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In a recent work, [Phys. Rev. D. 94, 104010 (2016)], hereafter Paper I, wehave numerically studied different prescriptions for the dynamics of a spinningparticle in circular motion around a Schwarzschild black hole. In the presentwork, we continue this line of investigation to the rotating Kerr black hole.We consider the Mathisson-Papapetrou formalism under three differentspin-supplementary-conditions (SSC), the Tulczyjew SSC, the Pirani SSC and theOhashi-Kyrian-Semerak SSC, and analyze the different circular dynamics in termsof the ISCO shifts and the frequency parameter ${x \equiv (M \Omega)^{2/3}}$,where $\Omega$ is the orbital frequency and $M$ is the Kerr black hole mass.Then, we solve numerically the inhomogeneous $(2+1)D$ Teukolsky equation tocontrast the asymptotic gravitational wave fluxes for the three cases. Ourcentral observation made in Paper I for the Schwarzschild limit is found tohold true for the Kerr background: the three SSCs reduce to the same circulardynamics and the same radiation fluxes for small frequency parameters butdifferences arise as $x$ grows close to the ISCO. For a positive Kerr parameter$a=0.9$ the energy fluxes mutually agree with each other within a $0.2\%$uncertainty up to $x<0.14$, while for $a=-0.9$ this level of agreement ispreserved up to $x<0.1$. For large frequencies ($x \gtrsim 0.1$), however, thespin coupling of the Kerr black hole and the spinning body results insignificant differences of the circular orbit parameters and the fluxes,especially for the $a=-0.9$ case. Instead, in the study of ISCO the negativeKerr parameter $a=-0.9$ results in less discrepancies in comparison with thepositive Kerr parameter $a=0.9$.

Spinning test-body orbiting around a Kerr black hole: circular dynamics and gravitational-wave fluxes