Photon-ring astrometry without an inclination prior: joint spin and inclination inference from the image center shift, generalized beyond equatorial emission

A recently introduced photon-ring astrometry technique measures black-hole spin from the displacement between the centers of the direct image and the first lensed photon-ring sub-image, decomposed into components parallel and transverse to the projected spin axis. As published, the technique requires an externally supplied observer inclination (obtained by assuming the M87* large-scale jet is aligned with the spin axis), is calibrated only on equatorial-emission models and magnetically arrested disk (MAD) simulations, and loses precision when the prograde/retrograde sense of the accretion flow is unknown. The open problem is to develop and validate a self-contained inference that simultaneously constrains spin and inclination from the center-shift observables, together with a center-shift model valid for off-equatorial (SANE-like) emission and a treatment of the flow-direction degeneracy and the dominant measurement systematics.

Background

The Event Horizon Telescope (EHT) produced the first horizon-scale image of the M87* black hole in 2019, showing a bright emission ring around a central depression. In the Kerr spacetime describing a rotating black hole of mass $M$ and dimensionless spin $a_\ast$ , strongly lensed light can orbit the hole before escaping to the observer, so the observed image contains a sequence of increasingly demagnified and distorted 'sub-images': the direct ( $n=0$ ) image and lensed photon-ring sub-images ( $n=1,2,\ldots$ ) that accumulate near the critical curve, the limiting locus on the observer's screen toward which the sub-rings converge. The photon-ring geometry depends on $a_\ast$ and on the inclination $i$ , the angle between the black-hole spin axis and the line of sight, which makes resolving the ring a promising route to precision spin measurement. The Black Hole Explorer (BHEX) is a proposed space-based very-long-baseline interferometry (VLBI) mission designed to resolve the $n=1$ sub-image of M87* and Sgr A* and to measure features of horizon-scale images with microarcsecond precision. Gates, Chang, Held and Palumbo (2026) identified a new astrometric observable: the displacement between the center of the direct image and the center of the $n=1$ sub-image, extracted with the mF-ring model (a family of elliptical-annulus brightness fits used in EHT image modeling) rather than the intensity-weighted centroid, which prior work showed to be comparably sensitive to plasma speed as to spin. Assuming the observed large-scale M87* jet is aligned with the spin axis (which fixes the projected spin-axis direction on the sky), the displacement is decomposed into components $S_{\parallel}$ and $S_{\perp}$ parallel and transverse to the projected spin axis, both normalized by the measured $n=1$ sub-image diameter. A semi-analytic model of emission confined to a thin equatorial ring, validated against ray-traced general-relativistic magnetohydrodynamic (GRMHD) simulations, shows that $S_{\parallel}$ is set mainly by inclination and emission radius while $S_{\perp}$ is tightly correlated with inclination and spin — a signature of frame dragging, the dragging of spacetime by the rotating hole that displaces the lensed image off-axis. Calibrating on GRMHD simulations of magnetically arrested disks (MADs, in which magnetic flux accumulates on the hole and emission is concentrated near the equatorial plane) with ion-to-electron temperature ratio parameter $R_{\rm high}$ , the authors find that a relative astrometric resolution of $\lesssim 0.1\,\mu{\rm as}$ constrains the spin of M87* to better than 9% if the accretion flow is known prograde, 22% if retrograde, and 26% if the flow direction is undetermined. The authors themselves identify what remains open: the jet–spin alignment (equivalently, inclination-prior) assumption must be removed to apply the method to Sgr A* and other sources; the equatorial-ring analytic model fails for standard and normal evolution (SANE) simulations, whose emission extends off the equatorial plane and whose center shifts deviate drastically from the equatorial predictions; the prograde/retrograde degeneracy limits the achievable spin precision; and the spin-constraint numbers inherit astrophysical uncertainty from the representativeness of the simulation ensemble and the weighting over $R_{\rm high}$ , as well as from the snapshot-alignment pipeline used to time-average images and from the rigidity of the mF-ring extraction model. They explicitly promise a sequel developing simultaneous spin and inclination constraints from the center displacement.

Problem Statement

Develop and validate an extension of relative photon-ring astrometry in which the displacement between the center of the direct ( $n=0$ ) image and the center of the first photon-ring sub-image ( $n=1$ ) — decomposed into components parallel ( $S_{\parallel}$ ) and transverse ( $S_{\perp}$ ) to the projected spin axis and normalized by the measured $n=1$ sub-image diameter — is used to simultaneously constrain the black-hole spin $a_\ast$ and the observer inclination $i$ , without requiring an externally supplied inclination or jet–spin alignment prior. Because the jet–spin alignment assumption in the source technique fixes both the observer inclination and the projected spin-axis direction on the sky that defines the $S_{\parallel}$ / $S_{\perp}$ decomposition, that reference direction must likewise be determined within the self-contained inference from the horizon-scale image (or otherwise measured independently of the jet–spin alignment assumption) rather than imported as an external prior. The single overarching objective is a self-contained spin (and inclination) inference from photon-ring astrometry that remains valid for realistic accretion flows, encompassing: (i) an analytic or calibrated empirical model of the center shifts for emission that is not confined to the equatorial plane, since the equatorial-ring model is demonstrably ill-suited to SANE-like off-equatorial flows whose center shifts deviate drastically from its predictions; (ii) an inference procedure that outputs joint constraints on $(a_\ast, i)$ from the shift observables (or their visibility-domain counterparts) and demonstrably does not import an inclination prior; (iii) a treatment — breaking or marginalizing — of the prograde/retrograde degeneracy that otherwise degrades the achievable spin precision (9% prograde vs. 22% retrograde vs. 26% undetermined at $\lesssim 0.1\,\mu{\rm as}$ astrometric resolution); and (iv) quantification of the dominant systematics the source technique inherits on realistic simulated images: snapshot alignment and time-averaging, the rigidity of the geometric ring-extraction model, and the representativeness and $R_{\rm high}$ -weighting of the GRMHD simulation ensemble used for calibration. Validation on realistic GRMHD simulations spanning both MAD and SANE accretion states, spins of both signs, and a range of inclinations is part of the problem, since the equatorial-model calibration underpinning the published precision forecasts does not transfer to SANE-like emission. Components (i)–(iv), together with this GRMHD validation, are the joint parts of that objective.

References

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