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

The compact region around the Galactic Centre supermassive black hole Sagittarius (Sgr) A* hosts a magnetized, turbulent accretion flow in extreme gravitational conditions that sporadically produces energetic flares across X-ray, infrared and radio bands. Understanding the physical nature, origin, and structure of these flares is key to constraining accretion processes near event horizons. One proposed explanation is the formation of compact, bright regions (hotspots, bubbles, flux tubes) within the disk, energized by mechanisms such as magnetic reconnection, and orbiting near the innermost stable circular orbit (ISCO). Prior observations by GRAVITY (near-infrared) and ALMA (millimetre polarimetry) have been interpreted as consistent with orbiting hotspots close to the event horizon. The Event Horizon Telescope (EHT) imaged Sgr A* on 6–7 April 2017, revealing a ring with a central brightness depression and also indicating rapid structural variability. On 11 April 2017, immediately following an X-ray flare, ALMA recorded highly variable, structured linear polarization light curves with periodic features on orbital timescales, suggestive of compact orbiting emission. The research question is whether time-variable polarimetric light curves from a single, unresolved line of sight can constrain and recover the three-dimensional (3D) structure and orbital geometry of a flare near Sgr A*. The study proposes and applies “orbital polarimetric tomography” to reconstruct a 3D emissivity distribution in orbit around Sgr A* from ALMA linear polarization light curves, aiming to locate and characterize flare morphology relative to the event horizon and to constrain viewing inclination and rotation direction.

Previous work modeled Sgr A* flares as parametric hotspots with few degrees of freedom, providing qualitative fits to polarimetric loops but lacking rigorous data fitting or flexibility to capture complex structures (e.g., GRAVITY detections of NIR orbital motion; ALMA polarimetric constraints suggesting an orbiting spot at r≈11M). EHT results from 6–7 April 2017 established a ring-like morphology with a central depression and favored low inclinations (~30°) when comparing to GRMHD simulations. Prior analyses largely neglected velocity shear and often assumed fixed simple morphologies. Foundational developments in dynamical imaging and spacetime tomography demonstrated feasibility in simulations for future EHT datasets. Theoretical GRMHD simulations predict flares arising from magnetic reconnection and flux eruption events, potentially forming compact, hotter, lower-density flux tubes that are optically thinner than the background flow. Prior polarimetric theory and ray-tracing codes provide the tools for polarized radiative transfer in Kerr spacetimes. Collectively, this literature motivates a flexible, physics-informed tomographic approach leveraging polarimetric variability to infer 3D flare structure and geometry.

Data and preprocessing: The analysis uses ~100 minutes of ALMA 229–230 GHz light curves on 11 April 2017 (MJD 57854), directly after an X-ray flare. Linear polarization (LP) time series Q(t)+iU(t) are time-averaged to ~1-minute cadence (~100 points per Stokes component). Following ref. 13, a constant LP component representing the background accretion flow is subtracted with magnitude P_disk=0.16 Jy and angle θ_disk=37°, and the electric vector position angle is derotated by 32.2° to account for Faraday rotation. Noise is modeled as homoscedastic with σ_Q=σ_U=0.01 Jy; total intensity (I) is not fitted but regularized to ~0.3±0.15 Jy. The initial time t0 for orbital evolution and reconstruction is set to 9:20 UT. Forward model and physics: The 4D emission e(t,x) is modeled as a time-propagated version of a canonical 3D emissivity e0(x) undergoing differential azimuthal rotation (shear) in a Keplerian orbit around a Kerr black hole of mass M≈4×10^6 M⊙ (spin weakly affects fits and is set to a=0 in the fiducial model). The transformation applies a radius-dependent angular displacement φ(t,r)=(t−t0)Ω(r), with Ω(r) following a Keplerian profile; radial/vertical velocities and complex dynamics (cooling, heating, expansion, turbulence) are neglected over the short (~1 orbit) timescale considered. The emission volume is constrained to 6M≤r≤20M and |z|≤4M (near the equatorial plane), consistent with orbits outside the ISCO of a nonspinning black hole. Polarized radiative transfer and ray tracing: For each image-plane coordinate, general relativistic geodesics are computed (kgeo), and polarized radiative transfer is performed for an optically thin medium to synthesize Stokes I,Q,U along each ray, including GR redshift g(x), slow-light time delays, and parallel transport rotation of the polarization basis. Synchrotron emissivity is modeled as a scalar field scaled by a Stokes vector with spectral index α≈1 and a volumetric linear polarization fraction q∈[0,1]. The fiducial magnetic field is homogeneous and vertical (Bz); alternative field geometries (radial, toroidal) are explored for sensitivity. Light curves are obtained by summing synthesized images over a large field of view at each time sample. Neural 3D representation: The unknown continuous 3D emissivity e(x) is represented by a coordinate-based MLP (“neural radiance field” style), with 4 fully connected layers of width 128 and ReLU activations. Inputs are positional encodings with maximum degree L=3 to favor smooth volumetric fields. This implicit representation provides regularization for the ill-posed inverse problem and can be sampled densely for visualization. Inverse problem and optimization: For each fixed viewing inclination θ (observer angle), the network weights w are optimized to minimize a χ² data misfit across Stokes components (primarily LP, with regularization on I), using ADAM with a polynomial learning-rate schedule from 1e−1 to 1e−4 over 50,000 iterations (implemented in JAX). Inclination is estimated by gridding θ from 4° to 80° in 2° steps; at each θ, five reconstructions are run from random initializations to assess stability. A validation-χ² metric mitigates overfitting to fixed ray positions by averaging χ² over 10 randomized sub-pixel ray perturbations within a small pixel area (3.125×3.125 µas²), using a 64×64 image-plane sampling over a 200 µas FOV. The preferred θ is the global minimum of validation-χ². Model assumptions and sensitivity tests: Key assumptions include: optically thin millimetre emission; vertical magnetic field (fiducial), with tests of radial and toroidal geometries; Keplerian clockwise (CW) rotation with shear; flare within the accretion disk (no jet component modeled). Sensitivity analyses explore rotation direction (CW vs CCW) and sub-Keplerian orbits (Ω=fΩ0, f=0.8–1.0), as well as magnetic-field configurations, assessing impacts on fit quality and recovered 3D structure. Synthetic experiments with known ground-truth morphologies (simple hotspot, flux tube, double source) at two inclinations evaluate reconstruction fidelity under background noise and model assumptions.

- The method accurately fits the ALMA linear polarization light curves over ~100 minutes starting at 9:20 UT, reproducing coherent Q–U loops and detailed features. - Validation-χ² favors low observer inclinations θ<18°, with a local/global minimum around θ≈12°, broadly consistent with prior EHT and GRAVITY inferences of low inclination. - The recovered 3D emission comprises two compact bright regions at radii r≈11M and r≈13M, with an azimuthally elongated bright structure at 11M trailed by a dimmer source at 13M. - Clockwise rotation (CW) and Keplerian orbital speeds provide better fits than counterclockwise (CCW) or sub-Keplerian orbits; moderate sub-Keplerian f=0.9 yields broadly consistent reconstructions but with degraded fit and a tendency toward slightly smaller recovered radii; larger deviations (f=0.8) further degrade fit (higher validation-χ²). - A vertical magnetic field (Bz) produces lower validation-χ² and compact hotspot-like reconstructions, preferred over radial or toroidal fields; a radial field yields more diffuse, fainter structures and poorer fits. - Results are largely insensitive to black-hole spin in the fitting; mass is fixed at M≈4×10^6 M⊙ from stellar dynamics. - Across multiple random initializations and nearby inclinations, the exact angular extents vary, but the radial and azimuthal locations, and the separation into two structures, remain stable. - Synthetic-data experiments demonstrate the approach can recover distinct underlying morphologies (simple hotspot, flux tube, double source) under similar observational conditions, supporting method validity.

The reconstructions provide the first 3D view of a flare orbiting near Sgr A* derived from unresolved polarimetric light curves, addressing whether single-pixel time-variable polarization can constrain the spatial structure and orbital geometry of flares. The findings support the hotspot/flux-tube interpretation of flares near the ISCO, with emission localized at r≈11–13M and exhibiting CW motion in a low-inclination plane. The preference for low inclination and CW rotation strengthens prior conclusions from GRAVITY and ALMA analyses and aligns with EHT constraints. The method’s ability to separate compact, highly polarized flare emission from partially depolarized background accretion enables robust fitting without relying on total intensity. By integrating GR ray tracing with neural 3D representations and physical constraints, the approach overcomes the extreme ill-posedness of reconstructing 3D structure from temporal LP signals alone. Sensitivity analyses clarify that certain physical assumptions (magnetic field geometry, orbital speed, rotation direction) materially influence fit quality and recovered structure, thereby providing a pathway to constrain these properties with higher-fidelity or multi-frequency data. The results demonstrate feasibility and motivate application to more informative datasets (e.g., spatially resolved EHT observations) to further probe black-hole and plasma physics (spin, magnetic topology, non-azimuthal flows).

This work introduces orbital polarimetric tomography—a computational framework that fuses polarimetric GR ray tracing with neural 3D representations—to reconstruct dynamic 3D flare structures around black holes from unresolved time-series polarization. Applied to ALMA observations on 11 April 2017 of Sgr A*, the method recovers two compact bright regions at ~11M and ~13M, consistent with a CW Keplerian orbit viewed at low inclination (~12°), and provides stable results under physically motivated assumptions. The framework successfully reproduces observed Q–U loops and performs well on synthetic datasets with varied morphologies. Future directions include incorporating spatially resolved EHT data and multi-frequency observations to relax assumptions (e.g., optically thin emission, vertical magnetic fields, purely azimuthal motion), constraining black-hole spin, orbital dynamics, and magnetic-field geometry. Extending the approach to other time-variable black-hole systems (quasars, microquasars) could enable population studies of flare physics and accretion processes.

- The inverse problem is highly ill-posed and non-convex; reconstructions are not unique and depend on model assumptions (inclination, magnetic-field configuration, orbital dynamics) and random initialization. - ALMA observations are unresolved at event-horizon scales (effectively single-pixel LP time series), limiting spatial information and requiring strong priors. - The model assumes emission confined near the equatorial disk, optically thin synchrotron radiation, and a homogeneous vertical magnetic field in the fiducial case; true magnetic topology and optical depth may differ. - Orbital dynamics are simplified to azimuthal Keplerian motion with shear, neglecting radial/vertical velocities and thermodynamic evolution (cooling, heating, expansion, turbulence), valid only over short (~1 orbit) timescales. - Only linear polarization is fitted; total intensity is not directly modeled (only weakly regularized) due to uncertain background accretion intensity. - Black-hole spin is not constrained because fits are weakly sensitive to spin with these data; more complex models (non-optically thin media, non-azimuthal flows) are not explored here. - Sensitivity tests show that strong deviations from Keplerian motion and certain magnetic-field geometries degrade fits and alter recovered structures, indicating model dependence. - The study does not provide a full posterior exploration; error bars from multiple runs reflect stability across initializations rather than rigorous uncertainties.