3D Steady Euler Equation for Description of Nebular Structures Formation

Abstract

Stereographic projection of Hopf field on the 3‐sphere into Euclidean 3‐space is used for description of flow of ideal fluid in accordance with Arnold’s structural theorem for 3D steady Euler equation. In such case, flow lines are Villarceau circles lying on tori corresponding to the levels of Bernoulli function. Existence of an optimal torus with minimal relative surface free energy is shown. Corresponding inclination of Villarceau circles to accretion plane is π/4. Beat of oscillations with wave numbers corresponding to structural radii of “optimal torus” lying in accretion plane leads to scaling of “optimal tori” with factor k = 1+sin(π/4). “Optimal tori” are considered as precursors of planetary orbits and ring systems. Scaling factors of Solar system, directly observed multi‐planet HR8799 system, planetesimal belts of inner ring of Pictoris system, in three ring system HD113766, in binary ring systems HD141569, HD98800B, Vega precisely correspond to optimal one. The probability distribution of ratios of semi‐major axes of neighbor planets in the indirectly observed multiple‐exoplanet systems has two maxima. First maximum close to optimal scaling factor, and second one is similar to squared optimal scaling factor. Scaling factors of satellite systems had deviations depending on central body axial tilt and local ratio of semi‐major axes of neighbor planets that explained by coincidence of flow lines involved in toroidal flow around protoplanets and proto‐Sun.