1 INTRODUCTION
Magnetic fields dominate most solar activities such as filaments eruption, flares, and coronal mass ejections. All these phenomena are energetic events due to explosive release of magnetic energy (Krall et al. Reference Krall, Smith, Hagyard, West and Cummings1982; Wang, Zhang & Xu Reference Wang, Ewell, Zirin and Ai1994; Shibata et al. Reference Shibata, Masuda, Shimojo, Hara, Yokoyama, Tsuneta, Kosugi and Ogawara1995; Tsuneta Reference Tsuneta1996; Bond et al. Reference Bond, Mullan, O’Brien and Sion2001; Priest & Forbes Reference Priest and Forbes2002; Lin et al. Reference Lin, Ko, Sui, Raymond, Stenborg, Jiang, Zhao and Mancuso2005; Nindos, Patsourakos & Wiegelmann Reference Nindos, Patsourakos and Wiegelmann2012). Thus, the solar magnetic field is the key for understanding the nature of solar activities. Since measurable magnetic sensitive lines are around the photosphere, reliable measurements of magnetic fields are nearly there (Stenflo Reference Stenflo1973; Harvey Reference Harvey and MULLER1977). However, the understanding of the magnetic field in the chromosphere and corona remains difficult due to both intrinsic physical difficulties and observational limitations (Gary & Hagyard Reference Gary and Hagyard1990; Gary & Hurford Reference Gary and Hurford1994; Liu, Kuhn & Coulter Reference Liu, Kuhn and Coulter2004). Generally, magnetic fields in the corona are regarded as force-free (Aly Reference Aly1989), because the plasma β (ratio of plasma pressure to magnetic pressure) is much less than unity. However, it is controversial in the photosphere, because two kinds of pressures are comparable (Demoulin et al. Reference Demoulin, Bagala, Mandrini, Henoux and Rovira1997). For the low-β corona where the plasma is tenuous (β≪1), the magnetic field satisfies the following force-free equations:
\begin{equation}
\nabla \times {\bm B} = \alpha ({\bm r}) {\bm B}, \vspace*{-20pt}
\end{equation}
\begin{equation}
\nabla \cdot {\bm B} = 0,
\end{equation}
implying that there is no Lorentz force in action and α is constant along magnetic field lines ( B ·∇α = 0).
At present, magnetic field extrapolation with a force-free assumption is a major method for studying the solar magnetic fields of active regions. Coronal fields can be reconstructed from a physical model (namely, the force-free model) in which the observed photospheric magnetic field is taken as a boundary condition (Sakurai Reference Sakurai1981; Wu et al. Reference Wu, Sun, Chang, Hagyard and Gary1990; Mikic & McClymont Reference Mikic, McClymont, Balasubramaniam and Simon1994; Amari et al. Reference Amari, Aly, Luciani, Boulmezaoud and Mikic1997; Wheatland, Sturrock & Roumeliotis Reference Wheatland, Sturrock and Roumeliotis2000; Yan & Sakurai Reference Yan and Sakurai2000; Wiegelmann Reference Wiegelmann2004; Song et al. Reference Song, Fang, Tang, Wu and Zhang2006; He & Wang Reference He and Wang2008; Liu, Zhang & Su Reference Liu, Zhang and Su2011a). This means that coronal magnetic fields are considered to be force-free, while at the boundary these are connected to the photospheric magnetic fields observed. The force-free extent of the photospheric magnetic field then becomes an important subject to study. Wiegelmann, Inhester, & Sakurai (Reference Wiegelmann, Inhester and Sakurai2006) proposed a preprocessing procedure to make a minor regulation within the allowable errors, so that the observed magnetic fields tend to a force-free field, which further indicates that the study of the force-free extent is significant and necessary for the field extrapolation. Metcalf et al. (Reference Metcalf, Jiao, McClymont, Canfield and Uitenbroek1995) calculated the dependence of the net Lorentz force in the photosphere and low chromosphere on the height using Mees Solar Observatory magnetograms and concluded that magnetic fields are not force-free in the photosphere, but become force-free roughly 400 km above the photosphere. Moon et al. (Reference Moon, Choe, Yun, Park and Mickey2002), studying the force-free extent in the photosphere using 12 vector magnetograms of three active regions, realised that the photospheric magnetic fields are not very far from the force-free case. Liu et al. (Reference Liu, Zhang, Su and Song2011b) tentatively applied the force-free extrapolation to reconstruct the magnetic fields above the quiet region and checked the force-free extent of this quiet region based on the high spatial resolution vector magnetograms observed by the Solar Optical Telescope/Spectro-Polarimeter on board Hinode. Tiwari (Reference Tiwari2012) found that sunspot magnetic fields are not so far from the force-free case. We conduct statistical research to make use of the vector magnetograms observed by the SMFT at HSOS from 1988 to 2001 in order to verify the force-freeness of the photospheric magnetic field.
This paper is organised as follows. The description of observations and data reduction is given in Section 2. The results are shown in Section 3. Finally, in Section 4, we present short discussions and conclusions.
2 OBSERVATIONS AND DATA REDUCTION
The observational data used here are 925 vector magnetograms corresponding to 925 active regions observed from 1988 to 2001 by the Solar Magnetic Field Telescope (SMFT) installed at Huairou Solar Observing Station (HSOS), located at the north shore of the Huairou reservoir. Magnetograms associated with the corresponding active region, which is nearest to the disk center, are chosen and calculated. Therefore, only one magnetogram is available for one active region. The SMFT consists of a 35-cm refractor with a vacuum tube, a birefringent filter, and a CCD camera, including an image processing system operated by a computer (Ai & Hu Reference Ai and Hu1986). The birefringent filter is tunable, working either at the photosphere line Fe i λ5324.19 Å, with a 0.150-Å bandpass, or at the chromosphere line, Hβ, with a 0.125-Å bandpass. The line of Fe i λ5324.19 Å (Lande factor g = 1.5), formed around the solar photosphere, is used for photospheric magnetic field observations. The bandpass of the birefringent filter is about 0.15 Å for the Fe i λ5324.19 Å line. The center wavelength of the filter can normally be shifted −0.075 Å relative to the center of Fe i λ5324.19 Å to measure the longitudinal magnetic field and then the line center is applied to measure the transverse one (Ai & Hu Reference Ai and Hu1986). Vector magnetograms are reconstructed from four narrow-band images of Stokes parameters (I, Q, U, and V). Note that V is the difference of the left and right circularly polarised images, Q and U are the differences between two orthogonal linearly polarised images for different azimuthal directions, I is the intensity derived from the sum either of two circularly polarised images in the line-of-sight field measurements or of two linearly polarised images in the transverse field measurement. When I, Q, U, and V are measured, the corresponding white light images are simultaneously obtained, which are employed to compensate for the time differences during the measurements of I, Q, U, and V. The sequence of obtaining Stokes images is as follows. First, acquire the V/I image, next the Q/I images, and then the U/I image. The time required to obtain a set of Stokes images was about 45 s. Each image is associated with 256 integrated frames. To reconstruct the vector magnetograms, a linear relation is necessary between the magnetic field and the Stokes parameters I, Q, U, and V, which is true under the weak-field approximation (Jefferies, Lites & Skumanich Reference Jefferies, Lites and Skumanich1989; Jefferies & Mickey Reference Jefferies and Mickey1991):
\begin{equation}
B_{\rm L}=C_{\rm L}V ,\vspace*{-10pt}
\end{equation}
\begin{equation}
B_{\rm T}=C_{\rm T}(Q^{2}+U^{2})^{1/4},\vspace*{-10pt}
\end{equation}
\begin{equation}
\theta ={\rm arctan}\bigg(\frac{B_{\rm L}}{B_{\bot }}\bigg), \vspace*{-10pt}
\end{equation}
\begin{equation}
\phi =\frac{1}{2}{\rm arctan}\bigg(\frac{U}{Q}\bigg),
\end{equation}
where B L and B T are the line-of-sight and transverse components of the photospheric field, respectively. Here, θ is the inclination between the vector magnetic field and the direction normal to the solar surface and φ is the field azimuth. Furthermore, C L and C T are the calibration coefficients for the longitudinal and transverse magnetic fields, respectively. Both theoretical and empirical methods are used to calibrate vector magnetograms (Wang, Ai, & Deng Reference Wang, Ai and Deng1996; Ai, Li & Zhang Reference Ai, Li and Zhang1982), so that two sets of calibration coefficients are available: the first set C L and C T are 8 381 and 6 790 G (Su & Zhang Reference Su and Zhang2004; Wang et al. Reference Wang, Ai and Deng1996), respectively, obtained by theoretical calibration; in the second set C L and C T are 10 000 and 9 730 G, respectively, which are deduced through empirical calibrations (Wang et al. Reference Wang, Shi, Wang and Lu1996). Faraday rotation and magneto-optical effects may affect the value of measured magnetic fields. Bao et al. (Reference Bao, Pevtsov, Wang and Zhang2000) analyzed the Faraday rotation in the HSOS magnetograms, which contributes about 12°. Gao et al. (Reference Gao, Su, Xu and Zhang2008) proposed a way of the statistical removal of Faraday rotation in vector magnetograms from HSOS. Zhang (Reference Zhang2000) found that the statistical mean azimuth error of the transverse field is 12.8° caused by magneto-optical effects. In addition, there may be other causes for the data uncertainty, for example saturation effects, filling factors, and stray light. After routine data processing of HSOS data, the spatial resolution of observational data is actually 2× 2 (arcsec pixel−1)2 and 3σ noise levels of vector magnetograms are 20 and 150 G for longitudinal and transverse components, respectively.
The acute angle method is employed to resolve the 180° ambiguity (Wang, Zhang, & Ai Reference Wang, Zhang and Xu1994; Wang Reference Wang1997; Wang, Yan, & Sakurai Reference Wang, Yan and Sakurai2001; Metcalf et al. Reference Metcalf2006), in which the observed field is compared with the extrapolated potential field in the photosphere. The orientation of the observed transverse component is chosen by requiring −90° ≤ Δθ ≤ 90°, where Δθ = θo−θe is the angle difference between the observed and extrapolated transverse components. To minimise the projection effects, the requirement that the horizontal width of an active region is less than 20° is added for each magnetogram.
3 RESULTS
In classical electromagnetic theory, the Lorentz force can be written as the divergence of the Maxwell stress:
\begin{equation}
{\bm F} = \frac{({\bm B}\cdot {\bm \nabla} ){\bm B}}{4\pi }-\frac{{\bm \nabla} ({\bm B}\cdot {\bm B})}{8\pi }.
\end{equation}
Assuming that the magnetic field above the plane z = 0 (namely, the photosphere) falls off fast enough as going to infinity, the net Lorentz force in the infinite half-space z> 0 is just the Maxwell stress integrated over the plane z = 0 (Chandrasekhar Reference Chandrasekhar1961; Molodensky Reference Molodensky1974; Aly Reference Aly1984; Low Reference Low and Hagyard1985). Then, the components of the net Lorentz force at the plane z = 0 can be expressed by the surface integrals as follows:
\begin{equation}
F_{x} = -\frac{1}{4\pi }\int B_{x}B_{z}{\rm d}x{\rm d}y,
\end{equation}
\begin{equation}
F_{y} = -\frac{1}{4\pi }\int B_{y}B_{z}{\rm d}x{\rm d}y,
\end{equation}
\begin{equation}
F_{z} = -\frac{1}{8\pi }\int \big(B_{z}^{2}-B_{x}^{2}-B_{y}^{2}\big){\rm d}x{\rm d}y.
\end{equation}
The necessary conditions of a force-free field are that all three components are much less than Fp (Low Reference Low and Hagyard1985), where Fp is a characteristic magnitude of the total Lorentz force and can be written as
\begin{equation}
F_{p} = \frac{1}{8\pi }\int \big(B_{z}^{2}+B_{x}^{2}+B_{y}^{2}\big){\rm d}x{\rm d}y.
\end{equation}
The values of Fx /Fp , Fy /Fp , and Fz /Fp then provide a measure of the force-free extent at the boundary plane z = 0 (the photosphere).
Following Metcalf et al. (Reference Metcalf, Jiao, McClymont, Canfield and Uitenbroek1995) and Moon et al. (Reference Moon, Choe, Yun, Park and Mickey2002), Fx /Fp , Fy /Fp , and Fz /Fp are utilised to check the force-free extent of the selected photospheric magnetograms. The necessary conditions of the force-free field are also expressed as |Fx |/Fp ≪1, |Fy |/Fp ≪1, and |Fz |/Fp ≪1 (Metcalf et al. Reference Metcalf, Jiao, McClymont, Canfield and Uitenbroek1995; Moon et al. Reference Moon, Choe, Yun, Park and Mickey2002), that is if three parameters Fx /Fp , Fy /Fp , and Fz /Fp are so small that they are negligible, then the magnetic field can be regarded as force-free completely. Metcalf et al. (Reference Metcalf, Jiao, McClymont, Canfield and Uitenbroek1995) suggested that the magnetic field is force-free if Fz /Fp is less than or equal to 0.1. The calibration coefficients C L and C T may affect the three parameters of Fx /Fp , Fy /Fp , and Fz /Fp . Thus, two sets of calibration coefficients mentioned above are applied to this study. In Case I, C L and C T are chosen as 8 381 and 6 790 G (Wang et al. Reference Wang, Ai and Deng1996; Su & Zhang Reference Su and Zhang2004), respectively. In Case II, C L and C T are chosen as 10 000 and 9 730 G (Wang et al. Reference Wang, Shi, Wang and Lu1996), respectively.
Figure 1 shows the possibility density function (PDF) and scatter diagrams of Fx /Fp , Fy /Fp , and Fz /Fp for the selected magnetograms (Case I). The mean values of absolute Fx /Fp , Fy /Fp , and Fz /Fp for all selected 925 magnetograms are 0.077, 0.109, and 0.302, respectively. The amplitudes of Fx /Fp and Fy /Fp are evidently smaller than that of Fz /Fp , which is the same as in the previous study (Metcalf et al. Reference Metcalf, Jiao, McClymont, Canfield and Uitenbroek1995). Therefore, Fz /Fp can work as a criterion indicating a force-free or non-force-free field more evidently. From the distribution of PDF and scatter diagrams, most of the magnetograms have the amplitudes of Fz /Fp distributed outside the zone consisting the width of ±0.1, which means that most of the photospheric magnetic fields deviate from a force-free field. There are about 17% of the magnetograms with values of Fz /Fp less than 0.1 for Case I (38% of the magnetograms with values of Fz /Fp less than 0.2). To see the relation between Fx /Fp , Fy /Fp , and Fz /Fp and the magnetic field strength, Figure 2 shows Fx /Fp , Fy /Fp , and Fz /Fp versus the magnetic components of Bx , By , and Bz for the selected magnetograms (Case I), where Bx , By , and Bz are the average of the absolute values of all pixels for each magnetogram. It can be seen only at the bottom right of Figure 2 that the amplitudes of Fz /Fp decrease roughly as Bz increases, and no evident correlation exists between the parameters (Fx /Fp , Fy /Fp , or Fz /Fp ) and magnetic components, but according to Equation (10), the value of Fz /Fp should decrease as Bz increases.
Figure 1. PDF and scatter diagrams of Fx /Fp , Fy /Fp , and Fz /Fp for the selected magnetograms. Mean values of absolute Fx /Fp , Fy /Fp , and Fz /Fp are plotted and indicated by <|Fx /Fp |>, <|Fy /Fp |>, and <|Fz /Fp |>, respectively (Case I).
Figure 2. Fx /Fp , Fy /Fp , and Fz /Fp vs. Bx , By , and Bz for the selected magnetograms (Case I).
To study the effect of calibration coefficients on these three parameters of Fx /Fp , Fy /Fp , and Fz /Fp , PDF and scatter diagrams of Fx /Fp , Fy /Fp , and Fz /Fp of the selected magnetograms (Case II) are plotted in Figure 3. Like case I, the amplitudes of Fx /Fp and Fy /Fp are also smaller than that of Fz /Fp in case II. For Case II, the mean values of absolute Fx /Fp , Fy /Fp , and Fz /Fp for all selected magnetograms are 0.078, 0.111, and 0.251, respectively, which are smaller than those of Case I on the whole. Moreover, for Case II there are about 25% of the magnetograms with the value of Fz /Fp less than 0.1 (49% of the magnetograms with the values of Fz /Fp less than 0.2). It should be noted that though more Fz /Fp in the zone mentioned above (consisting a width of ± 0.1), most of the magnetograms cannot be regarded as force-free. Besides, the widths of PDF are not narrow and the scatter of diagrams is diverge as well. In general, there is a deviation of Fz /Fp from the zone. For Case II, Fx /Fp , Fy /Fp , and Fz /Fp versus magnetic components of Bx , By , and Bz are plotted in Figure 4 to study their relations with the magnetic field strength. The results are consistent with those of Case I, only the amplitudes of Fz /Fp decrease as Bz increases.
Figure 3. PDF and scatter diagrams of Fx /Fp , Fy /Fp , and Fz /Fp for the selected magnetograms. Mean values of absolute Fx /Fp , Fy /Fp , and Fz /Fp are plotted and indicated by <|Fx /Fp |>, <|Fy /Fp |>, and <|Fz /Fp |>, respectively (Case II).
Figure 4. Fx /Fp , Fy /Fp , and Fz /Fp vs Bx , By , and Bz for the selected magnetograms (Case II).
To see the difference in results of Fx /Fp , Fy /Fp , and Fz /Fp between two cases, the scatterplots of Fx /Fp , Fy /Fp , and Fz /Fp of Case I versus the corresponding ones from Case II are shown in Figure 5. The correlations are calculated, which are 0.994, 0.995, and 0.980 for Fx /Fp , Fy /Fp , and Fz /Fp , respectively. Also, the linear fits (y = Ax+B) are done based on these scatterplots, the values of A are 0.0008, 0.0003, and 0.2034 and of B = 1.10, 1.10, and 0.95 for Fx /Fp , Fy /Fp , and Fz /Fp , respectively. The correlation shows that there exists high consistency between these two cases. Nevertheless, the correlation of Fz /Fp between two cases is not as good as those of Fx /Fp and Fy /Fp . This may imply that more attention should be focused on the amplitude of Fz /Fp in order to understand the extent of force-free.
Figure 5. Scatter diagrams of Fx /Fp , Fy /Fp , and Fz /Fp deduced from cases I and II, respectively, for the selected magnetograms.
4 DISCUSSIONS AND CONCLUSIONS
It is worth studying the force-free extent of the photospheric magnetic field together with the case of magnetic field extrapolation, since it has been assumed that the coronal magnetic field is force-free and the photospheric magnetic field should be matched observationally. In this paper, results of the force-free extent of the photospheric magnetic field are given by conducting statistical research, using 925 magnetograms corresponding to 925 active regions observed by the SMFT at HSOS.
A part of efforts to avoid the uncertainty of the data calibration is the employment of two sets of calibration coefficients to describe the force-free extent of the photospheric magnetic field. For Case I, the calibration coefficients C
L and C
T are 8 381 and 6 790 G, the mean values of absolute Fx
/Fp
, Fy
/Fp
, and Fz
/Fp
for all selected magnetograms are 0.077, 0.109, and 0.302, respectively; and for Case II, C
L and C
T are 10 000 and 9730 G, the mean values of absolute Fx
/Fp
, Fy
/Fp
, and Fz
/Fp
are 0.078, 0.111, and 0.251, respectively. There are 17% and 25% magnetograms with a value of Fz
/Fp
less than 0.1 for cases I and II, respectively; in other words, 17% of Case I and 25% of Case II are force-free. Although there are differences between the two cases, the correlation is as high as 0.994, 0.995, and 0.980 for Fx
/Fp
, Fy
/Fp
, and Fz
/Fp
, respectively. Consequently, we concluded that a large part of the photospheric magnetic fields does not belong to a force-free field. Therefore, before extrapolating a magnetic field, the force-free extent of the photospheric magnetic field should be adequately considered. We note that Fz
/Fp
decreases with the increase of Bz
, which is a more important parameter indicating whether a magnetic field is force-free or not, since the amplitude of Fz
/Fp
is larger than those of Fx
/Fp
and Fy
/Fp
. From Equation (10), it can be seen that Fz
/Fp
may be neglected when the amplitude of Bz
is comparable to
$\sqrt{B_{x}^{2}+B_{y}^{2}}$
, while they are apparently different between Bz
and
$\sqrt{B_{x}^{2}+B_{y}^{2}}$
, then the amplitude of Fz
/Fp
should be enlarged. According to Equations (8)–(10) and observatories as well, the correlations between Fx
/Fp
, Fy
/Fp
, and the magnetic field component are not evident.
The strengths of active region magnetic fields observed by the SMFT at HSOS are determined through calibration, under the conditions of a weak-field approximation and linear relations between the magnetic field and the Stokes parameters I, Q, U, and V. In addition, comparing the recent data obtained from a space satellite, the data of the SMFT at HSOS have lower resolution and more uncertainty of magnetic amplitudes. These disadvantages may affect the statistical results we have acquired. However, the statistical analysis associated with its results may have significance as a practical reference, because the SMFT has observed photospheric magnetic fields more than one solar cycles at HSOS and its reliability observations have been studied strictly and adequately. It is hoped that statistical results can be obtained on the basis of high-resolution data in the future.
ACKNOWLEDGMENTS
The authors thank the anonymous referee for helpful comments and suggestions. This work was partly supported by the National Natural Science Foundation of China (Grant Nos. 11203036, 10611120338, 10673016, 10733020, 10778723, 11003025, 11103037, 10878016, and 11178016), the National Basic Research Program of China (Grant No. 2011CB8114001), the Important Directional Projects of Chinese Academy of Sciences (Grant No. KLCX2-YW-T04), the Young Researcher Grant of National Astronomical Observations, Chinese Academy of Sciences, and the Key Laboratory of Solar Activity National Astronomical Observations, Chinese Academy of Sciences.
