Gravitationally Quantized Orbits in the Solar System: Computations Based on the Global Polytropic Model

Abstract

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 <i>n</i> and radius <i>R</i> 1 represents the central component <i>S</i> 1 (Sun or planet) of a polytropic configuration with further components the polytropic spherical shells <i>S</i> 2 , <i>S</i> 3 , ..., defined by the pairs of radi (<i>R</i> 1 , <i>R</i> 2 ), (<i>R</i> 2 , <i>R</i> 3 ), ..., respectively. <i>R</i> 1 , <i>R</i> 2 , <i>R</i> 3 , ..., are the roots of the real part Re(<i>θ</i> ) of the complex Lane-Emden function <i>θ</i>. 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.