Paper detail

Motion of spinning particles in the Kerr-Newman black hole exterior. I. Periodic orbits

The motion of a spinning particle in the exterior of a Kerr-Newman black hole is studied. The dynamics is governed by the Mathisson-Papapetrou equations in the pole-dipole approximation, including the spin-curvature coupling to leading order in the spin. In terms of conserved quantities, the dynamical equations in Mino time can be transformed into the integral form for both aligned and misaligned spins with respect to the orbital motion. These non-geodesic equations can be solved analytically with the solutions involving Jacobi elliptic functions. The radial potential is derived to study the particle's parameter space for various types of orbits, based on its roots corrected by the particle's spin. We consider equatorial motion oscillating between two turning points, which are the two outermost roots of the radial potential, in the misaligned case. In this case, there is an induced oscillatory motion out of the equatorial plane. In particular, the periods of the motion are obtained explicitly. When the orbits become a source of gravitational-wave emission, these periods of motion will play a key role in determining gravitational-wave signals in the frequency domain. Numerical kludge waveforms are constructed. The gravitational-wave amplitudes are found to be sensitive to the turning points of the orbits. The implications for gravitational-wave emission due to extreme mass-ratio inspirals (EMRIs) are discussed.