Reconciling Microwave-Polarization Coronal Magnetography of Solar Active Regions with Photospheric Magnetic-Field Extrapolations

The magnetic field of the solar corona above active regions is usually inferred indirectly, by extrapolating photospheric magnetograms under potential, linear force-free, or nonlinear force-free assumptions, because direct coronal measurements are scarce. Bogod and Yasnov (2016) combined one-dimensional solar-rotation stereoscopy of polarized microwave sources with gyroresonance spectral identification in RATAN-600 scans (56 frequencies across 5.98-15.95 GHz) to derive empirical field-strength-versus-height profiles $B(h)$ and longitude-height geometry for five active regions, finding kilogauss fields persisting to heights of 8-14 Mm and spiral-like reversals in the longitude-height diagrams. These empirical results frequently disagree with extrapolated fields, and the authors list five unresolved issues: reconciling radio-derived $B(h)$ and spiral/helical geometry with extrapolation models; ambiguous gyro-harmonic identification when free-free emission contributes or circular polarization is weak; the restriction of rotation stereoscopy to active regions stable over several days; the lack of full three-dimensional vector mapping; and the unexplained physical origin of the spiral/helical signatures.

Background

Quantitative knowledge of the coronal magnetic field is a prerequisite for understanding magnetohydrodynamic wave propagation, the storage and release of magnetic energy in flares and coronal mass ejections, and the structure of active-region plasma. Because the corona is too hot and diffuse for routine Zeeman measurements of its field, the standard practice is to extrapolate the photospheric field measured by magnetographs: a potential (current-free) extrapolation solves $\nabla\times\mathbf{B}=0$; linear and nonlinear force-free extrapolations solve $\nabla\times\mathbf{B}=\alpha\mathbf{B}$ with $\alpha$ constant or spatially varying, respectively, exploiting the low plasma beta of the corona (the ratio of gas pressure to magnetic pressure). All of these inherit well-documented systematics - noise and preprocessing of the boundary magnetograms, inconsistency of photospheric data with the force-free assumption, and non-uniqueness of the nonlinear reconstruction - and they lack direct coronal validation.

Radio observations provide the main direct diagnostic. Thermal free-free (bremsstrahlung) emission is weakly circularly polarized, with the polarization degree proportional to the line-of-sight field; gyroresonance emission, produced by thermal electrons gyrating in the magnetic field, is strongly circularly polarized and becomes optically thick at low harmonics $s=2$-$4$ of the electron gyrofrequency $f_B=2.8\,B$ MHz, where $B$ is the field strength in gauss. Because emission at frequency $f$ originates where $B\approx f/(2.8\,s)$, the highest frequency at which a gyroresonance source appears measures the strongest coronal field on the line of sight, and a spectral-polarization scan converts frequency into field strength once the harmonic number $s$ is identified (the third harmonic, $s=3$, is usually assumed). The microwave polarization method has a long lineage: Akhmedov et al. (1982) measured transition-region fields above sunspots of 80-90% of photospheric values from gyroresonance cutoffs; Iwai and Shibasaki (2013) mapped coronal and chromospheric fields from Nobeyama Radioheliograph polarization; Brosius and White (2006) measured 1750 G at a height of 8 Mm above a limb sunspot with the VLA.

Bogod and Yasnov (2016) added a geometric dimension. With one-dimensional scans of the RATAN-600 telescope (a 600-m ring antenna giving daily fan-beam scans) at 56 frequencies in the 5.98-15.95 GHz range, they applied solar-rotation stereoscopy: the apparent daily displacement of a stable source over successive observing days yields its height above the photosphere at each frequency, $h(f)$, and the gyroresonance relation converts $f$ into $B$, giving an empirical $B(h)$ together with the source heliographic longitude versus height, $\Delta\lambda(h)$. An improved coordinate accuracy of about $0.02^\circ$ (roughly 0.3 arcsec) made sub-megameter transverse structures detectable. For five active regions (NOAA 11665, 11289, 11423, 12436, 12209) they found transition-region fields of 1640-2250 G, fields of order 800-1000 G persisting to heights of 12-14 Mm, quasi-periodic longitudinal bendings implying transverse-to-longitudinal field-component ratios between about 0.05 and 0.9, and - in an active region with Stokes $V\approx 0$ (no net circular polarization) - a diverging spiral pattern in the longitude-height plane. This confirmed and extended earlier stereoscopic results (Nita et al. 2011; Bogod, Stupishin and Yasnov 2012) in which the stereoscopically derived heights and fields were inconsistent with photospheric-field extrapolations. The authors explicitly state that the measurements frequently disagree with extrapolated coronal fields, and they list the five unresolved issues that constitute this problem: the adequacy of potential, linear force-free, and nonlinear force-free extrapolations in reproducing the empirical $B(h)$ and the detected spiral/helical geometries; ambiguous spectral-mode (harmonic) identification when free-free emission contributes or Stokes $V$ is small or zero; the requirement of a sufficiently stable active region over sequential days, which limited the number of usable epochs; the absence of the full three-dimensional vector field (components perpendicular to the $(\Delta\lambda,h)$ plane are unmeasured); and the unestablished physical interpretation of the spiral/helical features (currents in coronal loops, stored helicity counteracting loop expansion, or other dynamic processes).

Problem Statement

Establish a validated, reconciled determination of the coronal magnetic-field structure of solar active regions from multi-frequency microwave polarization measurements. Completion is defined by the central question (1) below, which requires the systematic, quantitative comparison and attribution of the disagreements to identified causes that it states; the four further issues that Bogod and Yasnov (2016) left open (2-5) are independent extensions of the same program, not additional mandatory deliverables. Item 1 requires the spiral/helical signatures to be reconciled geometrically with the extrapolated fields; it does not require the physical-origin account of those signatures, which is item 5's separate question and remains an extension:

1. (Central question) Reconcile the empirically derived field-strength profiles $B(h)$ and the detected spiral/helical geometries with coronal fields reconstructed from photospheric magnetograms by potential, linear force-free, and nonlinear force-free extrapolation. This requires a systematic, quantitative comparison and an attribution of the frequently observed disagreements - stereoscopic and other radio-derived fields exceeding extrapolated values - to identified causes: radio-method systematics (harmonic number, free-free contamination, height assignment) versus extrapolation limitations (boundary-data quality, preprocessing, model class).

2. Make the frequency-to-field conversion robust by determining whether and how the gyroresonance harmonic can be identified in weakly polarized sources (small or zero Stokes $V$) and in the presence of free-free contamination, and by quantifying the resulting uncertainty in $B$ (which scales as $1/s$).

3. Extend the geometric measurement beyond active regions that stay stable over the multiple sequential days required by solar-rotation stereoscopy - to evolving regions, or via single-epoch two-dimensional alternatives such as multi-frequency imaging spectroscopy.

4. Extend the reconstruction from the heliographic longitude-height plane, which yields only $B(h)$ and longitudinal displacements, toward the full three-dimensional vector field, whose components perpendicular to the $(\Delta\lambda,h)$ plane are currently unmeasured.

5. Determine the physical origin of the spiral/helical longitude-height signatures - currents in coronal loops, stored magnetic helicity counteracting flux-tube expansion, or projection/sampling artifacts - which any adequate field model of these regions must either reproduce or exclude.

References

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