Magnetic field vector ambiguity resolution in a quiescent prominence observed on two following days

% context {Magnetic field vector measurements are always ambiguous, i.e., two or more field vectors are solutions of the observed polarisation.} % aims {The aim of the present paper is to solve the ambiguity by comparing the ambiguous field vectors obtained in the same prominence observed on two following days. The effect of the solar rotation is to modify the scattering angle of the prominence radiation, which modifies the symmetry of the ambiguous solutions. This method, which is a kind of tomography, was successfully applied in the past to the average magnetic field vector of 20 prominences observed at the Pic-du-Midi. The aim of the present paper is to apply this method to a prominence observed with spatial resolution at the TH\'EMIS telescope (European site at Iza\~na, Tenerife Island).} % methods {The magnetic field vector is measured by interpretation of the Hanle effect observed in the \ion{He}{i} D$_3$ 5876 \AA\, line, within the horizontal field vector hypothesis for simplicity. The ambiguity is first solved by comparing the two pairs of solutions obtained for a"big pixel"determined by averaging the observed Stokes parameters in a large region at the prominence center. Then, each pixel is disambiguated by selecting the closest solution in a propagation from the prominence center to the prominence boundary.} % results {The results previously obtained on averaged prominences are all recovered: Inverse Polarity, small angle of about $-21^{\circ}$ between the magnetic field vector and the filament long axis, within the Inverse Polarity scheme. The magnetic field strength, of about 6 G, is found to slightly increase with height, as previously observed also. The new result is the observed decreasing with height, of the absolute value of the angle between the magnetic field vector and the filament long axis.}


Introduction
Since the pioneering work by Hyder (1965) showing the possible existence of the Hanle effect in solar prominences, measurements of the effect were undertaken in order to diagnose the magnetic field in these objects. Leroy (1977) (see also Leroy et al. 1977) chose the He i D 3 5876 Å line for this purpose, also because this line is absent from the incident photospheric spectrum (as it is visible in Fig. 3 of this paper). Thus Doppler dimming or brightening is avoided by using this line. In addition, this line has the well-adapted Hanle sensitivity for determining the prominence magnetic field, which was found on the order of 6 G. Each spectral line has its Hanle sensitivity domain determined by the condition ωτ ≈ 1, where ω is the Larmor pulsation depending on the field strength (1.4 MHz/G ignoring the Landé factor), and τ the upper level radiative lifetime. A table of Hanle sensitivity for a series of spectral lines can be found in Sahal-Brechot (1981), where it can be seen that the He i D 3 line sensitivity fortunately lies around 6 G.
Leroy measured the Hanle effect in about 400 prominences at the Pic-du-Midi coronagraph, during the ascending phase of solar cycle XXI (1976)(1977)(1978)(1979)(1980)(1981)(1982), but without any spatial nor spectral resolution in these objects. One single spectral line is insufficient to determine the three coordinates of the field vector, because the Hanle effect is described by two parameters only: the linear polarization degree and direction. Two lines of different Hanle Send offprint requests to: V. Bommier, e-mail: V.Bommier@obspm.fr sensitivity are required for this purpose. Fortunately, the He i D 3 line is made of two partially overlapping components of different sensitivity, 3d 3 D 3,2,1 → 2P 3 P 2,1 and 3d 3 D 1 → 2P 3 P 0 , which is weaker and apart but more polarisable. Despite the difficulty to separate the components in the analysis Koza et al. 2017), Athay et al. (1983) analysed 13 prominences observed at Sacramento Peak with the Stokes I then II coronagraph spectropolarimeter, and clearly established the horizontality of the prominence magnetic field, with spatial resolution in prominences. Querfeld et al. (1985) obtained similar results for two prominences also observed with Stokes II. Leroy complemented his experiment with the quasi-simultaneous measurement of the Hydrogen Hα and/or Hβ line. From these data, Bommier et al. (1986b) also obtained the horizontality of the field vector in prominences, and also a fourth parameter, the electron density (Bommier et al. 1986a).
Observation of Hanle effect in solar prominences (Schmieder et al. 2013(Schmieder et al. , 2014a were then performed at the French-Italian Télescope Héliographique pour l'Étude du Magnétisme et des Instabilités Solaires (THÉMIS), installed at the European site at Izaña, Tenerife Island. The He i D 3 line was again observed, resolved in its two components. The horizontality of the field vector was recovered (Schmieder et al. 2014b), and further inverstigations were possible inside the prominence fine structure, with a better spatial resolution (Levens et al. 2016a(Levens et al. ,b, 2017Schmieder et al. 2017). The inversion method was based on Principal Component Analysis (PCA) technique (López Ariste & Casini 2002Casini et al. 2003Casini et al. , 2005Casini et al. , 2009. The inversion code HAZEL for simultaneous analysis of Hanle and Zeeman effects was developed besides (Asensio Ramos et al. 2008).
The analysis by Merenda et al. (2006) alternatively concludes to far from horizontal magnetic field vector in a prominence observed by the Tenerife Infrared Polarimeter (TIP) mounted on the German Vacuum Tower Telescope (VTT) of the Observatorio del Teide (Tenerife, Spain). The analysed line is the polarisable component of the He i 10,830 Å line. For typical prominence magnetic fields, this line is not far from the saturated Hanle effect regime. They conclude to not horizontal field from the fact that the observed linear polarisation degree and direction fall outside the horizontal field Hanle diagram, making the horizontal field incompatible with the observation. However, on the one hand the measurement inaccuracies and related box on the diagram are not reported, and on the other hand the observed polarisation is not far from the horizontal field diagram. It may then be remarked that if the inaccuracy box had been taken in consideration, horizontal field would have also been compatible with this observation. Moreover, the exact location of the filament on the Sun's surface remained undetermined, which was particularly difficult for this Polar Crown prominence of very high latitude, and which prevented the authors from properly evaluating the scattering angle, which acts upon the diagram.
However, magnetic field vector determinations are always ambiguous, and these recent investigations do not address this problem. Two or more field vectors are solution of the observed polarisation. The symmetries of scattering in the presence of a magnetic field were discussed in Bommier (1980). The first one is responsible for the same polarisation observed for two field vectors symmetrical with respect to the line-of-sight, in right-angle scattering. This symmetry is often called "fundamental ambiguity". When the scattering is not at right-angle, as in the present problem where we consider prominence observation modified by the solar rotation, the symmetry is modified also and the two ambiguous are not symmetrical with respect to the lineof-sight when the scattering is not at right-angle, but there are two ambiguous magnetic solutions. The two solutions may even correspond to slightly different field strengths. A second ambiguity is introduced by the Van Wleck ambiguity. In the present paper, we are not concerned by this ambiguity because do not try to separate the two partially overlapping components of the He i D 3 line and we analyse our data within the horizontal field hypothesis, following the previously cited results. The Van Wleck ambiguity is not the same for two lines of different magnetic sensitivity. Thus, this ambiguity is automatically solved when analysing such two lines. A third ambiguity appears in the strong field, or saturated regime of the Hanle effect, which is for instance the case of the Fe xiv 5303 Å line and of the Fe xiii 10,747 Å and 10,798 Å lines of the solar Corona. In the case of the saturated Hanle effect, the degeneracy is then of a factor 2 3 = 8.
Three methods were proposed in the past able to solve the fundamental ambiguity. The first one, which we will apply in the present paper, consists in comparing the two pairs of ambiguous solutions obtained for two different scattering angles, which are provided by observing the same prominence on two following days. Due to the solar rotation, the symmetry is modi-fied by the solar rotation, and the comparison of the two pairs is then able to discriminate between the "true" solution, which is common to the two days, and the "mirror" or "ghost" solution, which is different for the two days. This method was investigated in Bommier et al. (1981) and said to be successful, but the results were not given because too surprising at that time. Twenty prominences were analysed by this way. In most cases, the two ambiguous solutions are respectively directed on each side of the filament long axis, which lies along the photospheric neutral line below the prominence. As a result, in most of the cases the two ambiguous solutions each correspond to one of the two types of prominence magnetic structure: the Kippenhahn-Schlüter type, of Normal Polarity, and the Kuperus-Raadu type, of Inverse Polarity (see for instance Gibson 2018). The prominence polarity refers to the polarity, positive or negative, of the photospheric magnetic field on each side of the neutral line, which generally lies along the filament long axis. The magnetic field is horizontal in the prominence. When it crosses the neutral line from positive to negative, the magnetic model is of the Kippenhahn-Schlüter type and the Polarity is said "Normal". When, on the contrary, the prominence magnetic field crosses the the neutral line from negative to positive, the magnetic model is of the Kuperus-Raadu type and the Polarity is said "Inverse". In the sample of twenty prominences, two were found of Normal Polarity and eighteen of Inverse Polarity. A large majority of Inverse Polarity prominences was then found.
Two other methods were later on developed, which led to the same result: large majority of Inverse Polarity prominences. Leroy et al. (1984) developed a statistical analysis of the mirror symmetry of the two ambiguous solutions in a 256 quiescent prominence sample, and derived the statistical result of 75% Inverse Polarity prominences and 25% Normal Polarity prominences. The Normal Polarity prominences were also found to be lower, sharp-edged and with a stronger magnetic field. The third method is made of the comparison of the two pairs of solution provided by an optically thin line, like He i D 3 , and an optically thick line, like Hydrogen Hα. When the prominence internal absorption and radiation cannot be ignored in the line formation model, in the optically thick case, the symmetry of the two ambiguous solution is also modified. Bommier et al. (1994) analysed such observations in fourteen prominences and found twelve of them of the Inverse Polarity type, and only two of them of the Normal Polarity type.
In the present paper we apply the first method for solving the ambiguity in a prominence observed with the THÉMIS telescope, by comparing the magnetic solutions obtained on two following days. The previous ambiguity solutions of that type were achieved on Pic-du-Midi coronagraph data, where the observed polarisation was finally averaged on all the object, for accuracy purposes. The THÉMIS observations are accurate enough to avoid such an averaging and to obtain a field vector map of the prominence, as already published in the works cited above, but ambiguous in those works. In Sect. 2 we present the prominence observation, data treatment and Hanle inversion. We also detail our method for determining and propagating the ambiguity solution inside the prominence. We discuss the results in Sect. 3 and compare them with the MHD model of quiescent prominence by Aulanier & Démoulin (2003) in the concluding Sect. 4.

Observations
An observation campaign aimed to observe prominences on several following days, in order to disambiguate the observed magnetic field vector by Hanle effect, was led by V. Bommier at the THÉMIS telescope from 11 to 17 September 2008. The prominence under study in this paper was observed on 13 September from 11:48 UT to 13:40 UT, and on 14 September from 11:19 UT to 13:34 UT, at the Position Angle 229 • . On the Hα spectroheliograms available at the BASS2000 data basis, we identified the corresponding filament in the preceding days before passing at the limb and we verified that this filament/prominence lies far from active regions, thus enabling its quiescent character. However it is not a Polar Crown prominence, being located at low latitude.
The polarimetric analysis code is the one described in Bommier & Molodij (2002). The prominence was scanned with steps of 1.5 arcsec perpendicularly to the slit, also the solar limb direction, and 11 arcsec along the slit direction. The pixel size along the slit is 0.2 arcsec. Two lines were simultaneously observed: Hydrogen Hα and He i D 3 . Figs. 1 and 2 display the intensity at the center of the line, for the two days and the two lines. These images are corrected for the scattered light.
The polarimetric accuracy of the observations was determined by the standard deviation of the intensity in a continuum region of the spectrum. It was found of about 6 × 10 −3 in each pixel, for He i D 3 . This has to be compared to the theoretically required polarimetric accuracy for interpreting the Hanle effect, which we estimate to be 2×10 −3 , given the typical He i D 3 polarisation degree of 2-3% in prominences (see Table II of Bommier et al. 1994, with lower values in Hα). Thus, it is necessary to average at least over 10 pixels to reach the desired polarisation accuracy. The He i D 3 line in the prominence matter has the shape of Fig. 3, top. We took help of a gaussian fitting to localize the line, its maximum and its half-width, on each spectrum. The continuum was first subtracted by subtracting its linear fit. The line half-width was found of about 30 spectral pixels. Thus, by summing over these 30 spectral pixels, the required polarisation accuracy is largely reached and further averaging was not necessary for this purpose. We analysed line-integrated polarisations. Fig. 3 also shows how difficult it is to separate, in the analysis, the two overlapping components of the He i D 3 line. In a first step, we did not investigate this question, and we integrate the whole profile as a single line.
Outside the prominence matter, the spectrum has the shape of Fig. 3, bottom, where it can be seen that the He i D 3 line is totally absent. This spectrum results from the incident photospheric radiation. It is thus visible that the He i D 3 line is totally absent of the incident radiation. Thus, any Doppler dimming or brightening is avoided in He i D 3 .

Hanle inversion
The inversion was performed by linear interpolation in linear polarisation diagrams, as those shown for instance in Sahal-Brechot et al. (1977). An extensive application of this inversion method can be found in Bommier et al. (1981). As explained in this publication, one single line is insufficient to determine the three components of the magnetic field vector from the two linear polarisation parameters, which are the linear polarisation degree and direction, or the two Stokes parameters ratios Q/I and U/I. As explained above, the Stokes parameters have been integrated along the whole line profile. Advantage can be taken of the horizontality of the magnetic field in prominences, as investigated by several authors and methods as stated in the Introduction. By using this hypothesis, only two ambiguous field vectors are solution of the observed integrated polarisation. These two solutions lie respectively on each side of the filament long axis. The Carrington coordinates of the filament were found on the Hα spectroheliograms available at the BASS2000 data basis. Our inversion code makes use of these coordinates to determine the true value of the scattering angle, which may not be the right-angle. The inclination angle of the solar rotation axis with respect to the sky plane, in other words the disk center latitude, is also taken into account and was b 0 = 7.22 • at the time of the observations. Finally, in order to determine the filament long axis direction, we better localized the filament channel in the EIT image at 195 Å, on 6 September 2008 at 05:36. In this image, we determined the solar azimuth 193.5 • for this direction, with respect to the E-W oriented parallel direction.
From the polarimetric accuracy, our code is also able to derive the accuracy in field strength and direction determina-    tion. We thus obtained a mean inaccuracy of 0.3 G on the field strength and 1 • on the field azimuth, due to the polarimetric inaccuracy. We estimate as 2 • the inaccuracy on the determination of the longitude and latitude of the filament on the solar surface, and we derived the related inaccuracy of 0.25 G on the field strength and 0.7 • on the field azimuth. Finally, we estimate as 1 arcsec the inaccuracy on the solar limb position in our images. This induces an inaccuracy of 0.15 G on the field strength and an inaccuracy of 0.3 • on the field azimuth. Accordingly, we estimate the global inaccuracy as 0.7 G on the field strength and 2 • on the field azimuth.

Ambiguity resolution
The ambiguity was first solved by comparing the two pairs of solutions on the two following days, in a large averaged portion of the prominence, the "big pixel". Then, the solution was propagated from pixel to neighboring pixel, from the prominence center to the prominence boundary.
Initially, our two images contained 17290 pixels. For 2546 of them, no or very faint He i D 3 line was visible as shown in the bottom of Fig. 3. For 4615 pixels, the Hanle inversion provided no solution. The magnetic field may not be horizontal, or this is an effect of the above listed inaccuracies. For about 500 pixels only, the two ambiguous fields vectors surprisingly lied on the same side of the line-of-sight. We discarded this few pixels from our analysis.
We first averaged the Stokes parameters, integrated along the line profile, in 3903 (13 September) and 3187 (14 September) pixels representing the prominence body. We the performed the Hanle inversion on these averaged Stokes parameters. The azimuths of the two solutions were found as 131.55 • and 207.28 • on the 13 September, and 144.45 • and 215.21 • on the 14 September, leading respectively to differences of 12.91 • and 7.92 • . The difference between the differences is larger than our angular inaccuracy discussed above. The "true" solution is therefore the second one.
We then propagate this solution across the prominence. The eliminated pixels because of absence of line or of solutions, made the propagation sometimes uneasy. Finally, we solved the ambiguity in 4893 pixels of each image. The results are presented in Fig. 4, where the black zones correspond to eliminated pixels.

Results and discussion
In the following, we present our results as averaged at each height in the prominence. Only the prominence body, defined as the most intense Hα region of Fig. 1, bottom, left part, was retained for these averages. Figure 5 presents the variation with height of the α angle, which is the angle between the horizontal field vector and the filament long axis. As the filament long axis lies along the photospheric magnetic neutral lines, which separates two regions of opposite polarity, the sign of the α angle was assessed following the prominence field polarity with respect to these neighboring photospheric polarities. A negative α, as we obtain, means that the prominence is of the Inverse Polarity type.  Leroy et al. (1984) for 75% of the prominences, found to be of the Inverse Polarity type. In another paper devoted to Polar Crown prominences, where this angle can be determined without ambiguity solution when the prominences are seen edge-on, Leroy et al. (1983) obtain the same average value of α = −25 • . With the spatial resolution in the THÉMIS prominence, we obtain that this angle decreases with increasing height, in absolute value, at the rate of −3 × 10 −4 degrees/km. The lower part of the plot displays a quicker decrease of −4.5 × 10 −3 degrees/km. Such very low distances with respect to the solar limb were not accessible to the previous measurements, which were made with coronagraphs. Figure 6 displays the field strength variation. We obtain the 6 G typical field strength for prominences. In the upper part, we obtain a gradient of field strength of 0.25 × 10 −4 G/km, in good agreement with the value of 0.5 × 10 −4 G/km obtained by Leroy et al. (1983).
On the contrary, the lower part of the plot displays a decreasing of the field strengths, in those low altitudes, which were inaccessible to the coronagraphs. The decreasing rate we find is −0.5 × 10 −4 G/km. The limit between the two behaviors is 34 Mm, higher for the field strengths than for the field azimuth, where this limit is 20 Mm.

Conclusion
In the upper part (higher than 20 Mm) of this prominence observed with THÉMIS, we obtain an average field strength of 6.4 G and an average angle of −21 • between the field (assumed to be horizontal) and the filament long axis, in the Inverse Polarity scheme. This is in excellent agreement with the previous measurements listed in the Introduction. The new result concerns the decreasing of the absolute value of this angle with increas-ing height. We obtain a gradient of −3 × 10 −4 degrees/km for the absolute value of this angle, in the Inverse Polarity scheme for the prominence magnetic field (see Fig. 5). For the field strength, we obtain a positive gradient of 0.25 × 10 −4 G/km, in very good agreement with the previous determination by Leroy et al. (1983) (see Fig. 6).
This new result about the gradient of the horizontal field azimuth with respect to the filament long axis, is in excellent agreement with the MHD model of a quiescent filament by Aulanier & Démoulin (2003, see their Fig. 2). They also obtain a decreasing of the absolute value of this angle, with increasing height, in the upper part of the prominence.
With our THÉMIS observations, we are also able to determine the field a low altitudes in the prominence, inaccessible to the previous coronagraphic observations.
For the field azimuth with respect to the filament long axis, we get the quicker decreasing with height of the absolute value of this angle, of −4.5 × 10 −3 degrees/km. Fig. 2 of Aulanier & Démoulin (2003) displays more scattered values of this angle at lower altitudes, but the general trend is also a quicker decreasing with height of the absolute value of this angle, in agreement with our observations.
As for the field strength, Aulanier & Démoulin (2003) display increasing field strength with height also at low altitudes, even quicker than at higher altitudes. On the contrary we get an average decreasing of −0.54 × 10 −4 G/km in the lower regions of the prominence. In their paper, Aulanier & Démoulin (2003) assess that the highest field strengths values always lie at low altitude, as visible in their Fig. 2. Their obtained average gradient of the field strength with altitude is however positive. They point out that at low altitudes, the magnetic field is made of a mixing of independent structures, which may be complex, with not necessarily high He i D 3 emission. In our data, a larger number of pixels were eliminated in the low altitude region, due to failure in the inversion or to absence of He i D 3 emission. Our data are thus dominated by more intense He i D 3 emission, which may have biased the field strength we observe, with respect to the general field strength in this region. In addition, it is difficult to precisely locate the solar limb, and thus to obtain precise results at the very low altitudes.