Innermost stable circular orbits of spinning test particles in Schwarzschild and Kerr space-times

We consider the motion of classical spinning test particles in Schwarzschild and Kerr metrics and investigate innermost stable circular orbits (ISCO). The main goal of this work is to find analytically the small-spin corrections for the parameters of ISCO (radius, total angular momentum, energy, orbital angular frequency) of spinning test particles in the case of vectors of black hole spin, particle spin and orbital angular momentum being collinear to each other. We analytically derive the small-spin linear corrections for arbitrary Kerr parameter $a$. The cases of Schwarzschild, slowly rotating and extreme Kerr black hole are considered in detail. For a slowly rotating black hole, the ISCO parameters are obtained up to quadratic in $a$ and particle’s spin $s$ terms. From the formulas obtained it is seen that the spin-orbital coupling has attractive character when spin and angular momentum are parallel and repulsive when they are antiparallel. For the case of the extreme Kerr black hole with co-rotating particle we succeed to find the exact analytical solution for the limiting ISCO parameters for arbitrary spin. It has been shown that the limiting values of ISCO radius and frequency do not depend on the particle’s spin while values of energy and total angular momentum depend on it. We have also considered circular orbits of arbitrary radius and have found small-spin linear corrections for the total angular momentum, energy and frequency at given radius. System of equations for numerical calculation of ISCO parameters for arbitrary $a$ and $s$ is also explicitly written.

Figure 1

The influence of BH and test-body’s spin on the radii of ISCO. In case of Schwarzschild BH the ISCO radius for spinless particles equals to $6M$, see thick solid line. For a particle with spin, ISCO radius splits in two cases, with different mutual orientation of spin and orbital angular momentum (parallel or antiparallel), see dashed lines. In the case of Kerr BH and spinless particle, ISCO radius splits in two cases, with co- and counter-rotation to BH (angular momentum is parallel or antiparallel to BH spin), see thin solid lines. For a particle with spin each line additionally splits in two dashed lines corresponding to different spin orientation. The only exception is the case of extreme Kerr BH with a co-rotating particle: in this case the ISCO radius is equal to $M$ and does not depend on magnitude and direction of particle spin.

Figure 2

Dependence of the radii of the stable (upper branches) and unstable (lower branches) circular orbits on the total angular momentum, for different values of spin $s$. All parameters are given in units of $M$, see also [17].

Figure 3

Radii of ISCO for different directions of motion and spin of test particles. Six particles, $A$, $B$, $C$, $D$, $E$, $F$, with different directions of spin and orbital motion are presented, for different Kerr parameter $a$. For $a=0$ we use just letters (e.g., $A$), for $a\ll M$ we use primes (e.g., $A^{\prime }$), for $a=M$ we use double primes (e.g., $A^{\prime \prime }$). Direction of particle’s spin is shown by an arrow on a circle: clockwise arrow ($C$ and $F$) indicates particles co-rotating with black hole, anticlockwise arrow ($D$ and $E$) indicates counter-rotating particles, for spinless particles ($A$ and $B$) we use circles without arrow. Direction of orbital motion around BH is shown by straight arrow outside circles. Size of BH is shown for the case of the extreme Kerr space-time. The ISCO radii for $a\ll M$ are drawn on basis of only the first three terms in formula (71), with $s\ll a\ll M$. In the case of a Schwarzschild BH ($a=0$) the ISCO radius for spinless particles equals to $6M$, independently of direction of orbital motion, see $A$ and $B$. In the case of a Kerr BH the ISCO radius depends on direction of orbital motion. For slowly rotating BH, $a\ll M$: counter-rotating particle $A^{\prime }$ has orbit with $r_{\mathrm{ISCO}}>6M$, co-rotating particle $B^{\prime }$ has orbit with $r_{\mathrm{ISCO}}<6M$. For the extreme Kerr BH, $a=M$, a counter-rotating particle $A^{\prime \prime }$ has orbit with $r_{\mathrm{ISCO}}=9M$, a co-rotating particle $B^{\prime \prime }$ has orbit with $r_{\mathrm{ISCO}}=M$. Presence of spin increases or decreases ISCO radius in comparison with spinless case, see particles $C$, $D$, $E$, $F$ (also $C^{\prime }$, $C^{\prime \prime }$ and so on). We recall that the $z$-axis is directed along the axis of rotation of BH, so $a>0$. Total angular momentum $J=J_z$: $J>0$ corresponds to co-rotating of orbit, $J<0$ corresponds to counter-rotating orbit. Spin $s>0$ corresponds to spin vector parallel to $a$ (corotation with BH), spin $s<0$ is antiparallel to it. The ISCO radius of a spinning particle in the Schwarzschild case depends on $s_J$, projection of spin vector on total angular momentum vector. Therefore, particles $C$ and $D$ (also $E$ and $F$), which have different ISCO radii in the Kerr metric, have the same ISCO radius in Schwarzschild case. For the extreme Kerr BH particles $B^{\prime \prime }$, $D^{\prime \prime }$, $F^{\prime \prime }$ have the same ISCO radii.

Figure 4

The ISCO radius as a function of a particle’s spin for different values of $a$ close to $M$: $a/M=0.9$ (a), 0.99 (b), 0.999 (c), 0.9999 (d), 0.99999 (e), in the case of co-rotation. In limit $a\to M$ curves tend to the horizontal asymptote $r_{\mathrm{ISCO}}=1$.

Figure 5

The ISCO radii as a function of a particle’s spin for different values of $a$ close to $M$: $a/M=0.9$ (a), 0.99 (b), 0.999 (c), in the case of counter-rotation. In limit $a\to M$ the ISCO radius at $s=0$ tends to the $9M$.