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REVIEW 3 major objections 4 minor 83 references

The role of the Lorentz force in sunspot equilibrium

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Sunspots keep azimuthal force balance despite fine structure.

desk verdict First observational confirmation of the intraspine pressure excess, but the inference-dependent azimuthal correlation needs a synthetic-inversion null test before the 'conclusively' claim holds. read the letter →

arxiv 2505.23986 v1 pith:MLJOZ3KQ submitted 2025-05-29 astro-ph.SR

J.M. Borrero , A. Pastor Yabar , M. Schmassmann , M. Rempel , M. van Noort , M. Collados This is my paper · ORCID
classification astro-ph.SR
keywords sunspotsmagnetohydrostaticequilibriumLorentzforceazimuthaldirectionpenumbralfinestructureStokesinversionmagnetohydrodynamicsimulationsgaspressurefluctuations
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

Sunspots are long-lived, yet the standard theoretical models treat them as static, axially symmetric structures and ignore the strong azimuthal (around-the-spot) variations of the magnetic field produced by penumbral filaments, umbral dots, and light bridges. This paper asks whether magnetohydrostatic equilibrium still holds when those azimuthal fluctuations are included. By inverting spectropolarimetric data from two nearly disk-centered sunspots, one observed from the ground and one from space, and comparing with three-dimensional radiative magnetohydrodynamic simulations, it verifies the azimuthal force-balance equation $L_\phi = r^{-1}\,\partial P_g/\partial \phi$ at the $\tau_c=1$ level in both umbra and penumbra, with correlations near 0.9 and slopes consistent with unity. The paper concludes that sunspot fine structure is in almost perfect magnetohydrostatic equilibrium along the azimuthal direction, which helps explain why sunspots persist for days to months. It also reports the first observational confirmation that penumbral intraspines carry excess gas pressure and density relative to spines.

What carries the argument

The load-bearing object is the azimuthal component of the magnetohydrostatic force balance, Eq. (2): $L_\phi = r^{-1}\partial P_g/\partial\phi$, obtained from $\nabla P_g = \rho\mathbf{g} + \mathbf{L}$. This identity converts magnetohydrostatic equilibrium into a directly testable statement: wherever the magnetic field fluctuates along the azimuthal coordinate $\phi$, the gas pressure must fluctuate in lockstep, with the same amplitude and sign. The machinery that makes the test possible is a Stokes inversion under a magnetohydrostatic constraint, in which $\ln P_g$ is obtained from a Poisson-type equation involving the Lorentz force, combined with a PSF-coupled inversion that recovers fine structure, a decomposition of $L_\phi$ into magnetic tension and magnetic-pressure-gradient terms, and a radiative-MHD sunspot simulation degraded to the same spatial resolution as the observations.

What would settle it

Run the paper's magnetohydrostatic inversion on synthetic Stokes profiles generated from the radiative-MHD simulation, where the true gas pressure is known, and compare the inverted $r^{-1}\partial P_g/\partial\phi$ with the true value at $\tau_c=1$; if the inversion reproduces the true gradient only when the damped Lorentz force used inside the inversion is close to the actual Lorentz force, then the observed correlation is partly an artifact of the method.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the azimuthal component of the magnetohydrostatic momentum equation, $L_\phi = r^{-1}\partial P_g/\partial\phi$, is satisfied in the photosphere of both umbra and penumbra despite the strong fine-structure variations of the magnetic field. Here $L_\phi$ is the azimuthal component of the Lorentz force $\mathbf{L}=(4\pi)^{-1}(\nabla\times\mathbf{B})\times\mathbf{B}$ and $P_g$ is the gas pressure. The verification is quantitative: Pearson correlations between $L_\phi$ and $r^{-1}\partial P_g/\partial\phi$ reach about 0.93 in both penumbra and umbra for the ground-based sunspot, 0.87 and 0.88 for the space-based sunspot, and 0.97 and 0.996 for the degraded simulation, with linear-fit slopes close to one. The paper also decomposes $L_\phi$ into magnetic tension and magnetic pressure gradient, finding that the magnetic pressure gradient dominates and peaks at spine/intraspine boundaries. It further shows that these relations disappear when the inversion is run under vertical hydrostatic equilibrium instead of magnetohydrostatic equilibrium, indicating that the recovered $\partial P_g/\partial\phi$ fluctuations are tied to the Lorentz-force balance.

Load-bearing premise

The load-bearing premise is that the test measures an independent gas pressure, whereas in fact the inversion derives $P_g$ from a magnetohydrostatic Poisson equation that already contains the Lorentz force, so the near-unit correlation at $\tau_c=1$ may be partly manufactured by the inversion's own constraint.

Editorial extensions

If this is right

  • Azimuthal force balance can be treated as magnetohydrostatic to leading order, with the time-derivative, advective, and viscous terms negligible for the spine/intraspine structure at $\tau_c=1$.
  • Penumbral intraspines are confirmed to have excess gas pressure and density, supporting the horizontal-field, uncombed model of penumbral filaments.
  • Because the magnetic pressure gradient dominates the azimuthal Lorentz force, gas-pressure fluctuations along $\phi$ are approximately predictable from field-strength fluctuations via $\Delta P_g \simeq -\Delta\|\mathbf{B}\|^2/(8\pi)$ where that approximation applies.
  • The observed correlations vanish when vertical hydrostatic equilibrium is assumed instead of magnetohydrostatic equilibrium, showing that the inferred gas-pressure and density variations are signatures of the magnetic force balance.
  • The long lifetimes of sunspots are consistent with a leading-order magnetohydrostatic equilibrium that survives even when azimuthal magnetic inhomogeneities are included.

Reading between the lines

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

  • The near-unit slope at $\tau_c=1$ may be partly a consequence of the inversion's own magnetohydrostatic constraint: the gas pressure is derived from a Poisson equation that already contains the Lorentz force, so the correlation between $L_\phi$ and $r^{-1}\partial P_g/\partial\phi$ is not purely an independent measurement. The paper's Appendix A shows the relation is not enforced at all heights, b
  • A decisive next test would run the same inversion on synthetic Stokes profiles generated from the radiative-MHD simulation, where the true gas pressure is known, and compare the inferred $r^{-1}\partial P_g/\partial\phi$ with the true value at $\tau_c=1$ to separate physical equilibrium from inversion-imposed balance.
  • The paper establishes azimuthal equilibrium but does not address radial or vertical equilibrium; because the Evershed flow's advective term matters radially in the simulations, the overall sunspot state is better described as magnetohydrostationary, with azimuthal balance closer to static magnetohydrostatics than radial balance.
  • In the observations the equilibrium is verified only in a narrow layer around $\tau_c=1$, while the simulations satisfy it over a wider height range; observations formed at other heights could determine whether the conclusion is height-restricted.
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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

3 major / 4 minor

Summary. This paper tests whether the azimuthal component of the magnetohydrostatic (MHS) force balance, Eq. (2) (L_phi = r^-1 dP_g/dphi), holds in sunspot umbra and penumbra once azimuthal fine structure is considered. The authors invert spectropolarimetric data of two sunspots (GREGOR/GRIS and Hinode/SP) with the FIRTEZ code, using both pixel-wise and PSF-coupled inversions, and compare the inferred gas pressure, density, magnetic field, and Lorentz force with a degraded MuRAM MHD sunspot simulation. They report high Pearson correlations (c around 0.87-0.93 in observations, 0.97-0.996 in the simulation) and unit slopes for the scatter of L_phi versus r^-1 dP_g/dphi at tau_c=1, and conclude that the azimuthal direction is in almost perfect magnetohydrostatic equilibrium. The paper explicitly addresses the potential circularity of using an MHS-constrained inversion to test an MHS equation in Appendix A, and it provides control inversions under vertical hydrostatic equilibrium and without PSF coupling in Appendix C.

Significance. The physical conclusion would be significant: it extends sunspot equilibrium studies beyond axially symmetric models and provides a direct observational test of the predicted gas-pressure and density enhancements in penumbral intraspines. The paper has two clear strengths: the MuRAM simulation gives an inversion-free verification that the same azimuthal balance holds in a realistic numerical sunspot, and the authors are transparent about the circularity issue by devoting a dedicated appendix to it. However, the observational part of the claim is currently not as strong as the wording 'prove' suggests, because the gas pressure used in the correlation is not measured independently: it is obtained from a Poisson equation (Eq. A.4) that contains the Lorentz force. A quantitative synthetic-inversion null test is needed before the observational verification of Eq. (2) can be regarded as established.

major comments (3)
  1. [Section 6.4, Figure C.1] The hydrostatic-equilibrium control inversion does not test for inversion-induced circularity. In the HE run, P_g is computed from Eq. (A.5) with the Lorentz term set to zero, so the disappearance of the correlation between L_phi and r^-1 dP_g/dphi in Fig. C.1(d) is a direct consequence of the construction and cannot distinguish a genuine physical balance from an artifact of the MHS inversion. I request a synthetic null test: take a model atmosphere in which Eq. (2) is knowingly violated (for example, a MuRAM snapshot with an artificially imposed lateral pressure imbalance), synthesize Stokes profiles, run the FIRTEZ MHS inversion, and compare the inferred L_phi and r^-1 dP_g/dphi at tau_c=1. This test would quantify how much of the observed c about 0.87-0.93 and unit slope is manufactured by Eq. (A.4) and its boundary conditions at the height where the spectral lines are most sensitive.
  2. [Appendix A, Eqs. (A.4) and (A.8)] The argument that Eq. (A.7) is not strictly imposed by FIRTEZ is incomplete. Equation (A.8) shows that in the limit where the non-potential part W of F = L'/P_g is small, the inversion solves a Poisson equation for ln P_g whose source contains the potential part of the Lorentz force per unit pressure; whether this also enforces the phi-component of the force balance depends on the relative magnitude of W and on the boundary conditions. At tau_c=1, where the inversion has the strongest diagnostic sensitivity and where L' is intended to approximate the true Lorentz force, W may be small, and then a high correlation with unit slope could emerge even if the real atmosphere is not in MHS equilibrium along phi. Please provide maps of |nabla x W|/|nabla Psi| (or an equivalent measure of the non-potential residual) in the region where the slope is close to unity, and give the explicit damping law defining L' (the beta-dependence, the current threshold, and the resulting ratio L'/L at tau_c=1) so that the reader can judge how much of the observed correlation is built into the inversion.
  3. [Section 6, paragraph after Eq. (3)] The sentence 'Our combined results prove that, along the azimuthal phi direction, penumbral intraspines/spines are in almost perfect magnetohydrostatic equilibrium' is disproportionate to the present evidence. The MuRAM simulation alone does provide an inversion-free proof for a modeled sunspot, but the observational verification is contingent on the MHS assumption used to infer P_g. Until the synthetic-inversion null test is performed, the observational result should be phrased as 'consistent with' azimuthal MHS equilibrium under the FIRTEZ MHS assumption, rather than as a proof. This is a load-bearing formulation issue because the abstract and introduction make the observational result a central part of the paper's claim.
minor comments (4)
  1. [Throughout] The manuscript contains several typesetting or spelling artifacts such as 'di fferent', 'di ffraction', 'Alv ´en', and 'Recieved'; these should be corrected in the production version.
  2. [Figure 3 and Figure 5] The linear fits are displayed but the slope values are not printed in the panels; reporting the fitted slope m and its uncertainty together with the correlation coefficient c would make the 'unit slope' claim directly quantifiable for each dataset.
  3. [Section 6.2] The label 'Simulations (/5)' in Figures 3 and 5 is ambiguous; the caption should explicitly state that the MHD simulation values of L_phi and r^-1 dP_g/dphi are divided by five for display purposes.
  4. [Appendix A] The modified Lorentz force L' is described only verbally and never written as an explicit equation; since the paper's central caveat depends on the relation between L' and L, an explicit expression (or a precise equation reference to Borrero et al. 2019) should be provided.

Circularity Check

1 steps flagged · score 6.0 of 10

Observational verification of Eq. (2) is partly built into FIRTEZ's MHS-constrained gas pressure, though the azimuthal component is not directly enforced and MHD simulations provide independent support.

  1. fitted input called prediction [Appendix A, Eq. A.4; Section 6.4, Fig. C.1; Fig. 3d]
    "FIRTEZ does not directly solve the momentum equation in magneto-hydrostatics (Eq. 3). Instead, it takes the divergence of that equation and iteratively solves the following second-order Poisson equation: ... it is clear now that in the presence of a magnetic field, FIRTEZ infers a gas pressure Pg that balances the potential part Ψ of the force F = L′/Pg."

    The gas pressure used to build r^-1 dPg/dphi is not an independent observable; it is produced by FIRTEZ's Poisson equation whose source contains the divergence of L'/Pg, including the phi-derivative of L'_phi/Pg. The paper explicitly says Pg is inferred to balance the potential part of L'/Pg, so the curl-free part of L_phi is installed in the phi derivative of Pg by construction. At tau_c=1, beta is large and L' is close to the true Lorentz force L, so the high correlation and near-unity slope in Fig. 3d are partly a self-consistency check rather than an independent test. This is confirmed by the paper's own HE control: under vertical hydrostatic equilibrium the correlations are 'completely lost,' i.e. the correlation appears only when the MHS constraint is inserted.

full rationale

The paper's central diagnostic compares two quantities that are not independent: L_phi is computed from the inferred B, while Pg is inferred by FIRTEZ under an MHS constraint whose Poisson equation (Eq. A.4) contains L'/Pg in its source. The paper itself acknowledges this concern in Appendix A and shows that the relation is not imposed at all heights, which demonstrates that the test is not purely tautological. Nevertheless, the statement that the inferred Pg 'balances the potential part' of L'/Pg means a substantial part of the observed correlation and unit slope at tau_c=1 is built into the inversion. The HE control in Fig. C.1 reinforces this interpretation: removing the MHS constraint removes the correlation. No load-bearing self-citation chain or imported uniqueness theorem was found; the FIRTEZ method is described in the paper rather than merely delegated. The MuRAM simulations provide an independent, parameter-free confirmation of azimuthal magnetohydrostatic equilibrium in a numerical sunspot, so the overall conclusion does not rest solely on the partly circular observational inference. The score reflects partial circularity in the observational claim, not full equivalence of the derivation to its inputs.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The central claim rests on the MHS pressure solution from FIRTEZ, which is obtained from an equation that includes a modified Lorentz force. The free parameters are inversion node counts, the PSF model, the current damping law, and the simulation boundary parameter. No new physical entities are introduced except the numerical construct L'.

free parameters (4)
  • FIRTEZ inversion node counts = T: 4 and 2 nodes; vz: 2 and 4; Bx, By, Bz: 2 and 4 (Table 2)
    Chosen by hand for the hydrostatic and magnetohydrostatic inversions. Node numbers set the vertical resolution of the inferred pressure, field, and velocity, so they directly influence the derived L and pressure gradients.
  • GREGOR seeing PSF parameters = Gaussian sigma = 0.5 arcsec, alpha = 0.43
    The seeing term in the GREGOR PSF is an approximate model with hand-set parameters; a broader 5 arcsec Gaussian was dropped for poor convergence. The PSF affects the coupled inversion and the inferred azimuthal fluctuations.
  • FIRTEZ current damping law for L' = beta^2-scaled damping above a maximum current threshold
    The pressure equation uses a modified Lorentz force L' where electric currents are suppressed in low-beta regions. The threshold and damping factor are ad hoc; they alter the inferred pressure and can create the correlation tested in the paper.
  • MuRAM simulation boundary alpha = alpha = 2
    Boundary condition from Rempel (2012) that makes the top-boundary magnetic field more horizontal; this shapes the simulated penumbra and is not fitted to the present observations.
assumptions (6)
  • domain assumption Magnetohydrostatic equilibrium equation grad P_g = rho g + L holds in the photosphere at resolved scales.
    This is the target relation; the inversion also assumes it in modified form when computing gas pressure (Eq. 3 and A.4).
  • ad hoc to paper FIRTEZ obtains gas pressure by solving the divergence of the MHS equation (a Poisson equation), not the equation itself.
    Eq. A.4 gives grad^2(ln Pg) = -... + div(L'/Pg). This numerical approach imposes force balance only in an integrated sense and introduces a model-dependent pressure.
  • domain assumption The damped Lorentz force L' approximates the true Lorentz force L near tau_c = 1.
    In the photosphere beta is large, so the current damping should be small; the paper relies on this for the observational test to be meaningful, but it is not quantitatively demonstrated.
  • domain assumption Stokes inversion assumes LTE and vertical radiative transfer with horizontal PSF coupling.
    These are the FIRTEZ assumptions for mapping Stokes profiles to T, B, and vz; the inferred pressure and density inherit these assumptions.
  • standard math Fourth-order finite differences on discrete azimuthal arcs approximate the gradients needed for L and dPg/dphi.
    The authors state that using second or sixth order derivatives does not change results (Section 3). This is a numerical discretization assumption when evaluating Eq. 2.
  • domain assumption The MuRAM simulation is an adequate proxy for the real sunspot photosphere.
    The simulation is used as an independent check, but its umbra contains penumbra-like filaments at the chosen timestamp and boundary parameters are tuned (Appendix B, Section 6.3).
invented entities (1)
  • Modified Lorentz force L' with damped electric currents
    purpose: Used in FIRTEZ's pressure equation to prevent unreliable currents in the upper atmosphere from dominating the force balance.
    L' is not the measured Lorentz force; it is a numerical construct whose damping depends on beta and an unspecified maximum current threshold. Since the inferred gas pressure is computed from L', the subsequent comparison of L with dPg/dphi is not fully independent.

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Cite this review

Pith. "Pith review of The role of the Lorentz force in sunspot equilibrium." pith.science (2026). https://pith.science/paper/MLJOZ3KQ

@misc{pith2026250523986,
  author       = {Pith},
  title        = {Pith review of: The role of the Lorentz force in sunspot equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLJOZ3KQ}},
  note         = {Machine review of arXiv:2505.23986}
}
read the original abstract

Sunspots survive on the solar surface for time-scales ranging from days to months. This requires them to be in an equilibrium involving magnetic fields and hydrodynamic forces. Unfortunately, theoretical models of sunspot equilibrium are very simplified as they assume that spots are static and possess a self-similar and axially symmetric magnetic field. These assumptions neglect the role of small scale variations of the magnetic field along the azimuthal direction produced by umbral dots, light bridges, penumbral filaments, and so forth. We aim at studying whether sunspot equilibrium is maintained once azimuthal fluctuations in the magnetic field, produced by the sunspot fine structure, are taken into account. To this end we apply the FIRTEZ Stokes inversion code to spectropolarimetric observations to infer the magnetic and thermodynamic parameters in two sunspots located at disk center and observed with two different instruments: one observed from the ground with the 1.5-meter German GREGOR Telescope and another with the Japanese spacecraft Hinode. We compare our results with three dimensional radiative magnetohydrodynamic simulations of a sunspot carried out with the MuRAM code. We infer clear variations in the gas pressure and density of the plasma directly related to fluctuations in the Lorentz force and associated with the filamentary structure in the penumbra. Similar results are obtained in the umbra despite its lack of observed filamentary structure. Results from the two observed sunspots are in excellent qualitative and quantitative agreement with the numerical simulations. Our results indicate that the magnetic topology of sunspots along the azimuthal direction is very close to magnetohydrostatic equilibrium, thereby helping to explain why sunspots are such long-lived structures capable of surviving on the solar surface for days or even full solar rotations.

Figures

Figures reproduced from arXiv: 2505.23986 by the authors.

Figure 1
Figure 1. Point Spread Functions employed in the data analy￾sis. Left panel: PSF (diffraction limited plus seeing) used in the coupled-inversion of GREGOR/GRIS data. Middle panel: PSF (diffraction limited) used in the coupled-inversion of Hinode/SP data. For both GREGOR and Hinode data the full PSF (±1.6 Mm) is used in the synthesis, whereas in the inver￾sion only the square box is considered. Right panel: ideal PSF (Airy fun… view at source ↗
Figure 2
Figure 2. Summary of analyzed data and inversion results. Top row: continuum intensity normalized to the average quiet Sun intensity Ic/Ic,qs. Middle row: temperature T. Bottom row: inclination of the magnetic field γ with respect to the normal vector to the solar surface. All values are taken at the continuum optical depth τc = 1. Left column: sunspot observed with the ground telescope GREGOR and data inverted with the FIRTE… view at source ↗
Figure 3
Figure 3. Scatter plots of the physical quantities along the outer-most azimuthal arc (i.e. penumbra) for GREGOR data (red squares), Hinode data (blue circles) and MHD simulation (black triangles). Pearson correlations coefficients c for each of these data are provided in the legend. Linear fits to the data are shown in thick color solid lines. Upper-left: γ − ∥B∥; Upper-right: γ − Pg; Lower￾left: γ − ρ; Lower-right: Lϕ − r −… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Decomposition of the Lorentz force into magnetic pres￾sure gradient and magnetic tension. Variations of the magnetic pressure gradient Gϕ (blue) and magnetic tension Tϕ (red) along the azimuthal coordinate ϕ indicated in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Same as [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Works this paper leans on

83 extracted references · 66 canonical work pages

  1. [1]

    Anstee , S. D. & O'Mara , B. J. 1995, , 276, 859

    1995

  2. [2]

    1991, , 248, 315

    Bard , A., Kock , A., & Kock , M. 1991, , 248, 315

    1991

  3. [3]

    Barklem , P. S. & O'Mara , B. J. 1997, , 290, 102

    1997

  4. [4]

    S., O'Mara , B

    Barklem , P. S., O'Mara , B. J., & Ross , J. E. 1998, , 296, 1057

    work page 1998

  5. [5]

    S., Lagg , A., & Solanki , S

    Bloomfield , D. S., Lagg , A., & Solanki , S. K. 2007 a , , 671, 1005

    work page 2007

  6. [6]

    S., Solanki , S

    Bloomfield , D. S., Solanki , S. K., Lagg , A., Borrero , J. M., & Cally , P. S. 2007 b , , 469, 1155

    work page 2007

  7. [7]

    Borrero , J. M. 2007, , 471, 967

    work page 2007

  8. [8]

    M., Asensio Ramos , A., Collados , M., et al

    Borrero , J. M., Asensio Ramos , A., Collados , M., et al. 2016, , 596, A2

    work page 2016

Show all 83 references
  1. [9]

    M., Bellot Rubio , L

    Borrero , J. M., Bellot Rubio , L. R., Barklem , P. S., & del Toro Iniesta , J. C. 2003, , 404, 749

    2003

  2. [10]

    Borrero , J. M. & Ichimoto , K. 2011, Living Reviews in Solar Physics, 8, 4

    2011

  3. [11]

    M., Pastor Yabar , A., Rempel , M., & Ruiz Cobo , B

    Borrero , J. M., Pastor Yabar , A., Rempel , M., & Ruiz Cobo , B. 2019, , 632, A111

    2019

  4. [12]

    M., Pastor Yabar , A., & Ruiz Cobo , B

    Borrero , J. M., Pastor Yabar , A., & Ruiz Cobo , B. 2021, , 647, A190

    2021

  5. [13]

    Borrero , J. M. & Solanki , S. K. 2008, , 687, 668

    2008

  6. [14]

    R., Beck , C., & Del Toro Iniesta , J

    Cabrera Solana , D., Bellot Rubio , L. R., Beck , C., & Del Toro Iniesta , J. C. 2007, , 475, 1067

    2007

  7. [15]

    R., Borrero , J

    Cabrera Solana , D., Bellot Rubio , L. R., Borrero , J. M., & Del Toro Iniesta , J. C. 2008, , 477, 273

    2008

  8. [16]

    2012, Astronomische Nachrichten, 333, 872

    Collados , M., L \'o pez , R., P \'a ez , E., et al. 2012, Astronomische Nachrichten, 333, 872

    2012

  9. [17]

    2008, , 484, L17

    Danilovic , S., Gandorfer , A., Lagg , A., et al. 2008, , 484, L17

    2008

  10. [18]

    2019, , 623, A74

    de la Cruz Rodr \' guez , J., Leenaarts , J., Danilovic , S., & Uitenbroek , H. 2019, , 623, A74

    2019

  11. [19]

    Ebert , E., Mili \'c , I., & Borrero , J. M. 2024, , 691, A176

    2024

  12. [20]

    M., Guglielmino , S

    Falco , M., Borrero , J. M., Guglielmino , S. L., et al. 2016, , 291, 1939

    2016

  13. [21]

    2016, , 596, A4

    Franz , M., Collados , M., Bethge , C., et al. 2016, , 596, A4

    2016

  14. [22]

    & Schlichenmaier , R

    Franz , M. & Schlichenmaier , R. 2013, , 550, A97

    2013

  15. [23]

    K., Fligge , M., & Bruls , J

    Frutiger , C., Solanki , S. K., Fligge , M., & Bruls , J. H. M. J. 1999, in Astrophysics and Space Science Library, Vol. 243, Polarization, ed. K. N. Nagendra & J. O. Stenflo , 281--290

    1999

  16. [24]

    Georgoulis , M. K. 2005, , 629, L69

    2005

  17. [25]

    & Kahan , W

    Golub , G. & Kahan , W. 1965, SIAM Journal on Numerical Analysis, 2, 205

    1965

  18. [26]

    2007 a , in Astronomical Society of the Pacific Conference Series, Vol

    Ichimoto , K., Suematsu , Y., Shimizu , T., et al. 2007 a , in Astronomical Society of the Pacific Conference Series, Vol. 369, New Solar Physics with Solar-B Mission, ed. K. Shibata , S. Nagata , & T. Sakurai , 39

    2007

  19. [27]

    2007 b , Science, 318, 1597

    Ichimoto , K., Suematsu , Y., Tsuneta , S., et al. 2007 b , Science, 318, 1597

    2007

  20. [28]

    & Schmidt , H

    Jahn , K. & Schmidt , H. U. 1994, , 290, 295

    1994

  21. [29]

    2020, , 638, A28

    Jur c \'a k , J., Schmassmann , M., Rempel , M., Bello Gonz \'a lez , N., & Schlichenmaier , R. 2020, , 638, A28

    2020

  22. [30]

    & Collados , M

    Khomenko , E. & Collados , M. 2008, , 689, 1379

    2008

  23. [31]

    2007, , 243, 3

    Kosugi , T., Matsuzaki , K., Sakao , T., et al. 2007, , 243, 3

    2007

  24. [32]

    W., Ichimoto , K., et al

    Kubo , M., Lites , B. W., Ichimoto , K., et al. 2008 a , , 681, 1677

    2008

  25. [33]

    W., Shimizu , T., & Ichimoto , K

    Kubo , M., Lites , B. W., Shimizu , T., & Ichimoto , K. 2008 b , , 686, 1447

    2008

  26. [34]

    W., Elmore , D

    Lites , B. W., Elmore , D. F., Seagraves , P., & Skumanich , A. P. 1993 a , , 418, 928

    1993

  27. [35]

    W., Elmore , D

    Lites , B. W., Elmore , D. F., Seagraves , P., & Skumanich , A. P. 1993 b , , 418, 928

    1993

  28. [36]

    W., Elmore , D

    Lites , B. W., Elmore , D. F., & Streander , K. V. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 236, Advanced Solar Polarimetry -- Theory, Observation, and Instrumentation, ed. M. Sigwarth , 33

    2001

  29. [37]

    o hner-B \

    L \"o hner-B \"o ttcher , J. & Bello Gonz \'a lez , N. 2015, , 580, A53

    2015

  30. [38]

    Low , B. C. 1975, , 197, 251

    1975

  31. [39]

    Low , B. C. 1980 a , , 67, 57

    1980

  32. [40]

    Low , B. C. 1980 b , , 65, 147

    1980

  33. [41]

    1997, in Astronomical Society of the Pacific Conference Series, Vol

    Martinez Pillet , V. 1997, in Astronomical Society of the Pacific Conference Series, Vol. 118, 1st Advances in Solar Physics Euroconference. Advances in Physics of Sunspots, ed. B. Schmieder , J. C. del Toro Iniesta , & M. Vazquez , 212

    1997

  34. [42]

    2002, Astronomische Nachrichten, 323, 342

    Mart \' nez Pillet , V. 2002, Astronomische Nachrichten, 323, 342

    2002

  35. [43]

    K., Lagg , A., Solanki , S

    Mathew , S. K., Lagg , A., Solanki , S. K., et al. 2003, , 410, 695

    2003

  36. [44]

    R., Leka , K

    Metcalf , T. R., Leka , K. D., Barnes , G., et al. 2006, , 237, 267

    2006

  37. [45]

    U., & Weiss , N

    Meyer , F., Schmidt , H. U., & Weiss , N. O. 1977, , 179, 741

    1977

  38. [46]

    Nave , G., Johansson , S., Learner , R. C. M., Thorne , A. P., & Brault , J. W. 1994, , 94, 221

    1994

  39. [47]

    2017, , 607, A102

    Orozco Su \'a rez , D., Quintero Noda , C., Ruiz Cobo , B., Collados Vera , M., & Felipe , T. 2017, , 607, A102

    2017

  40. [48]

    R., & Rouppe van der Voort , L

    Ortiz , A., Bellot Rubio , L. R., & Rouppe van der Voort , L. 2010, , 713, 1282

    2010

  41. [49]

    Panja , M., Cameron , R., & Solanki , S. K. 2020, , 893, 113

    2020

  42. [50]

    Pizzo , V. J. 1986, , 302, 785

    1986

  43. [51]

    Pizzo , V. J. 1990, , 365, 764

    1990

  44. [52]

    H., Flannery , B

    Press , W. H., Flannery , B. P., & Teukolsky , S. A. 1986, Numerical recipes. The art of scientific computing (Cambridge: University Press, 1986)

    1986

  45. [53]

    2011, , 729, 5

    Rempel , M. 2011, , 729, 5

    2011

  46. [54]

    2012, , 750, 62

    Rempel , M. 2012, , 750, 62

    2012

  47. [55]

    & Borrero, J

    Rempel, M. & Borrero, J. 2021, Oxford Research Encyclopedia of Physics

    2021

  48. [56]

    & Schlichenmaier , R

    Rempel , M. & Schlichenmaier , R. 2011, Living Reviews in Solar Physics, 8, 3

    2011

  49. [57]

    u ssler , M., Cameron , R. H., & Kn \

    Rempel , M., Sch \"u ssler , M., Cameron , R. H., & Kn \"o lker , M. 2009 a , Science, 325, 171

    2009

  50. [58]

    u ssler , M., & Kn \

    Rempel , M., Sch \"u ssler , M., & Kn \"o lker , M. 2009 b , , 691, 640

    2009

  51. [59]

    J., Swenson , F

    Rogers , F. J., Swenson , F. J., & Iglesias , C. A. 1996, , 456, 902

    1996

  52. [60]

    & del Toro Iniesta , J

    Ruiz Cobo , B. & del Toro Iniesta , J. C. 1992, , 398, 375

    1992

  53. [61]

    B., Henriques , V

    Scharmer , G. B., Henriques , V. M. J., Kiselman , D., & de la Cruz Rodr \' guez , J. 2011, Science, 333, 316

    2011

  54. [62]

    Scharmer , G. B. & Spruit , H. C. 2006, , 460, 605

    2006

  55. [63]

    2016, , 596, A7

    Schlichenmaier , R., von der L \"u he , O., Hoch , S., et al. 2016, , 596, A7

    2016

  56. [64]

    & Temesv \'a ry , S

    Schl \"u ter , A. & Temesv \'a ry , S. 1958, in IAU Symposium, Vol. 6, Electromagnetic Phenomena in Cosmical Physics, ed. B. Lehnert , 263

    1958

  57. [65]

    2021, , 656, A92

    Schmassmann , M., Rempel , M., Bello Gonz \'a lez , N., Schlichenmaier , R., & Jur c \'a k , J. 2021, , 656, A92

    2021

  58. [66]

    2012, Astronomische Nachrichten, 333, 796

    Schmidt , W., von der L \"u he , O., Volkmer , R., et al. 2012, Astronomische Nachrichten, 333, 796

    2012

  59. [67]

    & Rempel , M

    Sch \"u ssler , M. & Rempel , M. 2005, , 441, 337

    2005

  60. [68]

    u ssler , M. & V \

    Sch \"u ssler , M. & V \"o gler , A. 2006, , 641, L73

    2006

  61. [69]

    2008, , 249, 221

    Shimizu , T., Nagata , S., Tsuneta , S., et al. 2008, , 249, 221

    2008

  62. [70]

    & Jur c \'a k , J

    Sobotka , M. & Jur c \'a k , J. 2009, , 694, 1080

    2009

  63. [71]

    2015, , 577, A7

    Socas-Navarro , H., de la Cruz Rodr \' guez , J., Asensio Ramos , A., Trujillo Bueno , J., & Ruiz Cobo , B. 2015, , 577, A7

    2015

  64. [72]

    Solanki , S. K. 2003, , 11, 153

    2003

  65. [73]

    Solanki , S. K. & Montavon , C. A. P. 1993, , 275, 283

    1993

  66. [74]

    Spruit , H. C. & Scharmer , G. B. 2006, , 447, 343

    2006

  67. [75]

    Stanchfield , II, D. C. H., Thomas , J. H., & Lites , B. W. 1997, , 477, 485

    1997

  68. [76]

    2008, , 249, 197

    Suematsu , Y., Tsuneta , S., Ichimoto , K., et al. 2008, , 249, 197

    2008

  69. [77]

    M., Frank , Z

    Title , A. M., Frank , Z. A., Shine , R. A., et al. 1993, , 403, 780

    1993

  70. [78]

    2008, , 249, 167

    Tsuneta , S., Ichimoto , K., Katsukawa , Y., et al. 2008, , 249, 167

    2008

  71. [79]

    van der Vorst, H. A. 1992, SIAM Journal on Scientific and Statistical Computing, 13, 631

    1992

  72. [80]

    2012, , 548, A5

    van Noort , M. 2012, , 548, A5

    2012

  73. [81]

    o gler , A., Shelyag , S., Sch \

    V \"o gler , A., Shelyag , S., Sch \"u ssler , M., et al. 2005, , 429, 335

    2005

  74. [82]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence a...

  75. [83]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.