The quiet solar atmosphere contains a spatially unresolved, mixed-polarity ('hidden' or turbulent) magnetic field whose Zeeman signals cancel but which depolarizes the resonantly scattered linear polarization of strong spectral lines near the solar limb through the Hanle effect. Inversions of this Hanle depolarization in the strong lines of the 'second solar spectrum' (Ca I 4227 Å, Sr I 4607 Å, Sr II 4078 Å, Ba II 4554 Å, Ca II K 3934 Å) currently return turbulent-field strengths spanning roughly 10–65 G for the same physical quantity: the same Ca I 4227 data yield 65 G or 20 G depending on the frequency-redistribution treatment assumed in the zero-field reference polarization, and Sr I 4607 inversions return 13 G, 46 G, or about 60 G depending on the assumed atmospheric model. The open problem is a model-validated determination of the hidden turbulent field strength and its height dependence from near-limb Hanle depolarization — deciding between the ~10–20 G and ~50–65 G regimes — with controlled model uncertainty for chromospheric lines (Ca II K). The source analysis also notes the systematic effects of single-beam, non-seeing-frozen polarimetry on the derived field, and proposes repeating such observations many times along the solar cycle to study fluctuations of the hidden turbulent field.

Background

Near the solar limb, sunlight scattered in spectral lines acquires a linear polarization signal Q/I (Stokes Q normalized by the intensity I), typically well below 1%, produced by the anisotropy of the radiation field in the solar atmosphere. This linearly polarized spectrum, the 'second solar spectrum', is strongest in resonance lines such as Ca I 4227 Å, Sr I 4607 Å, Sr II 4078 Å, Ba II 4554 Å, and Ca II K 3934 Å. It is observed as a function of the heliocentric angle, conventionally μ=cos⁡θ\mu=\cos\theta where θ\theta is the angle between the line of sight and the local vertical; the limb corresponds to μ=0\mu=0, and because of projection effects, line formation near the limb samples increasing atmospheric height as μ\mu decreases.

Weak magnetic fields modify scattering polarization through the Hanle effect: for field strengths such that the Larmor precession frequency in the excited atomic level is comparable to the inverse radiative lifetime (roughly 1–100 G for these lines), the coherent-scattering polarization is depolarized and its plane may rotate. Fields that are tangled at spatial scales below the telescope resolution ('turbulent' or 'hidden' fields, of mixed polarity so that Zeeman signals cancel) can therefore be diagnosed by the amount of Hanle depolarization, following the diagnostic theory introduced by Stenflo (1982) for an isotropic distribution of unresolved fields. To infer the field strength BturbB_{\rm turb} one compares the observed limb polarization p(μ)p(\mu) with the zero-field reference polarization p0(μ)p_0(\mu) — the polarization the same line would show with no magnetic field, computed from polarized non-local thermodynamic equilibrium (non-LTE) radiative transfer. The accuracy of BturbB_{\rm turb} is thus limited by the accuracy of p0(μ)p_0(\mu), which depends on the treatment of partial frequency redistribution (PRD, the correlation between absorbed and re-emitted photon frequencies, essential in strong resonance lines; approximations range from complete redistribution through angle-averaged and angle-dependent PRD), on the assumed atmospheric model, and on atomic collision rates.

This diagnostic has produced influential but mutually inconsistent results. Trujillo Bueno, Shchukina & Asensio Ramos (2004) concluded from Sr I 4607, modeled with a 3D magneto-convection simulation, that the quiet photosphere harbors a hidden field of about 60 G (volume filling), implying hidden magnetic energy comparable to the local thermal energy. Bommier et al. (2005), from THEMIS center-to-limb observations of Sr I 4607 interpreted with a plane-parallel semi-empirical atmosphere and the atomic density-matrix formalism, found 46 G on average with a vertical gradient of −0.12 G/km. Faurobert, Derouich, Bommier & Arnaud (2008) modeled the Ba II D2 line (the 4554 Å resonance line) with PRD, compared it with THEMIS observations, and derived 10–18 G in the low chromosphere after correction factors of 1.7–2 for collisional depolarization. Most recently, Malherbe (2025) analyzed a 2006 Pic du Midi dataset of five strong lines recorded with the slit orthogonal to the limb (continuous coverage 0≤μ≤0.450\le\mu\le0.45, with a single-beam liquid-crystal polarimeter at about 5 Hz cadence), deriving μ-averaged turbulent fields of 13–25 G, with the field strength decreasing toward the limb (i.e., with increasing height) in Ca I 4227 and Sr II 4078. Crucially, the same Ca I 4227 data give 65 G or 20 G depending on whether the zero-field envelope of Faurobert-Scholl (1992) is taken from its partial-redistribution or its intermediate-redistribution calculation — a factor-of-three model dependence within a single line — and the Sr I 4607 estimates span 13–60 G across analyses. For the Ba II 4554 even-isotope central peak (the component unaffected by the hyperfine structure of the odd isotope 137^{137}Ba) the source paper reports that no zero-field prediction was available for an inversion, and for the chromospheric Ca II K line the zero-field envelopes computed by Holzreuter & Stenflo (2007) with four different FAL-type (Fontenla–Avrett–Loeser semi-empirical) atmospheres differ enough to prevent an unambiguous altitude-gradient measurement. The modern modeling capability needed to resolve this exists — Janett et al. (2021) showed that an angle-dependent PRD treatment is essential to reproduce the observed Ca I 4227 line-core polarization peak — but it has not been carried through to a reconciled turbulent-field determination. Adjacent recent work has moved toward spatially resolved Hanle polarimetry at disk center (Zeuner et al. 2020; Bianda et al. 2018; del Pino Alemán & Trujillo Bueno 2021; Zeuner et al. 2024), quantifying systematic amplitude reductions of up to a factor of two from reconstruction and time-resolution effects, which underlines how analysis choices can bias Hanle-type inferences at precisely the level of the current field-strength discrepancies.

Problem Statement

Determine, from Hanle depolarization of the strong second-solar-spectrum lines observed near the solar limb (μ≲0.45\mu\lesssim0.45), the strength and height profile of the quiet Sun's spatially unresolved turbulent ('hidden') magnetic field — with quantified, controlled uncertainties — resolving the model dependence demonstrated by the source analysis (Malherbe 2025, arXiv:2505.17545) and deciding between the ~10–20 G and ~50–65 G regimes. This requires the following jointly linked results: (1) validated zero-field reference polarizations p0(μ)p_0(\mu) for the diagnostic lines (Ca I 4227 Å, Sr I 4607 Å, Sr II 4078 Å, Ca II K 3934 Å), computed under a controlled radiative-transfer treatment of frequency redistribution, atmospheric model, and collision rates, such that the present factor-of-three ambiguity (65 G versus 20 G for the same Ca I 4227 data, and 13–60 G for Sr I 4607 across the literature) is reduced to a stated uncertainty; (2) the corresponding inversions for Bturb(μ)B_{\rm turb}(\mu) across the line set, including the chromospheric Ca II K line with atmospheric-model uncertainty small enough to decide the altitude gradient. The completion boundary is a self-consistent, multi-line determination of the hidden turbulent field (strength, height dependence, uncertainty budget) that is stable against the plausible range of zero-field models and instrumental systematics.

The Ba II 4554 Å even-isotope central peak is a source-acknowledged gap: Malherbe (2025) performed no inversion because no predicted values of the line polarization with zero field were available to that analysis, whereas the PRD modeling of the Ba II D2 line by Faurobert et al. (2008) shows that a zero-field treatment does exist; integrating a community-accepted zero-field envelope for this peak into a multi-line inversion would therefore be a natural extension of the determination. In line with the source's own conclusion, such near-limb observations should further be repeated many times along the solar cycle to study the fluctuations of the hidden, weak and turbulent magnetic field.

References

  1. Malherbe, J.-M. 2025, 'Hidden magnetic fields of the quiet Sun derived from Hanle depolarization of lines of the "second solar spectrum" at the limb from Pic du Midi observations', arXiv:2505.17545, https://arxiv.org/abs/2505.17545
  2. Malherbe, J.-M. 2025, 'High polarization lines of the second solar spectrum of the Solar limb', arXiv:2505.04239, https://arxiv.org/abs/2505.04239
  3. Stenflo, J. O. 1982, 'The Hanle effect and the diagnostics of turbulent magnetic fields in the solar atmosphere', Solar Physics 80, 209–226, doi:10.1007/BF00147969, https://doi.org/10.1007/BF00147969
  4. Faurobert-Scholl, M. 1992, 'Hanle effect with partial frequency redistribution – II – Linear polarization of the solar CaI 4227 Å line', Astronomy & Astrophysics 258, 521–534, NASA ADS bibcode 1992A&A...258..521F, https://articles.adsabs.harvard.edu/pdf/1992A%26A...258..521F
  5. Faurobert-Scholl, M. 1993, 'Investigation of microturbulent magnetic fields in the solar photosphere by their Hanle effect in the SrI 4607 Å line', Astronomy & Astrophysics 268, 765–774, NASA ADS bibcode 1993A&A...268..765F, https://articles.adsabs.harvard.edu/pdf/1993A%26A...268..765F
  6. Bianda, M., Solanki, S. K., Stenflo, J. O. 1998, 'Hanle depolarisation in the solar chromosphere', Astronomy & Astrophysics 331, 760–770, NASA ADS bibcode 1998A&A...331..760B, https://articles.adsabs.harvard.edu/pdf/1998A%26A...331..760B
  7. Trujillo Bueno, J., Shchukina, N., Asensio Ramos, A. 2004, 'A substantial amount of hidden magnetic energy in the quiet Sun', Nature 430, 326–329, doi:10.1038/nature02669, https://doi.org/10.1038/nature02669
  8. Bommier, V., Derouich, M., Landi Degl'Innocenti, E., Molodij, G., Sahal-Bréchot, S. 2005, 'Interpretation of second solar spectrum observations of the Sr I 4607 Å line in a quiet region: turbulent magnetic field strength determination', Astronomy & Astrophysics 432, 295–305, doi:10.1051/0004-6361:20035773, https://doi.org/10.1051/0004-6361:20035773
  9. Derouich, M., Bommier, V., Malherbe, J. M., Landi Degl'Innocenti, E. 2006, 'Second solar spectrum of the Sr I 4607 Å line: depth probing of the turbulent magnetic field strength in a quiet region', Astronomy & Astrophysics 457, 1047–1052, doi:10.1051/0004-6361:20065782, https://doi.org/10.1051/0004-6361:20065782
  10. Holzreuter, R., Stenflo, J. O. 2007, 'Scattering polarization in strong chromospheric lines – II. Influence of the temperature curve on the Ca II K line', Astronomy & Astrophysics 467, 695–702, doi:10.1051/0004-6361:20066747, https://doi.org/10.1051/0004-6361:20066747
  11. Faurobert, M., Derouich, M., Bommier, V., Arnaud, J. 2008, 'Hanle effect in the solar Ba II D2 line: a diagnostic tool for chromospheric weak magnetic fields', Astronomy & Astrophysics 493, 201–206, doi:10.1051/0004-6361:200810474, https://doi.org/10.1051/0004-6361:200810474
  12. Bianda, M., Berdyugina, S., Gisler, D., Ramelli, R., Belluzzi, L., Carlin, E. S., Stenflo, J. O., Berkefeld, T. 2018, 'Spatial variations of the Sr I 4607 Å scattering polarization peak', Astronomy & Astrophysics 614, A89, doi:10.1051/0004-6361/201731887, https://doi.org/10.1051/0004-6361/201731887
  13. Zeuner, F., Manso Sainz, R., Feller, A., van Noort, M., Solanki, S. K., Iglesias, F. A., Reardon, K., Martínez Pillet, V. 2020, 'Solar Disk Center Shows Scattering Polarization in the Sr i 4607 Å Line', The Astrophysical Journal Letters 893, L44, doi:10.3847/2041-8213/ab86b8, https://doi.org/10.3847/2041-8213/ab86b8
  14. del Pino Alemán, T., Trujillo Bueno, J. 2021, 'The Impact of Limited Time Resolution on the Forward-scattering Polarization in the Solar Sr I 4607 Å Line', The Astrophysical Journal 909, 180, doi:10.3847/1538-4357/abdd25, https://doi.org/10.3847/1538-4357/abdd25
  15. Janett, G., Alsina Ballester, E., Guerreiro, N., Riva, S., Belluzzi, L., del Pino Alemán, T., Trujillo Bueno, J. 2021, 'Modeling the scattering polarization of the solar Ca I 4227 Å line with angle-dependent partial frequency redistribution', Astronomy & Astrophysics 655, A13, doi:10.1051/0004-6361/202141549, https://doi.org/10.1051/0004-6361/202141549
  16. Zeuner, F., del Pino Alemán, T., Trujillo Bueno, J., Solanki, S. K. 2024, 'Comparing Observed with Simulated Solar-disk-center Scattering Polarization in the Sr i 4607 Å Line', The Astrophysical Journal 964, 10, doi:10.3847/1538-4357/ad26f9, https://doi.org/10.3847/1538-4357/ad26f9
  17. Malherbe, J.-M., Moity, J., Arnaud, J., Roudier, Th. 2007, 'First observations of the second solar spectrum with spatial resolution at the Lunette Jean Rösch', Astronomy & Astrophysics 462, 753–762, doi:10.1051/0004-6361:20065910, https://doi.org/10.1051/0004-6361:20065910