A simple model of the lower corona which allows for a possible difference in the electron and proton temperatures is analyzed. With the introduction of a phenomenological heating term, temperature and density profiles are calculated for several different cases. It is found that, under certain circumstances, the electron and proton temperatures may differ significantly

Knight, J. W.

Newman, C. E.

Sturrock, P. A.

English

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Two-Fluid Model of the Solar Corona
by
J. W. Knight
C. E. Newman
P A. Sturrock
March 1974
SUIPR Report No. 566
National Aeronautics and Space Administration
Grant NG L 05-020-272
NASa-cR.f 3- 8-C- 6) TWO-FLUID MODEL OF THE-SOLAB CORONA (Stanford Univ.) 12 p C 
-2730
CSCL 03B
Unclas
G3/29 41304
INSTITUTE FOR PLASMA RESEARCH
STANFORD UNIVERSITY, STANFORD, CALIFORNIA
https://ntrs.nasa.gov/search.jsp?R=19740019194 2020-03-23T07:14:05+00:00Z
TWO-FLUID MODEL OF THE SOLAR CORONA
by
J.W. Knight, C.E. Newman and P.A. Sturrock
National Aeronautics and Space Administration
Grant NGL 05-020-272
SUIPR Report No. 566
March 1974
Institute for Plasma Research
Stanford University
Stanford, California
TWO-FLUID MODEL OF THE SOLAR CORONA
by
J.W. Knight, C.E. Newman and P.A. Sturrock
Institute for Plasma Research
Stanford University
Stanford, California
ABSTRACT
A simple model of the lower corona which allows for .a possible
difference in the electron and proton temperatures is analyzed. With
the introduction of a phenomenological heating term, temperature and
density profiles are calculated for several different cases. It is
found that, under certain circumstances, the electron and proton
temperatures may differ significantly.
pRECEDING PAGE BLANK NOT FILM
iii
I. INTRODUCTION
It is generally believed that the corona is heated by a flux of
non-thermal energy which originates in the photosphere and propagates
through the chromosphere, transition region, and possibly part of the
corona. D'Angelo (1969) suggests that if this non-thermal flux is in
the form of ion acoustic waves, Landau damping of these waves might
contribute significantly to the heating of the corona. D'Angelo also
noted that if this were the case, preferential heating of the protons
could cause the proton temperature to be higher than the electron
temperature.
There are several interdependent aspects of the problem of coronal
heating. These include the propagation of non-thermal energy, the
dissipation of this energy, and the effect of this dissipation on the
temperature and density structure of the corona. In this article, we
shall be concerned only with the last topic, and we shall consider
explicitly the possibility that a substantial fraction of the dissipated
energy preferentially heats the ions.
II. ANALYSIS
We adopt a simple planar model of the corona in which all quantities
are uniform in the horizontal coordinates and in which the gravitational
acceleration is independent of the vertical coordinate (z). The composition
is taken to be that of fully ionized hydrogen gas. Radiation losses and
magnetic fields are neglected. Since we are not treating the problem of
dissipation at this time, a phenomenological heating term is introduced.
We suppose that all of the dissipation occurs in a narrow region, of
width of order 10,000 kilometers, centered at a height of 15,000 kilometers
1
above the base of our model, which we take to be the level at which
T = 106.oK. The particular form adopted for the heating function
-3 -1
Q (erg cm s ) is
27 z -zl ]
Q(z) = Qo sin 2  z ), z ! z ! z + w (1)
where z = zw = 10,000 km. Q = 0 outside the range given in equation (1).
We further assume that a fraction f of the total energy input goes to
heating the protons, so that
Q = (-f) Q, Q = fQ (2)
where Q (z), Q (z) are the electron and proton heating functions,
- +
respectively.
The relevant equations will be taken from Braginskii (1965). The
equations of hydrostatic balance are
-dp+ + en E - mn g = 0 (3)
dz + + +
-dp_-dp - en E - mng = 0 , (4)
dz
where the symbols have their usual meaning. The subscripts + and -
refer to protons and electrons, respectively. We may set n n = n
+
in the above equations (but not in Poisson's equation). Then, neglecting
m in comparison with m and summing the above equations, we obtain
- +
d---z nk [T + T ] + nm+g = 0 . (5)
Two heat equations may be expressed as
d dT 3
-- -- + Q - nk (T - T) = 0 (6)dz + d + 2 E + -
and
2
d - 3dz dTz + + vE nk (T+ - T_) 0. (7)
The thermal conduction coefficients + , and the heat-exchange
coefficient VE are given by
-4.73 X-1 5/2 -6.12 -1 5/2 -2.41 -1 -3/2
= 10 T , - =10 X T , = 10 nT , (8)
- - + + E _
where X (the Coulomb logarithm) can be expressed with sufficient accuracy
as
X = 15.9 - 1.15 log n + 2.3 log T (9)
For the ranges of density and temperature of interest, we may with
sufficient approximation adopt X = 20. Then the coefficients in
equation (8) may be expressed as
7 5/2 7 5/2 2 -1 3/2
- = K T , + = K T , E =  Ak n T (10)
-6.57 7.96 -16.79
where. K = 10 ,K = 10 and A = 10
- +
With these substitutions, equations (6) and (7) become
2T 7/2
d = - K Q - AK - n 2 T - 3 / 2 (T - T ) , (11)
dz
d2T 7/2 -1 -1 2 -3/2
+ = - K Q + AK n T (T - T ) , (12)
+ + + - + -dz
from which we see that
2
d 7/2 7/2
- K T + KT+ = -Q (13)
dz
Equation (5) now becomes
1 dn (T+ T )-1  (C + d T + T ]) (14)
n dz - + dz - +
where
3
C = m gk - 1  10-3.48 (15)
+
III. RESULTS
Our aim is to calculate n(z), T (z) and T (z) for specified
+
forms of Q (z) and Q (z), and for given boundary conditions. One
- +
of the five required boundary conditions is as follows,
n = n at z = 0 (16)
Since equation (13) is immediately integrable, it is convenient to
express two of the conditions as follows,
7/2 7/2 7/2
K T + K T = (K_+ K ) T at z = 0 , (17)
- ++ + 0
and
z + z w Qdx=F.
d 7/2 7/2  (18)
-- 
(K T + K T+ =dz -++
Z 1
Equation (18) implies that the entire energy input flows back down as
a heat flux. The remaining conditions are taken to be the following:
T - T -0 , z - - , (19)
T - T 0 , z - a (20)
+ -
In selecting the boundary conditions at the base of the model, we
have considered estimates for the product nT in the transition region.
14
Athay (1965) estimated this quantity to be in the range of (3.7-6.0) X 10
14
later, Athay (1969) made the estimate 6 x 10 ; Moore and Fung (1972)
adopt the range (5-15.1) X 1014. From these estimates, we have chosen
6
the values log no = 8.50, 8.78, 9.00 at T = 10 . We also consider
-2 -1 5
two values of the downward heat flux F (erg cm s ): 6 X 10
4
(Dupree and Goldberg 1967), and 106 (Moore 1972).
The results of our calculations are displayed in Figures 1 and 2.
We see from Figure 1 that an increase in n leads to a similar increase
in n(z), so that the profile is insensitive to n . An increase of
0
n hardly affects T (z), but it leads to a reduction of T (z) since
o +
it tightens the coupling between the two species. (Note, however, that
the asymptotic value of T , as z - m , is not significantly affected
+
by n since T - T in this limit.)
o + -
Figure 1 also shows that, if n and F are fixed, an increase
in f leads to an increase in T in the heating region and to a
+
slight reduction in n in this region. The decrease in density tends
to offset the increase in pressure due to the increase in proton temperature.
The electron temperature is not significantly effected by a change in
f. We see from Figure 2 that, for fixed F and f, an increase in n
O
leads to an increase of n(z) and a reduction in T (z), as noted
+
above, but this diagram shows that there is a slight increase in T (z)
in the heating region. A comparison of Figures 1 and 2 shows that an
increase in F, for fixed n , f, leads.to increases in n(z), T (z)
O
and T (z).
+
8.5
It is notable that, if n is as small as 10 T may exceed
0 +
T , in the heating region, by up to 70%.
The above calculations show that, if the non-thermal energy goes
primarily into heating the proton gas, the proton temperature may be
substantially higher than the electron temperature in the heating region
of the corona. Hence one would need to be cautious in inferring the
coronal structure from observational data, since some observations
(such as radio-frequency brightness and coronal line strengths) depend
upon electron temperature, other observations (such as coronal-line widths)
5
depend upon the ion temperature, and some observations (such as the
variation of density with radius) depend upon a combination of the
temperatures.
In order to pursue this question, it is essential to investigate
in more detail the heating of the solar corona, to determine in particular
the allocation of energy between the electron gas and the proton gas.
ACKNOWLEDGEMENT
This work was supported by the National Aeronautics and Space
Administration under Grant NGL 05-020-272.
6
REFERENCES
Athay, R.G., 1965, Ap. J. 142, 755.
Athay, R.G., 1969, Solar Phys. 9, 51.
Braginskii, S.I., 1965, Reviews of Plasma Phys. 1, 205.
D'Angelo, N., 1969, Solar Phys. 7, 321.
Dupree, A.K. and Goldberg, L., 1967, Solar Phys. 1, 229.
Moore, R.L., 1972, Stanford University Institute for Plasma Research
Report No. 463.
Moore, R.L. and Fung, P.C.W., 1972, Solar Phys. 23, 78.
7
FIGURE CAPTIONS
Figure 1. Temperature and density profiles for a total energy input of
6 -2 -1
1.0 X 10 erg cm sec . Case I corresponds to an initial
9 -3
density of 10 cm and f = 1. Case II also corresponds to
9 -3
an initial density of 10 cm but with f = 0.5. Case III
8.78 -3
corresponds to an initial density of 10 cm and f = 1.
The T profiles are indistinguishable on this scale.
Figure 2. Temperature and density profiles for a total energy input of
5 -2 -1
6 x 10 erg cm sec and f = 1. Case I corresponds to an
9 -3
initial density of 10 cm Case II corresponds to an
8.78 -3initial density of 10 cm (The T curves for these
first two cases are indistinguishable.) Case III corresponds
8.5 -3
to an initial density of 10 cm-3
8
Figure 1
CASE I - I HEATING I
6.5 CASE II ........... REGION I
CASE M ---- I -- .% T+6.4 9.2
6.3- " .... .... . .... -- ,, , -9.2
2 6-9.0
(D 0 6.18.8 (
J 6.0 I T-8.6
5.9 ----. 4
S--8.2I I n8.2
0 10,000km 20,000 km 30,000 km
Z
Figure 2
CASE I ............. I HEATING
6.5- CASE II --- I REGIONCASE ---- -- T+6.4 --ow
6.5 // I ~-9.2
- -6.2 '--.- -- _- - 9.0
-6.1 - T 8.8
6.0 -............... 8.6
5.9 - "' ,-8.4
S--- 
- 8.2
0 10,000 km 20,000km 30,000 kmZ
9


Two-fluid model of the solar corona

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