Lisa V. Drummond, Scott A. Hughes, Viktor Skoupý, Philip Lynch, Gabriel Andres Piovano

We present a shifted-geodesic framework for computing gravitational-wave fluxes from spinning test bodies moving on bound orbits of Kerr black holes. The method provides a simple and efficient means of evaluating energy and angular momentum fluxes incorporating the leading effect of the smaller body's spin. Because post-adiabatic corrections, including secondary spin contributions, are subdominant to the leading adiabatic terms, this approximation is well justified. In particular, we find that oscillatory spin terms typically contribute very little to fluxes, but their contribution to the description of orbits is computationally expensive, making such terms a natural target for approximation. In our framework, orbital frequencies and integrals of the motion are perturbed to include spin effects, while the trajectory retains the global structure of geodesic motion. This simplifies the computation of gravitational radiation. The shifted-geodesic approximation is most reliable for orbits with lower eccentricity, lower inclination, and larger semi-latus recta. The approximation becomes less reliable as we approach the separatrix between stable and unstable orbits; fortunately, many inspirals spend less time in this region of parameter space. A diagnostic inspiral evolution shows very small dephasing due to use of the shifted-geodesic approximation ($\approx10^{-2}$ radians over 1 year), confirming that spin-induced flux corrections can be accurately included using this simple modification to a geodesic trajectory. This approximation provides a rapid and convenient way to compute spinning-body orbits, but is not intended to replace more accurate treatments. We propose it as a pragmatic alternative when speed and simplicity are prioritized, enabling efficient EMRI/IMRI flux calculations and supporting parameter-space studies for LISA. (Abridged)