Motion and gravitational wave forms of eccentric compact binaries with orbital-angular-momentum-aligned spins under next-to-leading order in spin–orbit and leading order in spin(1)–spin(2) and spin-squared couplings

Abstract

Abstract. A quasi-Keplerian parameterisation for the solutions of second post-Newtonian (PN) accurate equations of motion for spinning compact binaries is obtained including leading order spin-spin and nextto-leading order spin-orbit interactions. Rotational deformation of the compact objects is incorporated. For arbitrary mass ratios the spin orientations are taken to be parallel or anti-parallel to the orbital angular momentum vector. The emitted gravitational wave forms are given in analytic form up to 2PN point particle, 1.5PN spin-orbit and 1PN spin-spin contributions, whereby the spins are assumed to be of 0PN order.

Erratum: Effective-one-body Hamiltonian with next-to-leading order spin-spin coupling for two nonprecessing black holes with aligned spins [Phys. Rev. D87, 124036 (2013)]

“…In Ref. [47], this statement is shown using an approach which dates back to Dirac. The idea is to express the parallelism of S 1 , S 2 and J as a set of constraints where λ a := |S a | |l| −1 .…”

“…It has already been proved [47] that the conservative 2.5PN dynamics of maximally rotating compact binaries does not allow the spins to precess, if they are both initially aligned with the total angular momentum J . In Ref.…”

“…Ref. [47] shows that Eq. (A.2) is valid if one can express the time derivativeṠ b of the spins as a linear combination of the constraints.…”

Eccentric motion of spinning compact binaries

“…In order to express the time t as a function of the variable D in higher orders of c, we may use a generating function of the form W = Lφ + JΥ + r r+ f r (r )dr + W spin , (6.1) where r + denotes the radial distance at the periastron and f (r) is constructed in such a way that the new variable D is directly related to the time t as a derivative of W and closely related to the Kepler equation (see standard texts on Delaunay elements, e.g. [37], and also the quasi-Keplerian parameterisation for higher PN orders, for example [30,44,53]). Note that L is the orbital angular momentum and to be distinguished from the energy-related Delaunay element L D .…”

“…We work in dimension-less units as given in Eqs. (6)- (9) of [30] to obtain Eqs. (2.1-2.4) below, with the only exception of additionally imposing fast-spinning components, for convenience of the reader 3 .…”