First-post-Newtonian quadrupole tidal interactions in binary systems

TL;DR

This work develops the first-post-Newtonian framework for tidal interactions in binaries with a spin and a mass quadrupole, deriving consistent translational equations of motion that conserve total linear momentum and formulating them in the center-of-mass frame via a generalized orbital Lagrangian. Using the DSX formalism, it clarifies the role of global- and body-frame multipoles and provides explicit M1–M2–S2–Q2 truncations, including an action principle for the orbital dynamics. In the adiabatic limit where the quadrupole responds instantaneously to the tidal field, the authors obtain a reduced Lagrangian and a conserved energy, and derive the energy-frequency relation for circular orbits with tidal corrections. The results connect with and complement existing formalisms (e.g., Damour-Nagar EOB and Bini-Damour-Faye approaches) and are directly applicable to modeling gravitational-wave phasing in inspiralling neutron-star binaries by incorporating 1PN tidal effects through the deformability parameter lambda.

Abstract

We consider tidal coupling in a binary stellar system to first-post-Newtonian order. We derive the orbital equations of motion for bodies with spins and mass quadrupole moments and show that they conserve the total linear momentum of the binary. We note that spin-orbit coupling must be included in a 1PN treatment of tidal interactions in order to maintain consistency (except in the special case of adiabatically induced quadrupoles); inclusion of 1PN quadrupolar tidal effects while omitting spin effects would lead to a failure of momentum conservation for generic evolution of the quadrupoles. We use momentum conservation to specialize our analysis to the system's center-of-mass-energy frame; we find the binary's relative equation of motion in this frame and also present a generalized Lagrangian from which it can be derived. We then specialize to the case in which the quadrupole moment is adiabatically induced by the tidal field (in which case it is consistent to ignore spin effects). We show how the adiabatic dynamics for the quadrupole can be incorporated into our action principle and present the simplified orbital equations of motion and conserved energy for the adiabatic case. These results are relevant to gravitational wave signals from inspiralling binary neutron stars.