A simple model for extracting astrophysics from black hole images

TL;DR

The paper tackles how to extract astrophysical information from EHT black hole images by linking observed features to a simple two-parameter accretion-disk emission model with $r_{\rm cut}$ and $\beta$. It uses Schwarzschild ray-tracing to generate disk images and computes the observed flux via $I_{\nu'} = \sum_i [g]^3 I_\nu$, incorporating Doppler and gravitational redshift, then calibrates image features to angular scales with $\theta_g = GM/(c^2 D)$. By comparing to both the original EHT and PRIMO-reanalyzed images, the authors derive 68% mass ranges ($M$ around $(4.3$–$6.6)\times 10^9\,M_\odot$) and constraints on the inner disk edge ($r_{\rm cut}$ between $2M$ and several $M$) and the emission slope $\beta$, highlighting that the central brightness depression need not mark the horizon. The results are consistent with stellar-dynamics mass estimates but show that current resolution cannot uniquely determine all astrophysical parameters; they also indicate that PRIMO constraints tend to push $r_{\rm cut}$ inward only to a few $M$, implying nonzero spin when interpreted physically. The work motivates future Kerr/GRMHD extensions, thicker disk models, and visibility-domain analyses to sharpen constraints from EHT data.

Abstract

The Event Horizon Telescope (EHT) is providing unprecedented high-resolution images of supermassive black holes. These images are fundamentally related to properties of the luminous accretion disks, since black holes themselves produce no light. We develop a simple prescription to relate observational features of black hole images to a toy model for the intensity profile of the associated accretion disk. We apply our model to the original EHT image of M87*, as well as to the reanalyzed image from the PRIMO algorithm, providing generic, simultaneous constraints on the mass of the black hole and properties of the accretion disk emission. While current images lack the resolution to confidently detect the photon ring, the consideration of multiple image parameters are found to contain enough information to provide constraints on the inner edge of the accretion disk along with the black hole mass. Using observed features of the original EHT image, we constrain the mass of M87* to be $6.6^{+1.2}_{-1.0}\times 10^9 M_\odot$ to 68$\%$ confidence, and find that emission may extend all the way to the black hole horizon. When instead using constraints from the PRIMO algorithm's image along with constraints on the brightness asymmetry provided by the original EHT analysis, we find M87*'s mass to be $ 6.4^{+0.7}_{-0.7}\times 10^9 M_\odot$ to 68$\%$ confidence, with the inner edge of the accretion disk between $3M$ and $5.3M$. Both analyses rule out an inner edge of the accretion disk coinciding with the innermost stable circular orbit for a Schwarzschild black hole. Furthermore, the narrow ring width reported in the PRIMO image also confidently rules out emission increasing all the way down to the black hole horizon. Further assumptions on the mass of M87* and connections between the accretion disk cutoff and physical radii allow for rudimentary black hole spin estimates.

Figures (7)

• Figure 1: Two-dimensional cross-section of photon trajectories traced back from a distant observer located to the far right for the image-plane slice of $\alpha = \pi/2$. The left and bottom axes are in Schwarzschild radial coordinates, while the top right are in impact parameter coordinates. The black circle indicates the event horizon of the black hole and the green line indicates an accretion disk inclined at an angle $\theta_0 = 17^\circ$. Grey lines indicate the photons that never hit the accretion disk, while blue lines are the direct emission from the front of the disk, hitting the disk once. Yellow lines originate from the back of the disk and hit the disk twice. Red lines indicate the photon ring and hit the disk 3 or more times.
• Figure 2: Theoretical observational images of a Schwarzschild black hole with an accretion disk inclined at an angle $\theta_0 = 17^\circ$. The emission profile is given by Equation \ref{['eq:Ir']} for cases $\beta = \{-5,-3,-1\}$ and $r_{\rm cut} = \{2,3,6\}$M. Notice the left-right brightness asymmetry due to the Doppler effect and the top-bottom geometric asymmetry due to the inclination. The lensing and photon rings are very narrow. When visualizing our images, we use EHT's perceptually uniform colormap afmhot_us from the ehtplot library ehtplotCode.
• Figure 3: (Top left) Theoretical image generated with a disk cutoff of $r_{\rm cut} = 3M$ and emission profile power law of $\beta = -6$. (Top right) Same image blurred with a Gaussian filter to the equivalent 20 $\mu \rm as$ resolution of the DIFMAP algorithm. (Bottom left) Same image instead blurred with a Butterworth filter to compare with the PRIMO algorithm. (Bottom right) Emission profile of a horizontal cross-section passing through the origin for the theoretical (black solid line), Gaussian-blurred (blue dashed line), and Butterworth-blurred (red dotted line) images. We see that the lensing ring and photon ring present in the theoretical image are now completely smoothed out with either filtering procedure.
• Figure 4: (Top) Median inferred masses for each emission profile to be consistent with EHT's image for M87$^*$ when comparing against the reported diameter and width observations of the original EHT analysis. Emission profile combinations that are not viable are indicated by the exclusion region (gray hatched region). (Bottom) Same as the top panel, but now additionally considering constraints on the brightness asymmetry and central brightness depression ratio. Marginalizing over $\beta$ and $r_{\rm cut}$ yields a 68$\%$ credible mass range of $6.6^{+1.2}_{-1.0}\times 10^9 M_\odot$ when considering all four image features. We include contours from the stellar-dynamics mass estimate of $(6.6\pm0.4)\times 10^9 M_\odot$ for M87* from StellarDyn, demonstrating that the stellar-dynamics mass estimate is consistent with the viable emission profiles from analyzing the EHT image features.
• Figure 5: (Top) Median inferred masses for each emission profile to be consistent with the reported diameter and width from the PRIMO image analysis for M87$^*$. Emission profile combinations that are not viable are indicated by the exclusion region (gray hatched region). (Bottom) Same as the top panel, but additionally incorporating the reported brightness asymmetry and central brightness depression ratio measurements from the DIFMAP algorithm from the original EHT analysis. We also include contours from the stellar-dynamics estimate of $(6.6\pm0.4)\times 10^9 M_\odot$ for M87* from StellarDyn (black solid and dashed lines). When marginalizing over $r_{\rm cut}$ and $\beta$ and considering all four image features, we find a 68$\%$ credible mass of $6.4^{+0.7}_{-0.7}\times 10^9 M_\odot$. Notice that PRIMO's images imply that the emission is cut off at $\geq 3M$ regardless of brightness constraints, and does not continue all the way to the event horizon.