Measuring the Shape of Kerr Black Holes at the Photon Orbit

Abstract

The bright ring-like structures observed in the images of M87* and SgrA* captured by the Event Horizon Telescope strongly support the validity of general relativity. Lensed images of the emission region, often referred to as photon rings in this context, are a direct consequence of the unstable dynamics of null geodesics near the spherical photon orbit in the Kerr spacetime. The order of the lensed image can be characterized by the number of half-orbits the photons complete before reaching the observer, with higher-order photon rings produced by null geodesics that circle the black hole more times. However, low-order rings are significantly influenced by the astrophysical environment. Measuring the Lyapunov exponent requires probing the exponentially small differences between successive photon rings or between photon rings and the shadow. We investigate potential astrophysical sources of systematic error the estimation of Lyapunov exponent, including the location of the observed emission, and especially at low photon ring order. We show that it is nevertheless possible to measure this purely gravitational quantity to roughly 10% and 1% systematic uncertainty by resolving the n=2 and n=3 photon rings with the shadow size, respectively. Therefore, the forthcoming black hole imaging efforts to capture, even if indirectly, the n=2 photon ring can result in a measurement of the Lyapunov exponent that is not limited by astrophysical uncertainties.

Figures (9)

• Figure 1: Photons orbit towards a polar observer, located at $z= \infty$, projected into the x-z plane in Kerr spacetime with $a=0.5M$. The diagram shows the geometric picture of three distinct photon rings originating from the astrophysical disk around the black hole in the equtorial plane. On the right, corresponds to the direct image $n =0$ ring. The $n=1$ ring which is a null ray that completes half an orbit around the black hole before escaping to infinity is shown in the middle. The $n=2$ ring is depicted by the green line on the left. The dark circle at the center represents the event horizon for $a=0.5M$ in Kerr spacetime. The dots show the location of emission on the equtorial plane and the dashed lines show the continuation of the trajectory to infinity.
• Figure 2: Null trajectories that travel towards the polar observer at different emission radii in the equtorial plane is shown here. The dashed lines represent the direct emission $n=0$ coming from $r_{em}=10M, 26M, 55 M$ and the solid lines represent the $n=1$ photon rings. The dark disk in the center demonstrates the horizon for Kerr spacetime with $a=0.5 M$. This figure illustrates null trajectories originating from same location of emission can contribute to different subsequent rings depending on their initial conditions.
• Figure 3: This visualization illustrates null trajectories approaching a polar observer from various emission radii within the equatorial plane. All these rays contribute to the $n=1$ photon ring. The blue trajectory completes three-quarters of an orbit around the black hole, the red trajectory represents a half orbit, and the green trajectory completes one-quarter of an orbit. These different trajectories offer insight into the geometric and dynamical characteristics of photon rings.
• Figure 4: The y-axis represents the difference between the radius of the $n$th photon ring originating from $r_{em}$ and the corresponding geometric values of the apparent radius for various spin parameters and ring orders. The leftmost plot shows this difference for the secondary image with $n=1$, and as you move rightward, the value of $n$ increases, with the rightmost plot representing $n=5$
• Figure 5: The y-axis represents in this plot illustrates the difference between the geometric radius of each photon ring and the shadow size, presented on a logarithmic scale. The x-axis represents the order of the photon ring, causing the points to be scattered along the plane. The lines depict $\Delta R e^{n \gamma}$ for various spin values, and because the y-axis is logarithmic, these appear as straight lines.