Orbital polarimetric tomography of a flare near the Sagittarius A supermassive black hole

The supermassive black hole Sagittarius A* (Sgr A*) at the center of the Milky Way interacts with its accretion disk, producing high-energy flares observable across the electromagnetic spectrum (X-ray, infrared, and radio). A leading theory proposes that these flares originate from the formation of compact, bright regions (hotspots, bubbles, or flux tubes) within the accretion disk, close to the event horizon. These structures, predicted by sophisticated simulations involving magnetic reconnection, are hypothesized to orbit the black hole. Understanding the structure and dynamics of these flares is crucial for comprehending accretion processes around black holes. Prior near-infrared observations by the GRAVITY collaboration and radio observations by ALMA have hinted at the existence of orbiting hotspots close to the event horizon, but a detailed three-dimensional reconstruction has been lacking. The Event Horizon Telescope (EHT) has provided images of Sgr A*, revealing a ring-like structure with a central brightness depression, confirming the presence of a supermassive black hole. However, even in its quiescent state, Sgr A* shows considerable structural variability. This research aims to reconstruct the three-dimensional structure of an Sgr A* flare observed by ALMA on April 11, 2017, immediately following a high-energy X-ray flare, leveraging the high variability and linear polarization information contained within the ALMA light curves. This work goes beyond previous studies that employed simplified parametric hotspot models by attempting to recover the complex 3D structure of flares orbiting Sgr A*. The challenge lies in the fact that ALMA observations at the time did not have sufficient resolution to resolve the event horizon scales directly; therefore, a novel approach that uses the orbital motion of the flare to create temporal information that acts like multiple viewpoints is presented.

Numerous studies have investigated Sgr A* flares across various wavelengths. Genzel et al. (2003) reported near-infrared flares, while subsequent studies using Chandra and other telescopes characterized X-ray variability (Neilsen et al., 2013; Haggard et al., 2019). Fazio et al. (2018) and Marrone et al. (2008) presented multiwavelength light curves of Sgr A* flares. Dexter et al. (2020) and others proposed models linking flares to magnetic reconnection events within a magnetically arrested disk. Wielgus et al. (2022) analyzed ALMA millimeter light curves during the 2017 EHT campaign, providing initial evidence for orbital motion. The GRAVITY collaboration (GRAVITY Collaboration, 2018, 2020, 2023) reported near-infrared detections of orbital motions close to the last stable circular orbit (ISCO) of Sgr A*, offering further support for the orbiting hotspot hypothesis. Broderick and Loeb (2005) explored imaging bright spots in the accretion flow, and Ripperda et al. (2022) used simulations to model black hole flares and plasmoid-mediated reconnection events. The initial EHT imaging of Sgr A* (Event Horizon Telescope Collaboration, 2022a, 2022b, 2022c, 2022d, 2022e) provided a crucial framework by revealing the black hole shadow. Prior analyses of ALMA data (Wielgus et al., 2022) used a simplified model, while this study aims to capture the full 3D structure.

The researchers developed a novel computational approach called orbital polarimetric tomography to reconstruct the three-dimensional emission structure of the Sgr A* flare. Unlike traditional tomography, which relies on multiple viewpoints, this method leverages the orbital motion of the flare itself. As the emitting structure orbits the black hole, it is observed from a fixed viewpoint along different curved ray paths, effectively replacing multiple viewpoints. This approach builds on previous work on dynamical imaging and 3D tomography in curved spacetime. The methodology involves solving an underconstrained inverse problem, fitting a model to the ALMA data while incorporating the challenges of unresolved observations near the event horizon. To address the ill-posed nature of the problem, the researchers integrated neural 3D representations, which provide implicit regularization favoring smooth structures consistent with physical constraints. The data preprocessing involved time-averaging, subtraction of a constant linear polarization (LP) component representing the background accretion disk, and derotation of the electric vector polarization angle to correct for Faraday rotation. The core of the approach is a forward model that simulates the light curves given a 3D emission structure, its orbit, and black hole parameters. This model includes: 1. **Orbit Dynamics:** A Keplerian orbit model with an angular velocity that depends on the radial distance from the black hole and includes shearing due to differential rotation. 2. **Image Formation:** Relativistic ray tracing through curved spacetime to determine the light paths from the emission structure to the observer. This step calculates the polarized synchrotron emission from an optically thin disk, considering polarization effects and gravitational redshift. 3. **Light Curves:** A summation of the emission along ray paths over the image plane to synthesize light curves. 4. **Neural Representation:** A neural network (MLP) representing the 3D emission volume. The network inputs are coordinate positions, and the outputs are the corresponding emission intensities. The network weights are optimized to fit the observed light curves. The inverse problem is solved by minimizing a χ² loss function between the observed and modeled LP light curves, jointly estimating the 3D emission structure (represented by the neural network weights) and the observer's inclination angle. To avoid overfitting, a validation-χ² metric is used, which takes into account perturbations in ray positions. The optimization uses an ADAM optimizer in JAX with a specific learning rate schedule.

The orbital polarimetric tomography analysis of the ALMA data from April 11, 2017, revealed several key findings: 1. **Low Inclination Angle:** The analysis strongly prefers low inclination angles (θ < 18°), consistent with prior studies by EHT and GRAVITY. This is determined using the validation-χ² metric, which is more robust against overfitting than a standard χ². 2. **Compact, Bright Regions:** The 3D reconstruction shows two compact, bright regions located at radii of approximately 11M and 13M (where M is the black hole mass). This spatial distribution is consistent with previous qualitative assessments. 3. **Clockwise Rotation:** The analysis favors clockwise rotation of the emission structure in its orbital plane, supporting the findings of previous analyses using GRAVITY and ALMA data. 4. **Model Robustness:** The recovered 3D structure shows a degree of robustness to changes in the model's assumptions. While the detailed structure's sensitivity to inclination and initialization is observed, key features such as the azimuthal and radial positions remain consistent, indicating the stability of the method. This robustness was further evaluated using simulated data sets with different emission morphologies (simple hotspot, flux tube, and double source) and inclination angles, demonstrating successful recovery of the input structures. 5. **Magnetic Field Preference:** The analysis suggests a preference for a vertically oriented magnetic field, aligning with findings from both the near-infrared GRAVITY data and previous millimetre ALMA analyses. The results are however sensitive to the assumption of a vertical magnetic field. A radial magnetic field, for example, resulted in a significantly different reconstructed emission, suggesting a diffuse structure instead of a compact region.

This study successfully reconstructs the three-dimensional structure of an Sgr A* flare, directly addressing the limitations of previous studies that relied on simplified parametric models. The findings of compact, bright regions orbiting the black hole in a low-inclination, clockwise orbit provide strong observational support for theoretical models that posit the formation of flares through magnetic reconnection events within the accretion disk. The consistency between the results of this paper and those obtained using GRAVITY and EHT data further strengthens the reliability of the low-inclination and clockwise rotation assumptions. The utilization of neural radiance fields and polarimetric general relativistic ray tracing allowed the reconstruction to overcome the challenges posed by the underconstrained nature of the inverse problem and the lack of high-resolution observations. Although the reconstruction relies on several modeling choices (e.g., optically thin emission, Keplerian orbit), the robustness tests and analyses of different model parameters suggest that the main findings are stable. The methodology presented represents a significant advance in our ability to image dynamic structures around black holes.

Orbital polarimetric tomography provides a novel approach to 3D imaging of dynamic structures around black holes. Applying this method to ALMA observations of Sgr A* reveals a complex, dynamic emission structure, providing new insights into the physics of black hole accretion. Future work could involve extending the approach to higher-resolution data (e.g., EHT), multifrequency observations, and relaxing the assumptions of optically thin emission and strictly Keplerian orbits. Adaptations of this technique to other black hole systems could lead to valuable population statistics and enhance our understanding of accretion processes.

The study's key limitation is its dependence on several model assumptions. Although the reconstruction is shown to be reasonably robust, the results are sensitive to the choice of magnetic field configuration, orbital dynamics (Keplerian versus sub-Keplerian orbits), and the assumption of optically thin emission. The fact that the reconstruction is obtained from an effectively single-pixel time series naturally limits the level of detail that can be reliably recovered. A more extensive exploration of parameter space might also reveal further possible solutions or better constrain the uncertainty on parameters like inclination angle and the true 3D structure.