🔭 astrophysics

Modeling brightness temperature of sunspots using ALMA single-dish observations

This paper presents a modified 1D semi-empirical model for sunspot brightness temperature across the 0.3–10 mm range using ALMA observations, which reveals that while adjusting density and temperature improves the fit, significant discrepancies between the model and observations indicate that further refinements to the underlying physical assumptions are necessary.

Original authors: F. Matković, R. Brajša, A. O. Benz, H. -G. Ludwig, C. L. Selhorst, I. Skokić, D. Sudar, A. Hanslmeier

Published 2026-02-23

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CC BY 4.0

Original authors: F. Matković, R. Brajša, A. O. Benz, H. -G. Ludwig, C. L. Selhorst, I. Skokić, D. Sudar, A. Hanslmeier

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

The Big Picture: Trying to Take the Sun's Temperature

Imagine the Sun is a giant, glowing campfire. Usually, it's a steady, bright orange. But sometimes, dark spots appear on the fire. These are sunspots.

For a long time, scientists have known that these spots are "cool" (relatively speaking) in the visible light we see with our eyes. But what happens when we look at them with "radio eyes"? Do they stay cool, or do they heat up?

This paper is like a team of detectives trying to figure out the temperature of these sunspots using a very powerful new tool: ALMA (the Atacama Large Millimeter/submillimeter Array). Think of ALMA as a super-sensitive radio thermometer that can look at the Sun in different "colors" of radio waves, ranging from short waves (like a high-pitched whistle) to long waves (like a low bass drum).

The Mystery: The Sunspot's "Mood Swing"

The researchers had a specific theory (a model) about how sunspots should behave. It was like having a recipe book for a cake. They expected the sunspot to be:

Cooler than the surrounding fire at short radio wavelengths.
Cooler than the surrounding fire at long radio wavelengths, too.

But the reality was a surprise.

When they actually measured the sunspots, they found a "mood swing":

Short Waves (The "Cool" Phase): At short wavelengths, the sunspots were indeed darker and cooler than the rest of the Sun. This matched their expectations.
Long Waves (The "Hot" Phase): But as they tuned their radio eyes to longer wavelengths, the sunspots suddenly became brighter and hotter than the surrounding Sun!

It's as if you touched a patch of ice on a sidewalk, and it felt cold at first, but if you held your hand there a moment longer, it suddenly felt like it was burning your hand. The sunspots were flipping their behavior depending on how deep or high in the Sun's atmosphere you were looking.

The Investigation: Fixing the Recipe

The scientists realized their "recipe book" (the ATLCW model) was wrong. The model predicted the sunspot would be cool everywhere, but the data showed it was hot at the top.

So, they tried to fix the recipe. They didn't rewrite the whole book; they just tweaked two main ingredients:

Density: How crowded the particles are in the sunspot's atmosphere.
Temperature: How hot those particles are.

What they found:

The Density: The sunspot's atmosphere is actually much thinner (less dense) than the recipe book said. Imagine the air in the sunspot is like a thin fog, whereas the model thought it was a thick cloud.
The Temperature: The sunspot is actually hotter than the model predicted, especially in the upper layers.

Even after tweaking these ingredients, the new recipe still didn't perfectly match the measurements. The model still thought the sunspot was too cool at the longest radio wavelengths.

Why Was It So Hard to Get Right?

The paper explains a few reasons why this is tricky:

The "Blurry Camera" Effect: The radio telescopes used (like ALMA's single dish) are a bit like a camera with a slightly out-of-focus lens. They can't perfectly separate the dark center of the sunspot (the umbra) from the lighter, swirling edges (the penumbra). It's like trying to taste a soup where you can't tell if you're tasting the broth or the chunks of vegetables; you just get the average flavor.
Missing Ingredients: The model assumes the sunspot is a calm, static place. But the Sun is a dynamic, churning ball of gas. The model ignores things like magnetic fields acting like invisible springs or shocks that might heat things up unexpectedly.
The "Invisible" Heat: At the longest wavelengths, the sunspot might be glowing due to a special type of radiation caused by its strong magnetic field (called gyroresonance). The current model didn't include this "secret ingredient," which is why the model kept underestimating the heat at the long end of the spectrum.

The Takeaway

This study is a great example of science in action.

We have a theory: Sunspots are cool everywhere.
We get new data: New telescopes show sunspots get hot at long wavelengths.
We adjust the theory: We change the numbers for density and temperature to fit the data better.
We realize we need more work: Even with the changes, the theory isn't perfect. We need to build a better model that accounts for magnetic fields and the fact that the Sun is a messy, moving place, not a static picture.

In short: Sunspots are like chameleons. They look cool and dark up close, but if you look at them from a different "angle" (wavelength), they turn out to be surprisingly hot and bright. The scientists are now working on a better map to understand exactly why they change colors.

1. Problem Statement

Sunspots are compact regions of lower emission in visible light but exhibit complex thermal and density structures in the solar atmosphere. While semi-empirical models exist for the quiet Sun (QS) and active regions, accurately modeling sunspot brightness temperatures ( $T_b$ ) across the millimeter (mm) and sub-millimeter (sub-mm) range (0.3 – 10 mm) remains challenging.

Observational Gap: Previous studies were limited by insufficient data or failed to reproduce observed emission intensities. The advent of the Atacama Large Millimeter/submillimeter Array (ALMA) provided unprecedented spatial resolution, yet a comprehensive model fitting sunspot $T_b$ across the full ALMA range was lacking.
Model Discrepancy: Existing semi-empirical models, specifically the Avrett-Tian-Landi-Curdt-Wülser (ATLCW) sunspot model, were found to poorly reproduce observed brightness temperatures, particularly failing to capture the transition where sunspots appear darker than the QS at short wavelengths but brighter at longer wavelengths.
Physical Complexity: Sunspots involve strong magnetic fields, non-local thermodynamic equilibrium (NLTE) effects, and non-equilibrium (NEQ) ionization, which are often neglected in static 1D models.

2. Methodology

The authors employed a semi-empirical modeling approach to derive improved sunspot density and temperature profiles by fitting theoretical models to multi-instrument observational data.

Input Model: The study utilized the ATLCW sunspot model (Avrett et al., 2015) as the baseline. This is a static, 1D, non-LTE model specifying electron density ( $n_e$ ) and temperature ( $T_e$ ) profiles as a function of height.
Radiative Transfer Calculation:
- Assumed thermal bremsstrahlung as the dominant emission mechanism in the 0.3–10 mm range.
- Calculated brightness temperature ( $T_b$ ) using the Rayleigh-Jeans approximation and integrating optical depth ( $\tau$ ) over height (0 to 58,100 km).
- Neglected gyroresonance contributions for most wavelengths, assuming magnetic fields at chromospheric heights observable by ALMA are too weak to significantly affect mm emission, though this is discussed as a limitation.
Optimization Technique:
- Applied a $\chi^2$ -minimization technique to find the best fit between the model and observations.
- Modified the input ATLCW model by applying uniform multiplicative scaling factors ( $f_n$ for density and $f_T$ for temperature) across all heights.
- Avoided height-dependent scaling to prevent over-parameterization.
Observational Data:
- ALMA: Single-dish full-disk observations at Band 6 (1.21 mm) and Band 3 (2.80 mm).
- JCMT: Sub-mm observations at 0.35 mm and 0.85 mm (Lindsey & Kopp, 1995).
- Nobeyama: Radio telescope measurements at 2.60 mm, 3.50 mm, and 8.80 mm (Iwai et al., 2016; Iwai & Shimojo, 2015).
- Data Processing: Sunspot regions (umbra + penumbra) were extracted. Limb brightening effects were corrected by comparing sunspot $T_b$ to QS regions at similar radial distances. Two measurement sets were used: "a" (absolute $T_b$ ) and "b" (differential $T_b$ relative to the QS model).

3. Key Contributions

First Comprehensive ALMA Sunspot Fit: This work extends previous QS and Active Region modeling (Paper I & II) to sunspots, utilizing the full 0.3–10 mm range covered by ALMA and complementary radio telescopes.
Quantitative Parameter Correction: The study provides specific scaling factors required to correct the ATLCW sunspot model to match observations, revealing significant discrepancies in the original model's density and temperature assumptions.
Validation of Emission Mechanisms: Confirms that thermal bremsstrahlung is the dominant mechanism but highlights the limitations of static 1D models in capturing the full physics of sunspot emission, particularly at long wavelengths.
Methodological Framework: Demonstrates a robust workflow for combining single-dish data from multiple instruments (ALMA, JCMT, Nobeyama) to constrain atmospheric models where interferometric data is sparse.

4. Key Results

Observational Trends:
- Short Wavelengths (< 2 mm): Sunspots are significantly darker than the QS (e.g., ~3,600 K at 0.35 mm vs. ~4,800 K for QS).
- Long Wavelengths (> 2 mm): Sunspots become brighter than the QS, with the contrast increasing up to 8.8 mm (reaching ~9,000 K).
- This confirms a transition from photospheric/low-chromospheric cooling to high-chromospheric/coronal heating relative to the surrounding QS.
Model Fitting Outcomes:
- The original ATLCW sunspot model failed to reproduce observations, predicting sunspots to be darker than QS across the entire range and underestimating brightness at long wavelengths.
- Modified Model: To achieve the best fit, the authors derived the following scaling factors:
  - Electron Density ( $f_n$ ): The best-fit density is 3.6 – 4.0 times lower than the original ATLCW sunspot model. However, compared to the QS, the sunspot density is 1.4 – 1.8 times higher.
  - Temperature ( $f_T$ ): The best-fit temperature is ~1.2 times higher than the original ATLCW sunspot prediction, but ~1.2 times lower than the QS temperature.
Residual Discrepancies: Even with optimized density and temperature, the modified model still underestimates brightness temperatures at wavelengths > 5 mm (particularly at 8.8 mm), suggesting missing physics in the model.

5. Significance and Limitations

Significance:
- The study provides a critical benchmark for sunspot physics, demonstrating that while the ATLCW QS model is robust, the sunspot version requires fundamental revision.
- It establishes that sunspots are denser and hotter than the QS in the chromosphere, contrary to some simplified assumptions, likely due to magnetic confinement.
- The results support the use of single-dish data for global sunspot characterization when interferometric data is unavailable.
Limitations & Future Work:
- Static 1D Assumption: The model ignores time-dependent dynamics, NEQ ionization, and magnetohydrostatic balance, which can alter brightness temperatures by hundreds of Kelvin.
- Gyroresonance: The model neglects gyroresonance emission, which may become significant at longer wavelengths (e.g., 8.8 mm) where magnetic fields are strong enough.
- Spatial Resolution: Single-dish observations mix umbra and penumbra, potentially smoothing out substructure. The authors note that interferometric observations are needed to separate these components and refine 2D/3D models.
- Conclusion: The ATLCW sunspot model is insufficient for accurate absolute predictions. Future work must incorporate height-dependent scaling, NEQ effects, and magnetic field contributions (gyroresonance) to fully resolve the discrepancies, particularly at long mm wavelengths.