Photon Ring Astrometry I: A Simple Spin Measurement Technique for High-Resolution Images of M87*

Abstract

The central supermassive black hole of the galaxy M87 is currently a target for precision spin measurement using high-resolution, horizon-scale imaging. Such observations aim to resolve the first lensed (${n}~{=}~{1}$) sub-image of the photon ring from the broader direct image. In this work, we identify a concrete observable -- the displacement between the centers of the ${n}~{=}~{1}$ photon-ring sub-image and the direct image -- and propose its use in a simple spin-measurement technique. Leveraging the assumption that the observed large-scale jet of M87 is aligned with the black-hole spin axis, we separate the relative position of the photon ring into components parallel and transverse to the projected spin axis, normalizing both components with respect to the measured diameter of the ${n}~{=}~{1}$ sub-image. We show that the parallel shift is primarily determined by inclination and emission radius, while the transverse shift is tightly correlated with inclination and spin. We demonstrate these effects both in a simple geometric model (to explain the underlying physics) and in GRMHD simulations with magnetically arrested disks (to provide realistic instantiations of the effect). We find that a relative astrometric resolution of ${\lesssim}~{0.1\;μ\rm{as}}$ is sufficient to constrain the spin to better than 9% if the accretion flow is prograde or 22% if the flow is retrograde. If the direction of the accretion flow is undetermined, the spin can be constrained to within 26%. More generally, this identifies relative photon ring astrometry as a promising method to constrain the underlying spacetime geometry and introduces a spin-constraint technique that does not rely on geometric modeling of the observed emission.

Figures (7)

• Figure 1: Horizon-scale images of an emitting accretion disk around spinning black holes in a GRMHD simulation with a magnetically arrested disk (MAD) and in a geometric model of an equatorial disk. We show the direct (${n}~{=}~{0}$, shown pink) and secondary (${n}~{=}~{1}$, shown in blue) sub-image of the disk and explicitly mark their centers (correspondingly colored points). The sub-image centers are defined as the centroids of a model-dependent sub-image contours (correspondingly colored dashed contours), allowing us to compare the relative location (center shift, solid white lines) of the direct and secondary sub-images. In all images, the projection of the black hole's spin axis is oriented vertically. Left: Image of a MAD in a GRMHD simulation. The contour of each disk sub-image (${n}~{=}~{0}$ and ${n}~{=}~{1}$) is given by the ellipse whose Gaussian convolution best fits the brightness profile. The center shift can be decomposed into components parallel and transverse to the projected spin axis (see $S_\parallel$ and $S_\perp$ on zoomed-in inset plot of sub-image centers). The center shift can be normalized by the major diameter of the ${n}~{=}~{1}$ sub-image ($\hat{d}_1$, shown as blue double arrow) to make it independent of the black hole's mass-to-distance ratio. The critical curve (dotted yellow line) lies inside the ${n}~{=}~{1}$ sub-image contour. Right: Images of a geometric accretion disk model in the equatorial plane spanning radii ${r_s}~{\in}~{[3r_g,7r_g]}$. The black-hole spins are ${a_*}~{=}~{0.1}$, $0.7$, and $0.998$ (columns: left, center, right). The observer inclinations are $\theta_o=5^\circ$, $25^\circ$, and $45^\circ$ (rows: bottom, middle, top). Here each sub-image constitutes a projection of the equatorial disk (hence having a flat brightness profile) with the associated contour taken to be the sub-image projection of the disk's average radius (${r_s}~{=}~{5r_g}$).
• Figure 2: Feasibility regions in which fixed shift values can be achieved. We show the feasibility regions for shift values $\mathopen{}\mathclose{\left\{0.01,0.02,\dots,0.16\right\}$ (colored according to the shift values shown in color bar) for the normalized parallel shift $S_\parallel$ marginalized over all spins ${0}}~{\leq}~{a_*}~{\leq}~{1}$ (left) and for the normalized transverse shift $S_\perp$ marginalized over source radii ${2}~{\leq}~{r_s/r_g}~{\leq}~{10}$ (right).
• Figure 3: Normalized parallel (top) and transverse (bottom) center shifts relative to the projection of the black-hole spin axis for equatorial rings of emission viewed at ${\theta_o}~{=}~{17^\circ}$ as a function of spin. The center shifts of a source ring restricted to radii ${r_s/r_g}~{\in}~{\mathopen{}\mathclose{\left[2,5\right]}}$ are confined to the blue regions, while shifts of a source ring restricted to radii ${r_s/r_g}~{\in}~{\mathopen{}\mathclose{\left[2,10\right]}}$ may lie in the full shaded region (including both blue and pink regions). The mean values for the normalized parallel shift (\ref{['eq:vERMFit']}) and normalized transverse shift (\ref{['eq:hERMFit']}) are shown as dashed lines (blue for ${r_s/r_g}~{\in}~{\mathopen{}\mathclose{\left[2,5\right]}}$ and purple for ${r_s/r_g}~{\in}~{\mathopen{}\mathclose{\left[2,10\right]}}$).
• Figure 4: Normalized center shifts in horizon-scale images of GRMHD MAD simulations viewed at ${\theta_o}~{=}~{17^\circ}$ from the black-hole spin axis. The black-hole images were created from five simulations of spins ${a_*}~{\in}~{\mathopen{}\mathclose{\left\{0,\pm0.5,\pm0.94\right\}}}$ using six ion-electron temperature ratio ${R_{\rm high}}~{\in}~{\mathopen{}\mathclose{\left\{1,10,20,40,80,160\right\}}}$. The center shifts plotted against spin are colored by ion-electron temperature ratio $R_{\rm high}$ (red, orange, green, dark blue, light blue values in ascending order). The center shifts plotted against $R_{\rm high}$ (right column) are colored by spin (red, orange, aqua, blue, violet for values in ascending order). The simulation shifts are compared to the mean shift values predicted by the equatorial ring model, $\mathopen{}\mathclose{\left\langle S_{\parallel,\rm erm} \right\rangle$\ref{['eq:vERMFit']} (in top panels) and $\mathopen{}\mathclose{\left\langle S_{\perp,\rm erm} \right\rangle$\ref{['eq:hERMFit']} (in bottom panels). The parallel shift prediction is shown in gray, while the transverse prediction shift is shown in gray in the left panel and in spin-dependent colors in the right panel. Dashed and dotted lines correspond to source-radius ranges ${r_s/r_g}~{\in}~{[2,10]}$ and ${r_s/r_g}~{\in}~{[2,5]}$, respectively. We depicted the best-fit models of the simulation shifts: the prograde fits ($S_{\perp,+}$\ref{['eq:hProFit']} and $S_{\parallel, +}$\ref{['eq:vProFit']}) are shown as light blue bands with black dashed lines marking mean and the retrograde fits ($S_{\parallel,-}$\ref{['eq:vRetFit']} and $S_{\perp, -}$\ref{['eq:hRetFit']}) are shown as pink bands with black dash-dotted marking mean.
• Figure 5: Spin-constraint precision $\sigma_{|a_*|}$ for M87* with prograde (blue region) and retrograde (pink region) accretion flows as a function of relative astrometric error $\sigma_{\rm r.a.}$. In the darker regions, the ${n}~{=}~{1}$ ring diameter $\hat{d}_1$ is taken to be coincident with that of the ${n}~{=}~{0}$ ring and the spread in $\sigma_{|a_*|,i}$ accounts only for variation over all spin magnitudes ${0}~{\leq}~{|a_*|}~{\leq}~{1}$. In the lighter regions, the ${n}~{=}~{1}$ diameter is taken to lie anywhere within the ${n}~{=}~{0}$ ring. For comparison, the standard deviation corresponding to a flat distribution in spin is also shown (black-dashed line).