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Understanding the magnetic field and plasma-$\beta$ along umbral fan loops traced using 3-min slow waves

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Umbral fan loops traced by 3-min slow waves show magnetic field falling from about 2000 G at the photosphere to about 200 G in the corona, with plasma-beta below 1 along the whole loop.

desk verdict First full-height B and beta profiles along individual sunspot fan loops, with a credible but under-tested lower-atmosphere area calibration. read the letter →

arxiv 2507.16283 v1 pith:BHUU3ZKY submitted 2025-07-22 astro-ph.SR

Ananya Rawat , Girjesh Gupta , Tom Van Doorsselaere , S. Krishna Prasad , Robertus Erdélyi This is my paper · ORCID
classification astro-ph.SR
keywords SolaratmosphereSunspotsUmbralfanloops3-minslowwavesPlasmabetaMagneticfieldstrengthMagnetohydrodynamicsCoronalseismology
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that sunspot fan loops can be traced continuously from the photosphere to the corona by cross-correlating 3-minute slow oscillations across AIA, IRIS, and HMI passbands, and that this tracing makes it possible to map the magnetic field strength and plasma-beta along individual loops for the first time. Using flux conservation, the photospheric field measured by HMI (1596-2269 G across the sampled loops) is scaled by the measured loop cross-sectional area to obtain the field at every height, reaching 158-236 G at the coronal footpoints. The resulting plasma-beta is below unity all along the loops, with the beta=1 layer located 200-1000 km below the photosphere. If correct, the method turns the ubiquitous 3-min umbral oscillations into a seismological tool for probing the magnetic structure of sunspot atmospheres and for testing MHD wave models in expanding waveguides.

What carries the argument

The load-bearing identity is magnetic flux conservation, $B(h)A(h)=B_p A_p$, applied along a loop whose cross-sectional area $A(h)$ is measured at each atmospheric height. In the lower atmosphere the area comes from closed contours at $94\pm2\%$ of the maximum correlation-coefficient value in correlation images of 3-min filtered light curves, calibrated at the coronal footpoint against the FWHM-derived diameter; in the corona, $A(h)$ comes from Gaussian fits to cross-loop intensity profiles. The photospheric anchor $B_p$ is the RMS HMI magnetogram value, and the temperature and density needed for plasma-$\beta$ come from a sunspot model at lower heights and from differential emission measure analysis in the corona, with the loop inclination fixed by comparing observed and hydrostatic density scale heights.

What would settle it

Compare the predicted $B(h)$ with an independent magnetic field measurement at a chromospheric or transition-region height along the same loops, e.g., spectropolarimetry of a chromospheric line or microwave imaging; if the measured field deviates from $B_p A_p / A(h)$ by more than the error bars at any height, the area-tracing assumption fails.

Watch

Extended reading notes

Core claim

The authors establish that the magnetic field and plasma-beta vary along individual umbral fan loops from the photosphere to the corona, using the loops' own 3-min slow oscillations as tracers. For the representative Loop 2, the field drops from 1928 +/- 13 G at the photospheric footpoint to 158 +/- 50 G at the coronal footpoint; across all studied loops the photospheric range is 1596-2269 G and the coronal range 158-236 G. Plasma-beta at the footpoints is 0.2-0.5 at the photosphere and 0.0001-0.001 in the corona, and the loops remain below beta=1 throughout, implying the magnetized plasma is magnetic-pressure-dominated in the whole visible atmosphere. The obtained patterns resemble the previously established active-region model, but with lower chromospheric beta values, which the authors attribute to tracing isolated loops rather than the integrated umbra.

Load-bearing premise

The cross-sectional area of each loop at lower atmospheric heights is taken to be the 94 +/- 2% maximum-correlation contour of the 3-min oscillation signal, a level calibrated at only the coronal footpoint and assumed to hold at all heights.

Editorial extensions

If this is right

  • A height-resolved $B(h)$ and $\beta(h)$ profile is now available for individual umbral loops, not just for the global corona or for isolated loop segments, which is the input needed to model wave propagation in density-stratified, expanding flux tubes.
  • The sub-photospheric $\beta=1$ layer found here identifies the region where mode conversion of slow waves can occur, an important constraint for umbral helioseismology.
  • Coronal fields of 158-236 G along these loops imply Alfv\'en speeds and energy fluxes much larger than typical global coronal values, strengthening the case for Alfv\'en-wave heating models powered by umbral oscillations.
  • The same correlation-tracing recipe can be applied to other multi-passband sunspot observations to build a statistical sample of loop magnetic parameters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the $\beta=1$ depth extrapolation is sensitive to the chosen linear fit, the true transition could be several hundred kilometres higher or lower; testing this requires measuring the plasma-beta profile independently in the temperature-minimum region.
  • The assumption that 3-min correlation patches represent the flux-tube cross-section at every height could be checked by comparing correlation-contour areas at chromospheric heights with radio or spectropolarimetric maps of the same umbral loops; disagreement would indicate contamination by the background umbral oscillation field.
  • Applied to loops without visible coronal emission, the technique might extend $B(h)$ mapping to faint or newly forming loops, where coronal FWHM calibration would need to be replaced by a different anchor.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. This paper derives magnetic field strength and plasma-β along individual umbral fan loops from the photosphere to the corona. The authors use IRIS and SDO observations, trace loops in the lower atmosphere via 3-min slow-wave correlation analysis, obtain cross-sectional areas at multiple heights, and then apply flux conservation, B(h) = Bp Ap / A(h), with Bp from the HMI magnetogram. For the representative Loop 2 in AR 12470 they report a photospheric field of 1928 ± 13 G, a coronal footpoint field of 158 ± 50 G, β ≈ 0.3 at the photospheric footpoint and β ≈ 0.0006 at the coronal footpoint, and β < 1 along the whole loop with the β ≈ 1 layer at sub-photospheric heights. Similar results are summarized for six loops in two active regions, and the trends are compared with the Gary (2001) umbral model.

Significance. If the area estimates are reliable, this is a genuinely novel combination of wave seismology and flux-tube theory: it gives the first height-resolved magnetic-field and plasma-β profiles along individual umbral fan loops spanning photosphere to corona. The method is empirically grounded (Bp is measured, not fitted) and the paper reports error bars and a transparent propagation of uncertainties. The results are potentially important for wave-heating models and for constraining umbral atmosphere models. The main weakness is that the decisive lower-atmosphere cross-sectional areas rest on a correlation-contour calibration performed at a single height, so the headline B(h) and β(h) profiles require additional validation before the claims can be considered robust.

major comments (4)
  1. [Section 3.3, Appendix C, Eq. (5)] The lower-atmosphere cross-sectional areas, which enter Eq. (5) linearly, are defined by contours at 94 ± 2% of the maximum correlation coefficient, but the contour level is calibrated to the FWHM-based area only at the coronal footpoint (92%). The manuscript provides no sensitivity test of the chosen level and no independent verification of these areas. Because the 3-min band is dominated by umbral oscillations, the correlation patches could be broadened by background spatial coherence or by the 0.332 arcsec/pixel resolution rather than by the true flux-tube cross-section. A systematic error in A(h) propagates linearly into B(h) via Eq. (5) and quadratically into β via Eq. (2). I request a quantitative sensitivity study (e.g., repeating the analysis for contour levels from 90% to 98%) and, if possible, a comparison with an independent estimate such as magnetic flux conservation from HMI at the footpoint or a modeled flux-tube expansion.
  2. [Section 3.1, Fig. 2, Eq. (2)] The density in the lower atmosphere is taken from the Fontenla sunspot model and then arbitrarily divided by a factor of 2 to match the DEM density at the coronal footpoint. This scaling factor has no quoted uncertainty and is not propagated into the β estimates. Since β ∝ N, the reported βp values in Table 1 and the qualitative conclusion β < 1 along the whole loop depend directly on this ad hoc factor. The paper should either justify the scaling factor with a fitting procedure over the full height range or explicitly include it in the systematic-error budget and show how βp changes if the factor is varied over a plausible range.
  3. [Section 3.4, Table 1] The photospheric magnetic field Bp is obtained from HMI at a formation height of approximately 269 km, but the photospheric loop area Ap ≈ 0.22 arcsec² is smaller than one HMI pixel (0.5 arcsec ≈ 0.25 arcsec²). The manuscript does not state whether Bp is the unsigned line-of-sight field or a vector-field magnitude, nor how the coarse HMI spatial resolution and the reported µ ≈ 0.90 projection geometry affect the value inserted into Eq. (5). This is important because Bp sets the absolute normalization of B(h) and therefore of β(h). Please clarify the precise quantity used and assess the resulting systematic uncertainty.
  4. [Section 3.4, Fig. 4] The statement that the β ≈ 1 layer lies at sub-photospheric heights is an extrapolation, not a measurement: the authors fit a straight line to β values between the temperature-minimum region and the photosphere, and extend the line to β = 1 at negative heights. This extrapolation is model-dependent and no uncertainty is given for the quoted heights in Table 1 (e.g., −617 km for Loop 2). Since β < 1 at all measured heights, the sub-photospheric location of the β = 1 layer should be presented as a tentative inference, with an error estimate or at least a clear caveat.
minor comments (4)
  1. [Abstract] The final sentence of the abstract, 'we provide estimates on the magnetic field and plasma-β variations along the various fan loops traced from the photosphere to the corona using 3-min waves and along the corona', contains a redundant 'and along the corona' and should be rewritten for clarity.
  2. [Section 3.1] The paper states that the temperature sensitivity of each passband is 'well known' and then assigns a single formation height to each filter. Since IRIS/AIA passbands have broad response functions, a brief discussion of the systematic uncertainty in the assigned formation heights would strengthen the density and β profiles.
  3. [Section 3.1] The assumed 15% error in density and temperature is stated without justification or a reference for this specific value; given that these errors dominate the β uncertainty, a short justification would be helpful.
  4. [Figure 4] The black solid lines representing the Gary (2001) umbral model are useful, but the paper does not quantify the discrepancy between the model β values and the loop β values; adding a brief quantitative statement would make the comparison more informative.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: B(h) follows from flux conservation using measured HMI field and independently measured areas; the density scaling is a calibration, not a fitted prediction, and the beta=1 height is an extrapolation, not a circular reduction.

full rationale

The central magnetic-field derivation is not circular. The paper uses B(h) = Bp Ap / A(h) (Eq. 5), where Bp is an RMS field strength measured from HMI magnetograms and A(h) is a cross-sectional area obtained from independent measurements: FWHM Gaussian fits in the corona and correlation-contour areas in the lower atmosphere. No term in this equation is defined in terms of the target result. The lower-atmosphere density used for plasma-beta is taken from the Fontenla et al. (1999) sunspot model and scaled by a constant factor (1/2) to match the DEM density at the coronal footpoint; this is an explicit calibration of a model input, not a fit to the beta values themselves, and the qualitative conclusion beta<1 is insensitive to that factor. The beta=1 height is obtained by fitting a straight line between the temperature-minimum region and photospheric beta values and extrapolating; this is a modeling choice that carries uncertainty, but it is not a circular reduction of the kind where the output is equivalent to the input by construction. The tracing technique is cited from the authors' prior work (Rawat & Gupta 2023), but it is re-described in detail in Appendix C, supported by correlation images, and anchored to the FWHM measurement at the coronal footpoint, so the self-citation is not an unverified load-bearing premise. Systematic concerns about the 94 +/- 2% correlation-contour level representing true loop cross-sections, and about the density scaling, are correctness risks rather than circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The model rests on several external assumptions from prior literature and one fitted scaling. The new measurement is the area expansion and the photospheric field; everything else is standard formulas with those inputs.

free parameters (2)
  • Density scaling factor for the Fontenla model = 0.5 (density divided by 2)
    In Section 3.1, the sunspot model density is divided by 2 so that it matches the DEM-derived density at the coronal footpoint. This scaling directly affects all lower-atmosphere plasma-beta values.
  • Correlation contour level for loop cross-section = 94 +/- 2% of the maximum correlation coefficient (92% at coronal footpoint)
    In Section 3.3 and Appendix C, the contour level is chosen so that the contour matches the FWHM-derived cross-section at the coronal footpoint, then applied at all lower heights.
assumptions (5)
  • domain assumption Magnetic flux is conserved along the loop (B(h)A(h) = constant)
    Used in Eq. 5 to derive B(h) from the photospheric measurement and measured areas. Assumes no magnetic reconnection or flux loss along the loop.
  • domain assumption The Fontenla et al. (1999) sunspot model correctly provides temperature and density as a function of height in the umbral lower atmosphere
    Used in Section 3.1 to assign N and T from the photosphere to the transition region; beta values depend directly on these inputs.
  • domain assumption The 3-min filtered oscillations observed in different passbands trace the same magnetic loop at all heights
    Central to the tracing technique (Section 3.3, Appendix C). If correlation peaks capture plasma outside the loop, areas and footpoints are wrong.
  • domain assumption The DEM emission at the coronal footpoint is optically thin with a filling factor of 1
    Used for coronal density Ne = sqrt(EM/w) in Section 3.1.
  • standard math Hydrostatic equilibrium scale height formula is valid for the inclined coronal loop
    Used in Eq. 3 to derive the loop inclination from the density scale height ratio.

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Pith. "Pith review of Understanding the magnetic field and plasma-$\beta$ along umbral fan loops traced using 3-min slow waves." pith.science (2026). https://pith.science/paper/BHUU3ZKY

@misc{pith2026250716283,
  author       = {Pith},
  title        = {Pith review of: Understanding the magnetic field and plasma-$\beta$ along umbral fan loops traced using 3-min slow waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BHUU3ZKY}},
  note         = {Machine review of arXiv:2507.16283}
}
abstract

The plasma-$\beta$ is an important fundamental physical quantity in solar plasma physics, which determines the dominating process in the solar atmosphere, i.e., magnetic or thermodynamic processes. Here, for the first time, we provide variations of magnetic field and plasma-$\beta$ along magnetically structured loops from the photosphere to the corona. We have selected several fan loops rooted in sunspot umbra observed simultaneously by the Interface Region Imaging Spectrograph and Solar Dynamics Observatory. The 3-min slow waves enabled us to trace and analyze several fan loops with cross-sectional areas in the lower atmosphere and locate their footpoints at the photosphere. We find the RMS magnetic field strengths in the range 1596-2269 G at the photospheric footpoints of the fan loops, which decrease rapidly to 158-236 G at the coronal footpoints. We estimated the plasma-$\beta$ at the photospheric and coronal footpoints in the range 0.2-0.5 and 0.0001-0.001, respectively. We found plasma-$\beta$$<$$1$ along the whole loop, whereas the plasma-$\beta$$\approx$$1$ layer is found to be at sub-photospheric heights. We compared our findings for isolated individual fan loops with a previously established model for active regions and found an almost similar pattern in variations with height, but with different plasma-$\beta$ values. Our results demonstrate the seismological potential of 3-min slow waves omnipresent in the umbral sunspot atmosphere to probe and map isolated loops and determine magnetic field and plasma-$\beta$ along these loops. The obtained parameters provide crucial ingredients for the theoretical modeling of the umbral atmosphere and wave dynamics along loops.

Figures

Figures reproduced from arXiv: 2507.16283 by the authors.

Figure 1
Figure 1. Images of sunspot and fan loops belonging to AR 12470 obtained from different AIA, IRIS, and HMI passbands as labeled. The red lines on the AIA 171 ˚A image represent the manual tracing of coronal fan Loops 1 and 2, and asterisk symbols (*) represent their coronal footpoints. The yellow dashed lines represent the background regions for those loops. The sample slits across the coronal Loop 2 are marked with solid red… view at source ↗
Figure 2
Figure 2. Left: DEM profiles at the coronal footpoint of the Loop 2 shown by the asterisk in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Left: Cross-sectional area of the loop along its length. The area of the loop in the lower solar atmosphere is fitted with an exponentially rising function, shown in a blue solid line. The area along the coronal loop is also fitted with an exponentially rising function with a constant background, shown in the red solid line. Obtained area scale heights (λA) are printed in their respective color code at the top left … view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Left: Variation of plasma-β along the fan loops belonging to AR12470 where light and dark blue shaded regions are for loops 1 and 2, respectively. Right: Variation of plasma-β along the fan loops belonging to AR12553 where the light blue, dark blue, and pink shaded reg…
Figure 5
Figure 5. Figure 5: Image of the fan loop system belonging to AR 12553 obtained from AIA 171 ˚A passband studied in A. Rawat & G. Gupta (2023). The yellow solid lines represent the manual tracing of coronal Loops 3, 5, and 6 as labeled, and asterisk symbols (*) represent their coronal foo…
Figure 6
Figure 6. Figure 6: Intensity profiles along several slits across the coronal Loop 2 marked in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Correlation images obtained between various atmospheric heights as labeled. In each panel, the asterisk symbol (*) in the center refers to the coronal footpoint of Loop 2, and the red circle represents the cross-section of the loop obtained from the AIA 171 ˚A image us…

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Reviewed August 6, 2026 · model on record in the stance chip above.