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A deep learning algorithm for black hole spin estimation using hot-spot secondary images

This paper introduces STIHOS, a deep learning algorithm that accurately estimates black hole spin and inclination from the position angle difference between primary and secondary hot-spot images, demonstrating robust performance even under partial-orbit visibility and observational errors.

Original authors: Aristomenis I. Yfantis, Rami Al-Belmpeisi

Published 2026-08-20

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CC BY 4.0

Original authors: Aristomenis I. Yfantis, Rami Al-Belmpeisi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

At the very center of our Milky Way galaxy lies a monster of gravity, a supermassive black hole known as Sagittarius A*. For decades, astronomers have known it is there, but seeing it clearly has been a struggle against the blur of cosmic dust and the sheer speed of events happening so close to its edge. Recently, powerful new telescopes have finally begun to capture images of this dark heart, revealing a shadow surrounded by a ring of glowing gas. Within this swirling disk of superheated matter, astronomers have spotted brief, intense flashes of light. These are not random flickers but are believed to be "hot spots"—compact, bright regions of plasma that orbit the black hole like a lighthouse beam sweeping across the darkness. By tracking how these hot spots move and how their light bends around the black hole, scientists hope to measure the black hole's spin and the tilt of its orbit, two fundamental properties that define the shape of space and time around it.

A new study by researchers A. I. Yfantis and R. Al-Belmpeisi takes a fresh approach to unlocking these secrets. Instead of trying to solve complex equations by hand, they have built a sophisticated computer program, a type of artificial intelligence, to act as a translator between what we see and what is happening deep in the gravity well. The core idea relies on a specific visual effect: as a hot spot orbits the black hole, the intense gravity bends the light from that spot, creating not just one image, but a second, fainter image that appears on the opposite side of the black hole. This second image is a "secondary" view, a ghostly reflection of the same event. The researchers focused on the angle between the main spot and this ghostly twin. They found that this angle changes in a very specific way depending on how fast the black hole is spinning and how tilted the orbit is relative to our view.

To teach their computer program how to read these angles, the team generated a massive library of one hundred thousand simulated universes. In each simulation, they created a black hole with a different spin speed and a hot spot orbiting at a different distance and tilt. They used advanced physics software to calculate exactly how light would travel in each scenario, recording the angle between the primary spot and its secondary image over time. They then fed this data into their deep learning model, named STIHOS, training it to recognize the patterns that link the observed angles to the hidden properties of the black hole. The goal was to see if the computer could look at a set of angle measurements and instantly tell them the spin and tilt of the system, even if the data was incomplete or noisy.

The results were strikingly precise. When the simulations assumed the hot spot was orbiting perfectly flat, like a coin spinning on a table, the computer model could determine the black hole's spin with an accuracy of about 0.04 units and the tilt of the orbit within just two degrees. This level of precision held true even when the researchers tested the model with data that only showed half of the orbit, a common situation in real observations where a flare might fade before a full circle is completed. The model proved robust, successfully separating the effects of the black hole's spin from the geometry of the orbit, a task that is notoriously difficult to do with traditional methods.

However, the universe is rarely as simple as a flat table. The researchers also tested what happens when the hot spot orbits at a steep angle, tilted away from the central plane. In these more complex scenarios, the computer found that it became harder to distinguish between the spin of the black hole and the tilt of the orbit. Without extra information, the model could not always tell if a specific angle was caused by a fast-spinning black hole or a highly tilted orbit; the two effects could mimic each other. Despite this challenge, the model still provided useful estimates, narrowing down the possibilities significantly. The study suggests that if astronomers can combine these observations with other known facts, such as the general tilt of the galaxy's center, they can break this confusion and pin down the spin with high confidence.

The team also looked ahead to future telescopes, including the next generation of the Event Horizon Telescope and a proposed space-based mission called the Black Hole Explorer. They simulated how the model would perform with the sharper, clearer data these instruments are expected to provide. The results showed that with the superior resolution of future arrays, the uncertainty in measuring the black hole's spin could drop to less than 0.1 units, a level of precision that would allow for rigorous tests of Einstein's theory of gravity. Even with the current, slightly fuzzier data from existing telescopes, the model could still constrain the spin to within about 0.3 units, offering a meaningful step forward.

This work demonstrates that the fleeting flashes of light from hot spots are not just beautiful anomalies but are powerful tools for mapping the invisible geometry of space. By using artificial intelligence to decode the subtle angles between a hot spot and its gravitational reflection, astronomers can now extract the fundamental properties of the black hole at the center of our galaxy. While the method relies on simulations and requires careful handling of real-world data noise, it offers a clear and promising path toward understanding the nature of these cosmic giants. As our telescopes get better and our ability to track these hot spots improves, this technique could soon turn the first blurry glimpses of lensed light into a precise measurement of the black hole's spin, revealing the true shape of the spacetime around it.

Technical Summary: A Deep Learning Algorithm for Black Hole Spin Estimation Using Hot-Spot Secondary Images

Problem Statement
The supermassive black hole Sagittarius A* (Sgr A*) exhibits frequent flaring activity attributed to localized emission regions, or "hot spots," orbiting the black hole. While the Event Horizon Telescope (EHT) and GRAVITY Collaboration have provided high-precision constraints on Sgr A*'s mass and distance, isolating robust spacetime signatures (specifically the dimensionless spin parameter $a_*$ and inclination $i$ ) from these images remains challenging due to plasma-dependent emission, finite resolution, and rapid variability. Previous theoretical work (Yfantis et al. 2026, hereafter Y26) proposed a link between the angle difference ( $\Delta PA$ ) between primary and secondary images of a hot spot and the system's physical parameters. However, Y26 relied on averaging $\Delta PA$ over a full orbital period and assumed strictly equatorial orbits, limiting its applicability to realistic, partial-visibility observations and non-equatorial trajectories.

Methodology
The authors introduce STIHOS (Space-Time Inference via Hotspot Observations), a deep learning (DL) framework designed to solve the inverse problem of inferring black hole parameters from hot-spot observables.

Data Generation: Using the general relativistic radiative transfer code ipole, the authors generated a library of approximately 100,000 hot-spot models. These simulations varied key parameters: black hole spin ( $a_* \in [-1, 1]$ ), hot-spot emission radius ( $r_{hs} \in [4, 12] M$ ), orbital velocity coefficient ( $\Omega_{coef}$ ), system inclination ( $i \in [0, 60]^\circ$ ), and non-equatorial angular offset ( $\theta_z \in [-45, 45]^\circ$ ).
Observables: The models track the position angle difference ( $\Delta PA$ ) between the primary ( $n=0$ ) and secondary ( $n=1$ ) images over time, alongside the orbital period ( $T$ ) and emission radius ( $r_{hs}$ ).
Model Architecture: STIHOS utilizes a Multi-Layer Perceptron (MLP) with skip connections, batch normalization, and ReLU activations. The network is trained to map observables ( $r_{hs}$ , $T$ , and time-series $\Delta PA(t)$ ) to target parameters ( $a_*$ , $i$ , and $\theta_z$ ).
Training Strategy: The dataset was split 80/10/10 for training, validation, and testing. To handle observational noise, the authors employed two strategies: (1) injecting Gaussian noise (modeled via Radial Basis Function kernels for temporal continuity) during training, and (2) Monte Carlo dropout during inference to estimate prediction uncertainty.
Experimental Design: Five experimental setups were tested to evaluate performance under varying constraints:
- Exp I: Equatorial, full-orbit, time-averaged $\Delta PA$ (reproducing Y26).
- Exp II: Equatorial, full-orbit, time-resolved $\Delta PA(t)$ .
- Exp III: Equatorial, partial-orbit visibility (10–100% coverage).
- Exp IV: Non-equatorial, full-orbit visibility.
- Exp V: Non-equatorial, partial-orbit visibility.

Key Results
The algorithm demonstrates remarkable accuracy in recovering system parameters, particularly under idealized conditions:

Equatorial Orbits (Exp I & II): In noiseless scenarios, STIHOS recovers spin with a standard deviation of $\sigma_{a_*} \approx 0.04$ and inclination with $\sigma_i \approx 1.9^\circ$ . The time-resolved approach (Exp II) confirms that $\Delta PA(t)$ curves allow for simultaneous, high-precision inference of both spin and inclination across the parameter space.
Partial Visibility (Exp III): The model maintains high inference power even when only 50% of the orbit is visible, with errors increasing only modestly ( $\sigma_{a_*} \approx 0.08$ , $\sigma_i \approx 4^\circ$ ). Below 50% coverage, errors increase significantly.
Non-Equatorial Orbits (Exp IV & V): Introducing the off-equatorial parameter $\theta_z$ creates degeneracies between $a_*$ , $i$ , and $\theta_z$ . While spin and inclination constraints soften ( $\sigma_{a_*}$ rises to $\sim 0.22$ , $\sigma_i$ to $\sim 2.9^\circ$ ), the model provides only "soft constraints" on $\theta_z$ ( $\sigma_{\theta_z} \approx 9.55^\circ$ ) without prior information.
Observational Noise: The authors simulated errors expected from current (EHT), next-generation (ngEHT), and future (BHEX) arrays.
- EHT: Current noise levels degrade spin estimation to $\sigma_{a_*} \sim 0.3$ and inclination to $\sigma_i \sim 5^\circ$ (for equatorial cases).
- BHEX: With significantly improved resolution, BHEX is projected to constrain spin to $\sigma_{a_*} < 0.1$ and inclination to $\sigma_i \le 2^\circ$ , even with non-equatorial geometries.
Virtual Observations: Monte Carlo inference on virtual data points confirms that while EHT data yields broad confidence intervals, BHEX data allows for precise convergence on the true parameters, particularly for positive spins.

Significance and Claims
The paper claims that STIHOS provides a robust, non-degenerate method for extracting spacetime parameters from hot-spot secondary images, bypassing the need for complex mathematical disentanglement of direct and lensed emission.

Validation of Y26: The results validate the theoretical relation proposed in Y26, confirming that time-averaged $\Delta PA$ is a valid proxy for face-on inclination ( $i=0$ ) and that time-resolved curves offer superior constraints.
Feasibility for Near-Future Missions: The study suggests that detecting the secondary image of a hot spot requires less angular resolution than resolving the full photon ring, making it a viable target for current and near-future interferometric arrays (EHT, ngEHT, BHEX).
Degeneracy Management: The authors highlight a critical degeneracy between spin, inclination, and off-equatorial offset. They argue that while STIHOS can infer these parameters, breaking the degeneracy for non-equatorial orbits likely requires external priors on inclination (e.g., from other GRAVITY/EHT measurements) or future movie-based constraints on hot-spot deformation.
Methodological Contribution: The work demonstrates the utility of deep learning in accretion studies, offering a computationally efficient alternative to Gaussian processes for inferring highly non-linear relationships in black hole physics.

The authors conclude that while observational extraction of $\Delta PA$ from raw VLBI data remains a future challenge, STIHOS establishes a valuable framework for interpreting the first lensed emission observations as they become available.

Technical Summary: A Deep Learning Algorithm for Black Hole Spin Estimation Using Hot-Spot Secondary Images

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