Introduction
The existence of black holes (BH) is a key prediction of general relativity (GR) in the strong-field regime. More specifically, the theory posits that the spacetime geometry around these compact objects is described by the Kerr metric. This “Kerr hypothesis” underlies a considerable amount of astrophysics and also plays a driving role in theoretical physics, but it has yet to be directly tested. A number of experiments are now poised to change this. The recent successes obtained in gravitational-wave and radio interferometry by LIGO-Virgo, GRAVITY and EHT ([42 ] ; [30 ] ; [10 ] ) herald the advent of a new era of black hole astronomy, and their future extensions promise to deliver observations that will enable tests of the Kerr hypothesis with unprecedented precision.
In April 2019, the EHT collaboration published the very first images resolving the nucleus of the galaxy M87 ([10 ] ; [11 ] ; [12 ] ; [12 ] ; [14 ] ; [15 ] ). Reconstructed from 1.3 mm very-long-baseline interferometry (VLBI) observations, these images achieved an angular resolution comparable to the expected size of M87*—the supermassive compact object at the center of M87—and revealed a thick, non-uniformly bright ring with a diameter of ∼ 40 μ a s \sim40\,\mu\mathrm{as} ∼ 40 μ as [43 ] ; [14 ] ), and indeed, M87* may be consistently modeled as a Kerr BH surrounded by an accretion flow with emission originating within a few Schwarzschild radii of the event horizon. The bright ring may be interpreted as the direct (weakly lensed) image of this flow (e.g., [25 ] ; [39 ] ; [6 ] ), with relativistic beaming effects explaining its observed anisotropy.
Due to the limited resolution of these first EHT images of M87*, this consistency test of the Kerr hypothesis remains weak; instead, the [11 ] has thus far assumed the Kerr nature of M87*, so as to constrain its mass from observations. However, as technology continues to improve, increasingly better images will be obtained, and these should enable correspondingly sharper probes of the spacetime geometry around M87*. Nonetheless, it is not immediately clear how to best extract relativistic signatures from such probes, which raises the question: How could sharper images of M87* in principle lead to more stringent tests of the Kerr hypothesis?
Addressing this question is made particularly difficult by the inherent asymmetry of the imaging problem. Thanks to curved-spacetime ray tracing codes such as Gyoto ([? ]; [23 ] ), solving the forward problem has become a fairly routine task, and determining the observational appearance of a compact object with specified properties (spacetime metric, emission model) is now relatively straightforward. The inverse problem, on the other hand, remains nigh intractable: inferring the nature of an astrophysical source from its visual appearance requires one to disentangle effects of the spacetime geometry from properties of the emission, a problem subject to enormous degeneracy. A veritable zoo of theoretically proposed objects (besides the Kerr BH) have been shown to also produce simulated images with features compatible with the recent EHT observations,1 ? ]).
This present blurriness is not the only limitation: even with higher resolution, images that resolve only the direct emission will likely never separate astrophysical and geometrical effects. (This partly explains the present lack of robust constraints on the spin of M87*.) As such, these images may always be compatible with models of a Kerr BH surrounded by some (possibly exotic) emission, and hence they could never lead one to definitively conclude a violation of the Kerr hypothesis, as this would require achieving greater confidence in the astrophysics than in GR—this could of course break the degeneracy in principle, but seems very unlikely in practice (e.g., [24 , 2 ] ).
Recent work suggests a promising path forward. The strong gravity of a Kerr BH creates around it a photon shell —a region outside its event horizon in which its gravity is so strong that light rays can become trapped on unstably bound orbits ([? , ? ]). As a result, sources in the vicinity of the black hole can produce multiple relativistic images in the observer sky: provided that the emission is optically thin, the primary image (consisting of weakly lensed photons that travel directly to the observer after being emitted) is generically superimposed with a series of mirror images arising from highly bent photons that circumnavigated the photon shell multiple times on their way from source to observer ([? , 26 ]). If the emission is not spherical (so that it does not fully surround the hole), then these successive images form a discrete sequence of “photon rings” that are labeled by photon half-orbit number n n n γ \gamma γ δ \delta δ τ \tau τ ? , 26 ]). This suggests that observing this characteristic lensing pattern may provide a way to extract information about the black hole geometry, and to discriminate between different spacetimes (e.g., [? ]).
The photon ring has not yet been experimentally detected, but its GR-predicted substructure has already been numerically confirmed with state-of-the-art (general relativistic magnetohydrodynamics) GRMHD-simulated images of M87*, in which at least n = 3 n=3 n = 3 [40 ] ). Although these rings (which are observable ) are distinct from the theoretical critical curve (which is not observable ), they closely track its shape, which is insensitive to the astrophysical details of the surrounding emission and entirely fixed by the Kerr metric ([1 , 28 ] ). Critical curves have been extensively discussed in the literature and spacetimes other than Kerr are known to produce different shapes (e.g., [38 , ? , 7 , ? , 45 ]). Thus, measuring the shape of a high-n n n
This idea was recently made precise by [24 ] , who established that, in a certain class of models, the shape of the n = 2 n=2 n = 2 u ∼ 300 G λ u \sim300\,\mathrm{G}\lambda u ∼ 300 G λ 2 [40 , 24 ] ). They argued that a space mission targeting M87* could resolve the shape of its n = 2 n=2 n = 2
Of course, this first study was subject to multiple limitations, as it relied on a restricted class of emission models consisting of stationary, axisymmetric disks composed of circular-equatorial orbits. More precisely: 1) it only investigated a choice of 3 toy emission profiles, viewed from observer inclinations of 10 ∘ 10^\circ 1 0 ∘ 30 ∘ 30^\circ 3 0 ∘
It is crucial to assess the robustness of the GLM method in greater generality. The aim of this paper is to extend the GLM analysis to many more configurations; that is, to broaden its scope by performing a parameter survey of emission profiles, as well as BH spins and inclinations, in order to remove the first limitation outlined above. This work remains confined to the framework of thin, equatorial accretion disks; a parallel study of the impact of disk thickness is ongoing and will soon address the second limitation ([? ]). In the meantime, it is worth noting that recent work by [6 ] suggests that geometrically thin models yield effective approximations to numerically modeled GRMHD accretion flows.
Finally, another objective of this paper is to initiate the study of black-hole-spin estimates based on astrophysics-independent shape measurements of the n = 2 n=2 n = 2
The paper is organized as follows. First, Sec. 2 reviews the theoretical foundations of the GLM method: the critical curve is defined and contrasted with the concept of “black hole shadow,” the notion of “lensing bands” is introduced, and the expected interferometric signature of the photon ring is described. Next, Sec. 3 presents a numerical implementation of the GLM method that improves upon the original technique. Then, Sec. 4 describes how the choice of baseline may affect a shape measurement of the photon ring, especially if its thickness varies substantially with angle. After that, Sec. 5 presents the results of a survey over emission profiles while Sec. 6 examines the influence of BH spin and inclination. We find that a measurement is always possible at low inclinations, while new complications that may arise at high inclination necessitate further refinements to the method. We provide further discussion in Sec. 7, before ending with our conclusions and perspectives in Sec. 8.
Theoretical overview
Shadow, critical curve and photon rings
In the introduction, we asserted that the photon ring generically decomposes into a stack of discrete subrings. For completeness, here we list the (mild) astrophysical assumptions required for the presence of this substructure; this also provides us with an opportunity to distinguish the photon ring from the “black hole shadow” and critical curve, terms deserving of disambiguation.
First, the emission region must be optically thin, so that light may cross it multiple times before being reabsorbed; otherwise, one sees only a direct (n = 0 n = 0 n = 0 [3 ] ). The validity of this assumption depends on both the source and the observing frequency. As shown in Figure 3 of [40 ] , GRMHD predicts it to likely hold for 230 GHz photons collected from M87* by present-day EHT, and it is even more likely true at the higher 345 GHz frequency that ngEHT will target; by contrast, the core of Centaurus A is only expected to be optically thin past 1–5 THz ([36 ] ).
Figure 3. A visibility amplitude profile together with its envelope and best-fit model curve. Here, the naïve fitting method would fail because the envelope has complex, non-monotonic variations; yet, the periodicity is clearly visible and amenable to extraction via our refined fitting method.
Second, the emission region must have a gap rather than fully surround the hole; otherwise, the discrete structure visible in the intensity cross-sections displayed in [40 ] ; [6 ] , which resemble the layers of a wedding cake, will not be present. Instead, one will see a continuous cross-section with a mild (logarithmic) divergence near the the critical curve: the image-plane curve, first derived by [1 ] (who used the term apparent boundary ), corresponding to light rays that asymptote to unstably bound spherical photon orbits in the photon shell. The resulting image will then display a distinctive feature known as the black hole shadow : a central brightness deficit precisely bounded by the critical curve. This effect was first highlighted by [18 ] in the context of a BH surrounded by a spherical, radially infalling accretion flow, and later revisited by [38 ] .
However, as recently pointed out by [25 ] and [47 ] , this spherical-infall scenario is highly fine-tuned, and in fact no longer viewed as relevant for M87*: the latest EHT constraints derived from the polarimetric image of M87* ([16 ] , [17 ] ) strongly favor “magnetically arrested disk” (MAD) models ([46 ] ; [35 ] ) with emission concentrated near the midplane. Such models do not display the traditional “shadow” feature (i.e., a central brightness depression filling the critical curve), but rather a different “inner shadow” feature associated with the inner edge of the emission, which produces a smaller but even darker central brightness depression contained well within the critical curve ([6 ] ). For equatorial disks extending to the horizon, the inner edge of the emission in principle coincides with the lensed equatorial event horizon (though it may appear larger due to redshift effects).
To summarize: the boundary of the “black hole shadow” is a mathematical curve which is not in itself observable, and only happens to coincide with an image feature—the central brightness depression—in special configurations currently disfavored by EHT data on M87*. On the other hand, the photon ring is a visible feature that always arises in images of a BH surrounded by an optically thin emission region. Moreover, if this emission is gapped, then this photon ring decomposes into subrings that converge (exponentially in n n n n → ∞ n \to\infty n → ∞ n = 1 n = 1 n = 1 n = 2 n = 2 n = 2
Before turning to the study of the observable photon ring shape, we first elucidate the shape of the theoretical image-plane curve comprising the null geodesics that asymptote to unstably bound orbits around a Kerr BH ([1 ] ). The region of spacetime containing these orbits—the photon shell —is extensively reviewed in [? ]; [38 ] . Here we describe it with Boyer-Lindquist coordinates ( t , r , θ , ϕ ) (t,r,\theta,\phi) ( t , r , θ , ϕ ) M M M J = a ∗ M J = a_* M J = a ∗ M
d s 2 = Δ Σ ( d t − a ∗ sin 2 θ d ϕ ) 2 + Σ Δ d r 2 + Σ d θ 2 + sin 2 θ Σ [ ( r 2 + a ∗ 2 ) d ϕ − a ∗ d t ] 2 (1) \mathrm{d}s^2 = \frac{\Delta}{\Sigma}\left(\mathrm{d}t-a_* \sin^2\theta\,\mathrm{d}\phi\right)^2+\frac{\Sigma}{\Delta}\mathrm{d}r^2+\Sigma\mathrm{d}\theta^2+\frac{\sin^2\theta}{\Sigma}\left[\left(r^2+a_*^2\right)\mathrm{d}\phi-a_*\mathrm{d}t\right]^2 \tag*{(1)} d s 2 = Σ Δ ( d t − a ∗ sin 2 θ d ϕ ) 2 + Δ Σ d r 2 + Σ d θ 2 + Σ sin 2 θ [ ( r 2 + a ∗ 2 ) d ϕ − a ∗ d t ] 2 (1) with Δ ( r ) = r 2 − 2 M r + a ∗ 2 \Delta(r)=r^2-2Mr+a_*^2 Δ ( r ) = r 2 − 2 M r + a ∗ 2 Σ ( r , θ ) = r 2 + a ∗ 2 cos 2 θ \Sigma(r,\theta)=r^2+a_*^2\cos^2\theta Σ ( r , θ ) = r 2 + a ∗ 2 cos 2 θ a ~ = a ∗ / M ∈ [ − 1 , 1 ] \tilde{a}=a_*/M\in[-1,1] a ~ = a ∗ / M ∈ [ − 1 , 1 ]
A Kerr photon with 4-momentum p μ p^\mu p μ
λ = − p ϕ p t , η = p θ 2 p t 2 − a ∗ 2 cos 2 θ + λ 2 cot 2 θ . (2) \lambda=-\frac{p_\phi}{p_t},\qquad\eta=\frac{p_\theta^2}{p_t^2}-a_*^2\cos^2\theta+\lambda^2\cot^2\theta. \tag*{(2)} λ = − p t p ϕ , η = p t 2 p θ 2 − a ∗ 2 cos 2 θ + λ 2 cot 2 θ . (2) Bound photon orbits in Kerr are “spherical”: they evolve at a constant Boyer-Lindquist radius in the range r ~ − ≤ r ≤ r ~ + \tilde{r}_-\le r\le\tilde{r}_+ r ~ − ≤ r ≤ r ~ +
r ~ ± = 2 M [ 1 + cos ( 2 3 arccos ( ± a ~ ) ) ] (3) \tilde{r}_\pm=2M\left[1+\cos\left(\frac{2}{3}\arccos(\pm\tilde{a})\right)\right] \tag*{(3)} r ~ ± = 2 M [ 1 + cos ( 3 2 arccos ( ± a ~ ) ) ] (3) is the radius of the retrograde/prograde circular-equatorial orbit, according to whether one chooses the upper/lower sign. As they rotate around the BH, these bound orbits undergo polar librations between angles [ θ ~ − , θ ~ + ] [\tilde{\theta}_-,\tilde{\theta}_+] [ θ ~ − , θ ~ + ]
θ ~ ± = arccos ( ∓ u ~ + ) ≶ π 2 , (4) \tilde{\theta}_\pm=\arccos\left(\mp\sqrt{\tilde{u}_+}\right)\lessgtr\frac{\pi}{2}, \tag*{(4)} θ ~ ± = arccos ( ∓ u ~ + ) ≶ 2 π , (4) u ~ ± = r a ∗ 2 ( r − M ) 2 [ − r 3 + 3 M 2 r − 2 a ∗ 2 M ± 2 M Δ ( 2 r 3 − 3 M r 2 + a ∗ 2 M ) ] , (5) \tilde{u}_\pm=\frac{r}{a_*^2(r-M)^2}\left[-r^3+3M^2r-2a_*^2M\pm2\sqrt{M\Delta\left(2r^3-3Mr^2+a_*^2M\right)}\right], \tag*{(5)} u ~ ± = a ∗ 2 ( r − M ) 2 r [ − r 3 + 3 M 2 r − 2 a ∗ 2 M ± 2 M Δ ( 2 r 3 − 3 M r 2 + a ∗ 2 M ) ] , (5) and their characteristic conserved quantities are given by
λ ~ = M ( r 2 − a ∗ 2 ) − r Δ a ∗ ( r − M ) , η ~ = − a ∗ 2 u ~ + u ~ − . (6) \tilde{\lambda}=\frac{M(r^2-a_*^2)-r\Delta}{a_*(r-M)},\qquad\tilde{\eta}=-a_*^2\tilde{u}_+\tilde{u}_-. \tag*{(6)} λ ~ = a ∗ ( r − M ) M ( r 2 − a ∗ 2 ) − r Δ , η ~ = − a ∗ 2 u ~ + u ~ − . (6) Bound orbits are either prograde or retrograde, depending on the sign of λ ~ \tilde{\lambda} λ ~ r ∈ [ r ~ − , r ~ + ] r\in[\tilde{r}_-,\tilde{r}_+] r ∈ [ r ~ − , r ~ + ] Δ ϕ = 2 π \Delta\phi=2\pi Δ ϕ = 2 π r r r ? ]). The photon shell occupies a 3-dimensional region of space given at any time t t t r ~ − ≤ r ≤ r ~ + \tilde{r}_-\le r\le\tilde{r}_+ r ~ − ≤ r ≤ r ~ + θ − ≤ θ ≤ θ + \theta_-\le\theta\le\theta_+ θ − ≤ θ ≤ θ + 0 ≤ ϕ < 2 π 0\le\phi<2\pi 0 ≤ ϕ < 2 π a = 0 a=0 a = 0 r = 3 M r=3M r = 3 M
A photon that is initially outside the photon shell but has exactly the same conserved quantities ( λ , η ) = ( λ ~ , η ~ ) (\lambda,\eta)=(\tilde{\lambda},\tilde{\eta}) ( λ , η ) = ( λ ~ , η ~ ) r r r D ≫ M D \gg M D ≫ M i ≠ 0 i \ne0 i = 0 [1 ] showed that a photon with conserved quantities ( λ , η ) (\lambda,\eta) ( λ , η ) 3
ρ = 1 D η + λ 2 + a ∗ 2 cos 2 i , cos φ ρ = − λ D ρ sin i , (7) \rho= \frac{1}{D}\sqrt{\eta+\lambda^2+a_*^2\cos^2 i}, \qquad\cos\varphi_\rho=-\frac{\lambda}{D\rho\sin i}, \tag*{(7)} ρ = D 1 η + λ 2 + a ∗ 2 cos 2 i , cos φ ρ = − D ρ sin i λ , (7) where, following [40 ] , we replaced Bardeen’s original Cartesian coordinates ( α , β ) (\alpha,\beta) ( α , β ) ( ρ , φ ρ ) (\rho,\varphi_\rho) ( ρ , φ ρ ) φ ρ = 90 ∘ \varphi_\rho=90^\circ φ ρ = 9 0 ∘ C γ = ( ρ ~ , φ ~ ρ ) C_\gamma=(\tilde{\rho},\tilde{\varphi}_\rho) C γ = ( ρ ~ , φ ~ ρ )
ρ ~ = 1 D λ ~ 2 + a ∗ 2 ( cos 2 i − u ~ + u ~ − ) , (8) \tilde{\rho}=\frac{1}{D}\sqrt{\tilde{\lambda}^2+a_*^2\left(\cos^2 i-\tilde{u}_+\tilde{u}_-\right)}, \tag*{(8)} ρ ~ = D 1 λ ~ 2 + a ∗ 2 ( cos 2 i − u ~ + u ~ − ) , (8) cos φ ~ ρ = − λ ~ D ρ ~ sin i , (9) \cos\tilde{\varphi}_\rho=-\frac{\tilde{\lambda}}{D\tilde{\rho}\sin i}, \tag*{(9)} cos φ ~ ρ = − D ρ ~ sin i λ ~ , (9) over the radial extent r ∈ [ r ~ − , r ~ + ] r\in[\tilde{r}_-,\tilde{r}_+] r ∈ [ r ~ − , r ~ + ] φ ~ ρ \tilde{\varphi}_\rho φ ~ ρ ϕ \phi ϕ r r r
Since (9) admits two solutions for every choice of r r r φ ~ ρ \tilde{\varphi}_\rho φ ~ ρ i = 90 ∘ i=90^\circ i = 9 0 ∘ ρ ~ 2 ≥ 0 \tilde{\rho}^2\ge0 ρ ~ 2 ≥ 0 Figure 1 in [19 ] ).
Figure 1. The n = 1 n = 1 n = 1 n = 2 n = 2 n = 2 i = 45 ∘ i = 45^\circ i = 4 5 ∘ a = 0.99 a = 0.99 a = 0.99
Finally, we emphasize that the notion of a critical curve is not limited to observers far from the BH. For instance, the critical curve for observers on timelike circular-equatorial orbits around a Kerr BH was recently analyzed by [20 ] .
A photon with conserved quantities ( λ , η ) ≈ ( λ ~ , η ~ ) (\lambda,\eta)\approx(\tilde{\lambda},\tilde{\eta}) ( λ , η ) ≈ ( λ ~ , η ~ ) r ~ \tilde{r} r ~ θ ∈ [ θ ~ − , θ ~ + ] \theta\in[\tilde{\theta}_-,\tilde{\theta}_+] θ ∈ [ θ ~ − , θ ~ + ] N N N
∣ r − r ~ ∣ ∝ e γ N , (10) |r-\tilde{r}|\propto e^{\gamma N}, \tag*{(10)} ∣ r − r ~ ∣ ∝ e γ N , (10) where one full orbit is defined as a complete oscillation from a turning point θ ± \theta_\pm θ ± N N N (6) of [26 ] . The Lyapunov exponents governing the orbital instability depend on both the BH spin and photon shell radius ([40 ] ):
γ = 4 a ∗ r ~ 2 − M r Δ ( r − M ) 2 ∫ 0 1 d t ( 1 − t 2 ) ( u ~ + 2 − u ~ − 2 ) . (11) \gamma=\frac{4}{a_*}\sqrt{\tilde{r}^2-\frac{Mr\Delta}{(r-M)^2}\int_0^1\frac{dt}{\sqrt{(1-t^2)(\tilde{u}_+^2-\tilde{u}_-^2)}}.} \tag*{(11)} γ = a ∗ 4 r ~ 2 − ( r − M ) 2 M r Δ ∫ 0 1 ( 1 − t 2 ) ( u ~ + 2 − u ~ − 2 ) d t . (11) By definition, when a nearly bound photon reaches a distant observer, it appears at a position close to the critical curve (eq:8 ): ( ρ , φ ρ ) ≈ ( ρ ~ ( r ~ ) , φ ~ ρ ( r ~ ) ) (\rho,\varphi_\rho)\approx(\tilde{\rho}(\tilde{r}),\tilde{\varphi}_\rho(\tilde{r})) ( ρ , φ ρ ) ≈ ( ρ ~ ( r ~ ) , φ ~ ρ ( r ~ )) r ~ \tilde{r} r ~ ( r ~ , d ) (\tilde{r},d) ( r ~ , d ) d d d ( ρ ~ ( r ~ ) , φ ~ ρ ( r ~ ) ) (\tilde{\rho}(\tilde{r}),\tilde{\varphi}_\rho(\tilde{r})) ( ρ ~ ( r ~ ) , φ ~ ρ ( r ~ )) [40 ] heuristically showed, and then [26 ] rigorously proved, that a light ray shot backwards into the geometry from position ( r ~ , d ) (\tilde{r},d) ( r ~ , d )
N ( r ~ , d ) ≈ − 1 γ ( r ~ ) log ∣ d ∣ (12) N(\tilde{r},d)\approx-\frac{1}{\gamma(\tilde{r})}\log|d| \tag*{(12)} N ( r ~ , d ) ≈ − γ ( r ~ ) 1 log ∣ d ∣ (12) half-orbits around the BH before either returning to asymptotic infinity, if it was outside the critical curve ( d > 0 ) (d>0) ( d > 0 ) ( d < 0 ) (d<0) ( d < 0 ) [26 ] also computed subleading O ( d 0 ) O(d^0) O ( d 0 ) N ( r ~ , d ) N(\tilde{r},d) N ( r ~ , d )
While (12) is a purely geometric statement about the lensing behavior of a Kerr BH, it also constrains the structure of BH images. To describe these constraints, we will from now on restrict our attention to emission regions that are optically thin, stationary and axisymmetric. The assumption of stationarity and axisymmetry amounts to considering only time-averaged images (averaged over sufficiently long timescales), and the assumption of infinite optical depth allows one to neglect absorption effects; as discussed in Sec. 2.1, this approximation becomes exact in the limit of large observation frequency, and could already provide a good approximation at the frequencies of current or near-future observations of M87*.
Since the emission region around the BH is optically thin, a light ray that passes through it 2 N 2N 2 N N N N
I ( r , d ) ∝ N ( r ~ , d ) ≈ − 1 γ ( r ~ ) log ∣ d ∣ . (13) I(r,d)\propto N(\tilde{r},d)\approx-\frac{1}{\gamma(\tilde{r})}\log|d|. \tag*{(13)} I ( r , d ) ∝ N ( r ~ , d ) ≈ − γ ( r ~ ) 1 log ∣ d ∣. (13) As such, the lensing formula (11) implies a mild (logarithmic) divergence in the observed intensity I ( ρ , φ ρ ) I(\rho,\varphi_\rho) I ( ρ , φ ρ ) N N N
Due to (12) , any axisymmetric ring emitting isotropically produces an infinite sequence of lensed images within the photon ring ([26 ] ). Successive images within this sequence are produced by photons describing increasingly many half-orbits around the black hole: if the first (direct) image arises from photons with fractional half-orbit number 0 ≤ N 0 < 1 0\le N_0<1 0 ≤ N 0 < 1 n t h n^{\rm th} n th N ≈ N 0 + n N\approx N_0+n N ≈ N 0 + n n ∈ N n\in\mathbb{N} n ∈ N
Moreover, (12) implies that these successive images are increasingly lensed towards the critical curve by an exponential demagnification factor: a source ring whose n t h n^{\rm th} n th d n ( r ~ ) d_n(\tilde{r}) d n ( r ~ ) ( n + 1 ) t h (n+1)^{\rm th} ( n + 1 ) th
d n + 1 ( r ~ ) = e − γ ( r ~ ) d n ( r ~ ) . (14) d_{n+1}(\tilde{r})=e^{-\gamma(\tilde{r})}d_n(\tilde{r}). \tag*{(14)} d n + 1 ( r ~ ) = e − γ ( r ~ ) d n ( r ~ ) . (14) Depending on whether the emission region is gapped or not, this lensing pattern can result in one of two very distinctive image substructures within the photon ring.
If the BH is entirely immersed in its surrounding emission (as in the case of a spherically symmetric accretion flow onto the hole, for instance), then successive images of the emission region continuously blend together and the resulting intensity profile logarithmically rises to a smooth peak centered about the critical curve. Such an intensity profile is displayed in Figure 1 of [47 ] , with the corresponding image shown in Figure 2 therein: a bright photon ring encircling a central brightness deficit (the interior of the critical curve). Moreover, as shown in that paper, if the emitting matter is infalling, then beaming effects strongly attenuate the intensity inside the critical curve, creating the distinctive shadow effect reviewed in Sec. 2.1 above. The reason is that light rays appearing inside the critical curve cannot encounter a radial turning point, so any photons collected along such rays had to have been emitted opposite to the infalling matter’s direction of motion, thereby incurring a strong redshift.
Figure 2. Example of a visibility amplitude profile at fixed angle φ = 0 ∘ \varphi= 0^\circ φ = 0 ∘ (26) are visible: in the range [ 0 , 20 ] [0,20] [ 0 , 20 ] λ \lambda λ n = 0 n=0 n = 0 u ∈ [ 20 , 120 ] u \in[20,120] u ∈ [ 20 , 120 ] λ \lambda λ n = 1 n=1 n = 1 λ \lambda λ n = 2 n=2 n = 2
On the other hand, if the emission is gapped (as in the case of emission localized around the equator with gaps near the poles, for instance), then each successive image of the emission region displays a corresponding gap, and the resulting intensity profile no longer consists of a smooth logarithmic peak (13) , but rather a sequence of distinct peaks accumulating near the critical curve. Such an intensity profile is displayed in Figure 1 of [25 ] , with the corresponding image shown in Figure 5 therein. In such images, the photon ring decomposes into a stack of distinct photon subrings, each of which is labeled by n n n e − γ n e^{-\gamma n} e − γ n n = 3 n = 3 n = 3 [40 ] ), which is also present in all the images that we obtained from our toy models of M87*.
Figure 5. Example of a successful circlipse fit and Kerr hypothesis test. Left: Every 5 ∘ 5^\circ 5 ∘ d φ d_\varphi d φ n = 2 n = 2 n = 2 g ( d φ ) g(d_\varphi) g ( d φ ) φ \varphi φ Δ d φ ≈ 1 / u w \Delta d_\varphi\approx1/u_w Δ d φ ≈ 1/ u w g ( d φ ) g(d_\varphi) g ( d φ )
Photon subrings lie in lensing bands
To study the subring structure of the photon ring, it is helpful to introduce the (purely geometric) notion of a lensing band ([26 ] ; [6 ] ): a Kerr observer’s n t h n^{\mathrm{th}} n th n + 1 n + 1 n + 1 d > 0 d > 0 d > 0 r = r + r = r_{+} r = r + d < 0 d < 0 d < 0
r ± = M ± M 2 − a 2 . (15) r_{\pm} = M \pm\sqrt{M^{2} - a^{2}}. \tag*{(15)} r ± = M ± M 2 − a 2 . (15) We present a general method for computing Kerr lensing bands in App. A. Figure 1 illustrates the n = 1 n = 1 n = 1 n = 2 n = 2 n = 2 i = 45 ∘ i = 45^\circ i = 4 5 ∘ a = 0.99 a = 0.99 a = 0.99
Each n > 0 n > 0 n > 0 n t h n^{\mathrm{th}} n th n t h n^{\mathrm{th}} n th r = r + r = r_{+} r = r + r → ∞ r \to\infty r → ∞ (14) , these edges converge to the critical curve exponentially fast in n n n
w n + 1 ( r ⃗ ) ≈ e − γ ( r ⃗ ) w n ( r ⃗ ) ≈ e − n γ ( r ⃗ ) w 1 ( r ⃗ ) , (16) w_{n+1}(\vec{r}) \approx e^{-\gamma(\vec{r})}w_n(\vec{r}) \approx e^{-n\gamma(\vec{r})}w_1(\vec{r}), \tag*{(16)} w n + 1 ( r ) ≈ e − γ ( r ) w n ( r ) ≈ e − nγ ( r ) w 1 ( r ) , (16) where w 1 ( r ⃗ ) w_1(\vec{r}) w 1 ( r ) n = 1 n = 1 n = 1 n → ∞ n \to\infty n → ∞
To summarize: the lensing bands form a stacked sequence of annular shapes, each straddling the critical curve, with the ( n + 1 ) t h (n + 1)^{\mathrm{th}} ( n + 1 ) th n t h n^{\mathrm{th}} n th
From now on, we further restrict our attention to a thin disk of isotropic emitters filling the equatorial plane. In this setting, the lensing bands are particularly useful because they coincide with the photon subrings: the n t h n^{\mathrm{th}} n th n t h n^{\mathrm{th}} n th n t h n^{\mathrm{th}} n th (16) . Hence, measuring the angle-dependent ratio w n + 1 ( r ⃗ ) / w n ( r ⃗ ) ≈ e − γ ( r ⃗ ) w_{n+1}(\vec{r})/w_n(\vec{r}) \approx e^{-\gamma(\vec{r})} w n + 1 ( r ) / w n ( r ) ≈ e − γ ( r ) γ ( r ⃗ ) \gamma(\vec{r}) γ ( r ) [40 ] ).
Interferometric signature of a narrow curve in the sky
Having described the subring structure of BH images, we next turn to a description of the corresponding subring structure in the visibility (Fourier) domain. To this end, we first begin by reviewing some crucial facts about the interferometric signature of a narrow feature in the sky.
Consider an infinitely thin, closed light curve in the sky. Assuming this plane curve is convex, it may be parameterized by normal angle φ ∈ [ 0 , 2 π ) \varphi\in[0, 2\pi) φ ∈ [ 0 , 2 π ) r ⃗ ( φ ) = ( α ( φ ) , β ( φ ) ) \vec{r}(\varphi) = (\alpha(\varphi), \beta(\varphi)) r ( φ ) = ( α ( φ ) , β ( φ )) ( α , β ) (\alpha,\beta) ( α , β ) r ⃗ i ( φ ) \vec{r}_i(\varphi) r i ( φ ) n ^ ( φ ) = ( − cos φ , sin φ ) \hat{n}(\varphi) = (-\cos\varphi,\sin\varphi) n ^ ( φ ) = ( − cos φ , sin φ ) [26 ] , we define the projected position f ( φ ) f(\varphi) f ( φ )
f ( φ ) = − r ⃗ ( φ ) ⋅ n ^ ( φ ) = α ( φ ) cos φ + β ( φ ) sin φ . (17) f(\varphi) = -\vec{r}(\varphi) \cdot\hat{n}(\varphi) = \alpha(\varphi)\cos\varphi+ \beta(\varphi)\sin\varphi. \tag*{(17)} f ( φ ) = − r ( φ ) ⋅ n ^ ( φ ) = α ( φ ) cos φ + β ( φ ) sin φ . (17) This function encodes the whole curve r ( φ ) = ( α ( φ ) , β ( φ ) ) \boldsymbol{r}(\varphi) = (\alpha(\varphi),\beta(\varphi)) r ( φ ) = ( α ( φ ) , β ( φ ))
α ( φ ) = f ( φ ) cos φ − f ′ ( φ ) sin φ , (18) \alpha(\varphi) = f(\varphi)\cos\varphi- f'(\varphi)\sin\varphi, \tag*{(18)} α ( φ ) = f ( φ ) cos φ − f ′ ( φ ) sin φ , (18) β ( φ ) = f ( φ ) sin φ + f ′ ( φ ) cos φ . (19) \beta(\varphi) = f(\varphi)\sin\varphi+ f'(\varphi)\cos\varphi. \tag*{(19)} β ( φ ) = f ( φ ) sin φ + f ′ ( φ ) cos φ . (19) The projected position f ( φ ) f(\varphi) f ( φ ) R ( φ ) = f ( φ ) + f ′ ′ ( φ ) R(\varphi) = f(\varphi) + f''(\varphi) R ( φ ) = f ( φ ) + f ′′ ( φ ) f ( φ ) f(\varphi) f ( φ )
f ( φ ) = d φ 2 + C φ , (20) f(\varphi) = \frac{d_\varphi}{2} + C_\varphi, \tag*{(20)} f ( φ ) = 2 d φ + C φ , (20) where
d φ = f ( φ ) + f ( φ + π ) , (21) d_\varphi= f(\varphi) + f(\varphi+\pi), \tag*{(21)} d φ = f ( φ ) + f ( φ + π ) , (21) C φ = 1 2 [ f ( φ ) − f ( φ + π ) ] . (22) C_\varphi= \frac{1}{2}\left[f(\varphi) - f(\varphi+\pi)\right]. \tag*{(22)} C φ = 2 1 [ f ( φ ) − f ( φ + π ) ] . (22) The even part d φ = d φ + π d_\varphi= d_{\varphi+\pi} d φ = d φ + π φ \varphi φ C φ = − C φ + π C_\varphi= -C_{\varphi+\pi} C φ = − C φ + π [28 ] illustrate several examples of this shape decomposition.
We now arrive at the key point: if the curve is not infinitely thin, but still very narrow relative to its diameter (w ≪ d φ w \ll d_\varphi w ≪ d φ V ( u , φ ) V(u,\varphi) V ( u , φ ) ( u , φ ) (u,\varphi) ( u , φ )
1 d ≪ u ≪ 1 w , (23) \frac{1}{d} \ll u \ll\frac{1}{w}, \tag*{(23)} d 1 ≪ u ≪ w 1 , (23) where it adopts the “universal” form, valid to leading order in an expansion in 0 < w / d ≪ 1 0 < w/d \ll1 0 < w / d ≪ 1 [24 ] ,
V ( u , φ ) ≈ e − 2 π i C φ u u [ α L ( φ ) e − i π 4 e i π d φ u + α R ( φ ) e i π 4 e − i π d φ u ] , (24) V(u,\varphi) \approx\frac{e^{-2\pi i C_\varphi u}}{\sqrt{u}}\left[\alpha_L(\varphi)e^{-\frac{i\pi}{4}}e^{i\pi d_\varphi u}+\alpha_R(\varphi)e^{\frac{i\pi}{4}}e^{-i\pi d_\varphi u}\right], \tag*{(24)} V ( u , φ ) ≈ u e − 2 π i C φ u [ α L ( φ ) e − 4 iπ e iπ d φ u + α R ( φ ) e 4 iπ e − iπ d φ u ] , (24) with α L , R ( φ ) = α R , L ( φ + π ) > 0 \alpha_{L,R}(\varphi) = \alpha_{R,L}(\varphi+\pi) > 0 α L , R ( φ ) = α R , L ( φ + π ) > 0 V ( u , φ + π ) = V ∗ ( u , φ ) V(u,\varphi+\pi) = V^*(u,\varphi) V ( u , φ + π ) = V ∗ ( u , φ ) (33) below. In the context of interferometry, the complex function V ( u , φ ) V(u,\varphi) V ( u , φ ) ( u , φ ) (u,\varphi) ( u , φ ) u ≪ 1 / d u \ll1/d u ≪ 1/ d u ≫ 1 / w u \gg1/w u ≫ 1/ w (23) , the curve appears infinitely thin and as a result, its signal follows a very weak u − 1 / 2 u^{-1/2} u − 1/2 (24) .
In the context of realistic high-frequency VLBI observations, it is in practice significantly easier to measure the amplitude ∣ V ∣ |V| ∣ V ∣ visamp —rather than its absolute phase, as the latter is more susceptible to contamination by a variety of atmospheric and instrumental effects. While the full complex visibility (24) encodes both d φ d_\varphi d φ C φ C_\varphi C φ d φ d_\varphi d φ ∣ V ( u , φ + π ) ∣ = ∣ V ( u , φ ) ∣ |V(u,\varphi+\pi)| = |V(u,\varphi)| ∣ V ( u , φ + π ) ∣ = ∣ V ( u , φ ) ∣
∣ V ( u , φ ) ∣ ≈ α L 2 ( φ ) + α R 2 ( φ ) + 2 α L ( φ ) α R ( φ ) sin ( 2 π d φ u ) u . (25) |V(u,\varphi)| \approx\sqrt{\frac{\alpha_L^2(\varphi) + \alpha_R^2(\varphi) + 2\alpha_L(\varphi)\alpha_R(\varphi)\sin(2\pi d_\varphi u)}{u}}. \tag*{(25)} ∣ V ( u , φ ) ∣ ≈ u α L 2 ( φ ) + α R 2 ( φ ) + 2 α L ( φ ) α R ( φ ) sin ( 2 π d φ u ) . (25) To summarize: the clearest interferometric signature of a bright curve in the sky is its visibility amplitude (25) , which is strong in the universal regime (23) and only encodes partial information about the shape, namely its projected diameter d φ d_\varphi d φ C φ C_\varphi C φ
Interferometric signature of the photon ring
Combining the results of the previous sections, we are now in a position to describe the interferometric signature produced by the photon ring. We expect the photon subrings to be the only narrow features present in time-averaged BH images, since other fine features (such as emission ropes and other flares) should be transient and wash out after sufficient time-averaging. Thus, we expect that the n t h n^{\mathrm{th}} n th
1 w n − 1 ≪ u ≪ 1 w n , (26) \frac{1}{w_{n-1}} \ll u \ll\frac{1}{w_n}, \tag*{(26)} w n − 1 1 ≪ u ≪ w n 1 , (26) where, according to (16) , w n ( r ~ ) = e − γ ( r ~ ) w n − 1 ( r ~ ) w_n(\tilde r) = e^{-\gamma(\tilde r)}w_{n-1}(\tilde r) w n ( r ~ ) = e − γ ( r ~ ) w n − 1 ( r ~ ) ( n + 1 ) t h (n+1)^{\mathrm{th}} ( n + 1 ) th Figure 2 . In each range (26) , the n t h n^{\mathrm{th}} n th (25) governed by its projected diameter d φ ( n ) d_\varphi^{(n)} d φ ( n ) d φ ( n ) d_\varphi^{(n)} d φ ( n ) d ~ φ ≡ d φ ( ∞ ) \tilde d_\varphi\equiv d_\varphi^{(\infty)} d ~ φ ≡ d φ ( ∞ ) n → ∞ n \to\infty n → ∞
Theoretical shape of the critical curve
The phoval (for “photon ring oval”) is the geometric shape with
d φ 2 = R 0 + R 1 2 sin 2 φ + R 2 2 cos 2 φ , (27) \frac{d_\varphi}{2} = R_0 + \sqrt{R_1^2\sin^2\varphi+ R_2^2\cos^2\varphi}, \tag*{(27)} 2 d φ = R 0 + R 1 2 sin 2 φ + R 2 2 cos 2 φ , (27) C φ = ( X − χ ) cos φ + arcsin ( χ cos φ ) . (28) C_\varphi= (X-\chi)\cos\varphi+ \arcsin(\chi\cos\varphi). \tag*{(28)} C φ = ( X − χ ) cos φ + arcsin ( χ cos φ ) . (28) More precisely, this is a family of shapes parameterized by five parameters R 0 R_0 R 0 R 1 R_1 R 1 R 2 R_2 R 2 X X X χ \chi χ [28 ] .
We briefly summarize the geometric interpretation of these parameters. First, the projected position f c i r c l e ( φ ) = R 0 f_{\mathrm{circle}}(\varphi)=R_0 f circle ( φ ) = R 0 R 0 R_0 R 0 f e l l i p s e ( φ ) = R 1 2 sin 2 φ + R 2 2 cos 2 φ f_{\mathrm{ellipse}}(\varphi)=R_1^2\sin^2\varphi+R_2^2\cos^2\varphi f ellipse ( φ ) = R 1 2 sin 2 φ + R 2 2 cos 2 φ R 1 R_1 R 1 R 2 R_2 R 2 f c i r c l i p s e ( φ ) = d ~ φ / 2 f_{\mathrm{circlipse}}(\varphi)=\tilde d_\varphi/2 f circlipse ( φ ) = d ~ φ /2 (27) , which describes a shape known as a circlipse. It follows from the last paragraph that the projected diameter of the critical curve is always exquisitely close to that of a circlipse,
d ~ φ ≡ d φ ( ∞ ) ≈ 2 R 0 + 2 R 1 2 sin 2 φ + R 2 2 cos 2 φ , (29) \tilde d_\varphi\equiv d_\varphi^{(\infty)} \approx2R_0+2\sqrt{R_1^2\sin^2\varphi+R_2^2\cos^2\varphi}, \tag*{(29)} d ~ φ ≡ d φ ( ∞ ) ≈ 2 R 0 + 2 R 1 2 sin 2 φ + R 2 2 cos 2 φ , (29) with the best-fit parameters R 0 R_0 R 0 R 1 R_1 R 1 R 2 R_2 R 2 M / D M/D M / D a a a i i i (28) to the projected function produces the full phoval shape, which is translated along the α \alpha α X ∈ R X\in\mathbb{R} X ∈ R χ ∈ [ − 1 , 1 ] \chi\in[-1,1] χ ∈ [ − 1 , 1 ] 10 5 10^5 1 0 5 10 3 10^3 1 0 3 a → 1 a\to1 a → 1 [28 ] . Away from extremality, the (parity-odd) centroid motion gives a minor correction to the shape of the critical curve, which is almost entirely encoded in its (parity-even) circlipse diameter.
A critical curve with best-fit circlipse parameters R 0 R_0 R 0 R 1 R_1 R 1 R 2 R_2 R 2 d ⊥ ≡ d 0 d_\perp\equiv d_0 d ⊥ ≡ d 0 d ∥ ≡ d π / 2 d_\parallel\equiv d_{\pi/2} d ∥ ≡ d π /2 α \alpha α β \beta β
d ⊥ = 2 ( R 0 + R 2 ) , d ∥ = 2 ( R 0 + R 1 ) , (30) d_\perp=2(R_0+R_2),\qquad d_\parallel=2(R_0+R_1), \tag*{(30)} d ⊥ = 2 ( R 0 + R 2 ) , d ∥ = 2 ( R 0 + R 1 ) , (30) which define a fractional asymmetry [40 ]
f A ≡ 1 − d ⊥ d ∥ = R 1 − R 2 R 0 + R 1 ∈ [ 0 , 1 ] . (31) f_A\equiv1-\frac{d_\perp}{d_\parallel}=\frac{R_1-R_2}{R_0+R_1}\in[0,1]. \tag*{(31)} f A ≡ 1 − d ∥ d ⊥ = R 0 + R 1 R 1 − R 2 ∈ [ 0 , 1 ] . (31) The range 0 ≤ f A ≤ 1 0\le f_A\le1 0 ≤ f A ≤ 1 d ⊥ ≤ d ∥ d_\perp\le d_\parallel d ⊥ ≤ d ∥ R 2 ≤ R 1 R_2\le R_1 R 2 ≤ R 1 (27) that d ⊥ d_\perp d ⊥ d ∥ d_\parallel d ∥
Finally, we point out that the two quantities d ∥ d_\parallel d ∥ f A f_A f A a a a i i i Figure 7 of [40 ] . The map ( a , i ) ↦ ( d ∥ , f A ) (a,i)\mapsto(d_\parallel,f_A) ( a , i ) ↦ ( d ∥ , f A ) ( a , i ) ↦ ( a , i ) (a,i)\mapsto(a,i) ( a , i ) ↦ ( a , i ) ( − a , i + π ) (-a,i+\pi) ( − a , i + π ) ( d ∥ , f A ) (d_\parallel,f_A) ( d ∥ , f A ) d ~ φ \tilde d_\varphi d ~ φ (29) , or even just its minimum and maximum d ⊥ d_\perp d ⊥ d ∥ d_\parallel d ∥ X X X χ \chi χ d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) n = 2 n=2 n = 2
Figure 7. (Left) Gyoto log-scale image of a BH with spin a = 0.5 a=0.5 a = 0.5 i = 89 ∘ i=89^\circ i = 8 9 ∘ (54) with parameters μ = r − \mu=r_- μ = r − γ = − 2 \gamma=-2 γ = − 2 σ = 1.5 M \sigma=1.5M σ = 1.5 M n = 0 n=0 n = 0 β ≈ 0 \beta\approx0 β ≈ 0 n = 1 n=1 n = 1 β > 0 \beta>0 β > 0 n = 2 n=2 n = 2 β < 0 \beta<0 β < 0
Testing GR with the photon ring shape
As we have explained, the projected diameter of the Kerr critical curve depends only on the spacetime geometry: d ~ φ \tilde d_\varphi d ~ φ
However, this theoretical curve, and hence its diameter d ~ φ \tilde d_\varphi d ~ φ n n n n → ∞ n\to\infty n → ∞ d φ ( n ) → d φ ( ∞ ) = d ~ φ d_\varphi^{(n)}\to d_\varphi^{(\infty)}=\tilde d_\varphi d φ ( n ) → d φ ( ∞ ) = d ~ φ n n n d ~ φ \tilde d_\varphi d ~ φ n n n
The first (n = 1 n=1 n = 1 n ≥ 3 n\ge3 n ≥ 3 (26) , which (given present observation frequencies) could only be accessed using a space element at unrealistic separation from the Earth (e.g., Figure 5 of [39 ] ); second, this problem is compounded by the need for exceptional sensitivity, as the flux in each subring is also exponentially suppressed in n n n (16) . As such, the second subring seems to be the most promising target for a shape test of GR, as it may sit in the “sweet spot” where it remains accessible with current or near-future technology [24 ] , while n = 2 n=2 n = 2 d φ ( 2 ) ≈ d ~ φ d_\varphi^{(2)}\approx\tilde d_\varphi d φ ( 2 ) ≈ d ~ φ
[24 ] initiated a quantitative investigation of this idea and found that in a wide class of toy models of M87* (subject to the limitations listed in the introduction), the n = 2 n=2 n = 2 (25) . Moreover, they showed that it was possible to recover d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) n = 2 n=2 n = 2 n → ∞ n\to\infty n → ∞ n = 2 n=2 n = 2 d φ ( 2 ) ≠ d ~ φ d_\varphi^{(2)}\ne\tilde d_\varphi d φ ( 2 ) = d ~ φ d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) (29) .
We emphasize that, while the best-fit radii R 0 R_0 R 0 R 1 R_1 R 1 R 2 R_2 R 2 d ~ φ \tilde d_\varphi d ~ φ M / D M/D M / D a a a i i i R 0 R_0 R 0 R 1 R_1 R 1 R 2 R_2 R 2 d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) Figure 7 of [24 ] , which shows how these best-fit parameters vary with the astrophysical model, keeping the BH parameters fixed. Thus, the n = 2 n=2 n = 2 d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 )
d φ ( 2 ) 2 ≈ R 0 + R 1 2 sin 2 ( φ − φ 0 ) + R 2 2 cos 2 ( φ − φ 0 ) , (32) \frac{d_\varphi^{(2)}}{2}\approx R_0+\sqrt{R_1^2\sin^2(\varphi-\varphi_0)+R_2^2\cos^2(\varphi-\varphi_0)}, \tag*{(32)} 2 d φ ( 2 ) ≈ R 0 + R 1 2 sin 2 ( φ − φ 0 ) + R 2 2 cos 2 ( φ − φ 0 ) , (32) which includes an additional parameter φ 0 \varphi_0 φ 0 [24 ] to propose a novel GR test based on the n = 2 n=2 n = 2 d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) (32) .
If the deviation of d φ ( 2 ) d_{\varphi}^{(2)} d φ ( 2 ) d φ ( 2 ) d_{\varphi}^{(2)} d φ ( 2 ) [26 ] attained a stringent, sub-percent (0.04%) level of precision.
Finally, we wish to clarify in what sense we regard the GLM method as a test of the Kerr hypothesis. Such a test can either be
a consistency test aiming to confirm that some observable is consistent with the prediction of the Kerr spacetime, in which case there is no notion of model comparison; or,
a model-comparison test aiming to compare predictions of the Kerr hypothesis with alternative predictions derived from other spacetime geometries, so as to determine which one best explains the data (in the Bayesian sense).
In this article, we discuss the GLM method as a consistency test of the Kerr hypothesis and thus do not consider any other spacetime but that of Kerr. Performing an unambiguous model-comparison test of the Kerr hypothesis is a difficult challenge, as one must be careful to properly take into account all complex astrophysical degeneracies (e.g., [2 ] ).
Additional work is needed to establish the viability of this GLM method for a wider range of configurations. In this paper, we explore whether the method remains viable when some of the limitations of the original GLM analysis are removed; this will be the object of our parameter survey in Secs. 5 and 6 below.
Implementation
Our goal is to check the consistency, robustness, and precision of the Kerr hypothesis test reviewed in Sec. 2. To this end, we simulate high-resolution images of Kerr BHs using various types of emission and confirm that they each display a photon ring with n = 1 n = 1 n = 1 n = 2 n = 2 n = 2 d φ ( 2 ) d_{\varphi}^{(2)} d φ ( 2 ) n = 2 n = 2 n = 2 (25) in the visibility amplitude on long baselines, and verify that it follows the predicted functional form (32) , establishing the viability of the GLM method as a test of the Kerr hypothesis. We also study the discrepancy between the n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 Observatoire de Paris ), whose details are presented in [? ] .
Image simulation
Gyoto simulates an image, that is, a map I ( ρ , φ ρ ) I(\rho,\varphi_{\rho}) I ( ρ , φ ρ ) V ( u , φ ) V(u,\varphi) V ( u , φ )
V ( u , φ ) = ∫ 0 ∞ ∫ 0 2 π I ( ρ , φ ρ + φ ) e − 2 π i u ρ cos φ ρ ρ d φ ρ d ρ , (33) V(u,\varphi) = \int_{0}^{\infty}\int_{0}^{2\pi} I(\rho,\varphi_{\rho}+\varphi)e^{-2\pi i u\rho\cos\varphi_{\rho}} \rho\, \mathrm{d}\varphi_{\rho}\,\mathrm{d}\rho, \tag*{(33)} V ( u , φ ) = ∫ 0 ∞ ∫ 0 2 π I ( ρ , φ ρ + φ ) e − 2 π i u ρ c o s φ ρ ρ d φ ρ d ρ , (33) which by definition satisfies V ( u , φ + π ) = V ∗ ( u , φ ) V(u,\varphi+\pi) = V^{*}(u,\varphi) V ( u , φ + π ) = V ∗ ( u , φ ) [26 ] and make use of the projection-slice theorem to directly compute V ( u , φ ) V(u,\varphi) V ( u , φ ) φ \varphi φ φ \varphi φ I ( ρ , φ ) I(\rho,\varphi) I ( ρ , φ ) φ \varphi φ V ( u , φ ) V(u,\varphi) V ( u , φ ) φ \varphi φ
We make the following assumptions in our computations:
BH mass: M = 6.2 × 10 9 M ⊙ M = 6.2 \times10^{9} M_{\odot} M = 6.2 × 1 0 9 M ⊙ [21 ] ; [10 ] );
distance between the BH and observer: D = 16.9 D = 16.9 D = 16.9 [15 ] );
observation wavelength: λ = 1.3 \lambda= 1.3 λ = 1.3 [11 ] );
we neglect absorption effects, a suitable approximation for optically thin accretion flows, as discussed in Sec. 2 above;
we use Bardeen’s coordinates (7) in the sky (so the projected spin axis points in the direction φ ρ = 90 ∘ \varphi_{\rho} = 90^{\circ} φ ρ = 9 0 ∘
we terminate the ray tracing of geodesics after their second equatorial crossing, so as to avoid parasitic pixels produced by an under-resolved, partially imaged n = 3 n = 3 n = 3
From these values of M M M D D D M M M [15 ] ):
( M / D ) M 87 ∗ = 3.6212 μ a s . (34) (M/D)_{\mathrm{M87}^{*}} = 3.6212 \,\mu\mathrm{as}. \tag*{(34)} ( M / D ) M87 ∗ = 3.6212 μ as . (34) We take our field of view to be of 180 μ a s ≈ 50 M \mu\mathrm{as} \approx50M μ as ≈ 50 M × \times × n = 2 n = 2 n = 2
Image treatment and analysis
Apodization
We find that the visibility amplitude profile ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ n = 2 n = 2 n = 2 (25) because it is “polluted” by some other oscillation with higher period and lesser amplitude.
This effect arises from an artificial discontinuity in the Radon projection, which is set to zero outside our field of view but does not exactly vanish at its edges, where it instead decays to a tiny (but still nonzero) intensity that is ≲ 10 − 4 \lesssim10^{-4} ≲ 1 0 − 4
To eliminate this spurious artifact, one solution would be to increase the field of view until the intensity at the edges is so small that this effect vanishes. However, in order to still resolve the n = 2 n = 2 n = 2 [22 ] ), or else by decomposing the total image into a sum of multiple layers, each with its own resolution. The latter approach was adopted by [26 ] , who divided the image into layers labeled by half-orbit number n n n n t h n^{\mathrm{th}} n th n t h n^{\mathrm{th}} n th
Instead of computing images with an unnecessarily large field of view or using adaptive ray tracing, we dealt with this effect by simply multiplying the Radon projection by a window function. This method is known as apodization.
More specifically, we used the window function (also known as an apodization or tapering function)
W ( x ) = p ( r ( 1 − x ) ) p ( r ( 1 + x ) ) , (35) W(x)=p(r(1-x))p(r(1+x)), \tag*{(35)} W ( x ) = p ( r ( 1 − x )) p ( r ( 1 + x )) , (35) where x ∈ [ − 1 , 1 ] x\in[-1,1] x ∈ [ − 1 , 1 ] ϵ = 1 / r \epsilon=1/r ϵ = 1/ r p p p
p ( x ) = f ( x ) f ( x ) + f ( 1 − x ) , f ( x ) = { e − 1 / x 2 if x > 0 , 0 if x ≤ 0. (36) p(x)=\frac{f(x)}{f(x)+f(1-x)},\qquad f(x)=\begin{cases}e^{-1/x^2}&\text{if }x>0,\\0&\text{if }x\leq0.\end{cases} \tag*{(36)} p ( x ) = f ( x ) + f ( 1 − x ) f ( x ) , f ( x ) = { e − 1/ x 2 0 if x > 0 , if x ≤ 0. (36) This function is infinitely smooth (C ∞ C^\infty C ∞ x ∈ [ − 1 + ϵ , 1 − ϵ ] x\in[-1+\epsilon,1-\epsilon] x ∈ [ − 1 + ϵ , 1 − ϵ ] x = ± 1 x=\pm1 x = ± 1
Visibility amplitude profile fit
Once the visibility amplitude of a given image has been obtained, the next step in the GLM test of the Kerr hypothesis is to extract the diameter d φ ( 2 ) d_{\varphi}^{(2)} d φ ( 2 ) n = 2 n=2 n = 2 φ ρ \varphi_{\rho} φ ρ ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ φ = φ ρ \varphi=\varphi_{\rho} φ = φ ρ (25) in the baseline regime (26) appropriate to the n = 2 n=2 n = 2 α L \alpha_L α L α R \alpha_R α R d φ d_{\varphi} d φ d φ ( 2 ) d_{\varphi}^{(2)} d φ ( 2 )
In practice, however, a simple fitting method—such as the FindFit routine implemented in Mathematica—works only when the signal is extremely “clean,” in the sense that ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ (25) . This happens in ideal cases where the image perfectly resolves the photon ring and the baselines chosen for the fit correspond exactly to the domain (26) over which the contributions from the n = 2 n=2 n = 2 Figure 3 —the periodicity is still manifestly present. A more robust approach is clearly called for.
Markov chain Monte-Carlo (MCMC) methods offer such an approach and prove effective even when the signal is buried in noise. For instance, the data simulated by [26 ] for their experimental forecast contained so much noise so as to render the periodicity invisible to the eye (see panels (a) & (c) in their Figure 8 ), and yet an MCMC analysis enabled the true ring diameter to be extracted with great precision (see panels (b) & (d) in their Figure 8 ). While extremely robust, MCMC methods are also computationally intensive and thus a “middle-ground” approach is more desirable, especially for models without noise.
Figure 8. Visibility amplitude profiles ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ Figure 7 . At φ = 90 ∘ \varphi=90^\circ φ = 9 0 ∘ d φ d_\varphi d φ n = 1 n=1 n = 1 n = 2 n=2 n = 2 n = 0 n=0 n = 0 Figure 7 .
We devised such a procedure, which proved highly effective. We now describe it using Figure 3 as an illustrative example:
For each φ \varphi φ Figure 3 , we chose the range u ∈ [ 2500 , 2600 ] u\in[2500,2600] u ∈ [ 2500 , 2600 ] λ \lambda λ
Identify the local extrema (both maxima and minima) of the visibility amplitude profile within this window.
Interpolate between maxima to obtain the superior envelope e max ( u ) e_{\max}(u) e m a x ( u ) Figure 3 ).
Interpolate between minima to obtain the inferior envelope e min ( u ) e_{\min}(u) e m i n ( u ) Figure 3 ).
Using a simple fitting routine, obtain the ring diameter d φ d_{\varphi} d φ d d d
V f i t ( u ; d ) = α L 2 ( u ) + α R 2 ( u ) + 2 α L ( u ) α R ( u ) sin ( 2 π d u ) (37) V_{\mathrm{fit}}(u;d)=\sqrt{\alpha_L^2(u)+\alpha_R^2(u)+2\alpha_L(u)\alpha_R(u)\sin(2\pi d u)} \tag*{(37)} V fit ( u ; d ) = α L 2 ( u ) + α R 2 ( u ) + 2 α L ( u ) α R ( u ) sin ( 2 π d u ) (37) best fits the data, where the functions α L / R ( u ) \alpha_{L/R}(u) α L / R ( u )
α L / R ( u ) = e max ( u ) ± e min ( u ) 2 . (38) \alpha_{L/R}(u)=\frac{e_{\max}(u)\pm e_{\min}(u)}{2}. \tag*{(38)} α L / R ( u ) = 2 e m a x ( u ) ± e m i n ( u ) . (38) The precise interpolation method used to obtain the envelopes e max / min ( u ) e_{\max/\min}(u) e m a x / m i n ( u )
Some words of explanation are in order. First, the use of the refined model (37) allows us to consider signals that display the same periodic behavior dictated by the formula (25) , but with variable extrema e min e_{\min} e m i n e max e_{\max} e m a x λ \lambda λ ∼ 15 \sim15 ∼ 15
This method also derives significant power from the flexible fall-off rate built into the visamp model (37) , which generalizes the u − 1 / 2 u^{-1/2} u − 1/2 (25) to arbitrary damping rates. Such a step was already taken by [40 ] , who noticed that their signal presented damped oscillations with envelope e max ( u ) = u − 1 / 2 e − ( u w ) ζ e_{\max}(u)=u^{-1/2}e^{-\left(u w\right)^{\zeta}} e m a x ( u ) = u − 1/2 e − ( u w ) ζ w w w ζ \zeta ζ Figure 3 , this method works even when such beats produce a local increase of the visamp. Its drawback is that astrophysical noise (ignored here) may result in ill-defined envelope functions (38) .
We also remind the reader that Eq. (24) is only the leading term in an expansion in powers of 0 < w / d ≪ 1 0<w/d\ll1 0 < w / d ≪ 1 w ( φ ) w(\varphi) w ( φ )
Finally, note that while the model (37) is symmetric under L ↔ R L\leftrightarrow R L ↔ R (38) , according to which α L \alpha_L α L α R \alpha_R α R α R < α L \alpha_R<\alpha_L α R < α L
Multi-peaked distribution for the diameter
The above fitting procedure returns a “best-fit” diameter d φ d_{\varphi} d φ
RMSD u ( d ) = ⟨ [ V f i t ( u ; d ) − ∣ V ( u ) ∣ ] 2 ⟩ u ⟨ V f i t ( u ; d ) ⟩ u , (39) \operatorname{RMSD}_{u}(d)=\frac{\sqrt{\left\langle\left[V_{\mathrm{fit}}(u;d)-|V(u)|\right]^2\right\rangle_u}}{\left\langle V_{\mathrm{fit}}(u;d)\right\rangle_u}, \tag*{(39)} RMSD u ( d ) = ⟨ V fit ( u ; d ) ⟩ u ⟨ [ V fit ( u ; d ) − ∣ V ( u ) ∣ ] 2 ⟩ u , (39) where ⟨ ⋅ ⟩ u \langle\cdot\rangle_u ⟨ ⋅ ⟩ u d φ d_{\varphi} d φ RMSD u ( d ) \operatorname{RMSD}_{u}(d) RMSD u ( d ) 0 < RMSD u ( d φ ) ≪ 1 0<\operatorname{RMSD}_{u}(d_{\varphi})\ll1 0 < RMSD u ( d φ ) ≪ 1 Figure 4 , for which the above procedure yields a diameter d φ = 37.132 μ a s d_{\varphi}=37.132\,\mu\mathrm{as} d φ = 37.132 μ as (39) . Nevertheless, the resulting model V f i t ( u ; d φ ) V_{\mathrm{fit}}(u;d_{\varphi}) V fit ( u ; d φ ) ∣ V ( u ) ∣ |V(u)| ∣ V ( u ) ∣ RMSD u ( d φ ) \operatorname{RMSD}_{u}(d_{\varphi}) RMSD u ( d φ )
Figure 4. Left: Example in which naïvely fitting the model V f i t ( u ; d ) V_{\mathrm{fit}}(u; d) V fit ( u ; d ) (37) to the visamp profile ∣ V ( u ) ∣ |V(u)| ∣ V ( u ) ∣ g ( d φ ) g(d_\varphi) g ( d φ ) (40) . The absolute best-fit corresponds to the global maximum of g ( d φ ) g(d_\varphi) g ( d φ ) Δ d φ ≈ 0.2 μ a s \Delta d_\varphi\approx0.2\,\mu\mathrm{as} Δ d φ ≈ 0.2 μ as 1 / u w 1/u_w 1/ u w u w ∼ 1000 G λ u_w \sim1000\,\mathrm{G}\lambda u w ∼ 1000 G λ
Still, the global minimum of RMSD u ( d ) \operatorname{RMSD}_{u}(d) RMSD u ( d ) d ϕ = 37.115 μ a s d_{\phi}=37.115\,\mu\mathrm{as} d ϕ = 37.115 μ as V f i t ( u ; d ϕ ) V_{\mathrm{fit}}(u;d_{\phi}) V fit ( u ; d ϕ ) ∣ V ( u ) ∣ \lvert V(u)\rvert ∣ V ( u )∣ RMSD u ( d ϕ ) \operatorname{RMSD}_{u}(d_{\phi}) RMSD u ( d ϕ )
By definition, the global minimum of RMSD u ( d ) \operatorname{RMSD}_{u}(d) RMSD u ( d )
g ( d ϕ ) = e − RMSD u ( d ϕ ) . (40) g(d_{\phi})=e^{-\operatorname{RMSD}_{u}(d_{\phi})}. \tag*{(40)} g ( d ϕ ) = e − RMSD u ( d ϕ ) . (40) We show g ( d ϕ ) g(d_{\phi}) g ( d ϕ ) Figure 4 , where we recognize the tallest peak to be the global maximum d ϕ = 37.115 μ a s d_{\phi}=37.115\,\mu\mathrm{as} d ϕ = 37.115 μ as d ϕ = 37.312 μ a s d_{\phi}=37.312\,\mu\mathrm{as} d ϕ = 37.312 μ as Δ d ϕ ≈ 0.2 μ a s \Delta d_{\phi}\approx0.2\,\mu\mathrm{as} Δ d ϕ ≈ 0.2 μ as [26 , 27 , 28 ] in their experimental forecast. Its physical origin can be intuitively understood as follows.
Over a sufficiently narrow baseline window, one may view u ≈ u w ≳ 1 / d u\approx u_{w}\gtrsim1/d u ≈ u w ≳ 1/ d ∣ V ( u ) ∣ \lvert V(u)\rvert ∣ V ( u )∣ d ϕ → d ϕ + k / u w d_{\phi}\to d_{\phi}+k/u_{w} d ϕ → d ϕ + k / u w k ∈ Z k\in\mathbb{Z} k ∈ Z d ϕ d_{\phi} d ϕ Δ d ϕ ≈ 1 / u w \Delta d_{\phi}\approx1/u_{w} Δ d ϕ ≈ 1/ u w Figure 4 confirms this argument: the observed gap of Δ d ϕ ≈ 0.2 μ a s \Delta d_{\phi}\approx0.2\,\mu\mathrm{as} Δ d ϕ ≈ 0.2 μ as
u w ≈ 1 2 × 10 − 7 a r c s e c 1.03 × 10 12 r a d ≈ 1030 G λ , (41) u_{w}\approx\frac{1}{2\times10^{-7}\,\mathrm{arcsec}}\frac{1.03\times10^{12}\,\mathrm{rad}}{}\approx1030\,\mathrm{G}\lambda, \tag*{(41)} u w ≈ 2 × 1 0 − 7 arcsec 1 1.03 × 1 0 12 rad ≈ 1030 G λ , (41) which, as expected, lies well within the chosen baseline window of u ∈ [ 1000 , 1045 ] G λ u\in[1000,1045]\,\mathrm{G}\lambda u ∈ [ 1000 , 1045 ] G λ
Another intuitive way to understand this degeneracy comes from the observation that the number of damped oscillations (or “hops”) from the origin u = 0 u=0 u = 0 u ≈ u w u\approx u_{w} u ≈ u w
N w ≈ u w d ϕ . (42) N_{w}\approx u_{w}d_{\phi}. \tag*{(42)} N w ≈ u w d ϕ . (42) Hence, shifts d ϕ → d ϕ + k / u w d_{\phi}\to d_{\phi}+k/u_{w} d ϕ → d ϕ + k / u w N w → N w + k N_{w}\to N_{w}+k N w → N w + k d ϕ d_{\phi} d ϕ N w N_{w} N w u ≈ u w u\approx u_{w} u ≈ u w N w N_{w} N w V f i t ( u , d ϕ + k / u w ) V_{\mathrm{fit}}(u,d_{\phi}+k/u_{w}) V fit ( u , d ϕ + k / u w ) k ∈ Z k\in\mathbb{Z} k ∈ Z
In practice, we account for this degeneracy by adjuting our fitting procedure to keep track of several of the most prominent peaks of the multi-peak distribution, as follows:
For each ϕ \phi ϕ d ϕ d_{\phi} d ϕ ∣ V ( u ) ∣ \lvert V(u)\rvert ∣ V ( u )∣
Pick an interval of some width Δ d ϕ ≳ 10 / u w \Delta d_{\phi}\gtrsim10/u_{w} Δ d ϕ ≳ 10/ u w g ( d ϕ ) g(d_{\phi}) g ( d ϕ ) Figure 4 .
Find the peaks of g ( d ϕ ) g(d_{\phi}) g ( d ϕ ) d ϕ d_{\phi} d ϕ g ( d ϕ ) g(d_{\phi}) g ( d ϕ )
The resulting list should contain several diameters d ϕ d_{\phi} d ϕ g ( d ϕ ) g(d_{\phi}) g ( d ϕ ) d ϕ d_{\phi} d ϕ g ( d ϕ ) g(d_{\phi}) g ( d ϕ ) g ( d ϕ ) g(d_{\phi}) g ( d ϕ ) ϕ \phi ϕ d ϕ d_{\phi} d ϕ
Circlipse fit and test of the Kerr hypothesis
Having determined the absolute best-fit ring diameter d ϕ d_{\phi} d ϕ ϕ \phi ϕ
d f i t ( ϕ ; R ) = 2 R 0 + 2 R 1 2 sin 2 ( ϕ − ϕ 0 ) + R 2 2 cos 2 ( ϕ − ϕ 0 ) , (43) d_{\mathrm{fit}}(\phi;\mathbf{R})=2R_{0}+2\sqrt{R_{1}^{2}\sin^{2}(\phi-\phi_{0})+R_{2}^{2}\cos^{2}(\phi-\phi_{0})}, \tag*{(43)} d fit ( ϕ ; R ) = 2 R 0 + 2 R 1 2 sin 2 ( ϕ − ϕ 0 ) + R 2 2 cos 2 ( ϕ − ϕ 0 ) , (43) with parameters R = [ R 0 , R 1 , R 2 , ϕ 0 ] \mathbf{R}=\left[R_{0},R_{1},R_{2},\phi_{0}\right] R = [ R 0 , R 1 , R 2 , ϕ 0 ] [ 0 ∘ , 180 ∘ ) [0^\circ,180^\circ) [ 0 ∘ , 18 0 ∘ ) 5 ∘ 5^\circ 5 ∘ ϕ ∈ { 0 ∘ , 5 ∘ , … , 175 ∘ } \phi\in\{0^\circ,5^\circ,\ldots,175^\circ\} ϕ ∈ { 0 ∘ , 5 ∘ , … , 17 5 ∘ } R f i t \mathbf{R}_{\mathrm{fit}} R fit
RMSD ϕ ( R ) = ⟨ [ d f i t ( ϕ ; R ) − d ϕ ] 2 ⟩ ϕ ⟨ d f i t ( ϕ ; R ) ⟩ ϕ , (44) \operatorname{RMSD}_{\phi}(\mathbf{R})=\frac{\sqrt{\left\langle\left[d_{\mathrm{fit}}(\phi;\mathbf{R})-d_{\phi}\right]^{2}\right\rangle_{\phi}}}{\left\langle d_{\mathrm{fit}}(\phi;\mathbf{R})\right\rangle_{\phi}}, \tag*{(44)} RMSD ϕ ( R ) = ⟨ d fit ( ϕ ; R ) ⟩ ϕ ⟨ [ d fit ( ϕ ; R ) − d ϕ ] 2 ⟩ ϕ , (44) where ⟨ ⋅ ⟩ ϕ \langle\cdot\rangle_{\phi} ⟨ ⋅ ⟩ ϕ 0 < RMSD ϕ ( R f i t ) ≪ 1 0<\operatorname{RMSD}_{\phi}(\mathbf{R}_{\mathrm{fit}})\ll1 0 < RMSD ϕ ( R fit ) ≪ 1 RMSD ϕ ( R f i t ) \operatorname{RMSD}_{\phi}(\mathbf{R}_{\mathrm{fit}}) RMSD ϕ ( R fit ) 4
As explained in the last section, this naïve approach is not robust because of the multi-peaked distribution for d ϕ d_{\phi} d ϕ d ϕ d_{\phi} d ϕ ϕ \phi ϕ
However, there is a simple way to remedy this by using the data collected at all angles simultaneously, as follows:
For each angle ϕ \phi ϕ g ( d ϕ ) g(d_{\phi}) g ( d ϕ ) d ϕ d_{\phi} d ϕ Figure 5 .
These peaks can then be separated into multiple circlipse-shaped subsets C i C_{i} C i
g ( C i ) ≡ ∏ ϕ g ( d ϕ ) . (45) g(C_{i})\equiv\prod_{\phi}g(d_{\phi}). \tag*{(45)} g ( C i ) ≡ ϕ ∏ g ( d ϕ ) . (45) Finally, each subset can be fit to a circlipse shape (39), with normalized RMSD ( C i ) ≡ RMSD ϕ ( R f i t ) \operatorname{RMSD}(C_{i})\equiv\operatorname{RMSD}_{\phi}(\mathbf{R}_{\mathrm{fit}}) RMSD ( C i ) ≡ RMSD ϕ ( R fit )
This “multi-fit method” results in several circlipses C i C_{i} C i g ( C i ) g(C_{i}) g ( C i ) Figure 5 , the likeliest C i C_{i} C i g ( C i ) g(C_{i}) g ( C i ) n = 2 n=2 n = 2 5
That said, there is no reason for this to be the case in general, and in some cases the “true” circlipse in the underlying image may well differ from the likeliest circlipse C max C^{\max} C m a x C i max C_{i}^{\max} C i m a x RMSD ( C i max ) ≠ min i RMSD ( C i ) \operatorname{RMSD}(C_{i}^{\max})\ne\min_{i}\operatorname{RMSD}(C_{i}) RMSD ( C i m a x ) = min i RMSD ( C i )
In practice, the least favorable examples present a handful of circlipses C i max C_{i}^{\max} C i m a x g ( C i ) g(C_{i}) g ( C i ) Δ d ϕ ≈ 1 / u w \Delta d_{\phi}\approx1/u_{w} Δ d ϕ ≈ 1/ u w RMSD ( C i max ) \operatorname{RMSD}(C_{i}^{\max}) RMSD ( C i m a x ) C i max C_{i}^{\max} C i m a x
Importance of the choice of baseline window
Before reporting the results of our parameter surveys, we first describe some new and important features of the GLM method that we noticed in the course of our investigation.
In order to determine the diameter d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) n = 2 n = 2 n = 2 ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ (26) , i.e., on baselines long enough to resolve the width of the n = 1 n = 1 n = 1 n = 2 n = 2 n = 2 φ ρ = φ \varphi_\rho= \varphi φ ρ = φ
1 w 1 ( φ ρ ) ≪ u ≪ 1 w 2 ( φ ρ ) . (46) \frac{1}{w_1(\varphi_\rho)} \ll u \ll\frac{1}{w_2(\varphi_\rho)}. \tag*{(46)} w 1 ( φ ρ ) 1 ≪ u ≪ w 2 ( φ ρ ) 1 . (46) It is only in this regime that the signature of the n = 2 n = 2 n = 2 (25) that encode the ring diameter d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) Figure 2 , this requires us to choose a baseline window u ≳ 100 G λ u \gtrsim100\ \mathrm{G}\lambda u ≳ 100 G λ
The angle-dependence of condition (46) is important. At low inclinations, this dependence is relatively weak, and the baseline threshold b 2 ( φ ) ≈ 1 / w 1 ( φ ) b_2(\varphi) \approx1/w_1(\varphi) b 2 ( φ ) ≈ 1/ w 1 ( φ ) n = 2 n = 2 n = 2 ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ φ \varphi φ φ \varphi φ b 2 ( φ ) b_2(\varphi) b 2 ( φ ) (46) for all angles simultaneously; worse, the visamp may even cease to encode the n = 2 n = 2 n = 2
These difficulties can be quantitatively fleshed out as follows. The GLM method can only succeed for a fixed choice of baseline window u ≈ u w u \approx u_w u ≈ u w (46) for all angles, i.e., such that
max φ b 2 ( φ ) ≪ u w ≪ min φ b 3 ( φ ) . (47) \max_{\varphi} b_2(\varphi) \ll u_w \ll\min_{\varphi} b_3(\varphi). \tag*{(47)} φ max b 2 ( φ ) ≪ u w ≪ φ min b 3 ( φ ) . (47) Here, b n ( φ ) ≈ 1 / w n − 1 ( φ ) b_n(\varphi) \approx1/w_{n-1}(\varphi) b n ( φ ) ≈ 1/ w n − 1 ( φ ) n t h n^{\mathrm{th}} n th ∣ V ( u , φ + π ) ∣ = ∣ V ( u , φ ) ∣ |V(u,\varphi+\pi)| = |V(u,\varphi)| ∣ V ( u , φ + π ) ∣ = ∣ V ( u , φ ) ∣
b n ( φ + π ) = b n ( φ ) . (48) b_n(\varphi+\pi) = b_n(\varphi). \tag*{(48)} b n ( φ + π ) = b n ( φ ) . (48) On the other hand, the subring widths w n ( φ ρ ) w_n(\varphi_\rho) w n ( φ ρ ) w n ( φ ρ + 2 π ) = w n ( φ ρ ) w_n(\varphi_\rho+ 2\pi) = w_n(\varphi_\rho) w n ( φ ρ + 2 π ) = w n ( φ ρ ) φ ρ ∈ [ 0 , 2 π ) \varphi_\rho\in[0,2\pi) φ ρ ∈ [ 0 , 2 π ) w 1 ( φ ρ ) w_1(\varphi_\rho) w 1 ( φ ρ ) w 1 ( φ ρ + π ) w_1(\varphi_\rho+\pi) w 1 ( φ ρ + π ) b 2 ( φ ρ ) b_2(\varphi_\rho) b 2 ( φ ρ ) w 2 ( φ ρ ) < w 1 ( φ ρ ) w_2(\varphi_\rho) < w_1(\varphi_\rho) w 2 ( φ ρ ) < w 1 ( φ ρ ) w 2 ( φ ρ ) ∼ w 1 ( φ ρ + π ) w_2(\varphi_\rho) \sim w_1(\varphi_\rho+\pi) w 2 ( φ ρ ) ∼ w 1 ( φ ρ + π ) φ ρ \varphi_\rho φ ρ ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ φ = φ ρ \varphi= \varphi_\rho φ = φ ρ n = 1 n = 1 n = 1 n = 2 n = 2 n = 2
Low to moderate inclinations
At low-to-moderate inclinations, each subring produces a distinct signature in its own baseline regime, and so the threshold b 2 ( φ ) b_2(\varphi) b 2 ( φ ) Figure 2 and Figure 6 . According to Eq. (16) , the next thresholds are related to it by
Figure 6. Measurements of the photon ring shape carried out over insufficiently long baseline windows display characteristic transition effects. In this example, the BH spin is a = 0.5 a = 0.5 a = 0.5 i = 45 ∘ i = 45^\circ i = 4 5 ∘ μ = 3 r + / 2 \mu= 3r_{+}/2 μ = 3 r + /2 γ = − 1 \gamma= -1 γ = − 1 σ = 0.5 M \sigma= 0.5M σ = 0.5 M d φ d_{\varphi} d φ [ 550 , 650 ] [550, 650] [ 550 , 650 ] λ \lambda λ d φ d_{\varphi} d φ d φ ( 2 ) d_{\varphi}^{(2)} d φ ( 2 ) d φ ( 1 ) d_{\varphi}^{(1)} d φ ( 1 ) ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ n = 2 n = 2 n = 2 φ = 60 ∘ \varphi= 60^\circ φ = 6 0 ∘ n = 1 n = 1 n = 1 φ = 140 ∘ \varphi= 140^\circ φ = 14 0 ∘ φ = 100 ∘ \varphi= 100^\circ φ = 10 0 ∘
b n + 1 ( φ ) ≈ e γ ( φ ) b n ( φ ) , (49) b_{n+1}(\varphi) \approx e^{\gamma(\varphi)}b_n(\varphi), \tag*{(49)} b n + 1 ( φ ) ≈ e γ ( φ ) b n ( φ ) , (49) an approximate Kerr-lens equation that becomes exact as n → ∞ n \to\infty n → ∞ n ≳ 2 n \gtrsim2 n ≳ 2
On the other hand, one cannot determine b 2 ( φ ) b_2(\varphi) b 2 ( φ ) w 1 ( φ ρ ) w_1(\varphi_\rho) w 1 ( φ ρ ) (47) as
max φ b 2 ( φ ) ≪ u w ≪ min φ e γ ( φ ) b 2 ( φ ) , (50) \max_{\varphi} b_2(\varphi) \ll u_w \ll\min_{\varphi} e^{\gamma(\varphi)} b_2(\varphi), \tag*{(50)} φ max b 2 ( φ ) ≪ u w ≪ φ min e γ ( φ ) b 2 ( φ ) , (50) where b 2 ( φ ) b_2(\varphi) b 2 ( φ ) e γ ( φ ) e^{\gamma(\varphi)} e γ ( φ ) γ 0 = min φ γ ( φ ρ ) \gamma_0 = \min_{\varphi} \gamma(\varphi_\rho) γ 0 = min φ γ ( φ ρ ) u w u_w u w (50) is
max φ b 2 ( φ ) ≪ e γ 0 min φ b 2 ( φ ) . (51) \max_{\varphi} b_2(\varphi) \ll e^{\gamma_0} \min_{\varphi} b_2(\varphi). \tag*{(51)} φ max b 2 ( φ ) ≪ e γ 0 φ min b 2 ( φ ) . (51) At low inclinations, where b 2 ( φ ) ≈ b 0 b_2(\varphi) \approx b_0 b 2 ( φ ) ≈ b 0 e γ 0 e^{\gamma_0} e γ 0 10 ∼ 20 10 \sim20 10 ∼ 20 e γ 0 ∼ 1 e^{\gamma_0} \sim1 e γ 0 ∼ 1 a = 1 a = 1 a = 1 b 2 ( φ ) b_2(\varphi) b 2 ( φ ) (47) cannot be satisfied, but it does mean that some choices of baseline window u w u_w u w φ \varphi φ
This effect is illustrated in Figure 6 , which displays a model from our parameter survey with BH spin a = 0.5 a = 0.5 a = 0.5 i = 45 ∘ i = 45^\circ i = 4 5 ∘ b 2 ( φ ) b_2(\varphi) b 2 ( φ ) b 2 ( 60 ∘ ) ≈ 200 b_2(60^\circ) \approx200 b 2 ( 6 0 ∘ ) ≈ 200 λ \lambda λ b 2 ( 140 ∘ ) ≈ 700 b_2(140^\circ) \approx700 b 2 ( 14 0 ∘ ) ≈ 700 λ \lambda λ d φ d_\varphi d φ u w ≈ 600 u_w \approx600 u w ≈ 600 λ \lambda λ n = 2 n = 2 n = 2 d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) φ = 60 ∘ \varphi= 60^\circ φ = 6 0 ∘ n = 1 n = 1 n = 1 d φ ( 1 ) d_\varphi^{(1)} d φ ( 1 ) φ = 140 ∘ \varphi= 140^\circ φ = 14 0 ∘ φ = 102 ∘ \varphi= 102^\circ φ = 10 2 ∘ n = 1 n = 1 n = 1 n = 2 n = 2 n = 2
Of course, at moderate inclinations, this issue can always be avoided by selecting a larger baseline window. For the example of Figure 6 , any choice u w ≳ 1000 u_w \gtrsim1000 u w ≳ 1000 λ \lambda λ n = 2 n = 2 n = 2 n = 3 n = 3 n = 3 u ≳ 2000 u \gtrsim2000 u ≳ 2000 λ \lambda λ d φ d_\varphi d φ d φ ( 1 ) d_\varphi^{(1)} d φ ( 1 ) d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) Figure 6 .
The reason is the following. For the angles in the blue zone, d φ = d φ ( 2 ) d_\varphi= d_\varphi^{(2)} d φ = d φ ( 2 ) n = 2 n = 2 n = 2 d φ = d φ ( 1 ) d_\varphi= d_\varphi^{(1)} d φ = d φ ( 1 ) n = 1 n = 1 n = 1 d φ d_\varphi d φ d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) d φ ( 1 ) d_\varphi^{(1)} d φ ( 1 ) d φ d_\varphi d φ d φ d_\varphi d φ (40) —is allowed to “jump” across green zones (and indeed, one can see two such jumps in the second green zone 150 ∘ < φ < 175 ∘ 150^\circ< \varphi< 175^\circ 15 0 ∘ < φ < 17 5 ∘ d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) φ → φ + π \varphi\to\varphi+ \pi φ → φ + π
On the other hand, the continuous curves C i C_i C i φ → φ + π \varphi\to\varphi+ \pi φ → φ + π { C i } \{C_i\} { C i } Figure 6 . While these curves C i C_i C i Δ φ ρ ≈ 1 / u w \Delta\varphi_\rho\approx1/u_w Δ φ ρ ≈ 1/ u w d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 )
To summarize: at low inclinations i ≲ 20 ∘ i \lesssim20^\circ i ≲ 2 0 ∘ b 2 ( φ ) b_2(\varphi) b 2 ( φ ) u w u_w u w n = 2 n = 2 n = 2 d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) φ \varphi φ i ≲ 50 ∘ i \lesssim50^\circ i ≲ 5 0 ∘ b 2 ( φ ) b_2(\varphi) b 2 ( φ ) u w u_w u w n = 2 n = 2 n = 2 d φ ( 2 ) d_\varphi^{(2)} d φ ( 2 ) φ \varphi φ n = 1 n = 1 n = 1 d φ ( 1 ) d_\varphi^{(1)} d φ ( 1 ) d φ d_\varphi d φ Figure 6 . Moreover, at these inclinations, this difficulty can always be avoided by selecting a sufficiently large baseline window.
High inclinations
At higher inclinations i ≳ 60 ∘ i \gtrsim60^\circ i ≳ 6 0 ∘ b 2 ( φ ) b_2(\varphi) b 2 ( φ ) φ \varphi φ π \pi π 2 π 2\pi 2 π w 2 ( φ ρ ) < w 1 ( φ ρ ) w_2(\varphi_\rho) < w_1(\varphi_\rho) w 2 ( φ ρ ) < w 1 ( φ ρ ) φ ρ \varphi_\rho φ ρ w 2 ( φ ρ ) ∼ w 1 ( φ ρ + π ) w_2(\varphi_\rho) \sim w_1(\varphi_\rho+ \pi) w 2 ( φ ρ ) ∼ w 1 ( φ ρ + π ) φ ρ \varphi_\rho φ ρ V ( u , φ ) V(u,\varphi) V ( u , φ ) φ = φ ρ \varphi= \varphi_\rho φ = φ ρ φ = φ ρ + π \varphi= \varphi_\rho+ \pi φ = φ ρ + π n = 1 n = 1 n = 1 n = 2 n = 2 n = 2
This phenomenon is clearly visible in the example shown in Figure 7 , which corresponds to a BH of spin a = 0.5 a = 0.5 a = 0.5 i = 89 ∘ i = 89^\circ i = 8 9 ∘ w 2 ( φ ρ ) ≈ w 1 ( φ ρ + π ) w_2(\varphi_\rho) \approx w_1(\varphi_\rho+ \pi) w 2 ( φ ρ ) ≈ w 1 ( φ ρ + π ) φ ρ = 90 ∘ \varphi_\rho= 90^\circ φ ρ = 9 0 ∘ ∣ V ( u , 90 ∘ ) ∣ |V(u,90^\circ)| ∣ V ( u , 9 0 ∘ ) ∣ Figure 8 , cannot display a sharply delineation between regimes dominated by the n = 1 n = 1 n = 1 n = 2 n = 2 n = 2 (24) , their periodicity ought to be set by the diameter of a ring of fixed width in the image, which in this case must be the composite ring consisting of the upper half of the n = 1 n = 1 n = 1 n = 2 n = 2 n = 2 (25) on long baselines, but with a projected diameter d φ d_\varphi d φ n = 1&2 ring of uniform width. In practice, this expectation is not quite realized in this model because of the direct (n = 0 n = 0 n = 0 n = 1&2 ring, the vertical intensity cross-section consists of three spikes; as a result, the visamp profile at angles φ ≈ 90 ∘ \varphi\approx90^\circ φ ≈ 9 0 ∘ n = 1 n = 1 n = 1 n = 0 n = 0 n = 0 n = 0 n = 0 n = 0 n = 2 n = 2 n = 2
Parameter survey for the emission profile
Description of the models
In this paper, we restrict our attention to the “thin-disk” emission models presented in Sec. III.A of [26 ] and Sec. 3.2 of [6 ] , which consist of emission profiles that are stationary, axisymmetric, and confined to the equatorial plane.
The emitters forming the disk are assumed to describe stable, circular-equatorial orbits all the way down to the radius of the innermost stable circular orbit (ISCO), past which they follow the prescription of [8 ] for infall to the horizon. This specifies the redshift factor g g g
I ( α , β ) = ∑ n = 0 N ( α , β ) f n J ( r s ( n ) ) g 4 ( r s ( n ) , α ) , (52) I(\alpha,\beta)=\sum_{n=0}^{N(\alpha,\beta)}f_nJ\left(r_s^{(n)}\right)g^4\left(r_s^{(n)},\alpha\right), \tag*{(52)} I ( α , β ) = n = 0 ∑ N ( α , β ) f n J ( r s ( n ) ) g 4 ( r s ( n ) , α ) , (52) where J ( r ) J(r) J ( r ) r s ( n ) = r s ( n ) ( α , β ) r_s^{(n)}=r_s^{(n)}(\alpha,\beta) r s ( n ) = r s ( n ) ( α , β ) n t h n^{\mathrm{th}} n th ( α , β ) (\alpha,\beta) ( α , β ) N + 1 N+1 N + 1 6
f n = { 1 n = 0 , 1.5 n ∈ { 1 , 2 } , 0 n ≥ 3 , (53) f_n=
\begin{cases}
1 & n=0,\\
1.5 & n\in\{1,2\},\\
0 & n\ge3,
\end{cases} \tag*{(53)} f n = ⎩ ⎨ ⎧ 1 1.5 0 n = 0 , n ∈ { 1 , 2 } , n ≥ 3 , (53) whose effect is to remove n ≥ 3 n\ge3 n ≥ 3 n = 1 n=1 n = 1 n = 2 n=2 n = 2 n = 0 n=0 n = 0 [39 ] ; [6 ] ). In this sense, the equatorial model (52) provides a good approximation for realistic accretion flows, and its simplicity and relatively low computational cost make it ideal for the survey of emission profiles undertaken here.
While the equatorial source profile J ( r ) J(r) J ( r )
J ( r ; μ , γ , σ ) = e − 1 2 [ γ + arcsinh ( r − μ σ ) ] 2 ( r − μ ) 2 + σ 2 . (54) J(r;\mu,\gamma,\sigma)=\frac{e^{-\frac{1}{2}[\gamma+\operatorname{arcsinh}(\frac{r-\mu}{\sigma})]^{2}}}{\sqrt{(r-\mu)^{2}+\sigma^{2}}}. \tag*{(54)} J ( r ; μ , γ , σ ) = ( r − μ ) 2 + σ 2 e − 2 1 [ γ + arcsinh ( σ r − μ ) ] 2 . (54) This is a 3-parameter subset of Johnson’s S U S_U S U δ \delta δ [29 ] for its versatility: this functional form provides a simple way to generate smooth profiles with emission typically concentrated near the event horizon of the BH, and displaying substantial variation with changes in μ \mu μ γ \gamma γ σ \sigma σ
μ \mu μ [ r − , 2 r + ] [r_{-},2r_{+}] [ r − , 2 r + ] r ± r_{\pm} r ± (15) . The peak is very close to or behind the event horizon near the lower end of this range (left column of Figure 9 ), but it moves farther out near the upper end (right column of Figure 9 ). This variation does not significantly affect the observed thickness of the photon subrings.
γ \gamma γ [ − 2 , 2 ] [-2,2] [ − 2 , 2 ] γ < 0 \gamma<0 γ < 0 Figure 9 ), the steep end of the distribution is to the left and is (mostly) cut off by the horizon, leaving only a gentle-sloping tail in the emission profile. When γ > 0 \gamma>0 γ > 0 Figure 9 ), the situation is reversed: the steep end lies on the right and remains visible outside the horizon, while the gentle-sloping side is cut off. Hence, the rings tend to become thinner as γ \gamma γ
σ \sigma σ [ 0.25 M , 1.5 M ] [0.25M,1.5M] [ 0.25 M , 1.5 M ] Figure 9 ), the profile is broad enough to produce thick rings, which makes the GLM test easy to carry out on a baseline window u ∈ [ 450 , 550 ] u\in[450,550] u ∈ [ 450 , 550 ] λ \lambda λ μ \mu μ γ \gamma γ Figure 9 ), the profile is a steep spike (provided that the distribution’s peak lies outside the horizon) that produces narrower rings: as a result, the n = 1 n=1 n = 1 n = 2 n=2 n = 2 u w ≳ 2000 u_{w}\gtrsim2000 u w ≳ 2000 λ \lambda λ
Figure 9. Impact of the emission profile on the observed ring thickness and baseline domains. We show two extreme cases: (Left) A broad emission profile, leading to large rings and an early transition between n = 1 n = 1 n = 1 n = 2 n = 2 n = 2 b 2 ( 0 ∘ ) ≈ 300 b_2(0^\circ) \approx300 b 2 ( 0 ∘ ) ≈ 300 λ \lambda λ n = 2 n = 2 n = 2 n = 1 n = 1 n = 1 n = 2 n = 2 n = 2 b 2 ( 145 ∘ ) ≳ 1800 b_2(145^\circ) \gtrsim1800 b 2 ( 14 5 ∘ ) ≳ 1800 λ \lambda λ
Images with thinner rings have a larger baseline threshold b 2 ( φ ) b_{2}(\varphi) b 2 ( φ ) n = 2 n=2 n = 2 d φ ( 2 ) d_{\varphi}^{(2)} d φ ( 2 ) Figure 9 with two extreme examples that display maximally thick rings (left column) and maximally thin rings (right column) within our considered parameter range.
Choice of parameters for the survey
In order to assess the robustness of the GLM method described in Sects. 2 and 3, we conducted a survey over the emission-profile parameter space { μ , γ , σ } \{\mu,\gamma,\sigma\} { μ , γ , σ } a = 0.5 a=0.5 a = 0.5 i = 45 ∘ i=45^\circ i = 4 5 ∘ Table 1 .
Table 1. Parameter values considered in our survey of emission profiles (54) , with r ± r_{\pm} r ± (15) of the outer/inner horizon.
As a cross-check, we repeated the survey using the currently favored values of BH spin and inclination for M87*. The power in the relativistic jet produced by M87* suggests that this BH is highly spinning, since most low-spin GRMHD models do not give rise to a sufficiently powerful jet ([10 ] ). We therefore take a = 0.94 a=0.94 a = 0.94 [14 ] ). We also assume that the spin axis of the BH and of its accretion disk are aligned with its forward jet, which has a measured inclination of i ≈ 17 ∘ i\approx17^\circ i ≈ 1 7 ∘ ? ]). Since r − ≈ r + / 2 r_{-}\approx r_{+}/2 r − ≈ r + /2 μ = r + / 2 \mu=r_{+}/2 μ = r + /2
Results
We investigated these models to address the following questions:
Q1. Is the GLM test possible in principle? In particular, does d φ ( 2 ) d_{\varphi}^{(2)} d φ ( 2 ) (43) of a circlipse?
Q2. If so, how do the best-fitting circlipse parameters R f i t R_{\mathrm{fit}} R fit
Our survey led us to the following answers:
A1. Yes. In all the models considered here, we identified a regime in which the n = 2 n=2 n = 2 d φ ( 2 ) d_{\varphi}^{(2)} d φ ( 2 ) u ≳ 150 u\gtrsim150 u ≳ 150 λ \lambda λ Figure 2 ) or as high as u ≳ 2000 u\gtrsim2000 u ≳ 2000 λ \lambda λ Figure 9 ). We always found an excellent fit to the circlipse shape (43) as measured by the normalized RMSD (44) : in all cases,
RMSD φ ( R f i t ) < 10 − 4 . (55) \operatorname{RMSD}_{\varphi}(R_{\mathrm{fit}})<10^{-4}. \tag*{(55)} RMSD φ ( R fit ) < 1 0 − 4 . (55) A2. By analogy with (30) , it is helpful to repackage the best-fit parameters R f i t = { R 0 , R 1 , R 2 , φ 0 } R_{\mathrm{fit}}=\{R_{0},R_{1},R_{2},\varphi_{0}\} R fit = { R 0 , R 1 , R 2 , φ 0 } n = 2 n=2 n = 2 7
d + = d φ 0 + π / 2 ( 2 ) = 2 ( R 0 + R 1 ) , d − = d φ 0 ( 2 ) = 2 ( R 0 + R 2 ) (56) d_{+}=d_{\varphi_{0}+\pi/2}^{(2)}=2(R_{0}+R_{1}),\qquad d_{-}=d_{\varphi_{0}}^{(2)}=2(R_{0}+R_{2}) \tag*{(56)} d + = d φ 0 + π /2 ( 2 ) = 2 ( R 0 + R 1 ) , d − = d φ 0 ( 2 ) = 2 ( R 0 + R 2 ) (56) of the associated circlipse shape (d − ≤ d + d_{-}\le d_{+} d − ≤ d + R 1 ≥ R 2 R_{1}\ge R_{2} R 1 ≥ R 2 ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) Table 2 and Table 3 . These results give a good overview of the variability in the circlipse shape obtained from different astrophysical source profiles. We return to the question of spin inference in Table 4 below.
Table 2. Mean values and standard deviations for the inferred n = 2 n=2 n = 2 d ± d_{\pm} d ± a = 0.5 a=0.5 a = 0.5 i = 45 ∘ i=45^\circ i = 4 5 ∘ d ∥ = 10.319 M d_{\parallel}=10.319M d ∥ = 10.319 M d ⊥ = 10.236 M d_{\perp}=10.236M d ⊥ = 10.236 M
Table 3. Mean value and standard deviations for the inferred n = 2 n = 2 n = 2 d ± d_{\pm} d ± a = 0.94 a = 0.94 a = 0.94 i = 17 ∘ i = 17^\circ i = 1 7 ∘ d ∥ = 9.834 M d_{\parallel} = 9.834M d ∥ = 9.834 M d ⊥ = 9.733 M d_{\perp} = 9.733M d ⊥ = 9.733 M
Table 4. Lower bounds on the uncertainties that we can expect from a measurement of BH spin and inclination using the n = 2 n = 2 n = 2
As discussed in Sec. 2.9, we expected the n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 d + > d ∥ d_{+} > d_{\parallel} d + > d ∥ d − > d ⊥ d_{-} > d_{\perp} d − > d ⊥ n = 2 n = 2 n = 2 n = 2 n = 2 n = 2
Nevertheless, it is possible to approximate the critical curve diameters ( d ∥ , d ⊥ ) (d_{\parallel}, d_{\perp}) ( d ∥ , d ⊥ ) ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) n = 2 n = 2 n = 2 ( a , i ) (a,i) ( a , i ) n = 2 n = 2 n = 2 ( a , i ) (a,i) ( a , i ) n = 2 n = 2 n = 2 Table 4 .
Range of allowed phovals within the n = 2 n = 2 n = 2
These results highlight that the critical curve is not necessarily the most relevant concept for the analysis of the n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 n = 2
Since Sec. 2.9, we have used “photon ring shape” to refer to the simplest observational target for a measurement of the n = 2 n = 2 n = 2 d ϕ 2 d_{\phi}^{2} d ϕ 2 n = 2 n = 2 n = 2 n = 2 n = 2 n = 2
f p h ( φ ; R ) = R 0 + R 1 2 sin 2 ( φ − φ 0 ) + R 2 2 cos 2 ( φ − φ 0 ) + ( X − χ ) cos ( φ − φ 0 ) + arcsin [ χ cos ( φ − φ 0 ) ] (57) f_{\mathrm{ph}}(\varphi;\boldsymbol{R}) = R_{0} + \sqrt{R_{1}^{2}\sin^{2}(\varphi-\varphi_{0}) + R_{2}^{2}\cos^{2}(\varphi-\varphi_{0})}
+ (X-\chi)\cos(\varphi-\varphi_{0}) + \arcsin[\chi\cos(\varphi-\varphi_{0})] \tag*{(57)} f ph ( φ ; R ) = R 0 + R 1 2 sin 2 ( φ − φ 0 ) + R 2 2 cos 2 ( φ − φ 0 ) + ( X − χ ) cos ( φ − φ 0 ) + arcsin [ χ cos ( φ − φ 0 )] (57) for some choice of parameters R = { R 0 , R 1 , R 2 , X , χ , φ 0 } \boldsymbol{R} = \{R_{0},R_{1},R_{2},X,\chi,\varphi_{0}\} R = { R 0 , R 1 , R 2 , X , χ , φ 0 } φ 0 \varphi_{0} φ 0 n n n
Since we expect the geometric shape of the n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) n = 2 n = 2 n = 2
For the two choices of BH spin and inclination in our survey of emission profiles, this projection results in the blue regions displayed in Figs. 10 and 11. Their numerical computation is nontrivial and we relegate the details to App. B. Note that since d − ≤ d + d_{-} \leq d_{+} d − ≤ d + 8
As expected, all the emission profiles in our survey produce n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) ( d + , d − ) (d_{+},d_{-}) ( d + , d − )
More precisely, these diameters (the gold dots in Figs. 10 and 11) form a relatively straight line connecting the corners of the blue region that correspond to the outer and inner edges of the lensing band. In addition, they are not uniformly scattered along this line; rather, they are tightly grouped near its center. Both of these facts may be intuitively understood as follows. Recall from Sec. 2.5 that the inner and outer edges of the n t h n^{\mathrm{th}} n th n t h n^{\mathrm{th}} n th r = r + r = r_{+} r = r + r → ∞ r \to\infty r → ∞ r + < r < ∞ r_{+} < r < \infty r + < r < ∞ [26 ] , these curves roughly appear to be dilated versions of each other (with γ ( φ ) \gamma(\varphi) γ ( φ ) ( d + , d − ) (d_{+},d_{-}) ( d + , d − )
Since an equatorial disk is a superposition of equatorial rings, the n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 ( d + , d − ) (d_{+},d_{-}) ( d + , d − )
Parameter survey for BH spin and inclination
Choice of parameters for the survey
To better assess the robustness of the GLM method described in Secs. 2 and 3, we also conducted a parameter survey over BH spins and inclinations ( a , i ) (a,i) ( a , i ) ( a , i ) (a,i) ( a , i )
a “typical 1” profile for which the diameters ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) n = 2 n=2 n = 2
an “overestimated” profile for which the diameters ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) n = 2 n=2 n = 2
an “underestimated” profile for which the diameters ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) n = 2 n=2 n = 2
a “typical 2” profile for which they were slightly subaverage;
a “narrow” profile with a sharp peak (small σ \sigma σ ( d + , d − ) (d_{+},d_{-}) ( d + , d − )
Table 5 lists the parameters { μ , γ , σ } \{\mu,\gamma,\sigma\} { μ , γ , σ }
Table 5. Parameters of the five representative emission profiles selected for the survey over BH spins and inclinations.
In this survey, we considered all the combinations ( a , i ) (a,i) ( a , i ) a ∈ { 0 , 0.25 , 0.5 , 0.75 , 0.99 } a \in\{0,0.25,0.5,0.75,0.99\} a ∈ { 0 , 0.25 , 0.5 , 0.75 , 0.99 } i ∈ { 1 ∘ , 22 ∘ , 45 ∘ , 67 ∘ , 89 ∘ } i \in\{1^\circ,22^\circ,45^\circ,67^\circ,89^\circ\} i ∈ { 1 ∘ , 2 2 ∘ , 4 5 ∘ , 6 7 ∘ , 8 9 ∘ } 0 ∘ 0^\circ 0 ∘ 90 ∘ 90^\circ 9 0 ∘ a = 0 a=0 a = 0
Results
We investigated these models to address the two questions posed in Sec. 5.3. Our survey led us to the following answers:
A1. Yes, up to i ≲ 45 ∘ i \lesssim45^\circ i ≲ 4 5 ∘ In all the models considered here, we identified a regime in which the n = 2 n=2 n = 2 d φ ( 2 ) d_{\varphi}^{(2)} d φ ( 2 ) i = 67 ∘ i=67^\circ i = 6 7 ∘ i = 89 ∘ i=89^\circ i = 8 9 ∘ i ≲ 45 ∘ i \lesssim45^\circ i ≲ 4 5 ∘ (43) as measured by the normalized RMSD (44) : in all such cases,
RMSD φ ( R f i t ) < 5 × 10 − 4 . (58) \operatorname{RMSD}_{\varphi}(R_{\mathrm{fit}})<5\times10^{-4}. \tag*{(58)} RMSD φ ( R fit ) < 5 × 1 0 − 4 . (58) For most configurations, these best fits were obtained in a window u ∈ [ 1000 , 1100 ] G λ u\in[1000,1100]\,\mathrm{G}\lambda u ∈ [ 1000 , 1100 ] G λ n = 2 n=2 n = 2 u w ≈ 2000 G λ u_{w}\approx2000\,\mathrm{G}\lambda u w ≈ 2000 G λ i = 45 ∘ i=45^\circ i = 4 5 ∘
A2. As in Sec. 5.3, we extracted a pair of diameters ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) Figure 12 for the case of spin a = 0.5 a=0.5 a = 0.5 i = 45 ∘ i=45^\circ i = 4 5 ∘ ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) Figure 12 ) and then measure its size. The dimensions of these bounding boxes are listed for each ( a , i ) (a,i) ( a , i ) Table 6 below. As we explain in Sec. 6.3, this analysis bears relevance for the “inverse problem” of how to infer BH spin and inclination from the n = 2 n=2 n = 2
Figure 12. Parametrization of the region in the ( d + , d − ) (d_+,d_-) ( d + , d − ) a = 0.5 a = 0.5 a = 0.5 i = 45 ∘ i = 45^\circ i = 4 5 ∘
Table 6. Triplets ( s min , s max , ∣ δ ∣ max ) (s_{\min}, s_{\max}, |\delta|_{\max}) ( s m i n , s m a x , ∣ δ ∣ m a x ) ( d + , d − ) (d_+,d_-) ( d + , d − ) s min s_{\min} s m i n s max s_{\max} s m a x δ \delta δ M M M
We first note that this survey over BH spins and inclinations confirms one of our crucial observations from Sec. 5.4: for each choice of ( a , i ) (a,i) ( a , i ) n = 2 n=2 n = 2 ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) n = 2 n=2 n = 2 Table 5 occupy a very small subregion (green box in Figure 12 ) of the full region of allowed phoval shapes (blue box). As explained in Sec. 5.4, diameters along this line correspond to n = 2 n=2 n = 2 r s ∈ [ r + , ∞ ) r_{s}\in[r_{+},\infty) r s ∈ [ r + , ∞ ) n = 2 n=2 n = 2 ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) r s r_{s} r s
Rather than pursue this association with “effective” emission radii, here we employ a more agnostic parameterization of this line in the ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) s s s s = 0 s=0 s = 0 s = 1 s=1 s = 1 δ \delta δ ( d + , d − ) ↔ ( s , δ ) (d_{+},d_{-})\leftrightarrow(s,\delta) ( d + , d − ) ↔ ( s , δ ) s min s_{\min} s m i n s max s_{\max} s m a x ∣ δ ∣ max |\delta|_{\max} ∣ δ ∣ m a x ( a , i ) (a,i) ( a , i ) ( s min , s max , ∣ δ ∣ max ) (s_{\min},s_{\max},|\delta|_{\max}) ( s m i n , s m a x , ∣ δ ∣ m a x ) Table 6 .
Note that, with the exception of the extreme case a = 0.99 a=0.99 a = 0.99 i = 45 ∘ i=45^\circ i = 4 5 ∘ s min ≳ 0.18 s_{\min}\gtrsim0.18 s m i n ≳ 0.18 s max ≲ 0.75 s_{\max}\lesssim0.75 s m a x ≲ 0.75 2 M ≲ r s ≲ 7 M 2M\lesssim r_{s}\lesssim7M 2 M ≲ r s ≲ 7 M Table 5 .
Inferring BH spin and inclination from photon ring shape
In this survey, we have solved the “forward problem” of how to compute the diameters ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) n = 2 n=2 n = 2 ( a , i ) (a,i) ( a , i ) n = 2 n=2 n = 2 Figure 10 , Figure 11 and Figure 12 . Perhaps surprisingly, we have found that the ring shapes produced by our emission models occupy a much smaller subregion (the green bounding box in Figure 12 ) of this full region of geometrically allowed shapes.
Figure 10. Results of the survey of emission profiles in the “intermediate” case with spin a = 0.5 a=0.5 a = 0.5 i = 45 ∘ i=45^\circ i = 4 5 ∘
Figure 11. Results of the survey of emission profiles in the best-guess case for M87* with spin a = 0.94 a=0.94 a = 0.94 i = 17 ∘ i=17^\circ i = 1 7 ∘
This observation is highly relevant to the “inverse problem” of how to infer BH spin and inclination from a measurement of the n = 2 n=2 n = 2 ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) n = 2 n=2 n = 2 ( a , i ) (a,i) ( a , i )
Evidently, given the considerable size of these blue regions, these purely geometric constraints on ( a , i ) (a,i) ( a , i ) ( a , i ) (a,i) ( a , i )
Since our five models were picked to span the full range of s s s s min s_{\min} s m i n s max s_{\max} s m a x ∣ δ ∣ max |\delta|_{\max} ∣ δ ∣ m a x
Thus, we do not claim here to have definitively computed the bounding boxes for “realistic” choices of astrophysical profiles. Rather, we hope to have provided useful organizing principles for tackling the problem of spin inference: the choice between blue and green boxes clarifies the interplay between geometric and astrophysical assumptions, with the additional constraining power provided by the latter made manifest in the much smaller size of the green regions relative to the blue ones.
An additional benefit of this framework is that it enables us to quantify the uncertainty in a given spin estimate by computing the range of ( a , i ) (a,i) ( a , i ) ( d + , d − ) (d_+,d_-) ( d + , d − ) n = 2 n = 2 n = 2
Discussion
Limitations of this work
We have revisited the proposal made by [29 ] for a test of the Kerr hypothesis based on the photon ring shape. We have established the robustness of the proposed test in a large class of models, provided that the BH inclination is not too high. This confirms that M87* holds great promise as an observational target for this test. However, as pointed out in the introduction, this conclusion remains subject to some caveats:
We have considered only simple equatorial emission models. It is important to confirm whether the GLM method remains viable in the presence of more realistic accretion flows (such as fluctuating, geometrically thick structures), where effects such as absorption can become important (for instance, the interferometric signature of the photon ring is suppressed as the emission region grows thicker and more opaque). Some encouraging preliminary results in this direction will soon be reported elsewhere ([? ]).
We have not presented a complete, quantitative study of the baseline lengths for which the method works. In particular, we have not identified an “ideal” baseline window in which the GLM test succeeds for the largest set of configurations (astrophysical profiles and BH spins and inclinations), and we have largely ignored practical concerns in our theoretical study of the baseline domain. With the current wavelength of observations (1.3 mm), a space antenna would have to be almost 1.5 million km away from Earth—close to L 2 L_2 L 2 u w ∼ 1000 u_w \sim1000 u w ∼ 1000 λ \lambda λ [40 ] .
We have not included any source of noise in our simulations. Generating mock data with instrumental and astrophysical noise is crucial, not just for demonstrating the feasibility of this proposed test, but also for providing a rigorous, Bayesian approach to the multi-fit method described in Sec. 3. In their original analysis, [26 ] included instrumental noise but did not model astrophysical fluctuations, which can affect the photon ring signature in instantaneous snapshots and inform the requirements for (incoherent) time-averaging. A new code—AART, for Analytic Adaptive Ray Tracing —tailored to this specific problem has recently been developed [5 ] to examine these issues.
While amplitude gain errors are very problematic for VLBI data analyses (e.g., [14 , 15 ] ), it is worth noting that they may not pose a significant problem for a measurement of the photon ring diameter, which only requires a measurement of the periodicity of oscillations in the visibility amplitude (rather than a measurement of its absolute scale).
Perspectives: inferring black hole spin and inclination
If a future space mission targeting M87* successfully measured the shape of its n = 2 n = 2 n = 2 (43) , then the natural next step would be to assume the Kerr nature of the source and try to infer the BH spin and inclination ( a , i ) (a, i) ( a , i )
Our survey over emission profiles in Sec. 5 shows that this shape displays a residual dependence on the astrophysical source profile (even for the n = 2 n = 2 n = 2 n = 2 n = 2 n = 2 Table 4 . (This uncertainty is undoubtedly much greater for spin estimates based on the n = 1 n = 1 n = 1
Spin estimates can be tightened with additional assumptions about the nature of the emission. As discussed in Secs. 5.4 and 6.2, the photon ring shapes produced by our emission models inhabit only a small subset of the full range of geometric shapes allowed by BH lensing. Thus, assuming that an observed photon ring shape is produced by a model within this class considerably narrows down the set of BH spins and inclinations compatible with the measurement, excluding many more values of ( a , i ) (a, i) ( a , i )
It would be extremely interesting to quantify the reduction in uncertainty enabled by different astrophysical assumptions. This requires one to tackle the inverse problem of how to determine the values of ( a , i ) (a, i) ( a , i )
Finally, spin estimates can also be substantially improved by combining the constraints derived from multiple image features. Early work suggests that measuring both the photon ring shape and that of the central brightness depression could significantly constrain BH spin and inclination [6 ] . Of course, simultaneously resolving more than one ring (such as the n = 1 n = 1 n = 1 n = 2 n = 2 n = 2 [4 ] .
Perspectives: other photon ring tests
The prospect of measuring multiple subrings naturally opens the door to even more stringent tests of the Kerr hypothesis (or more precise techniques for determining the BH parameters).
For instance, comparing the widths of successive subrings at different angles around the image could in principle provide a measurement of the angle-dependent demagnification factors e − γ ( φ ) e^{-\gamma(\varphi)} e − γ ( φ ) γ ( r ~ ) \gamma(\tilde r) γ ( r ~ ) [40 , 27 ] . If the demagnification factors were not e − γ ( φ ) e^{-\gamma(\varphi)} e − γ ( φ ) ? ]. Conversely, if γ ( φ ) \gamma(\varphi) γ ( φ ) ( a , i ) (a, i) ( a , i )
Likewise, polarimetric imaging of successive subrings could also be used to test the Kerr hypothesis, or conversely, to yield an estimate of the BH parameters ( a , i ) (a, i) ( a , i ) [34 ] , while also providing a probe of the plasma and magnetic field around the BH [17 ] . Lastly, the photon ring substructure predicts a characteristic pattern of autocorrelations whose detection could also provide a window into the Lyapunov exponents γ \gamma γ τ \tau τ δ \delta δ [32 ] . A related promising idea involves timing signatures such as quasi-periodic oscillations [38 ] .
Finally, a wide variety of other techniques for testing the Kerr hypothesis and strong-field GR via VLBI observations have been proposed (e.g., [? ]), many of which do not rely upon the photon ring. Similarly, some constraints on BH inclination can be derived from other image features, including the brightness asymmetry of the direct (n = 0 n = 0 n = 0 [44 ] .
Open questions and future directions
We have thoroughly investigated the shape of the n = 2 n = 2 n = 2
Reversing the logic, one can assume the Kerr nature of the source and try to infer its parameters from the photon ring shape. We hope to further explore the question of spin inference by solving the inverse problem described in Sec. 6.3. This approach could most likely also be used to infer the BH mass with much greater precision than the EHT has so far achieved using only a direct (n = 0 n = 0 n = 0 ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) n = 2 n=2 n = 2 n = 2 n=2 n = 2
Finally, the most pressing task is to revisit all these questions in the context of the first (n = 1 n=1 n = 1 n = 1 n=1 n = 1
From a theoretical standpoint, these questions are harder to tackle because the n = 1 n=1 n = 1 [27 ] ; that is, its width-to-diameter ratio w / d w/d w / d [27 ] to arise. If so, then there is no baseline regime in which its interferometric signature follows the universal form [28 ] , and it may well be necessary to compute the subleading correction in w / d w/d w / d
Summary
Images of a BH are typically dominated by a bright, narrow ring encircling a central brightness depression. This “photon ring” decomposes into a sequence of self-similar subrings indexed by half-orbit number n n n [28 ] to test the Kerr hypothesis via an interferometric measurement of the n = 2 n=2 n = 2 i ≲ 45 ∘ i \lesssim45^\circ i ≲ 4 5 ∘ i ≈ 17 ∘ i \approx17^\circ i ≈ 1 7 ∘
At higher inclinations i ≳ 45 ∘ i \gtrsim45^\circ i ≳ 4 5 ∘ n = 1 n=1 n = 1 n = 2 n=2 n = 2 Figure 6 ), or else pick up the signature of a “hybrid n = 1 & 2 n=1\&2 n = 1&2 Figure 7 ). A more refined, sophisticated shape test has yet to be devised for these situations.
Our investigation also revealed that for a given choice of BH spin and inclination, the observable photon ring shape depends noticeably on the astrophysical properties of the emitting source. In particular, the most directly accessible (n = 1 n=1 n = 1 n = 2 n=2 n = 2 Figure 10 & Figure 11 ). As such (and as discussed throughout Secs. 5 and 6), the lensing bands introduced in Sec. 2.5 are more relevant than the critical curve to the problem of BH spin inference from the photon ring shape.
We have outlined a promising approach for tackling this problem in Sec. 6.3, with some encouraging preliminary results (Figure 12 ).
Acknowledgements
All of the images in this paper were ray traced on the tycho computer cluster of the Paris Observatory. We thank Samuel Gralla, Alejandro Cárdenas-Avendaño, Andrei Lobanov, and Hengrui Zhu for helpful comments and discussions. A.L. also gratefully acknowledges support from Will and Kacie Snellings, and M.W. thanks Alexandra Elbakyan for her contributions to the open science initiative.
Computing Kerr lensing bands
Let us consider, within the framework reviewed in Sec. 2, a null geodesic connecting a point source at ( t s , r s , θ s , φ s ) (t_s,r_s,\theta_s,\varphi_s) ( t s , r s , θ s , φ s ) ( t o , r o , θ o , φ o ) (t_o,r_o,\theta_o,\varphi_o) ( t o , r o , θ o , φ o ) ( λ , η ) (\lambda,\eta) ( λ , η ) (2) via the radial and angular potentials (e.g., [27 ] )
R ( r ) = ( r 2 + a ∗ 2 − a ∗ λ ) 2 − Δ ( r ) [ η + ( λ − a ∗ ) 2 ] , (59) \mathcal{R}(r) = \left(r^2+a_*^2-a_*\lambda\right)^2-\Delta(r)\left[\eta+(\lambda-a_*)^2\right], \tag*{(59)} R ( r ) = ( r 2 + a ∗ 2 − a ∗ λ ) 2 − Δ ( r ) [ η + ( λ − a ∗ ) 2 ] , (59) Θ ( θ ) = η + a ∗ 2 cos 2 θ − λ 2 cot 2 θ , (60) \Theta(\theta)=\eta+a_*^2\cos^2\theta-\lambda^2\cot^2\theta, \tag*{(60)} Θ ( θ ) = η + a ∗ 2 cos 2 θ − λ 2 cot 2 θ , (60) whose zeros define the radial and angular turning points of the ray. Integrating the geodesic equation along the trajectory gives
I r ≡ ∫ r s r o d r ± r R ( r ) = ∫ θ s θ o d θ ± θ Θ ( θ ) = G θ , (61) I_r\equiv\int_{r_s}^{r_o}\frac{\mathrm{d}r}{\pm_r\sqrt{\mathcal{R}(r)}}=\int_{\theta_s}^{\theta_o}\frac{\mathrm{d}\theta}{\pm_\theta\sqrt{\Theta(\theta)}}=G_\theta, \tag*{(61)} I r ≡ ∫ r s r o ± r R ( r ) d r = ∫ θ s θ o ± θ Θ ( θ ) d θ = G θ , (61) where the signs ± r \pm_r ± r ± θ \pm_\theta ± θ τ = I r = G θ \tau=I_r=G_\theta τ = I r = G θ
We consider the case of a distant observer ( r o = D ≫ M ) (r_o=D\gg M) ( r o = D ≫ M ) θ o = i \theta_o=i θ o = i ( ρ , φ ρ ) (\rho,\varphi_\rho) ( ρ , φ ρ )
λ = − D ρ cos φ ρ sin i , η = ( D ρ ) 2 − λ 2 − a ∗ 2 cos 2 i , (62) \lambda=-D\rho\cos\varphi_\rho\sin i,\qquad\eta=(D\rho)^2-\lambda^2-a_*^2\cos^2i, \tag*{(62)} λ = − D ρ cos φ ρ sin i , η = ( D ρ ) 2 − λ 2 − a ∗ 2 cos 2 i , (62) in terms of which the four roots r j r_j r j R ( r ) \mathcal{R}(r) R ( r ) [27 ] ]:
r j = − z − ϵ j A 2 − z 2 + ϵ ~ j B 4 z , r_j=-z-\epsilon_j\sqrt{\frac{\mathcal{A}}{2}-z^2+\widetilde{\epsilon}_j\frac{\mathcal{B}}{4z}}, r j = − z − ϵ j 2 A − z 2 + ϵ j 4 z B , ϵ j = { 1 if j ∈ { 1 , 3 } , − 1 if j ∈ { 2 , 4 } , ϵ ~ j = { 1 if j ∈ { 1 , 2 } , − 1 if j ∈ { 3 , 4 } , \epsilon_j=\begin{cases}1&\text{if }j\in\{1,3\},\\-1&\text{if }j\in\{2,4\},\end{cases}
\qquad
\widetilde{\epsilon}_j=\begin{cases}1&\text{if }j\in\{1,2\},\\-1&\text{if }j\in\{3,4\},\end{cases} ϵ j = { 1 − 1 if j ∈ { 1 , 3 } , if j ∈ { 2 , 4 } , ϵ j = { 1 − 1 if j ∈ { 1 , 2 } , if j ∈ { 3 , 4 } , A = a ∗ 2 − η − λ 2 , B = 2 M [ η + ( λ − a ∗ ) 2 ] , C = − a ∗ 2 η , \mathcal{A}=a_*^2-\eta-\lambda^2,\qquad\mathcal{B}=2M\left[\eta+(\lambda-a_*)^2\right],\qquad\mathcal{C}=-a_*^2\eta, A = a ∗ 2 − η − λ 2 , B = 2 M [ η + ( λ − a ∗ ) 2 ] , C = − a ∗ 2 η , P = − A 2 12 − C , Q = − A 3 [ ( A 6 ) 2 − C ] − B 2 8 , \mathcal{P}=-\frac{\mathcal{A}^2}{12}-\mathcal{C},\qquad
\mathcal{Q}=-\frac{\mathcal{A}}{3}\left[\left(\frac{\mathcal{A}}{6}\right)^2-\mathcal{C}\right]-\frac{\mathcal{B}^2}{8}, P = − 12 A 2 − C , Q = − 3 A [ ( 6 A ) 2 − C ] − 8 B 2 , ω ± = ( − Q 2 ± P 3 27 + Q 2 4 ) 1 / 3 , z = ω + + ω − − A / 3 2 . \omega_\pm=\left(-\frac{\mathcal{Q}}{2}\pm\sqrt{\frac{\mathcal{P}^3}{27}+\frac{\mathcal{Q}^2}{4}}\right)^{1/3},
\qquad
z=\sqrt{\frac{\omega_++\omega_--\mathcal{A}/3}{2}}. ω ± = ( − 2 Q ± 27 P 3 + 4 Q 2 ) 1/3 , z = 2 ω + + ω − − A /3 . Letting r i j = r i − r j r_{ij}=r_i-r_j r ij = r i − r j A = r 32 r 42 \mathcal{A}=\sqrt{r_{32}r_{42}} A = r 32 r 42 B = r 31 r 41 \mathcal{B}=\sqrt{r_{31}r_{41}} B = r 31 r 41
I r + ( r ) = 2 r 31 r 42 F ( arcsin r − r 4 r − r 3 r 31 r 41 , r 32 r 41 r 31 r 42 ) , (63) I_r^+(r)=\frac{2}{\sqrt{r_{31}r_{42}}}F\left(\arcsin\sqrt{\frac{r-r_4}{r-r_3}\frac{r_{31}}{r_{41}}},\frac{r_{32}r_{41}}{r_{31}r_{42}}\right), \tag*{(63)} I r + ( r ) = r 31 r 42 2 F ( arcsin r − r 3 r − r 4 r 41 r 31 , r 31 r 42 r 32 r 41 ) , (63) I r − ( r ) = 1 A B F ( arccos ( 1 − r − r 2 r − r 1 B A 1 + r − r 2 r − r 1 B A ) ( A + B ) 2 − r 21 2 4 A B ) , (64) I_r^-(r)=\frac{1}{\sqrt{\mathcal{A}\mathcal{B}}}F\left(\arccos\left(\frac{1-\frac{r-r_2}{r-r_1}\frac{\mathcal{B}}{\mathcal{A}}}{1+\frac{r-r_2}{r-r_1}\frac{\mathcal{B}}{\mathcal{A}}}\right)\sqrt{\frac{(\mathcal{A}+\mathcal{B})^2-r_{21}^2}{4\mathcal{A}\mathcal{B}}}\right), \tag*{(64)} I r − ( r ) = A B 1 F ( arccos ( 1 + r − r 1 r − r 2 A B 1 − r − r 1 r − r 2 A B ) 4 A B ( A + B ) 2 − r 21 2 ) , (64) which are antiderivatives of I r I_r I r [26 ] ).
If ρ > ρ ~ ( φ ρ ) \rho>\tilde{\rho}(\varphi_\rho) ρ > ρ ~ ( φ ρ ) r 4 > r 3 > r + > r − > r 2 > r 1 r_4>r_3>r_+>r_->r_2>r_1 r 4 > r 3 > r + > r − > r 2 > r 1 r = r 4 r=r_4 r = r 4 τ ( r ) \tau(r) τ ( r )
I r C + ( r ) = I r + ( r o ) ∓ I r + ( r ) , (65) I_r^{C^+}(r)=I_r^+(r_o)\mp I_r^+(r), \tag*{(65)} I r C + ( r ) = I r + ( r o ) ∓ I r + ( r ) , (65) with upper/lower sign before/after reaching the turning point. The total Mino time elapsed along the full light ray is
I r t o t a l = 4 r 31 r 42 F ( arcsin r 31 r 41 | r 32 r 41 r 31 r 42 ) . (66) I_r^{\mathrm{total}}=\frac{4}{\sqrt{r_{31}r_{42}}}F\left(\arcsin\sqrt{\frac{r_{31}}{r_{41}}}\,\middle|\,\frac{r_{32}r_{41}}{r_{31}r_{42}}\right). \tag*{(66)} I r total = r 31 r 42 4 F ( arcsin r 41 r 31 r 31 r 42 r 32 r 41 ) . (66) If ρ < ρ ~ ( φ ρ ) \rho<\tilde{\rho}(\varphi_\rho) ρ < ρ ~ ( φ ρ ) r = r + r=r_+ r = r + τ ( r ) \tau(r) τ ( r )
I r C − ( r ) = I r ± ( r o ) − I r ± ( r ) , (67) I_r^{C^-}(r)=I_r^\pm(r_o)-I_r^\pm(r), \tag*{(67)} I r C − ( r ) = I r ± ( r o ) − I r ± ( r ) , (67) with the upper sign when all roots are real (in which case r + > r 4 > r 3 > r 2 > r 1 r_+>r_4>r_3>r_2>r_1 r + > r 4 > r 3 > r 2 > r 1 r 3 = r ˉ 4 r_3=\bar r_4 r 3 = r ˉ 4 r + > r − > r 2 > r 1 r_+>r_->r_2>r_1 r + > r − > r 2 > r 1 9
I r t o t a l = I r C − ( r + ) . (68) I_r^{\mathrm{total}}=I_r^{C^-}(r_+). \tag*{(68)} I r total = I r C − ( r + ) . (68) Regardless of where the photon is shot back from, it crosses the equatorial plane a maximum number N + 1 N+1 N + 1
N = ⌊ I r t o t a l − u ± a ∗ 2 + sign ( sin φ ρ ) F o 2 K ⌋ − H ( sin φ ρ ) , (69) N=\left\lfloor\frac{I_r^{\mathrm{total}}\sqrt{-u_\pm a_*^2}+\operatorname{sign}(\sin\varphi_\rho)F_o}{2K}\right\rfloor-H(\sin\varphi_\rho), \tag*{(69)} N = ⌊ 2 K I r total − u ± a ∗ 2 + sign ( sin φ ρ ) F o ⌋ − H ( sin φ ρ ) , (69) u ± = Δ θ ± Δ θ 2 + η a ∗ 2 , Δ θ = 1 2 ( 1 − η + λ 2 a ∗ 2 ) , u_\pm=\Delta_\theta\pm\sqrt{\Delta_\theta^2+\frac{\eta}{a_*^2}},
\qquad
\Delta_\theta=\frac{1}{2}\left(1-\frac{\eta+\lambda^2}{a_*^2}\right), u ± = Δ θ ± Δ θ 2 + a ∗ 2 η , Δ θ = 2 1 ( 1 − a ∗ 2 η + λ 2 ) , K = K ( u + u − ) , F o = F ( arcsin ( cos ( i ) u + ) | u + u − ) , K=K\left(\frac{u_+}{u_-}\right),
\qquad
F_o=F\left(\arcsin\left(\frac{\cos(i)}{\sqrt{u_+}}\right)\middle|\frac{u_+}{u_-}\right), K = K ( u − u + ) , F o = F ( arcsin ( u + cos ( i ) ) u − u + ) , where H H H F F F K K K F ( u ∣ k ) F(u|k) F ( u ∣ k ) K ( k ) ≡ F ( π / 2 ∣ k ) K(k)\equiv F(\pi/2|k) K ( k ) ≡ F ( π /2∣ k ) u ± u_\pm u ± u = cos 2 θ u=\cos^2\theta u = cos 2 θ Θ ( θ ) \Theta(\theta) Θ ( θ ) ( λ , η ) = ( λ ~ , η ~ ) (\lambda,\eta)=(\tilde{\lambda},\tilde{\eta}) ( λ , η ) = ( λ ~ , η ~ ) (6) , they reproduce u ~ ± \tilde{u}_\pm u ~ ± (5) .
The photon intersects the equatorial plane for the ( n + 1 ) t h (n+1)^{\mathrm{th}} ( n + 1 ) th n ≤ N n\leq N n ≤ N τ ( θ ) \tau(\theta) τ ( θ ) [26 ] ]
G θ ( n ) = 2 m K − sign ( sin φ ρ ) F o − u − a ∗ 2 , (70) G_\theta(n)=\frac{2mK-\operatorname{sign}(\sin\varphi_\rho)F_o}{\sqrt{-u_-a_*^2}}, \tag*{(70)} G θ ( n ) = − u − a ∗ 2 2 m K − sign ( sin φ ρ ) F o , (70) where m = n + H ( sin φ ρ ) m=n+H(\sin\varphi_\rho) m = n + H ( sin φ ρ ) N N N G θ ( N ) ≤ I r t o t a l ≤ G θ ( N + 1 ) G_\theta(N)\leq I_r^{\mathrm{total}}\leq G_\theta(N+1) G θ ( N ) ≤ I r total ≤ G θ ( N + 1 )
The n t h n^{\mathrm{th}} n th r s r_s r s ( ρ , φ ρ ) (\rho,\varphi_\rho) ( ρ , φ ρ )
I r C ± ( r s ) = G θ ( n ) (71) \boxed{I_r^{C^\pm}(r_s)=G_\theta(n)} \tag*{(71)} I r C ± ( r s ) = G θ ( n ) (71) with G θ ( n ) G_\theta(n) G θ ( n ) (70) and I r C ± ( r s ) I_r^{C^\pm}(r_s) I r C ± ( r s ) (65) and (67) for points outside and inside the critical curve. These points always trace a closed, continuous curve C n ( r s ) C_n(r_s) C n ( r s )
The contours C 0 ( r s ) C_0(r_s) C 0 ( r s ) C 1 ( r s ) C_1(r_s) C 1 ( r s ) r s r_s r s Figure 6 of [26 ] . Note that some of them can cross the critical curve: these continuous curves are thus stitched together from segments solving Eq. (71) with opposite signs. Aside from this minor inconvenience, it is a straightforward task to numerically solve Eq. (71) for any C n ( r s ) C_n(r_s) C n ( r s )
In particular, as explained in Sec. 2.5, the n t h n^{\mathrm{th}} n th C n ( r + ) C_n(r_+) C n ( r + ) C n ( ∞ ) C_n(\infty) C n ( ∞ ) (65) , so the outer edge C n ( ∞ ) C_n(\infty) C n ( ∞ ) (71) with a + + + (67) , so the inner edge C n ( r + ) C_n(r_+) C n ( r + ) (71) with a − - −
We note that when crossing the inner/outer edges C n ( r + ) C_n(r_+) C n ( r + ) C n ( ∞ ) C_n(\infty) C n ( ∞ ) n t h n^{\mathrm{th}} n th N + 1 N+1 N + 1 (69) ) must by definition change by 1. This provides a means of checking that C n ( r + ) C_n(r_+) C n ( r + ) C n ( ∞ ) C_n(\infty) C n ( ∞ )
Finally, we note that Eq. (61) may also be directly solved for r s ( I r ) r_s(I_r) r s ( I r ) [27 ] . Then r e q ( n ) ( ρ , φ ρ ) ≡ r s ( G θ ( n ) ) r_{\rm eq}^{(n)}(\rho,\varphi_\rho)\equiv r_s(G_\theta(n)) r eq ( n ) ( ρ , φ ρ ) ≡ r s ( G θ ( n )) ( ρ , φ ρ ) (\rho,\varphi_\rho) ( ρ , φ ρ ) ( n + 1 ) t h (n+1)^{\mathrm{th}} ( n + 1 ) th C n ( r s ) C_n(r_s) C n ( r s ) [6 ] .
B Computing the range of allowed phovals within a lensing band
In this appendix, we explain in detail how to solve this problem: Given a lensing band B ⊂ R 2 \mathcal{B}\subset\mathbb{R}^2 B ⊂ R 2 A \mathcal{A} A ( d + , d − ) (d_+,d_-) ( d + , d − ) B \mathcal{B} B
We glossed over this question after introducing it in Sec. 5.4, but its answer is crucial for obtaining the blue regions in Figure 10 , Figure 11 and Figure 12 . At first glance, this problem seems straightforward: we could scan the 6-dimensional space of phovals (53) and ask, for each R = { R 0 , R 1 , R 2 , X , χ , φ 0 } \boldsymbol{R}=\{R_0,R_1,R_2,X,\chi,\varphi_0\} R = { R 0 , R 1 , R 2 , X , χ , φ 0 } P 6 \mathcal{P}_6 P 6 B \mathcal{B} B A 6 ⊂ P 6 \mathcal{A}_6\subset\mathcal{P}_6 A 6 ⊂ P 6 Π : P 6 → R 2 \Pi:\mathcal{P}_6\to\mathbb{R}^2 Π : P 6 → R 2
Π : R ↦ ( d + , d − ) = 2 ( R 0 + R 1 , R 0 + R 2 ) . (72) \Pi:\boldsymbol{R}\mapsto(d_+,d_-)=2(R_0+R_1,R_0+R_2). \tag*{(72)} Π : R ↦ ( d + , d − ) = 2 ( R 0 + R 1 , R 0 + R 2 ) . (72) would give precisely the sought-after region in the ( d + , d − ) (d_+,d_-) ( d + , d − )
A = Π ( A 6 ) . (73) \mathcal{A}=\Pi(\mathcal{A}_6). \tag*{(73)} A = Π ( A 6 ) . (73) In practice, however, the computational cost of this “brute force” approach is too high to achieve sufficient precision in reasonable time: the high-dimensionality of the parameter space P 6 \mathcal{P}_6 P 6
The tractable method we came up with consists of two parts:
First, we recast the question of whether a given point ( d + , d − ) (d_+,d_-) ( d + , d − ) A \mathcal{A} A
Second, we devised an efficient way to scan across regions of the ( d + , d − ) (d_+,d_-) ( d + , d − )
Our solution is quite general. We describe each part separately:
Part 1: Minimization method
Let B ⊂ R 2 \mathcal{B}\subset\mathbb{R}^2 B ⊂ R 2 I \mathcal{I} I O \mathcal{O} O C \mathcal{C} C B \mathcal{B} B
δ B ( C ) = max z ∈ C d ( z , B ) , (74) \delta_{\mathcal{B}}(\mathcal{C})=\max_{z\in\mathcal{C}}d(z,\mathcal{B}), \tag*{(74)} δ B ( C ) = z ∈ C max d ( z , B ) , (74) where d d d
d ( z , B ) = { min a ∈ I ∣ z − a ∣ if z lies inside of I , min a ∈ O ∣ z − a ∣ if z lies outside of O , 0 if z ∈ B . (75) d(z,\mathcal{B})=
\begin{cases}
\min_{a\in\mathcal{I}}\lvert z-a\rvert& \text{if }z\text{ lies inside of }\mathcal{I},\\
\min_{a\in\mathcal{O}}\lvert z-a\rvert& \text{if }z\text{ lies outside of }\mathcal{O},\\
0 & \text{if }z\in\mathcal{B}.
\end{cases} \tag*{(75)} d ( z , B ) = ⎩ ⎨ ⎧ min a ∈ I ∣ z − a ∣ min a ∈ O ∣ z − a ∣ 0 if z lies inside of I , if z lies outside of O , if z ∈ B . (75) For us, C \mathcal{C} C B \mathcal{B} B I = C n ( r + ) \mathcal{I}=C_n(r_+) I = C n ( r + ) O = C n ( ∞ ) \mathcal{O}=C_n(\infty) O = C n ( ∞ )
With this definition, δ B ( C ) = 0 \delta_{\mathcal{B}}(\mathcal{C})=0 δ B ( C ) = 0 C \mathcal{C} C B \mathcal{B} B ( d + , d − ) (d_+,d_-) ( d + , d − ) C \mathcal{C} C B \mathcal{B} B
Δ : ( d + , d − , r ) ↦ δ B ( C ( R 0 , d + 2 − R 0 , d − 2 − R 0 , X , χ , φ 0 ) ) , \Delta:(d_+,d_-,\boldsymbol{r})\mapsto\delta_{\mathcal{B}}\left(\mathcal{C}\left(R_0,\frac{d_+}{2}-R_0,\frac{d_-}{2}-R_0,X,\chi,\varphi_0\right)\right), Δ : ( d + , d − , r ) ↦ δ B ( C ( R 0 , 2 d + − R 0 , 2 d − − R 0 , X , χ , φ 0 ) ) , where C ( R ) \mathcal{C}(\boldsymbol{R}) C ( R ) P 4 \mathcal{P}_4 P 4 r = ( R 0 , X , χ , φ 0 ) \boldsymbol{r}=(R_0,X,\chi,\varphi_0) r = ( R 0 , X , χ , φ 0 ) m : R 2 → R \boldsymbol{m}:\mathbb{R}^2\to\mathbb{R} m : R 2 → R
m : ( d + , d − ) ↦ min r ∈ P 4 Δ ( d + , d − , r ) . (76) m:(d_+,d_-)\mapsto\min_{\boldsymbol{r}\in\mathcal{P}_4}\Delta(d_+,d_-,\boldsymbol{r}). \tag*{(76)} m : ( d + , d − ) ↦ r ∈ P 4 min Δ ( d + , d − , r ) . (76) If m ( d + , d − ) m(d_+,d_-) m ( d + , d − ) ( d + , d − ) (d_+,d_-) ( d + , d − )
A = { ( d + , d − ) ∈ R 2 : m ( d + , d − ) = 0 } . (77) \mathcal{A}=\{(d_+,d_-)\in\mathbb{R}^2:m(d_+,d_-)=0\}. \tag*{(77)} A = {( d + , d − ) ∈ R 2 : m ( d + , d − ) = 0 } . (77) We have thus recast the problem of determining whether a given ( d + , d − ) (d_+,d_-) ( d + , d − ) A \mathcal{A} A P 4 \mathcal{P}_4 P 4
Part 2: Random walk
To compute A \mathcal{A} A ( d + , d − ) (d_+,d_-) ( d + , d − ) I \mathcal{I} I O \mathcal{O} O m ( d + , d − ) m(d_+,d_-) m ( d + , d − ) m m m
We start from the parameters R ∈ A 6 ⊂ P 6 \boldsymbol{R}\in\mathcal{A}_6\subset\mathcal{P}_6 R ∈ A 6 ⊂ P 6 I \mathcal{I} I O \mathcal{O} O
We randomly pick one of the radii { R 0 , R 1 , R 2 } \{R_0,R_1,R_2\} { R 0 , R 1 , R 2 } ± b \pm b ± b R 1 ≥ R 2 R_1\ge R_2 R 1 ≥ R 2 R 1 ↔ R 2 R_1\leftrightarrow R_2 R 1 ↔ R 2
After the increment, we test if the resulting phoval (with the same values of X , χ X,\chi X , χ φ 0 \varphi_0 φ 0 ( d + , d − ) = 2 ( R 0 + R 1 , R 0 + R 2 ) (d_+,d_-)=2(R_0+R_1,R_0+R_2) ( d + , d − ) = 2 ( R 0 + R 1 , R 0 + R 2 ) A \mathcal{A} A m ( d + , d − ) m(d_+,d_-) m ( d + , d − )
m ( d + , d − ) = 0 m(d_{+},d_{-}) = 0 m ( d + , d − ) = 0 ( d + , d − ) = 2 ( R 0 + R 1 , R 0 + R 2 ) (d_{+},d_{-}) = 2(R_{0}+R_{1},R_{0}+R_{2}) ( d + , d − ) = 2 ( R 0 + R 1 , R 0 + R 2 ) A \mathcal{A} A
m ( d + , d − ) > 0 m(d_{+},d_{-}) > 0 m ( d + , d − ) > 0 ( d + , d − ) = 2 ( R 0 + R 1 , R 0 + R 2 ) (d_{+},d_{-}) = 2(R_{0}+R_{1},R_{0}+R_{2}) ( d + , d − ) = 2 ( R 0 + R 1 , R 0 + R 2 ) { R 0 , R 1 , R 2 } \{R_{0},R_{1},R_{2}\} { R 0 , R 1 , R 2 }
Steps 2 and 3 can be repeated any number of times, resulting in a set of rejected values { ( d + , d − ) } r e j e c t e d \{(d_{+},d_{-})\}_{\mathrm{rejected}} {( d + , d − ) } rejected
A = { ( d + , d − ) } a c c e p t e d . (78) \mathcal{A} = \{(d_{+},d_{-})\}_{\mathrm{accepted}} . \tag*{(78)} A = {( d + , d − ) } accepted . (78) One advantage is that the minimization routine m m m ( d + , d − ) (d_{+},d_{-}) ( d + , d − ) A \mathcal{A} A
The boundaries of the region A \mathcal{A} A Figure 10 , Figure 11 and Figure 12 ) can be computed to arbitrary precision by tuning the value of the increment b b b
Shape of the n = 1 n=1 n = 1
The n = 1 n=1 n = 1
Does the n = 1 n=1 n = 1 (43) )? If so, to what precision? Can its projected diameter d φ ( 1 ) d_{\varphi}^{(1)} d φ ( 1 ) ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣
Answering these questions in detail would be the subject of a whole other paper. Here, we merely record some observations that arose from our study and should inform future analyses:
Only narrow rings with a very small width-to-diameter ratio w / d ≪ 1 w/d \ll1 w / d ≪ 1 (23) ). For thicker rings, there is not a large enough separation between the scales 1 / d 1/d 1/ d 1 / w 1/w 1/ w n = 1 n=1 n = 1 (50) and in Sec. 5), with a typical width of w 1 ≈ 1 M w_{1} \approx1M w 1 ≈ 1 M d ≈ 10 M d \approx10M d ≈ 10 M w 1 / d ≈ 10 % w_{1}/d \approx10\% w 1 / d ≈ 10% (25) ) may be substantial. Indeed, in most of our n = 1 n=1 n = 1 u − 1 / 2 u^{-1/2} u − 1/2
Geometrically, this also means that the periodicity d φ ( 1 ) d_{\varphi}^{(1)} d φ ( 1 ) n = 1 n=1 n = 1 n = 1 n=1 n = 1 w 1 / d ≈ 10 % w_{1}/d \approx10\% w 1 / d ≈ 10%
Even in the absence of a universal regime 1 / d ≪ u ≪ 1 / w 1/d \ll u \ll1/w 1/ d ≪ u ≪ 1/ w (25) ), the n = 1 n=1 n = 1 b 1 ≪ u ≪ b 2 b_{1} \ll u \ll b_{2} b 1 ≪ u ≪ b 2 ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ d φ ( 1 ) d_{\varphi}^{(1)} d φ ( 1 ) Δ u = b 2 − b 1 \Delta u = b_{2}-b_{1} Δ u = b 2 − b 1
While these issues may complicate a measurement of the n = 1 n=1 n = 1 Figure C.1 ), and we briefly address them in reverse order.
Figure C.1. Kerr hypothesis test using the GLM method applied to the n = 1 n=1 n = 1 a = 0.94 a=0.94 a = 0.94 i = 17 ∘ i=17^\circ i = 1 7 ∘ μ = r − \mu=r_{-} μ = r − γ = − 3 2 \gamma=-\frac{3}{2} γ = − 2 3 σ = 0.5 \sigma=0.5 σ = 0.5 ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ φ = 135 ∘ \varphi=135^\circ φ = 13 5 ∘ n = 1 n=1 n = 1 u w ∈ [ 30 , 70 ] u_{\mathrm{w}}\in[30,70] u w ∈ [ 30 , 70 ] λ \lambda λ (40) as a function of fitting diameter d φ ( 1 ) d_{\varphi}^{(1)} d φ ( 1 ) g max = e − R M S D u , min g_{\max}=e^{-\mathrm{RMSD}_{u,\min}} g m a x = e − RMSD u , m i n R M S D u ( d φ ) ≤ 2 R M S D u , min \mathrm{RMSD}_{u}(d_{\varphi})\leq2\mathrm{RMSD}_{u,\min} RMSD u ( d φ ) ≤ 2 RMSD u , m i n g ( d φ ) ≥ g max g(d_{\varphi})\geq g_{\max} g ( d φ ) ≥ g m a x d φ ( 1 ) d_{\varphi}^{(1)} d φ ( 1 ) φ = 125 ∘ \varphi=125^\circ φ = 12 5 ∘ 130 ∘ 130^\circ 13 0 ∘
Regarding point 3: a baseline region of width Δ u \Delta u Δ u N = Δ u / T N = \Delta u/T N = Δ u / T T T T T = 1 / d φ T = 1/d_{\varphi} T = 1/ d φ n = 1 n=1 n = 1 b 1 ≪ u ≪ b 2 b_{1} \ll u \ll b_{2} b 1 ≪ u ≪ b 2
N 1 ( φ ) ≈ [ b 2 ( φ ) − b 1 ( φ ) ] d φ . (79) N_{1}(\varphi) \approx[b_{2}(\varphi)-b_{1}(\varphi)]d_{\varphi}. \tag*{(79)} N 1 ( φ ) ≈ [ b 2 ( φ ) − b 1 ( φ )] d φ . (79) Likewise, for higher-order rings n ≳ 2 n \gtrsim2 n ≳ 2 (49) , we have
N n ( φ ) ≈ [ b n + 1 ( φ ) − b n ( φ ) ] d φ ≈ e γ ( φ ) N n − 1 ( φ ) , (80) N_{n}(\varphi) \approx[b_{n+1}(\varphi)-b_{n}(\varphi)]d_{\varphi} \approx e^{\gamma(\varphi)}N_{n-1}(\varphi), \tag*{(80)} N n ( φ ) ≈ [ b n + 1 ( φ ) − b n ( φ )] d φ ≈ e γ ( φ ) N n − 1 ( φ ) , (80) so that the number of visamp hops within the regime dominated by the n t h n^{\mathrm{th}} n th n n n e γ e^{\gamma} e γ N n ( φ ) N_{n}(\varphi) N n ( φ ) N 1 ( φ ) N_{1}(\varphi) N 1 ( φ ) Figure C.1 , we have Δ u = b 2 − b 1 ≈ 80 G λ \Delta u = b_{2}-b_{1} \approx80\,\mathrm{G}\lambda Δ u = b 2 − b 1 ≈ 80 G λ φ = 135 ∘ \varphi= 135^\circ φ = 13 5 ∘
d φ ∼ 40 μ a s ≈ 1 5 G λ , (81) d_{\varphi} \sim40\,\mu\mathrm{as} \approx\frac{1}{5\,\mathrm{G}\lambda}, \tag*{(81)} d φ ∼ 40 μ as ≈ 5 G λ 1 , (81) where we manipulated units as in Eq. (41) , it follows that
N 1 ( 135 ∘ ) ≈ 16 , (82) N_{1}(135^\circ) \approx16, \tag*{(82)} N 1 ( 13 5 ∘ ) ≈ 16 , (82) and indeed, we count 16 visamp nulls within the gray region in Figure C.1 . This is in principle a sufficient number of hops to obtain a good estimate of the periodicity, and hence of d φ ( 1 ) d_{\varphi}^{(1)} d φ ( 1 ) Figure C.1 follows a circlipse shape, albeit with larger error bars than for the n = 2 n=2 n = 2
Regarding point 2: in the limit w / d → 0 w/d \to0 w / d → 0 d φ d_{\varphi} d φ w / d w/d w / d n = 1 n=1 n = 1 1 / d φ ( 1 ) 1/d_{\varphi}^{(1)} 1/ d φ ( 1 ) d φ p e a k d_{\varphi}^{\mathrm{peak}} d φ peak d φ ( 1 ) < d φ p e a k d_{\varphi}^{(1)} < d_{\varphi}^{\mathrm{peak}} d φ ( 1 ) < d φ peak w / d → 0 w/d \to0 w / d → 0 ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ d φ ( 1 ) d_{\varphi}^{(1)} d φ ( 1 )
Regarding point 1: it is helpful to consider an axisymmetric ring with purely radial intensity profile I ( ρ , φ ) = I ( ρ ) I(\rho,\varphi)=I(\rho) I ( ρ , φ ) = I ( ρ )
I ( ρ ) = 1 2 π w 2 e − d 2 8 w 2 I 0 ( d ρ 2 w 2 ) e − ρ 2 2 w 2 , (83) I(\rho)=\frac{1}{2\pi w^{2}}e^{-\frac{d^{2}}{8w^{2}}}I_{0}\left(\frac{d\rho}{2w^{2}}\right)e^{-\frac{\rho^{2}}{2w^{2}}}, \tag*{(83)} I ( ρ ) = 2 π w 2 1 e − 8 w 2 d 2 I 0 ( 2 w 2 d ρ ) e − 2 w 2 ρ 2 , (83) with I 0 I_{0} I 0 I 0 ( x ) ≈ x → ∞ ( 2 π x ) − 1 / 2 e x I_{0}(x) \mathrel{\mathop{\approx}_{x\to\infty}} (2\pi x)^{-1/2}e^{x} I 0 ( x ) ≈ x → ∞ ( 2 π x ) − 1/2 e x d → ∞ d\to\infty d → ∞ w w w ρ = d / 2 \rho=d/2 ρ = d /2 V ( 0 ) = 1 V(0)=1 V ( 0 ) = 1
V ( u , φ ) = V ( u ) = J 0 ( π d u ) e − 2 π 2 w 2 u 2 . (84) V(u,\varphi)=V(u)=J_{0}(\pi d u)e^{-2\pi^{2}w^{2}u^{2}}. \tag*{(84)} V ( u , φ ) = V ( u ) = J 0 ( π d u ) e − 2 π 2 w 2 u 2 . (84) The key point is the following. If the ring is very thin, then there is a wide range of baselines u ≪ 1 / w u\ll1/w u ≪ 1/ w ∣ V ( u ) ∣ ∼ J 0 ( π d u ) |V(u)|\sim J_{0}(\pi d u) ∣ V ( u ) ∣ ∼ J 0 ( π d u ) (25) ) as soon as d u ≳ 1 ≫ w u du\gtrsim1\gg wu d u ≳ 1 ≫ w u (25) ), only multiplied by that exponential; in general, expect ∣ V ( u , φ ) ∣ |V(u,\varphi)| ∣ V ( u , φ ) ∣ (25) ) times e − c ( φ ) u p e^{-c(\varphi)u^{p}} e − c ( φ ) u p p = 2 p=2 p = 2
