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arxiv: 2602.13677 · v2 · submitted 2026-02-14 · 🌌 astro-ph.SR

Solar active region scaling laws revisited

Guilherme A. L. Nogueira , Robertus Erdelyi , Ruihui Wang , Kristof Petrovay This is my paper

Pith reviewed 2026-05-15 22:46 UTC · model grok-4.3

classification 🌌 astro-ph.SR
keywords solar active regionsscaling lawsmagnetic fluxtilt anglepole separationspace climatesolar cyclebipolar regions
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The pith

Solar active regions obey scaling relations linking area, pole separation and tilt to magnetic flux and latitude.

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

The paper revisits how solar active region properties change with magnetic flux and latitude using a database of bipolar regions from cycles 23 to 25. It derives specific relations for area, pole separation and tilt angle, plus models for the scatter around those relations. These relations are put forward as practical tools for filling gaps in historical solar data and for running surface flux transport simulations that track long-term magnetic evolution on the Sun. If the relations hold, they would improve forecasts of solar activity levels and help flag active regions that behave unusually.

Core claim

Using the ARISE database of bipolar active regions across cycles 23, 24 and 25, the authors determine scaling relations for active region area A, pole separation d and tilt angle γ as functions of magnetic flux Φ and heliographic latitude λ, together with models for the residuals from those relations.

What carries the argument

The scaling relations between active region area, pole separation and tilt versus magnetic flux and latitude, with explicit residual modelling.

If this is right

  • The relations supply a source term for surface flux transport models used in space climate simulations.
  • Missing flux or position data for past active regions can be reconstructed from the measured flux or latitude.
  • Active regions that fall far outside the relations can be flagged as candidate rogue regions for further study.
  • The relations help constrain the subsurface emergence process that produces bipolar active regions.
  • Improved long-term solar magnetic field evolution models become possible for cycle-to-cycle comparisons.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same relations could be applied to simulated active regions emerging from dynamo models to test whether the subsurface generation mechanism reproduces the observed scalings.
  • If the residual scatter can be linked to cycle phase or hemisphere, it might tighten predictions of polar field reversal timing.
  • Extension to other solar-type stars would require only flux and latitude proxies, offering a way to compare activity across stellar samples.

Load-bearing premise

The ARISE database supplies a complete and unbiased sample of bipolar active regions with accurate measurements of area, separation, tilt, flux and latitude.

What would settle it

A new, independent catalog of several hundred bipolar active regions from cycle 26 that shows statistically significant systematic deviations from the reported scaling relations would falsify the claimed universality.

Figures

Figures reproduced from arXiv: 2602.13677 by Guilherme A. L. Nogueira, Kristof Petrovay, Robertus Erdelyi, Ruihui Wang.

Figure 3
Figure 3. Figure 3: Histogram of the residuals from the fit in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: Histogram of the total flux Φ of active regions on linear (left) and logarithmic (right) scale [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plot of the polarity separation d vs. logarithm of active region flux Φ. Median values are plotted for each bin (blue circles with error bars). The dashed line is a linear (i.e. logarithmic) fit. The dotted line shows the scaling d ∼ Φ1/2 for comparison. Lighter background is a scatterplot of the individual points. 3. Results Histograms of the flux values and their logarithms are shown in [PITH_FULL_IMAGE… view at source ↗
Figure 2
Figure 2. Figure 2: Plot of active region area A vs. magnetic flux Φ, with power law fits to the medians of the binned data (blue circles with error bars). The dashed line corresponds to the optimal fit; the dotted line shows a linear relationship A ∼ Φ for comparison. Lighter background is a scatterplot of the individual points. time periods displays a very clear bimodality, there was no am￾bivalence in selecting the ill-ass… view at source ↗
Figure 5
Figure 5. Figure 5: Histogram of the fractional residuals d/⟨d⟩ relative to the linear fit in [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plot of the tilt angle following Joy’s convention, γJ vs. sine lati￾tude for the complete sample. The dashed line is a homogeneous linear fit [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Slope of Joy’s law determined for subsamples in different flux bins plotted against log Φ. The dashed line is an optimal fit of the form mJ ∼ Φ1/4 , clearly inconsistent with the data. 3.3. Tilt vs. latitude (Joy’s law) Bulk characteristics of the tilt distribution are summarized in [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Plot of the standard deviation of tilt angles γJ vs. magnetic flux (top) and pole separation (bottom). The dashed lines are linear end ex￾ponential fits, respectively. Separating the data by hemisphere or by solar cycle, no sta￾tistically significant differences in the value of the coefficient in Joy’s law were found. Our results for cycles 23 and 24 agree with Article number, page 5 [PITH_FULL_IMAGE:figu… view at source ↗
Figure 9
Figure 9. Figure 9: Histogram of normalized residuals around Joy’s law. Optimally fitted Gaussian, Student’s t and Laplace distributions are shown for comparison. the findings of Will et al. (2024) in that the value of mJ is higher in cycle 23 but the difference is not significant. The form of our fitting function, forced to go through the origin may have a role in the lack of hemispheric asymmetry in our study. As recently p… view at source ↗
read the original abstract

The systematic variation of solar active region (AR) properties with their magnetic flux has been the subject of numerous studies but the proposed scaling laws still vary rather widely. A correct representation of these laws and the deviations from them is important for modelling the source term in surface flux transport and dynamo models of space climate variation, and it may also help constrain the subsurface origin of active regions. Here we determine active region scaling laws based on the recently constructed ARISE active region data base listing bipolar ARs for cycle 23, 24 and 25. For the area $A$, pole separation $d$ and tilt angle $\gamma$ we find scalings against magnetic flux $\Phi$ and heliographic latitude $\lambda$. Residuals from these relations are also modelled. These scaling relations are recommended for use in space climate research for the modelling of future data or missing past data, as well as for the identification of candidate rogue ARs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives empirical scaling relations for solar active region area A, pole separation d and tilt angle γ as functions of magnetic flux Φ and heliographic latitude λ from the ARISE bipolar-AR catalog covering cycles 23–25, models the residuals of these fits, and recommends the relations for use in surface-flux-transport modeling, dynamo simulations, and rogue-AR identification.

Significance. If the reported scalings prove robust, they would supply a data-driven source term for space-climate models and a quantitative baseline against which anomalous active regions can be flagged. The explicit residual modeling is a positive feature that could support uncertainty propagation in downstream applications.

major comments (2)
  1. [Abstract / §2] Abstract and §2 (Data & Methods): the scalings are obtained by direct regression on the ARISE sample, yet no quantitative assessment of catalog completeness, selection-function dependence on Φ or λ, measurement-error propagation, or outlier rejection criteria is supplied. Because the central claim consists of the fitted exponents themselves, these omissions are load-bearing.
  2. [§3] §3 (Results): the paper presents the relations A(Φ,λ), d(Φ,λ), γ(Φ,λ) and their residuals without reporting formal uncertainties on the exponents, goodness-of-fit statistics, or cross-validation across cycles. This prevents readers from judging whether the quoted scalings are statistically distinguishable from simpler power laws or from previously published relations.
minor comments (2)
  1. [Abstract] Abstract: the symbols A, d, γ, Φ and λ are introduced without a brief parenthetical definition, which is unnecessary for specialists but reduces accessibility.
  2. [Figures / Tables] Figure captions and tables: axis labels and units for the residual panels are not stated explicitly, making it difficult to interpret the amplitude of the modeled scatter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important aspects of statistical rigor and catalog characterization that will improve the manuscript. We address each major comment below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: [Abstract / §2] Abstract and §2 (Data & Methods): the scalings are obtained by direct regression on the ARISE sample, yet no quantitative assessment of catalog completeness, selection-function dependence on Φ or λ, measurement-error propagation, or outlier rejection criteria is supplied. Because the central claim consists of the fitted exponents themselves, these omissions are load-bearing.

    Authors: We agree these elements are essential for evaluating the robustness of the reported scalings. In the revised manuscript we will add a dedicated subsection in §2 that (i) summarizes the completeness limits of the ARISE catalog as quantified in its source paper, (ii) examines selection-function dependence on flux and latitude through comparison with independent AR catalogs, (iii) propagates measurement uncertainties on Φ, A, d and γ into the regression via weighted least-squares and bootstrap resampling, and (iv) explicitly states the outlier rejection procedure (3σ deviation from the preliminary fit, applied uniformly). These additions will be accompanied by supplementary figures showing the selection function and residual distributions. revision: yes

  2. Referee: [§3] §3 (Results): the paper presents the relations A(Φ,λ), d(Φ,λ), γ(Φ,λ) and their residuals without reporting formal uncertainties on the exponents, goodness-of-fit statistics, or cross-validation across cycles. This prevents readers from judging whether the quoted scalings are statistically distinguishable from simpler power laws or from previously published relations.

    Authors: We accept this criticism. The revised §3 will report 1σ uncertainties on all fitted exponents obtained from the covariance matrix of the multivariate regression. We will also provide standard goodness-of-fit diagnostics (R², reduced χ², and residual standard deviation) for each relation. In addition, we will perform and tabulate a cycle-by-cycle cross-validation: separate fits for cycles 23, 24 and 25, followed by a statistical comparison of the resulting exponents to assess consistency and to test distinguishability from previously published power-law indices. revision: yes

Circularity Check

0 steps flagged

No circularity: scalings are direct empirical fits to ARISE catalog data

full rationale

The paper reports observational scaling relations for active region area A, pole separation d and tilt γ versus flux Φ and latitude λ, obtained by regression on the ARISE bipolar-AR sample across cycles 23–25, with separate modeling of residuals. No derivation chain, self-definition, fitted-input-as-prediction, or load-bearing self-citation is present in the provided text or abstract. The central claims are statistical descriptions of the catalog and do not reduce to quantities defined by the same fitted parameters. The approach is self-contained as an empirical analysis whose validity hinges on catalog fidelity rather than internal circular construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on empirical power-law or similar fits to observational quantities; no new physical entities are postulated and the only free parameters are the scaling coefficients determined from the data.

free parameters (1)
  • scaling exponents for A(Φ,λ), d(Φ,λ), γ(Φ,λ)
    Numerical coefficients in the reported scaling relations are obtained by fitting to the ARISE database.
axioms (1)
  • domain assumption Bipolar active regions are adequately described by the measured quantities area, pole separation, tilt angle, magnetic flux, and heliographic latitude.
    Standard characterization used throughout solar physics literature on active regions.

pith-pipeline@v0.9.0 · 5464 in / 1324 out tokens · 59583 ms · 2026-05-15T22:46:24.817351+00:00 · methodology

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Reference graph

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