Radii of spherical photon orbits around Kerr-Newman black holes

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.

Figure 1

Two positive solutions $x_{\mathrm{in}}$, $x_{\mathrm{out}}$ in the polar plane and the event horizon radius $x_h$ are plotted as functions of $u$ and $w$.

Figure 2

${\tilde{R}}_o^{(2)}$ and ${\tilde{R}}_i^{(2)}$ are positive as functions of $u$ and $w$.

Figure 3

In the critical case, equatorial orbit solutions $t_1$, $t_2$, $t_3$ and event horizon $t_h$ are plotted as functions of charge parameter $w$.

Figure 4

The critical curve (green) and the two parts of the physical $(u,w)$ plane (lower left triangle). When $(u,w)$ is in the shaded part, two roots exist for the equatorial orbit equation. When $(u,w)$ is in the other part, four roots exist for the equatorial orbit equation.

Figure 5

The roots of the equation of equatorial orbits $x_1$, $x_2$, $x_3$, $x_4$ and the black hole event horizon $x_h$ are shown as functions of $(u,w)$. There are always two orbits outside the event horizons.

Figure 6

The dimensionless $z$ component of the orbital angular momentum per energy $\tilde{L}$ for the orbits. ${\tilde{L}}_1$, ${\tilde{L}}_2$, ${\tilde{L}}_3$, ${\tilde{L}}_4$ correspond to the orbits $x_1$, $x_2$, $x_3$, $x_4$, respectively. ${\tilde{L}}_3>0$, the inner orbit $x_3$ is prograde. ${\tilde{L}}_4<0$, the outer orbit $x_4$ is retrograde.

Figure 7

The second derivatives ${\tilde{R}}_i^{(2)}$ as functions of $u$ and $w$ for equatorial solutions $x_i$. The solutions are radially unstable since ${\tilde{R}}_i^{(2)}>0$.

Figure 8

In the eKN case, two positive real roots $x_1$, $x_2$ and the event horizon $x_h$ are plotted as functions of rotation parameter $u$ and effective inclination angle $v$.

Figure 9

The critical surface $v_s$ is plotted as a function of the rotation parameter $u$ and the charge parameter $w$.