Faraday rotation and measurements of magnetic fields in sunspots

Vl.Tta$ln Astronomy,

Vol. 31, pp.

59-62,

0083-6656/88 $0.00+ .50 Copyright © 1988 Science Press & Pergamon Journals Ltd.

1988

-FARADAY R O T A T I O N AND M E A S U R E M E N T S OF M A G N E T I C F I E L D S IN S U N S P O T S

Ye Shihui Jin Jiehai Purple Mountain Observatory, Nanjing, China ABSTRACT In this paper the numerical solutions of Unno-Beckers' equations for the magneto-sensitive line Fel ~5250.216 are used to demonstrate the influence of Faraday rotation on the measurements of the azimuth of the transverse field. We propose a method to determine the intrinsic direction of the transverse field with the observed azimuthal angle of the plane of linear polarization. INTRODUCTION One of the difficulties

of the measurements

fields lies in the determination component.

Direct observations

plane of polarization may deviate transverse observed

(~).

significantly

of solar vector magnetic

of the direction of the transverse

can give us only the azimuth of the

Under the influence from the intrinsic

of Faraday rotation, azimuth of the

field ( X ). In order to get the true value of ~

it is necessary

to establish

the theoretical

X

from the

relation

between these two angles. CALCULATION

OF THE AZIMUTH OF THE PLANE OF POLARIZATION

When magneto-optical

effects are taken into account,

parameters I, Q, U, V of magneto-sensitive governed by the following Unno-Beckers'

spectral

equations

the Stokes lines are

of transfer: q

iI

l*nI qQ

cosO

qq

%

1 + nI

-O R

qv

QI

0wsin 2X

!

d

By numerical

qU

PR

I + ql

-Pwcos 2×

U

qV

-Owsin 2X

Pwcos 2X

I +rl

V

solution

I

J

•

S

of this system of equations we may get the 59

(l)

60

Ye Shihui and Jin Jiehai

linear polarization of

@ according

parameters

Q and U and then calculate

the value

to the formula:

l -i ~I 61 = ~ tan [ J Ud(AI)/ ~ 0 0

Qd(AI)]~

]

]

tan-

(2) Q

RELATION BETWEEN THE AZIMUTHAL ANGLES OF THE PLANE OF POLARIZATION AND OF THE MAGNETIC FIELD We have performed numerical

solutions

various values of the field strength line and line of sight ( y ) Y

= 45°,

various

X

= I0° and

the Faraday effect

61

= 0.OIA, calculations

is largest

are made for

from line center ( AI

are shown in Figure

AI

- 0 ) and

= 0.20~. The

I. It can be seen that

in line core where

~

>>

X

but quite

10 4

3

C t Jr....

5250 with

(B), the angle between the field

to the far red wing until

results of computations

I

etc. As an example, when B-]000G,

spectral regions starting

moving consecutively

of Eq. (]) for

P

L

s

!

I

-2

f

0

-l

1

2

3

4 10' (~

Fig.

l Change of

weak in far wings where curve approaches

@

~ ~

from line center to wing. X. Concretely

speaking,

the asymptote OL, we have 2 ~

when the solid

approximately

equal

to 20 ° , i.e. the value of 2 X • The dashed line is the parabola-like curve suggested by Makita (1986). Let us denote the difference between

~

and

X

by

@

, then our calculations

show that

0 varies

Magnetic with both

B

and

y

Fields

(see Figs.

in Sunspots

61

2 and 3). 3

7=45 ° x=O

B=IO00 G

.....

B=2000 G

\ o--- o - - - .

\

B=3000 G

>~.--.,c--.~. B=4000 G

'..

\ \

\\

\. •

\

N

/'"

\"-IL ,1 -I0

-5

,,..//

./

.

""

/i]

10

S

0

~o~i~ and

0

Fig. 2 R e l a t i o n b e t w e e n

B •

lo'0 B=2000 G,

5

.....

X= 10 °

7 =300

.d, a•~ ,

--x---x---7---45°

, • t e

s S

8~,=0.01 A

,~_.00o

4

.

s s t

w~l~

ILl • d

e¢

IS#

t°ss~ sS •

2

~s

I# ##

•

•

issss s ss • s • • os • • s

~

•

,,• It~

#s

. , f o # St_ o~s ° -,-.~p L

•

.s ° .o.o

. - " 'A

I

-lO

~ ....

"

°

./

.,*

,~,0. #

#

°--°°,1

-5

o

W'"

0

I

I

5

]0

Jr"

O,"

-2

•

Fig. 3

Relation

between

0

and

y.

62

Ye Shihui and Jin Jiehai

CORRECTION FOR THE INFLUENCE OF FARADAY ROTATION One method is to make observations sensitive magnetic

in the far wings of a magneto-

line. These spectral regions are not sensitive enough to fields and so errors of measurement may be large. A better

method proposed by us is to perform numerical solutions of UnnoBeckers' equations

for the magneto-sensitive

region used. When B and

y

line and the spectral

are known from independent observations,

one may decide which curve on the U ~ Q diagram should be used to derive

X

from the directly measured

~ .

ERRORS IN THE MEASUREMENTS OF THE AZIMUTH OF POLARIZATION If various spectral passband widths are used, the theoretical curve on the U- Q diagram exhibits some peculiar features (see Fig. 4). When

6%

increases,

@

steadily decreases.

large passband widths the value of

~

But for sufficiently

approaches a certain limit.

This allows us to estimate the errors of the measured values of when large

6% is adopted.

20

*- * --"

B=2000 G, 7"---45°, X=0

.--o--*

! ~ 4 0 0 0 G , 7 = 3 0 °, X=0

~_~_-~=" ,. . . . . . . .

~

%

__

--,.- ~ , c . v ~--'I~'~

6x=J~OA

-% L

A

-30

-20

6x~.O1 A l -I 0

0

l 10

I 20

l 30

I 40

I 50

I 60

~o4~ Fig.4

Relation between @ and 6%.

REFERENCE Maklta, M. (1986), Solar Phys.,

103, I