The spherical photon orbits around a black hole with constant radii are particularly important in astrophysical observations of the black hole. In this paper, the equatorial and nonequatorial 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 $v$. For orbits in the polar plane ($v=1$), we find that there are two positive solutions to the orbit equation, one is inside and the other is outside the event horizon. Particularly, we obtain the analytical expressions for the radii of these two orbits that are functions of $u$ and $w$. For orbits in the equatorial plane ($v=0$) around an extremal Kerr-Newman black hole ($u+w=1$), we find three positive solutions to the equation and provide analytical formulas for the radii of the three orbits. We also find 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 around a nonextremal Kerr-Newman black hole, there are four or two positive solutions to the corresponding sextic equation, which we show numerically. The number of solutions depends on the choice of parameters in different regions of the $(u,w)$ plane. A critical curve in the $(u,w)$ plane that separates these regions is also obtained. On this critical curve, the explicit formulas of three solutions are found. There always exist two equatorial photon orbits outside the event horizon in the nonextremal Kerr-Newman 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, we find 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, an 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 in the nonextremal black hole case.

Ying-Xuan Chen

• Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China

Jia-Hui Huang*

• Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China; Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China; and School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China

Haoxiang Jiang

• Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China