Spinning test body orbiting around a Kerr black hole: Comparing spin supplementary conditions for circular equatorial orbits

The worldline of a spinning test body moving in curved spacetime can be provided by the Mathisson-Papapetrou-Dixon equations when its centroid, i.e., center mass, is fixed spin supplementary condition (SSC). In present study, we continue exploration shifts between different centroids started recently published work [Iason Timogiannis et al., Phys. Rev. D 104, 024042 (2021).], henceforth paper I, for Schwarzschild spacetime, examining frequencies circular equatorial orbits under change SSC Kerr spacetime. particular, examine convergence terms prograde and retrograde orbital frequencies, these are expanded power series measure centroid shifted from Mathisson-Pirani or Ohashi-Kyrian-Semer\'ak frame to Tulczyjew-Dixon one. Since I have seen that innermost stable (ISCOs) hold special place this comparison process, focus on them rigorously work. We introduce novel method finding ISCOs any employ it formalisms. resort numerical investigation SSCs ISCO case, due technical difficulties not allowing I's analytical treatment. Our conclusion, as there appears SSCs, which improved proper taken into account, but exists limit fact approximation consider only first two lower multipoles extended ignore all higher ones.