Canonical angles in a compact binary star system with spinning components: Approximative solution through next-to-leading-order spin-orbit interaction for circular orbits

This paper 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 [Theoreme uber Bewegungsintegrale und ihre Anwendung in Bahntheorien, A Theoretische Geodasie Vol. 121 (Verlag der Bayerischen Akademie der Wissenschaften, Munchen, Germany, 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 J and the amplitudes of the orbital angular momentum and of the spins, L, S sub(1), and S sub(2). In contrast, S [: =] S sub(1) + S sub(2) is not conserved, and we wish to shift its nonconservation 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 e that may be truncated appropriately by considering the physical properties of the regarded system.