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

TL;DR

This paper analytically investigates the effects of small spin on the innermost stable circular orbits of test particles in Schwarzschild and Kerr space-times, revealing spin-dependent corrections and behaviors.

Contribution

It derives analytical small-spin corrections for ISCO parameters in Kerr space-times, including special cases like Schwarzschild and extreme Kerr black holes.

Findings

Spin-orbital coupling is attractive when spins are parallel.

Limiting ISCO radius and frequency are independent of particle spin in extreme Kerr.

Explicit formulas for small-spin corrections to energy and angular momentum are provided.

Abstract

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 details. For a slowly rotating black hole the ISCO parameters are obtained up to quadratic in $a$ and particle's spin $s$ terms. From the formulae 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 and energy at given radius. System of equations for numerical calculation of ISCO parameters for arbitrary $a$ and $s$ is also explicitly written.