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Issue
A&A
Volume 640, August 2020
Article Number A71
Number of page(s) 12
Section The Sun and the Heliosphere
DOI https://doi.org/10.1051/0004-6361/202038408
Published online 13 August 2020

© ESO 2020

1. Introduction

Filaments belong to the largest structures seen in the chromosphere and corona of the Sun. They are composed of fine threads of plasma which are confined by magnetic field lines. Filaments, also called prominences when observed off-disk above the limb, are easily identified using spectral lines that map the chromosphere, such as Hα or the He I 10830 Å triplet. Quiescent filaments are large and long-lived phenomena in the solar atmosphere. They are located outside of active regions and typically have lifetimes from days to weeks. On the contrary, active region (AR) filaments are always found inside active regions; they are smaller in size and more dynamic than their quiescent counterparts. Their lifetimes are usually between hours and days. For a review on the main properties of filaments, we refer to, for example, Tandberg-Hanssen (1995), Mackay et al. (2010), Parenti (2014), and Vial & Engvold (2015).

Filaments are found on top of photospheric polarity inversion lines (Babcock & Babcock 1955) and it is commonly accepted that they are sustained against gravity by the magnetic field lines. The topology of these field lines is often reported as a flux rope, where field lines twist around the plasma at the main axis or spine of the filament. There are numerous studies that support this scenario through modeling (see, e.g., van Ballegooijen & Martens 1989; Amari et al. 1999), extrapolations (see, e.g., Guo et al. 2010; Canou & Amari 2010; Jing et al. 2010; Yelles Chaouche et al. 2012), and observations (see, e.g., Kuckein et al. 2012; Xu et al. 2012; Wang et al. 2020). Magnetic field strengths of 20−40 G are typically found in quiescent filaments (see, e.g., Trujillo Bueno et al. 2002; Merenda et al. 2006), whereas AR filaments can host fields up to 600−800 G (Kuckein et al. 2009; Guo et al. 2010; Xu et al. 2012). However, we point out that AR filaments might also harbor weaker fields (Díaz Baso et al. 2016) because they are partially transparent to the radiation coming from the active region below. This severely complicates the inference of magnetic fields in AR filaments (Díaz Baso et al. 2019a, b). Disruptions of the magnetic field can produce the loss of equilibrium of the filament’s structure, thus producing an eruption. For instance, the reconnection of the magnetic field lines leads to dramatic changes in the magnetic field configuration, involving the release of energy and opening of field lines (see, e.g., Antiochos et al. 1999; Moore et al. 2001). Conversely, magnetic reconnection can also be a consequence of ideal magnetohydrodynamic (MHD) instabilities (Low 1996). These MHD simulations have shown that instabilities, such as the kink instability (Török & Kliem 2005), can also trigger large-scale eruptions. In addition, a torus instability can lead to a flux rope eruption (Kliem & Török 2006).

In a statistical study based on 106 major filament-eruption events observed at the Big Bear Solar Observatory (BBSO), Jing et al. (2004) concluded that a bit more than half of the events were associated with coronal mass ejections (CMEs). Furthermore, AR filament eruptions are more likely to be associated with flares (95%) than quiescent filament eruptions (27%). In their study, 64% of the disk events were attributed to new flux emergence. The study revealed that different mechanisms may be responsible for the triggering of filament eruptions. Similar eruptive events, such as coronal mass ejections (CMEs), have been associated with both an increase or decrease of photospheric magnetic flux (Zhang et al. 2008). Flux cancellation plays an important role for low-altitude eruptions, but it additionally requires the removal of the overlying containing field (Yardley et al. 2018).

In the present study, we infer the plasma properties of an ejected filament. From individual case studies using the He I 10830 Å triplet, it has been reported that ejected plasma can reach line-of-sight (LOS) velocities between 200 and 300 km s−1 (Penn 2000). An additional difficulty lies in the determination of physical quantities under such strong spectral-line shifts, which typically come along with multiple displaced Doppler components (see, e.g., Muglach et al. 1997; Penn & Kuhn 1995; González Manrique et al. 2016). Sasso et al. (2011, 2014) needed up to five different atmospheric components to reproduce the observed He I line profiles with an inversion code and to extract the LOS velocities and magnetic field properties. In their study, they tracked an activated filament in a flaring environment, showing upflows of up to 60 km s−1. Moreover, their blueshifted spectral-line components were mainly associated with transverse magnetic fields in the body of the filament. Multiple Gaussian fits were also necessary to fit the Hα-line profiles in a filament eruption observed with the Swedish Solar Telescope (SST, La Palma, Spain) by Doyle et al. (2019). Their strongly blueshifted Hα profiles reached at least 60 km s−1, although the real velocities were likely to be higher, but outside of the spectral range of the instrument. Recently, Wang et al. (2020) followed a quiescent filament eruption with the Dunn Solar Telescope (DST, New Mexico, USA). The authors concentrated on the magnetic properties of the filament and found homogeneous linear polarization signals in the He I 10830 Å triplet, which they interpret as a magnetic flux rope.

Here we report on the eruption of a quiescent filament where the expelled plasma crossed the slit spectrograph of the 1.5-meter GREGOR telescope (Schmidt et al. 2012) located on the island of Tenerife, Spain. We perform an analysis of the four Stokes profiles arising from the chromospheric He I 10830 Å triplet detected during the eruption process. The goal is to characterize the ejected plasma of the filament.

2. Observations

2.1. Ground-based observations

The filament eruption was observed on 2016 July 3 with the ground-based GREGOR telescope and the full-disk imager Chromospheric Telescope (ChroTel, Kentischer et al. 2008; Bethge et al. 2011), both located at the Observatorio del Teide, Tenerife, Spain. To our knowledge, this is the first filament eruption observed by GREGOR.

Figure 1 shows an Hα full-disk image of the Sun before the eruption started. Full-Stokes slit-spectropolarimetry with the GREGOR Infrared Spectrograph (GRIS, Collados et al. 2012) was acquired close to disk center (μ = 0.99) during the eruption of the filament. Three short raster scans of 20 steps each during the maximum of the event assured a proper tracking of the eruption (see Table 1). The first two raster scans, A and B, were taken between 10:02 UT and 10:07 UT, whereas the third one C was recorded between 10:08 UT and 10:10 UT, with a slightly shifted field-of-view (FOV) toward South with respect to the first two scans. The step size of the slit was 0 . ″ 135 $ 0{{\overset{\prime\prime}{.}}}135 $ and each step consisted of ten accumulations with an exposure time of 100 ms each. The spatial sampling along the slit was 0 . ″ 136 $ 0{{\overset{\prime\prime}{.}}}136 $ and the orientation is shown as yellow (maps A and B) and an orange (map C) lines in Figs. 2 and 3. The GREGOR polarimetric unit (Hofmann et al. 2012) was used to carry out the polarimetric calibration of the data. Dark and flat-field corrections were performed following the standard procedures described by Collados (1999, 2003). The observations significantly benefited from the adaptive optics system (Berkefeld et al. 2010) which was locked on granulation. According to the open data policy of GREGOR and SOLARNET, the observations can be freely downloaded from the GRIS data archive1.

thumbnail Fig. 1.

ChroTel Hα filtergram on 2016 July 3 at 09:39 UT before the eruption. Solar north is up and solar west is right. The red rectangle outlines the region-of-interest shown in Fig. 2. An animation of the ChroTel images during the filament eruption is available as an online movie.

Table 1.

Spectropolarimetric raster scans of GRIS at GREGOR on 2016 July 3.

thumbnail Fig. 2.

Deep HMI magnetogram of 160 summed-up, de-rotated individual magnetograms between 09:00 and 11:00 UT on 2016 July 3. The magnetogram is clipped between ±100 G to enhance weak magnetic fields. The filled red contour corresponds to the filament in Hα at rest extracted from a ChroTel filtergram before the eruption at 09:39 UT. The yellow and orange lines represent the two slit positions of the spectrograph at about 10:06 UT and 10:09 UT, corresponding to maps A–C.

thumbnail Fig. 3.

SDO overview images with an overlap of the slit for maps A and B (yellow line, at about 10:06 UT), and map C (orange line at about 10:09 UT) from the spectrograph at GREGOR. Clockwise starting top left: HMI continuum and magnetogram followed by several AIA channels. The cyan and green arrows in the upper right 304 Å filtergram point to the erupted and stable filament, respectively. This figure is available as an online movie between 08:30 UT and 13:00 UT.

The spectral range spanned between 10824.3 Å and 10842.5 Å, with a spectral sampling of 18 mÅ px−1. This wavelength range includes, among other lines, the photospheric Si I 10827 Å line and the He I 10830 Å triplet, the latter being formed in the upper chromosphere (Avrett et al. 1994). The triplet comprises the so-called “blue” component at 10829.09 Å (JL = 1 → JU = 0) and a blended “red” component at ∼10830.30 Å (JL = 1 → JU = 1, 2).

In addition to the high-resolution GREGOR images, the auxiliary ChroTel was running in a non-standard observing mode, acquiring full-disk Hα images with a higher cadence of one minute. ChroTel is a small full-disk imager located on the flat roof of the Vacuum Tower Telescope (VTT, von der Lühe 1998) next to the GREGOR telescope. The images were corrected for dark current, flat-field, and limb darkening. An overview image of the scene with ChroTel before the eruption started is shown in Fig. 1. The full time series, including the filament eruption, is available as an online movie.

2.2. Space-borne observations

Filtergrams of the Atmospheric Imaging Assembly (AIA, Lemen et al. 2012) and magnetograms of the Helioseismic and Magnetic Imager (HMI, Scherrer et al. 2012) on board of the Solar Dynamics Observatory (SDO, Pesnell et al. 2012) were used as large context images. They were aligned and calibrated using aia_prep and they show the scene in the different layers from the photosphere to the corona (Fig. 3).

3. Data analysis

3.1. Alignment of SDO and GREGOR data

For maps A and B, we aligned the GREGOR slitjaw data to the calibrated SDO data. We used simultaneous data from 10:06 UT taken by the slitjaw camera at GREGOR in the 7770 Å continuum and matched it to SDO/HMI continuum data by shifting and rolling the GREGOR image. This was performed manually and we estimate the precision to be within 0 . ″ 3 $ 0{{\overset{\prime\prime}{.}}}3 $. Because the slitjaw data show the spectrograph slit, its location and orientation could be retrieved easily and it is shown as a yellow line in Figs. 2 and 3. For map C, unfortunately, no slitjaw data were available, but there are simultaneous observations in a broadband channel at 4307 Å with the High-resolution Fast Imager (HiFI; Kuckein et al. 2017) instrument at GREGOR. We therefore had to carry out two steps to determine the slit alignment: (1) We matched the HiFI data at 08:31 UT to SDO/HMI and also matched slitjaw data at that time to SDO/HMI, because all three instruments had recorded at that time and because the target was focused on pores, which simplified the alignment. In this way, we could determine the pixel coordinates to which the GRIS slit position corresponds in HiFI images. We then drew an artificial slit on the HiFI image at 10:09 (the time of map C) and aligned this HiFI image to SDO/HMI by rotating, resampling, and shifting. This allowed us to determine the slit position and orientation in solar coordinates. Because this method includes more steps, we estimate its precision to about 1″. The slit of map C is shown as an orange line in Figs. 2 and 3.

3.2. k-means clustering for the Stokes profiles

A wide variety of Stokes profiles with different shapes were found in the data. This is naturally explained by the extreme atmospheric conditions present during eruptions and flares. The presence of such different spectral profiles creates difficulties when setting up the model and initial value of the parameters of an inversion code whose goal is to determine the physical parameters of the observed atmosphere. For spectral-line inversions, it is helpful to provide an initial-guess atmosphere, which is fairly close to the real conditions. This avoids local minima, speeds up the inversion process, and improves correct convergence of the code.

We used k-means clustering, an unsupervised machine learning algorithm, to classify the different types of Stokes I profiles into similar groups. This technique is often used to group common spectral profiles in large data sets (see, e.g., Pietarila et al. 2007; Viticchié & Sánchez Almeida 2011; Panos et al. 2018; Robustini et al. 2019; Sainz Dalda et al. 2019). The k-means algorithm implementation used in this work is part of the scikit-learn library for python. We found that 30 groups cover well the diversity of the spectra. The average representative spectral profile for each group is shown in Appendix A (Fig. A.1). The gray profiles represent all individual profiles belonging to the same group. A selected group of relevant intensity profiles of map A is depicted in Fig. 4. Profiles corresponding to the central dark areas in the slit-reconstructed image, namely, groups 2, 6, 10, 13, and 28, have a strong blueshift of the He I 10830 Å multiplet in common. These profiles are associated with the expelled plasma of the filament, the dark areas in Fig. 4. On the contrary, groups 4 and 25 show strong He I absorption and are located where the filament is found at rest. This is also the reason why the filament at rest is not seen in the upper part of the slit-reconstructed image because only the blueshifted He I is shown.

thumbnail Fig. 4.

Selected groups of similar Stokes I profiles determined using k-means classification. Left: slit-reconstructed He I image centered at 10828.5 Å, in the blue wing of He I, corresponding to a Doppler shift of −50 km s−1 of the He I red component. The vertical direction is along the slit. The horizontal direction is the raster-scan direction. The dark pixels represent blueshifted intensity profiles related to the erupted filament. The profiles inside the color-coded contours are shown in the right-hand panels. Right: colored spectral profiles show the average intensity profile of each group, whereas the gray profiles depict all individual profiles of each group.

3.3. HAZEL inversions

One of the reasons for clustering the profiles into groups is to facilitate their inversions with the inversion code HAZEL2. It is impossible to carry out an inversion of the whole FOV with a single configuration for the input parameters. Profiles clearly showing only a single component in the He I 10830 Å multiplet should be inverted with a single chromospheric component. Likewise, those that present more components should contain the appropriate number of atmospheric components, so that the inversion code can reliably determine the optical depths and velocities of the components. Up to three atmospheric components were necessary to fit the extreme blueshifted spectral profiles.

The first task is to align the Q >  0 reference directions in both the observations (defined by the polarimeter) and HAZEL (defined by the γ angle). To this end, we follow the description of the original paper (Asensio Ramos et al. 2008) and the examples shown in the manual3. The next step is the selection of atmospheric models that should match the observations. These models contain the photosphere, treated in local thermodynamic equilibrium (LTE), which produces the Si I 10827 Å line and up to three chromospheres that produce the absorption in the He I 10830 Å multiplets blended with the wing of the Si I line, along with a very simple parameterized Voigt function to fit the telluric line at ∼10832 Å. This last component is added to end up with good fits in those cases in which the He I triplet partially blends with the telluric contamination due to a strong redshift. Concerning the photosphere, we have found that five nodes in temperature, one node in microturbulent velocity and two nodes in bulk velocity produce sufficiently good fits.

Inversions are carried out in two cycles. The first cycle simply fits the Stokes I profile, with all the components in the He I multiplet. The magnetic field is set to zero in this cycle. We found it necessary to put hard limits on the LOS velocity for each component to force the consistent isolation of all velocity components. In a second cycle, all Stokes parameters are fitted by freeing the magnetic field in the photosphere and all the chromospheres. As has already been put forward by Díaz Baso et al. (2019b), the inversions of more than one single component in the He I 10830 Å multiplet leads to a panoply of ambiguous solutions. This is especially relevant for cases in which all components have very similar velocities. In those cases, the solutions we get are probably of limited interest. However, this might not be the case when the components are clearly segregated in velocity.

Contrary to the fitting of Stokes I, fitting the Stokes Q, U, and V profiles turned out to be a complex task. After analyzing a few representative pixels, we found that the weighting scheme [5,5,1] for Stokes Q, U, and V works decently well for the cases with a single component, while the scheme [5, 1/2, 1] worked best for the cases with more than one component. The reduced weight for Stokes U is a consequence of the apparent incompatibility of the synthetic Stokes Q and U signals with the observed ones. This is an issue that requires a deep investigation and that we defer for the future. Reducing the weight in Stokes U produced a relatively good fit both in Stokes Q and U on average.

4. Results

4.1. Temporal evolution as seen by the full-disk instruments

4.1.1. Before the eruption

We scrutinized Hα full-disk images from the Kanzelhöhe Observatory (Austria) data archive (Pötzi et al. 2013), which showed the filament for the first time on 2016 June 28. At the beginning, the filament channel is not completely filled, as several fragments of the spine do not show Hα absorption. Over the next days, the fragments expand and merge to form a larger filament. It is worth mentioning that the full-disk Hα movies (not shown) demonstrate that the filament is very dynamic, that is, different pieces of the filament’s spine merge, change their shape, split, etc, over the subsequent days.

As expected, the filament lays on top of the photospheric polarity inversion line (PIL) as depicted in Fig. 2. Since the filament is very long, it is located on top of broader and narrower areas of the PIL. Of particular interest is the very narrow PIL located at (x, y = 60″, 220″). The HMI magnetogram (Fig. 2) shows an “abutted” opposite polarity plage, which, at some locations, almost touch. Here the filament is more compact and confined. The different AIA channels (Fig. 3, see also the online movie) witness intensity enhancements, arising from the compact PIL, starting at about 9:27 UT. The enhancements are seen from the chromosphere (AIA 304 Å), via the transition region, up to the corona (AIA 171 Å, 193 Å, 94 Å, and 131 Å). In particular, the channel at 131 Å shows an increment of intensity south of the PIL, nearby the eastern part of the filament, five minutes later. The brightening occurs at the location where the material of the erupted filament is later detected.

4.1.2. During the eruption

The eruption originates at the above-mentioned abutted PIL around (x, y = 60″, 220″). The event was associated to a Geostationary Operational Environmental Satellite (GOES) B1.6 flare, which originated at coordinates N15 W04 at 09:52 UT, peaked at 09:58 UT, and ended at 10:03 UT. The full-disk Hα images from ChroTel exhibit two well-defined brightenings on 2016 July 3 at 09:55 UT, next to the narrow PIL, one toward the north and the other one toward the south (see online movie). These are likely the ribbons of the B-class flare. The stationary character of these ribbons indicate that the eruption remained confined. The Hα filtergrams show bright plasma, indicating heating, expelled toward opposite directions (north and south), which, about 20 min later, appears as cooler dark material that later recombines again with the filament channel. Further away from the origin of the eruption; its having first bright and then dark Hα features is ascribed to the strongly wavelength-shifted spectral line during this event. When the eruption starts, plasma is violently expelled toward higher layers of the atmosphere. Hence, the Hα line is heavily blueshifted and the ChroTel filter, with the full width at half maximum (FWHM) of the passband filter between 0.5 Å and 1.0 Å (Bethge et al. 2011), likely only sees the wing of the line (higher intensities as compared to the line core). However, plasma later drifts back toward the solar surface again. Therefore, we see Hα absorption (dark material) which apparently flows into the filament channel. The trajectory of the plasma therefore starts at the abutted PIL, where the filament eruption starts, and partially ends southward recombining with the stable part of the filament.

The AIA 94 Å, 171 Å, 193 Å, and 304 Å filtergrams nicely trace the ejected plasma, which appears as intensity enhancements in the data and is highlighted with a cyan-colored arrow in Fig. 3. Furthermore, the time evolution of AIA 171 Å images in Fig. 5 and the animated Fig. 3 show that the slit from GREGOR was ideally located to record the outward moving material of the filament. The green arrow in Fig. 3 points to the stable part of the filament. The AIA 171 Å filtergrams (Fig. 5) depict the evolution of the filament eruption. At the beginning, the eruption consists of a rapid rising phase, which is well-traced in the first three panels between 09:55 UT and 10:05 UT. Afterward, the material slows down and expands, as seen in the last three panels.

thumbnail Fig. 5.

AIA 171 Å time evolution of the filament eruption between 09:55 and 10:40 UT. The red arrow marks the erupted material. The first three panels show a very dynamic rising phase of the filament, whereas the last three panels suggest a slower evolution, which is compatible with the untwisting of field lines.

4.2. High-resolution analysis of the ejected plasma

The slit-spectrograph GRIS at GREGOR recorded three short time series when the ejected plasma of the filament crossed the slit, between 10:02 UT and 10:10 UT (Table 1), about 170″ south from the triggering of the eruption. According to the full-disk Hα filtergrams, the filament eruption had just begun a few minutes earlier (09:55 UT) and the expelled plasma was just crossing the scanned area with the slit. The top part of the raster scans partially overlap with the spine of the original filament (best seen in Fig. 2), which remained stable. A significant amount of plasma from the eruption recombines later with this part of the filament.

The results from the inversions are illustrated in Figs. 6, 7, 9, and 10. The y-direction is along the slit whereas the x-direction is the scan direction of the slit. Maps A and B are co-spatial, but map C shows a different FOV, slightly shifted to South with respect to the former maps (see yellow and orange slits in Figs. 2 and 3). Maps A–C are temporally consecutive.

thumbnail Fig. 6.

LOS velocity maps inferred from the inversions with HAZEL. From left to right: temporally consecutive maps A, B, and C. Maps A and B roughly represent the same FOV, whereas map C is slightly shifted toward South with respect to the first two maps. Up to three atmospheric components where necessary to fit the observed Stokes profiles. A composition of the three components is exhibited here, where the fastest LOS velocity is always shown.

thumbnail Fig. 7.

Same as Fig. 6 but for the optical depth τ. The maps are a composition of up to three atmospheres. Only the optical depth corresponding to the fastest LOS velocity component is shown.

4.2.1. Line-of-sight velocity

The LOS velocities are shown in Fig. 6, which displays a composition of the three inferred atmospheres. Each pixel shows the LOS velocity arising from the fastest atmosphere. The very rapid flows are associated to the ejected plasma of the filament. In general, all three maps reveal strong upflows, with maximum values of −65.4 km s−1, −73.0 km s−1, and −66.1 km s−1 in maps A–C, respectively. This is expected from the strong blueshifts seen of the He I triplet in Fig. 4. The triplet sometimes invades significantly the red wing of the Si I line. The average LOS velocities inside the erupted filament material, between y ∈ (18″, 40″), are −48.1 km s−1, −53.5 km s−1, and −42.0 km s−1, for maps A, B, and C, respectively. The dispersion of the velocities is high, yielding a standard deviation of 19.8 km s−1, 12.1 km s−1, and 26.1 km s−1, for maps A–C. Conversely, the plasma flows are slow and downward directed at the stable filament in the upper part of the FOV (above 52″) in maps A and B. We used a high optical depth (τ >  1.2) to isolate the pixels belonging to the stable filament. The average LOS velocities within this threshold correspond to 1.6 km s−1 and 0.5 km s−1 for maps A and B, respectively. The standard deviation is 0.9 km s−1 for both maps. The stable filament is not seen in the upper part of map C because the FOV is shifted with respect to maps A and B.

4.2.2. Optical depth

Figure 7 exhibits the optical depth of He I 10830 Å inferred from the inversions. As mentioned above, each pixel shows the optical depth associated to the fastest LOS velocity atmosphere. Maps A and B depict an increased optical depth τ above 53″ in the y-axis. This area corresponds to the spine of the stable filament, which is at rest and remains stable during the partial eruption. There the optical depth reaches up to 2.5. An example of the strong absorption profiles, which almost reach the line depth of the much deeper photospheric Si I 10827 Å line is shown in Fig. 4 (clusters 4 and 25). An individual profile example from the stable filament together with its inversion result is represented in Fig. 11. The ejected plasma from the filament eruption is faintly seen as dark-violet colors all over the FOV (Fig. 7). Between y = 27″ − 28″, τ is slightly enhanced in map A and continues to increase 2 min later in map B. The optical depth in this area increases from about 0.7 to 1.1. Map C reaches optical-depth values up to 1.0.

4.2.3. Polarization and magnetic field

The Stokes Q/Ic, U/Ic, and V/Ic signals within the three scans A, B, and C are generally small (∼10−3). This can be seen, for example, in the average polarization values in map B (Fig. 8), which were computed generously across the entire He I triplet, between 10828.5 Å and 10831.5 Å. We note that these numbers are just broad averages meant to identify areas of high polarization signals. The largest signals arise from the stable filament portion in the upper part of the FOV and belong to Stokes Q/Ic and U/Ic. The Stokes Q/Ic and U/Ic profiles in the filament (bottom four panels of Fig. 11) have a typical one-lobe Hanle effect signature, which is also expected in quiet-Sun filaments (see, e.g., Trujillo Bueno et al. 2002). The Stokes V/Ic signals are negligible across the whole FOV (Fig. 8). Interestingly, the erupted filament material at around y ∼ 28″ in Fig. 8 only exhibits Stokes Q polarization signals. We discuss this further in Sect. 5.

thumbnail Fig. 8.

Mean intensity and mean polarization signals of He I 10830 Å in map B, between 10828.5 Å and 10831.5 Å. The absolute values of Stokes |Q|/Ic, |U|/Ic, and |V|/Ic were used and averaged to quantify the amount of polarization signals.

The inferred magnetic fields in Figs. 9 and 10 refer to the local solar reference frame. Up to three atmospheric components are needed for the He I inversions. Here we also represent a combined map for each physical quantity. The reference for choosing one of the three components in each pixel is again given by the highest LOS velocity. We used a mask to exclude low-polarization signals. To that end, we computed the average polarization signal P QS ¯ = Q ¯ 2 + U ¯ 2 + V ¯ 2 $ \overline{P_{\mathrm{QS}}} = \sqrt{\bar{Q}^2+\bar{U}^2+\bar{V}^2} $ in a quiet-Sun area, between x ∈ ( 0 ″ − 1 . ″ 3 $ 0{{\prime\prime}}{-}1{{\overset{\prime\prime}{.}}}3 $) and y ∈ ( 0 ″ − 13 . ″ 5 $ 0{{\prime\prime}}{-}13{{\overset{\prime\prime}{.}}}5 $), in each map. Since the polarization signals are generally small, we find that pixels with a P ¯ $ \overline{P} $ signal 20% above the P QS ¯ $ \overline{P_{\mathrm{QS}}} $, for maps A and B, represent well relevant pixels. However, for map C we lowered the threshold, as the degree of polarization is slightly lower in the entire FOV compared to maps A and B. This is likely because the erupted filament material is moving away. Thus, the threshold was set to include pixels that have P ¯ $ \overline{P} $ signals 15% above the P QS ¯ $ \overline{P_{\mathrm{QS}}} $. The horizontal fields in Fig. 9 show a salt-and-pepper pattern and the results have to be interpreted with caution. This reveals the complexity of the inversion process with such tangled Stokes profiles. Most of these complex profiles require two or three atmospheric components in the inversion process, which, in turn, leads to ambiguities and multiple solutions for the same observed profiles. For example, the four top panels of Fig. 11 show the observations (dots) and best fit (red solid line) from the inversions. In particular, Stokes Q shows an extended faint lobe at the position of the He I triplet (three rightmost vertical dashed lines), and since Stokes U is below the noise level, the result has to be interpreted with caution due to the arising ambiguities. Therefore, we do not rely on individual inversions and, rather, we provide a statistical study of the horizontal and vertical fields inside the erupted filament material, which is shown in Table 2. For the statistics, we used the results from the most blueshifted atmospheric component for each pixel. Independent of the distribution, the P50% percentile provides the median. The P16% and P84% percentiles are shown to indicate the equivalent range of ±1 standard deviation σ under the assumption of a normal distribution. Maps A and B show on average stronger horizontal fields B hor ¯ $ \overline{B_{\mathrm{hor}}} $ of 254 G and 262 G, respectively, than map C, which yields on average 173 G, but in the latter map the area of the erupted material is much smaller.

thumbnail Fig. 9.

Same as Fig. 6 but for the horizontal component of the magnetic field Bhor. The maps are a composition of up to three atmospheres. Only the magnetic field corresponding to the fastest LOS velocity component is shown. A mask was used to exclude pixels with low polarization signals (light gray pixels).

thumbnail Fig. 10.

Same as Fig. 6 but for the vertical magnetic field Bver. The maps are a composition of up to three atmospheres. Only the magnetic field corresponding to the fastest LOS velocity component is shown. A mask was used to exclude pixels with low polarization signals.

thumbnail Fig. 11.

Example of normalized Stokes I/Ic, Q/Ic, U/Ic, and V/Ic profiles. Top: dots correspond to the observations from a pixel within the erupted filament material in map B, at coordinates (x, y) = (1 . ″ $ {{\overset{\prime\prime}{.}}} $7, 28 . ″ $ {{\overset{\prime\prime}{.}}} $4). The red solid line represents the fit of the HAZEL inversions with two atmospheric components. The inferred horizontal magnetic field is 150 G and 23 G, for the first and second components, respectively. The vertical magnetic field is negligible and the LOS velocities are −4.3 km s−1 and −27.8 km s−1, respectively. Bottom: dots refer to a pixel within the filament at rest in the upper FOV of map B, at coordinates (x, y) = (0 . ″ $ {{\overset{\prime\prime}{.}}} $8, 53 . ″ $ {{\overset{\prime\prime}{.}}} $5). The red solid line represents the fit of the HAZEL inversions with one atmospheric components. The inferred horizontal magnetic field is 9 G. The vertical magnetic field is negligible and the LOS velocity is −2.1 km s−1. Vertical dotted lines indicate the wavelength positions at rest of Si I and the He I triplet.

Table 2.

Horizontal and vertical magnetic field statistics.

The vertical magnetic fields are smaller than the horizontal ones, which is not surprising as the Stokes V signals are almost absent (Fig. 8). We find on average a | B ver | ¯ $ \overline{|B_{\mathrm{ver}}|} $ of 58 G and 78 G for maps A and B, respectively, within the area of the erupted filament, and 39 G for map C (Table 2).

Regarding the area of the stable filament, that is, above y = 53″ in maps A and B, we find as well salt-and-pepper-like horizontal fields. However, the Stokes Q and U signals are significantly different from the Stokes profiles found in the erupted material. Figure 11 compares one example of Stokes profiles of the stable filament with the erupting material, together with its best inversion fit. The linear polarization profiles, that is, Stokes Q and U, are larger and tighter than in the erupted material. Stokes I shows a deep and unshifted He I triplet. The vertical fields in the filament are almost absent (Fig. 10), as expected in the spine of the filament. This can be seen in the lack of Stokes V in the bottom panel of Fig. 11.

We compute the azimuth ϕ of the magnetic field lines based on

ϕ = arctan ( B y B x ) , $$ \begin{aligned} \phi = \arctan \left( \frac{B_{ y}}{B_x} \right), \end{aligned} $$(1)

where By and Bx are the inferred horizontal components of the magnetic field from the inversions. Due to the potential ambiguities which arise from the inference of the magnetic field using the Stokes Q, U, and V profiles, we have different possible solutions for the azimuth (±180° and ±90° in the azimuth in the LOS reference frame). By assuming that the field lines have an homogeneous distribution, rather than an unorganized one, we find a smooth solution for the azimuth by rotating all angles to the first quadrant (Q1 where ϕ ∈ [0° ,90°]) of a Cartesian two-dimensional system. This rotation was achieved by showing ϕmod90°. We use the fact that the observations are very close to disk center, so that the inferred azimuth is very similar to the azimuth in the LOS. The results are shown in Fig. 12. The average azimuth ϕ ¯ $ \overline{\phi} $ and its standard deviation σ appear in Table 3. The other possible solutions for all quadrants are also mentioned in the table. Any combinations between quadrants Q1–Q4 of the erupted filament and the stable filament are possible. Maps A and B show almost identical numbers (within decimals), therefore we merged both results in the table. Unfortunately, map C does not show the stable filament in the upper part of the slit because of its shifted FOV (Fig. 7).

thumbnail Fig. 12.

Same as Fig. 6 but for the azimuth of the magnetic field ϕ. One solution, corresponding to the first quadrant ϕ ∈ [0° ,90°], is shown.

Table 3.

Possible average azimuth ϕ ¯ $ \overline{\phi} $ configurations for the erupted and stable filament areas within maps A and B.

5. Discussion

Our high spatial- and spectral-resolution data from the infrared spectrograph attached to the 1.5-meter GREGOR telescope show intriguing observations of the material of a filament eruption. The triggering of the eruption itself was not covered by the high-resolution observations. However, full-disk Hα images (Fig. 1) and AIA/SDO images (Fig. 3) reveal that the origin was located next to an abutted PIL and was associated to a B1.6 flare. A segment of the filament then rose, moved southward, and crossed our FOV about ten minutes later, while the remaining filament remained stable. The ejected plasma is well seen in the hotter AIA channels, for instance, 94 Å, 171 Å, and 193 Å, and in addition in the chromospheric He II 304 Å filtergrams (Figs. 3 and 5). Fragmentary eruptions are not uncommon and often partially expelled material recombines with the original filament (see, e.g., Jenkins et al. 2018; Yan et al. 2020). This is also happening in our case and is best seen in the context Hα full disk images.

5.1. Variety of spectral profiles

A large variety of spectral profiles were found in the observations, which demonstrates the complexity of analyzing very dynamic and eruptive events on the Sun. Sasso et al. (2011) and Schad et al. (2016) presented strongly redshifted Stokes profiles in their He I 10830 Å observations, with LOS velocities of up to 100 km s−1 and 185 km s−1, respectively. Both works used the inversion code He-Line Information Extractor (HELIX, Lagg et al. 2004) with several atmospheric components. However, we took a different inversion approach. We first classified and put into groups all similar Stokes I profiles across the FOV using the unsupervised machine-learning algorithm k-means. We came up with 30 representative clusters (Appendix A), which cover the spectral variety. Then, according to the shape of the average intensity profile of each group, we chose up to three atmospheric components, each one having a different initial set of velocity ranges. This was crucial for the successful convergence of the inversion code because the initial guess atmospheres comprised velocity ranges which were already close to the observed ones. In addition, it allowed us to disentangle the different features within the observed FOV. There were clearly two important groups of spectral profiles inside the FOV (Fig. 4): (1) the erupted material (for example, clusters 2 and 10) and (2) the filament at rest (clusters 4 and 25). An example of the four Stokes parameters of each group is shown in Fig. 11. The first group illustrates similar profiles as presented by Sasso et al. (2011). However, their profiles exhibit a deeper absorption of He I, because they were optically thicker than ours. Surprisingly, compared to the filament eruption shown by Sasso et al. (2011 2014), there are no downflows in our erupted plasma. The authors showed Stokes I profiles which have both blueshifted and redshifted He I absorption lines. Since we do not detect that in our observations, a plausible explanation is that our captured material must have belonged to a strongly-rising phase of the eruption process. The Hα and SDO movies show that the rising filament material partially falls back to the filament channel. Unfortunately, these times were not covered by our GREGOR observations. Strongly redshifted He I profiles indicating a draining of the plasma may be expected there.

5.2. High velocities of the erupted material

We analyzed three consecutive maps (Table 1), taken over a short time scale (<8 min). Maps A and B are roughly co-spatial and they cover about four minutes of the event. Within this short timescale, we see significant changes in the optical depth of the erupted filament material (Fig. 7). However, the inferred magnetic fields display very similar values (Figs. 9 and 10). The changes in the shape of the He I cloud are ascribed to the fast crossing of the material throughout our FOV. The velocities reported for filament eruptions are widespread. It is often distinguished between a slow rising phase (for example, 10−15 km s−1) followed by a rapid ejection of the material (for example, 100−200 km s−1) (Sterling & Moore 2004, 2005; Penn 2000; Dhara et al. 2017). The velocities also depend on the height up to which the filament eruption can be traced, for example, 600 km s−1 were found in transition-region spectral-line Doppler shifts (Kleint et al. 2015), and even higher velocities when tracing CME material (see, e.g., Moon et al. 2002; Cheng et al. 2020).

At the height of He I 10830 Å formation we find LOS velocities of up to ∼73 km s−1 (Fig. 6), which are upflows given the proximity to the disk center. We interpret this value as a lower limit, since the He I triplet runs into the neighboring Si I line and cannot be reliably distinguished anymore by the inversion code. The largest amount of erupted material is seen in map A, followed by map B, while map C shows the smallest amount (Fig. 7). This is consistent with the assumption of a cloud of He I rapidly moving horizontally from left to right across the FOV. The AIA/SDO filtergrams confirm this motion of the plasma, which we roughly estimate at ∼100 km s−1, from the tracking of bright features of the filament in SDO/AIA images. This motion is roughly perpendicular to the slit orientation in the movie. Combining the fastest LOS velocities inferred from our He I observations with the estimated projected velocities from SDO/AIA we compute the total velocity as

v = v He 2 + v AIA 2 , $$ \begin{aligned} { v} = \sqrt{{ v}_{\rm He}^2+{ v}_{\rm AIA}^2}, \end{aligned} $$(2)

which yields a lower limit of v ∼ 124 km s−1 for the erupted plasma of our filament.

5.3. Magnetic field configuration

The erupted material shows faint but coherent polarization signals, which result from the Hanle effect, indicating an organized orientation of the magnetic field lines (Fig. 8). Moreover, there is no indication of circular polarization or Zeeman-effect patterns. Only Stokes Q is found in the area of high He I absorption belonging to the erupted material of the quiescent filament. Conversely, inside the stable filament, Stokes Q and U is prominently present, indicating that a different orientation of the magnetic field lines exists. We quantified this orientation by computing the average azimuth of the magnetic field lines, obtained with Eq. (1), in the erupted and in the stable filament (Table 3). When rotating the azimuth angles to the first quadrant Q1, we discover a smooth solution (Fig. 12) as expected from the homogeneous polarization signals. However, this solution is not unique. To search for differences between the magnetic structure of the erupting material and the stable filament, we rotate the azimuth solution by multiples of 90°. The outcome is that there is no possible combination of the several quadrants that satisfy that the stable filament is aligned with the erupted plasma of the filament. This demonstrates that both magnetic systems have a different orientation of the field lines. The closest possible orientation appears if both the erupted material and the stable filament lie in the first quadrant Q1. Then the difference of the azimuth angle between both structures is only ∼18°.

We cannot provide the absolute orientation of the magnetic field lines with respect to the solar surface. The reason is that the intrinsic ambiguities are bound to the Stokes profiles themselves. In addition, more ambiguities arise from the use of up to three atmospheric components, which are necessary to fit the observed Stokes profiles. The use of multiple components was already reported by Sasso et al. (2011) and was ascribed to different layers along the LOS, highlighting the importance of the fine structure in filaments. When combining the results of our three atmospheric components – taking into account only the most blueshifted atmosphere, as it best represents the erupted plasma – we find predominantly horizontal magnetic fields of 254 G and 262 G, (Table 2), in the first two maps, respectively. The average vertical component of the magnetic field in the erupted material for these two maps is lower, 58 G and 78 G, respectively. Hence, the total field strength lies between typical values inferred for quiescent filaments (20−40 G, e.g., Trujillo Bueno et al. 2002; Merenda et al. 2006) and AR filaments (600−800 G, Kuckein et al. 2009; Guo et al. 2010; Xu et al. 2012). The retrieved number is more than twice higher than the average total magnetic field of ∼119 G inferred by Sasso et al. (2014), which the authors found in their He I absorption cloud associated to their active region filament eruption. We note that our filament does not belong to an active region and, therefore, it shows uncommonly high field strengths compared to stable quiescent filaments. Whether the strong fields might be a consequence of an ambiguity cannot be ruled out at this point since no other chromospheric lines with polarimetry are available.

Although the erupted plasma has an organized orientation (Fig. 8 and movie of Fig. 3), based on the analyzed data, we cannot trace individual field lines. Wang et al. (2020) interpreted in their observations a flux rope topology for the filament, which does not change during the initial phase of their eruption. Furthermore, Xue et al. (2016) reported an untwisting of the flux rope in an erupted filament. Our inferred field strengths and observed polarization maps are not incompatible with such a flux rope topology. However, we do not find evidence for rotating motions of the erupting plasma, as described by Li et al. (2017). If there was such a rotation, we would find flows going in opposite directions on both sides close to the main axis of the flux rope of the erupted material. According to the SDO/AIA (Fig. 3) and the full-disk Hα filtergrams, the main axis of the material should be approximately perpendicular to our y-axis (along the slit), in maps A and B. The redshifts, which are found in our observations, are outside of the erupted material, represented by higher optical depth (Fig. 7). In addition, the average azimuth angle of the magnetic field between the two consecutive maps A and B is virtually the same. Both maps were observed within 4 min and 13 s. In this time range, no changes in the field orientation are seen. The AIA 171 Å filtergrams in Fig. 5 (see also movie of Fig. 3), suggest an untwisting of the erupted filament in the time range 10:10–10:40 UT. Hence, this happens after our high-resolution observations have indicated that we just observed the fast rising phase of the filament. The untwisting stage was observed here only about 15 min after the start of the filament eruption, which we pinpointed at 09:55 UT, according to the full-disk Hα images.

6. Conclusions

We characterized the plasma of an erupting filament using full-Stokes spectral-line inversions. We found fast-moving plasma with a total velocity of ∼124 km s−1, as a lower limit. Furthermore, predominantly horizontal magnetic fields, on average between 173 and 262 G, were found, which is untypically high for quiescent filaments. The field lines were smoothly organized, showing a low dispersion, of the order of 11°, in the azimuth angle. The orientation remained stable within two consecutive observed maps. This indicates no rotation of the field lines at fast time scales below ∼2.5 min, during the fast-eruption phase. In our case, the untwisting phase started about 15 min after the filament eruption started.

Erupting-filament observations with polarimetry from ground-based high-resolution telescopes are rare and, to our knowledge, this is the first filament eruption observed by GREGOR. Difficulties arise mainly due to the unpredictability of such spontaneous events. The present data have shown the complexity of Stokes profiles associated to eruptive solar events such as filament eruptions. There are no current inversion codes that would automatically deal with the numerous ambiguities arising from the observed polarization signals. One important step to disentangling the various signals is using machine-learning algorithms, such as the k-means used in this work. In the future, a combination of such classification algorithms, together with complex inversion codes that also provide the probability of the possible solutions, are necessary. The use of extensive multiwavelength polarimetric observations, as planned, for example, for the Daniel K. Inouye Solar Telescope (DKIST; Tritschler et al. 2016) and the European Solar Telescope (EST; Jurčák et al. 2019), will also significantly reduce the number of ambiguities in this area of study.

Movies

Movie of Fig. 1 Access here

Movie of Fig. 3 Access here

2

We used HAZEL2, the current version of the code can be found at https://github.com/aasensio/hazel2

3

The HAZEL manual, with a description of how to choose the reference direction in the code can be found in https://aasensio.github.io/hazel2/index.html

Acknowledgments

The 1.5-meter GREGOR solar telescope was built by a German consortium under the leadership of the Leibniz-Institut für Sonnenphysik in Freiburg (KIS) with the Leibniz-Institut für Astrophysik Potsdam (AIP), the Institut für Astrophysik Göttingen (IAG), the Max-Planck-Institut für Sonnensystemforschung in Göttingen (MPS), and the Instituto de Astrofísica de Canarias (IAC), and with contributions by the Astronomical Institute of the Academy of Sciences of the Czech Republic (ASCR). The ChroTel filtergraph has been developed by the KIS in cooperation with the High Altitude Observatory (HAO) in Boulder, Colorado, USA. We thank B. Kliem for helpful comments on the manuscript. We thank the referee for providing helpful suggestions to improve the manuscript. The observations were supported by the SOLARNET access time, which belonged to the European Commission’s 7th Framework Programme under grant agreement No. 312495. CK and SJGM thank the German Academic Exchange Service (DAAD) for funds from the German Federal Ministry of Education & Research and the Slovak Academy of Science under project No. 57449420. CK acknowledges KIS for travel support within this collaboration. Funding from the Horizon 2020 projects SOLARNET (No 824135) and ESCAPE (No 824064) is gratefully acknowledged. SJGM acknowledges the support of the project VEGA 2/0048/20. SJGM also is grateful for the support of the Stefan Schwarz grant of the Slovak Academy of Sciences and the support by the Erasmus+ programme of the European Union under grant number 2017-1-CZ01-KA203-035562 during his 2019 stay at the IAC. SJGM was partially supported by the Spanish Ministry of Science, Innovation and Universities through the grant PGC2018- 095832-B-I00, and by the European Research Council through the grant ERC- 2017-CoG771310-PI2FA. AAR acknowledges financial support from the Spanish Ministerio de Ciencia, Innovación y Universidades through project PGC2018-102108-B-I00 and FEDER funds. This research has made use of NASA’s Astrophysics Data System.

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Appendix A: k-means clustering

An overview of all the 30 groups obtained from the k-means clustering is shown in Fig. A.1.

thumbnail Fig. A.1.

Classification of the Stokes I profiles within map A using k-means. A total amount of 30 cluster represent well the variety of intensity profiles across the map. The blue profile is the average of all the individual profiles, which appear in gray color, within one cluster. Each cluster is identified by a number, which appears in the lower left corner of each panel.

All Tables

Table 1.

Spectropolarimetric raster scans of GRIS at GREGOR on 2016 July 3.

In the text
Table 2.

Horizontal and vertical magnetic field statistics.

In the text
Table 3.

Possible average azimuth ϕ ¯ $ \overline{\phi} $ configurations for the erupted and stable filament areas within maps A and B.

In the text

All Figures

thumbnail Fig. 1.

ChroTel Hα filtergram on 2016 July 3 at 09:39 UT before the eruption. Solar north is up and solar west is right. The red rectangle outlines the region-of-interest shown in Fig. 2. An animation of the ChroTel images during the filament eruption is available as an online movie.
In the text
thumbnail Fig. 2.

Deep HMI magnetogram of 160 summed-up, de-rotated individual magnetograms between 09:00 and 11:00 UT on 2016 July 3. The magnetogram is clipped between ±100 G to enhance weak magnetic fields. The filled red contour corresponds to the filament in Hα at rest extracted from a ChroTel filtergram before the eruption at 09:39 UT. The yellow and orange lines represent the two slit positions of the spectrograph at about 10:06 UT and 10:09 UT, corresponding to maps A–C.
In the text
thumbnail Fig. 3.

SDO overview images with an overlap of the slit for maps A and B (yellow line, at about 10:06 UT), and map C (orange line at about 10:09 UT) from the spectrograph at GREGOR. Clockwise starting top left: HMI continuum and magnetogram followed by several AIA channels. The cyan and green arrows in the upper right 304 Å filtergram point to the erupted and stable filament, respectively. This figure is available as an online movie between 08:30 UT and 13:00 UT.
In the text
thumbnail Fig. 4.

Selected groups of similar Stokes I profiles determined using k-means classification. Left: slit-reconstructed He I image centered at 10828.5 Å, in the blue wing of He I, corresponding to a Doppler shift of −50 km s−1 of the He I red component. The vertical direction is along the slit. The horizontal direction is the raster-scan direction. The dark pixels represent blueshifted intensity profiles related to the erupted filament. The profiles inside the color-coded contours are shown in the right-hand panels. Right: colored spectral profiles show the average intensity profile of each group, whereas the gray profiles depict all individual profiles of each group.
In the text
thumbnail Fig. 5.

AIA 171 Å time evolution of the filament eruption between 09:55 and 10:40 UT. The red arrow marks the erupted material. The first three panels show a very dynamic rising phase of the filament, whereas the last three panels suggest a slower evolution, which is compatible with the untwisting of field lines.
In the text
thumbnail Fig. 6.

LOS velocity maps inferred from the inversions with HAZEL. From left to right: temporally consecutive maps A, B, and C. Maps A and B roughly represent the same FOV, whereas map C is slightly shifted toward South with respect to the first two maps. Up to three atmospheric components where necessary to fit the observed Stokes profiles. A composition of the three components is exhibited here, where the fastest LOS velocity is always shown.
In the text
thumbnail Fig. 7.

Same as Fig. 6 but for the optical depth τ. The maps are a composition of up to three atmospheres. Only the optical depth corresponding to the fastest LOS velocity component is shown.
In the text
thumbnail Fig. 8.

Mean intensity and mean polarization signals of He I 10830 Å in map B, between 10828.5 Å and 10831.5 Å. The absolute values of Stokes |Q|/Ic, |U|/Ic, and |V|/Ic were used and averaged to quantify the amount of polarization signals.
In the text
thumbnail Fig. 9.

Same as Fig. 6 but for the horizontal component of the magnetic field Bhor. The maps are a composition of up to three atmospheres. Only the magnetic field corresponding to the fastest LOS velocity component is shown. A mask was used to exclude pixels with low polarization signals (light gray pixels).
In the text
thumbnail Fig. 10.

Same as Fig. 6 but for the vertical magnetic field Bver. The maps are a composition of up to three atmospheres. Only the magnetic field corresponding to the fastest LOS velocity component is shown. A mask was used to exclude pixels with low polarization signals.
In the text
thumbnail Fig. 11.

Example of normalized Stokes I/Ic, Q/Ic, U/Ic, and V/Ic profiles. Top: dots correspond to the observations from a pixel within the erupted filament material in map B, at coordinates (x, y) = (1 . ″ $ {{\overset{\prime\prime}{.}}} $7, 28 . ″ $ {{\overset{\prime\prime}{.}}} $4). The red solid line represents the fit of the HAZEL inversions with two atmospheric components. The inferred horizontal magnetic field is 150 G and 23 G, for the first and second components, respectively. The vertical magnetic field is negligible and the LOS velocities are −4.3 km s−1 and −27.8 km s−1, respectively. Bottom: dots refer to a pixel within the filament at rest in the upper FOV of map B, at coordinates (x, y) = (0 . ″ $ {{\overset{\prime\prime}{.}}} $8, 53 . ″ $ {{\overset{\prime\prime}{.}}} $5). The red solid line represents the fit of the HAZEL inversions with one atmospheric components. The inferred horizontal magnetic field is 9 G. The vertical magnetic field is negligible and the LOS velocity is −2.1 km s−1. Vertical dotted lines indicate the wavelength positions at rest of Si I and the He I triplet.
In the text
thumbnail Fig. 12.

Same as Fig. 6 but for the azimuth of the magnetic field ϕ. One solution, corresponding to the first quadrant ϕ ∈ [0° ,90°], is shown.
In the text
thumbnail Fig. A.1.

Classification of the Stokes I profiles within map A using k-means. A total amount of 30 cluster represent well the variety of intensity profiles across the map. The blue profile is the average of all the individual profiles, which appear in gray color, within one cluster. Each cluster is identified by a number, which appears in the lower left corner of each panel.
In the text

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