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Magnetic Field Measurements in the Solar Chromosphere Using the H $_{\beta}$ 4861\AA~Line I: Forward Modeling Based on 1D Models

This study validates a non-LTE magnetic field inversion method for the H $_{\beta}$ 4861\AA~line using 1D semi-empirical models, demonstrating its ability to accurately retrieve line-of-sight magnetic fields across a wide range of strengths (0–5000 G) and paving the way for full-disk chromospheric magnetic field observations with the GanYu Solar Station.

Original authors: Jiasheng Wang, Wenxian Li, Xianyong Bai, Yingzi Sun, Yuanyong Deng, Jiaben Lin

Published 2026-03-24

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CC BY 4.0

Original authors: Jiasheng Wang, Wenxian Li, Xianyong Bai, Yingzi Sun, Yuanyong Deng, Jiaben Lin

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

Imagine the Sun not just as a giant, burning ball of light, but as a complex, multi-layered onion. The outermost layer we can see is the photosphere (the "surface"). Above that lies the chromosphere, a turbulent, fiery atmosphere that acts like a bridge, transporting massive amounts of energy upward to heat the Sun's outermost layer, the corona.

The big mystery scientists are trying to solve is: What does the magnetic field look like in this middle layer (the chromosphere)?

Magnetic fields are the "invisible wires" that control solar flares and space weather. While we can map them easily on the Sun's surface, seeing them clearly in the chromosphere is like trying to read a book through a thick, foggy window. The light is messy, and the tools we usually use aren't sensitive enough.

The New Tool: The "H-beta" Lens

In this paper, the researchers are testing a new way to see through that fog using a specific color of light called H-beta (a shade of blue-green). Think of the H-beta line as a special pair of glasses that can peek into the chromosphere.

However, these glasses are tricky. The light they capture is distorted by the Sun's heat and movement. To fix this, the scientists used a technique called Non-LTE Inversion.

The Analogy: Imagine you are trying to guess the temperature of a room by listening to the echo of a clap. If the room is empty, the echo is clear. But if the room is full of people moving around (like the Sun's atmosphere), the echo gets muddled.
The Solution: The researchers built a super-accurate computer simulation (a "digital twin" of the Sun) to understand exactly how the sound (light) gets muddled. Then, they used that knowledge to "un-muddle" the real data and figure out the original temperature and magnetic field.

The Two-Step Detective Method

The team tested two different detective strategies to decode the magnetic field:

The "Weak-Field" Shortcut (WFA): This is like using a quick, rough rule of thumb. It's fast and works well when the magnetic field is weak or moderate.
- The Problem: It gets confused when the magnetic field changes rapidly as you go higher up, or when the field is extremely strong (like in sunspots).
The "Deep Dive" (Non-LTE Inversion): This is the slow, heavy-duty method. It simulates the physics of every single atom to get a precise answer.
- The Innovation: The researchers found that if they use the Quick Shortcut first to get a "best guess," and then feed that guess into the Deep Dive method, the final result is much better. It's like using a rough sketch to help an artist finish a masterpiece; the sketch gives the artist a great starting point, saving time and reducing mistakes.

What They Found

The researchers ran thousands of simulations to see how well their method worked:

Two Heights, One Line: They discovered that the H-beta light actually comes from two different places at once: a lower layer (near the surface) and a higher layer (in the middle of the atmosphere). It's like a radio signal picking up two stations at the same time. Their method successfully separated these signals to map the magnetic field at the higher, more interesting layer.
Quiet Sun vs. Sunspots:
- In Quiet Sun areas (calm regions), the method worked beautifully, accurately mapping magnetic fields from very weak to very strong.
- In Sunspots (intense magnetic storms), it was harder. The "fog" was thicker, and the magnetic fields were so strong they confused the quick shortcut. However, even here, the "Quick + Deep Dive" combo was the best tool available.
The Future: They proved that this method is ready for real-world use. They are planning to apply it to data from a new, powerful telescope in China (the GanYu Solar Station) that will take pictures of the entire Sun's magnetic field, not just small patches.

Why This Matters

Understanding the Sun's magnetic field in the chromosphere is crucial for predicting space weather. Just as we need to know about hurricanes to protect our coasts, we need to know about solar magnetic storms to protect our satellites, power grids, and astronauts.

By refining this "H-beta lens" and the "Quick + Deep Dive" method, the scientists are giving us a sharper, clearer view of the Sun's invisible magnetic skeleton, helping us better predict when the Sun might throw a tantrum.

1. Problem Statement

The solar chromosphere is a critical layer for understanding energy transport and magnetic reconnection, yet its magnetic field configuration remains poorly understood. Current limitations include:

Diagnostic Challenges: The low density of the chromosphere leads to non-local thermodynamic equilibrium (non-LTE) conditions, making standard LTE inversion techniques inaccurate.
Spectral Complexity: Chromospheric lines are broad, and polarization signals (especially linear polarization) are weak and often dominated by scattering effects rather than Zeeman splitting.
Instrumental Limits: High-resolution vector magnetic field measurements above the photosphere are difficult due to limited sensitivity and the complex interplay between velocity fields, magnetic fields, and non-LTE radiative transfer.
Specific Gap: While hydrogen Balmer lines (H $\alpha$ , H $\beta$ ) are excellent for diagnosing dynamic and thermal properties, their potential for precise magnetic field measurement via polarization has been underutilized due to technical difficulties.

This study aims to validate the H $\beta$ 4861 ˚A line as a diagnostic tool for chromospheric magnetic fields using non-LTE inversion techniques and to evaluate the efficacy of using the Weak-Field Approximation (WFA) as an initial guess for these inversions.

2. Methodology

2.1 Forward Modeling and Synthetic Data Generation

Atmospheric Models: The authors used semi-empirical 1D models: FAL-C for the quiet Sun and FAL-S for sunspots.
Radiative Transfer: Synthetic Stokes profiles ( $I, Q, U, V$ $I, Q, U, V$ ) were generated using the RH code (Uitenbroek 2001) to solve the coupled Radiative Transfer Equation (RTE) and Statistical Equilibrium (SE) equations under non-LTE conditions.
- The simulation included 9 energy levels of hydrogen (Lyman, Balmer, Paschen series).
- Partial Frequency Redistribution (PRD) was applied to the H $\beta$ absorption profiles.
- Collisional broadening parameters (Anstee, Barklem, and O'Mara) and collisional rates (Johnson) were incorporated.
Magnetic Configurations: Two configurations were tested:
1. Constant Magnetic Field: Uniform strength along the height.
2. Linear Gradient Magnetic Field: Strength decreasing linearly with height (vanishing at $\log \tau_{500nm} = -7.0$ ).
Observational Mimicry: Synthetic profiles were convolved with instrumental broadening (85 mÅ, simulating the Solar Full-disk Multi-layer Magnetograph, SFMM) and added with Gaussian noise ( $\sigma_n = 1.5 \times 10^{-3}$ ).

2.2 Inversion Techniques

Two distinct methods were employed to retrieve magnetic fields from the synthetic data:

Weak-Field Approximation (WFA):
- Assumes Zeeman splitting is smaller than the Doppler width.
- Calculates Line-of-Sight (LOS) field ( $B_{LOS}$ ) from Stokes $V$ and the first derivative of Stokes $I$ .
- Calculates Transverse field ( $B_{\perp}$ ) from Stokes $Q/U$ and the second derivative of Stokes $I$ .
- Limitation: Assumes atmospheric parameters are constant with optical depth; sensitive to line broadening mechanisms.
Non-LTE Inversion (DeSIRe):
- Uses the DeSIRe code (Departure coefficient aided Stokes Inversion based on Response functions).
- A hybrid approach combining the SIR code (LTE) with the RH solver (non-LTE).
- Utilizes fixed departure coefficients to modify analytical response functions, drastically reducing the number of full RTE iterations required.
- Strategy: The study tested two initialization strategies for DeSIRe:
  - Fixed Guess: Arbitrary initial values for field strength and inclination.
  - WFA-Guided: Using WFA-derived estimates as the initial guess for the non-LTE inversion.

3. Key Contributions

Dual Formation Height Characterization: Confirmed that the H $\beta$ line center forms at two distinct heights: the lower photosphere ( $\sim$ 300–400 km) and the middle chromosphere ( $\sim$ 1100–1300 km), with the specific height depending on the atmospheric model (Quiet Sun vs. Sunspot).
Validation of Hybrid Inversion Strategy: Demonstrated that using WFA results as an initial guess for the non-LTE DeSIRe inversion significantly improves convergence and accuracy, particularly for weak to moderate magnetic fields.
Quantitative Performance Metrics: Provided rigorous statistical validation (Pearson correlation coefficients and linear regression slopes) of H $\beta$ inversions across different field strengths (0–5000 G) and atmospheric conditions.

4. Key Results

4.1 WFA Performance

LOS Component: WFA showed a high correlation with input models but exhibited a systematic bias (mean offset of ~60 G) and failed in very weak fields (<250 G).
Transverse Component: Performance was poor due to the weakness of linear polarization signals and sensitivity to noise. The inferred transverse fields often saturated at ~250 G for weak inputs.
Conclusion: WFA is useful for estimating LOS fields but insufficient for full vector reconstruction without further refinement.

4.2 Non-LTE Inversion (DeSIRe) Results

Constant Field Models:
- Inversions using WFA-derived initial guesses showed significantly better agreement with model inputs than those with fixed guesses.
- Correlation: Pearson correlation coefficients reached 0.96 for the Quiet Sun (FAL-C) and 0.87 for Sunspots (FAL-S) in the constant field scenario.
- Outliers: The number of outliers (deviations >3 $\sigma$ ) was drastically reduced in the 0–500 G range when using WFA initialization.
Linear Gradient Models:
- Quiet Sun: At the chromospheric formation height ( $\sim$ 1121 km, $\log \tau_{500nm} \approx -4.88$ ), the correlation improved to 0.93 with WFA initialization. However, a systematic overestimation of ~24–61% persisted due to the height-dependent gradient.
- Sunspots: The formation height shifted deeper ( $\sim$ 398 km) due to cooler temperatures. The best correlation was 0.77, with field strengths underestimated by ~18%. Stronger gradients and higher field strengths (>2000 G) led to larger deviations.
Spectral Window Sensitivity: Using a fixed spectral window ( $\pm 0.13$ ˚A) centered on the line core yielded more accurate results for gradient fields compared to windows defined solely by Voigt profile fitting.

4.3 Height Sensitivity

Quiet Sun: Best inversion accuracy occurred at $\log \tau_{500nm} \approx -5.0$ (chromosphere) and $\approx -1.0$ (photosphere).
Sunspots: The peak sensitivity shifted to $\log \tau_{500nm} \approx -4.16$ (chromosphere) and $\approx -1.16$ (photosphere), reflecting the deeper formation of H $\beta$ in cooler umbral atmospheres.

5. Significance and Future Implications

Feasibility for Full-Disk Monitoring: The study validates that H $\beta$ polarization signals are measurable even with realistic noise and instrumental broadening, supporting the deployment of this technique on the Solar Full-disk Multi-layer Magnetograph (SFMM) at the GanYu Solar Station.
Methodological Advancement: The "WFA-initialized non-LTE inversion" approach offers a computationally efficient path to retrieving chromospheric magnetic vectors, bridging the gap between fast approximations and rigorous non-LTE physics.
Limitations: The reliance on 1D models limits the ability to capture 3D structures (shocks, fibrils) and leads to degeneracies in gradient fields. Future work must extend to 3D MHD simulations (e.g., Bifrost, MURaM) and multi-line inversions (combining H $\beta$ with H $\alpha$ or Ca II) to resolve height ambiguities.
Astrophysical Application: The techniques developed are applicable to stellar magnetometry, where hydrogen Balmer lines are crucial diagnostics for non-LTE, turbulent, and magnetized stellar atmospheres.

In summary, this paper establishes that the H $\beta$ 4861 ˚A line, when analyzed with non-LTE inversion techniques initialized by WFA estimates, is a robust tool for measuring line-of-sight magnetic fields in the solar chromosphere, offering a promising avenue for future full-disk solar magnetic monitoring.