Photon spheres and spherical accretion image of a hairy black hole

In this paper, we first consider null geodesics of a class of charged, spherical and asymptotically flat hairy black holes in an Einstein-Maxwell-scalar theory with a nonminimal coupling for the scalar and electromagnetic fields. Remarkably, we show that there are two unstable circular orbits for a photon in a certain parameter regime, corresponding to two unstable photon spheres of different sizes outside the event horizon. To illustrate the optical appearance of photon spheres, we then consider a simple spherical model of optically thin accretion on the hairy black hole, and obtain the accretion image seen by a distant observer. In the single photon sphere case, only one bright ring appears in the image, and is identified as the edge of the black hole shadow, whereas in the case with two photon spheres, there can be two concentric bright rings of different radii in the image, and the smaller one serves as the boundary of the shadow, whose radius goes to zero at the critical charge.

Figure 1

Plots of hairy black hole solutions with $\alpha =0.8$ and $M=1$ and the associated effective potentials $V_{\mathrm{eff}}(r)$. Left: metric functions $m(r)$ (dotted), $\varphi (r)$ (dashed) and $\delta (r)$ (solid) for the hairy black holes with different electric charges $Q=1.04$ (blue), $Q=1.045$ (green), $Q=1.048$ (orange) and $Q=1.05$ (red). Vertical lines denote the corresponding event horizons. Right: for $Q=1.04$ (blue) and $Q=1.045$ (green), the black hole effective potential $V_{\mathrm{eff}}(r)$ has a single-peak structure with a single maximum. Nevertheless, a double-peak structure with two local maxima is observed for $V_{\mathrm{eff}}(r)$ when $Q=1.048$ (orange) and $Q=1.05$ (red). Note that the global maximum of $V_{\mathrm{eff}}(r)$ is responsible for determining the size of the black hole shadow.

Figure 2

The profile of the effective potential $V_{\mathrm{eff}}(r)$ (left) and trajectories of light rays with different impact parameters $b$ (right) for the hairy black hole with $\alpha =0.8$, $Q=1.04$ and $M=1$. The effective potential has a single maximum of $1/b_{ph}^2$ at $r=r_{ph}$, where the photon sphere is located. The gray and red lines correspond to light rays with $b<b_{ph}$ and $b>b_{ph}$, respectively, and the dashed blue circle represents the photon sphere. The black hole is shown as a solid black disk.

Figure 3

Plots of the total number of light ray orbits $n$, and the intensity $F_o$ and accretion image seen by a distant observer for the hairy black hole with $\alpha =0.8$, $Q=1.04$, and $M=1$. Left: the total number of orbits $n\equiv \mathrm{\Phi }/(2\pi )$ as a function of $b$, where $\mathrm{\Phi }=\mathrm{\Delta }\phi$ is the total change of the azimuthal angle of light rays traveling outside the event horizon. The red (gray) segment is determined by light rays with $b>b_{ph}$ ($b<b_{ph}$). Middle: the total photon intensity $F_o(b)$ as a function of $b$, which has a sharp peak at $b=b_{ph}$. Light rays with $b>b_{ph}$ ($b<b_{ph}$) contribute to the red (gray) segment. Right: the 2D image of the accretion flow viewed in the observer’s sky. The bright ring is determined by the photon sphere, and serves as the boundary of the black hole shadow.

Figure 4

Upper: plots of the effective potential and trajectories of light rays for the hairy black hole with $\alpha =0.8$, $Q=1.04$ and $M=1$. The effective potential has two maxima at $r=r_{ph1}$ and $r=r_{ph2}$, corresponding to two photon spheres. Two dashed blue concentric circles of radii $r_{ph1}$ and $r_{ph2}$ denote the two photon spheres. The gray, red and orange lines represent light rays in Region 1, Region 2 and Region 3, respectively. Lower: plots of the fractional number of photon orbits $\phi (r)/(2\pi )$ and the corresponding trajectories for four light rays with different $b$, which come from infinity and revolve around the photon spheres many times before escaping. The dashed blue vertical lines in the left panel correspond to the photon spheres.

Figure 5

The total number of orbits $n(b)=\mathrm{\Phi }/(2\pi )$ (left), the observed total photon intensity $F_o(b)$ (middle) and the observed 2D accretion image (right) for the hairy black hole with $\alpha =0.8$, $Q=1.05$ and $M=1$. The gray, red and orange segments of $n(b)$ and $F_o(b)$ are determined by light rays in Regions 1, 2 and 3, respectively, in accordance with colors used in Fig. 4. Due to the presence of two photon spheres, $n(b)$ and $F_o(b)$ have two sharp peaks at $b=b_{ph1}$ and $b=b_{ph2}$, corresponding to light rays that revolve around the photon spheres at $r=r_{ph1}$ and $r=r_{ph2}$, respectively. Accordingly, the accretion image viewed in the observer’s sky has two concentric bright rings of radii $b_{ph1}$ and $b_{ph2}$. The bright ring with the smaller radius ($b_{ph1}$) is identified as the edge of the shadow, and barely visible due to the narrow width of the peak of $F_o(b)$ at $b=b_{ph1}$.

Figure 6

Dependence of the photon sphere radii, $r_{ph}$, $r_{ph1}$ and $r_{ph2}$, the associated impact parameters, $b_{ph}$, $b_{ph1}$ and $b_{ph1}$, and the horizon radius $r_h$ on the black hole charge $Q$ for RN black holes with $M=1$ and hairy black holes with $\alpha =0.8$ and $M=1$. The RN and hairy black holes coexist in the yellow region. In the orange and pink regions, the hairy black holes have two photon spheres of radii $r_{ph1}$ and $r_{ph2}$ with $r_{ph1}<r_{ph2}$, and $b_{ph1}$ and $b_{ph2}$ are the impact parameters associated with the corresponding photon spheres. The minimum of $b_{ph1}$ and $b_{ph2}$ is identified as the radius of the black hole shadow. In the orange region, $b_{ph2}<b_{ph1}$, and hence the shadow radius is $b_{ph2}$, i.e., the impact parameter of the larger photon sphere. Moreover, the smaller photon sphere of radius $r_{ph1}$ is not responsible for imaging the accretion flow since it is invisible to the distant observer. In the pink region, the shadow radius is the impact parameter $b_{ph1}$ of the smaller photon sphere, and both photon spheres play a role in the observed accretion image. The right panel highlights the impact parameters of the photon spheres for the RN and hairy black holes in the coexisting region, and shows that, for a given $Q$, the hairy black hole has a smaller shadow.