Radii of spherical timelike geodesics in Kerr-Newman black holes

Abstract

The existence, radii and radial stability of the equatorial and non-equatorial (particularly, the polar) spherical orbits are discussed for particles with different conserved energy. The radii of these orbits generally are solutions of a quintic polynomial equation with four dimensionless parameters. For the case with $γ=1$, we obtain the analytical expressions for the radii of the polar, equatorial and general orbits. The radial stability of the orbits outside the event horizon is also discussed. 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 there is always one spherical orbit outside the event horizon. For the cases with $γ\neq1$, we focus on the study of polar and equatorial orbits. For polar orbits with $0<γ<1$, a boundary surface in $(u, w, γ)$ space is identified which determines the existence of spherical polar orbits outside the event horizon. Numerical results of the radii and radial stability of the polar orbits are shown for examples with specific values of $γ$. For polar orbits with $γ>1$, it is found that there is always one unstable orbit outside the event horizon. For equatorial orbits with $0<γ<1$, in each rotating case (prograde case and retrograde case), a boundary surface in $(u, w, γ)$ space is also 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 orbits with $γ>1$, it is found that there is always one unstable orbit outside the event horizon.