Whether the weak internetwork magnetic fields that cover most of the quiet solar surface are predominantly horizontal or vertical — and how that angular preference varies with atmospheric height — remains an open, actively debated question. Stenflo (2013) established a model-independent, sign-based result from center-to-limb polarimetry of the 5247–5250 Å lines: the doubly differential signed linear-polarization amplitude changes sign at μ≈0.22, implying predominantly vertical fields in the deeper photosphere and predominantly horizontal fields above a transition layer. Stenflo explicitly deferred the quantitative continuation of two conversions, each of which would require radiative-transfer modelling and make the results model dependent: converting μ to geometrical height (§5.1), and converting polarization amplitudes into intrinsic field strengths and explicit angular-distribution parameters at those heights (§5.3). His interpretation section (§5.2) discusses, as physical context rather than as a deferred deliverable, how the transition height should depend on field strength and on the polarity-mixing scale of the flux footpoints, and it notes the complementary Hanle-effect expectation that tangled subresolution fields approach the isotropic case in the small-scale limit. Work published since — independent sign-level confirmation, height-node inversions, DKIST and Sunrise III multiline inversions, MHD forward-model comparisons, and two recent reviews — supplies corroborating evidence at individual optical depths, while the height-calibrated angular distribution, its strength dependence, and the transition-height physics remain open.

Background

The quiet Sun — the solar surface outside sunspots, pores, and active regions — is carpeted by small-scale magnetic fields that dominate the Sun's surface magnetic-flux budget by coverage. The stronger kilogauss concentrations collect in the 'network' that outlines supergranular convection cells; the cell interiors contain the much weaker 'internetwork' (also called intranetwork) fields, the most ubiquitous magnetic structures on the Sun. Spectropolarimetry with the Japanese Hinode satellite showed that these internetwork regions carry a surprising amount of horizontal magnetic flux in the photosphere (the Sun's visible surface layer) (Orozco Suárez et al. 2007; Lites et al. 2008), a finding that has been debated against simulation-based pictures of a small-scale dynamo generating more intermittent, more vertical fields ever since.

Spectropolarimetry measures the four Stokes parameters: I (intensity), Q and U (the two linear-polarization components), and V (circular polarization). In the Zeeman effect, spectral lines split in a magnetic field, and the polarization of the split components encodes field direction: a field component along the line of sight produces circular polarization, while a transverse field component produces linear polarization. Converting observed amplitudes into intrinsic field inclinations and strengths, however, requires radiative-transfer modeling of how and where the line photons escape, and it is sensitive to noise and to magnetic structure below the telescope resolution element.

Stenflo (2013) addressed the horizontal-versus-vertical question with a deliberately model-independent, sign-based diagnostic. From polarimetric observations of the Fe I 5247–5250 Å line pair in seven facula-free internetwork regions drawn from a data set of 14 disk positions between disk center and limb, he measured the center-to-limb variation (CLV) — the dependence on μ\mu, the cosine of the heliocentric angle (μ=1\mu=1 at disk center, μ=0\mu=0 at the limb) — of the signed linear-polarization amplitude. That amplitude is a doubly differential measure: for each line, the amplitude of the central π\pi component minus the mean of the blue and red σ\sigma-lobe amplitudes (each lobe amplitude extracted per pixel by least-squares template fitting and then averaged along the spectrograph slit), with the resulting π−σ\pi-\sigma difference then taken between the 5250.22 Å and the 5247.06 Å line; positive values correspond to an angular distribution more peaked around the vertical than the isotropic one, negative values to a distribution favoring the horizontal plane. Stenflo stresses that non-differential measures of the same amplitudes are unreliable, because they are susceptible to errors in the polarization zero level that may differ between the two lines. Because line formation shifts to higher atmospheric layers for observations nearer the limb, μ\mu maps monotonically (though not linearly) to the height sampled. The measured amplitude changes sign around μ≈0.22\mu\approx0.22 (the zero crossing of the straight-line fit), indicating predominantly horizontal fields in the higher layers sampled near the limb and predominantly vertical fields in the deeper layers sampled near disk center. As a quantitative language for such results, Stenflo used the angular-distribution parametrization fa∝μBaf_a\propto\mu_B^a, with μB\mu_B the cosine of the field inclination to the vertical: a=0a=0 is isotropic, a=−1a=-1 a 'pancake' distribution of horizontal fields, and a→∞a\to\infty a delta function of vertical fields; but he deliberately fitted no value of aa, converted no amplitudes into field strengths, and translated no μ\mu values into heights, because each such conversion introduces model dependence. A complementary diagnostic, the Hanle effect — the modification of scattering linear polarization in spectral lines by weak magnetic fields — is sensitive to tangled, subresolution fields of roughly a few to a few hundred gauss that Zeeman diagnostics cannot resolve, and thus constrains the small-scale limit of the angular distribution.

The paper is explicit about the two conversions it does not attempt: a quantitative translation of μ\mu into a height scale 'would require radiative-transfer modelling, which is outside the scope of the present work' (§5.1), and the paper does not convert polarization values into field strength or μ\mu values into geometrical height, since such a conversion 'would require radiative-transfer modelling and make the results model dependent' (§5.3). Its interpretation section (§5.2) discusses the physics expected to set the transition height — the competition between magnetic buoyancy, which favors vertical fields, flux-tube expansion, and the loop topology of mixed polarities, whose summits are horizontal, with the maximum loop height scaling with the polarity-mixing scale of the footpoints — and, as complementary context, notes that Hanle-effect observations of tangled fields on scales of a few km or less lead one to expect an approach to the isotropic case in the small-scale limit. A larger statistical sample is mentioned only as the way to improve the ensemble averages (§2.4). The present problem is the quantitative continuation of the two deferred conversions, together with a physical determination of the transition height that §5.2 treats qualitatively.

Problem Statement

Determine the height-resolved angular distribution of quiet-Sun internetwork magnetic fields and what sets it, replacing the sign-only center-to-limb picture of Stenflo (2013) with a quantitative one. Taking the observed sign change at μ≈0.22\mu\approx0.22 of the doubly differential signed linear-polarization amplitude defined in the Background — the π\pi-minus-mean-σ\sigma amplitude difference formed for each of the Fe I 5250.22 and 5247.06 Å lines and differenced between them, spatially averaged over facula-free internetwork slit positions — as the model-independent anchor (predominantly vertical fields in deep layers, predominantly horizontal fields above), a complete answer provides, on a common calibrated height scale:

  1. Height calibration: a quantitative mapping from the center-to-limb observing coordinate μ\mu (or from an equivalent multi-height set of spectral lines) to geometrical atmospheric height for the relevant Zeeman diagnostics, such that the observed zero crossing maps to a physical transition height with its model dependence quantified.
  2. Angular distribution and strengths: the intrinsic angular distribution of internetwork fields as a function of height — and, where the data permit, field strength — for example the exponent a(height,B)a(\text{height},B) in the parametrization fa∝μBaf_a\propto\mu_B^a (μB\mu_B the cosine of the field inclination to the vertical; a=0a=0 isotropic, a=−1a=-1 horizontal, a→∞a\to\infty vertical) or an equivalent explicit inclination-distribution function — together with intrinsic field strengths obtained from the measured polarization amplitudes, with a demonstrated treatment of the linear-polarization retrieval biases documented in the literature.
  3. Transition physics: a quantitative determination of how the horizontal–vertical transition height varies with magnetic flux density and with the spatial scale of polarity mixing.

These components jointly serve the single objective stated in the first sentence: each supplies a piece — the height scale, the distribution itself, and its physical dependence — without which the quantitative picture is incomplete. A complete answer must make the components mutually consistent and must state uncertainties, including any quantified revision of the μ≈0.22\mu\approx0.22 zero crossing itself from a larger or multi-line dataset, and including any significant dependence of the recovered distribution on region, center-to-limb azimuth, or solar-cycle phase, so that it is explicit whether one height-resolved distribution applies to quiet-Sun internetwork fields in general or how it varies across them.

Hanle-effect constraints on tangled, subresolution fields (the 'hidden' flux of order 10–100 G) are complementary background rather than a required component of this problem: Stenflo (2013, §5.2) records the expectation that the angular distribution approaches the isotropic case in the small-scale limit, and a quantitative Zeeman-based distribution can be compared with such constraints, but no Zeeman–Hanle unification is demanded here.

References

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