Authors: Ryan Campbell and Michail Mathioudakis Queen’s University Belfast, Ricardo Gafeira Instituto de Astrofísica de Andalucía, Carlos Quintero Noda and Manuel Collados Instituto de Astrofísica de Canarias
For decades we have been making significant progress in developing our understanding of quiet Sun magnetism (see  for a review). At disk centre, we can access the horizontal component of the magnetic vector by measuring linear polarisation (LP), and the vertical component by measuring circular polarisation (CP). When trying to extract information about the magnetic vector from spectropolarimetric measurements in the quiet Sun, we are attempting to measure signals which are at the limit of our polarimetric sensitivity. At disk centre, for weak fields, accessing the horizontal component is more difficult than accessing the vertical component, because a vertical field produces a larger amplitude CP profile than the amplitude of a LP profile produced by a horizontal field of equal strength. Ultimately, this means there is a vital need to understand the influence of noise on our results.
High spatiotemporal resolution observations and visualisation tools
In a new study  we present optimised quiet Sun observations of the Fe I 15648.5 Å spectral line in the deep photosphere using the GREGOR telescope equipped with the GRIS-IFU instrument. Our goal was to reveal as much polarisation as possible in a dead-quiet region of the solar surface, while still obtaining a time-series with a good cadence. Our observations show exceptionally high polarization fractions, with polarization confidently detected above a 5σ threshold in 65% of the field of view (FOV). We refer to this 65% as the “magnetised” area. This includes 61% of the full FOV with CP signals, 23% with LP, and, significantly, 18% with both. Although we anticipated higher fractions of polarisation to be revealed, not least because our modelling with radiative hydrodynamic simulations predicted it was possible , these results exceeded our expectations.
We find small-scale magnetic loops, like that in Figure 1, which seem to be embedded in a sea of magnetism, and we are able to observe their evolution. We used the Stokes Inversion based on Response functions (SIR) code  to retrieve physical information about the thermodynamics, kinematics and magnetism of the plasma, performing over 45 million inversions. We find the median magnetic field strength is weaker than previous GRIS-IFU observations , indicating that we cannot explain the higher polarisation fractions by supposing that we observed a more active target.
Shown in Figure 2, we also developed an open-source tool called SIR Explorer (SIRE) for exploring SIR inversion outputs, which is available on GitHub: https://github.com/r-j-campbell/SIRExplorer.
A noisy problem
We interrogated the impact of two conflicting approaches to the treatment of noise:
- Approach 1: the inversion proceeds without any attempt to remove Stokes Q, U, or V profiles that contain pure noise (no signal).
- Approach 2: before inversion, for each pixel, any Stokes Q, U, or V profile which does not reach the 5σ threshold is set to zero at every wavelength. However, if one linear polarisation parameter was measured confidently, the other was not set to zero.
In both approaches, we only include magnetised pixels. This means we do not include the 35% of the scan which had no polarisation. A threshold this high is already quite stringent.
As Figure 3 shows, there are significant differences. The magnetic field strength distributions of each approach diverge below 600 G, with noise potentially resulting in the over-estimation of B. Further, in approach 1, the distribution of inclinations is overwhelmingly horizontal, with only about 10% of the pixels having γ >164 or γ <16 . One might conclude that the internetwork photosphere is evidently horizontal in nature. However, given that only 23% of the FOV had confidently measured LP signals while 60% had CP signals, we are circumspect about this interpretation. We argue this γ distribution is created by the inversion, as SIR interprets noise in Stokes Q and U as real signal. In the case where noisy profiles are removed (approach 2), the distribution looks very different – there are peaks at 0° and 180°, when a pixel has only Stokes V, and 90° when only Stokes Q and/or U were measured. The remainder have intermediate inclinations for those pixels with both LP and CP. Stokes I is also sensitive to γ, so this is an oversimplification of the problem, but it is nevertheless what is happening in many pixels.
Despite our very stringent approach to noise treatment, our analysis in approach 2 revealed an internetwork region with a majority (>60%) of magnetized pixels displaying a clear transverse component of the magnetic field, with an inclination in the range 15<γ<165. This result is in stark contrast to previous observations at disk centre, including the first GRIS-IFU observations of this spectral line , which had predominantly vertical magnetic fields in the deep photosphere. The result agrees with previous studies which examined the centre-to-limb variation in linear polarisation only, and therefore also circumvented the problem of measuring LP and CP simultaneously in an unbiased way, and found that the field orientation for the weakest magnetic fields is predominantly horizontal throughout the photosphere . However, using a large 1.5m telescope, a highly magnetically sensitive spectral line, and with outstanding seeing conditions, we have been able to achieve this by analysing full-Stokes inversions at disk centre only (and despite having taken very stringent steps to remove noise contamination from our inversions!).
Our study highlights the importance of accounting for noise when conducting analyses of magnetic fields in the solar photosphere. The Daniel K. Inouye Solar Telescope (DKIST) and European Solar Telescope (EST) will be essential for advancing our understanding of the Sun’s magnetic field, because the higher spatial resolution provided could reveal even higher fractions of polarisation, or, at least, polarisation profiles with larger amplitudes. When it comes to the statistical analysis of data from these next-gen telescopes, we hope that our study will stir debate on how best to deal with noise in inversions.
-  Bellot Rubio & Orozco Suarez – Quiet Sun magnetic fields: an observational view
-  Campbell et al. 2023 – Exploring Magnetic Loops and Serpentine Fields in the Quiet Sun with the GRIS-IFU – 10.3847/1538-4357/acb33e
-  Campbell et al. 2021b – Constraining the magnetic vector in the quiet solar photosphere and the impact of instrumental degradation – 10.1051/0004-6361/202141421
-  Ruiz Cobo & del Toro Iniesta – Inversion of Stokes Profiles – 10.1086/171862
-  Campbell et al. 2021a – Temporal evolution of small-scale internetwork magnetic fields in the solar photosphere – 10.1051/0004-6361/202040028
-  Lites et al. 2017 – Are Internetwork Magnetic Fields in the Solar Photosphere Horizontal or Vertical? – 10.3847/1538-4357/835/1/14
This research has received financial support from the European Unionʼs Horizon 2020 research and innovation program under grant agreement No. 824135 (SOLARNET).