on heating and cooling in some active region loops

O N H E A T I N G A N D C O O L I N G I N S O M E A C T I V E R E G I O N

L O O P S

UDIT NARAIN and MUKUL KUMAR

Astrophysics Research Group, Meerut College, Meerut-250 001, India

(Received 15 February, 1985)

Abstract. It is shown that the neglect of radiative losses by Antiochos and Sturrock (1976) in investigating conduction cooling is not justified. It is further shown that the anomalous current dissipation leads to substantial amount of heating contrary to remarks made by Ionson (1982).

1. Introduction

Antiochos and Sturrock (1976) have solved the energy equation in coronal structures

by taking into account conduction and ignoring radiation. Here we use their approach for obtaining conductive losses in three coronal features. To know whether radiative

losses are important or not we calculate these losses, too. Assuming anomalous current dissipation to be responsible for the energy input to the loop, we find a substantial

amount of heating.

2. Theoretical Formulation

Active regions consist of loops whose footpoints have smaller cross sections than their tops. The arc length s of such a loop in line dipole geometry is given by (Antiochos and Sturrock, 1976)

s = h O , (1)

where h is the vertical height of the loop from the centre of the line dipole (situated in

the photosphere), and 0is the angle which a line joining a point on the loop to the centre of dipole makes with the vertical. Let the cross section at the top (s = 0) be A o, so that the cross section A ( O ) at any other point of the loop is given by

A ( O ) = A o cos 2 0. (2)

If the cross-sectional radius of the loop at the top be 'a ' then A o = rca 2.

Let 0 b be the angle corresponding to the base of the loop (Antiochos and Sturrock, 1976). Then the total length L of the loop, in view of Equation (1), is given by

L / 2 = s b = hO b . (3)

The vertical height of the base above the line dipole may be obtained from

h b = h cos 2 0 b . (4)

Solar Physics 99 (1985) 111-118. 0038-0938/85.15. �9 1985 by D. Reidel Publishing Company

112 UDIT NARAIN AND MUKUL KUMAR

In view of Equations (1)-(4), the total volume of the loop may be given as

vL = 0.5 AoL(O + sin20 /2). (5)

In line dipole geometry the total energy loss rate parallel to field lines via conduction per unit volume, as a function of 0 and time t, is given by (Antiochos and Sturrock, 1976)

S(O, t) = 2 A(O)F(O, t)/V(O), (6)

where

with

V(O) = Aoh(O + 0.5 sin20)

F( O, t) = Fo(t ) (cos Ob/ O b sin Ob)( O + sin 0cos 0)/2 cos 2 0

Fo(t) = (4/7) • 1 0 - 6 h - l Z 3 5 ( t ) ,

and

To(t ) = Too(1 + t/tc) - ~

The quantity tc in Equation (9) is a constant given by

t C = 1.05 • 106poT3o65h20btanOb,

where Po is the pressure at t = 0.

(7)

(8)

(9)

(10)

Conduction across the field lines may be ignored (Kumar and Narain, 1985: Rosner et aL, 1978).

The temperature T and electron density 1% in this model are given by

T = To(t ) [ 1 - (cos 0b/0 b sin Ob)(O sin 0/cos 0)] 2/7 , (11)

and

where

fie = 3.5G(0)- 2/7/2k, (12)

G(O) = (2/7)(2kno) - 3 5 - 3 x 105h2Po2"St~-lOtanO. (13)

In order to know whether radiative losses are important or not, this energy loss rate per unit volume may be obtained from

UR = P(T )n 2 , (14)

where the quantity P ( T ) has been evaluated by Cox and Tucker (1969), Tucker and Koren (1971), and Kato (1976).

Following Rosner et aL (1978), we assume that the heating due to coronal current dissipation is confined to a sheath of thickness Aa given by

Aa = 1.6 x 1035 rlane2Th - l(Ab)2, (15)

ON HEATING AND COOLING IN SOME ACTIVE REGION LOOPS 113

where t/a is the anomalous resistivity given as

t/a = 2 x 10-8ne ~ s. (16)

The quantity Ab is the total change in the non-potential component of the magnetic field and may be evaluated from

Ab = 1.0 x 10- 13an~gT- 1.5 (17)

Now the heat-generation rate per unit volume may be obtained from

Utr E . J = 1.1 • 10-I8ne~'ST, (18)

in which E, the induced electric field, and J, the current density, are:

E = tl, J , J = cAb/47rAa. (19)

Here c is the velocity of light in vacuum.

Since the magnetic flux along the loop is conserved (Parker, 1975), there is

B(O, t)A(O) : 8 (0 , t )A(0 ) .

Assuming time variation of magnetic field to be the same as that of the square root of the pressure, we find

B(O, t) = Bo(1 + t/tc) ~ (20)

where Bo[ = (16rcnokToo) 1/2 ] is the magnetic field at the top of the loop at the start of cooling phase. This expression enables us to find the induced voltage produced by the time variation of the magnetic flux. The corresponding average induced electric field is

E = 3 • 102Vo(1 + t / t~)-12/(h/2) V cm -1 , (21)

where

V o = 47ra2(rcnokToo)~

In order to have an idea about turbulence, it is worthwhile to evaluate Dreicer field

E D ~ 3 . S x lO-8ne T - 1 Vcm -1 (22)

3. Data, Method, and Results

Table I exhibits the input data, taken from Parkinson (1973). In order to know the total length of the loop through Equation (3), one requires 0 b which may be obtained from (Krieger, 1978)

sec2 0b = 7.6 X 102(n9T6) - 0.5, (23)

where n 9 and T 6 are electron density and temperature in units of 10 9 c m - 3 and 10 6 K,

respectively. Now Table I, together with Equation (5), enables us to obtain the radius 'a' of the loop at its top.

114 UDIT NARAIN AND MUKUL KUMAR

TABLE I Observational data a

Feature h no Too V L Bmi n (cm) (em- 3) (K) (cm 3) (gauss)

Loop I 2.0 (9) b 1.0 (10) 5.5 (6) 1.0 (27) 16.6

Loop II 5.0 (9) 4.3 (9) 3.5 (6) 1.0 (29) 8.8

Loop II! 1.0 (10) 2.3 (9) 2.5 (6) 1.0 (30) 6.5

a Taken from Parkinson (1973). b Number in the parenthesis represents the power of ten, e.g. 2.0 (9) = 2.0 x 10 9.

TABLE II Generated data

Feature 0 b L a hb t c Po Bo (radian) (em) (cm) (cm) (sec) (dyne cm - 2) (gauss)

Loop I 1.47 5.9 (9) a 2.6 (8) 2.0 (7) 2.4 (3) 1.5 (1) 19.5

Loop II 1.50 1.5 (10) 1.6 (9) 2.6 (7) 2.8 (4) 4.2 10.2

Loop III 1.52 3.0 (10) 3.7 (9) 3.2 (7) 1.8 (5) 1.6 6.3

a Number in the parenthesis represents the power of ten.

Table II exhibits some of the generated da ta for a ready reference. Energy-loss rate per unit volume via conduction has been evaluated through

Equations (6)-(10) and Table I. The variable t is allowed to take values 0, to~4, to~2, and

3 tS4 whereas 0 assumes values 0, 0.4, 0.8, and 1.2 radian. For obtaining UR (Equation (14)) we require P ( T ) , which has been taken f rom Tucker

and Koren (1971), and n e which has been evaluated using Equations (12), (13) and

Table I. Only aforesaid values of 0 have been considered. The heat-generation rate per unit volume has been obtained through Equations

(11)-(13), (15)-(19), and Table I. In Figure 1, Ab and Aa are exhibited as a function of

0. Figure 2 exhibits Dreicer field alongwith the induced electric field E (cf. Equations (21) and (22)) as a function of 0 and t. These data enable us to evaluate rate of heat

generation per unit volume. The energy loss rate due to conduction and radiation per unit volume alongwith the heat generation rate per unit volume are exhibited in Table III .

4. Discussion and Conclusions

The electron density, temperature and flux distribution calculated using Equations (6)-(13) do not seem to be accurate beyond 0 = 1.2 in view of assumptions neglecting radiative loss and exchange of p lasma with chromosphere. Consequently our investi-

gations are not extended beyond 0 = 1.2. Figure 1 shows the variation of non-potential component of magnetic field Ab and


16 I 4 .0 I I

Fig. 1.

r-.

$

v

.0

12

8

- • Loop Ill ...................................

- 3.0 r, E O

..................................

Loop 'I'I~ ~ v

- 2 . 0 <3

.9o

Loop I~

I I I 0 0 0.4 0-8 1.2

e (rad)

I.O

Non-potential component of the magnetic field (solid lines) and sheath thickness (dashed lines) as a function of angular distance along loops I, II, and III.

I I I 0 I I ',\

" Loop Z \ Loop I ~ 3 0

O > 0 ~ ~, -~

Ln m > '~ ~.. 20 > I O [ 0 ~ ~ ~'-

_ -~,. ~ ._ "'.- 6 LOOp rl' . . . . - ~ '- 6 Loop ~ .~ .~....~. c

2 2 ,,, - IO

Loop Ill / Loop~

4 - 4 ~

I I I I O 0 0 . 4 0. 8 1.2 0 r-c/4 to /2 3to ~1.

e ( rod ) t ( $ec )

( a ) ( b ' ) Fig. 2. (a) Dreicer field as a function of angular distance along loops I, II, and III at t = 4/2. (b) Calculated electric fields as a function of time for loops I, II, and III at 0 = 0.8. Solid curves refer to Dreicer fields and

dashed curves to induced electric fields.

TA

BL

E

III

He

ati

ng

a

nd

c

oo

lin

g r

ate

s (i

n e

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m-

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Fe

atu

re

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1 ~

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1 2

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2 3

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op

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.00

0

.00

0

.00

1

.99

3

t.9

9 -

3

1.9

9 -

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5.6

03

5

.20

3

b 2

.78

"- 3

b 2

.17

- 3

1

.84

- 3

2.0

1 -

3

2.0

1 -

3

2.0

1 -

3

5.6

03

5

.20

3

c 2

.82

- 3

2

.19

- 3

1.7

6-

3 2

.08

- 3

2.0

8 -

3

2.0

8 -

3

5.6

03

5

.20

3

d 2

.79

- 3

2

.17

~ 3

1

.75

- 3

2

.29

- 3

2

.29

- 3

2

.29

- 3

5

,60

3

5.2

03

Lo

op

II

a 0

.00

0

.00

0

.00

3

.70

- 4

3

.70

- 4

4

.60

- 4

1

.00

3

9.4

02

b 6

.42

- 5

5

.02

- s

4

.03

- 5

3

.70

- 4

3

.70

- 4

4

.60

- 4

1

.00

3

9.4

0 z

c 6

.49

- 5

5

.02

- 5

4

.08

- s

3

.78

- 4

3

.78

- 4

4

.70

- 4

1

.00

3

9.3

0 z

d 6

.42

- 5

4

.98

- 5

4

.04

- 5

4

.05

4

4.0

5 -

4

5.0

6-

4 1.

003

9.4

02

4.9

03

4.9

03

4.9

03

4.9

03

8.8

02

8.8

02

8.8

02

8.8

02

Lo

op

III

a

0.0

0

0,0

0

0.0

0

1.5

9-4

1

.59

-4

2.1

2-4

2

.80

2

2.6

0 z

2.5

02

b 3

.83

- 6

2.9

7--

6

2.3

9--

6

1.6

0--

4

1.6

0--

4

2.1

3--

4

2.8

02

2

.60

2

2,5

02

C

3.8

7 -

6

3.0

0 -

6

2.4

0 -

6

1.61

4

1,61

--4

2

.15

--4

2

.80

2

2.6

0 z

2.4

52

d 3

.84

- 6

2

.96

- 6

2

.37

- 6

1

.70

- 4

1

.70

- 4

2

.27

- 4

2

.80

2

2.6

02

2

.45

2

:z

~J

" 1,

2,

an

d

3 st

an

d

for

t =

te

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an

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4, r

esp

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.

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rip

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by

wh

ich

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to

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ult

ipli

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.

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c,

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d

d st

an

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for

0 =

0

.0,

0.4

, 0

.8,

an

d

1.2

, re

spe

cti

ve

ly.


sheath thickness Aa with 0. The quantity Ab increases with 0. Equation (17) shows that this variation is due to faster decrease in temperature, and the slower increase in density with increased 0. Further, the temperature descreases with time also whereas the density remains constant. The increase of Ab with 0 seems justified in the sense that if we assume that the coronal current rsponsible for the non-potential component of the magnetic field throughout the loop is the same then the field towards the base would be larger than that at the top because of difference in the area of cross section at various points of the loop.

The sheath thickness, Aa, has a tendency to increase with 0. This can be explained through Equations (15)-(17) which show that Aa o 5 -2.5 F/e" T . Since /7 e increases slightly and T decreases appreciably with 0 therefore Aa should increase slightly with 0. This seems physical for the heating depends directly on current density which in turn varies inversely with sheath thickness consequently the heating should be more near the top than it is away from the top of the loop.

The answer to the question whether we are justified in using turbulent resistivity, is provided by Figure 2. Throughout the loop the Dreicer field is of order 10 - 5 V cm - 1 whereas the average induced electric field strengths are of order 10 4 g cm- 1. There- fore the currents induced by these electric fields will be strongly turbulent (Rosner et al.,

1978). Equation (20) determines magnetic field as a function of 0 and t. Our assumption,

of taking time variation of magnetic field equal to the square root of that of pressure, boils down to taking gas pressure and magnetic pressure as equal. Even in this case we get substantial amount of anomalous heating (cf. Table III). If we do not assume equipartition with pressure (i.e. increase the magnetic field) the amount of heating will increase further. This is in contrast to the point made by Ionson (1982).

It is obvious from Table III that conductive loss rates per unit volume decrease with time. This is as expected. The losses are directly proportional to temperature gradients which decrease with time. These losses are zero at the top (0 = 0), increase, become maximum and then decrease. This is due to the combined effect of Temperature gradients and the decrease in area of cross-sectoin (funneling effect) of the loop.

The radiative loss rates calculated on the basis of electron densities and temperatures of Antiochos and Sturrock (1976) are either approximately equal or greater by an order or two than the conductive model, it is not valid to claim accuracy for these losses but they do show the trend that radiative losses are not negligible for the loops under consideration.

It may therefore be concluded that the neglect of radiative losses by Antiochos and Sturrock is not justified, in general. Further the anomalous heating is several orders of magnitude higher than the cooling by conduction or (and) radiation.

Acknowledgements

We wish to thank Profs. R. M. Misra, S. M. R. Ansari, and Dr. P. K. Raju for their interest in the present investigation. One of us (M.K.) gratefully acknowledges the

118 uDrr NARAIN AND MUKUL KUMAR

financial assis tance rendered by Depa r tmen t of Atomic Energy in the form of a Junior

Research Fel lowship during the course of these investigations. Some facilities provided

by Meerut College, Meerut are also acknowledged.

References

Antiochos, S. K. and Sturrock, P. A.: 1976, Solar Phys. 49, 359. Cox, D.P. and Tucker, W. H.: 1969, Astrophys. J. 157, 1157. Ionson, J. A.: 1982, Astrophys. J. 254, 318. Kato, T.: 1976, Astrophys. J. Suppl. 30, 397. Krieger, A. S.: 1978, Solar Phys. 56, 107. Kumar, M. and Narain, U.: 1985, Solar Phys. 95, 69. Parker, E. N.: 1975, Astrophys. J. 201,494. Parkinson, J. H.: 1973, Solar Phys. 28, 487. Rosner, R., Golub, L., Coppi, B., and Vaiana, G. S.: 1978, Astrophys. J. 222, 317. Tucker, W. H. and Koren, M.: 1971, Astrophys. J. 168, 283.

on heating and cooling in some active region loops

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