Universal interferometric signatures of a black hole’s photon ring

Photon shell. The photon shell, illustrated in Fig. 2, is the region of a black hole spacetime containing bound null geodesics or “bound orbits” that neither escape to infinity nor fall across the event horizon. For Schwarzschild, the photon shell is the two-dimensional sphere at r = 3M and any θ, ϕ, and t. For Kerr, this two-dimensional sphere fattens to a three-dimensional spherical shell. It is best described using Boyer-Lindquist coordinates, in which the metric of a Kerr black hole of mass M and angular momentum J = aM (with 0 ≤ a ≤ M) isds2=−ΔΣ(dt−a sin2θdϕ)2+ΣΔdr2+Σdθ2+sin2θΣ[(r2+a2)dϕ−adt]2(1A)Δ=r2−2Mr+a2, Σ=r2+a2cos2θ(1B)

• Download high-res image
• Open in new tab
• Download Powerpoint

These coordinates have the special property that each bound orbit lies at some fixed value of r in the ranger−γ≤r≤r+γ(2A)r±γ=2M[1+cos (23arccos (±aM))](2B)

Every point in the equatorial annulus r−γ≤r≤r+γ, θ = π/2, has a unique bound orbit passing through it. On the boundaries r=r±γ, the orbits reside entirely in the equatorial plane. At generic points, on the other hand, they oscillate in the θ direction between polar anglesθ±=arccos(∓u+)(3)whereu±=ra2(r−M)2[−r3+3M2r−2a2M±2MΔ(2r3−3Mr2+a2M)](4)

We will refer to one such complete oscillation (e. g., from θ₋ back to itself) as one orbit, since the photon typically returns to a point near, but not identical to (since the azimuthal angle ϕ also shifts), its initial position.

To summarize, the photon shell is the spacetime regionr−γ≤r≤r+γ, θ−≤θ≤θ+, 0≤ϕ<2π(5)(depicted in Fig. 2) for all times − ∞ ≤ t ≤ ∞.

The bound orbit at radius r has the energy-rescaled angular momentumℓ=M(r2−a2)−rΔa(r−M)(6)

The inner circular equatorial orbit at r−γ is prograde, while the outer one at r+γ is retrograde: ℓ(r∓γ)≷0. The overall direction of the orbits reverses at the intermediate value r0γ for which 𝓁 vanishes. At that radius, [θ₋, θ₊] equals [0, π], and the orbits can pass over the poles.

The bound geodesics are unstable in the sense that, if perturbed slightly, they either fall into the black hole or escape to infinity where they can reach a telescope. The observed photon ring image arises from photons traveling on such “nearly bound” geodesics. Consider two geodesics, one of which is bound, with the other initially differing only by an infinitesimal radial separation δr₀. The equation of geodesic deviation shows that, after n half-orbits between θ_±, their separation grows toδrn=eγnδr0(7)

Here, the so-called Lyapunov exponent γ is a function on the space of bound orbits given by (see the Supplementary Materials)γ=4ar2−MrΔ(r−M)2∫01dt(1−t2)(u+t2−u−)(8)

A closely related formula appears in (14). Hence, the nearly bound geodesic will typically cross the equatorial plane a number of times of ordern≈1γln |δrnδr0|(9)until δr_n ≫ δr₀, when the geodesic is well separated from the bound orbit and it shoots off to infinity (or crosses the event horizon if δr₀ < 0). These Lyapunov exponents are central and potentially observable quantities that characterize the geometry of the Kerr photon shell.

Photon ring and subrings. The photon ring is the image on the observer screen produced by photons on nearly bound geodesics (7). In the limit in which the photons become fully bound, it may be shown that their images approach a closed curve C_γ given byρ=D−1a2(cos2θobs−u+u−)+ℓ2(10A)φρ=arccos(−ℓρ D sin θobs)(10B)where (ρ, φ_ρ) are dimensionless polar coordinates on the observer screen, while (D, θ_obs) denote the observer’s distance and inclination from the Kerr spin axis, respectively. We can view C_γ as parameterized by the shell radius r−γ≤r≤r+γ from which the photon originated. For each value of r, Eq. 10B has two solutions for φ_ρ in the range 0 ≤ φ_ρ ≤ 2π, so each radius in the photon shell appears at two positions on C_γ. A notable consequence of Eqs. 10A and 10B is that for θ_obs ≠ 0, both 𝓁 and ρ, and hence φ_ρ, are functions only of r, θ_obs, and D. Hence, a measurement at a specific angle φ_ρ along the ring probes a specific radius r of the Kerr geometry and not, as might have been expected, a specific angle around the black hole!

Astrophysically observed photon intensities I_ring(ρ, φ_ρ) at the screen can be computed by backward ray tracing. One follows the null geodesics from the observer screen back into the Kerr spacetime, integrating the Doppler-shifted strength J of matter sources along the geodesic, with attenuation factors accounting for the optical depth. Scattering effects are negligible because the expected plasma frequency and electron gyroradius are in the megahertz range, several orders of magnitude below the observing frequencies that we consider. For the images in this paper, we used ipole (15). A light ray aimed exactly at the curve C_γ is captured by the photon shell and (unstably) orbits the black hole forever. Those aimed inside C_γ fall into the black hole, while those aimed outside escape to infinity. Therefore, C_γ is the edge of the black hole shadow.

If we shoot a light ray very near, a distance δρ from the shadow edge at ρ_c, it will circle many times through the emission region before falling into the black hole or escaping to infinity. The affine length of the ray and its number of half-orbits accordingly diverge as δρ → 0n≈−1γln |δρρc|(11)

This follows from Eq. 9 together with a computed relation between δρ and δr₀. For optically thin matter distributions, Eq. 11 implies a mild divergence in the observed ring intensity I_ring ∼ n as the shadow edge is approached, since a light ray that completes n half-orbits through the emission region can collect ∼n times more photons along its path. The photon ring is then the bump in the photon intensity containing this logarithmic divergence at the shadow edge. Although the divergence is cut off by a finite optical depth, this notable feature remains visually prominent in many ray-traced images of GRMHD simulations, as in Fig. 1.

The photon ring can be subdivided into subrings arising from photons that have completed n half-orbits between their source and the screen. This definition for the photon ring agrees with that in (16) but differs from the later usage in (9) and (10) by the inclusion of the n = 1 and 2 contributions. These low n contributions fully account for the thin ring image visible in Fig. 1. To orbit at least n/2 times around the black hole, the photon must be aimed within an exponentially narrowing windowδρnρc≈e−γn(12)around the shadow edge. Hence, the subrings occupy a sequence of exponentially nested intervals centered around C_γ.

Each subring consists of photons lensed toward the observer screen after having been collected by the photon shell from anywhere in the universe. Hence, in an idealized setting with no absorption, each subring contains a separate, exponentially demagnified image of the entire universe, with each subsequent subring capturing the visible universe at an earlier time. Together, the set of subrings are akin to the frames of a movie, capturing the history of the visible universe as seen from the black hole. In an astrophysical setting, these images are dominated by the luminous matter around the black hole. For a black hole surrounded by a uniform distribution extending over the poles, the contributions made by each subring to the total intensity profile cannot be told apart, and the individual subrings cannot be distinguished on the image. However, for a realistic disk or jet with emission peaked in a conical region, the subrings are visibly distinct: the nth subring is approximately a smooth peak of width e^−γn. Summing these smooth peaks, like layers in a tiered wedding cake (see Fig. 3), reproduces the leading logarithmic divergence in the intensity (Eq. 11).

• Download high-res image
• Open in new tab
• Download Powerpoint

The photons comprising successive subrings for the same angle φ_ρ traverse essentially the same orbits and hence encounter the same matter distribution around the black hole. Apart from source variations on the time scale of an orbit, intensities of the nth and (n + 1)th subring differ only because they correspond to windows whose widths δρ_n and δρ_{n + 1} differ by a factor of e^−γ. Hence, for large enough n, the intensities are related byIringn+1(ρc+δρ,φρ)≈Iringn(ρc+eγδρ,φρ)(13)

We therefore find the angle-dependent subring flux ratioFringn+1Fringn≈e−γ(14)

Equations 13 and 14 are matter-independent predictions for the photon ring structure that involve only general relativity. The prediction holds only for “large enough” n: At small n, there are nonuniversal matter-dependent effects from photons that do not traverse exactly the same region around the black hole. Insight into when n is large enough might be obtained from GRMHD simulations.

Since the exponent γ depends on a, θ_obs, and φ_ρ (see the Supplementary Materials), the flux ratio asymmetry in Eq. 14 provides a new method for determination of the spin. For Schwarzschild, γ = π (8), corresponding to a demagnification factor of e^−π ≈ 4%. For a black hole of maximal spin a/M = 1 viewed from an inclination θ_obs = 17^∘ [as estimated for M87; (17)], the factor e^−γ is as large as 13% on the part of the ring where the black hole spins toward the observer. Although Eq. 12 breaks down for n = 0, this suppression factor suggests that the leading n = 1 subring should provide ∼10% of the total luminosity, in order-of-magnitude agreement with GRMHD simulations.

Visibilities for a thin, uniform, and circular ring. Each baseline joining two elements of an interferometer samples a complex visibility V(u), which corresponds to a single Fourier component of the sky image I(x) (18)V(u)=∫I(x)e−2πiu·x d2x(15)

Here, u is the dimensionless vector baseline projected orthogonal to the line of sight and measured in units of the observation wavelength λ, while x is a dimensionless image coordinate measured in radians.

In terms of polar coordinates (ρ, φ_ρ) on the observer screen (Eqs. 10A and 10B), the image and corresponding visibility function of an infinitesimally thin, uniform, and circular ring areI(ρ,φρ)=1πdδ(ρ−d2)(16A)V(u,φu)=J0(πdu)(16B)where d is the ring diameter in radians and the image is normalized to have a total flux density of unity, V(0) = 1. J_m denotes the mth Bessel function of the first kind, which admits the asymptotic expansionJm(πdu)≈1π2ducos[π(du−2m+14)](17)valid for πdu ≫ m². Hence, V(u) is a weakly damped pure frequency with period Δu = 2/d inside an envelope that falls as 1/u.

Visibilities for a nonuniform ring. The image of a thin ring with nonuniform brightness in φ_ρ decomposes into a sum over angular Fourier modesI(ρ,φρ)=1πdδ(ρ−d2)∑m=−∞∞βmeimφρ(18)where β−m=βm* since the image is real. The total image flux density is given by β₀ > 0.

The corresponding visibility function isV(u,φu)=∑m=−∞∞βmJm(πdu)eim(φu−π/2)(19)

Using Eq. 17, for long baselines, we may approximateV(u,φu)≈α+(φu)cos (πdu)+α−(φu)sin (πdu)duα±(φu)≡1π∑m=−∞∞βmeim[φu+π2(m−1±1)](20)

Thus, for sufficiently long baselines, the radial visibility function of a nonuniform thin ring is determined by a single pair of weakly damped, orthogonal modes α_±(φ_u). Their envelope still fall as ∣V(u)∣∼1/u, and the modes have a common period of Δu = 2/d in complex visibilities (or Δu = 1/d in visibility amplitudes). The angular spectrum of the image {β_m} is easily retrieved from the angular spectrum of the visibilities (see the Supplementary Materials).

Visibilities for a thick ring. Baselines of length u ≳ 1/L are required to resolve image features of size ≲L. Hence, the visibility function of any ring with diameter d and thickness w ≪ d has two asymptotic regimes(I):1d≪ u ≪1w,(II):1d≪ 1w ≪u(21)

Baselines in regime (I) resolve the diameter of the ring but not its thickness, while longer baselines in regime (II) resolve both. Hence, the visibility function in regime (I) behaves like that of a thin ring (a damped periodicity with envelope ∣V(u)∣∼1/u), while the envelope of the visibility function in regime (II) is sensitive to the radial profile of the ring. In general, the visibility of any smooth ring decays exponentially in regime (II), although images with discontinuous derivatives, such as a uniform disk or annulus, can have slower, power-law falloffs.

The validity of the approximation of Eq. 17 in regime (I) depends on the amount of power at high values of m. Specifically, it requires mmax≲πd/w. Under this condition, ∣V(u)∣ has ≲d/w periods in regime (I).

Visibilities for a noncircular ring. Although the photon ring is nearly circular for all black hole spins and inclinations, the primary interferometric signatures discussed thus far do not require an image with perfectly circular structure. For instance, if an image is stretched, I(x, y) → I′(x, y) = I(ax, by); then, its visibility function is correspondingly compressed, V(u, v) → V′(u, v) = ∣ab∣⁻¹V(u/a, v/b). Thus, the visibility profiles of a stretched ring share the properties and asymptotic expansions derived for a circular ring (e. g., Eq. 20), except that the radial periodicities become a function of position angle. To leading order in the asymmetry 1 − a/b, the diameter corresponding to a damped radial periodicity in the visibility domain matches that of the stretched ring along the baseline’s position angle. For the black hole in M87, the asymmetry is expected to be a few percent at most, even for a maximally rotating black hole (see fig. S2).

Visibilities of the photon subrings. As discussed earlier, the photon ring decomposes into subrings labeled by the photon half-orbit number n. According to Eq. 14, the width of the radial intensity profile produced by the nth subring is w_n ∼ w₀e^−γn, while the brightness remains approximately constant with n (until some n_max determined by the optical depth). Each subring thus contributes a periodically modulated visibility, Vn(u)∼wn/u, which falls more steeply for baselines u > 1/w_n. Hence, the nth subring dominates the signal in the regime1wn−1≪ u ≪1wn(22)

This implies that the totality of subring contributions has an envelope defined by this turnover behaviorV(u)≈∑nwn<1/uwnu∼1u3/2(23)

Together, the subrings then form a cascade of damped oscillations on progressively longer baselines, each dominated by the image of a single subring and conveying precise information about its diameter, thickness, and angular profile. Figure 4 displays visibilities of the time-averaged GRMHD image in Fig. 1, which exhibit the expected damped periodicity, as well as clear contributions on long baselines from distinct subrings (see Fig. 5 for a schematic illustration of this cascade).

• Download high-res image
• Open in new tab
• Download Powerpoint

• Download high-res image
• Open in new tab
• Download Powerpoint