Spinning test body orbiting around a Schwarzschild black hole: Circular dynamics and gravitational-wave fluxes

We consider a spinning test-body in circular motion around a nonrotating black hole and analyze different prescriptions for the body's dynamics. We compare, for the first time, the Mathisson-Papapetrou formalism under the Tulczyjew spin-supplementary condition (SSC), the Pirani SSC, and the Ohashi-Kyrian-Semerak SSC, and the spinning particle limit of the effective-one-body Hamiltonian of Damour and Nagar [Phys. Rev. D 90, 044018 (2014).]. We analyze the four different dynamics in terms of the innermost stable circular orbit (ISCO) shifts and in terms of the coordinate-invariant binding energies, separating higher-order spin contributions from spin-orbit contributions. The asymptotic gravitational-wave fluxes produced by the spinning body are computed by solving the inhomogeneous $(2+1)D$ Teukolsky equation and contrasted for the different cases. For small orbital frequencies $\mathrm{\ensuremath{\Omega}}$, all the prescriptions reduce to the same dynamics and the same radiation fluxes. For large frequencies, $x\ensuremath{\equiv}(M\mathrm{\ensuremath{\Omega}}{)}^{2/3}>0.1$, where $M$ is the black hole mass, and especially for positive spins (aligned with the orbital angular momentum) a significant disagreement between the different dynamics is observed. The ISCO shifts can differ by up to a factor of 2 for large positive spins; for the Ohashi-Kyrian-Semerak and the Pirani SSC the ISCO diverges around dimensionless spins $\ensuremath{\sim}0.52$ and $\ensuremath{\sim}0.94$, respectively. In the spin-orbit part of the energetics the deviation from the Hamiltonian dynamics is largest for the Ohashi-Kyrian-Semerak SSC; it exceeds 10% for $x>0.17$. The Tulczyjew and the Pirani SSCs are compatible across almost the whole spin and frequency range. Our results will have direct applications in including spin effects in effective-one-body waveform models for circularized binaries in the extreme-mass-ratio limit.

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