Analytical calculation of black hole spin using deformation of the shadow

We succeed to find compact analytical expressions which allow one to easily extract the black hole spin from observations of its shadow, without need to construct or model the entire curve of the shadow. The deformation of the Kerr black hole shadow can be characterized in a simple way by oblateness (the ratio of the horizontal and vertical angular diameters which are supposed to be measured by an observer). The deformation is significant in case the black hole is nearly extreme and the observer is not so far from the equatorial plane. In this approximation, we present: (i) the spin lower limit via oblateness, (ii) the spin via oblateness and viewing angle, in case the latter is known from other observations.

Figure 1

The simplest way to characterize the deformation of the shadow is to use oblateness, the ratio of horizontal ($\mathrm{\Delta }x$) and vertical ($\mathrm{\Delta }y$) diameters of the shadow which are supposed to be measured by an observer ($\mathrm{\Delta }x\le \mathrm{\Delta }y$). The oblateness $k=\mathrm{\Delta }x/\mathrm{\Delta }y$ ranges from 1 (Schwarzschild, no deformation) to $\sqrt{3}/2$ (extreme Kerr, the strongest deformation). For analytical calculation of diameters, we need to know the left and the right horizontal borders of the shadow, $x_L$ and $x_R$, and the vertical border, $y_m$.

Figure 2

Geometry of the problem. (a) Position of the observer and the black hole. We assume that $r_{\mathrm{O}}\gg m$. We denote ${\vartheta }_{\mathrm{O}}$ the inclination angle, and $i$ the viewing angle. The main results are obtained for the nearly equatorial observer, which means that $i\ll 1$. (b) and (c) Celestial coordinates of the observer. $\mathrm{\Theta }$ is the colatitude, and $\mathrm{\Psi }$ is the azimuthal angle, see also [1, 6]. $x$ and $y$ are Cartesian coordinates calculated for $r_{\mathrm{O}}\gg m$ by the formulas (9). Note that in our coordinates, the origin corresponds to a principal null ray, and in figures in [5] the origin is determined by zero impact parameters. Therefore in our case the origin is horizontally shifted by the value $a\sin {\vartheta }_{\mathrm{O}}$ in comparison with [5]. For example, for the extreme Kerr BH, the shadow is situated between $-m/r_{\mathrm{O}}$ and $8m/r_{\mathrm{O}}$ in our paper, whereas in [5] it lies between $-2m$ and $7m$. For details see [24].

Figure 3

Left: The shadow curves for the distant equatorial observer for (from the leftmost to the rightmost) $a=0$, 0.1m, 0.6m, 0.9999m. Right: The shadow curves for $a=0.97m$, 0.99m, 0.9999m. There is a notable difference in location of left borders, whereas the right borders are approximately at the same place; see (10) and (11).

Figure 4

Left: The left-hand side of (11) is plotted numerically as a function of $\delta$. The change in position of the right border of the shadow is small. Right: The left-hand side of (10) is plotted numerically as a function of $\sqrt{\delta }$, at the same range of $\delta$ as on the left figure. There is a significant change in position of the left border of the shadow corresponding to the small change of $\delta$.

Figure 5

Accuracy of determination of black hole spin by formulas (28) and (30) in comparison with exact calculation, see details in the text.