Innermost stable circular orbits around a spinning black hole binary

Using the exact solution that describes multi-centered rotating black holes, recently discovered by Teo and Wan, we investigate the innermost stable circular orbit (ISCO) for massive particles and the circular orbit for massless particles moving around a spinning black hole binary. We assume equal masses $M_1 = M_2=m$ and equal spin angular momenta $|J_1| = |J_2|$ for both black holes. Firstly, we examine the case where two black holes are spinning in the same direction ($J_1=J_2$). We clarify that that for particles rotating in the same direction as (opposite directions to) black holes' spin, the greater the spin angular momenta of the black holes, the more the radii of the ISCO for massive particles and the circular orbit for massless particles decrease (increase). We show that distinct ISCO transitions occur for particles rotating in the same direction as the black holes in three ranges of spin angular momenta: $0<J_1/m^2=J_2/m^2< 0.395...$, $0.395...<J_1/m^2=J_2/m^2< 0.483...$, and $0.483...<J_1/m^2=J_2/m^2<0.5$. Conversely, particles rotating in the opposite direction to the black holes exhibit a consistent transition pattern for the case $0<J_1/m^2=J_2/m^2<0.5$. Secondly, we study the situation where binary black holes are spinning in opposite directions ($J_1=-J_2$). We clarify that for large (small) separations between black holes, the ISCO appears near the black hole that is spinning in the same (opposite) direction as particles' rotation. Additionally, we show that different ISCO transitions occur in the three angular momentum ranges: $0<J_1/m^2=-J_2/m^2< 0.160...$, $0.160...<J_1/m^2=-J_2/m^2< 0.467...$, and $0.467...<J_1/m^2=-J_2/m^2<0.5$.