₃, ..., defined by the pairs of radi (R₁, R₂), (R₂, R₃), ..., respectively. R₁, R₂, R₃, ..., are the roots of the real part Re(θ) of the complex Lane-Emden function θ. Each polytropic shell is assumed to be an appropriate place for a planet, or a planet’s 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.