Radii of spherical photon orbits around Kerr-Newman black holes

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The spherical photon orbits around a black hole with constant radii are particular important in astrophysical observations of the black hole. In this paper, the equatorial and non-equatorial spherical photon orbits around Kerr-Newman black holes are studied. The radii of these orbits satisfy a sextic polynomial equation with three parameters, the rotation parameter $u$, charge parameter $w$ and effective inclination angle $v$. It is found that there are two polar orbits and one is inside and the other is outside the event horizon. For orbits in the equatorial plane around an extremal Kerr-Newman black hole , we find three positive solutions to the equation and provide analytical formulas for the radii of the three orbits. It is also found that a critical value of rotation parameter $u=\frac{1}{4}$ exists, above which only one retrograde orbit exists outside the event horizon and below which two orbits (one prograde and one retrograde) exist outside the event horizon. For orbits in the equatorial plane of a nonextremal Kerr-Newman black hole, there are four or two positive solutions. A critical curve in $(u,w)$-plane which separates the regions is also found. On this critical curve, the explicit formulas of three solutions are found. There always exist two equatorial photon orbits outside the event horizon in this case. For orbits between the above two special planes, there are four or two solutions to the general sextic equation. When the black hole is extremal, it is found that there is a critical inclination angle $v_{cr}$, below which only one orbit exists and above which two orbits exist outside the event horizon for a rapidly rotating black hole $u>\frac{1}{4}$. At this critical inclination angle, exact formula for the radii is also derived. Finally, a critical surface in parameter space of ($u,w,v$) is shown that separates regions with four or two solutions.