The quiet solar photosphere is pervaded by a tangled, mixed-polarity magnetic field at scales far below the resolution of observations. Because opposite polarities cancel in Zeeman polarization signals, this 'hidden' field is invisible to conventional magnetometry; instead it is diagnosed through the Hanle effect — the magnetic modification of the linear polarization produced by resonant scattering in spectral lines such as Sr I 4607 Å. Building on the inference that the quiet-Sun internetwork harbors a substantial hidden magnetic energy density (Trujillo Bueno, Shchukina & Asensio Ramos 2004), Trujillo Bueno (2006) identified as unresolved the determination of the sub-resolution field itself: the shape of the field-strength probability distribution cannot be constrained from a single spectral line, so the inferred mean strength and energy density depend on the assumed distribution and on the microturbulent (isotropic, volume-filling) modeling approximation, and deciding which field strength and strength distribution best agrees with the observed scattering-polarization signals requires computing the Hanle depolarization produced by the actual magnetic-field configurations of magneto-convection simulations. That work presented the concentration of the hidden field in intergranular (downflowing) plasma and the sufficiency of its energy density to compensate the radiative energy losses of the solar outer atmosphere as conclusions of the microturbulent modeling — conclusions that remain conditional on completing the field determination. The microturbulent approximation has been superseded by direct 3D polarized radiative-transfer modeling in small-scale-dynamo magnetoconvection simulations (Shchukina & Trujillo Bueno 2011; del Pino Alemán et al. 2018), disk-center scattering polarization has been detected and shown consistent with such simulations (Zeuner et al. 2020, 2024), and DKIST has resolved sub-arcsecond scattering-polarization structure (Zeuner et al. 2026). Nevertheless, the quantitative determination remains open: atomic-line Hanle estimates (~100–160 G), molecular-line Hanle estimates (~10 G), multiline Zeeman inversions (~46 G), and limb Hanle depolarization (13–25 G) still disagree, and the strength PDF, its height dependence, and the mean magnetic energy density have not been determined within a single self-consistent field model.

Background

The quiet Sun — the ~99% of the solar surface outside active regions — is threaded by magnetic fields of mixed polarity on scales from the granulation scale (~1000 km) down to unresolved, sub-resolution scales. The circular and linear polarization signatures of the Zeeman effect (the splitting and polarization of spectral lines in a magnetic field) cancel almost completely when opposite polarities coexist within one spatial resolution element, so conventional magnetograms reveal only the strongest, largest structures — the 'tip of the iceberg'. A complementary diagnostic is scattering polarization: spectral lines formed by scattering of anisotropic radiation acquire linear polarization (the 'second solar spectrum', the spectrum of linear polarization observed near the solar limb). This scattering polarization is modified by weak magnetic fields through the Hanle effect, which rotates and depolarizes the scattered signal for field strengths roughly between a few gauss and a few hundred gauss. Crucially, the Hanle effect is insensitive to field polarity, so it survives polarity cancellation and can detect tangled, mixed-polarity ('turbulent' or 'hidden') fields that Zeeman diagnostics average away. In the simplest modeling, the unresolved field is treated as 'microturbulent' — isotropic, volume-filling, and uncorrelated over scales below the photon mean free path — so the resulting depolarization depends only on the field-strength probability density function (PDF). A recurring convention is the exponential PDF; the equipartition field (where magnetic and convective kinetic energy densities are equal, of order a few hundred gauss in the photosphere) sets a natural strength scale. Using detailed 3D radiative transfer with the Hanle effect in the Sr I 4607 Å line under the microturbulent and exponential-PDF assumptions, Trujillo Bueno, Shchukina & Asensio Ramos (2004, Nature 430, 326) concluded that internetwork regions harbor a vast amount of hidden magnetic energy and unsigned flux, carried by tangled sub-resolution fields with strengths between equipartition values and ~1 kG, concentrated in intergranular (downflowing) plasma, and in principle sufficient to compensate the radiative energy losses of the solar outer atmosphere. In a 2006 review summarizing this work, Trujillo Bueno stressed the model dependence of these conclusions: with a single spectral line there is not enough information to constrain the shape of the sub-resolution strength PDF, whose functional form must therefore be assumed, and different assumed PDFs yield different mean and RMS fields and hence different energetic conclusions; moreover, because the inference was based on the microturbulent field model, a definitive determination requires computing the Hanle depolarization produced by the actual magnetic-field configurations of magneto-convection simulations and comparing it with the observed scattering-polarization signals — the program the author described as under way, with the aim of determining which field strength and strength distribution lead to the best agreement with observations. The review presented the intergranular concentration of the hidden field and the sufficiency of its energy to compensate outer-atmosphere radiative losses as conclusions of the modeling, conclusions that remain conditional on completing that determination. Uncertainties in collisional depolarization rates (which set the non-magnetic polarization baseline) and in the non-LTE atomic and molecular physics of the diagnostic lines further limit the precision of Hanle inferences, and observational comparison at higher spatial resolution — center-to-limb and disk-center scattering polarization in lines such as Sr I 4607 Å — is the means by which any such determination must be validated. These threads share one objective: establish the hidden magnetic field of the quiet photosphere — its strength, strength distribution, height dependence, and topology — and settle its energetic role. Subsequent developments define the modern form of the problem. Magnetoconvection simulations with small-scale dynamo (SSD) action — self-excited conversion of turbulent kinetic energy into magnetic energy by stretching, folding, and twisting of field lines, operating independently of the 11-year solar cycle — now provide realistic 3D field configurations, and a new generation of telescopes (notably DKIST, the Daniel K. Inouye Solar Telescope, and its Visible Spectro-Polarimeter, ViSP) delivers sub-arcsecond polarimetry. Yet the quantitative estimates from different diagnostics still disagree: atomic-line Hanle modeling points to a mean field of order 100–160 G in the photosphere, molecular-line Hanle diagnostics (e.g., C2 Swan lines) suggest fields roughly ten times weaker, multiline Zeeman inversions of infrared lines yield ~46 G on average, and limb Hanle depolarization of several strong lines gives 13–25 G with an apparent altitude gradient. No single, self-consistent field model has been shown to reproduce all of these simultaneously.

Problem Statement

Determine the sub-resolution, tangled ('hidden') magnetic field of the quiet solar photosphere — its mean strength, its field-strength probability density function, and its detailed spatial and height variation between granular (upflowing) and intergranular (downflowing) plasma — through Hanle-effect diagnostics that go beyond the microturbulent, volume-filling field model: compute the Hanle depolarization and Zeeman signatures produced by the actual magnetic-field configurations of 3D magneto-convection simulations (in current practice, time-dependent models with small-scale-dynamo action) through 3D polarized radiative transfer, and determine which field strength and strength distribution lead to the best agreement with the observed scattering-polarization signals. The determination should be validated against spatially resolved scattering-polarization observations (e.g., center-to-limb and disk-center measurements in Sr I 4607 Å) and should confront the currently discrepant magnetization estimates from the different Hanle and Zeeman diagnostics (summarized under Background). From the determined field, establish the mean magnetic energy density of the quiet photosphere and whether it suffices to compensate the radiative energy losses of the solar chromosphere (of order 10710^{7} erg cm−2^{-2} s−1^{-1}). The determination is complete when one self-consistent field model, spanning the granular (upflowing) and intergranular (downflowing) photosphere, simultaneously accounts within the observational uncertainties for the observed scattering-polarization signals and for the currently discrepant Hanle and Zeeman magnetization estimates, and yields the strength distribution with its height variation together with the mean magnetic energy density and its sufficiency verdict.

References

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