Magnetic fields in a coronal condensation

Chinese Astronomy ~ (1978) 238-247

~Pergamon Press. Printed in Great Britain

Aclx( Astr. Siniou 18 (1977)158-165

0146-6364/78/1201-0238-$07.50/0

MAGNETIC FIELDS IN A CORONAL CONDENSATION

Cao Tian-jun P~z,ple Moz.~,~t~:zinObsert~:zto~,,~/,Ae.czde,,r~aSini~ (Received 1977 May 24)

ABSTRACT Using the potential field model, we have calculated the magnetic structure of the coronal condensation, which appeared on the east llmb of the solar disk at the 22 September 1968 eclipse. The comparison between calculated magnetic field geometry and observed contour of the coronal condensation shows general agreement. From this analysis, we see that the magnetic field extending to the lower coronal region is approximately the potential field during the steady period of an active region. The observed contour of the coronal condensation may be considered as the projection of the magnetic tubes of the active region on the plane perpendicular to the line of sight. The matter flows along the magnetic field lines.

I. INTRODUCTION While the magnetic field in the solar atmosphere exerts a dominating and controlling influence on the occurrence and development of various solar physical phenomena, the coronal atmosphere, being a dilute and fully ionised gas, is particularly sensitive to the magnetic field, hence the effect of the coronal magnetic field is especially important.

Small scale,

non-homogeneous coronal structures, such as the streamers, condensations, caves, polar plumes and helmets are mostly the results of the magnetic field distribution.

At present, the study

of the coronal magnetic field is not only a comprehensive and unified investigation including the photospheric and chromospheric magnetic fields, but more importantly, it aims at an essential understanding of the production of an entire series of events in the solar atmosphere. The coronal field is unlike the photospheric fields in that, except in the prominences where one can observe directly the Zeeman splitting, there is very little actually obseved data.

We rely at present almost entirely on theoretical calculations.

As is well known, the

coronal magnetic field is closely related to the coronal matter density.

Hence, nearly all

the methods of study at present utillse this mutual relation for verification and improvement. They can be roughly divided into the following two types: I. To make a comparison between the theoretically calculated field and the observationally known profiles of the corona.

Here one does not take into account the electric current in the

corona caused by the solar wind or transient coronal disturbances and one makes the calculatlon of a potential field model under the assumption of no current.

This method was

Coronal Magnetic F i e l d s

initiated

by Newkirk e# u Z . [ 1 ] ,

calculated field

structures

giving satisfactory

and was u s e d i n many s u b s e q u e n t works f o r comparing t h e

with the observed small scale structures

in the coronal density,

results.

2. Taking i n t o a c c o u n t t h e i n t e r a c t i o n using the f i e l d d i s t r i b u t i o n conditions,

239

between t h e s o l a r wind and t h e m a g n e t i c f i e l d ,

and t h e f l o w o f k i n e t i c

t o s o l v e t h e m a g n e t o h y d r o d y n a m i c a l and t h e s o l a r wind e q u a t i o n s f o r t h e

distributions

of matter,

f i e l d and v e l o c i t y i n t h e u p p e r c o r o n a and t h e i n t e r p l a n e t a r y

Pneuman and Kopp [2] worked a l o n g t h i s l i n e o v e r t h e y e a r s 1968-70. problem i s

medium.

But t h e t h e o r e t i c a l

r a t h e r complex.

In r e g a r d Co s m a l l s c a l e s t r u c t u r e potential

and

e n e r g y a t t h e s u r f a c e as boundary

f i e l d model t o c a l c u l a t e

compared w i t h t h e i r p r o f i l e s out a t h e o r e t i c a l

in the coronal field,

and o b t a i n e d s a t i s f a c t o r y

calculation of the magnetic field

and t h e r e s u l t was e q u a l l y s a t i s f a c t o r y . magnetic field distribution to s a t i s f a c t o r y calculations

results.

and d e t a i l e d

These r e s u l t s

results.

i n 1966 used t h e

i n s o l a r p r o m i n e n c e s , and Harvey [4] i n 1969 c a r r i e d

i n t h e c o r o n a l e m i s s i o n r e g i o n (X 5303)

Comparison between s i m i l a r l y c a l c u l a t e d c o r o n a l structures

such as s t r e a m e r s , a r c s , plumes, a l l

f i e l d model and t h e r e s u l t s

the magnetic l i n e s in the p o t e n t i a l

have shown t h a t i n t h e c o r o n a , t h e field.

show t h a t , ~ h i l e f o r t h e c o r o n a l r e g i o n g r e a t e r t h a n 2.5 Re (R e i s t h e r a d i u s

of the San), the p o t e n t i a l

field

approximation is obviously unrealistic

d i s t u r b a n c e s by t h e s o l a r wind on t h e m a g n e t i c f i e l d , the d e t a i l e d s t r u c t u r e

because of the

t h e model i s a good a p p r o x i m a t i o n f o r

i n an a c t i v e r e g i o n i n t h e i n n e r c o r o n a a t d i s t a n c e s l e s s t h a n 2.5 Re.

R e c e n t l y , L e v i n e and A l t s c h u l e r [6] i n e s t i m a t i n g t h e c h a n g e s i n t h e p o t e n t i a l b r o u g h t a b o u t by an e l e c t r i c irrotational

led

Even f o r s u r g e s , Rust and Roy [5] have more t h a n once made

with the potential

surges follow strictly

Rust [3] f i r s t

the magnetic field distribution

f i e l d diagram

c u r r e n t i n t h e c o r o n a , h y p o t h e s i s e d two t y p e s o f c u r r e n t s ,

current distribution

and B) a f o r c e - f r e e

field

distribution,

A) an

and t h e i r r e s u l t s

showed t h a t i t

i s o n l y when t h e r e i s a r a t h e r s t r o n g c u r r e n t i n t h e c o r o n a t h a t t h e c o r o n a l

magnetic field

shows an o b v i o u s d e v i a t i o n from t h e p o t e n t i a l

agreement between the p o t e n t i a l possible existence of electric

field

c o n t o u r and t h e d e n s i t y s t r u c t u r e

does n o t e x c l u d e t h e

of mechanical energy or large gradients in temperature

Harvey had a l r e a d y p o i n t e d o u t t h a t i t

i s only during the r a p i d l y changing phase

o f an a c t i v e r e g i o n t h a t s t r o n g c u r r e n t s c a u s e t h e c o r o n a l m a g n e t i c f i e l d o b v i o u s l y from t h e p o t e n t i a l During t h e t o t a l

d i m e n s i o n a l form? theoretlcal

to deviate

field.

e c l i p s e o f 1968 September 22, t h e r e was on t h e e a s t limb a c o n d e n s a t i o n

which was i n t h e shape o f a r e g u l a r b u n d l e w i t h v e r y c l e a r c o n t o u r s . physical factors

Therefore, the

c u r r e n t s i n t h e lower c o r o n a l e s s t h a n 2 . 5 Re and t h e

c h r o m o s p h e r e p r o d u c e d by d i s s i p a t i o n and d e n s i t y .

field picture.

J u s t what a r e t h e

t h a t b i n d t h e c o r o n a l m a t t e r i n t o such a r e g u l a r shape? To c l a r i f y

calculatlon

What i s i t s

three-

t h e s e q u e s t i o n s , we c a r r i e d o u t t h e f o l l o w l n g t e n t a t i v e ,

of its magnetic field

structure.

2. CALCULATION OF THE MAONETIC FIELD IN THE CORONAL CONDENSATION AND RESULTS The m a g n e t i c f i e l d

intensity

H in a current free field

V x H,=, O,

v.H--O.

satisfies

the following equations:

(I)

240

Coronal Magnetic F i e l d s

Thatis, H =

-V~, ¥

is the potentialfunctionwhichsatisfiest h e Laplaceequation V~

T h e r e f o r e , we a r e e s s e n t i a l l y magnetic field intensity

0

=

s o l v i n g a Laplace equation with the observed photospheric

as boundary c o n d i t i o n s .

Ne t a k e t h e a v e r a g e r a d i a l component o f t h e f i e l d

o v e r e a c h e l e m e n t o f a r e a on t h e p h o t o s p h e r e t o d e t e r m i n e t h e Legendre c o e f f i c i e n t s ,

from which we t h e n f i n d t h e magnitude and d i r e c t i o n o f t h e f i e l d a t any p o i n t i n t h e s p a c e above t h e p h o t o s p h e r e : t h i s i s c a l l e d a s p h e r i c a l s o l u t i o n o f t h e p o t e n t i a l

fleld

I f the h e i g h t of the r e g i o n considered i s not g r e a t e r than the c h a r a c t e r i s t i c

[I]. s i z e of the

r e g i o n , t h e n we can use an a p p r o x l m a t e s o l u t i o n p r o p o s e d by Schmldt [7] i n 1964 and c a l l e d t h e plane surface solution of the potential

field.

I t i s as f o l l o w s .

When t h e s a i d c o n d i t i o n h o l d s , we can r e g a r d t h e p h o t o s p h e r i c boundary as n e a r l y a p l a n e surface.

Then f o r each e l e m e n t a l a r e a t h e p e r p e n d i c u l a r component o f t h e p h o t o s p h e r i c f i e l d

can b e by a m e g n e t l c " s o u r c e " and t h e f i e l d be made up o f t h e c o n t r i b u t i o n s

from a l l

s t r e n g t h a t any p o i n t i n t h e s p a c e above w i l l t h e n

these photospheric sources (Fig. l).

completely e q u i v a l e n t to s o l v i n g the Laplace equation with the v e r t i c a l

This i s

component o f t h e f i e l d

as boundary c o n d i t i o n s .

i fB~(z,1~p~ov,,.,"

/ ~photosphere

Y Fig. I Let t h e s e i d e a l s o u r c e s on t h e p h o t o s p h e r i c l a y e r have a d e n s i t y o t , and a f i e l d i n t e n s i t y H'

and l e t n b e t h e u n i t normal v e c t o r t o t h e l a y e r .

point

Then t h e p o t e n t i a l

f u n c t i o n T(r) a t any

P above t h e l a y e r i s g i v e n by

"*>

I °'

•

P

I

.

where p i s t h e d i s t a n c e o f P from t h e a r e a e l e m e n t .

Ir

°

r'[

da°

Also we have t h e f o l l o w i n g r e l a t i o n

b e t w e e n o ' and H ' :

Hence

2,~,' =, H'..

~P'(') -

± I I, -H:,,'1

2,,

Js'

(2)

Coronal ~gnetic

a'

d e n o t i n g t h e s i z e and r '

distribution

the coordinate of the area element.

i n s p a c e c a n be f o u n d f r o m £ h e f o l l o w l n g

H(r) and t h e d i s t r i b u t i o n

of magnetic lines ,ix

The s u n s p o t d i a g r a m ( F i g . 2 ) observational

Finally,

the field

expression:

-- V~F(r).

- -

(3)

from

.=

dy

nx(x, y, z)

vlsual

241

Fields

d=

..=

H,(., y, z)

n.(x, y, z)

(4)

c o r r e s p o n d i n g t o t h e p r e s e n t c o n d e n s a t i o n i s b a s e d on t h e

d a t a o b t a i n e d a t t h e P u r p l e I~ountaln O b s e r v a t o r y a t t h e t i m e o f

meridian passage of the active

r e g i o n on S e p t e m b e r 29.

The f i e l d

intensities

a r e t h e mean

v a l u e s g i v e n i n " M a g n l t n y e P o l y a S o l n e c h n y x P y a t e n " and " S o l a r Phenomena ~ n t h l y p u b l l s h e d by t h e Rome O b s e r v a t o r y , I t a l y .

In selecting

t h e d a t a , we t r i e d

Bulletin"

to use only those

t h a t a r e w i t h i n 45 ° o f t h e c e n t r a l

meridian llne of the solar disk, while taking into

consideration

In g e n e r a l ,

their

when d a t a on t h a t

completeness.

we t o o k t h e d a t a o f S e p t e m b e r 29, i t

day i s l a c k l n g t h a t we went on t o 28 o r e v e n 26.

is only

The whole p h o t o s p h e r i c

r e g i o n s p a n n e d by t h e s u n s p o t g r o u p was d i v i d e d i n t o 25 × 30 = 750 e l e m e n t s o f s i z e I ° x I ° . Relatlve

to the magnetic f i e l d

of spots,

the field

o f t h e p h o t o s p h e r e i s v e r y s m a l l and can

be a p p r o x i m a t e d t o z e r o . 7

No.388

Z0 9% ,%'0.'~g4

~o.387

./20 -N N ILS • •

)

" • ~

~

mo m

°

15N

I

10NI

235 175

! --20 ~

,,

tl0

I --10 °

Fig. 2 Disposition In our c a l c u l a t l o n s , shown i n F i g . 2 , the solar

No.390

I 10°

0

o f S u n s p o t g r o u p s and t h e i r

magnetic

we t o o k a s r e f e r e n c e s y s t e m t h e r e c t a n g u l a r

with the z axis perpendicular

|r

20' field

coordinate system as

to the plane of the paper.

limb c o r r e s p o n d e d t o a C a r r i n g t o n l o n g i t u d e o f 200~36 and i t s

e q u a t o r was t a k e n a s t h e c o o r d i n a t e o r i g i n .

strengths.

The m e r l d i a n on

intersection

with the

The s c a l e o f t h e z - a x i s was i n u n i t s o f

0.01745 Re . U s i n g (2) and ( 3 ) , we c a l c u l a t e d

the field

strengths

at the points

c o r r e s p o n d i n g to

Coronal Magnetic F i e l d s

242

x = -12

-

+4, y

0

-

30, z =

-

s e p a r a t e diagrams f o r d i f f e r e n t F i g . 3.

From F i g . 3,

15.

1-

For c l a r i t y ,

t h e i s o g a u s s f i e l d l i n e s were drawn on

x - v a l u e s , a s e l e c t i o n o f t h e s e diagrams a r e d i s p l a y e d i n

we can s e e d i r e c t l y

the fleld

distribution

at various sections along the

HI Jr=~ --2

.

.

.

.

.

.

.

J# Z m --2,

4

4

/

4

W

rw--10

AO0

.200

r

iS'

~P

--

ly

Fig. 3

Distribution

o f i s o g a u s s c u r v e s on d i f f e r e n t

sections along

the l i n e of s i g h t . line of sight.

The s e r i e s o f l s o g a u s s d l a g r a ~

r e g i o n i n which t h e f i e l d

is nearly zero.

show t h a t around x = - 7 and - 8 . t h e r e e x i s t s

This r e g i o n c o r r e s p o n d s t o t h e boundary b e t w e e n

a

C o r o n a l Mmgnetic F i e l d s

c l o s e d and open m a g n e t i c s t r u c t u r e s ,

its

243

h e i g h t i s b e t w e e n 0 . 0 5 2 Re and 0 . 1 4 Re

(36000-100000 km). The d i s t r i b u t i o n

of magnetic llnes

in the three-dlmensional

space as calculated

drawn i n F i g . 4.

Here we c a n o n l y draw t h e c l o s e d m a g n e t i c l l n e s o f t h i s

some o f t h e f i e l d

llnes

terrestrial top.

t h a t j o i n on t o o t h e r a c t i v e

projections

we a l s o f o u n d t h e o u t e r m o s t f i e l d

on t h e p l a n e p e r p e n d i c u l a r

to the line of sight

field

model i s c o r r e c t ,

then the last

a s an open s t r u c t u r e

on t h e

lines of the closed bundle, their a r e shown i n F i g . 5, where t h e

d o t - d a s h c u r v e marks t h e maximum e n v e l o p e o f t h e c l o s e d f i e l d If the potential

f r o m (4) i s

region, while

regions or extend into the solar-

s p a c e c a n o n l y be r o u g h l y s k e t c h e d i n o u r c a l c u l a t i o n

In our c a l c u l a t i o n ,

active

lines

a s s e e n by t h e o b s e r v e r .

should coincide with the obseved

density envelope.

0 'J

:

"~'

....i,;

/ fl-..,T#NY/

L[/[I//J/,o

,,,''.

5

0

,ix do, @

(-1 Fig. 4

Distribution of magnetic lines in space.

3. DISCUSSIONS

The greatest shortcoming of a calculation based on the potential field model is the fact that it does not show the field distribution at the time.

This is because, whether in the

spherical solution of the Laplace equation, or in the Schmidt's plane surface solution, we need the vertical component of the photospheric field as boundary conditions.

But the

actually observed quantity in a solar magnetic observation is the component along the line of sight, in other words, it is only around the central meridian of the solar disk that what we observe is the vertical component.

Therefore, in the case of the spherical solution, which

requires the vertical components over the whole surface, we need to have observations over an

244

Coronal Magnetic Fields

entire rotation period.

This shows that we must assume that the entire photospheric field

does not change throughout a whole rotational period, and this is clearly impossible. Similarly, in our calculation, in order to have the form and field of the active region corresponding to the observed coronal condensation, we must use the data obtained 7 days later when the active region crosses the central meridian. can only be an approximation.

Even for a steady active region, this

However, as pointed out by Altschuler and Newkirk [8], if we

exclude rapid or eruptive types of coronal disturbances, a long-period coronal disturbance is generally evolutionary in character, and it can last several days or longer.

If an electric

current appears for a short time,then after its disappearance the state of the potential field will be restored.

Even after the occurrence of flares with the rapid release of magnetic

energy stored in twisted magnetic lines of the lower corona and chromosphere, a current-free state will immediately re-establish itself.

Therefore, so long as the active region is not in

the stage of rapid increase or decrease, the approximation is still suitable. If the potential field model is correct and the changes in the morphology and field of the active region are not large, then the optical density contours of the present coronal condensation should be the same as the calculated closed bundle of magnetic lines projected on the plane perpendicular to the line of sight.

Transforming the coordinate system of Fig. 5 in

which the photosphere is a plane to the solar disk system of Fig. 6,

the corresponding

closed bundle of field lines are marked with black dots on the latter.

~ No.390

388

No389/i"

15'o.39o

I

1968.9.23

If we divide the

a88

/

No. 387

1968.9.24

1968.9.25

No. 390

No.389

No. 390

No.387

No. 387 t,

*

No. 389 1968.9.26 Fig. 7

1968.9.27 E v o l u t i o n of s u n s p o t morphology of the a c t i v e region, September 23 - 27.

Coronal Magnetlc Fields

245

i

f0

'

Fig. 5

I

l

•

,

I

The outermost ~iosed magnetxc ixnes pro3ected on the plane normal to the llght of slght. The dotdash curve is the maximum envelope of the bundle of closed magnetic lines.

Fig. 6

Coronal Mmgnetlc Fields

247

optical contour of thls condensation on the east limb into three parts I, II and III (as shown), then it is clear that Part II agrees well with the projection of the calculated maximum envelope. To clarlfy further, we give in Fig. 7 the sunspot diagram of thls active region for the days September 23 - 27.

These dlagrams show clearly that Sunspot Group 388 decllned and

vanished on 23 - 24 so that no magnetic data were obtained before its dlsappearance.

From its

N polarlty glven In "Solar Phenomena Monthly Bulletin" of the Rome Observatory, and its positlon relative to the envelope of III, this group on September 21 or 22 must have been forming the boundary condltions of the field enveloping III.

That is, on 22, a large part of

the fleld llnes from Groups 390 and 387 joined on to Group 388 and formed the envelope of III, but this would have dlsappeared after September 25. Next, Fig.7 shows also that, on 23, there was another spot south of Group 387, which rapldly vanished at the tlme of the total eclipse.

This could also have reacted on the

agreement between the theoretical envelope and part I. In summary, conslderlng the above mentioned shortcoming in the theoretical calculation based on the potential fleld model, there is a general agreement between the theoretical envelope and the density contour of the condensation.

Thls shows that the active regzon was

in a stage of steady development and that the magnetic field that extended into the corona was baslcally a potential fleld.

Motion of matter was along the fleld lines shown in Fig. 4, and

partlcles were trapped in the closed bundle of lines to form

materlal concentrations.

Because in the present active reglon, spots of opposlte polarities arranged themselves parallel to the solar equator, the observed density contours are the projections of the closed bundle on the plane perpendlcular to the line'of sight. I wish to thank Comrade Wang Chang-bin for wrlting the computer programs used in thls paper.

REFERENCES [I] [2] [3] [4] [5] [6] [7] [8]

Newkirk, Gordon, Jr. et al., Solar Phys., 3 (1968), 321. Pneuman, G.W., Solar Phys., 3 (1968), 578. Rust, D., Ph.D., Thesis Department of Astro-Geophysics, University of Colorado. (1968). Harvey, J.W., Ph.D. Thesis Department of Astro-C~ophysies, University of Colorado. (1969). Roy, J.R., Solar Phys.,28 (1973), 95. Levine, R.H. & Altschuler, M.P., Solar Phys., 36 (1974), 345. Sehmidt, H.U., NASA Symp. On Physics of Solar Flares, (1964), 107. Altschuler, M.P., In G.J. Newklrk (ed) "Coronal Disturbances" I.A.U. Symp.,57 (1974), 35.