Power-law relationship between the semi-major axis and the rotation period ratio in an eccentric system: Its physical meaning and applications

Abstract This study aims to address longstanding questions about orbital motion, introduces a synchronous orbit radius ratio and investigates the relationship between the semi-major axes and synchronous orbit radius ratios of known systems with the same mass ratio orders of magnitude, the same spin–orbit angles and eccentric or circular orbits, including binaries, star–planet systems and planet–moon systems. The results of this work show that in a system with an eccentric orbit (e > 0.01) and a mass ratio order with a magnitude higher than −7.5217, the semi-major axis exhibits a power-law distribution relationship with the product of the mass ratio and square of the rotation period ratio. Furthermore, under spin–orbit alignment conditions, the power-law exponent exhibits a positive linear relationship with the mass ratio order of magnitude and is greater than zero and less than or equal to one. Moreover, the power-law exponent for a spin–orbit misaligned system correlates negatively with its spin–orbit angle. This power-law relationship was derived based on a drag force caused by the gravitational field rotation proposed by extending Einstein’s equivalence principle. This study concludes that the power-law relationship and drag force dominate the formation of various orbits.