Measuring the Black Hole and Accretion Parameters of Sagittarius A* from EHT Observations using a Semi-Analytic Model

TL;DR

Sgr A* presents rapid, intraday variability that complicates traditional VLBI imaging. The authors develop a snapshot-based Bayesian framework that fits a fast, semi-analytic dual-cone emission model to sparse visibilities and then stacks snapshot posteriors in a hierarchical model to recover both the time-averaged structure and variability. They find that, for 2017 EHT data, the black hole spin $a_{*}$ and magnetic-field parameters are poorly constrained, while the observer inclination $\theta_{\rm o}$ is near face-on, the emission peak lies near the horizon ($R_{\rm peak}$), and the spin position angle is relatively well constrained; a significant near-horizon emission component is inferred. Tests on synthetic GRMHD data show no strong biases in inferring $a_{*}$, $\theta_{\rm o}$, or $p.a.$, supporting the method's robustness, though model misspecification and axisymmetry remain important caveats. Overall, the approach demonstrates a viable path to extract BH and accretion physics from sparse mm-VLBI data and provides a framework for applying to future EHT and ngEHT datasets, including potential polarization and multi-component emission modeling.

Abstract

The Event Horizon Telescope (EHT) Collaboration produced the first image of the apparent shadow of the central black hole of Sagittarius\,A$^*$ (\sgra). \sgra source structure varies significantly on timescales shorter than the duration of an observation, preventing improved data coverage through Earth rotation aperture synthesis. This rapid variability provides the opportunity to quantify intrinsic variability and separate time-variable emission features from stable signatures of strong gravity and the accretion environment. To infer the properties \sgra and its surrounding accretion flow, we perform Bayesian inference on a series of EHT data segments (``snapshots''). We directly fit parameters of a semi-analytic emission model jointly with complex station gains to snapshot visibilities, then extract estimates of the time-averaged, persistent source structure and temporal variability by stacking snapshots in a Bayesian hierarchical model. This approach successfully reproduces parameters of General Relativistic Magnetohydrodynamics simulations using synthetic EHT observations. Even with physically motivated assumptions about the \sgra environment, black hole spin and magnetic field parameters are poorly constrained by 2017 EHT observations. Our inference constrains other parameters, favoring a nearly face-on observer inclination ($θ_{\rm o} = 9.2\degree \pm 3.6 \degree \pm_{\rm v} 11.6\degree$), an emission peak near the horizon ($R_{\rm peak} = 4.9 \pm 0.1 \pm_{\rm v} 0.5\,GM/c^2$), near-vertical projected spin position angle ($p.a. = 7.3\degree \pm 7.08 \degree \pm_{\rm v} 43.5\degree$ counterclockwise from vertical), and dominant emission $43.4\degree \pm 2.0\degree \pm_{\rm v} 5.9\degree$ above the equatorial plane, where we separate average structure uncertainty ($\pm$) from the impacts of temporal variability and model misspecification ($\pm_{\rm v}$).

Figures (11)

• Figure 1: Schematic depiction of the snapshot modeling methodology. We perform inference on complex visibilities from individual snapshots of varying coverage, jointly inferring physical model parameters and complex station gains. The resulting posteriors are then stacked in a global Bayesian hierarchical model, giving estimates for the average source structure and variable features. Top row, left to right: example ($u,v$)-coverage for individual snapshots, with snapshots ranging between poor coverage (a minimum of four baselines), such as the snapshot indexed as scan 27 (April 7, 2017 at 06:03); medium coverage, such as scan 114 (April 7, 2017 at 11:23); and good coverage (a maximum of 9 baselines), such as scan 157 (April 7, 2017 at 13:49 UTC). Even for the best-sampled snapshots, coverage is very sparse. Data taken at $227.07$ GHz are shown in blue and baselines of 3 G$\lambda$ and 6 G$\lambda$ are shown in green and red, respectively. Middle row, left to right: images drawn from the inferred posterior distributions corresponding to snapshot visibility data (scan 27, scan 114, and scan 157, respectively). Due to the sparsity of data in a single snapshot, many parameters are unconstrained and myriad different image morphologies fit the data. These morphologies contribute to structures in the mean and standard deviation images. Bottom row, left to right: mean and standard deviation (uncertainty) images from the global posterior, a Bayesian hierarchical model stacking the posteriors from every snapshot, with an average source structure distribution and a variability distribution. The mean image from the average source structure distribution indicates persistent morphological features across snapshots. The mean variability map shows the image features most subject to intra-snapshot variability.
• Figure 2: Images produced from Sgr A$^*$ hierarchical averaging results for both the average source structure distribution $\boldsymbol{\mu}$ (top row) and the variability distribution $\boldsymbol{\sigma}$ (bottom row). The first column shows the mean of image samples, the second column the scattered mean image, the third column the scattered standard deviation or uncertainty about the mean, and the fourth column samples from the marginal hierarchical posterior. Scattered images are produced by blurring the nominal image with the diffractive scattering kernel from 2018ApJ...865..104J. The mean image from the average source structure posterior is highly symmetric, with a visible inner shadow and a bright photon ring. Individual image draws in the average posterior display less variable morphologies than image samples for a single snapshot posterior; the standard deviation image suggests this variation is confined primarily to near-horizon emission interior to the photon ring. The mean variability map displays the images features most subject to variability across snapshots.
• Figure 3: Snapshot inference applied to the Sgr A$^*$ 2017 data set. The parameter posteriors from every tenth snapshot are stacked vertically. Results for (a) the black hole spin parameter $a_{\rm *}$ and (b) the observer inclination $\theta_{\rm o}$ are shown. Spin posteriors are wide, with occasional preference towards positive or negative spin in some snapshots. Inclination posteriors, however, show a strong and stable tendency for lower observer inclinations across snapshots.
• Figure 4: Joint marginal probability densities for a subset of the averaged model parameters. The black hole spin and azimuthal angle of the fluid flow $\chi$ are unconstrained. The spin position angle, black hole spin inclination, cone opening angle, and characteristic emission radius are well constrained by our analysis. Up to model specification, the hierarchical model suggests a nearly face-on observer inclination, a nearly vertical spin axis, strong near-horizon emission, dominant non-equatorial emission, and slow-moving plasma relative to the innermost stable circular orbit (ISCO).
• Figure 5: Hierarchical averaging image results for the GRMHD synthetic data test. Left to right: the true average image used to construct the data, including the impact of both diffractive blurring and refractive scintillation; mean image from the average source structure posterior, blurred with the diffractive scattering kernel for Sgr A$^*$; the scattered image uncertainty/standard deviation; samples from the average source structure posterior. While the mean posterior image angular structure differs from the ground truth, the posterior image structure is highly uncertain, and some samples are similar to the true image.