Canonical Angles In A Compact Binary Star System With Spinning Components: Approximative Solution Through Next-To-Leading-Order Spin-Orbit Interaction for Circular Orbits

TL;DR

This paper develops an approximate analytical solution for the evolution of spin orientations and orbital phase in circular binary star systems with spinning components, incorporating next-to-leading order spin-orbit interactions.

Contribution

It introduces a modified iterative method to remove oscillatory terms for unequal masses and derives an asymptotic series for the system's evolution.

Findings

Explicit first-order iteration solution obtained.

Method handles unequal mass systems with spin-orbit coupling.

Asymptotic series allows controlled approximation of the dynamics.

Abstract

This publication will deal with an explicit determination of the time evolution of the spin orientation axes and the evolution of the orbital phase in the case of circular orbits under next-to-leading order spin-orbit interactions. We modify the method of Schneider and Cui proposed in ["Theoreme \"uber Bewegungsintegrale und ihre Anwendungen in Bahntheorien", Verlag der Bayerischen Akademie der Wissenschaften, volume 212, 2005.] to iteratively remove oscillatory terms in the equations of motion for different masses that were not present in the case of equal masses. Our smallness parameter is chosen to be the difference of the symmetric mass ratio to the value 1/4. Before the first Lie transformation, the set of conserved quantities consists of the total angular momentum, the amplitudes of the orbital angular momentum and of the spins, $L, S_1,$ and $S_2$. In contrary, the magnitude of the total spin $S=|S_1+S_2|$ is not conserved and we wish to shift its non-conservation to higher orders of the smallness parameter. We perform the iterations explicitly to first order, while performing higher orders would mean no structural difference or harder mathematical difficulties. To apply this method, we develop a canonical system of spin variables reduced by the conservation law of total angular momentum, which is imposed on the phase space as a constraint. The result is an asymptotic series in $\epsilon$ that may be truncated appropriately considering the physical properties of the regarded system.