A new solution to the ambiguity problem 

(Solar Orbiter nugget #21 by G. Valori¹, D. Calchetti¹ , A. Moreno Vacas² , É. Pariat³ , S.K. Solanki¹ , P. Löschl¹ , J. Hirzberger¹ et al.​)

Introduction

Zeeman spectropolarimetry provides a method to infer the properties of the photospheric magnetic field, however, this method is hampered by the 180°-ambiguity in the orientation of the transverse component, specifically: Two equally-intense, oppositely-directed transverse fields produce the very same spectro-polarimetric signal. The removal of such an ambiguity is necessary to fully determine the magnetic field vector, which regulates much of the solar activity, and it is essential to progressing our understanding of solar active regions, their energetics, and their capacity to produce solar eruptions.

When viewing from only one line-of-sight (LoS), assumptions are needed in order to remove the 180°-ambiguity, which makes the disambiguation model-dependent. Solar Orbiter's Polarimetric and Helioseismic Imager (SO/PHI), by providing observations from out of the Sun-Earth line, can now be used to unequivocally and fully determine the photospheric vector magnetic field using observations only (see Figure 1).

Figure 1. Stereoscopic disambiguation concept: The magnetic field, B, is observed by two telescopes, namely SO/PHI-HRT and SDO/HMI, separated by some finite angle. Then, the (unambiguous) B_LoS^HMI reveals the correct orientation of the ambiguous B_tr^HRT (see [1] for details).

A new opportunity: the Solar Orbiter Mission and the SO/PHI Instrument

The SO/PHI instrument typically provides maps of the continuum intensity, the magnetic field vector (B_LoS, γ, φ), and the LoS-velocity with a cadence that is variable. SO/PHI is composed of 2 separate telescopes, namely: The Full-Disc Telescope (FDT) that images the entire solar disk and the High-Resolution Telescope (HRT) that observes a smaller part of the solar disk at higher resolution. The full Stokes vector is recorded at each pixel across the 2kx2k detector that the instrument employs. This full Stokes vector is recorded using a tunable filter system that scans the photospheric Fe I 617.3nm absorption line and the nearby continuum at 6 wavelengths [2].

The full spectropolarimetric capabilities of SO/PHI and the unique orbit of SO, combined with observations from Earth-bound observatories, allow for remote-sensing observations from different vantage points of the same area on the Sun, thereby providing the required observational constraints to remove the ambiguity in the transverse field (Figure 1).

Stereoscopic disambiguation method (SDM), in details

Radiative-transfer inversions of spectro-polarimetric observations provide estimates of the magnetic field B⃗ as, e.g.,

B⃗=B⃗_LoS + σB_tt⃗.

From this equation, one can infer the amplitude and orientation of the LoS component of the magnetic field (B⃗_LoS) as well as the amplitude and direction t⃗ of the transverse component of the magnetic field (B_t> 0). The orientation σ = ±1 of the transverse component along t⃗ cannot, however, be determined.

The ambiguity in the orientation of the transverse component B_r is a parity problem [3] in each pixel of the detector image plane. Hence, the disambiguation of a vector magnetogram is equivalent to fixing σ in each pixel of the image plane.

Figure 2. The SDM reference system. Here, A=SDO/HMI and B=SO/PHI-HRT.

SDM employs a special reference system (Figure 2) to remap SO/PHI-HRT and SDO/HMI observations such that:

• σ is computed as a combination of observed field components and the separation angle, ɣ, between the telescopes as: σ=(B_LoS^B - B_LoS^A cosɣ) / (B_w^A sinɣ);
• σ is geometrically equal to either +1 or -1 (nominal values);
• The components n∙B on the two telescopes are nominally identical by construction.

The SDM was developed and successfully tested on numerical simulations in [1]. The SDM is basically a pixel-by-pixel combination of observations from two telescopes onto the detector plane of one of the two. Therefore, an accurate remapping of the (ambiguous) vector field of, say, telescope A onto the detector plane of telescope B is required (see Figure 2). After such a remapping, the above equation provides σ as a simple analytical combination of measured quantities. 

SDM application results

As an application example of SDM, we consider observations of AR12965, obtained by SO/PHI-HRT and SDO/HMI on 17/03/2022, when the separation angle between the two spacecraft was γ=27°. While the method can resolve simultaneously the ambiguity on both detectors, here we only show the ambiguity removal in B⃗_t as observed by SO/PHI-HRT. The input to the method are the ambiguous SO/PHI and SDO/HMI vector magnetograms of AR12965. Figure 3 shows the successful application of the SDM to the disambiguation of observed data: the vector magnetic field on SO/PHI-HRT is fully determined, for the first time by using observations only. The transverse component is smooth, and pointing radially outwards from a positive flux concentration, as expected.

Figure 3. First stereoscopic-disambiguated HRT vector magnetogram. BLoS is plotted in grayscale. The red and blue arrows show the direction of the transverse component of the magnetic field on the positive and negative components of B_LoS, respectively. The yellow isoline contours |B_LoS|=400 G.

On the other hand, a few, unexpected orientations in localized, small regions are present (e.g. the left-pointing arrows around [330,80] in Figure 3). Such locations are positively identified by diagnostic metrics that SDM provides. In particular, the SDM is such that, nominally, σ = ±1, and that n∙B⃗ is the same for both telescopes. Such quantities can be used as a diagnostic of SDM results, meaning that departures from nominal values identify pixels where the SDM should not be applied because of differences in calibration and inversion between the employed data sets, optical distortion, or co-registration errors (Figure 4). Indeed, the unexpected orientations in Figure 3 correspond to locations where departures from nominal values in Figure 4 are present.

Figure 4. Computed σ (left) and normalized difference in n∙B⃗ (right) showing departures from nominal values.

Future applications

This first application proves that the SDM can accurately resolve the ambiguity in the transverse component of the magnetic field. Future applications will include:

• Combining SO/PHI-HRT with SDO/HMI at different distances and separation angles;
• Including other ground- and space-borne-telescopes, to study the dependence of SDM accuracy from the details of the observed spectral line;
• Benchmarking of traditional, single-view-point disambiguation methods;
• Incorporating data from the Vigil mission at L5 (at constant 1AU, γ=60° separation), whose Photospheric Magnetic field Imager (PMI) will provide vector magnetograms at 2.4" resolution and 30 minute cadence.

This study has been published in Gherardo Valori, et al., 2023, A&A, 677, A25, https://doi.org/10.1051/0004-6361/202345859 

Further information on SO/PHI data is given at: https://www.mps.mpg.de/solar-physics/solar-orbiter-phi 

Acknowledgements

Solar Orbiter is a space mission of international collaboration between ESA and NASA, operated by ESA. We are grateful to the ESA SOC and MOC teams for their support. The German contribution to SO/PHI is funded by the BMWi through DLR and by MPG central funds. The Spanish contribution is funded by AEI/MCIN/10.13039/501100011033/ and European Union “NextGenerationEU/PRTR” (RTI2018-096886-C5, PID2021-125325OB-C5, PCI2022-135009-2,PCI2022-135029-2) and ERDF “A way of making Europe”; ”Center of Excellence Severo Ochoa" awards to IAA-CSIC (SEV-2017-0709, CEX2021-001131-S); and a Ramón y Cajal fellowship awarded to DOS. The French contribution is funded by CNES. The HMI data are courtesy of NASA/SDO and the HMI science team.

Affiliations

¹ Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany e-mail: valori@mps.mpg.de
² Instituto de Astrofísica de Andalucía (IAA-CSIC), Apartado de Correos 3004, E-18080 Granada, Spain
³ Sorbonne Université, École polytechnique, Institut Polytechnique de Paris, Université Paris Saclay, Observatoire de Paris-PSL, CNRS, Laboratoire de Physique des Plasmas (LPP), 75005 Paris, France

References

[1] Valori, Loeschl, Stansby, Pariat, Hirzberger, Chen. 2022, Sol.Phys., 297, 1, https://doi.org/10.1007/s11207-021-01942-x

[2] Solanki, S. K., del Toro Iniesta, J. C., Woch, J., et al. 2020, A&A, 642, A11, https://doi.org/10.1051/0004-6361/201935325 

[3] Semel, Skumanich, 1998, A&A, 331