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

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 $R_1$ represents the central component $S_1$ (Sun or planet) 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 $\mathrm{Re}(\theta)$ of the complex Lane-Emden function $\theta$. 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.