arXiv:1912.06557v1 [astro-ph.SR] 9 Dec 2019

. h esrdslrmgei ed u lotebs fthe of base the also but field, magnetic solar measured accuracy only the the not understanding for atmosphere estimation magnetic basic a solar provides in formation e Zeeman line the the of of measurement means by the light on the polarized based and of is field field magnetic magnetic vector chromospheric photospheric solar the informa of sola general the the tion that in implies transfer This radiation atmosphere. of magnetic theory the on based usually regions. active solar in helicity the corresponding a the current field, the also magnetic field, magnetic the non-potential of the of evolution formation structure, basic large the incl activities, a ing magnetic solar present the on observations information These of mount Sciences. of Chines the Academy Observing of Observatory Solar Astronomical National Huairou of mag at Station vector observed solar of been series have a century, netograms last the of 80s the Since Introduction 1 c Rview cec hn rs n pigrVra elnHeidelberg Berlin Springer-Verlag and Press China Science ti oie htorrsac fslrmgei ed is fields magnetic solar of research our that noticed is It rmPlrmtyt eiiy tde fSlrMgei Fie Magnetic Solar of Studies Helicity: to Polarimetry From ASnumber(s): PACS Magneti parameters, Stokes - Polarization spectrum, Solar ques studies. basic future some the t discuss for concerning also challenges We especially helicity. Sciences, and S current of Huairou shear, Academy at Chinese measurement transfer the field radiation of magnetic of in theory progress the the of research in approximation questions the basic to the introduce due briefly we paper, this In Citation: . e aoaoyo oa ciiy ainlAtooia Ob Astronomical National Activity, Solar of Laboratory Key ogiZag rmPlrmtyt eiiy c.China-Phy Sci. Helicity, to Polarimetry From Zhang, Hongqi h uio oa bevn Station Observing Solar Huairou the hsc,Mcais&Astronomy & Mechanics Physics, ff c.Testudy The ect. 2017 CEC CHINA SCIENCE ogiZhang Hongqi eevd;accepted ; Received ud- .Mc.Ato.,(,doi: (), , Astron. Mech. s. nd of evtre fteCieeAaeyo cecs 011 Bei 100101, Sciences, of Academy Chinese the of servatories in o h esrmnso h antcfilsadcorresp and fields magnetic the of measurements the for tions esrmnso oa antcfilsadtepsil error possible the and fields magnetic solar of measurements e r - - fseta ie nteslramshr.W nrdc some introduce We atmosphere. solar the in lines spectral of ed urn n helicity and current - fields c ennptnilt nslratv ein,sc stemag the as such regions, active solar in non-potentiality he lrOsrigSaino ainlAtooia Observato Astronomical National of Station Observing olar a antcfilsbsdo hs antclysniieli sensitive magnetically these so- on of based diagnostic fields magnetic The lar process. lines complex sensitive a is magnetically atmosphere the solar of transfer radiative The Transfer Radiative of Description General 2 researched be future. to the in and remained questions are there fundamental Nevertheless, some aspects. all do to and references fields, give fundamental magnetic not solar concerning of topics observations the from questions of Sta- some Observing discuss onl We Solar briefly achievements. scientific Huairou some at highlighting and mag- obtained tion, on data based field research field netic magnetic solar of background ignored. been have the instruments if optical even by deviations caused observations, or errors field errors magnetic of these sources lead from possible obtained This the know fields. to magnetic need solar us the of analysis theoretical nti ae,w r orve,eaieadaayesome analyze and examine review, to try we paper, this In * hssihn.o link.springer.com phys.scichina.com igChina jing d at lds sources onding basic netic o.N.: No. Vol. ries Print-CrossMark still doi: nes in y s Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -2 normally accompanies with some assumptions and approxi- (7) mations of the theoretical analysis, which probably are some error sources of the inversion of solar magnetic fields from The symbols have their usual meanings given by Landi = − observations. The important is to prevent or avoid the influ- DeglInnocenti (1976) [13]. InEqs. (7), dτc κcds, and ffi ence of these errors and trying to obtain relative accurated ηI , ηQ, ηU , ηV are Stokes absorption coe cient parameters ff information of solar magnetic fields. As the relevant forms of and ρQ, ρU and ρV are related to the magneto-optical e ects. the solution of the radiative transfer equations have been used In the analytical solution of the radiative transfer equations to detect the solar magnetic fields, the corresponding spectral for polarized spectral lines [4-6], several limitations are intro- line formation in solar magnetic fields need to be analyzed in duced: (a) Uni-dimensional plane-parallel atmosphere witha more detail. constant magnetic field; (b) linear dependence of the source In the stellar magnetic atmosphere, the non local thermo- function with optical depth; (c) constant ratio of the line and ffi dynamic equilibrium (non-LTE) equation of Stokes vector continuous absorption coe cients. The transfer equations for polarized radiation can be solved to give Iν = (Iν, Qν, Uν, Vν) for the transfer of polarized radiation can be written in a general matrix form (Stenflo (1994) [1]) −1 2 2 2 2 I = S0 + µS1△ [(1 + ηI )((1 + ηI ) + ρQ + ρU + ρV )], −1 2 d Iν hν Q = −µS △ [(1 + η ) η + (1 + η )(η ρ − η ρ ) = −(N B Φ − N B Φ ) I + j, (1) 1 I Q I V U U V ds Jl JlJu Jl Ju Ju Jl Ju 4π ν +ρQ(ηQ̺Q + ηU ̺U + ηV ̺V )], where in the case of non-scattering process the emission vec- −1 2 U = −µS1△ [(1 + ηI ) ηU + (1 + ηI )(ηQρV − ηV ρQ) (8) tor is approximately by +ρQ(ηQ̺Q + ηU ̺U + ηV ̺V )], hν −1 2 j = Φ 1N A , (2) V = −µS1△ [(1 + ηI ) ηV + ρV (ηQ̺Q + ηU ̺U + ηV ̺V )]. 4π Ju Ju Ju Jl and the relationship with where 2 2 2 2 2 2 2 2 2hν3 △ = (1 + ηI ) [(1 + ηI ) − η − η − η + ρ + ρ + ρ ] A = B , N C = N C , Q U V Q U V ji 2 ji i i j j ji (3) 2 c −(ηQ̺Q + ηU ̺U + ηV ̺V ) , Ci j g j Eij/kT = e and g jB ji = giBi j, (4) C ji gi and where the source function S (τ) = S0 + S1τ. In the approximation of the weak magnetic field (Stix, where B , B and A are Einstein coefficients, C and C are i j ji ji i j ji 2002) [14], one can obtain the following formulas collisional coefficients [2,3]. Non-LTE population departure 2 coefficients bi are defined as: ηI = ηp + O(vb), 2 LT E LT E 1 ∂ ηp 2 2 4 bl = Nl/N , bu = Nu/N (5) η = − v sin ψ cos2ϕ + O(v ), (9) l u Q 4 ∂v2 b b LT E 2 with N the actual population and N the Saha- 1 ∂ ηp (l,u) (l,u) η = − v2 sin2 ψ sin 2ϕ + O(v4), Boltzmann values for the lower and upper levels, respec- U 4 ∂v2 b b tively. The expanded form of Φ is the matrix in the right- ∂ηp 3 ηV = vb cos ψ + O(v ), hand side of Eq. (7), and we introduce the source function in ∂v b the form where vb = △λb/△λD, and △λb is the corresponding splitting

N jA ji of different subcomponents of the magneto-sensitive spectral S ν = . (6) Ni Bi j − N j B ji lines due to Zeeman effects and △λD is the Doppler width. With Eq. (9) and the neglect of magneto-optic effects, we When we studythe formationof polarizedlight in the mag- can get the simple relationship between the magnetic field netic field with Stokes parameters, Unno-Rachkovsky equa- and Stokes parameters. It is found that tions of polarized radiative transfer of spectral lines can be taken the form (Unno (1956) [4] and Rachkovsky (1962a, b) µS1 I ≈ S0 + , [5,6]) (1 + ηI ) µS ≈ − 1 ≈ ′ 2 2 = ′ 2 Q 2 ηQ CT B sin ψ cos2ϕ CT BT cos2ϕ, I η0 + ηI ηQ ηU ηV I − S (1 + ηI ) µS1 ′ 2 2 ′ 2 d  Q   ηQ η0 + ηI ρV −ρU   Q  U ≈ − ηU ≈ C B sin ψ sin 2ϕ = C B sin 2ϕ, µ = . (1 + η )2 T T T dτ       I c  U   ηU −ρV η0 + ηI ρQ   U        µS1 ′ ′       V ≈ − ηV ≈ C B cos ψ = C BL, (10)  V   ηV ρU −ρQ η0 + ηI   V  + η 2 L L       (1 I )             Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -3 where in National Astronomical Observatories of China (Ai & Hu,

2 2 2 1986) [15]. The FeIλ5324.19Å line in Figure 1 is a nor- µS ∂ ηp eλ C′ = 1 0 (M g − M g ) (11), mal triplet in the magnetic field and the Lande factor g=1.5, T 4(1 + η )2 ∂v2 4πm c2 △λ 1 1 2 2 I  e D  the excitation potential of the low energy level of this line  2  µS ∂ηp  eλ  C′ = − 1  0 M g − M g .  is 3.197eV.The equivalent width of the line is 0.33Å and the L 2 2 ( 1 1 2 2) (12) (1 + ηI ) ∂v 4πmec △λD residual intensity at the core is 0.17(Kurucz et al., 1984) [16]. This means that the longitudinal component of magnetic field This is a relatively wide line than other normal photospheric can be written in the form lines, such as the working lines used at the ground-based vector magnetographs, such as the FeIλ5250.2Å line (Lande BL = CLV, (13) factor g=3) used by the Video Vector Magnetograph at Mar- shall Space Flight Center (West & Hagyard 1983) [17] and where Cl is the calibration parameter of longitudinal compo- the FeIλ 6301.5Å (Lande factor g=1.667) and FeIλ 6302.5Å nent of field and is a function of the wavelength of a spectral (Lande factor g=2.5) lines used by the Solar Polarimeter at line. The intensity BT and azimuth angle ϕ of the transverse Mees Solar Observatory (Ronan et al., (1987) [18]). magnetic field can be obtained The bandpass of the birefringent filter of the Huairou Mag- 4 2 2 1 −1 U BT = CT Q + U , ϕ = tg . (14) netograph with 3 sets KDP crystal modulators is about 0.15Å 2 Q for the FeIλ5324.19Å line. The center wavelength of the fil- p ! ter can be shifted and is normally at -0.075Å from the line 3 Measurements of Photospheric Magnetic center of FeIλ5324.19Å for the measurement of longitudinal Field magnetic field and at the line center for the transverse one [15]. The theoretical calibration of Huairou Magnetograph The studies of radiative transfer of spectral lines without was presented by Ai, Li & Zhang (1982) [7] and observa- consideration of scattering in solar magnetic field have been tional one by Wang, Ai, & Deng (1996) [19] and Su & Zhang achieved, such as by Ai, Li & Zhang (1982),Jin & Ye (1983), (2004) [20]. Zhang (1986), Song et al. (1990, 1992) and Qu et al. (1997) [7-12] in China.

3.2 Numerical Calculation of Radiative Transfer of Stokes Parameters

The theoretical analysis of Stokes parameters in the magnetic

Photosphere field is important for the diagnostic of the magnetic field in 83/10/18#1 the solar atmosphere. Now, we firstly study the Stokes pro-

Umbra files of the FeIλ5324.19Å line under the VAL-C quiet Sun 82/11/13#1 model atmosphere (Vernazza et al., 1981) [21] by the numer- ical calculation code of Ai, Li & Zhang (1982) [7] and fit the observed profile I (Kurucz, et al., 1984 [16])in thecase ofno Umbra 81/03/25#1 magnetic field. Figure 2 shows the Stokes parameters I, Q, U, V of the FeIλ5324.19Å line, calculated with the VAL-C atmospheric model and a homogeneousmagnetic field B from 500 to 4000 gauss, inclination of the field ψ=30◦, azimuth ϕ=22.5◦ and µ = 1. Figure 3 shows the comparison with the SW umbral model atmosphere [Stellmacher & Wiehr (1970), Stellmacher Figure 1 The spectra of FeI 5324.19 Å obtained from Kitt Peak observa- & Wiehr, (1975) [22,23]]. Because we chose the azimuth an- tional results. From [16] gle of the field is 22.5◦, the Stokes parameter Q is equal to U if the magneto-optical effect is ignored. Thus, the difference 3.1 Formation of FeIλ5324.19Å Line in Photospheric between Stokes parameters Q and U in Figure 2 and 3 pro- Magnetic Field vides the information of the Faraday rotation of the plane of The FeIλ5324.19Åline is a workingline for the Huairou Vec- polarization, in the real situation that that the magneto-optical tor Magnetograph (Solar Magnetic Field Telescope - SMFT) effect cannot be ignored. Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -4

1.0 4

a3 b bral [22, 23] model atmosphere. The solid lines in Figures 0.8 2 4000G 4a and 4b show Stokes V/Ic in the wing of FeIλ5324.19Å at 1 0.6 ) 0.075Å from the line center, while other solid lines in Figure - 2 i

r 4000G 0

(10 500G /

q 2 2 1 4 0.4 r 4c-f show [(Q/I ) +(U/I ) ] and “error azimuthal angles” -1 c c

-2 of wavelength 0.005Å from the line center (i.e. almost at the 0.2 00G -3 line center). 0.0 -4 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 Wavelength (0.1nm) Wavelength (0.1nm) 0.3 0.3 4 0.3 Quiet Sun (VAL-C) Umbra (SW) 4000G a b 3 c4000G d

2 0.2 0.2 0.2 1 ) v - 2 r v

0 r

(10 500G u r -1 0.1 0.1 0.1 -2

-3 500G -4 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500 Wavelength (0.1nm) Wavelength (0.1nm)

0.3 0.3 Figure 2 The Stokes parameters ri, rq, ru, and rv (i.e. I/Ic, Q/Ic, U/Ic c d and V/Ic) of the FeIλ5324.19Å line numerically calculated under the VAL-C model atmosphere (red solid lines) [21] with the inclination ψ=30◦, azimuth ◦ 0.2 0.2 1/4

ϕ = 22.5 , and µ = 1 for the strength from 500 to 4000 gauss with the in- ) 2 um

terval of 500 gauss, but Stokes I/Ic from 0 Gs. The blue dotted lines (in +r 2 qm

Figures b and c) mark r and r in the case that the magneto-optical effect (r q u 0.1 0.1 has been ignored, respectively.

0.0 0.0 0 400 800 1200 1600 2000 0 400 800 1200 1600 2000 1.0 2 a b 4000G Quiet Sun (VAL-C) 1 40 40 0.8 4000G 0 e f 500G

0.6 ) 20 20

- 2 -1 Umbra (SW) i r (10 qm

r -2 0.4 0 0 -3 Degree 0.2 0G -4 -20 -20 0.0 -5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 Wavelength (0.1nm) Wavelength (0.1nm)

4 0.30 -40 -40 4000G 0 400 800 1200 1600 2000 0 400 800 1200 1600 2000 3 c4000G d Gauss Gauss

2 2 Figure 4 The relationship amount the Stokes parameters rv and (r + ) qm - 2 1 2 1/4 2 2 1/4 v r 0.15 r ) (i.e. V/I , [(Q/I ) +(U/I ) ] ) of the FeIλ5324.19Å line and mag- (10 um c c c um

r 0 500G netic strength in the wing from 0.005 to 0.185Å with the interval of 0.01Å -1 in the quiet Sun (left) and umbral (right) model atmospheres. e-f) The “error -2 azimuthal angles” of the transverse field inferred due to the magneto-optical -3 0.00 500G effect. The others are the same with Figure 3. The V/Ic relates to the lon- 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 Wavelength (0.1nm) Wavelength (0.1nm) gitudinal component of the field, while Q/Ic and U/Ic to the transverse one.

Figure 3 The Stokes parameters ri, rq, ru, and rv (i.e. I/Ic, Q/Ic, U/Ic and V/Ic) of the FeIλ5324.19Å line numerically calculated under the VAL-C model atmosphere (red solid lines) [21] with the inclination ψ=30◦, azimuth ϕ = 22.5◦, and µ = 1 for the strength from 500 to 4000 gauss with the in- Figure 4 shows that the quiet Sun and sunspot atmosphere terval of 500 gauss, but Stokes I/Ic from 0 Gs, and the same case under the have different sensitivities of Stokes parameters of the spec- SW umbral model atmosphere (blue solid or dotted lines) [22, 23]. tral line. The ratio between the Stokes parameter rv (i.e. V/Ic) of the quiet Sun and sunspot umbra is about 1.8, and Figure 4 shows the the Stokes parameters Q, U and V cal- the linear approximation to Stokes V can be used for rela- culated in Figure 3 in a almost equivalent form. It presents tively weak magnetic field only below 2000G for the quiet the curve clusters on the relationship between the strength of Sun and 1000G for the umbra, respectively. Similarly, the 2 2 1/4 magnetic fields and the Stokes parameters Q, U and V, and ratio between the Stokes parameter (rqm + rum) (i.e. 2 2 1/4 the “error azimuthal angles” of the transverse field inferred [(Q/Ic) + (U/Ic) ] ) of the quiet Sun and sunspot umbra due to the magneto-optical effect with different wavelengths is about 1.5 normally and it changes with wavelengths. The of the FeIλ5324.19Å in the quiet Sun (VAL-C) [21] and um- linear approximation can be used for relative weak magnetic Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -5

field only below 1000G for the quiet Sun and 600G for the where the subscript f marks that the Faraday rotation of the umbra, respectively. These reflect the nonlinearity between polarized spectral line has been considered. It is found that the strength of magnetic fields and the measured polarized the “error azimuthal angle” of the transverse field obviously components of the spectral line. relates to the wavelength from the line center to the wing of It is presented from the observations [24] that the magneto- the FeIλ5324.19Å line [24]. It is found that the “error az- optical effect is a notable problem for the measurement of the imuthal angle” of the transverse field is about 10◦ near the transverse field with the FeIλ5324.19Å line. Although the line center and about 5◦ at -0.15Å from the line center as the magneto-optical effects have been neglected in some stud- magnetic field is 1000 Gauss. Similar result for Stokes pa- ies of the transverse magnetograms obtained by the Huairou rameters I, Q, U of the FeIλ5324.19Å line under the cool magnetograph (Wang et al. 1992) [25], its influence is more umbra model (Stellmacher & Wiehr, 1970) [22] can be found obvious near the line center than in the far wing. also. The “error azimuthal angle” of the transverse field is ◦ ◦ The magneto-optical effect is a notable problem for the di- an order of about 20 near the line center and about 10 at - agnosis of magnetic field in solar active regions using mag- 0.15Å. It is noticed that the maximum errors of the azimuthal netic sensitive lines by vector magnetographs (Landolfi and angles of transverse components of the magnetic field can ◦ Landi Degl’Innocenti, 1982; West and Hagyard, 1983; Sku- reach ±20 due to the magneto-optical effect in Figure 4. manich and Lites, 1987) [17,26,27], because it influencesthe Although we display some examples of the magneto- determination of the azimuthal angle of the transverse mag- optical effects on the FeIλ5324.19Å line, but these provide netic field. The corresponding rotation of the azimuthal an- a basic estimation on the order of “error azimuthal angles” gles of polarized light related to the transverse components for the measurement of the transverse magnetic field by the of the fields can be found in Figure 4. FeIλ5324.19Å line only. The “error azimuthal angle” relates to the inclination angle and intensity of the magnetic field 4 10 0.005 from the calculations. The magneto-optical effects with the a0.035 8 b 3 0.075 0.155 FeIλ5250.22Å line were calculated by Solanki (1993) [28] 0.2 6

2 ) ff

) and also West & Hagyard (1983) [17]. It seems the di er- 1

4 4

Ci (10 ff Cq (10 2 ent sensitivity of the magneto-optical e ect relative to the 1

0 FeIλ5324.19Åline. This is consistent with the result obtained 0 -2 by Landolfi & Landi DeglInnocenti (1982) [26], i.e. the error

-1 -4 -4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 in the azimuthal angle is larger for intermediate values of the Log optical depth (500nm) Log optical depth (500nm) 6 6 10 3 Zeeman splitting (0.5 vH 2.5), if we keep in mind that the

8 c d land´efactor of the FeIλ5250.22Å line is 3.0 and that of the 2 6 FeIλ5324.19Å line is 1.5, and the equivalent widths of these ) ) 1 4 3 ff 1 lines are di erent. We also notice that the calculated ampli- Cv (10 Cu (10 2 tude of the magneto-opticaleffect relates to the choices of the 0 0 parameters of solar model atmospheres and the simplification -2 of radiative transfer equations etc. -4 -1 -4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 Log optical depth (500nm) Log optical depth (500nm)

3.3 Formation Layers of Spectral Lines

Figure 5 The contribution functions Ci, Cq, Cu, and Cv of Stokes parame- It is important to know where in the solar atmosphere a given ters I, Q, U and V of the FeIλ5324.19Å line are numerically calculated under a VAL-C model atmosphere at the wavelengths △λ=0.2 (cross), 0.155 (star), spectral line formed. This information is providedby the con- 0.075 (diamond), 0.035 (delta) and 0.005 (black) Å from the line center, with tribution function C I defined by Stenflo (1994) [1] the magnetic field B=1000 Gauss, the inclination ψ=30◦, azimuth ϕ=22.5◦, and µ = 1. ∞ ∞ I(0) = C I(τc)dτc = C I(x)dx (16) Z0 Z−∞ The “error azimuthal angle” caused by the magneto- for the Stokes vector I = (I, Q, U, V), where C (x) = optical effect is defined as the difference between that of the I (ln10)τ C (τ ) and x = logτ , τ is the continuum optical magneto-optical effect and that of the non magneto-optical c I c c c depth at 5000Å. It provides the contribution to the emergent effect, Stokes parameters from the different layers of the solar atmo-

U f U sphere. △ϕ = 0.5 tg−1 − tg−1 , (15) Q Q Now, we can define the equivalent source functions S ⋆ " f ! !# I Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -6

(Zhang, 1986)[9] 3.4 Measurements of Full Solar Disk Magnetic Field

⋆ 1 S I = S − [ηQQ + ηU U + ηV V] η0 + ηI Solar Magnetism and Activity Telescope (SMAT) is operated ⋆ 1 at Huairou Solar Observing Station, National astronomical S Q = − [(ηQ(I − S ) + ρV U − ρU V)] η0 + ηI Observatories of China started at 2003 (Zhang et al., 2007) 1 (17) S ⋆ = − [(η (I − S ) − ρ Q + ρ V)] [31]. A birefringent filter is centered at 5324.19Å for the U η + η U V Q 0 I measurement of vector magnetic field and the bandpass of ⋆ 1 S V = − [(ηV (I − S ) + ρU Q − ρQU)]. η0 + ηI the filter is 0.1Å. Figure 6 shows the comparison between both longitudinal magnetograms obtained by MDI of SOHO Similar studies have been done by Jin (1981), Song et al. satellite and SMAT on 2006 May 22. It can be found the ba- (1990), Qu et al. (1997) [10, 12, 29]. These represent for- sic morphological correlation between both magnetograms. mal radiative sources of Stokes parameters contributing to Some slight differences of both magnetograms are probably the emergent Stokes parameters at different layers of the so- caused by the different seeing condition, observing noise, and lar atmosphere. The Unno-Rachkovskyequations of radiative also the data reducing methods. transfer of polarized light can be written in a compact form A local area of the full disk vector magnetogram is shown d I ⋆ µ = (η0 + ηI )( I − S I). (18) in Figure 7. The strong transverse componentsof the field ex- dτc tend from the strong longitudinal field in the active regions. It is the same as eq. (11.129) of [1] and µ is the cosine of It is estimated that in this region the sensitivity of the longitu- the heliocenter angle. In the following, we analyze the line dinal component of magnetic field is about or lesser than the formation in the center of the solar disk, where µ = 1. We order of 5 gauss and the transverse one is about 100 gauss in notice that the contribution functions in the far wings also Figure 8. bring some information of the continuum, while we also no- tice that the continuum does not significantly contribute to the Figure 9 shows the correlation between the vector magne- Stokes parameter V of the lines near the working wavelengths tograms observed by SMFT and SMAT in an active region of the Huairou Magnetograph to influence the estimation on (Su & Zhang, 2007) [32]. The relationships of the longi- the formation layers of the Stokes parameters. The contribu- tudinal components, the transverse components, and the az- tion function C I is imuthal angles of both vector magnetic fields are shown in the figure. It is found a relative high correlation between 1 1 τci C (τ ) = S ⋆ exp − (η + η )dτ (η + η ), (19) both vector magnetogams observed from SMFT and SMAT. I ci µ I µ 0 I c 0 I " Z0 # The study on the spatial integration scanning spectra of filter where τci is the continuum optical depth at the i-th depth magnetographis also important for the calibration of the full- point. We can find that this contribution function obviously disk vector magnetic fields, which was made by Wang et al. relates to the simple form of the radiative transfer equation (2010) [33]. (18) and can be used in the numerical calculation of the ra- diative transfer equations conveniently. Figure 5 shows the formation depths of the FeIλ5324.19Å line in the solar VAL-C model atmosphere with different wavelengths within 0.005Å - 0.2Å from the line center. In the far wing △λ = 0.2Å, Stokes parameters I, Q, U, V mainly form near the relative deep solar photosphere logτ5 ∼ (−1, 0), while near the line center △λ = 0.005Å Stokes parame- ters mainly form in the relative high photosphere logτ5 ∼ (−3, −1), where τ5 means the optical depth of continuous spectrum at 5000Å. The calculation provides that the Stokes parameters of the FeIλ5324.19Å line form below the solar temperature min- −4 −3 imum region (about τ500nm = 10 ∼ 10 ) in the model atmosphere (Vernazza et al., 1976) [30]. This means that Figure 8 The intensity distribution of longitudinal and transverse magnetic FeIλ5324.19Å is a typical photospheric line. The forma- field along the dashed beeline in the local area of full disk magnetogram on tion layer of the spectral lines in sunspots is different from 2006 April 28 (bottom left), which vector magnetogram is shown in Figure that in the quiet Sun. 7. From [31] Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -7

Figure 6 The comparison between the full disk longitudinal magnetograms observed form MDI of SOHO satellite (left) and vector magnetograph at Huairou (right) on 2006 May 22. From [31]

4 Formation of Hβ Line in Solar Chromo- the line is formed at a height of about 1900 km (Allen, 1973) spheric Magnetic Field [40], while different formation heights of the Hβ line were proposed by various authors. Its oscillator strength is 0.1193 It is believed that solar active phenomena often relate to and the residual intensity at the core is 0.128 (Grossmann- the complex configurations of coronal and chromospheric Doerth & Uexkull, 1975) [41]. It is normally believed that magnetic fields. The observations of chromospheric mag- the Doppler broadening forms in the core of the Hβ line, and netograms were made, such as, at the Crimean, Kitt Peak resonance damping and Stark broadening in the wing. The and Huairou Observatories (cf. Severny & Bumba (1958), Hβ line is composed of 7 lines and all located within a width Tsap (1971), Giovanelli (1980), Zhang et al. (1991) [34-37]). of nearly 0.1Å. We can use the formulae given by Bethe & The Hβ line is a working line of SMFT at Huairou Solar Salpeter (1957) [39] to calculate individual component lines, Observing Station of National Astronomical Observatories the wavelength shift, the normalized oscillator strength and of China for the measurements of chromospheric magnetic damping constant, which are shown in Table 1. In compari- fields (Zhang & Ai, 1987) [38]. son with the results given by Allen (1973) [40] and Garcia & Mack(1965) [42], these available parameters showed slight difference in the mean wavelengths. When the magnetic field Table 1 Parameters of H line. γ is the damping parameter, g f is the β R alone is present, the Hβ line shows the anomalous Zeeman oscillator strength and g fw is the normalized oscillator strength. [38, 39] effect. In the solar atmosphere, both magnetic field and inter- atomic microscopic electric field are present, the wave func- 8 Transition WL (Å) γR × 10 gf gfw tions of the hydrogen atom energy levels become degener- ff 2p1/2 − 4d3/2 4861.279 6.5452 0.2436 0.2552 ate, and di erent wave functions relate to complicated energy shifts. 2s1/2 − 4p3/2 4861.287 0.8328 0.13702 0.1435 2p − 4s 4861.289 6.3122 0.00609 0.0068 1/2 1/2 4.1 Radiative Transfer of Hβ Line 2s1/2 − 4p1/2 4861.298 0.8328 0.0685 0.0718 We study the formation of polarized light of the Hβ Line with 2p / − 4d / 4861.362 6.5452 0.4385 0.45935 3 2 5 2 Eq. (7), Unno-Rachkovsky equations of polarized radiative − 2p3/2 4d3/2 4861.365 6.5452 0.0487 0.0510 transfer, in the solar atmospheric magnetic field. The source 2p3/2 − 4s1/2 4861.375 6.3122 0.0122 0.01278 function S ν inEq. (6) for the line spectrum is departure from LTE with coefficients bl and bu in Eq. (5). We also assume the continuum to be LTE. Figure 10 shows the solar Hβ line profile and some photo- As a magnetic field is present, the broadeningof the hydro- spheric lines overlap in the wing. The wavelength of the Hβ genline, suchas Hβ, shouldbe thejointeffect of the magnetic line is 4861.34Åand its equivalent width is 4.2Å. The core of field and the microscopic electric field distributed according Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -8

Figure 7 Local area of full disk vector magnetogram on 2006 April 28. The bars mark the transverse magnetic field. From [31] to the Holtsmark statistics. It is assumed that the microscopic For Hβ line, we can choose βl = 0.1189/F0 = 9.512 × 12 2/3 electric field is isotropic in the solar atmosphere, and their 10 Ne . Then, the second terms on the right of formulae contribution to the polarization of the emergent light of the (20) and (21) for the lower solar atmosphere are the main Hβ line is negligible, thus the angular dependence and polar- terms and the effect of the interatomic electric field is all im- ization properties of the π and σ components of the transi- portant. As we go up in the chromosphere, the density of tions of the Hβ line can be obtained from the classical theory. charged ions falls and the first term becomes more and more In comparison with formulae of the non-polarized hydro- important. gen line proposed by Zelenka (1975) [43], Zhang & Ai The first terms in the formulae (20) and (21) reflect the (1987) [38] introduced the absorption coefficients of Hβ line contribution under the weak microscopic electric field, thus in the magnetic field atmosphere in the form the approximation of Zeeman splitting is suitable. There

βl vi(△m = 1), vi(△m = 0) and vi(△m =−1) connected with Zee- ηp,l,r = η0 fwiH(ai, vi(△m)) W(β, r0/D)dβ man splitting are following, for k = 1, 2, 3, vi(△m = 2 − k) = i Z0 (k) X v − vm , and 1 2π π ∞ + η0 fw j H S (a j, λ) (20) 2 4π (k) eλ H ′ ′ j 0 0 βl v = G − G M − G − k . Z Z Z m 2 [( ) (2 )] (22) X   4πme△λD W(β, r0/D)dβ sin θdθdχ, and Here, M is the magnetic quantum number of the lower en- ergy level, G and G′ are the land´efactors for the lower and βl ρp,l,r = 2η0 fwiF(ai, vi(△m)) W(β, r0/D)dβ upper levels. fwi is the normalized oscillator strengths of the i 0 X Z Zeeman sub-components of the Hβ line. 1 2π π ∞ While the second terms in the formulae (20) and (21) + η f F S (a , λ) (21) 2π 0 w j j j 0 0 βl reflect the common contribution of the magnetic field and X Z Z Z   strong microscopic electric field to the line broadening. △λ j W(β, r0/D)dβ sin θdθdχ, is the shift of the polarized j-component of the spectral line △λ−△λ j(θ,χ,β,△m) where S (a j, λ) = a j, and fwi and fw j are nor- △λD caused by the magnetic and strong statistical electric field. malized oscillator strengths of polarized subcomponents of   fw j is the normalized oscillator strength of the polarized sub- the Hβ line. W(β, r0/D) is the Holtsmark distribution func- components of the Hβ line in the magnetic and microscopic tion of the microscopic electric field. βl is a criterion to mea- electric field. sure the relative size of the shift caused by the electric field, The damping parameter of i-th component of the line is relative to the fine structure of the hydrogen line. If βl is very 2 γiλ0 large, then the effect of the electric field can be negligible. ai = , where γi = (γradiation + γresonance + γelectron)i. 4πc△λD Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -9

Figure 9 The relationship between the vector magnetograms observed by the SMFT and the SMAT in the region marked with the white box in the Stokes V image at the bottom-right corner of the first panel. The relationships of the longitudinal components, the transverse components and the azimuthal angles of the vector magnetic fields of both instruments are in the first two rows, the vector magnetograms in the last row. The wavelengths of both instruments are marked in each panel. In the scatter correlation plots, a 2.5 factor has been applied to the circularly polarized signal and a 3.0 factor to the linearly polarized signals of the SMAT. From [32]

Now, a hydrogen atom and a perturbing, charged particle of the perturbing Hamiltonian was made by Casini & Landi have been considered in the external magnetic and electric DeglInnocent (1993) [44]. field. The hamiltonian of the hydrogen atom can be written 4.2 Numerical Calculation of Hβ Line Hˆ = Hˆ0 + Vˆ H + Vˆ F, (23) For analyzing the general properties on the formation of the where the first term on the right is the unperturbed hamilto- Hβ line, we used the VAL mean quiet chromospheric model nian, the second and third terms represent the perturbation of C (Vernazza et al., 1981) [21] and relevant non-LTE departure the magnetic field and electric field, respectively. The pertur- coefficients. We assumed that, when the degeneracy of en- bation equation is ergy levels disappears under the action of the magnetic field ′ and microscopic electric field, each magneton energy level (Hˆ0 + Vˆ H + Vˆ F) | ψi = (E0 + E ) | ψi, (24) keeps its original departure coefficient. It is only an approxi- ′ where E0 and E are the energy eigen value and perturba- mation. The emergent Stokes profiles of Hβ line are shown in tion, respectively. One can calculate the shifts △λ j of the sub- Figure 11. The calculation results show that the influence of components of the spectral line by means of Eq. (24). Nor- the magneto-optical effect for Stokes parameter V is insignif- mally, the general solution of the eigenvalue problem for the icant, but that for Stokes parameters Q and U in the Hβ line hydrogen lines does not depend on the direction of the mag- center is significant, for example the error angle of the trans- netic field and microscopic electric field. The choice of the verse field is about 7◦. Even though this means that the in- coordinate system for the calculation of the Hermitian matrix fluence of the magneto-optical effect for Hβ line is weak than Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -10

Figure 10 Left: Hβ lines in the quiet Sun (solid line) and sunspot umbra (dotted line) observed by the National Solar Observatory of USA (L. Wallace, K. Hinkle, and W. C. Livingston, http://diglib.nso.edu/ftp.html). Right: dI/dλ inferred by the corresponding Hβ lines in above.

1.0 3 that of the photospheric lines, such as FeIλ5324.19Å line. 0.00 a0.06 b 0.8 0.15 0.25 2 0.45

1.0 12 0.6 ) 4 1 a10 b Cq 0.8 Ci (10 0.4 8 3000 0

0.6 ) 6 0.2 - 4 ri 2000

rq (10 4 0.4 0.0 -1 -7 -6 -5 -4 -3 -2 -1 0 -7 -6 -5 -4 -3 -2 -1 0 2 Log optical depth (500nm) Log optical depth (500nm) 1000 & 3000 0.2 1000 3 4 0 c d 0.0 -2 3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 2 Wavelength (0.1nm) Wavelength (0.1nm)

2 12 4 ) 2 1 10 c d Cu 3000 3000 Cv (10 1 3 8 0 ) 6 ) 0 - 4 2000 - 2 2 2000 rv (10 ru (10 4 -1 -1 -7 -6 -5 -4 -3 -2 -1 0 -7 -6 -5 -4 -3 -2 -1 0 2 1000 Log optical depth (500nm) Log optical depth (500nm) 1 1000 0 500 -2 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Wavelength (0.1nm) Wavelength (0.1nm) Figure 12 Contribution functions Ci, Cq, Cu, and Cv of Stokes parameters I, Q, U and V of the numerical solution of Hβλ4861.34Å line calculated for the Figure 11 Profiles of Stokes parameters r , r , r , and r (i.e. I/I , i q u v c VAL model atmosphere at the wavelengths △λ=0.45 (cross), 0.25 (star), 0.15 Q/I , U/I and V/I ) of the Hβ line calculated for the VAL model atmo- c c c (diamond), 0.06 (delta) and 0.0 (black) Å from the Hβ line center. B=1000 sphere, a homogeneous magnetic field intensity 1000-3000 gauss of inclina- gauss, ψ=30◦, azimuth ϕ=22.5◦ and µ = 1. τ is continuum optical depth at tion ψ=30◦, azimuth ϕ=22.5◦ and µ = 1. I is the continuum. c c 5000Å. The horizontal coordinate is in the logarithmic scale.

By comparing the calculated signals of Hβ line for the The contribution functions of Stokes parameters of the measurement of chromospheric magnetic field in Figure 11 Hβ line calculated with the VAL model atmosphere [21] are with the photospheric one with the FeIλ5324.19Åline in Fig- shown in Figure 12. We can see that in our calculation the ure 3, it is foundthat theirratio onthe peakvaluesof Stokes V emergent Stokes parameters at Hβ line center almost form in the wing of both lines is about a order of 0.18 (0.04/0.22), in the higher atmosphere (1500-1600 km), but that in the while the that of Stokes Q is 0.06 (0.0007/0.012) for 3000 wing, for example, at -0.45Å away from the Hβ line center gauss magnetic field. Because the the core of the Hβ line the emergent Stokes parameters reflects the information of is too flat, this is consistent with the calculated result of the the photospheric field (300 km). The formation height of the weak signals of Stokes Q and U about a order of 10−4 in Balmer lines in the solar atmosphere excluding the magnetic Figure 11. This implies that the longitudinal components of field was analyzed also by several authors cf. Gibson (1973) the Hβ chromospheric magnetic field are detectable, while [45]. From the calculated results, we can see that the forma- the challenge with the high accuracy diagnostic occur in the tion heights of Stokes parameters Q, U and V are almost the measurements of the transverse components of the chromo- same to I. spheric fields. It is needed to pointout that real formationlayers of Stokes Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -11 parameters of the Hβ line are more complex than theoretical served at Huairou Solar Observing Station. It confirms the cases. Numerical results of the equations of radiative transfer disturbance of FeI 4860.98Å in the measurements of chro- depend on the selection of atmospheric models and parame- mospheric magnetic fields. ters of spectral lines. For example, the formation height near It is noticed that in the quiet, plage regions, and the line center obviouslydependson the selection of the value even penumbrae, the influence of the photospheric blended of the absorption coefficient of the line; and in the wing of FeIλ4860.98Å line is generally insignificant. As regards the the line that also depends on the amplitude of the line broad- Hβ chromospheric magnetograms, we can select the work- ening. The formation layers of different kind of structures ing wavelength between -0.20 and -0.24Å from the line core observed in the Hβ images in the solar atmosphere may be of Hβ to avoid the wavelengths of the photospheric blended different, such as sunspot umbrae, penumbrae and dark fila- linesin thewingof Hβ. A similar evidence on the disturbance ments, even though these features are observed at the same of blended lines in the wing of Hα is presented by Hanaoka wavelength in the wing of the chromospheric line. However, (2005) [52]. the computation of the formation layers of the line provides us a reference to estimate the possible spatial distribution of 04:09 UT, May 10, 1991 01:54 UT the detected magnetic field.

4.3 Possibility of Reversal Features in Hβ Chromo- spheric Magnetograms

Figure 13 shows active region NOAA6619 on May 10 1991. ◦ The 180 ambiguity of transverse components of the vector magnetic field probably is an obstinate question in the mea- 04:05 UT 04:46 UT surements of magnetic fields with the polarized spectral lines. In Figure 13, the 180◦ ambiguity of transverse components of the photospheric vector magnetic field has been resolved un- der the assumption of the minimizing the vertical gradient of the magnetic pressure and minimizing some approximation to the field’s divergence etc. cf. also Metcalf et al. (2006)

[46]. It is found the twisted photospheric magnetic field formed Figure 13 Active region NOAA 6619 on May 10 1991. Top left: Pho- in the compacted delta active region, and the vector mag- tospheric filtergram. Bottom left: Photospheric vector magnetogram. Top netic field extends form photosphere into the chromosphere right: Hβ filtergram. Bottom right: Hβ longitudinal magnetogram. White in the fibril-like features in the active region. The chromo- (black) is positive (negative) polarity in the magnetograms. The size of each figure is 2.′8 × 2.′8. spheric reversal signal structure in the center of sunspot rela- tive to positive polarity in the photosphere can be found. The similar results have been presented, such as, by Chen et al. (1989), Zhang et al. (1991), Almeida (1997), Zhang (2006) [37,47-49]. Due to the disturbance of the photospheric blended lines, such as FeIλ4860.98Å, nearby the Hβ line center, the rever- sal signal in the chromospheric magnetograms relative to the photospheric ones occurs in the sunspot umbrae in Figure 13. It is the same with the reversal of Stokes V nearby the line center in Figure 10. In the approximation of the week mag- netic field (Eqs. (12)and(13)), the sensitivity of longitudinal magnetic field is proportional to the variation of the line pro- file V ∼ dI/dλ with the wavelength (Stenflo et al., 1984) [50]. Figure 14 shows the statistical distribution of Stokes pa- Figure 14 Statistical distribution of Stokes parameter V in the blue wing rameter V at the different wavelengths in the blue wing from of Hβ line in the umbrae of a solar active region. From [51] the Hβ line center [51] at the umbra of a solar active region. It is based on a series of the longitudinal magnetograms ob- A comparison between photospheric and chromospheric Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -12

Hβ magnetograms in the quiet-Sun was provided by Zhang Severny, (1958), Hagyard et al. (1984), Chen et al. (1989), & Zhang(2000) [53], who found the similar patterns of the Chen et al. (1994), L¨u, Wang, & Wang (1993), Schmieder magnetic elements in the photosphere and chromosphere. It et al. (1994), Wang et al. (1994), Zhang et al. (1994), Li probably brings a message that the magnetic field extends et al. (2000), Wang et al. (2000) [47,56-64]), while the non- in the form of the fibril-like configuration in the quiet Sun. potential field can also be measured from the strong magnetic The fibril-like magnetic features can also be found around the gradient of active regions, which is strongly correlated with sunspots in the chromospheric magnetogram of Figure 13. active region flare-CME productivity (cf. Severny (1958), From the presentation above, it is found the complexity Falconer (2001) [56, 65]. The change of vector magnetic and difficulty on the diagnostic of chromospheric magnetic fields during the solar flares was observationally presented, fields relative to the photospheric ones. such as, by Chen et al. (1989) [47]. For analyzing the relationship between the non-potential 5 Magnetic fields in solar active regions magnetic field and electric current in solar active regions, the photospheric vector magnetograms, after the resolution ◦ 5.1 Non-potentiality of magnetic fields from observa- of 180 -ambiguity of transverse components of field, in the tions δ active region NOAA 6659 in June 1991 are shown in Fig- ure 16, which observed at Huairou Solar Observing Station. Figure 15 shows the vector magnetogram and Dopplergram A series of powerful flares occurred in this active region as of active region NOAA5395 on March 11 1989, observed at studied by Zhang (1996) [66]. It is found that the high shear the Huairou Solar Observing Station of the National Astro- of transverse magnetic field and gradient of longitudinal mag- nomical Observatories of China. netic field formed near the magnetic neutral line of the active Ai et al. (1991) [54] indicated that the flares in active re- region, where the transverse magnetic field rotated counter- gion NOAA5395 tend to occur on the redshift side of the in- clockwise around the magnetic main pole of negative polar- version lines of the Hβ Doppler velocity fields observed half ity. to two hours before. A similar case can be confirmed in Fig- The shear angle can be weighted by the transverse mag- ure 15, although in this Dopplergram the disturbance from netic field cf. Hagyard et al. (1984) [57] the flare sites cannot be neglected, due to the almost same observing times between this Dopplergram and the flare. − Boh · Bph θ = B · cos 1 , (25) T oh B B Some evidence in this powerful flare-producing region oh ph ! also can be found (Zhang (1995) [55]): 1) The twisted where B and B are the observed transverse field and that transverse magnetic field occurred in bay-like magnetic main oh ph calculated from the magnetic charges in the approximation of poles of opposite polarity in the active region. The main poles potential field. The amplitude of the shear angle in eq. (25) of opposite polarity in this region moved apart at a speed reflects the non-potentiality of the active region. of about 0.15km/s. 2) The newly-sheared vector magnetic The horizontal gradient of the photospheric longitudinal structures formed near the magnetic neutral line between the magneticfield in active regionscan be inferredfrom (cf. Leka collided magnetic main poles of opposite polarity. With the & Barnes (2003) [67]) emergence of new magnetic flux, the change of sheared an- gles of the horizontal magnetic field between the magnetic 2 2 ∂Bz ∂Bz main poles of opposite polarity is insignificant in comparison |▽ (Bz)| = + . (26) s ∂x ∂y with daily vector magnetograms, but the distribution of the ! ! intensity of the vertical current inferred from the horizontal The main contribution of magnetic shear in the active regions magnetic field evolved only gradually. 3) The flare sites oc- comes from the deviation of the transverse field from the po- curred near the magnetic islands and bays of oppositepolarity tential field inferred by magnetic charges in the photosphere, and were associated with the change of the vector magnetic while the magnetic gradient comes from the non-uniformity field. Although some of the flare sites are located near the of the longitudinal field. peak areas of the vertical electrical current density, their cor- As one knows the non-potential magnetic field formed in responding relationship is insignificant. solar active regions means the existence of electric current (cf. Moreton & Severny, (1968) [68]). The relationship be- 5.2 Magnetic Shear and Gradient tween the magnetic field and electric current is

The magnetic shear is an important parameter to measure the 1 non-potentiality of magnetic field in solar active regions (cf. J = ▽× B, (27) µ0 Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -13

March 11, 1989 March 11, 1989 01:52 01:53

01:57 01:57

Figure 15 Left: A resolved the 180◦-ambiguity of transverse components of photospheric vector magnetic field in the active region (NOAA 5395) at 01:57 UT on March 11, 1989, which is overlaid by a Hβ filtergram at 01:52 UT. The arrows mark the directions of transverse magnetic field. Right: A corresponding Hβ Dopplergram of the active region at 01:53 UT with longitudinal components of magnetic field. The red (blue) contours correspond to positive (negative) fields of ±50, 200, 500, 1000, 1800, 3000 gauss. The white (black) indicates up-(down-)ward follow. The size of magnetogram is 5.′23 × 3.′63. The north is top and the east is at left.

Figure 16 Ambiguity-resolved vector magnetograms at 0050 UT on 1991 June 6, at 0353 UT on 1991 June 9, and at 0438 UT on 1991 June 12 (from left to right) transformed from the image plane to heliographic coordinates. The contours indicate the longitudinal magnetic field distribution, ±20, 160, 640, 1280, 1920, 2240, 2560, and 2880 G. From [66]

−2 −3 −1 where J is in units of Am and µ0 = 4π × 10 GmA is the Eq. (28) permeability in free space. As letting B = B b and b is the unit vector along the di- B ∂by ∂b 1 ∂B ∂B rection of magnetic field, the current may be written in the J = − x + b − b , (29) z µ ∂x ∂y µ y ∂x x ∂y form [69] 0 ! 0 !

B 1 J = ▽× b + (▽B) × b. (28) where bx = Bx/B and by = By/B. The distribution of the two µ0 µ0 current components in active region NOAA6659 is shown in Figure 17. It is found that the electric current in solar active regions re- lates to the properties of chirality and gradient of magnetic Similar studies by means the two current components of field. The first term in eq. (28) connects with the twist of active regions were taken, such as, on the resolution of unit magnetic lines of force and intensity of the field. The the azimuthal ambiguity of vector magnetograms of so- second term in eq. (28) connects with the heterogeneity and lar active regions (Georgoulis et al. (2004) [70]), and orientation of magnetic field. the non-potentiality of solar active regions (Vemareddy The vertical electric current can be written according to (2017) [71]). Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -14 05:29 UT, June 9, 1991 a b c

Figure 17 (a) Photospheric vector magnetic field in the active region (NOAA 6659) at 05:29 UT on June 9, 1991. The white (black) shows the positive (negative) polarity and arrows mark the transverse field. The corresponding vertical electric current (b) inferred by the term of the magnetic chirality of the magnetic field and (c) by the term of the magnetic gradient and shear term in Eq. (29). The solid (dashed) contours correspond to the upward (downward) flows of vertical currents of ± 0.002, 0.008, 0.02, 0.04, 0.075, 0.12 Am−2. The size of view is 2.′72 × 2.′72.

5.3 Free Energy Contributed from Non-potential mag- netic fields

Now we analyze the free magnetic energy contributed from different components of magnetic fields (which is similar to Priest (2014) [72], p117)

1 1 W = B2dV = (B2 + 2 B · B + B2)dV o 2µ o 2µ n n p p 0 Z 0 Z 1 = (B2 + 2 B · B )dV + W , (30) 2µ n n p p 0 Z where Wo is the observed magnetic energy, Wp is the poten- Figure 18 Butterfly diagram of the electric current helicity. The mean den- tial magnetic energy, Bo is the observed magnetic field, Bn is sity of the current helicity of active regions is marked by size of the circles the non-potential component of magnetic field and Bp is the for grades: 0, 1, 3, 5, 7 (10−3G2m−1). The filled and hollow circles mark the potential component of magnetic field, respectively. negative and positive sign of helicity. The dash-dotted line marks the average value of current helicity and the dashed line marks the average value of the As we set Bp = ∇ψ, it is fund the free energy imbalance of current helicity after the data smooth. From [81]

1 1 W = W − W = B2dV + ψ B · d s. 6 Distribution of Magnetic Helicity with Solar f o p 2µ n µ n 0 Z 0 IS Active Cycles

If the vertical component Bns = 0 in the surface S of the close According to the formula of current helicity density (Bao & volume, it is found Zhang, 1998) [74], the vertical component of current helicity density in the photospheric lyaer can be inferred by photo-

1 2 spheric vector magnetograms W f = BndV. (31) 2µ0 2 Z h⊥c = ( B ·▽× B)⊥ ∼ αB , (32)

While Zhang (2016) [73] noticed that the contribution of the where α = b ·▽× b. 1 parameter Bn · Bp in the solar surface cannot be neglected In addition to the solar flare activities with helicity, such µ0 normally in the evolution of the flare-productive regions. This as, Bao et al. (1999), Zhang et al. (2008), Yang et al. (2012) probably means the non-potential components of magnetic [75-77], it is found that the mean magnetic helicity density fields extend from the subatmosphere and bring the free en- in the solar active regions tends to show negative signs in ergy into the atmosphere. the northern hemisphere and positive ones in the southern Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -15 hemisphere (Seehafer (1990), Pevtsov, Canfield, & Metcalf still some questions on the theories of the solar observations (1995), Abramenkoet al. (1996), Bao & Zhang(1998),Zhang relative to the diagnostic with the radiative transfer of spectral & Bao (1998), Hagino & Sakurai (2004), Hao & Zhang lines, even if some problems of solar instrumental techniques (2011) [74,78-83] ). Figure 18 shows the distribution of mean have been ignored. current helicity density of active regions in 1988 - 1997 in- ferred from vector magnetograms observed at Huairou Solar 8 Discussions for some new challenges on mea- Observing Station. Moreover, the reversal sign of helicity in surements of solar magnetic fields these statistical work has also attracted much attention [84]. The change of magnetic helicity with solar cycles, such as We have introduced the fundamental background of the ra- helicity butterfly diagram, also bring some new thoughts on diative transfer process of the magnetic sensitive lines in the the generations of magnetic fields with the dynamo process solar magnetic atmosphere, and correspondinganalysis of so- in the subatmosphere (such as, Kleeorin et al. (2003), Zhang lar vector magnetograms based on the study at Huairou Solar et al. (2012), Pipin et al. (2013) [85-87]). Observing Station of National Astronomical Observatories, Chinese Academy of Sciences. 7 The question on accuracy of measured vec- As a series of vector magnetograms have been observed tor magnetic fields and some relevant interesting results have been presented from these observations, we should notice that the meticu- In addition to some fundamental achievements on discover- lous fine analysis based on the relative complex calculated ing some important properties of solar magnetic activities, it from observed magnetograms probably is still needed, even is also needed to notice that the magnetic fields and relevant if some results probably are more interesting, such as the parameters, such as helicity etc., observed with different in- meticulous configuration and variation of the magnetic shear, struments not only have shown the basic same tendency, but free energy density, electric current and helicity in the so- also with some differences [25,88-90]. As a sample, the dis- lar active regions. We probably also need to keep in mind crepancies on the statistical distribution of the current helicity that the quantitative analysis of solar magnetic field from ob- parameters hc and α of solar active regions obtained by dif- served magnetograms contains some intrinsic difficulties. ferent solar vector magnetographs at Huairou in China and This means that there are still some basic questions on the Mitaka in Japan can be found by Xu et al. (2016) [90] in measurements of solar magnetic fields need to be carefully Figure 19. analyzed. Some of them have been presented in the follow- ing: • Radiative transfer of magnetic sensitive lines: We have presented the radiative transfer equations of mag- neticlly sensitive lines under the different approximations whether ignoring the scattering or the assumption of thermo- dynamic equilibrium, such as the numerical and analytical forms (in subsec. ??), which bring some ambiguity for the accurate results. The analytical solution has normally been used to inverse the photospheric vector magnetic fields. This means that the va- rieties of solar atmosphere models, such as the sunspot, fac- Figure 19 Correlations in (a) hc and (b) αav obtained from SMFT (Huairou) and SFT (Mitaka) data. The values of correlation coefficients (CC) are shown ula and quiet Sun, have been ignored. From the calculated on the panels. From [90] results, we can find the different magnetic sensitivities of po- larized spectral lines for the different kinds of solar atmo- This relates to a basic question on the accuracy for the spheres, such as the quiet Sun and sunspots in Figure 4. It measurements of the solar magnetic fields by means of solar is also noticed that the different sensitivity for the measure- vector magnetographs and the corresponding observing the- ments of the longitudinal and transverse components of mag- ories of solar magnetic fields. Some restrictions on the cali- netic field can be easily found from Eqs. (10), i.e. the lon- bration of magnetic field from Stokes parameters have been gitudinal and transverse fields relate to the first and second presented due the nonlinearity and simplicity in the above order derivative of the profiles of spectral lines, respectively. calculation of the radiative transfer equations of the spectral In the condition Eqs. (13) and (14) of the approximation of lines in solar magnetic atmosphere. It means that there are the weak fields, the influence of magneto-optical effects has Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -16 been neglected for the transverse components of the magnetic netic fields: fields obviously. We can find that these provide the constraint The resolution of the 180◦ ambiguity of transverse fields is a condition on the observations of solar magnetic fields by dif- difficult question due to the property of polarized light with ferent approximate methods or situations. Zeeman effects. • Solar model atmosphere: Several basic assumptions and approaches have been used to We have presented the observations of photospheric vector resolve the 180 ambiguity of transverse components of mag- magnetic fields with the FeIλ5324.19Å line and chromo- netic fields, such as comparing the observed field to a refer- spheric magnetic fields with the Hβλ4861.34Å line based ence field or direction, minimizing the vertical gradient of the on the analysis of the radiative transfer equations for polar- magnetic pressure, minimizing the vertical current density, ized radiation by Unno (1956) [4] and Rachkovsky (1962a, minimizing some approximation to the total current density, b) [5,6] under the assumption of local thermodynamic equi- and minimizing some approximation to the field’s divergence librium or their extension to non-local thermodynamic equi- (Metcalf et al. (2006) [46], Georgoulis (2012) [95]). Which librium at Huairou Solar Observing Station. of these treatments is prioritized is still debatable or question- Even if the numerical calculation can be used in the analysis able. of the radiative transfer equations (7) of Stokes parameters The projection of vector magnetic fields from solar disk to and provides some important information on the formation the heliospheric coordinates relates the transform of the dif- of polarized spectral lines in the different atmosphere mod- ferent components of magnetic field [96], such as in Figure els, it still rarely has been used in the real inversion of the 16, while the different Stokes parameters relate to different magnetic field, due to some difficulty and arbitrariness in the sensitivities and noise levels (cf. Eqs. (10)). This leads to accurate selections of inputed atomic and solar parameters. the inconsistency in the transformation of different compo- It is noticed that the different formation heights of magnetic nents of magnetic fields. It also causes degradation in spatial sensitive lines occur in the quiet Sun and sunspots, such as resolution of the vector magnetic fields after the inevitable the Wilson effect of sunspots, due to the the transparency smoothing fields from the image plane to heliographic coor- of sunspots, for the measurements of photospheric magnetic dinates in Figure 16. field, and the different heights for different chromospheric • Diagnostics of electric fields in solar atmosphere: features, such as the prominence, fibrils and plages etc. This The Stark effect is notable in the broadened wings of hydro- means that the observed magentograms probably does not al- gen and helium lines. It has been used in the analysis of the ways provide the information of magnetic fields at the same radiative transfer of Hβ line for the measurements of mag- height in the solar atmosphere, even if at the similar optical netic fields at Huairou Solar Observing Station, see Eqs. (20) depths of the working spectral lines. and (21). A notable question is the detection of the configuration of The Stark effect is the shifting and splitting of spectral lines of magnetic fields in the solar eruptive process, cf. Chen et al. atoms and molecules due to the presence of an external elec- (1989) [47]. Some study with Stokes parameters of spec- tric field. Normally, the isotropic plasma is statistically elec- tral lines can be found (Hong et al.(2018) [91]). Due to the trically neutral, due to the Debye effect. The electric fields distortion of the spectral lines with the variation of solar at- in the solar flares were presented by Zhang & Smartt (1986) mosphere in the solar eruptive process [3], the analysis of [97] based on the spectral analysis of the linear and quadratic relevant Stokes parameters of the spectral lines is a notable components of the Stark broadening. topic. A dynamic and quantitative depiction of the induction elec- • Measurements of magnetic fields in the corona: tric field E = u × B on the changes of the active regions was The Hanle effect is a reduction in the polarization of light described by Liu et al. (2008) [98], where u is the velocity when the atoms emitting the light are subject to a magnetic of the footpoint motion of the magnetic field lines and B is field in a particular direction, and when they have themselves the magnetic field. E represents the dynamic evolution of the been excited by polarized light. The Hanle effects with polar- velocity field and the magnetic field, shows the sweeping mo- ized light in the corona is notable question for the measure- tions of magnetic footpoints, exhibits the buildup process of ments of the coronal magnetic fields, cf. Liu & Lin (2008), current, and relates to the changes in nonpotentiality of the Qu et al. (2009), Li et al. (2017) [92-94]. active region in the photosphere. The analysis of the polarized lights due to the Hanle effect The measurements of the polarized light due to the Stark ef- for the diagnostic of the coronal magnetic field is still a chal- fect (or electric fields) in the anisotropic plasma is also a chal- lenging topic [1]. lenging topic. • 180◦ ambiguity on the determination of transverse mag- • Horizontal component of electric current and corre- Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -17 sponding helicity: visable methods (Wang et al. (1992), Bao et al. (2000), The observations of chromospheric magnetic field are im- Zhang et al. (2003), Liang et al. (2006), Xu et al .(2016) portant for diagnosing the spatial configuration of magnetic [25,88,90,103,104]). lines of force as compared with the photospheric one, while In this paper, we havetaken more attention on the measure- the disturbance of photospheric blended lines in the wing of ments of magnetic fields at Huairou Solar Observing Station. Hβ line also bothers the detection of the magnetic field in the Of course, there are still more questions on the measurements high solar atmosphere in Figure 14. The lower sensitivity of of solar magnetic fields that we have not covered yet or dis- Stokes Q and U of the Hβ line cripples the observations of cussed in detail. the transverse components of chromospheric magnetic fields. = h i The electric current helicity density hc εi jkbi∂bk/∂x j con- 9 Acknowledgements tains six terms, where bi are components of the magnetic field. Due to the observational limitations, only four of the This study is supported by grants from the National Natu- above six terms can be inferred from solar photospheric vec- ral Science Foundation (NNSF) of China under the project tor magnetograms. By comparing the results for simulation, grants 11673033, 11427803, 11427901 and Huairou Solar Xu et al. (2015) [99] distinguished the statistical difference Observing Station, Chinese Academy of Sciences. of above six terms for isotropic and anisotropic cases. This means that the electric current and magnetic (current) helicity 1 Stenflo, J. O., 1994, Solar Magnetic Field, Polarized Radiation Diag- inferred from the photospheric vector magnetograms do not nostics, Kluwer Academic Publishers, Dordrecht. contain the completeness in theoretical sense. 2 Rutten, R. J., 2003, Radiative Transfer in Stellar Atmospheres, Ster- • The limited spatial resolution for the measurements of rekunding Instituut Uterecht, Institute of Theoretical Astrophysics magnetic fields: Oslo. 3 Fang, C., Ding, M. D., & Chen P. F., Physics of Solar Active Regions It is normally believed that the mean free path of pho- (in Chinese), Nanjing University Press. tons emerging from the solar photosphere and chromosphere 4 Unno, W., 1956, Line Formation of a Normal Zeeman Triplet, Publ. is about 100 km (Zuccarello (2012), Judge et al. (2015) Astron. Soc. Jap., 8, 108-125. 5 Rachkovsky, D.N., 1962a, A system of radiative transfer equations in [100,101]). This relates to the detection of the magnetic flux the presence of a magnetic field Izv. Krymsk. Astrofiz. Obs., 26, tules in the solar atmosphere and their possible distribution. 63-73. • Influence of the Doppler motion for the measurements 6 Rachkovsky, D.N., 1962b, Magnetic rotation effects in spectral lines of solar magnetic fields: Izv. Krymsk. Astrofiz. Obs., 28, 259-270. 7 Ai, G. X., Li, W., & Zhang, H. Q., 1982, FeIλ5324.19Å line forms The influence for the measurements of magnetic fields from in the solar magnetic field and the theoretical calibration of the solar the Doppler velocity field due to the solar rotation was dis- magnetic field telescope, Acta Astron. Sinica 23, 39-48. cussed (such as by Wang, Ai, & Deng (1996) [19]), and 8 Jin, J. & Ye, S., 1983, A study of the magnetically sensitive line Fe I 5324.191, Acta Astrophysica Sinica, 3, 183-188. some for Doppler measurements of the velocity field in the 9 Zhang, H., 1986, FeIλ5324.19Å line formation depth in the solar mag- solar photosphere and implications for helioseismology can netic field, Acta Astrophysica Sinica, 6, 295-302. be found (Rajaguru et al. (2006) [102]). However, the influ- 10 Song, W. H., Ai, G. X., Zhang, H. Q., & Li, X.C., 1990, An analysis of ence on the accuracy of the magnetic field measurement for the formation in magnetic field and the formation depth of seven solar photospheric FeI lines. (I), ChA&A, 14, 155-166. the moving objects in the solar atmosphere is still a subject 11 Song, W., Ai, G., Zhang, H. & Li, X., 1992, The longitudinal magnetic worth exploring. field observation calibration of several solar photospheric Fe I lines for multichannel birefringence filter solar magnetic field telescope. (II), Based on the above discussions, we can find that the di- Chin. J. At. Mol. Phys., 9, 2311-2321 agnostic of solar magnetic fields in the solar atmosphere 12 Qu, Z. Q., Ding, Y. J., Zhang, X. Y., Chen, X. K, 1997, Theory of line still is a fundamental topic. Although we have made formation depth and its application. (I): The contribution function and 15 some achievements on the analysis of solar vector magnetic the line formation depth, PABei, , 98-111. 13 Landi Degl’Innocenti, E., 1976, MALIP - a programme to calculate the fields based on the theory of the radiative transfer of spectral Stokes parameters profiles of magnetoactive Fraunhofer lines, A. Ap. lines from observations, the quantitative study and the cor- Suppl. 25, 379-390. responding assessment on its accuracy still remain some to 14 Stix M., 2002, The Sun: An Introduction, 2nd ed., Berlin, Springer. 15 Ai, G.X. & Hu, Y.F., 1986, Propose for a solar magnetic field telescope be made, especially on the aspects of exact inference from and its working theorem, Publ. Beijing Astron. Obs. 8, 1-10. vector magnetograms, such as the electric current and helic- 16 Kurucz, R., Furenlid, I., Brault, J. & Testerman, L., 1984, National ity etc, due to some theoretical ambiguities and errors, even Solar Observatory Atlas No.1 Solar Flux Atlas from 296 to 1300 nm, if one excludes the questions caused from the observational Printed by the University Publisher, Harvard University. 17 West, E.A. & Hagyard, M.J., 1983, Interpretation of vector magneto- techniques. Comparison on the vector magnetograms ob- graph data including magneto-optic effects. I - Azimuth angle of the served by different magnetographs probably is one of the ad- transverse field, Solar Physics, 88, 51-64. Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -18

18 Ronan, R. S., Mickey D. L. and Orrall, F. Q., 1987 The derivation 42 Garcia, J.P. & Mack, J.E., 1965, Energy Level and Line Tables for One- of vector magnetic fields from Stokes profiles - Integral versus least Electron Atomic Spectra, J. Opt. Soc. Am., 55, 654-685. squares fitting techniques, Solar Physics, 133, 353-359. 43 Zelenka, A., 1975, The asymmetry of the H-alpha absorption coeffi- 19 Wang, T. J., Ai, G. X., & Deng, Y. Y., 1996, Calibration of Nine- cient, Solar Physics, 40, 39-52. channel Solar Magnetic field Telescope. I. The methods of the obser- 44 Casini, R. & Landi Degl’Innocenti, E., 1993, The polarized spectrum vational calibration, Astrophys. Reports, Publ. of Beijing Astro. Obs., of hydrogen in the presence of electric and magnetic fields, Astron. 28, 31-40. Astrophys., 276, 289-302. 20 Su, J.T. & Zhang, H.Q. 2004, Calibration of Vector Magnetogram with 45 Gibson, E.G., 1973, The Quiet Sun, Scie. Tech. Inf. Off., NASA. the Nonlinear Least-squares Fitting Technique, ChJAA 4, 365-376 46 Metcalf, T. R., Leka, K. D., Barnes, Graham, Lites, B. W., Georgoulis, 21 Vernazza, J.E., Avrett, E.H. & Loeser, R., 1981, Structure of the solar M. K., Pevtsov, A. A.; Balasubramaniam, K. S., Gary, G. A., Jing, J., chromosphere. III - Models of the EUV brightness components of the Li, J., Liu, Y., Wang, H. N., Abramenko, V., Yurchyshyn, V., Moon, Y. quiet-sun, Astrophys. J. Suppl. Ser., 45, 635-725. J., 2006, An Overview of Existing Algorithms for Resolving the 180 22 Stellmacher, G. & Wiehr, E., 1970, Magnetically Non Split Lines in Ambiguity in Vector Magnetic Fields: Quantitative Tests with Syn- Sunspots, Astron. Astrophys., 7, 432-442. thetic Data Solar Physics, 237, 267-296. 23 Stellmacher, G. & Wiehr, E., 1975, The deep layers of sunspot umbrae, 47 Chen, J., Ai, G., Zhang, H. & Jiang, S., 1989, Analysis of solar flare in Astron. Astrophys., 45, 69-76. relation to the magnetic fields on Jan 14, 1989, Publ. Yunnan Astron. 24 Zhang, H., 2000, Analysis of the Transverse Magnetic Field in Solar Obs. Suppl., 108-113. Active Regions by the Huairou Vector Magnetograph, Solar Physics, 48 Almeida, J. S., 1997, Chromospheric polarity reversals on 197, 235-251. sunspots. Are they consistent with weak line emission? Astron. 25 Wang, H. M., Varsik, J., Zirin, H., Canfiled, R.C., Leka, K.D., & Astrophys., 324, 763-769. Wang, J.X., 1992, Joint vector magnetograph observations at BBSO, 49 Zhang, H., 2006, Magnetic Fields and Solar Activities, Astrophys. Huairou Station and Mees Solar Observatory, Solar Physics, 142, 11- Space Sci., 305, 211-224. 20. 50 Stenflo, J. O., Harvey, J. W., Brault, J. W., & Solanki, S. K., 1984, Di- 26 Landolfi, M. & Landi Degl’Innocenti, E., 1982, Magneto-optical ef- agnostics of solar magnetic fluxtubes using a Fourier transform spec- fects and the determination of vector magnetic fields from Stokes pro- trometer, Astron. Astrophys., 131, 333-346. files, Solar Physics, 78, 355-364. 51 Zhang, H., 1993, Solar chromospheric magnetic fields in active regions 27 Skumanich, A. & Lites, B. W., 1987, Stokes profile analysis and vector inferred by monochromatic images of the Stokes’ parameter V of the magnetic fields. I - Inversion of photospheric lines, Astrophys. J., 322, H-beta line, Solar Physics, 146, 75-92. 473-482. 52 Hanaoka, Y., 2005, Hα Stokes V/I Features Observed in a Solar Active 28 Solanki, S. K., 1993, Smallscale Solar Magnetic Fields - an Overview, Region, Publ. Astron. Soc. Jap., 57, 235-244. Space Sci. Rev., 63, 1-188. 53 Zhang, M. & Zhang, H. 2000, A comparison between photospheric and 29 Jin, J.H., 1981, The depth of formation of spectral lines in a magnetic chromospheric quiet-Sun magnetograms, Solar Physics, 194, 19-28. field, ChA&A, 5, 49-53. 54 Ai, G.X., Zhang, H.Q., Li, J., Li, W. & Chen, J.M., 1991, The observa- 30 Vernazza, J.E., Avrett, E.H. & Loeser, R., 1976, Structure of the so- tional evidence of the solar flares occurring in the redshift area of Hβ lar chromosphere. II - The underlying photosphere and temperature- Dopplergrams, ChSBu, 36, 1275-1278. minimum region, Astrophys. J. Suppl. Ser., 30, 1-60. 55 Zhang, H.Q., 1995, Configuration of magnetic shear and vertical cur- 31 Zhang, H.Q., Wang, D.G., Deng, Y.Y., Hu, K.L., Su, J.T., Lin, J.B., rent in active region (NOAA 5395) in 1989 March, Astrophys. J. Suppl. Lin, G.H., Yang, S.M., Mao, W.J., Wang, Y.N., Hu, Q.Q., Xue, J.S., Ser., 111, 27-40. Lu, H.T., Ni, H.K., Chen, H.L., Zhou, X.J., Zhu, Q.S., Yuan, L.J. & 56 Severny, A. B., 1958, Izv. Krim. Astrophys. Obs., 20, 22. Zhu, Y., 2007, Chin. J. Astron. Astrophys., 7, 281-288. 57 Hagyard, M.J., Smith, J.B., Jr., Teuber, D., & West, E.A., 1984, A quan- 32 Su, J.T. & Zhang H. 2007, The Effects of Polarization Crosstalk and So- titative study relating observed shear in photospheric magnetic fields to lar Rotation on Measuring Full-Disk Solar Photospheric Vector Mag- repeated flaring, Solar Physics, 91, 115-126. netic Fields, Astrophys. J., 666, 559-575. 58 Chen, J. M., Wang, H. M., and Zirin, H., 1994, Observations of vector 33 Wang, X., Su, J. & Zhang, H., 2010, Calibration of a full-disc longitu- magnetic fields in flaring active regions, Solar Physics, 154, 261-273. dinal magnetogram at the Huairou Solar Observing Station, Mon. Not. 59 L¨u, Y. P., Wang, J. X., & Wang, H. N., 1993, Shear angle of magnetic R. Astron. Soc., 406, 1166-1176. fields, Solar Physics, 148, 119-132. 34 Severny, A. B. & Bumba, V., 1958, On the penetration of solar mag- 60 Schmieder, B., Hagyard, Ai, G., Zhang, H., Kalman, B., Gyori, L., netic fields into the chromosphere, Obs., 78, 33-35. Rompolt, B., Demoulin, P., M., & Machado, M.E., Relationship be- 35 Tsap, T., 1971, The Magnetic Fields at Different Levels in the Active tween magnetic field evolution and flaring sites in AR 6659 in June Regions of the Solar Atmosphere (presented by N. V. Steshenko), in 1991, Solar Physics, 1994, 150, 199-219. Solar Magnetic field, Ed by R. Howard, IAU Symp. 43, 223-230. 61 Wang, H., Ewell, M., Zirin, H. & Ai, G., 1994, Vector magnetic field 36 Giovanelli, R. G., 1980, An exploratory two-dimensional study of the changes associated with X-class flares, Astrophys. J., 424, 436-443. coarse structure of network magnetic fields, Solar Physics, 68, 49-69. 62 Zhang H., Ai G., Yan X., Li W. & Liu Y., 1994, Evolution of Vec- 37 Zhang, H., Ai, G., Sakurai, T. & Kurokawa, H., 1991, Fine structures of tor Magnetic Field and White-Light Flares in a Solar Active Region chromospheric magnetic field and material flow in a solar active region, (NOAA 6659) in 1991 June, Astrophys. J., 423, 828-846. Solar Physics, 136, 269-293. 63 Li, H., Sakurai, T., Ichimoto, K., UeNo, S. 2000, Magnetic Field Evo- 38 Zhang, H. & Ai, G., 1987, Formation of the H-beta line in solar chro- lution Leading to Solar Flares I. Cases with High Magnetic Shear and mosphere magnetic field, Chin. Astron. Astrophys., 11, 42-48. Flare-Related Shear Change, Publ. Astron. Soc. Jap., 52, 465-481. 39 Bethe, H.A. & Salpeter, E.E., 1957, Quantum Mechanics of One-and 64 Wang, H., Yan, Y., Sakurai, T. & Zhang, M., 2000, Topology of Mag- Two-electron Atoms, Springer-Verlag. netic Field and Coronal Heating in Solar Active Regions - II. The Role 40 Allen, C.W., 1973, Astrophysics Quantities, The Athlone press. of Quasi-Separatrix Layers, Solar Physics, 197, 263-273. 41 Grossmann-Doerth, U. & Uexkull, M.V., 1975, Spectral investigation 65 Falconer, D.A., 2001, A prospective method for predicting coronal of the chromosphere. V - Observation and analysis of H beta, Solar mass ejections from vector magnetograms, J. Geophys. Res., 106, Physics, 42, 303-309. 25185-25190. Hongqi Zhang Sci. China-Phys. Mech. Astron. () Vol. No. -19

66 Zhang, H., 1996, Spatial Configuration of Highly Sheared Magnetic J., 751, 47-57. Structures in Active Region (NOAA 6659) in 1991 June, Astrophys. 87 Pipin, V. V., Zhang, H., Sokoloff, D. D., Kuzanyan, K. M., & Gao, J., 471, 1049-1057. Y., 2013 The origin of the helicity hemispheric sign rule reversals in 67 Leka, K.D. & Barnes, G., 2003, Photospheric Magnetic Field Prop- the mean-field solar-type dynamo, Mon. Not. R. Astron. Soc., 435, erties of Flaring versus Flare-quiet Active Regions. I. Data, General 2581-2588. Approach, and Sample Results, Astrophys. J., 595, 1277-1295. 88 Bao, S. D., Pevtsov, A. A., Wang, T. J. & Zhang, H. Q., 2000, Helic- 68 Moreton, G. E. & Severny, A. B., 1968, Magnetic Fields and Flares in ity Computation Using Observations From two Different Polarimetric the Region CMP 20 September 1963, Solar Physics, 3, 282-297. Instruments, Solar Physics, 195, 75-87. 69 Zhang, H. Q., 2001, Electric Current and Magnetic Shear in Solar Ac- 89 Pevtsov, A.A., Dun, J.P., Zhang, H.: 2006, Helicity Measurements tive Regions, Astrophys. J., 557, L71-L74. from Two Magnetographs, Solar Physics, 234, 203-212. 70 Georgoulis, M. K., LaBonte, B. J. & Metcalf, Thomas R., 2004, On 90 Xu, H., Zhang, H., Kuzanyan, K. & Sakurai, T., 2016, On the Origin the Resolution of the Azimuthal Ambiguity in Vector Magnetograms of Differences in Helicity Parameters Derived from Data of Two Solar of Solar Active Regions, Astrophys. J., 602, 446-467. Magnetographs, Solar Physics, 291, 2253-2267. 71 Vemareddy, P., 2017, Contribution of Field Strength Gradients to the 91 Hong, J., Ding, M. D., Li, Y., & Carlsson, M., 2018, Non-LTE Calcu- Net Vertical Current of Active Regions, Astrophys. J., 851, 3-12. lations of the Fe I 6173 Line in a Flaring Atmosphere, Astrophys. J., 72 Priest, E. R., 2014, Magnetohydrodynamics of the Sun, Cambridge Uni- 857, L2-L8. versity Press. 92 Liu, Y. & Lin, H.S., 2008, Observational Test of Coronal Magnetic 73 Zhang, H., 2016, Photospheric Magnetic Free Energy Density of Solar Field Models. I. Comparison with Potential Field Model, Astrophys. Active Regions, Solar Physics, 291, 3501-3517. J., 680, 1496-1507. 74 Bao, S. D., & Zhang, H. Q. 1998, Patterns of Current Helicity for the 93 Qu, Z.Q., Zhang, X.Y., Xue, Z.K., Dun, G.T., Zhong, S.H., Liang, Twenty-second Solar Cycle, Astrophys. J., 496, L43-L46. H.F., Yan, X.L., & Xu, C.L., 2009, Linear Polarization of Flash Spec- 75 Bao, S.D., Zhang, H.Q., Ai, G.X., and Zhang, M., 1999, A survey of trum Observed from a Total Solar Eclipse in 2008, Astrophys. J., 695, flares and current helicity in active regions, Astron. Astrophys. Suppl., L194-L197. 139, 311-320. 94 Li, H., Landi Degl’Innocenti, E. & Qu, Z.Q., 2017, Polarization of 76 Zhang, Y., Liu, J.H. & Zhang, H.Q., 2008, Relationship between Ro- Coronal Forbidden Lines, Astrophys. J., 838, 69-83. tating Sunspots and Flares, Solar Physics, 247, 39-52. 95 Georgoulis, M.K. 2012, Comment on “Resolving the 180 Ambiguity 77 Yang, X., Zhang, H., Gao, Y., Guo, J. & Lin, G., 2012, A Statistical in Solar Vector Magnetic Field Data: Evaluating the Effects of Noise, Study on Photospheric Magnetic Nonpotentiality of Active Regions Spatial Resolution, and Method Assumptions”, Solar Physics276, 423- and Its Relationship with Flares During Solar Cycles 22 - 23, Solar 440. Physics, 280, 165-181. 96 Hagyard, M. J., 1987, Changes in measured vector magnetic fields 78 Seehafer N., 1990, Electric current helicity in the solar atmosphere, when transformed into heliographic coordinates, Solar Physics, 107, Solar Physics, 125, 219-232. 239-246. 79 Pevtsov, A.A., Canfield, R.C., & Metcalf, T.R., 1995, Latitudinal vari- 97 Zhang, Z. & Smartt, R. N., 1986, Electric field measurements in solar ation of helicity of photospheric magnetic fields, Astrophys. J., 440, flares, Solar Physics, 105, 355-363. L109-L112. 98 Liu, J.H., Zhang, Y. & Zhang, H.Q., 2008, Relationship between Pow- 80 Abramenko, V.I., Wang, T. J., & Yurchishin, V.B., 1996, Analysis of erful Flares and Dynamic Evolution of the Magnetic Field at the Solar Electric Current Helicity in Active Regions on the Basis of Vector Mag- Surface, Solar Physics, 248, 67-84. netograms, Solar Physics, 168, 75-89. 99 Xu, H., Stepanov, R., Kuzanyan, K., Sokoloff, D., Zhang, H., & Gao, 81 Zhang, H. & Bao, S., 1998, Latitudinal distribution of photospheric Y. 2015, Current helicity and magnetic field anisotropy in solar active current helicity and solar activities, Astron. Astrophys., 339, 880-886. regions, Mon. Not. R. Astron. Soc., 454, 1921-1930. 82 Hagino, M. & Sakurai, T.: 2004, Latitude Variation of Helicity in Solar 100 Zuccarello, F.; 2012, EST Team The European Solar Telescope: Active Regions, Publ. Astron. Soc. Jap., 56, 831-843. project status, Mon. Not. R. Astron. Soc., 19, 67-74. 83 Hao, J. & Zhang, M., 2011, Hemispheric Helicity Trend for Solar Cy- 101 Judge, P. G., Kleint, L., Uitenbroek, H., Rempel, M., Suematsu, Y., cle 24, Astrophys. J., 733, L27-L32. & Tsuneta, S. 2015, Photon Mean Free Paths, Scattering, and Ever- 84 Zhang, H., Sakurai, T., Pevtsov, A., Gao, Y., Xu, H., Sokoloff, D., & Increasing Telescope Resolution, Solar Physics290, 979-996. Kuzanyan, K. 2010, A new dynamo pattern revealed by solar helical 102 Rajaguru, S. P., Wachter R. & Hasan, S. S., 2006, Influence of mag- magnetic fields, Mon. Not. R. Astron. Soc., 402, L30-L33. netic field on the Doppler measurements of velocity field in the solar 85 Kleeorin, N., Kuzanyan, K., Moss, D., Rogachevskii, I., Sokoloff, D., photosphere and implications for helioseismology, in Proceedings of & Zhang, H. 2003, Magnetic helicity evolution during the solar activ- the ILWS Workshop. Goa, India. February 19-24, p. 21. ity cycle: Observations and dynamo theory, Astron. Astrophys., 409, 103 Zhang, H., Labonte, B., Li, J., & Sakurai, T. 2003, Analysis of Vector 1097-1105. Magnetic Fields in Solar Active Regions by Huairou, Mees and Mitaka 86 Zhang, H., Moss, D., Kleeorin, N., Kuzanyan, K., Rogachevskii, I., Vector Magnetographs, Solar Physics, 213, 87-102. Sokoloff, D., Gao, Y., & Xu, H. 2012, Current Helicity of Active Re- 104 Liang, H., Zhao, H., & Xiang, F., 2006, Vector Magnetic Field Mea- gions as a Tracer of Large-scale Solar Magnetic Helicity, Astrophys. surement of NOAA AR 10197, ChJAA, 6, 470-476.