Intrinsic radiality of the solar polar photospheric magnetic field: separating true geometry from inversion and resolution bias

In the solar polar regions, vector magnetograms (maps of the solar magnetic-field vector) from moderate-resolution instruments such as SDO/HMI (the Helioseismic and Magnetic Imager on NASA's Solar Dynamics Observatory, at 1-arcsecond resolution) and SOLIS/VSM (the Vector Stokes Magnetograph on the Synoptic Optical Long-term Investigations of the Sun facility) show photospheric magnetic fields that appear systematically non-radial: vectors tilt toward the pole and are more inclined to the plane of sky than the local radial direction. Sun et al. (2021) showed that this signal depends strongly on the magnetic filling factor $f$ (the unresolved sub-pixel fractional magnetic area): freeing $f$ moves the inferred vectors toward radial but leaves a residual bias with a severe $f$–$B$ degeneracy. It remains open whether the polar photospheric field is intrinsically radial at sub-arcsecond scales and, quantitatively, how much of the observed non-radial inclination is true sub-pixel geometry versus inversion- and resolution-induced bias. Candidate paths toward resolving this include bias-controlled, validated measurements (DKIST, the Daniel K. Inouye Solar Telescope, a 4-meter ground-based solar telescope; out-of-ecliptic observations from Solar Orbiter/PHI, the Polarimetric and Helioseismic Imager aboard the Solar Orbiter spacecraft), realistic synthetic-observation benchmarks, informative priors for the polar filling-factor distribution, and reconciliation of inter-instrument differences such as the opposite HMI/VSM inclination biases. The problem is central to the reliability of polar-field measurements, which anchor solar-cycle prediction and open-flux estimates.

Background

The Sun's polar magnetic fields (heliographic latitudes above about 60 degrees) are the reservoir of poloidal flux that seeds each new activity cycle, and they anchor models of the global corona and solar wind. Because the poles cannot be viewed head-on from Earth, synoptic maps (full-surface maps assembled from near-Earth views as the Sun rotates) and coronal models routinely assume the polar field is radial (pointing along the local surface normal), so that the radial component follows from the line-of-sight component as $B_r=B_l\,\mu^{-1}$, where $\mu$ is the cosine of the heliocentric angle. Prior work supports radiality only on spatial scales of several heliographic degrees; on small scales the polar field is concentrated in isolated patches a few megameters wide with intrinsic kilo-Gauss strengths, as shown by the 0.3-arcsecond Spectropolarimeter (SP) aboard the Japanese Hinode solar-observatory satellite (Tsuneta et al. 2008).

The relevant measurements come from spectropolarimetry: the Stokes parameters $(I,Q,U,V)$ — the total intensity and the two linear- and one circular-polarization components of the light — of a magnetically sensitive spectral line are inverted to recover the field vector, described by strength $B$, inclination $\gamma$ (0 degrees toward the observer, 180 degrees away), and azimuth. Two instruments dominate polar studies. The Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory provides full-disk Stokes images of the Fe I 617.3 nm line at 1-arcsecond resolution, sampled at only six wavelengths; its pipeline uses a Milne-Eddington inversion (an inversion assuming a simple stratified atmosphere) with the magnetic filling factor held at unity. The filling factor $f$ is the fraction of a resolution element actually occupied by magnetic field; since polar patches cover only a few percent of the polar area, $f$ in individual HMI pixels is plausibly well below 1. In the weak-field approximation (the regime in which the circular-polarization signal is proportional to the line-of-sight field and the linear-polarization signal to the square of the transverse field), $fB\cos\gamma\propto V$ and $fB^{2}\sin^{2}\gamma\propto(Q^{2}+U^{2})^{1/2}$, so $\tan\gamma\propto f^{1/2}(Q^{2}+U^{2})^{1/4}/V$: assuming $f=1$ when the true $f<1$ biases the inferred vector toward the plane of sky. At HMI's moderate resolution, $f$ is strongly degenerate with $B$. A further complication is the 180-degree azimuth ambiguity intrinsic to Zeeman polarimetry (polarized line profiles cannot distinguish a field azimuth from the opposite direction), whose standard resolution methods degrade at the large heliocentric angles of the poles. The Vector Stokes Magnetograph (VSM) on the Synoptic Optical Long-term Investigations of the Sun (SOLIS) facility provides the competing full-disk vector data.

Sun, Liu, Milić and Griñón Marín (2021, Research Notes of the AAS 5, 134) analyzed about 1000 HMI pixels of significant polarization in the southern polar region and found that the inferred vectors are always more inclined toward the plane of sky than the radial direction ($|\cos\gamma|<\mu$ in every pixel), 93 percent tilt toward the pole, with a median deviation from radial of 18 degrees. Re-inverting with $f$ free (which peaks near 0.25, versus roughly 0.15 implied by Hinode) makes the vectors more radial, but a residual bias persists, the $f$–$B$ correlation coefficients fall mostly between $-1$ and $-0.8$, and the formal parameter uncertainties grow. Meanwhile VSM polar data show the opposite bias (vectors tilting toward the equator), and the HMI–VSM orientation discrepancy persists even in plage regions (bright areas of moderate magnetic strength outside sunspots). The authors conclude the observed non-radiality may be a systematic bias inherent to limited resolution, and identify higher-resolution observations from the Daniel K. Inouye Solar Telescope (DKIST, a 4-meter ground telescope), out-of-ecliptic viewing-angle-dependent measurements from the Polarimetric and Helioseismic Imager (PHI) aboard Solar Orbiter, and synthetic-observation benchmarks from numerical simulations as the path to an answer. The stakes are community-wide: magnetogram-based estimates of the interplanetary open magnetic flux (field anchored at the Sun but extending into interplanetary space) underestimate in-situ measurements by a factor of two or more, a shortfall plausibly tied to unresolved, weakly filled polar fields.

Problem Statement

Determine whether the photospheric magnetic field of the solar polar regions is intrinsically radial at (sub-)arcsecond scales, by quantitatively decomposing the apparent non-radial inclinations measured at moderate resolution into true sub-pixel field geometry and inversion-/resolution-induced bias. Concretely, an answer must state the intrinsic radial or non-radial character of the polar photospheric field — the intrinsic inclination distribution of polar magnetic patches relative to the local radial direction at these scales — with defensible, validated error bounds, in which the relevant inversion and resolution biases (filling-factor treatment, spatial degradation, spectral sampling, photon noise, azimuth disambiguation at large heliocentric angles) are quantified and controlled. Routes identified in the literature as useful toward this end, but not themselves mandatory components of an answer, include end-to-end validation against synthetic observations from known-input simulations, an informative prior for the distribution of the magnetic filling factor $f$ in polar patches that breaks the $f$–$B$ degeneracy, and a reconciliation of the opposing orientation biases between instruments (e.g., HMI versus VSM) that identifies which measured properties are instrumental and which are solar.

References

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