Radii of spherical timelike geodesics in Kerr-Newman black holes

Abstract The spherical geodesics of a neutral massive particle around Kerr-Newman black holes are investigated. The radii of these orbits generally satisfy a quintic polynomial equation with four dimensionless parameters, i.e. the spin u and charge w of the black hole and the conserved angular momentum β and conserved energy γ of the particle. For the γ = 1 case, we obtain analytical expressions of the radii for the polar, equatorial and general orbits. In the ( u , w , β ) space, a no-orbit surface is found. When the parameters lies on this surface there is no orbit outside the event horizon, otherwise one unstable orbit exists outside the event horizon. For polar orbits with 0 < γ < 1, a boundary surface in ( u , w , γ ) space is identified which determines the existence of polar orbits outside the event horizon. For polar orbits with γ > 1, there is always one unstable orbit outside the event horizon. For equatorial circular orbits with 0 < γ < 1, in each rotating case (prograde and retrograde case), a boundary surface in ( u , w , γ ) space is identified which divides the parameter space into two regions: one region with two orbits (one stable and the other unstable) and the other with no orbit outside the event horizon. Parameters on the boundary surface correspond to ISCOs. An analytical formula for the ISCOs is derived by choosing ( w , γ ) as independent variables. For equatorial circular orbits with γ > 1, one unstable orbit always exists outside the event horizon. The radial stability of the various orbits outside the event horizon is also discussed.