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Sep 15, 2006 - J. T. SU,1 H. Q. ZHANG,1 Y. Y. DENG,1 X. J. MAO,2 Y. GAO,1 AND G. H. LIN1. Received 2006 April 26; accepted 2006 August 18; published ...
The Astrophysical Journal, 649: L141–L144, 2006 October 1 䉷 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.

THE INFLUENCE OF FARADAY ROTATION ON THE VERTICAL ELECTRIC CURRENT DENSITY J. T. Su,1 H. Q. Zhang,1 Y. Y. Deng,1 X. J. Mao,2 Y. Gao,1 and G. H. Lin1 Received 2006 April 26; accepted 2006 August 18; published 2006 September 15

ABSTRACT In this Letter we analyze the effects of Faraday rotation on the vertical electric current density in solar active regions. ˚ line at On the basis of the numerical solutions of the transfer equations of Stokes parameters for the Fe i 5324.19 A the solar disk center, we derive the variations of the vertical current with wavelengths, which are caused by Faraday rotation. The simulations show that Faraday rotation changes both the sign and the magnitude of the vertical current at line center, but it changes only its magnitude at the far wings of the line. At the selected locations on a simple sunspot, the observed variations of the vertical current with wavelengths are compared with the model. We find that ˚ line: the sign the azimuth rotation is significant when observations are taken near the center of the Fe i 5324.19 A change of the vertical current occurs in about 13.5% of the total selected locations, and in about 26% of those the magnitude increases the vertical current more than 100% of the “real” value. We also make a preliminary attempt to correct a case of the vector magnetic field observations for Faraday rotation. For NOAA AR 10162, in particular, it appears that Faraday rotation has a significant effect on the vertical current. Subject heading: Sun: magnetic fields among them (Wang et al. 1992; Bao et al. 2000; Zhang et al. 2003). However, Hagyard & Pevtsov (1999) pointed out the presence of significant effects of Faraday rotation in HSOS vector magnetograms. The solar hemispheric asymmetry in the helicity sign seems to be more evident in the HSOS data than in Mees Solar Observatory data (Pevtsov et al. 1995; Longcope et al. 1998; Bao & Zhang 1998). Yet the vertical electric current density, Jz ∼ (⵱ ⴛ B)z calculated from the data taken at line center directly affects the values for the helicity proxy a since a p m 0 Jz /Bz. This Letter adopts and extends the analysis of Hagyard et al. (2000), aiming to show the influence of Faraday rotation on the vertical current density, especially at or near ˚ line. Analyzing the variations the center of the Fe i 5324.19 A of Jz with wavelength, we present evidence that Faraday rotation not only affects the magnitude of Jz but can also affect its sign if observations are taken near the center of the line.

1. INTRODUCTION

It is known that measurements of the orientation of linearly polarized light yield the azimuth of the transverse magnetic field via the inverse Zeeman effect. However, the linearly polarized light can be modulated by Faraday rotation, which is a result of the anomalous dispersion of light in a magnetized medium near an absorption wavelength. When the Faraday effect produces the rotation of the linear polarization plane, it will further produce the rotation of the azimuth of the transverse magnetic field. The azimuthal angle measured near the line center will be relatively more seriously affected than that measured in the far wings of the line. On the other hand, the linear polarized light at the line center is perpendicular to the direction of the linearly polarized light in the wings. So when we measure the azimuth of the transverse field in the line wings, it will rotate abruptly by 90⬚ from its orientation measured near the line center. We call this a p-j rotation effect. Since the two effects could not be clearly separated in actual observations, Hagyard et al. (2000) designated the net effect as ZeemanFaraday (Z-F) rotation. Using numerical simulations, Su & Zhang (2004) found that the interval between two symmetric wings of a spectral line (where is the p-j effect) may be diminished, eliminated completely, or increased when the Stokes profiles Q and U are convolved with the filter transmission function. In the case of the interval increasing, the azimuth suffers from Z-F rotation more seriously. Moreover, a line with a larger Lande` g factor has a better chance to be confronted with the increased interval than the line with a smaller Lande` g factor. Generally speaking, the azimuth error produced by the Z-F rotation effect is less signif˚ icant in observations near the center of the Fe i 5324.19 A (g p 1.5) line, which is the working line employed by the Huairou Solar Observatory Station (HSOS), than that near the center ˚ (g p 3) line. Several studies have made of Fe i 5250.22 A comparisons of HSOS magnetograms (near the line center) with those from other observatories and showed qualitative agreement

2. NUMERICAL SIMULATIONS

The difference equation for the vertical electric current density Jz is written as

Jz p

{

1 1 [B (x ⫹ 䉭x) ⫺ By (x ⫺ 䉭x)] m0 y 2䉭x

⫺ [Bx (y ⫹ 䉭y) ⫺ Bx (y ⫺ 䉭y)]

}

1 , 2䉭y

(1)

where 䉭x and 䉭y are the intervals of two adjacent points and are taken as 0⬙. 35 (the scale of each pixel of CCD camera) in the present work. To calculate the value of Jz at a point, equation (1) indicates that we need to know all the magnetic field parameters (B: field strength; f: inclination; x: azimuth) of the four spatial points around that particular point. The simulations are carried out with the following parameters: four points with the same B and x, but different f. Once the values are assigned for the above four sets of parameters, the value of Jz(real) (as the “real” value) at a point can be obtained using equation (1). Moreover, to study the impact of Faraday rotation on Jz, we

1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China. 2 Astronomy Department, Beijing Normal University, Beijing 100875, China.

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Fig. 1.—Different types of curves correspond to different field strength. Left: Jz as a function of wavelength. At the bottom, the values of the first column are field strengths and those of the second are the real Jz values. Right: dx(l) as a function of wavelength.

˚ express it as a function of wavelengths at the Fe i 5324.19 A line in the center of the solar disk. Taking the given values of magnetic field parameters above as the initial conditions, we numerically solve the Unno-Rachkovsky equations (including anomalous dispersion) of the polarized radiative transfer of the line in the solar atmospheric magnetic field to obtain Stokes Q, U, and V profiles. Thus, these are the profiles modified by the Faraday rotation. A penumbra model atmosphere (Ding & Fang 1989) is also adopted. With the Stokes Q and U profiles, we can obtain the variations of the field azimuth (or Bx and By) with wavelengths. Now placing the four sets of such data into equation (1), we obtain the variations of Jz with wavelengths at this particular point, which include the effect of Faraday rotation. The field inclinations of fa p 15⬚, fb p 15⬚, fc p 16⬚, and fd p 16⬚ are used for the four points A, B, C, and D around the position of o(x, y) aligned in anticlockwise direction (see the plot at the top right of Fig. 1). The field azimuths of the four points are the same, x a, b, c, d p 30⬚. Their magnetic field strength is also the same. To study the variations of Jz with magnetic field strength, we assign the values of Ba, b, c, d ranging from 1000 to 3000 G. For the sake of the following discussions, we give the definition of the average azimuth error of four points as

冘 4

dx(l) p

[x i (l) ⫺ x i(real) ]/4

(2)

ip1

and the error ratio of Jz as RJz(l) p 100%F [Jz (l) ⫺ Jz(real)] /Jz(real)F,

(3)

where x i(real) is the field azimuth given or fitted (see § 3). The

Fig. 2.—Filtergram (left) and longitudinal magnetogram (right) of the sun˚ of Fe i 5324.19 A ˚ line. The map of the spot taken at the offset ⫺0.075 A filtergram has been divided by radial lines and concentric circles used in analysis. The size of the maps is 112 # 84. The white rectangle in the right image marks the field of view where the data are corrected for Faraday rotation in Fig. 6.

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results of the simulations can be seen in the left panel of Figure 1, where different types of curves correspond to different field strength, and the Jz(real) values for different field strength are marked in the same plot. The right panel shows dx(l) curves at different field strength. It is clear from Figure 1 that the Jz value increases from the line wings to its center. For the case of B p 3000 G, the ratio of RJz strongly varies from 55% ˚ ) to 260% (at the line center). At Ⳳ0.10 A ˚ , when (Ⳳ0.15 A B 1 1000 G, Jz changes the sign from minus to plus. The average azimuth error of dx(l) at the line center varies quickly with the increase of the field strength from 10⬚ to 38⬚. These suggest that Faraday rotation could affect both the magnitude and sign of Jz at line center, but it only affects its magnitude at the far wings of the line. 3. COMPARISON OF THE OBSERVED Jz PROFILE WITH THE MODEL

3.1. Observations and Data Reduction Observational data were obtained on 2002 October 24 by the HSOS vector magnetograph for a relatively simple sunspot in AR 10162. It was a positive polarity spot located at N26⬚, E04⬚ at 03:00 UT. The filtergram and magnetogram taken at ˚ in the blue wing of the Fe i 5324.19 A ˚ line are shown 0.075 A in Figure 2. In observations, 31 vector magnetograms were obtained in about 2 hr. The first vector magnetogram was taken ˚ at the red of the line with the spectral filter tuned to 150 mA center. Subsequent magnetograms were taken at an increment ˚ toward the line center, ending at 150 mA ˚ in the blue of 10 mA wing. To avoid the effect of the 180⬚ ambiguity in the azimuth, we compared azimuths at each successive filter position with those of the previous filter setting, starting with the azimuth at ˚ in the red wing, and resolved the ambiguity by taking 150 mA the direction yielding the smallest difference between the azimuths at the two wavelengths (Hagyard et al. 2000). The detailed processes of data reduction have been described by Su & Zhang (2004). The map of the filtergram in Figure 2 has been divided by 12 radial lines coming out from the center of the umbra at an interval of 30⬚, and seven concentric circles in an equal step along the radii (Hagyard et al. 2000). We select 84 pixels distributed on the cross points of the radial lines and the circles. For these selected 84 pixels, around each we take four additional neighboring pixels of A, B, C, and D aligned in anticlockwise direction. The distance between A and C or B and D is 0⬙. 7 (the scale of 2 pixels). The least-squares fitting technique (Balasubramaniam & West 1991) has been used to get the best fit of the observational Stokes Q, U, and V profiles to

Fig. 3.—Left: Observed Jz profile (plus signs) in the sunspot compared with the model (solid line). Right: Same as the left, but for the profile of the average azimuth error. The fitted values for both Jz and average azimuth rotation are shown in the plot by dashed lines.

No. 2, 2006

VERTICAL CURRENT PROFILE ANALYSIS

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TABLE 1 Parameters for 4 Pixels Obtained by Fitting Pixel A B C D

...... ...... ...... ......

a

l0 ˚) (A

h0

lD ˚) (A

uB1

B (G)

f (deg)

x (deg)

0.25 0.26 0.37 0.30

5324.188 5324.188 5324.188 5324.188

6.91 6.96 6.57 6.02

41.4 40.4 38.5 42.0

⫺1.39 ⫺1.07 ⫺1.19 ⫺1.36

1847 1846 1886 1878

56.8 57.7 56.8 55.7

75.6 77.0 78.0 76.0

the analytical profiles (including anomalous dispersion; Landolfi & Landi Degl’Innocenti 1982) for 420 pixels. Then the eight parameters are obtained for each pixel. They are the line center (l 0), the Doppler width (l D), the damping constant (a), the slope of the source function (uB1), the opacity ratio (h0), the total magnetic field strength (B), the inclination (f), and azimuth (x) of the magnetic field vector. The data of 10 pixels (inner umbra) were omitted due to a low signal-to-noise ratio there (Hagyard et al. 2000). 3.2. Results With the observed Stokes Q and U profiles as well as the fitted magnetic field parameters, we can obtain the variations of the transversal components of magnetic fields with wavelengths, Bx(obs) (l) and By(obs) (l). Similarly, those of the models Bx(model) (l) and By(model) (l) can also be obtained. Using the method mentioned in § 2, we can plot two Jz curves at each cross point. One curve is the observed, while the other is the model. Figure 3 shows a sample of the observed (plus sign) in comparison with the model (solid line). The fitted value of Jz marked in the left panel is calculated from the fitted magnetic field parameters, which are shown in Table 1. In Figure 3 it is evident that at the line center, the average azimuth error of dx(l) is more than 20⬚, and yet the value of Jz is up to more than twice of the fitted value (taken as the real value). For this case, the effect of Faraday rotation tends to increase the magnitude of the vertical current density, but in some other cases, we find that this magnitude is decreased because of the rotation. The observed Jz profile is not so symmetric, which may be for ˚ line two reasons. First, the two wings of the Fe i 5324.19 A are asymmetric when observed in the umbra (Wallace et al. 2002). Second, the cross-talk effect may not be eliminated completely. We discuss this further in the next section. Using equation (3), we calculate the error ratio of RJz near the line center at each selected pixel. Instead of the observed values of Jz, we employ the model values to calculate all the error ratios (74 pixels). The correlations between the error ratios and the average azimuth errors, and between the error ratios and longitudinal magnetic fields are shown in the left two panels of Figure 4. The general trends can be deduced from the correlations: the error ratio increases as both the azimuth rotation and the longitudinal field strength increase. To compare the results at the line center mentioned above with those at the line wings, we make similar plots at the wavelength offset of

Fig. 4.—Left: Correlations between the error ratio RJz at the line center and the average azimuth error (top), and between RJz and the longitudinal magnetic fields (bottom). Right: Same as the left, but for the line wing.

˚ shown in the right two panels of Figure 4. The trends ⫺0.12 A seem to be similar with those taken near the line center. However, with only two pixels, their error ratios are more than 100%. There is no pixel changing its electric current sign at this wavelength. Table 2 shows the parameter statistics of the fitted Jz s of 74 pixels, observed at both the line center and the ˚ . The numbers of error ratios of RJz larger line wing of ⫺0.12 A than 100%, observed at the line center and at the line wing, are 19 and 2 pixels, respectively. The number of electric currents changing sign at the line center are 10 pixels, eight of which change sign from plus to minus while the rest do a reverse change. The fitted average values of plus and minus Jz values are nearly equal to those observed at the line wing ˚ . However, it is quite clear that both of them are of ⫺0.12 A weaker than those observed at the line center. 4. DISCUSSION AND CONCLUSIONS

We have derived the variations of the vertical electric current density Jz with wavelengths from both numerical simulations and observations over the viewing field of a simple sunspot. The calculations based on the model are compared with the observed profiles of Jz. The results reveal that the error ratio of Jz is closely associated with the azimuth rotation. At the line center the electric current is greatly affected: its sign can be changed and its magnitude increased up to several times more than its real value. Certainly, its magnitude can be decreased as well because of the same effect. Generally speaking, Jz at the far line wings is not so seriously affected. In the observations at the line center, there were 10 pixels (13.5% of 74 pixels) with the electric current sign changing for this simple sunspot, and 19 pixels (26% of 74 pixels) with the electric current magnitude growing up to more than 100% of the fitted values, which have been taken as the real values (see Table 2). To avoid

TABLE 2 Statistics of 74 Selected Pixels on Jz Obs.

RJz 1 100% (pixels)

Jz p FJzF (pixels)

Jz p ⫺FJzF (pixels)

J¯ z,Jz p FJzF (Am⫺2)

J¯ z,Jz p ⫺FJzF (Am⫺2)

Fitted . . . . . . . . . ˚ ........ 0.00 A ˚ ...... ⫺0.12 A

0 19 2

37 33 37

37 41 37

0.0079 0.0107 0.0080

⫺0.0100 ⫺0.0147 ⫺0.0103

Note.—Jz p FJzF and Jz p ⫺FJzF indicate the use of positive and negative values, respectively, and J¯ z denotes the averaged value.

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Fig. 5.—Numerical simulations of 0%–5% Stokes V signal added onto ˚ line. Left: Jz profiles. Right: Average Stokes Q and U signals for Fe i 5324.19 A azimuth error profiles.

the problem caused by Faraday rotation, observations should be ˚ . However, another problem taken in the far wings of Fe i 5324.19 A of the circular-to-linear polarization cross-talk effect is raised due to the polarimeter. The numerical simulations for Jz and azimuth rotation are shown in Figure 5, in which we have added 0%–5% of Stokes V signal onto the signals of Stokes Q and U. Magnetic field parameters are marked in the left panel. We can see that at the far wings, the Jz profiles were seriously disturbed. The redwing data show larger deviation from the real values than those of the blue wing. It is because the azimuths in the red wings deviate more from the real values than those in the blue wings. Comparing the left panel of Figure 3 with that of Figure 5, we think that even after the linear corrections, the cross-talk effect still exists in the reduced data. It is not easy to correct a vector magnetogram for Faraday rotation since the field azimuth rotation depends on not only the total magnetic field strength but also the field inclination. Moreover, it is even related to other fitted parameters, such as the opacity ratio (h0). On the basis of the study of the correlations between the azimuth rotation dx and the longitudinal component of magnetic fields Bz (Su & Zhang 2004), we try to provide an average

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Fig. 6.—Uncorrected and corrected Jz derived from observations at the line center in (a) and (b), respectively. Contours correspond to 4, 6, 8, 12, 15, 30, 50, and 80 in units of mA m⫺2. Solid contours indicate a positive sign and dashed countours a negative sign for the vertical electric current density.

correcting formula for the magnetogram taken at the line center, dx(l) p 0.412 ⫹ 0.015FBzF. If FBzF ! 200 G, we do not correct the azimuths for the actual magnetogram. We employ it to the part of AR 10162 shown in the white box in Figure 2. Figures 6a and 6b show the uncorrected and corrected Jz images, respectively. Figure 6a shows a stronger positive Jz intensity than that of Figure 6b. The Jz-averaged values before and after correction are 5.7 and ⫺14.0 mA m⫺2, respectively. The corrected value is closer to the value of ⫺30.0 mA m⫺2 observed at the line wing ˚ . This indicates that the correcting formula seems to of ⫺0.12 A work at least for this particular active region. The authors very much thank the anonymous referee for helpful suggestions and comments on the manuscript, which were extremely significant to improve this work. This work is supported by the NSFC projects (10233050, 10228307, 10311120115, 10473016, 10273002, and 2006CB806300) and the BRPC project (TG2000078401).

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