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Measurements of Vector Magnetic Fields

Jiang-Tao Su and Hong-Qi Zhang

National Astronomical Observatories Chinese Academy of Sciences

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Outline

Foundation for measuring of vector magnetic fields ----Zeeman effect

Magnetogram data reductionCalibrations for vector magnetic fields

and Faraday rotation Influence of stray-light on longitudinal magn

etic signal

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I. Zeeman effect

Intensity differences ofZeeman splitting polarizedcomponents diagnosed bymagnetographs are StokesI 、 Q 、 U 、 V.

The relationships among them as the followings ：

VV

VVN1

UU

UUN1

QQ

QQN1

II

II

I

V

II

II

I

U

II

II

I

Q

Stokes V ---- BL

Stokes Q ---- BT

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Q/I

U/I

V/I

Longitudinal magnetic field BL

Transverse magnetic field BT

Calibrated

Calibrated

Diagnosis of Stokes parameters Q/I, U/I, V/I by Video Vector Magnetograph at Huairou

With line FeIλ5324.19 Å

-0.075 Å

(Zhang 2000)

0.0 Å

0.0 Å

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II. Magnetogram data reduction

1 、 Linear bias present in Stokes Q/I and U/I

-0.12 Å

-0.12 Å

Estimate of bias in areas outside the AR

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00325.0/ fitIU

00103.0/ fitIQ

GB fitt 630

Distribution plots

of the U/I and Q/I I

intensities

Transverse field bias distribution

Subtract linear biases from original Stokes Q/I and U/I data:

fittedobs

fittedobs

IUIUIU

IQIQIQ

)/()/(/

)/()/(/

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Measurements of transverse magnetic fields with

filter tuned to line wings are to be suffered the

affections of cross-talk between circular and

linear polarized lights.

Measurements of transverse magnetic fields with

filter tuned to line center are to be suffered the

more serious affections of Faraday rotations

2 、 Circular-to-linear cross talk

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b: -0.06Å

r: +0.06Å

)(050.000084.0

)(0149.00036.0

rbrb

rbrb

VVUU

VVQQ

The cross-talk is estimated by making a scatter plot of the difference between the red (r) and blue (b) wings [(Qb-Qr)/I vs. (Vb-Vr)/I, and (Ub-Ur)/I vs. (Vb-Vr)/I] at the ±60 mÅ filter positions, respectively, over the entire field of view.

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VKCIUIU

VKCIQIQobs

obs

22

1.1

)/(/

)/(/

The formulae for correction

of cross-talk:

Two azimuths observed at ±120mÅ

from the line center

After the cross-talk corrections

Before the cross-talk corrections

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A explanation for magnitude differences in sunspot umbra region is:

There is the asymmetry in two wings of line FeIλ5324.19 Å in umbra region A rough ratio of polarized signal Q or U at 150mÅ red to at 150mÅ blue is 1.5, estimated from the line of FeIλ5324.19 Å.

The line of FeIλ5324.19 Å in umbra region

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III. Calibrations of full vector magnetic fields and Faraday rotation

Calibration method: With nonlinear least-square techniques, the observed Stokes V

/I, Q/I and U/I profiles were compared with the model profiles and the parameters of magnetic strength H, inclination γ and azimuth χ were obtained. Then we got the coefficients of CL and CT for longitudinal and transverse components of magnetic field by the approximate relations:

BL=CL(V/I) ， BT=CT [(Q/I)2+(U/I) 2] 1/4.

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Preceding calibration works

Theoretical calibrations of vector magnetic fields (Ai, Li and Zhang, 1982)

Empirical calibration by comparison with Kitt Peak data (Ai, 1994-1995)

Calibration of longitudinal magnetic field with observational and empirical methods using Huairou data (Wang, Ai and Deng, 1996)

8882;9606 LELO CC

9730;101 4 TL CC

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The Stokes profiles data were obtained on 2002-10-24 and 2003-10-23, with the HSOS vector magnetograph for two relativesimple sunspots of AR 10162 (N26 E04) and AR 10484 located (N04 E12.4). The former data are used to diagnose the Faraday rotation the latter data to calibrate vector magnetic fields.

The spectral scan data were Stokes images extending from 150mÅin the blue wing to +150mÅ in the red wing of the FeIλ5324.19 Åline at steps of 10mÅ.

The standard deviation data were Stokes images observed at ±60mÅ from the line center.

Observed data (Stokes profiles)

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Analytical solutions for Stokes parameters

Landolfi & Landi Del’Innocenti (1982)

])1[(

]))(1()1[(

]))(1()1[(

]})1)[(1{(

2

2

2

22220

1

1

1

1

RV

RU

RQ

BI

VVIB

UQVVQIUIB

QVUUVIIB

VUQIIB

222222222 ])1[()1( RVUQVUQII

VVUUQQR

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Definition for :2

ijii

jiii

jiii

fitaVobsV

fitaUobsUfitaQobsQ

iV

iUiQ

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21212

)];()([

)];()([)];()([

2

22

,,,,,,, 100 HBaD

All parameters being treated as independent parameters(Balasubramaniam and West 1991).

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Sunspot filtergram

of AR 10162

Map of the sunspot’s filtergram showing radii

and circles for the selection of the pixels used in

the analysis (Hagyard, et al. 2000).

**

*

*

****

Azimuth rotation>60º

*

Azimuth rotation about 40º

*

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Q/I U/I

V/I

=0.5actan(Q/U)

The 8 parameters obtained are as the followings ：

AR 10162

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a

b

c

d

Q/I U/I V/I

AR 10484

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Some abnormal pixels as the minus Doppler width obtained are deleted in data reduction.

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Calibration of transverse field ：

Calibrations at the routine observational line positions

Calibrations at line wing -0.12Å

Calibration of transverse field ：

Calibration of longitudinal field ：

Calibration of longitudinal field ：

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Calibration Coefficients Based on Least-square Fitting

Contents Band-pass location

(Å)Coefficient (G)

BT -0.12 10550±92

# 0.00 6790±52

BL -0.12 10075.5±134.3

# -0.075 8381.0±159.1

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Least-square fitting:

Virtues: more accurate total magnetic field strength, declination and azimuth of magnetic field vector can be obtained.

Shortcomings: (1)only the strong fields of sunspots can be derived (during the process of data reduction, we get the minimum transverse magnetic field strength 220 Gauss, the minimum longitudinal magnetic field strength 49 Gauss).

(2) the process of obtaining the data is time-consuming.

Discussions

The fitting errors for a sampled pixel

parameters

H γ χ 0 D η0 α μB

errors 3.2% 2.2% 0.3% 1.6×10-6% 3.8% 32% 12% 5%

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Orientations of three linearly polarized light (nбL ， nπ ， nбR)

1/2tg-1 （ U/Q ）

Rotation of polarization

direction

Relative retardation

Top of solar atmosphere

Faraday rotation

Azimuth of magnetic field

1/2tg-1 （ U’/Q’ ）Orientations of three linearly polarized

components (nбL ， nπ ， nбR)

1/2tg-1 （ U’/Q’ ）

Diagnostic of magnetograph

Azimuth difference

Δ =ф-ф’

Propagation

Faraday rotation

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A controversy on Faraday rotation

Hagyard, et al. (2000) found that Faraday rotation of azimuth will be a significant problem in observations taken near the center of a spectral line for fields as low as 1200 G and inclinations of the fields in the range of 20 º ~ 80 º degree.

Bao, et al. (2000) and Zhang (2000) found a mean azimuth rotation only ~12 º for Huairou transverse field data taken at the center of FeIλ5324.19 Å.

Zhang, et al. (2003), by comparisons of the data from Huairou, Mees and Mitaka observatories, found that there is a basic agreement on the transversal fields. Why is there such controversy on azimuth rotation?

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π-σ rotation effect

The linearly polarized component at line center is perpendicular to thedirection of the linearly polarized component in the wings. The azimuth will abruptly change 900 obtained with different positions of a spectral line from line wings to line center.

B= 1500 G ， γ=30o ，χ=22.5o

Azi

mut

h ф

Wavelength

We cannot separate the two effects (Hagyard et al. 2000). So the combination effect of them affects the measurements of azimuth of magnetic field vector.

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o5.364

=0.

5act

an(Q

/U)

o902,1 o5.1433

centerlineinput _

3.π-σand Faraday rotations

1. 、 2. π-σ rotation

4. Elimination of 180º ambiguity

π-σ rotation eliminated partially by convolution

with filter profile

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The spectral line with a relatively larger Landé factor g has the more possibility to confront “splitting” enlarged due to π-σ rotation effect

The Landé factor g of line FeIλ5324.19 Å is 1.5 and FeIλ5250.22 Å 3. Thus, the former will confront less large (say, > 60º) azimuth rotation

A deduction about azimuth rotation

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Definition of azimuth rotation ： fittedobscenter_line

AR 10162

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Azimuth rotation>60º

About 40º

Azimuth rotation >60º

Azimuth rotation

about 40º

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The results show that the rotation of azimuth is less significant in the observations taken near the center of the Fe I 5324.19Å line than those taken near the center of the Fe I 5250.22 Å line.

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Observational phenomenon

Magnetic signal V/I in umbra center is weaker than that of penumbra. The causes may be: (1) Stray light disturb polarized light. (2) There is serious magnetic saturation effect when measuring magnetic fields in the center of sunspot. (3) There is CCD nonlinear response to weaker polarized light intensity.

VI. Influence of stray-light on longitudinal magnetic signal

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Employed different solar atmosphere

model ， the theoretical calibrating curve

for longitudinal magnetic field (LMF) can

be obtained at -0.75 Å from line center.

Numerical simulation of magnetic saturation

Linear deviation for LMF calibration

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Numerical simulations show that when BL = 3000 G, the saturation

effect can only cause a maximum error of ~ 6% for the band-pass

position of -0.075 Å to the center of Fe Iλ5324.19 Å.

Moreover, in umbra region, Stokes V/I profiles are approximately

complete and not distorted.

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So magnetic saturation effect is not the main contributor to this phenomena.

Stray light Stray light include two parts (Martinez Pillet 1992; Chae et al. 1998):

(1) small-spread-angle (SSA) stray light (also named blurring part) caused by atmospheric seeing (affecting scale: several arcsecs) .

(2) large-spread-angle (LSA) stray light (also named scattering part) originating from the instrument and the Earth’s atmosphere (affecting scale: larger than 10 arcsecs).

The observational Stokes I obs image result from spatial re-distributions of disturbance-free Stokes I image :

)1(*I*I)1(I scblobs

)2())((

)(exp

)1()(

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2

rb

AM

b

ram

r

j jjj

scbl

A point-spread-function (PSF) (SSA part and LSA part) is taken as in the form:

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So the observed Iobs can be rewritten:)3()b,r(c)b,r(c)r(I LLij

jj

obs

Фj Фj ФL

the limb

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Methods of PSF and LSA stray light correcting curve determinations

Accurate determination of the solar limb

Aureole dataStokes I image

Derivative of I with respect to distance

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)4()))(()((( 22 i

fittedi

bgiobs

bg rIrI

LSA stray light integral(correcting curve)

Intensity profile across the limb

We make a linear least-square fit to the aureole data

and the chi-square is defined as:

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Variation of scattered light fraction in a day

Ruan Guiping (2004)

Polynomial fit

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In routine observations, what we should first is to subtract stray light intensity

from polarized light intensity, which should be large-spread-angle stray light.

The intensity to be subtracted is named as BLACK LEVEL (BL). However,

it is clear that the stray light intensity near an active region (AR) is not equal

to that of Sun limb.

The intensities of left and right circularly polarized light versus pixel location. IS refers to the actual background noise signal. Ii is a series of BL intensities and i = 1, 2, 3, 4, and 5.

Large-spread-angle stray light correction

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The magnetic signal V/I can be expressed as:

?2 LSRSi III

)3(2

)2(2

)1(2

,2)(

/

3

4

5

III

III

III

IIIII

IIIIIV

LSRS

LSRS

RSLS

RSLSiLR

LSRSLR

Where IR and IL are right and left circularly polarized light intensities, IRS

and ILS are right and left circularly polarized background noise intensities.

(1)Stokes V/I is smaller than the real value.(2)Stokes V/I is nearly equal to the real value.(3)Stokes V/I is larger than the real value.

In practice, how to select a reasonable BL to make:

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The observed sunspot was located r=0.5 from solar disk center. The intensity ratio between distance from disk center r=1. and r=0.5 is 1.93. We also know BL= 65 in the limb. So The BL value (r=0.5) can be obtained from this expression:

1111 /*)()/(* rzerorrrrrr ILIIIILL

Where L zero≈110 (the position stray light signal is zer0). Then BL (r=0.5) is 43.

The magnetograms for BL of 40, 55 and 65 are:

Correction method of large-spread-angle stray light

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Affection of strayed light on magnetic field azimuths

)/()arctan(21

STureSTrue UUQQ

noise signal introduced noise signal introduced by strayed lightby strayed light

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Thanks!