Second-Post-Newtonian Energy and Angular Momentum Fluxes for Eccentric Inspirals in Effective-One-Body Formalism via Coordinate Transformation

The effective-one-body (EOB) formalism accurately describes the conservative dynamics of general binary orbits, but current implementations of radiation reaction remain largely limited to quasi-circular inspirals. Extending EOB to eccentric orbits currently requires the corresponding post-Newtonian (PN) energy fluxes in EOB coordinates, which are only known to 1PN order. In this paper, we compute the instantaneous gravitational-wave energy flux in EOB coordinates to 2PN accuracy using a systematic coordinate transformation between the Arnowitt–Deser–Misner (ADM) and EOB phase-space variables. We derive the 2PN-accurate transformation laws for the relative coordinates and velocities between the two coordinate systems and re-express the 2PN instantaneous energy flux entirely in EOB variables. Working within the EOB test-particle framework (with finite mass ratio ν) for an equatorial elliptic orbit, we adopt a Keplerian reparameterization in terms of the semilatus rectum p, eccentricity e, and two phase variables (ξ,ϕ) associated with the radial and azimuthal motion, which makes the calculations more transparent and facilitates the subsequent computation of gravitational waveforms. Using the conservative orbital angular frequency, we compute the orbit-averaged energy flux. In addition to the energy flux, we also compute the corresponding 2PN angular momentum flux in EOB coordinates using the same transformation method. Our results, expressed in gauge-invariant variables x=(Mω)2/3 and eccentricity et, agree with known PN results and show improved accuracy compared to the 1PN EOB fluxes of Hinderer and Babak.