Power-law relationship between the semi-major axis and the product of the mass ratio and the square of the rotation period ratio in an eccentric system: Its physical meaning and roles
Authors: Dai Yongfeng 
Published: November 2023

Abstract
This study introduces the concepts of mass ratio order of magnitude and synchronous orbit radius ratio (i.e., the cube root of the product of the mass ratio and the square of the rotation period ratio) of systems (the ratio refers to 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 greater than −7.5217, the semi-major axis has a power-law distribution relationship with the product of the mass ratio and the 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 0 and less than or equal to 1. Based on analyses of Einstein's equivalence principle and Newton's law of gravity, it is also proposed that in addition to a freely falling frame that can replace gravity, a uniformly accelerated frame exists in a local physical space of a gravitational field that rotates along with the host celestial body, with acceleration from an initial velocity of zero to the current velocity of the gravitational field, which can be replaced by the drag force. Moreover, in a system, when the movement of one component reaches a circular orbital velocity under the action of the drag force caused by the rotation of the gravitational field of the other, this component is no longer subjected to the drag force; however, is constrained by the ratio of the orbital radius of the primary body to that of the secondary body, the movement of the primary body fails to attain a circular orbital velocity unless the orbital radii are similar. When the two drag forces appear within the same system, the orbital velocity caused by the drag force owing to the rotation of the gravitational field of the secondary body transformsinto a semi-major axis in the presence of the circular orbital velocity caused by the drag force owing to the rotation of the gravitational field of the primary body (here we consider one of the two components as a reference). Furthermore, this power-law indicates a relationship between the orbital velocity forming the semi-major axis and the ratio of the two drag forces. Finally, this study concludes that the power-law relationship plays a crucial role in the formation of eccentric orbits, spin-orbit misalignment, single pulsars and runaway OB stars, orbits in multiplanet systems (and the deflection of the spin axes of a number of such planets), planet-moon systems and hierarchical triple systems; the migration of planets; planet-planet scattering (including the formation of hot Jupiters, warm Jupiters and distant planets); the lack of close-in brown dwarfs and the fate of the Earth during the late evolution of the Sun.