Energy equilibrium in coronal magnetic loops -reinvestigated

ENERGY EQUILIBRIUM IN CORONAL MAGNETIC LOOPS -

REINVESTIGATED

SURESH CHANDRA and LALAN PRASAD Department of Physics, University of Gorakhpur,

Gorakhpur 273 009, India

(Received 16 March, 1994; accepted 12 May, 1994)

Abstract. Temperature distribution in cylindrically symmetric coronal magnetic loops has been reinvestigated under various conditions: (a) loop with the pressure varying along the radial distance, and (b) loop with constant pressure, for cooler apex loops and hotter apex loops. This work is reinvestigation of our previous work published in Astrophysics and Space Science (Chandra and Prasad, 1993b).

1. Introduction

Temperature distribution in coronal magnetic loops has been discussed by a number of authors (e.g., Chandra, 1985, 1987; Chandra and Prasad, 1993b; Craig et al., 1978; Galeev et al., 1981; Hood and Priest 1980; Levine and Withbroe, 1977; Roberts and Frankenthal, 1980; Rosner et al., 1978; Serio et al., 1981; Wragg and Priest, 1981; and references cited therein). A common approach followed by most of them is the consideration of the equilibrium between various processes for the transfer of energy in a loop.

In the present investigation, the temperature distribution in the coronal magnetic loop: (i) with pressure varying with radial distance, and (ii) with constant pressure, has been discussed for (a) cooler and (b) hotter apex loops, by considering the equilibrium between heat conduction, radiation loss and external source of heating. With including the variation of the magnetic field along the loop, the temperature distribution has been discussed by considering the geometries of the loop proposed by Antiochos and Sturrock (1976).

This reinvestigation of our previous work (Chandra and Prasad, 1993b) also shows that no loops with equal apex and base temperatures can exist, but a small temperature gradient supports the existence of the loop.

2. Energy Equilibrium between Heat Conduction and Radiation Loss in Coronal Magnetic Loops

Assuming radiation loss as the only process competing with conduction in a coronal magnetic loop, the energy equilibrium is given by

V . F c q - V . F r a d = 0 . (2ol)

Here, V • Frad is the radiation loss and can be expressed by

Astrophysics and Space Science 222: 9-38, 1994. © 1994 KluwerAcademic Publishers. Printed in Belgium.

10 SURESH CHANDRA AND LALAN PRASAD

V . Frad = nEf(T), (2.2)

where n is the electron density and the function f (T) is called radiation loss function. The radiation loss function f (T) for an optically thin plasma has been calculated by a number of authors (Pottasch, 1965; Cox and Tucker, 1969; Tucker and Koren, 1971; McWhirter et al., 1975). However, in the present investigation the results of Cox and Tucker (1969) have been fitted in the form

f (T) = X Ta, (2.3)

where X and a are the parameters having different values for different temperature ranges. In the presence of a strong magnetic field, the thermal conduction of heat in the directions at right angles to the lines of force is greatly suppressed and the heat conducts nearly along the field lines. Under such a condition, the thermal heat flux vector is given by (Kopp and Kuperus, 1968)

Fc(z) = -koT 5/20T B(z) Oz I n ( z ) l ' (2.4)

where z is the length measured along the field line and k0 is a constant ~ 10 -6 c.g.s, units. When using Equation (2.4), one should conveniently choose the origin (z = 0) for the length along the field line so that the temperature gradient is positive. Coronal loops of two varieties are discussed in the literature:

(i) Cooler apex coronal magnetic loops, where the temperature on the apex T,~ is smaller than that on the base Tb.

(ii) Hotter apex coronal magnetic loops, where the temperature on the apex Ta is larger than that on the base Tb.

To the best of our knowledge, loops with the cooler apex are not yet observed. But, since these have also been discussed by some authors (e.g., Roberts and Frankenthal, 1980), they have been taken into account for theoretical investigations. The stationary state equilibrium, in the absence of an external heating source and given by Equation (2.1) cannot be applicable for hotter apex loops, but it may be valid for the cooler apex loops. In the later sections, we shall account for the external source for heating, and will also discuss the equilibrium in hotter apex loops.

2.1. COLLAR APEX CORONAL MAGNETIC LOOPS

In a cooler apex loop, where the temperature on the apex Ta is smaller than that on the base (foot-point) Tb, it would be convenient to consider the origin (z = 0) of the magnetic flux length on the apex of the loop, so that the gradient of the temperature is positive. Now the foot-point is at z = L. We assume a cylindrically symmetric structure of the loop, so that the magnetic field is constant along the loop and the Equations (2.2) through (2.4) can be rearranged into the form

ko 0 (TSI20T~ = nExT '~. (2.5) k Oz ) o z

ENERGY EQUILIBRIUM IN CORONAL MAGNETIC LOOPS - REINVESTIGATED 1 1

Equation (2.5) is equally valid if the physical parameters, such as temperature T, electron density n, and pressure p, are assumed to be functions of z only or of both z and r, (r is the radial distance from the axis of the loop). In the present investigation, the pressure along the flux line will be taken as constant, i.e., independent of z. Then the study in this section can be further divided into the following cases:

(a) loop with the pressure varying along the radial distance r, and (b) loop with constant pressure.

2.1.1. Cooler Apex Loops with Non-Constant Pressure For the loop of non-constant pressure (with the magnetohydrostatic equilibrium j x B = Vp(r)) , the magnetic lines of force are helical in shape and Equation (2.5) is the basic equation, where the temperature and electron density are functions of both z and r, while the pressure p is a function of r only. Now we assume that the physical parameters can be written as the products of their r and z dependent components,

T = T(r , z) = Tr(r )Tz(z ) ,

n = n(r, z) = nr(r )nz(Z) , (2.6)

p = p(r) = pr(r)pz(z) ;

hydrogen is in the fully ionized state so that

o r

p = 2 n k T

pr(r ) = k lnr ( r )Tr ( r ) , (2.7a)

Pz = k2nz(z )Tz(z ) , (2.7b)

where the constants kl and k2 are such that kl • k2 = 2k. By using Equation (2.6) in Equation (2.5), the equations depending on r and z variables, separately, can be obtained as

T7/2 2 a = C l n r T ~ , (2.8a)

0 (TSz/2OTz) 2 c, Oz Oz -= C2nzT~ , (2.8b)

where the constants C1 and C2 are such that C1C2 = x /ko . With the help of Equations (2.7a) and (2.8a) one can easily obtain the relations

Tr = ( P-~R)

and

n.._~r ~_~ (p....S._r "~ ( 7 - 2 a ) / ( l l - 2 a )

\pR] nR

(2.9)

(2.1o)


Here, the suffix R refers to the value of r at the outer cylindrical surface of the loop. The variation of pressure along the radial distance r in the magnetic loops has recently been discussed by Chiuderi et al., (1977). They represented Pr by the first three terms of the Fourier expansion (Chandra and Prasad, 1993a).

1Or = P R ~ I -- 192 C O S ( ' h ' I ' / R ) - - 193 cos(27rr/R)], r < R , (2.11)

where Pl, P2 and P3 are the parameters obtained on the basis of pressure structure taken from Harvard/Skylab EUV data (Foukal, 1975, 1976). The values of the parameters reported by Chiuderi et al. (1977) are

ModelI Pl =1-82 p2=0.35 P3= 1.17 ModellI P1=0.55 p2=0.45 P3=0.0

Model I represents a pressure maximum with (P/Pn)max = 3 at r / R ~ 0.52, consistent with preliminary EUV results of Foukal (1975), whereas Model II describes a pressure minimum at the origin, P0 = 0.1pn, as indicated by Foukal's (1976) data.

For the sake of simplicity, we introduce two new dimensionless variables

( Tz ~ 7/2 z (2.12) ' s = 2 ,

where L is the half length of the field line and Tzb is the z-dependent part of the temperature at the foot point. For constant pressure Pz, by using Equation (2.7b) in Equation (2.8b) we get

d2rz --2 :~-1 (2.13) d S 2 = A z T z ,

where

A 2 2 a - 3 . 5 2 = (3.5C2nzbT~b )L , (2.14a)

2 2). (2.14b) ,k = 1 + f f ( a -

The boundary conditions for Equation (2.13) are

at the foot point

S - - 1, ~'z = 1 (2.15a)

at the apex

drz S = 0, dS = 0. (2.15b)

ENERGY EQUILIBRIUM IN CORONAL MAGNETIC LOOPS- REINVESTIGATED 13

The first boundary condition is obtained on the basis of Equation (2.11), while the second boundary condition is obtained by considering the symmetrical variation of the physical parameters about the apex of the loop. Equation (2.13) is a second order differential equation and its integration gives

where the integration constant C can be obtained by considering the situation on the apex (S = O, drz /d , . .q = O, r z - - Tza ~-- (Tza/Tzb) 7/2) as C' = -2A2Tz~a/A and one can get

By using the transformation Tz = "rza cosh 2/A u, where u is a new variable, in Equation (2.17), we finally get

where C u is a constant of integration, and the integral on the left side of the Equation (2.18) can be solved analytically for the integer values of (2 - A)/A. If (2 - A)/A is an odd integer, say (2 - A)/A = 2m + 1, where m is an integer, then

where the function FUM1 is defined by

Obviously, FUMI(1) = 0. If (2 - A)/A is an even !nteger, say (2 - A)/A = 2/, where I is an integer, then

where the function FUM2 is defined by

14 SURESn CHANDRA AND LALAN PRASAD

Obviously, at T = T ' the value of FUM2(1) = 0. Finally, the integration constant C" is calculated by considering the condition on the apex (z = 0) and is found to be zero. Thus, the final relation between the z-dependent temperature and distance is obtained as

/ - - - - . - -

FOlVl(Tz/Tza) = Az V,~/2(Tza/Tzb ) 7('~-2)/4 z__ L '

(2.23)

where FUM stands for FUM1 or FUM2 depending upon the odd or even value of the integer (2 - A)/A, respectively. Now at the foot-point z = L, Tz = Tzb and Equation (2.23) is given by

FUM(Tza/Tza) = Az ~ / -~ (Tza /Tzb ) 7(~-2)14 • (2.24)

Using Equations (2.23) and (2.24), we get

FUM(Tz/Tza) z FUM(TMTza) Z

(2.25)

Finally, with the help of Equations (2.14a) and (2.24), the solution between the length 2L of the field line and the parameters on the apex and the base is given by

( 1 )'/2 7-2o)/4 (Tzb)7("/' (Tzb) 2L = 4 \ 7--C-2~2)~ J ~ zb FUM .

nzb \ Tza ] \ Tza ] (2.26)

2.1.2. Cooler Apex Loops with Constant Pressure For a loop of constant pressure, the magnetic lines of force are parallel to the loop axis. Obviously, it is a special case of the loop with non-constant pressure. Therefore, the expressions for the constant pressure loops can be obtained from those for non constant pressure loops by removing the dependence of the physical parameters on the radial distance r. It can be done by putting Tr (r) = 1; nr (r) = 1; pr(r ) = 1; kl = 1; C1 = 1. It leads to k 2 : 2k; C2 = X/kO, Tz : T; nz = n; Pz = P. The final relations for constant pressure loop corresponding to Equations (2.25) and (2.26) are

FUM(T/Ta) z FUM(Tb/Ta) = L ' (2.27)

2 L = 4 \ ~ X ~ J nb ~aa FUM ~aa " (2.28)

ENERGY EQUILIBRIUM IN CORONAL MAGNETIC LOOPS - REINVESTIGATED 15

TABLE I Values of the parameters X and c~ obtained by fitting the radiation loss values of Cox and Tucker (1969)

Temperature ranges X a Error range Region (erg cm 3 s -2 K-~)

1.2 x 104 < T < 5 x 104 2.216 × 10 -26 5/6 5 x 104 < T < 2.5 x 105 3.604 X 10 -26 5/6 2.5 x 105 < T < 1 x 106 6.869 x 10 -15 -5/4 1 x 106 < T < 1 x 107 1.988 x 10 -is -13/18 1 x 107 < T < 1 x l0 s 3.086 x 10 -25 1/4 1.2 x 104 < T < 2.5 x l0 s 2.63 × 10 -26 5]6

-25% to 20% 1 -20% to 20% 2 -25% to 25% 3 -15% to 15% 4 -15% to 20% 5 -42% to 42% 1+2

3. Calculations and Results

The basic requirement for the present investigation for getting the relation between the temperature and length of the magnetic lines of force (or the length of the loop for constant pressure) analytically, is that (2 - A)/A must be an integer. Therefore, we tried to fit the values of Cox and Tucker (1989) for f ( T ) by the relation (2.3) so that c~ is

1 - 1 o~ = 2 - 3 - 5 ~ 1 , (3.1)

where I is an integer. The values of f ( T ) in the similar form (Equation 2.3) have been fitted independently by Hildner (1974) and Rosner et al. (1978), and the values of X and a of Hildner (1974) are used by other authors (e.g., Hood and Priest, 1980; Roberts and Frankenthal, 1980). But these values of t~ could not fulfill our basic requirement (Equation 3.1 ) and we tried to find out the values of X and c~ so that c~ satisfies the Equation (3.1). Present values of X and c~ for different temperature ranges are given in Table I. These values of the parameters can reproduce the data of Cox and Tucker (1969) within the error range reported in column 4 of Table I. In order to show the fitting explicitly, the present calculated values for different temperature ranges are plotted with the values of Cox and Tucker (1969) in Figure 3.1.

As is obvious from the Equations (2.9), (2.10), (2.25), and (2.27) that the relative variations of physical parameters depend upon o~ only, and are independent of the parameter X. Therefore, in the present section, only four different variations would be obtained. The variation of r-dependent components of the physical parameters for the models of Chiuderi et al., (1977) is shown in Figure 3.2. It is obvious from the Equations (2.25) and (2.27) that the variation of z dependent part of the temperature for the non-constant pressure loop is similar to that of the variation of


2~

© o2~

I

I I I

2/~'~- ' ' ' ' o lo lo S lo 6 ,0 7 S

TEMPERATURE(K)

Fig. 3.1. Variation of the radiation loss function f(T) versus temperature T. Solid line (--) indicates the results of Cox and Tucker (1969); dashed lines (- - -) are the results obtained by using the present values of the parameters X and c~ (Table I) for different regions.

temperature for the constant pressure loop. The variation of temperature with the length along the magnetic flux for the cooler apex loop is shown in Figure 3.3. The values of the length of the loop 2L for the constant pressure [Equation (2.28)] is calculated as a function of the ratio of the base to theapex temperatures Tb/Ta and are plotted in Figure 3.4 for the cooler apex loops.

4. Numerical Integration

The investigations in Sections 2 and 3 restrict the loop temperature to lie within one of the temperature ranges given in the Table I. For a loop, whose temperature spreads over into two successive temperature ranges, another way to find out the temperature distribution in the coronal magnetic loop is to solve the energy equilibrium equation,

k°-&z \ Oz ) = (4.1)

numerically, with the boundary conditions


(a)

r ' r "

1. i

F-"-

I I I ,L I I

0.2 0.~ 0.6 0,8 1.0 r/R

(b)

~22 c

2

1

o-o ' ' ' ' ' ' ' o ' 8 ' . 0.2 o.~ o.~ ~.o :-I~

Fig. 3.2. Variation of T,/TR [plot AI for the model I and A2 for model II of Chiuderi et al. (1977)], and n~/r~A [plots Bl and BE for the same models I and II, respectively, versus r /R in the cylindrical coronal magnetic loops. Figure (a) is for c~ = 5/6; (b) for c~ = - 5 / 4 ; (c) for a = - 1 3 / 1 8 ; and (d) for a = 1/4.

18

(c)

(d)

rr

1

i . . _ ~

0@

SURESH CHANDRA AND LALAN PRASAD

i [ i I I I I I i

,~=-13/lg B1

I I I l I I I I I

0.2 0.4. 0.6 0.8 1.0 r/R

PF

I

i_, ~-

J I I I I i i r I

4= 1/4 B I

0.4 0.6 0.8 r/~

1.0

Fig. 3.2. Continued.

ENERGY EQUILIBRIUM IN CORONAL MAGNETIC LOOPS - REINVESTIGATED 1 9

(a)

c~

! ,(= 5/6 L

1si

1

I I 1 I I

( b )

'7' 0.2 0.& 0.6 0.8 1.0

APEX Z/L BASE

S 2 ~ N

A P E X

I I I ~ J I 1 I

' 012 ' ' ' d I I I L 0.4 .6 0.~ 1.0

Z/L BASE

Fig. 3.3. Variation of Tz/Tza (for non-constant pressure loops) and T/Ta (for constant pressure loops) is similar and is plotted against z/L for the cooler apex magnetic loops. Figure (a) is for c~ = 5/6; (b) for o~ = - 5 / 4 ; (c) for c~ = - 1 3 / 1 8 ; and (d) for t~ = 1/4.

20

(c)


i 0 i - - , - , I ~ =-13/1B

r6

t _ ~

I I I 1 I, I I . . . . .

I 0 0.2 0,/~ 0 6 0.• 1.0

APEX Z/L BASE

(d)

f~

" f l

0

APEX

i t i i ~ i i

0.2 0.L 0.6 0.2~ 1.0

Z/L BASE

Fig. 3.3.

ENERGY EQUILIBRIUM 1N CORONAL MAGNETIC LOOPS - REINVESTIGATED 21

t-- 0 z LU

©

2 5

5

i [ I I ] I I I I I 1 ~

rb--1,<1 #K 4:____~!ms %= lxlOTK

~:-sI4 %-- 1Xl°6K .<-ilC %= zs×~°sK

, • , , , , , • , , , , , , , 2 3 4 5 6 7 8 9 10 12 16 20

%1%.

Fig. 3.4. Variation of the length of the loop 2L versus Tb/Ta for the cooler apex loops.

at the base

z = L T = Tb n = n b (4.2a)

at the apex

d T / d z = 0 T = Ta z ----- 0 (4.2b)

d n / d z = 0 n = na

For the constant pressure p = 2 k n T = 2knbTb and by using the relation S : z / L , where L is the half length of the loop, the final equation to be investigated, numerically, for the cooler apex loops, is

d2T 2.5

dS 2 T \ d S ] + koT9/2 I ( T ) . (4.3)

For a cooler apex loop we define the values of the apex and base temperatures as (Tb, Ta) = (4X 106K, 6x 105 K), (3x 106K, 4x 105 K) and (1 x 106K, 3× 105K) and solved Equation (4.3), numerically, for the base density n b = 1 × 109 cm -3. During the calculations it is noticed that the numerical integration for the cooler apex loops is very sensitive for the value of L. Therefore, by the method of


2} ~+- - i i I i I I t l . . . . . .

~6.0 J

b

5.5 i 1

o 0:2 o.; o:s o:7 0.'9 ,~PEX s IRAgF

Fig. 4.1. Variation of temperature in the cooler apex loops: (a) Tb = 4 x 106 K, Ta = 6 x 105 K, L = 1.234 x 101° cm; (b) Tb = 3 x 106 K, Ta = 4 x 105 K, L = 5.856 x 109 cm; (c) Tb = l x 106 K, T a = 3 x l0 s K , L = 5 . 2 7 1 x 108cm.

numerical integration we find out the value of L so that the given temperatures on the apex and the base are reproduced. The variation of temperature in the loop for the above defined combinations is shown in Figure 4.1.

5. Effect of Heating Due to an External Source

On considering the heating due to an external source, the energy equilibrium is given by

0 (T5/2 0ri k°"~z [, ' --~-z ) = xn2T~ - hn~T?. (5.1)

Here, the term hn~T~ is assumed to represent a general nature of heating due to an external source, which has been adopted by a number of authors (e.g., Roberts and Frankenthal, 1980), but sometimes for simplicity, either fl = 3' = 0, or/3 = 1 and 7 = 0 has been taken into account (Hood and Priest, 1980; Wragg and Priest, 1981). Here, we are starting with the general values of fl and % but for discussing the effect of the external heating, analytically, we shall restrict them to a certain condition. Now the stationary state equilibrium would be possible for both the


hotter apex loops as well as the cooler apex loops. For the hotter apex loops, z = 0 is at the base of the loop, whereas for the cooler apex loops, z = 0 is at the apex of the loop, so that the temperature gradient is positive in both cases. Further, we assume that the pressure is constant so that the magnetic flux lines are parallel to the axis of the cylindrical loop, and the temperature and electron density are functions of length, along the magnetic flux, only. For the constant pressure Pl = 2kniT1 = 2knblTbl , where rib1 and Tbl are the electron density and temperature on the base of the loop. Now, by introducing the new dimensionless variables

[ Ta '~ 7/2 z (5.2) 7-1 : \ ~ j , S l = L-['

where L1 is the half length of the magnetic line or of the loop, Equation (5.1) reduces to the form

d27-1 = AI #-I - ( 5 . 3 )

where 7 _ 2 , T a - 3 . 5 r 2

)("t%l'Zbl "tJl , (5.4a) Al2 = 2 k0

/3 , -~7-3 .5 /" 2 7 anbl-tbl ~1 (5.4b)

B~ = 2 k0 '

2 2), (5.4c) = 1 + ~ ( ~ -

2 0 = 1 -t- ff (3' - / 3 ) (5.4d)

and the boundary conditions for the Equation (5.3) are

at the base

7-1 = 1,

(5.5) at the apex

d7-1 = O, 71 = 7"al [=- (Tal /Tbl )7/2] (say)

dS1

Equation (5.3) is a second order differential equation and its integration gives

d7"1 ~2 2A~ ~ 2B12 0 dSl J = -"~--7"1 ff--7"1 + C~. (5.6)

The value of the constant of integration C~ can be obtained, by applying the boundary condition on the apex, as C'~ = -2A17"hl/A2 ,x + 2Bird12 e / 0 and finally we get

24 SURESrl CHANDRA AND LALAN PRASAD _ _

2A 2 2B 2 (5.7) dS1 / = X

For investigating the effect of external heating, analytically, let us simplify the approach by putting a restriction to the parameters:/3 = 2 - a + '7, so that A = 0. Then Equation (5.7) reduces to the forms:

dr1

and

dT1

= CBI 2 - A 2 ~/'~A dS1

= C A 2 - B 2 k / / ~ dS1

(for hotter apex loop) (5.8)

(for cooler apex loop). (5.9)

5.1. HO'TTER APEX LOOPS

Equation (5.8) can be solved by using the transformation 7-1 = "ral sin 2/~ u, where u is a new variable. We finally get

(Tai ~7(X-2)/4 z f udu = v/B2-A 2 V/~\-~bl/ z+C~', (5.10)

where C~ ~ is a constant of integration. The integral on the left side of Equation (5.10) can be solved analytically for the integer values of (2 - A)/A. If (2 - A)/A is an odd integer, say (2 - A)/A = 2m + 1, where m is an integer, then

fsin(2-~"P'udu=fsin2m+ludu=-FUNl(f-~l). (5.11)

Here, the relations between u, rl and T1 are used, and the function FUN1 is defined by

FUNI (T) = [1-(T/T')7)q2]I/2 1

X [ ( T ~ 7)tm/2~--~] + m-1 22k+2(Tr~[)2~ (2TRY) ! [ (--m -~- (2T/Z- 2k- 2)]g_~ l)l.~ ( T ) 7"k(2m-2k-2)/4 ] k---0

(5.12)

Obviously, FUNI(1) = 0. If (2 - A)/A is an even integer say (2 - A)/A = 21, where 1 is an integer, then

fsin(2-~')/:'udu=fsin2Zudu=-FUN2'(~a~ ) , (5.13)


where the function FUN2 ~ is defined by

T ) [1 - (T/T')TM2] 1/2 FUN21 ~7 = 21 ×

× [ (T~7A(21-1)/4 \TT] + Z 22k~i ~'2-k -- l J ~[~ -- 1) !] 2 t-I (21_l)![(l_k_l)!]2 (T)7A(21-2k-1)/4] ~7 -- k=l

(2/)! ( T '~ 7~/4 22t(i!)2 sin-1 \~-7] • (5.14)

Now, we define

F U N 2 T ) (2/)! rr

= 1N2' + z ' ( 5 . 1 5 )

so that FUN2(1 ) = 0. Finally, the constant of integration C~ ~ is calculated at the apex (z = L) and the final relation between the z-dependent part of the temperature T1 and the distance z is obtained as

FUN (~Ial) = ~/B2 - A 2 1 ~ ' ~ (Tal ~7('x-2)/4 ( 1 - L ) (5.16) \ Ybl ]

where FUN stands for FUN1 or FUN2 depending upon the odd or even value of the integer (2 - A)/A, respectively. Now, at the foot point z = O, T1 = Tbj and Equation (5.16) is given by

7()~-2)/4 FUN ( Tb, ) = ~ B 2 _ A2 ~ F (To' 7

\Tal ) \T-~b~ ] (5.17)

and from Equations (5.16) and (5.17) we get

FUN(T1/Tal ) z = 1 - - ( 5 . 1 8 ) FUN(Tbl/Tal) L

Finally, with the help of Equations (2.13) and (5.17) the relation between the length 2L of the field line and the physical parameters on the apex and the base is given by

: l ~ 1/2 T(7--2°~)/4 :Zb 1 )7(A-2)/4 (Tb 1 2L = 4 \7--'~] ~bl nbl k~al] FUN \ T-~al ] (5.19)

5.1.1. Calculations and Results The basic requirement for the present investigation for getting the analytical relations is the same as discussed in the Section 3. The values of Cox and Tucker (1989) for f (T) were fitted by the relation (2.3) so that a satisfies the relation


(3.1). In this case also, Equations (5.18) and (5.19) show that the relative variations of physical parameters depend upon a only, and are independent of the parameter X. Therefore, in the present section also, only four different variations would be obtained. The variation of temperature with the length along the magnetic flux for the hotter apex loop is shown in Figure 5.1. The values of the lengths of the loops 2L [Equation (5.19)] are calculated as a function of the ratio of the base to the apex temperature Tb/Tc, and are plotted in Figure 5.2 for the hotter apex loops.

5.2. COOLER APEX LOOPS

Equation (5.9) is similar to the cases discussed in Section (2.1.2) and can be solved in a similar manner, which leads to (corresponding to Equation 2.24)

- = 'V/2"fi k' a- alal j FUM k' a- alal j (5.20)

In Section 2, where the external source of heating is not taken into account, the expression for A, corresponding to the Equation (5.20) can be summarized as

( Tb ~ 7()~-2)/4 A = v / ~ \ ' ~ a a j FUM (TT---~ba) . (5.21)

If we assume that in the above two cases [Equations (5.20) and (5.21)] the ratio of the two temperatures, on the apex and the base, is the same, i.e. Tb/Ta = Tbl/Tal, then we get

A~ - B~ = A 2. (5.22)

By using Equations (5.4a,b) and (2.24) (under the condition of cooler apex loops with constant pressure) in Equation (5.22) we get

Zl = (_~lbl)(Tb~(2°l-7)/4 [1 h , ,~ / J -2 rT '7 - °q -1 /2 / ' . ~ b l J - - X ' % I "tbl .I ~"

(5.23)

Again, if the values of the physical parameters on the base in both the cases are the same, i.e. nb= nbl and Tb ----- Tbl, then Equation (5.23) shows that for the same values of physical parameters on the base and the apex of the loop, the length of the loop increases by a factor of [1 - (h/x)n~2T~-'~] -1/2 due to the external heating.

Physically, also it can be shown that in the presence of an external source of heating for maintaining the energy equilibrium with the same values of physical parameters on the base and the apex, as in absence of external heating, a large volume is required so that the radiation loss is increased by the same amount as is the heating due to the external source. Further, for the constant area of cross-section, the volume of the loop can only be increased by increasing its length.

(a)

~- 0.6

i 4 = 5/6 . ~

I I I

i

0./~ I

0.2

(b)

I 1 P t 1 I t t I I

0 0,2 0. Z, 0.6 0.~ 1.0 BASE Z/L APEX

1,0 " ' - - ~ I T f I I L I i

0.~

~-~ 0.6 i.--

0.L


0,2

0 0.2 0.4 0.6 0.B 1.0 BASE Z/L APEX

Fig. 5.1. Variation of T/Ta is plotted against z/L for the hotter apex magnetic loops. Figure (a) is for c~ = 5 /6 ; (b) for c~ = - 5 / 4 ; (c) for c~ = - 1 3 / 1 8 ; and (d) for c~ = 1/4.

28

(c)


i.o i . . . . . . ~ = - 1 3 / 1 8 ~

0.8

0.6

0,4

t I _ _ 1 I ~ I L I - T

~ 0.2 9,4 0.6 0.~5 1. ~, BASE Z/L APEX

( d )

t -

~-~ 0.6

0.4

0.2

I , / I I f

O 0.2 0. 4 0.~ 0.6 1.0 BASE Z/L APEX

Fig. 5.1.


15

I I I I I l I I 1 I

~10 i - , - (.9 Z LLI J

(D $

5

~ / 4 Tb= txtO7 H,

Tb= 1×106K Tb= 2.5x I 0 5 ~

0.2 0.4 0.6 0.8 1.0

Fig. 5.2. Variation of the length of the loop 2L versus Tb/T,~ for the hotter apex loops.

6. Numerical Integration (Hotter Apex Loops with Heating Due to an External Source)

The investigation in Section 5 restricts the loop structure to lie within one of the temperature ranges given in Table I. For the loop, whose temperature spreads over into two successive temperature ranges, another way to find out the temperature distribution in the coronal magnetic loop is to solve the energy equilibrium equation

-:-d (TsI2dT ~ = nZ f ( T ) _ hnZT.y k° dz \ dz ]

numerically. The boundary conditions for the Equation (6. l) are

at the apex

dT s = L , T = T a and ~ z = 0 ,

at the base

s = 0 , T = Tb.

(6.1)

(6.2a)

(6.2b)


10

9

8

B

7 -2/,

i • i

C

' -:;2 _20 LOG(h)

Fig. 6.1. Variation of l o g h versus l o g L for the hotter apex coronal loops. Curve A is for Ta = 1 x 10 6 K, Tb = 3 x 10 5 K; curve B is for T,, = 3 × 10 6 K, Tb = 4 x 10 5 K; curve C is for Ta = 4 × 10 6 K, Tb = 6 x 10 5 K.

For the constant pressure in the loop, and using the relation s = z/L, Equation (6.1) can be rearranged as

2.5 hL n r: L nbl~, T ~-z-2"5, (6.3) ds -----$ - T - \ d s ] + k0T 4"5 f (T) ko

We defined the values of the apex and base temperatures as (Ta, Tb) = ( 1 X 106 K, 3 x 105 K), (3 x 106 K, 4 x 105 K) and (4 x 106 K, 6 × 105 K), and solved Equation (6.3), numerically. During the computation, it is noticed that the solution of Equation (6.3) is very sensitive to the values of h and L. Thus, by the method of integration, we found a set of the values for h and L, so that the defined temperatures on the apex and the base are reproduced. A variation of L versus h for g = 1 and "7 = 2 is shown in Figure 6.1. The figure shows that the value of L decreases with the increase of the value of h.

It can be interpreted that by increasing the heating due to an external source, for the same values of the apex and base temperatures, the length of the loop decreases.


7. Cooler Apex Loop with Non-Constant Area of Cross-Section

Experimentally it is observed that the magnetic field strength on the base of the loop is larger than that on the apex. For the conservation of the magnetic flux, it shows that the area of cross-section of the loop changes along the length. Antiochos and Sturrock (1976) discussed the loop models for the current-free magnetic field above the chromosphere. One of these models has been used by Elwert and Narain (1980) for estimating the amount of heating due to an external source. The models of Antiochos and Sturrock (1976) would be applied for the constant pressure loop of force-free magnetic filed. Following Antiochos and Sturrock (1976), the magnetic field above the chromosphere may be approximated either by the field of horizontal line-dipole (L) or by horizontal point dipole (P) situated at the depth D below the chromosphere. The height of the flux tube above the chromosphere is H.

For the cooler apex loops, we choose the origin (z = 0) of the length of the field line on the apex of the loop and define the angle ¢ such that ¢ = 0 is on the apex and ¢ = Cb is on the base. Then, the area of cross-section A of the loop, normalized to unity on the apex, is given by Antiochos and Sturrock (1976) and Chandra and Narain (1982)

A(¢) = cos 2 (~, (7.1L)

A(¢) = COS 6 ¢[1 + 3 sin 2 ¢]-1/2 (7.1P)

and the energy equilibrium equation, for the hotter apex loop is given by

kodT ( d T ) "-A -~z ATS/2 = nEf(T)" (7.2)

By using Equation (7.1) and the relation between z and ¢ for the geometries discussed by Antiochos and Sturrock (1976) in Equation (7.2), for the constant pressure loop, we get

dET 2.5 ( d T ) 2 c d T ~Erp2 ]t~2,Ob.tb . ~ ,

de E - T ~ - + 2tan de + koT9/E f(T),

d¢E -- T ~d-¢) + tan ¢ 1 + 3 sin E ¢ d¢ 4-

m n T:R E koT9/E cos 2 ¢(1 + 3 sin E ¢)f(T) ,

where R = H + D and the boundary conditions are

(7.3L)

(7.3P)

at the apex

dT ¢ = 0 , T=Ta, d-C=0 ' (7.4a)


at the base

¢ = Cb, T = Tb. (7.4b)

Now, from Equation (7.1), the area of cross-section on the base of the loop is given by

Ab = cos 2 Cb, (7.5L)

At, = cos 6 Cb(1 + 3 sin 2 ~b) -1 /2 (7.5P)

and the ratio of the two areas, defined as the compression factor 1", is given by

F = magnetic field on the base _-

magnetic field on the apex

area on the apex 1

area on the base Ab

Therefore,

1 ~ = sec2¢b, (7.6L)

F : sec6¢b(1 -q- 3 sin 2 q~b) 1/2. (7.6P)

Obviously, for the known magnetic field strengths on the apex and on the base of the loop, the compression factor is known and the value of the angle ¢b can be calculated with the help of Equation (7.6). Finally, Equation (7.3) is solved numerically for the temperature combinations and base density discussed in Section 4. During the computation, it is noticed that the solution of Equation (7.3) is very sensitive for the value of R. Therefore, by the method of integration we find out the values of R so that the defined temperatures on the base and the apex are reproduced. In the present investigation the values of ¢b taken are from 50 to 80 degrees. The calculated values of R as a function of ¢b are plotted in Figure 7.1. As is obvious from Figure 7.1 that the value of R decreases with the increase of the value of ¢b.

Physically, it can be explained that for the greater compression factor (larger value for ¢b) with the constant area of cross-section on the base, the area of cross- section on the apex is larger. Therefore, the radiation loss from the apex would be increased. Further, for maintaining the base and the apex temperatures, the only possibility is to decrease in the height of the loop so that the net volume is such that the radiation loss is conserved.

8. Hotter Apex Loops with Non-Constant Area of Cross-Section, Constant Pressure and Heating Due to an External Source

As discussed in the Section 7 that the area of cross-section of the loop changes along the length. For the hotter apex loops, we choose the origin (z = 0) of the length of the field line on the base of the loop and define the angle ¢ such that


I0

9

_ . . I

7

6

~ - s • , - i - i - s

!

I i i i I I I

50 55 60 65 70 75 gO 35 Oa td eclrees)

Fig. 7.1. Variation of R (= H + D) versus the angle at the base eb. The plots L are for the horizontal line dipole geometry, while the plots P are for the horizontal point-dipole geometry. Plots with the suffix 1 (as in Ll, P1) are for Ta = 3 × 10 5 K, Tb = 1 × 10 6 K; suffix 2 for Ta = 4 x 10 5 K, Tb = 3 x 10 6 K; and suffix 3 forTa = 6 x l0 5 K, Tb = 4 x 10 6 K;

¢ = 0 is on the base and ¢ = Ca is on the apex. Then, the area of cross-section A of the loop, normalized to unity on the apex, is given by Antiochos and Sturrock (1976) and Chandra and Narain (1982)

A(¢ ) = cos2(¢a - ¢), (8.1L)

A(¢) = COS6(¢a -- ¢)[1 + 3 sin2(¢a -- ¢)] -1/2 (8.1P)

and the energy equilibrium equation, after including the heating due to an external source to the hotter apex loop of non-constant area of cross-section, is given by

-Adz ATS/2 = n2f(T) - hn~T 7. (8.2)

For the constant pressure, and following the steps of Section 7, Equation (8.2) reduces to (corresponding to Equation 7.3)

d2T (~¢2 ~ 2.5 ( d T ) 2 ', / d-¢ - - d T n2T2R 2 koT 9/-----T hn~b~R 2 - - - 2tan(Ca - ¢ ) ~--~ + f (T )

k0TZ-7+2.5 '

(8.3L)


d2T 2.5

d e 2 T

(dT'~ 2 - t a n ( C o - ¢)

11 + 9 sin2(¢a - ¢) dT

1 + 3sin2('¢a - ¢) d-¢ +

j" + c o s 2 ( ¢ a - ¢ ) [ l + 3 s i n 2 ( ¢ a - ¢ ) ] [ ~ f(T)

The boundary conditions for the Equation (8.3) are

hn R2

(8.3P)

at the apex

¢=¢. ,

at the base

dT T = Ta, = 0, (8.4a)

dO

¢ = 0, T = Tb. (8.4b)

Now, from Equation (8.1), the area of cross-section on the base of the loop is given by

Ab = c o s 2 (~a, (8.5L)

Ab = c o s 6 Ca(1 + 3 sin 2 t~a) - i / 2 (8.5P) and the ratio of the two areas, defined as the compression factor P, is given by

F = magnetic field on the base

magnetic field on the apex

Therefore,

P = seC2¢a~

area on the apex 1 ~ ,

area on the base Ab

(8.6L)

I" = sec6¢a(1 + 3 sin 2 Ca) 1/2. (8.6P)

Obviously, for the known magnetic field strengths on the apex and on the base of the loop, the compression factor is known and the value of the angle Ca can be calculated with the help of Equation (8.6). Finally, Equation (8.3) is solved numerically for/3 = 1 and '7 = 2, and for the temperature combinations and base density discussed in the previous sections. The results are shown in Figures 8.1 and 8.2. Figure 8.1, like Section 6, shows that by increasing the heating due to an external source, the length of the loop decreases. Figure 8.2 shows that, for a given value of h, with the increase of the value of ¢~, the value of R decreases. The effect is more pronounced for the point dipole geometry than the line dipole geometry.

ENERGY EQUILIBRIUM 1N CORONAL MAGNETIC LOOPS - REINVESTIGATED 3 5

10

9

$

I i I i

-2/+ - 3 -22 -21 -20 LOG{h)

' • | . . . . • ! , i - - - ~

9

$ A

6

5

-2L, - 2:3 z22:~ -21 -20 LOG{h)

Fig. 8.1. Variation of log R versus log h: (a) for ~,~ = 50 ° and (b) ff,~ = 88 °. The plots L are for the horizontal line-dipole geometry, while the plots P are for the horizontal point-dipole geometry. Plots with the suffix 1 are for Ta = 1 x 10 6 K, Zb = 3 x 105 K; suffix 2 for Ta = 3 × 106 K, Tb = 4 x 105 K; and suffix 3 for Ta = 4 x 10 6 K, Tb = 6 × 105 K;


1 0 - • " , i

- - ,,,

L2 "~

q ~ h=1~ 2~ ~~ % o ' #~ ' ~o ~ s ' 7 b 7~ ~o ' ~s

7

6

5

! l i

.- L3

L _ l _

PI

b

i i i • ,

%0 ~ 60 ~ 70 7s ~o 8~

Fig. 8.2. Variation o f l o g R versus $,,: (a) for h = 10 -24 and (b) h = 10 -20, The plots L are for the horizontal line-dipole geometry, while the plots P are for the horizontal point-dipole geometry. Plots with the suffix 1 are for Ta = 1 x 106 K, T~ = 3 x 105 K; suffix 2 for Ta = 3 x 106 K, T b = 4 x 1 0 5 K ; a n d s u f f i x 3 f o r T ~ = 4 x 10 ~ K , T b = 6 x 105K;

ENERGY EQUILIBRIUM 1N CORONAL MAGNETIC LOOPS- REINVESTIGATED

9. Discuss ion and Conc lus ion

37

Investigations for the cooler apex loops, in the Sections 2 and 3, and Figure 3.2 show that the variation of temperature along the radial distance from the axis of the cylindrical loop depends upon the nature of the variation of the pressure inside the loop. In the loop of non-constant pressure, the variation of the temperature along the magnetic flux line Tz/Tza is similar to the temperature variation T/Ta along the length of the constant pressure loop. For the confirmation of the temperature structure inside the non-constant pressure loop, one has to wait for the observations with better resolution. For the hotter apex loops with external source of heating (Section 5), for a particular choice of parameters, analytical expressions for the variation of temperature along the loop length are derived.

Figures 3.4 and 5.2 show that no loop can exist for the equal temperatures on the apex and the base, but a little difference in the two temperatures supports the existence of the loop. It supports the observations that no loops with equal apex and base temperatures have been observed, while some loops are observed where the difference in the two temperatures is very small. Further increase in the difference between the apex and base temperatures shows that the length of the loop increases in the case of a hotter apex loop (in presence of the external source of heating), while it remains almost unchanged for the cooler apex loop. Although, to the best of our knowledge, no loop with cooler apex has so far been experimentally observed, but these also have been included in the present study as these have also been discussed by some authors. If the temperature of the loop does not lie within one of the temperature ranges discussed in the Sections 2, 3 and 5, then the temperature distribution in the loop can be obtained by solving the equilibrium equation, with the boundary conditions, numerically (Sections 4 and 6). The temperature distribution in the cooler apex loop, for the base and the apex temperatures lying in two above mentioned successive temperature ranges, is shown in Figure 4.1. It shows that from the base to the region near the apex, the variation of temperature is very slow while around the base it varies rapidly. Investigations of Section 6, show that by increasing the heating due to an external source, for the same values for the apex and base temperatures, the length of the loop decreases.

Cylindrically symmetric loop structures, assumed so far, cannot in general be true because the magnetic field strength on the base of the loop is greater than that on the apex. For the conservations of magnetic flux, it leads to the idea that the area of cross-section of the loop changes along the length. In Section 7, for cooler apex loops, the temperature distribution is investigated by including the loop geometries proposed by Antiochos and Sturrock (1976) for a horizontal line- dipole and for a horizontal point-dipole situated below the chromosphere. It is found (Figure 7.1) that for the same value for Cb, the height of the loop for the line-dipole geometry is larger than that for the point-dipole geometry for the same apex and base temperatures. Further, on increasing the value of Cb, the height of the loop decreases. The variation is found more pronounced in case of the point-dipole geometry, as expected.

38 SURESla CHANDRA AND LALAN PRASAD

Finally, in Section 8, we investigated the case of hotter apex loops with non- constant area of cross-section and heating due to an external source. Here also we found that, for a constant value of ~b,,, by increasing the heating due to an external source, the height of the loop decreases. This feature is similar to that obtained in Section 6. Further, for a given value of h, with the increase of the value of ~ba, the height of the loop decreases. The variation is, here also, found more pronounced in case of the point-dipole geometry than the line-dipole geometry. This behavior is very similar to that obtained in Section 7 for cooler apex loops.

Acknowledgements

We are grateful to Prof. Dr. W.H. Kegel for pointing out a mistake of sign in Equation 2.1. Financial supports from Council of Scientific and Industrial Research, New Delhi and Indian Space Research Organisation, Bangalore are thankfully acknowledged.

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Energy equilibrium in coronal magnetic loops -reinvestigated