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 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.