Measuring Spin from Relative Photon-ring Sizes

The images of M87* on horizon scales closely match the theoretical expectations from accreting supermassive black holes, providing the best evidence thus far for the identification of black holes with active galactic nuclei (Event Horizon Telescope Collaboration et al. 2019a, 2019b, 2019c, 2019d, 2019e, 2019f, hereafter Papers I‒VI). The size of the annular emission region provided a mass estimate for M87* of (6.5 ± 0.7) × 10⁹ M_⊙ from the dynamics of radio waves on scales of the photon orbit, the near-horizon spherical region associated with photons that execute unstable closed orbits (Paper VI). This mass is consistent with that implied by stellar motions at ≳50 pc scales, (6.6 ± 0.4) × 10⁹ M_⊙ (Gebhardt et al. 2011), confirming that gravity operates as predicted by general relativity about supermassive black holes on scales ranging from the event horizon to interstellar distances, covering more than five orders of magnitude.

The measurement of the mass of M87* by the Event Horizon Telescope (EHT) was predicated on the identification of strongly lensed emission with the ring-like structures that surround the silhouette of the horizon, the black hole "shadow" (Hilbert 1917; Luminet 1979; Falcke et al. 2000; Broderick & Loeb 2009). The bulk of this emission is associated with two components: (1) diffuse emission formed by those photons that follow the most direct path to the observer and (2) a ring-like structure dominated by those photons that detour around the back side of the black hole and are therefore more strongly lensed. These components—which we enumerate as n = 0 and n = 1—represent the first two in an infinite series of lensed images, corresponding to the primary and secondary lensed images. The n ≥ 1 components are now commonly referred to as "photon rings" associated with successively tighter photon trajectories about the black hole (see, e.g., Darwin 1959; Luminet 1979; Johnson et al. 2020, and references therein). The n = ∞, or asymptotic, photon ring defines the boundary of the shadow. Examples of the photon trajectories that produce direct emission (n = 0) and the n = 1 and n = 2 photon rings are shown in Figure 1.

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When individual photon rings are spatially unresolved, the relationship between their combined emission and the size of the shadow is dependent on the location of the emission region. Thus, the mass measurements presented in Paper VI rely on calibrations that employed numerical simulations of the emission region about the black hole. The astrophysical uncertainties surrounding the structure of the emission region dominate the systematic uncertainty on the mass.

Measurements of the black hole spin are similarly complicated. Many authors have noted the modification in the shape of the shadow with spin for oblique observers and the attendant possibility for a mass-independent spin measurement (Bardeen 1973; Chandrasekhar 1983; Takahashi 2004). However, for the nearly polar-viewing geometry of M87*, with the spin of the black hole at most 20° to the line of sight (Walker et al. 2018; Paper V), there is nearly no deformation in the photon-ring shape, rendering such shape measurements exceedingly difficult. Spin also produces a roughly 6% variation in the size of the shadow for polar observers, but the ≳ 10% precision of the current best alternative mass measurement precludes this avenue as well. Both of these limitations prevent precision tests of general relativity at present using M87*.

However, two fortuitous developments create an opportunity to exploit the time-variability of M87* to overcome the dominant systematic uncertainties in measuring mass, spin, and the emission region. The first is that novel techniques (Broderick et al. 2020a, 2020b) and future instruments (Doeleman et al. 2019) promise to disentangle the direct emission component (n = 0) and n = 1 photon ring, and the possibility of space-based stations, with their associated long baselines, may lead to the direct detection of the n = 2 photon ring (Johnson et al. 2020).

The second is that the emission within the general relativistic magnetohydrodynamic (GRMHD) models of M87* tend to be dominated by that from the equatorial plane (Paper V), providing a substantial simplification of the geometry of the emission process.

Therefore, encoded within the photon rings and their relationships are the properties of both the emission region and the spacetime (see, e.g., Figure 10 of Broderick & Loeb 2006a). Here we show that with multiple measurements of the photon rings of M87*, made when the image is dominated by emission at different radii, it is possible to break the degeneracy between where the emission occurs and how large the photon rings appear. This procedure is fundamentally a lensing measurement, similar to those described by Broderick & Loeb (2006a) and Tiede et al. (2020), that leverages the evolving relationship between the different image components.

The analysis we present here should also be understood to exist within the context of more general modeling exercises. For M87*, viewed very nearly along the polar axis and with the strong priors that the emission is optically thin and dominated by emission near the equatorial plane (e.g., MAD models), the procedure described here distills the key observables and their associated physical implications for the spacetime. However, we neglect the additional information available in the large-scale flux distribution and the implications that jet kinematics has for this. Similarly, we ignore significantly non-equatorial emission regions, e.g., from the jet funnel wall or within the jet itself. Thus, in this sense, the analysis presented here is a subset of the joint emission region‒spacetime modeling that is possible with semianalytic analyses (e.g., Broderick et al. 2014, 2016; Johannsen et al. 2016; Tiede et al. 2020) and already implemented within Themis (Broderick et al. 2020a).

In Section 2 we describe the measurement and the underlying lensing properties that make it possible to break the degeneracies between the mass, spin, and emission region. Section 3 contains the implications for detection of the n = 1 or n = 2 photon ring for mass estimates. Section 4 presents the spin and mass measurements possible for a toy problem in which the emission is confined to a ring within the equatorial plane. Section 5 explores the impact of a radial emission distribution and finite emission region height. Section 6 describes two tests of general relativity made possible by observing multiple photon rings. Finally, conclusions are collected in Section 7.

In addition to presuming that the black hole is well described by the Kerr spacetime, we make three additional underlying assumptions:

• 1.  
  The emission is confined to the equatorial plane.
• 2.  
  The emission is axisymmetric, i.e., produced by rings, and reaches a maximum at a single radius.
• 3.  
  The observer is positioned at infinity along the polar axis.

The first two assumptions are well justified for GRMHD models and especially so for models in the MAD state. The large magnetic fluxes captured by the black hole produce a high-pressure funnel region that compresses even virial accretion flows to disk heights of h ∼ 0.1r, where r is the Boyer–Lindquist radius (see Paper V, and references therein). The accreting gas is strongly differentially rotating, rapidly shearing features into ring structures on timescales short in comparison to the accretion time. ⁷ We will further idealize the emission as arising from a single equatorial ring; in Section 5 we show that the radius of the emission-ring idealization may be replaced with the location of the emission maximum for models with extended emission. The third assumption is well justified by the near-polar-viewing geometry of M87*.

Given the above assumptions, the direct emission ⁸ and subsequent lensed images form discrete, concentric circular rings, i.e., what are commonly called photon rings. The observed radii of these rings are functions of the spacetime and location of the emission region, as illustrated in Figure 1. We begin by exploring the latter dependence.

Consider the emission from a single, azimuthally symmetric ring located in the equatorial plane with radius r_em. All emission components are constrained to connect with the same radial position in the equatorial plane, which induces a relationship between them (see Figure 1). The n = 0 ring is associated with those photon trajectories that connect with the emitting ring without executing any orbits about the black hole prior to reaching the observer at infinity. The n = 1 photon ring is composed of photons that execute a half orbit, the n = 2 photon ring those that undergo a full orbit, etc. The n = ∞ photon ring defines the boundary of the black hole shadow (Bardeen 1973).

Though the radius of each photon ring depends on the black hole spin in a nontrivial fashion, the black hole mass simply acts as an overall scaling factor. We thus have the decomposition

$$\begin{eqnarray}\begin{array}{rcl}{\theta }_{n=0} & = & ({GM}/{c}^{2}D){\vartheta }_{n=0}(a,{r}_{\mathrm{em}}{c}^{2}/{GM})\\ {\theta }_{n=1} & = & ({GM}/{c}^{2}D){\vartheta }_{n=1}(a,{r}_{\mathrm{em}}{c}^{2}/{GM})\\ {\theta }_{n=2} & = & ({GM}/{c}^{2}D){\vartheta }_{n=2}(a,{r}_{\mathrm{em}}{c}^{2}/{GM}),\end{array}\end{eqnarray} \tag{ 1 }$$

where D is the distance to the source and GM/c² D is the angular size of the gravitational radius, ⁹ and the ϑ are dimensionless functions that encode the dependence on a and r_em, where the latter is explicitly measured in gravitational radii. We compute these functions numerically via the procedure described in Appendix A.

A dependence on mass, spin, and emission location is also retained, and is perhaps more observationally accessible, through relative measurements of different-order photon rings. The top panel of Figure 2 shows θ_n=1 plotted against θ_n=0 for all r_em and M and for three different fixed values of a. While the horizon provides a natural minimum r_em for all orders of photon ring, θ_n=0 is unbounded from above, and thus even after fixing a (and effectively fixing M by choice of units) the observed photon-ring radii are a one-dimensional family traversed by r_em.

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These families are not, however, degenerate. In the top panel of Figure 2 we show versions of the a = 0.75 and a = 0.998 curves that have had their masses rescaled in such a way as to match the a = 0 curve at θ_n=0 = 5.5GM/c² D (note that the underlying r_em at the intersection point do not match). These rescaled curves are shown by the dashed lines, and they are clearly discrepant from the a = 0 curve for all other values of θ_n=0. That is, the measurement of θ_n=0 and θ_n=1 from emission at two different r_em is sufficient to break the degeneracy between M and a.

The measurement of θ_n=2 provides yet another independent constraint on M and a. The bottom panel of Figure 2 shows θ_n=2 as a function of θ_n=1 for the same three values of a as in the top panel. Because the effect of spin on the functional relationships between the different ring sizes is not a simple scaling, the θ_n=1 and θ_n=2 obtained by rescaling the mass do not intersect in the bottom panel of Figure 2. Therefore, the mass and spin may be disentangled if the n = 2 photon ring can be measured.

In the following sections we explore the ability to leverage these overconstrained photon-ring measurements to produce corresponding constraints on M and a.

For a given black hole mass and spin, and under the assumptions specified in Section 2, the radius of a photon ring depends monotonically on the location of its emitting region within the equatorial plane, r_em. All orders of photon ring have a minimum radius imposed by the equatorial location of the black hole horizon. The n = 0 ring radius is unbounded from above, but for photon rings (n ≥ 1), there exists a maximum radius because all contributing rays are necessarily strongly lensed. For example, the maximum n = 1 photon-ring radius for a spin-zero black hole is simply the impact parameter for 90° photon scattering, corresponding to photon trajectories incoming from r_em → ∞. Across all values of spin and r_em, the n = 1 photon ring is bounded on 4.30 < ϑ_n=1 < 6.17 and the n = 2 photon ring is bounded on 4.77 < ϑ_n=2 < 5.22 (see Figure 3); these bounds can be compared against the minimum and maximum radii for the shadow itself of 4.83 < ϑ_n=∞ < 5.20.

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Because each photon-ring radius is related to the black hole mass by a simple scaling (see Equation (1)), the boundedness of n > 0 photon rings affords any measurement of such a photon ring a corresponding hard constraint on the black hole mass. The size of the n = 0 ring is unconstrained—i.e., larger values of r_em correspond to larger rings, without bound—so any attempt to constrain the black hole mass using a measurement of the n = 0 ring must rely on some prior expectation of plausible values for r_em. ¹⁰ But for measurements of photon rings with n > 0, any "astrophysical uncertainty" associated with an unknown or poorly known r_em can introduce only a finite, bounded bias on the corresponding black hole mass. These limits are shown for the n = 1 and n = 2 photon rings in the right-hand panels of Figure 3, for which the mass is bounded on 0.162θ_n=1 < GM/c² D < 0.232θ_n=1 and 0.191θ_n=2 < GM/c² D < 0.209θ_n=2, respectively. ¹¹ By n = 2, the dominant systematic impacting the translation between photon-ring radius and black hole mass is no longer r_em, but rather the black hole spin.

The degeneracy between M and D is common to all static angular measurements of the near-horizon emission. In principle, this degeneracy may be broken by the detection and coherent characterization of time varying structures (see, e.g., Broderick & Loeb 2005, 2006a; Moriyama et al. 2019; Tiede et al. 2020; Wong 2021). In what follows, however, we will focus only on what may be inferred from measurements of emission structures that are stationary throughout an observing night, ignoring any potential additional temporal information.

The horizon-scale structure of M87* is variable. This is seen via the presence of month-timescale ejections (see, e.g., Ly et al. 2007; Hada et al. 2016; Walker et al. 2016) and the variations across the 2017 EHT campaign (Paper I, Paper IV, Paper VI). This variability is anticipated by the variable horizon-scale structures seen in GRMHD simulations (see, e.g., Davis & Tchekhovskoy 2020; Ripperda et al. 2020). Insofar as this variability can manifest as a change in the peak emission radius with time, it provides an opportunity to break the spin‒mass degeneracy in Figure 2.

We begin again with the idealized case in which we imagine having measurements of θ_n=0 and θ_n=1 from an emitting ring located at some equatorial radius r_em. A single pair {θ_n=0, θ_n=1} of such measurements produces a degenerate joint constraint on M and a, traversed by (the unknown) r_em; Figure 4 illustrates this joint constraint for two choices of the underlying r_em. The widths of the bands in Figure 4 are set almost entirely by the uncertainty in the θ_n=1 measurement, and they are only weakly sensitive to the uncertainty of θ_n=0. For the purposes of Figure 4 we have assumed ring radius measurement uncertainties of 0.25% and 0.5% for the narrow and wide bands, respectively. ¹²

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The degeneracy between M and a may be broken by measuring {θ_n=0, θ_n=1} pairs originating from different r_em. We will assume that these observations occur at different times, and therefore are uniquely identifiable, i.e., each {θ_n=0, θ_n=1} pair may be determined. However, should multiple emission rings be present, corresponding to multiple emission maxima (see Section 5), these may produce sufficient information to break the M–a degeneracy in a single observation epoch.

An example of this degeneracy breaking is illustrated in Figure 4, which shows two bands corresponding to photon rings emitted from Boyer–Lindquist radii of 2GM/c² and 6GM/c² around a black hole with a = 0.85. ¹³ The overlap region between these two bands covers the true mass and spin values. The degree to which two such measurements produce useful mass or spin constraints depends primarily on the uncertainties in the measurements of θ_n=1. It depends further on the difference in r_em between the two observations; two measurements with the same emission location are naturally degenerate with each other and provide no leverage with which to separate M and a.

A practical demonstration of the constraints on mass and spin that can be obtained from a pair of {θ_n=0, θ_n=1} measurements with 0.25% Gaussian uncertainties is shown in Figure 5. From these synthetic measurements we constructed a Gaussian likelihood and sampled the corresponding posterior using the ensemble Markov Chain Monte Carlo (MCMC) method provided by the emcee Python package (Foreman-Mackey et al. 2013). We use 64 independent walkers and run for 10⁵ steps, discarding the first half of each chain. Explorations with fewer walkers and steps indicate that by this time the MCMC chains are well converged. We see that the posterior matches well with the general shape of the overlap region in Figure 4.

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If θ₂ can be measured in addition to θ₁ and θ₀, then emission from a single r_em can be used to break the degeneracy between a and M. An example of such a joint constraint is shown in Figure 6. The degree to which the degeneracy is broken is very sensitive to the precision with which θ₂ can be measured. For the purposes of Figure 6 we have assumed a measurement precision of 0.05%. ¹⁴

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We note that because the mass manifests as an overall, omnipresent scale, the measurements of all other quantities are dependent only on relative angular measurements. Therefore, the black hole spin and location of the emission regions (measured in gravitational radii) can be estimated with much greater accuracies than would be implied by the systematic uncertainties associated with M, e.g., the uncertain distance to M87*.

The demonstrations in the previous section are highly idealized, associated with a single ring-like emission region confined to the equatorial plane. More commonly, the emission is expected to be distributed radially throughout the equatorial plane and vertically off of the equatorial plane, and the emission within the photon rings will be modestly optically thick. Even subject to the remaining assumption that the viewer is polar, these complications present the possibility of biases in the ring positions and corresponding spacetime-parameter estimates.

To assess these biases we consider two images from slim radiatively inefficient accretion flow simulations about an a = 0.8 black hole, similar to those described in Broderick & Loeb (2006b). The mass, distance, and total flux are chosen to be consistent with those of M87* (see, e.g., Paper VI). The emission is generated via synchrotron from thermal and nonthermal electron populations. The temperature is virial, and both electron populations are power laws in radius and Gaussian in height, with h = 0.1r, roughly consistent with the MAD models in GRMHD simulations (Narayan et al. 2003; McKinney et al. 2012, Paper V). In addition, an internal cutoff in the electron densities was imposed to modify the typical location of the emission region. That is, for each electron species s, the density is assumed to be

$$\begin{eqnarray}{n}_{s}\propto {r}^{-{\alpha }_{s}}{e}^{-{z}^{2}/2{h}^{2}}\left\{\begin{array}{ll}{e}^{-{\left(r-{r}_{\mathrm{cut}}\right)}^{2}/2} & r\lt {r}_{\mathrm{cut}}\\ 1 & \mathrm{otherwise}.\end{array}\right.\end{eqnarray} \tag{ 2 }$$

The orbital motion is assumed to be on circular geodesics outside of the innermost stable circular orbit, and on plunging ballistic inside with the specific energy and specific angular momentum set by that at the ISCO. The magnetic field was set by choosing a fixed plasma β = 10. The resulting images are shown in Figure 7 when r_cut = 6GM/c² (top) and 2GM/c² (bottom). The radiative transfer computation includes synchrotron self-absorption and follows the full complement of Stokes parameters (Broderick & Blandford 2004). Optical depths at the peak of the n = 0 ring and n = 1 photon ring are τ ≈ 1, and in the n = 2 photon ring they can reach as high as τ ≈ 2.

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The radial flux profiles, shown in Figure 8, clearly illustrate the distributed emission. It is asymmetric about the n = 0 peak, a direct consequence of the asymmetric radial distribution of the emitting electrons. In addition, the subsequent higher-order photon rings are superimposed upon the emission of the n = 0 ring, and in the case of the smaller radial cutoff n = 1, photon ring(s).

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We identify the location of the photon-ring peaks from the radial flux profiles via an iterative procedure that begins with n = 0 ring and successively attempts to remove the lowest-order photon-ring emission. This effectively removes the biases in the peak locations due to the overlap of the various image components. More detail on this procedure is presented in Appendix B. The resulting estimates of the n = 0 ring and n = 1 and n = 2 photon-ring positions are listed in Table 1.

We also list the fraction of the total flux contained within each ring component, f_n , in Table 1. For the cases considered here, the n = 1 photon ring contains between 7.5% and 17.3% of the total flux depending on the location of the emission region. The fractional flux drops precipitously as the order of the photon ring increases. However, we note that the flux in the photon rings is strongly dependent on the dynamical state of the material and could be substantially brighter in the presence of large radial motion.

For comparison, we also determine a set of idealized photon-ring position estimates. We do this by determining the equatorial emission radius associated with the measured θ_n=0 (listed in the second column of Table 1) and then infer the locations of the remaining photon rings assuming all of the emission is confined to a ring in the equatorial plane with this radius. These estimates and the fractional difference are also listed in the fifth and sixth columns of Table 1. The identification of the flux profile peaks with the position of the photon rings from an equatorial emission ring at some radius appears to be an excellent approximation.

We perform the same demonstration analyses as in Section 4. The implication for mass and spin of measuring (θ_n=0, θ_n=1) to 0.25% on two epochs with different r_em is shown in Figure 9. This is comparable to Figure 5, albeit at slightly different choices of the emission radii. When (θ_n=0, θ_n=1, θ_n=2) are measured to 0.05% in a single epoch, the constraints on mass and spin are shown in Figure 10.

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For the simplified simulated images explored in this section, we have ignored a number of additional potential complications. Our simulations reach only modest optical depths (τ ≲ 2) in the images; much higher optical depths may modify the relationship between θ_n=1,2 and θ_n=0. We have also considered only face-on systems; nonzero inclination and the consequent breaking of axisymmetry would make the identification of the ring radii more difficult. Both of these limitations are most naturally addressed through more comprehensive forward-modeling of the image structure (e.g., Broderick et al. 2016). Nevertheless, it is clear that emission regions extended in radius and modestly in height do not provide a significant impediment to spin and mass measurements.

Careful measurement of the asymptotic photon ring provides a number of opportunities to probe general relativity more generally (Takahashi 2004; Johannsen & Psaltis 2010; Psaltis et al. 2015; Johannsen et al. 2016; Medeiros et al. 2020). These have been applied to the existing M87* EHT images (Psaltis et al. 2020, though see Völkel et al. 2021; Kocherlakota et al. 2021). These methods require either a careful measurement of the photon-ring shape, additional ancillary measurements of mass and spin, or are otherwise limited in their implications. Here we describe a possible precision constraint produced from the radio interferometry alone.

Where two independent measurements of {θ_n=0, θ_n=1} from distinct emission locations during different epochs provide a measurement of mass and spin, any additional constraint in the M–a plane would render the problem overconstrained and would therefore permit a test of general relativity. While this supplementary measurement may be the result of ancillary observations (e.g., mass estimates from stellar orbits), it need not be: a measurement during a third epoch with a differing r_em is sufficient.

In such a test, the constancy of the underlying spacetime parameters, spin and mass, implies that all constraints in the M–a plane must be consistent, i.e., all bands in the M–a plane must intersect at a single point as long as the spacetime is described by the Kerr metric. This test is conceptually identical to the tests of general relativity presented in pulsar studies (see, e.g., Kramer et al. 2006), where the various constraints on the masses of the members of the binary are overconstrained by various metric-dependent observables. An example multimeasurement test is shown in Figure 11 (left panel), which shows three hypothetical observations at emission radii ranging from 2GM/c² to 6GM/c² for a black hole with a = 0.80. As anticipated, all bands converge at a single point, both improving the estimates of M and a and demonstrating the self-consistency of the Kerr metric.

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In contrast, we show two additional simulated comparisons, with {θ_n=0, θ_n=1} measured from images in a manner identical to that in Section 5 but generated in a Kerr–Newman spacetime with charges Q = 0.03 and Q = 0.25 in the center and right panels of Figure 11, respectively. In practice, Kerr–Newman is a proxy for any non-Kerr metric that permits a well-defined and simple demonstration of the impact of deviations from Kerr of the photon-ring properties; practical applications to specific metrics would require constructing the low-order analogs of the Kerr photon rings. Nevertheless, we note that Q = 0.25 could be excluded by observations of similar quality to those that can be performed in the near future.

If the n = 2 photon ring can be detected and θ_n=2 measured, it is possible to test general relativity in a pair of observations. Moreover, the detection of the n = 2 photon ring implies a significantly improved precision in the measurement of θ_n=2 over that required to measure θ_n=1. Each pair of photon-ring radii implies a joint constraint in the M–a plane, yielding three such bands for each measurement. However, only two of these constraints are typically distinct; those arising from θ_n=1 and θ_n=2 are nearly degenerate with those due to θ_n=0 and θ_n=2. A second epoch of observations during a period with differing r_em produces a second, independent measurement of a and M associated with the two additional constraints As before, the constancy of the spacetime requires that these are consistent with the prior values, i.e., all bands must cross at a single location.

Figure 12 shows example constraints in the M–a plane when {θ_n=0, θ_n=1, θ_n=2} are measured from images. As before, when the spacetime is Kerr, all constraints intersect at a single mass and spin (a = 0.8). Even small deviations (Q = 0.01) could be excluded in this instance, though the precision with which this could be done depends sensitively on the precision of the measurements of θ_n=2 and θ_n=1.

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To lowest order, the additional spacetime structure encoded in Q modifies the relationship between spin and the quadrupole moment that the methods presented here exploit. These lensing-based signatures are, in essence, what semianalytic RIAF models are probing (see, e.g., Broderick et al. 2014; Johannsen et al. 2016). Methods relying on the shadow shape invariably face the unfortunate effect of partial cancellation of spin and quadrupole effects in the lensed image seen at infinity, leaving only mild sensitivity on black hole spin and, by extension, deviations from Kerr. Like shape measurements, tests based on multiple photon-ring size measurements are independent of the overall mass scale, simplifying their comparison. In addition, the impact of differing systematic uncertainties is limited, because they employ only high-resolution radio imaging data, instead of relying on ancillary measurements (such as stellar dynamical masses)—the precision of the gravitational test depends solely on the precision of the ring size measurements.

While we focus on M87* here, we note that in systems with significant auxiliary independent measurements of mass and spin, it is possible to supplement the constraints presented here directly. This is the case, e.g., for Sgr A* for which independent high-precision mass measurements exist (Boehle et al. 2016; Gravity Collaboration et al. 2019). Such posterior distributions are narrow in mass but (currently) uninformative in spin and are therefore partially complementary to the contours discussed here. As a result, the mass measurements can be improved and help to further constrain spin and test general relativity beyond what is shown in Figures 11 and 12. The situation would further improve with the detection of a pulsar orbiting a supermassive black hole where mass and perhaps spin could be reconstructed with better accuracy than Boehle et al. (2016) and Gravity Collaboration et al. (2019) (Wex & Kramer 2020).

In this paper we have presented a number of idealized analyses demonstrating that relative size measurements for different orders of photon-ring permits can be used to place constraints on black hole mass and spin without having to resolve the shape of the asymptotic photon ring. Specifically, we have focused on the opportunities afforded by multiepoch measurements of θ_n=0 and θ_n=1—such as are expected to be possible using multiple EHT observing campaigns—or from a single-epoch measurement of θ_n=0, θ_n=1, and θ_n=2, such as may be obtainable using future space missions. Both of these methods leverage the relationship between the sizes of the different-order photon rings to eliminate the degeneracy that is otherwise present between the location of the emitting region (e.g., orbital radius of the emitting material) and the spacetime properties (i.e., black hole mass and spin).

These are fundamentally lensing measurements and are similar in spirit to simultaneous reconstructions of dynamical and stationary parameterized emission regions (e.g., Broderick et al. 2016; Tiede et al. 2020). These are significantly simplified in the case of M87* by the near-polar-viewing angle and the approximately azimuthally symmetric, near-equatorial emission for MAD-type accretion flows Paper V.

Just two measurement epochs of the size of the n = 0 ring and n = 1 photon ring, associated with epochs in which the emission region has significantly varied, is sufficient to produce a joint mass and spin measurement. This is now possible with new image analysis methods (e.g., Broderick et al. 2020b). The accuracy of the spin measurement is strongly dependent on those of the angular radii of the n = 1 photon ring, requiring a precision better than 1% in practice. Assuming that this precision can be reached in practice, a joint analysis of the 2017 and 2018 EHT observations campaign may present the first opportunity to use horizon-resolving images to measure a black hole spin.

Observations with proposed future Earth-based arrays (e.g., Doeleman et al. 2019; Raymond et al. 2021) will provide much higher signal-to-noise ratios and much better baseline coverage, yielding correspondingly better estimates of the n = 0 ring and n = 1 photon-ring sizes. As a result, these future experiments hold great promise to repeatedly measure the black hole spin and mass in M87* with high accuracy. Similarly, future space-based millimeter interferometers (e.g., Johnson et al. 2020) may enable the first detection of the n = 2 photon ring, and thus a single-epoch spin measurement, and extend the science motivation for doing so. This requires a similar effective imaging resolution to that required to unambiguously detect the n = 2 photon ring.

This immediately raises the prospect of high-precision tests of general relativity. A set of measurements of θ_n=0 and θ_n=1 over more than two epochs with differing emission locations is overconstrained. That is, the bands of all must intersect at a single position in the M–a plane. This is distinct from prior constraints on general relativity in two ways. First, it is based solely on high-resolution radio imaging data. It does not require comparison with an ancillary mass estimate, i.e., that from the stellar dynamics observations at much larger radii. As a result, the precision of the measurement is solely dependent on the precision of the ring size measurements. Second, for the same reason that the spin measurement is independent of the systematic uncertainties on the mass described in Paper VI, the comparison is similarly devoid of these systematic uncertainties. In this way, M87* may generate high-precision tests of general relativity in the near future.

Additional complications, e.g., inclination, optical depth, kinematics, asymmetry, etc., may be naturally included via more complete semianalytical model comparisons in which the gravitational lensing and source emission are modeled directly (see, e.g., Broderick & Loeb 2006b; Broderick et al. 2016). Such models have already been deployed on Sgr A^* in multiple analyses that recover spin and disk inclination (Broderick et al. 2009; Huang et al. 2009; Broderick et al. 2011, 2016). State-of-the-art comparison frameworks include natively implemented examples of these sorts of physically motivated parameterized emission models, e.g., that in Themis (Broderick et al. 2020a). The analysis presented here is a particularly simple example of such a comparison, selecting a handful of salient features based on the simplicity of their physical interpretation. Therefore, its success provides a reason for significant optimism regarding the prospects of these more detailed, though more expensive, modeling schemes.

We thank Michael D. Johnson and Thomas Bronzwaer for helpful comments. This work was supported in part by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade. A.E.B. thanks the Delaney Family for their generous financial support via the Delaney Family John A. Wheeler Chair at Perimeter Institute. A.E.B. and P.T. receive additional financial support from the Natural Sciences and Engineering Research Council of Canada through a Discovery Grant. R.G. receives additional support from the ERC synergy grant BlackHoleCam: Imaging the Event Horizon of Black Holes (grant No. 610058). D.W.P. is supported by the NSF through grants AST-1952099, AST-1935980, AST-1828513, and AST-1440254; by the Gordon and Betty Moore Foundation through grant GBMF-5278; and in part by the black hole Initiative at Harvard University, which is funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation to Harvard University.

The relationships between photon-ring radii when n ≫ 1 are derived in Johnson et al. (2020). However, these asymptotic formulae are unsuitable for the low-n photon rings that are expected to be detected in the coming years. Therefore, we numerically construct the mapping ϑ_n=0,1,2(a, r_em). These functions are shown in Figure 13.

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For a number of values of a, we begin by generating a collection of null geodesics propagated backward in time from a set of adaptively spaced radial positions on a distant screen, located at a Boyer–Lindquist radius of 2 × 10⁴ GM/c² and centered on the polar axis (see the upper part of Figure 1) using the radiative transfer and ray-tracing code VRT2 (Broderick & Blandford 2004; see Gold et al. 2020 for performance and comparison with alternate codes). The resulting trajectories are inspected for crossings of the equatorial plane, at which the radial coordinate position is recorded, labeling each crossing by its order (setting n = 0 for the last crossing, n = 1 for the penultimate crossing, etc.). That is, a tabulated set of r_em,n (a, ϑ) is generated. Finally, this table is numerically inverted to produce a set of tabulated ϑ_n (a, r_em). Repeating this procedure for many a produces a set of two-dimensional tables from which values at arbitrary a and r_em are obtained by interpolation.

Given a radial flux profile, I(θ), we seek to reconstruct the θ_n=0,1,2. This is modestly complicated by the overlap between photon-ring fluxes. Therefore, we implement an iterative procedure in which we identify the contribution to the flux profile of the lowest remaining order photon ring and remove it. Here we describe this process in detail.

The procedure begins with the total I(θ) and a list of window sizes ${({\theta }_{\min },{\theta }_{\max })}_{n}$, which encompass the associated order of photon rings. We then construct an approximation of the lowest-order photon-ring profile:

$$\begin{eqnarray}&&\begin{array}{c}{I}_{{\rm{ring}},n}(\theta )\\ =\left\{\begin{array}{ll}I(\theta ) & \theta \lt {\theta }_{max,n+1}\,{\rm{and}}\,\theta \gt {\theta }_{min,n+1}\\ {\rm{cubic}}\,{\rm{spline}}\,{\rm{interpolation}} & {\rm{otherwise}},\end{array}\right.\end{array}\end{eqnarray} \tag{ B1 }$$

where the spline is performed on the values of I(θ) outside the window. The relevant photon-ring radius is set to the maximum of I(θ). Finally, we set I(θ) → I(θ) − I_ring,n and iterate the process. This is illustrated in Figure 14.

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The accuracy of this procedure depends on the degree of overlap and the accuracy of the cubic spline interpolations. The latter is dependent on the widths of the photon rings (narrow interpolation regions are more accurate than broad regions) and the relative location of the rings (where rings are on top of each other, the cubic spline approximation becomes less well defined).