Power-law relationship between the semi-major axis and rotation period ratio in an eccentric system: Its physical meaning and various applications
Authors: Dai Yongfeng 
Published: February 2024

Abstract
This study aims to address long-term questions about orbital motion, introduces a synchronous orbit radius ratio (cube root of the product of the mass ratio and square of the rotation period ratio; secondary/primary) 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 study proposes that in a local physical space of a gravitational field rotating along with the host celestial body, there exists a frame that replaces gravity and exhibits a free-fall acceleration towards the host and an acceleration from an initial zero velocity to the current velocity of the gravitational field based on Einstein's equivalence principle. This frame can be divided into two subframes: one with the acceleration towards the host and the other with the acceleration in the direction of gravitational field movement; additionally, the latter can be replaced by the drag force. The analysis of Newton's law of gravity using such a frame not only supports the proposal of gravitational field movement but also obtains an expression for drag force. In a system, when the movement of the secondary body reaches a circular orbital velocity under the action of the drag force caused by the gravitational field rotation of the primary body, the secondary body is no longer subjected to the drag force. Moreover, considering the mass centre of the system as the axis under the action of the reaction force, the primary body moves in a circular orbit. However, as the drag force acts on the primary body owing to the gravitational field rotation of the secondary body, the velocity is constrained by the mass ratio unless their masses are similar. This power-law describes a relation between the orbital velocity forming the semi-major axis caused by the drag force due to the gravitational field rotation of the secondary body and the ratio of the two drag forces when they appear within the same system and the primary and secondary bodies rotate in the same direction. Simultaneously, the drag force drives the eccentric orbital precession in the apogee region; if the spin–orbit misalignment exists, the orbital plane precession is driven by the drag force caused by the gravitational field rotation of the primary body in the direction of its rotation.