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The so-called "global polytropic model" is based on the assumption ofhydrostatic equilibrium for the solar system, or for a planet's system ofstatellites (like the jovian system), described by the Lane-Emden differentialequation. A polytropic sphere of polytropic index $n$ and radius $R_1$represents the central component $S_1$ (Sun or planet) of a polytropicconfiguration 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$\mathrm{Re}(\theta)$ of the complex Lane-Emden function $\theta$. Eachpolytropic shell is assumed to be an appropriate place for a planet, or aplanet's satellite, to be "born" and "Live". This scenario has been studiednumerically for the cases of the solar and the jovian systems. In the presentpaper, the Lane-Emden differential equation is solved numerically in thecomplex plane by using the Fortran code DCRKF54 (modified Runge-Kutta-Fehlbergcode of fourth and fifth order for solving initial value problems in thecomplex plane along complex paths). We include in our numerical study sometrans-Neptunian objects.

Gravitationally quantized orbits in the solar system: computations based on the global polytropic model