Measurement of the Kerr spin parameter by observation of a compact object’s shadow

A black hole casts a shadow as an optical appearance because of its strong gravitational field. We study how to determine the spin parameter and the inclination angle by observing the apparent shape of the shadow, which is distorted mainly by those two parameters. Defining some observables characterizing the apparent shape (its radius and distortion parameter), we find that the spin parameter and inclination angle of a Kerr black hole can be determined by the observation. This technique is also extended to the case of a Kerr naked singularity.

Figure 1

The shadows of Kerr black holes. The celestial coordinates $(\alpha ,\beta )$ are measured in the unit of the black hole mass $M$.

Figure 2

The shadows of Kerr naked singularities. The arc (the black solid curves in (a)–(d)) is constructed by unstable spherical photon orbits in positive radius $r_{\mathrm{sph}}$. The small distorted disk (the shaded region in (a), (b)) is constructed by the null-geodesics which escape into the other infinity ($r=-\infty$). The straight line in (c), (d) is constructed by null-geodesics which plunge into a ring singularity. The left endpoints of those lines correspond to the principal null-geodesics.

Figure 3

The observables for the apparent shape of a Kerr black hole are the radius $R_{\mathrm{s}}$ and the distortion parameter ${\delta }_{\mathrm{s}}≔D_{\mathrm{cs}}/R_{\mathrm{s}}$, approximating it by a distorted circle, where $D_{\mathrm{cs}}$ is the difference between the left endpoints of the circle and of the shadow.

Figure 4

(a) The contour map of radii of the shadows of a Kerr black hole. (b) The contour map of their distortion parameters.

Figure 5

The contour maps of the radius of the arc $R_{\mathrm{s}}$ (the red solid curves) and the distorted parameter ${\delta }_{\mathrm{s}}$ (the green dashed curves). The contours are those for $R_{\mathrm{s}}/M=4.84,\cdots ,5.18$, with the contour interval being 0.02, and ${\delta }_{\mathrm{s}}=0.01,\cdots ,0.1$ and $0.12,\cdots ,0.22$, with the contour intervals being 0.01 and 0.02, respectively. We can evaluate the spin parameter $a$ of the Kerr black hole and the inclination angle $i$ of the observer.

Figure 6

The observables for the apparent shape of a Kerr naked singularity are the radius $R_{\mathrm{a}}$ and the central angle ${\vartheta }_{\mathrm{a}}$, approximating it by an arc.

Figure 7

(a) The contour map of radii of the arcs of a Kerr naked singularity. (b) The contour map of central angles of the arcs. In the blank spaces (white areas) in (a) and (b), the arc does not exist.

Figure 8

The contour map of the radius of the arc $R_{\mathrm{a}}$ (the red solid curves) and its central angle ${\vartheta }_{\mathrm{a}}$ (the green dashed curves). The contours are those for $R_{\mathrm{a}}/M=4.2,\cdots ,5.1$, with the contour interval being 0.1 and ${\vartheta }_{\mathrm{a}}=30°,\cdots ,330°$, with the contour interval being 30°. If one can observe ${\vartheta }_{\mathrm{a}}$ and $R_{\mathrm{a}}$, the spin parameter $a$ of the Kerr naked singularity and the inclination angle $i$ could be evaluated, although there exists a degenerate region near the blank space shaded in gray.

Figure 9

The separation distance between the left endpoint of the dark spot and the center of the arc, which we denote $D_{\mathrm{sa}}$. We then define the aspect ratio of the shadow of a Kerr naked singularity; ${\mathcal{A}}_{\mathrm{sa}}≔D_{\mathrm{sa}}/R_{\mathrm{a}}$, where $R_{\mathrm{a}}$ is the arc radius.

Figure 10

The contour map of the aspect ratio of the shadow of a Kerr naked singularity, ${\mathcal{A}}_{\mathrm{sa}}$.

Figure 11

The contour maps of the aspect ratio ${\mathcal{A}}_{\mathrm{sa}}$ (the blue dashed curves) and of the radius of the arc $R_{\mathrm{a}}$ (the red solid curves). The contours are those for $R_{\mathrm{a}}/M=4.2,\cdots ,5.1$, with the contour interval being 0.1 and ${\mathcal{A}}_{\mathrm{sa}}=1.2,\cdots ,2.0$, with the contour interval being 0.1. The one-to-one correspondence between ($R_{\mathrm{a}}$ and ${\mathcal{A}}_{\mathrm{sa}}$) and ($a$ and $i$) is very clear. We can use this contour map to determine the spin parameter and the inclination angle.