Does the degree of circular polarisation at millimetre wavelengths diagnose the magnetic footpoint asymmetry of solar flare loops?

Solar flares accelerate mildly relativistic electrons that emit gyrosynchrotron radiation, whose circular polarisation at optically thin millimetre wavelengths has been proposed as a diagnostic of the magnetic asymmetry between the two footpoints of a flaring loop. da Silva et al. (2020) fitted the 45 and 90 GHz flux and polarisation spectra of two flares with a three-dimensional gyrosynchrotron loop model assuming a uniform, symmetrically injected electron distribution, and found that the observed polarisation degrees map onto the modelled footpoint asymmetry $s = (|B_2| - |B_1|)/|B_1|$. Whether this mapping is unique — or whether asymmetric electron injection, anisotropic pitch-angle distributions, and non-uniform transport can mimic it — is unconstrained, and its confirmation requires spatially resolved interferometric millimetre polarimetry of flares, which still does not exist: ALMA full-Stokes solar polarimetry was commissioned only in 2023 (Band 3, 12-m array) and solar flare observing with ALMA remains uncommissioned. The problem is to settle the status of this diagnostic: quantify the magnetic-versus-injection/transport degeneracy by forward modelling, and confront the mapping with resolved millimetre polarimetric flare observations combined with multiwavelength constraints, thereby validating and calibrating it or refuting it.

Background

Solar flares release magnetic energy in the Sun's corona and accelerate electrons to mildly relativistic energies. Trapped in coronal magnetic loops, these electrons emit gyrosynchrotron radiation — a broadband continuum produced by mildly relativistic electrons spiralling around magnetic field lines — whose circular polarisation (degree of polarisation $p = V/I$, the ratio of the circularly polarised Stokes flux $V$ to the total flux $I$) depends on the magnetic field strength and direction, the viewing angle, and the energy and pitch-angle distribution of the radiating electrons (the pitch angle being the angle between an electron's velocity and the magnetic field). At centimetre wavelengths this emission is often optically thick, i.e. self-absorbed, so its interpretation in terms of coronal magnetic fields is entangled with source geometry and projection effects. At millimetre wavelengths (45–90 GHz and above) the gyrosynchrotron emission is typically optically thin and therefore, in principle, a more direct probe of both the accelerated electrons and the magnetic field in the flaring loop.

Spatially unresolved millimetre polarimetry was provided by the POEMAS two-telescope system, which measured right- and left-circular polarisation of the full Sun at 45 and 90 GHz (Valio et al. 2013). da Silva et al. (2020) analysed two flares, SOL2013-02-17 ($p \approx 4$–$7\%$) and SOL2013-11-05 ($p \approx 20$–$30\%$, of opposite sign), observed by POEMAS together with microwave (1–15 GHz), 212 GHz, and hard X-ray data. They fitted the flux and polarisation spectra with a three-dimensional gyrosynchrotron model of a single magnetic loop of fixed geometry, assuming a uniform, isotropic, energetic-electron distribution, with free parameters comprising the loop-top field strength, the mirror ratios at the two footpoints (the ratios of footpoint to loop-top field strength, which govern the magnetic mirroring of trapped electrons), the non-thermal electron density, and the electron spectral index. The model reproduces the observed spectra, and its central result is a modelled relation between the polarisation degree and the magnetic footpoint asymmetry $s = (|B_2| - |B_1|)/|B_1|$, where $B_2$ and $B_1$ are the stronger and weaker footpoint fields: large $p$ arises from highly asymmetric loops ($s \approx 0.9$ for SOL2013-11-05), small $p$ from nearly symmetric ones ($s \approx 0.04$ for SOL2013-02-17). The authors therefore proposed that $p$ could diagnose the footpoint asymmetry even without spatial resolution.

This diagnostic rests on assumptions the authors themselves flag as unverified. The model attributes all brightness and polarisation asymmetry to the magnetic field, because it injects electrons symmetrically and transports them uniformly. Yet earlier work showed that the initial pitch-angle distribution of accelerated electrons (Lee and Gary 2000) and an asymmetric injection point (Melnikov, Shibasaki and Reznikova 2002; Reznikova et al. 2009) reshape the electron distribution along flare loops and hence the polarisation, and electron-transport models contain many parameters that unresolved data cannot constrain; the fitted field strengths additionally depend on the assumed source height and geometry and on the limited spectral sampling. Confirming the mapping observationally requires spatially resolved interferometric millimetre polarimetry of flares. The Atacama Large Millimeter/submillimeter Array (ALMA), a millimetre interferometer that has observed the Sun at 3 mm since 2016, began offering full-Stokes (all four Stokes parameters) solar polarimetry only in Cycle 10 (2023), in Band 3 (around 100 GHz) and only on the 12-m array, and solar flare observing has not been commissioned (Shimojo et al. 2024); ALMA flare detections to date are single-dish, total-intensity-only images (Sković et al. 2023).

Problem Statement

Determine whether the degree of circular polarisation of the millimetre-wavelength gyrosynchrotron emission of a solar flare — the broadband radiation of mildly relativistic electrons spiralling around the coronal magnetic field lines of the flaring loop — is a robust diagnostic of the magnetic asymmetry between the two footpoints (the loop's two endpoints anchored in the solar surface) of that loop, as proposed by da Silva et al. (2020) through the mapping between the polarisation degree $p = V/I$ (ratio of the circularly polarised Stokes flux $V$ to the total flux $I$) and the footpoint asymmetry $s = (|B_2| - |B_1|)/|B_1|$, where $B_2$ and $B_1$ are the magnetic field strengths at the stronger and weaker footpoint. The objective can be settled along either of two scientifically linked routes, which serve one question rather than separate deliverables:

1. Degeneracy (forward-modelling route): quantify whether asymmetric injection of non-thermal electrons into the loop, anisotropic initial pitch-angle distributions, or spatially non-uniform electron transport can produce the same polarisation degree that the uniform-electron model attributes to magnetic footpoint asymmetry. This requires gyrosynchrotron transfer in three-dimensional asymmetric loop models coupled to realistic electron transport, a systematic exploration of the parameter space (including the systematic effects of assumed source height, loop geometry, and limited spectral sampling on the fitted field strengths), and a determination of whether the $p \to s$ mapping is unique once injection and transport asymmetries are admitted, and whether spectral, temporal, and multiwavelength constraints (hard X-ray imaging and photospheric magnetograms, i.e. maps of the Sun's surface magnetic field) can break any residual degeneracy.

2. Confirmation (observational route): confront the mapping with spatially resolved interferometric millimetre polarimetric imaging of flares (e.g. ALMA Band 3 full-Stokes observations — measuring all four Stokes parameters at around 100 GHz with the Atacama Large Millimeter/submillimeter Array — or an equivalent interferometric millimetre polarimeter), analysed together with multiwavelength imaging spectroscopy and comprehensive electron-transport modelling, and establish whether the observed relation between resolved polarisation degree and independently constrained footpoint asymmetry confirms, recalibrates, or refutes the unresolved diagnostic of the source paper.

A complete answer establishes the diagnostic as validated and calibrated with quantified uncertainties, or as degenerate or refuted; either outcome for the stated mapping closes the problem.

References

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