The transition from a supersonic to a subsonic corona was investigated by increasing the density N sub O at the base of the corona while keeping the temperature T sub O constant. For the initial values of N sub O and T sub O, the solution of the inviscid solar wind equations is of the Parker type. As N sub O is increased, Parker type supersonic solutions cease to exist. No subsonic solutions exist

Durney, B.

English

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TRANSITION FROM A SUPERSONIC TO A SUBSONIC SOLAR WIND B. Durney 
ABSTRACT The transition from a supersonic to a subsonic corona was investigated by increasing the 
density No at the base of the corona (initially No = 9.3X 1O7/cm3) while keeping the 
temperature To there constant (To = 2.1X lo6 O K ) .  For the initial values of N and To, 
the solution of the inviscid solar wind equations is of the Parker type. As No is increased, 
Parker type supersonic solutions cease to exist for No = NOp - 1.1 7X lo8 /em3. No sub- 
sonic solutions exist, however, for No < Nos where Nos > 3X lo8 /cm3. For NOp < No 
<Nos the solar wind equations permit supersonic solutions which are not of the Parker 
type, with the temperature varying as ( l/r)4/3 for large distances. 
9 
The transition region separating supersonic expansions 
of the solar wind [Parker, 19581 from subsonic 
expansions [Chamberlain, 19611 has been considered by 
Parker [ 19651 . Here we study this transition region for 
a given value of the temperature at the base of the 
corona. Chamberlain's notation [ 19613 will be used 
throughout this paper; that is, the dimensionless values 
of the temperature, square of the velocity, and the 
inverse of the radial distance will be defined as 
r = TIT1 = mw2/kT1 h = GM, mlkTlr (1) 
where T1 is a reference temperature taken equal to 
2X 106"K, w the radial expansion velocity, m the average 
mass (for a hydrogen-helium mixture 10: l),M, the mass 
of the sun, and G and k are the gravitational and 
Boltzmann constants, respectively. The usual momen- 
tum and energy equation can then be written as 
( 2 4  
& =  1 -2r /h-dr /dh  
dh O S (  1 - r / $ )  
For A we take [Chamberlain, 19611 : A = 5.8X 106/C 
where C is the mass flow given in terms of N ,  $, and h 
by C = N $ I i 2  A-2 (N is the total particle density). In 
equation (2b) eW is the residual energy per particle at 
infinity. For ew # 0 a well-known solution of equations 
(2a) and (2b) is the Parker supersonic solution. Then for 
small values of A 
The asymptotic expansion for $ is obtained from 
equation (2b). The parameter Do in equation (3) is left 
undetermined; this allows equations (2) to be solved as 
follows. Given Cand eW, the integration of (2) and (3) is 
started from small values of X 
<< 1) 
toward the sun; the condition that the numerator and 
denominator of equation (2a) 
The author is with the National Center for Atmospheric 
Research, Boulder, Colorado. The National Center for 
Atmospheric Research is sponsored by the National Science 
Foundation. = 1-2r/X-dr/dh 175~ = 0.5(1-~/$)  (4) 
232 
https://ntrs.nasa.gov/search.jsp?R=19730002052 2020-03-17T07:22:19+00:00Z
1.2E8- I I I I I I I I I I I I I I I I I I I I I I I 
- 
- l.lE8 
\ 
calculations. No subsonic solutions were found for 
The above considerations led to the search for a 
supersonic solution of equations (2) with an asymptotic 
expansion different than that given by equation (3). It 
was indeed found that the full equations (2) also admit a 
solution with the following asymptotic expansions 
N~ < 109/cm3. 
T = Doh4 " [ 1 + D1 A + D2 A4' t D3 X5 t D4 X2 1 
I I I I I i I I I I I I I I I I I I I I I I I I I with 
0 .IO .20 9. E? 
1 2 
De E,=- ~ o o ; C o = - 5 D o ; C 1 = D 1 = -  
2 39, Figure 1. 
The dotted lines are interpolated values. 
Density at the base of the corona versus D,. 
3.0 
\u" 1.5 
0 
D 5 = - ? [ E 2  t 22  AD^^^^ ] ; 
63 9, 
As in the Parker solutions the parameter Do in equation 
(5) is left undetermined, and equations (2) can again be 
solved by integrating from small values of A toward the 
sun. An iteration procedure was used to determine Do so 
that Ig,(A) = Ig2(h) = 0 (cf. eq. (4) for the same value of 
X = AJ. This determines the critical point. The iteration 
Figure 2. 
To =2.1XI06 "K. 
Residual energy at infinity versus No for procedure is based on the fact that if Igl vanishes first 
(Igl (A,) = 0 and I g 2  (A) f 0 for A < Ac), then Do is too 
233 
io 
f io- 
io- 
10 
I I I 
io" 
x 
Figure 3. rversush forC=5.47X104,e,=0.887. 
10-3 
A 
Figure 4. + versus h for C = 5.4 7X 1 04, ew = 0.88 7. 
large and it should be decreased; if on the other hand rPz 
vanishes first, then Do is too small and it should be 
increased. This method allows the critical point he to be 
kiiown as accurately as desired. Once A, is determined a 
Taylor expansion of r and J /  in the neighborhood of he 
can be used to integrate across the critical point. In 
figures 3 and 4, r and $ are plotted versus h for 
C =  5.47X lo4, ew = 0.887. With the help of the asymp- 
totic expansions (5) the integrations were started from 
h = 4.7X toward the sun. The arrow indicates the 
position of the sonic point. For the above values of C 
and ew the values of the density and temperature at the 
base of the corona are To = 2.1X106'K and 
No = 2.88X 10' /em" respectively. Therefore, to sum- 
marize, if the temperature at the base of the corona is 
assumed constant and equal to 2.1X 106'K, the solution 
of the solar wind equations is of the Parker type for 
values of No < 1.17X 108/cm3. For values of No such 
that 1.17X 108/cm3 <N<N,S (N,S >3X108/cm3) the 
flow continues to be supersonic but differs from the 
Parker solution. The asymptotic expansions of the 
velocity and temperature are now given by equations 
(5). For N > N z  the flow becomes subsonic. The value 
of N: was not determined with great accuracy; the 
numerical calculations suggest that NJ -2X 1 o9/cm3. 
Since the integrations are time consuming no supersonic 
solutions were evaluated for No > 3X 10' /cm3. 
As the density at the base of the corona is increased 
from an initial value of No = 9.3X 107/cm3 and the 
temperature there is held constant (To = 2.1X 106"K) 
there is not a direct transition from a Parker type 
supersonic flow to a subsonic flow. AsN,  is increased 
the conductive flux at infinity e&, decreases sharply and 
vanishes for afinite value of the total energy per particle 
at infinity e,. This is not surprising, since as No increases 
the mass flow C also increases, and since To has been 
kept constant e,  must decrease. The sharp decrease in 
€2 arises as more and more of the conductive flux at 
infinity is converted into kinetic energy of the particles. 
The Parker type supersonic solutions cease to exist when 
e& vanishes. The total energy flux at infinity remaining, 
however, different from zero, it is to be expected from 
physical grounds that if No increases still further there 
will not be an immediate transition to a subsonic flow 
but rather a transition to a different type of supersonic 
flow (cf. eqs. (5) and (6)). The expansion becomes 
adiabatic [Chamberlain, 19611 and the heat conduction 
flux is much smaller than the internal energy flux 
[Hundhausen, 19711. Any further increase in No must 
now be entirely accounted for by a decrease in the 
kinetic energy flux at infinity. The velocity reaches a 
maximum and decreases towards i: finite value as r + a. 
Finally, for values of No large enough, e ,  vanishes and 
the flow becomes subsonic. 
ACKNOWLEDGMENTS 
The author is indebted to Drs. R. A. Kopp, C .  W. 
Pneuman, and E. J. Weber for many illuminating 
discussions and to Dr. G .  Vickers presently at Sheffield 
234 
University for carrying out the analytical and numerical 
calculations for the subsonic solutions. 
REFERENCES 
Chamberlain, J. W.: Interplanetary Gas, 111, A Hydro- 
dynamic Model of the Corona. Astrophys. J., Vol. 
133,1961, p. 675. 
Hundhausen, A. J.: Dynamics of the Outer Solar Atmo- 
sphere. Physics of the Solar System, edited by S .  I. 
Rasool, Goddard Space Flight Center publication 
X-630-7 1-380,197 1 .  
Kopp, R.A.: The Equilibrium Structure of a Shock- 
Heated Corona. Thesis, Astronomy Department, 
Harvard University, 1968. 
Parker, E. N.: Dynamics of the Interplanetary Gas and 
Magnetic Fields. Astrophys. J., Vol. 128, 1958, 
p. 664. 
Parker, E. N.: Dynamics Properties of Stellar Coronas 
and Stellar Winds, IV, The Separate Existence of 
Subsonic and Supersonic Solutions. Astrophys. J., 
Vol. 141 ,1965 ,~ .  1463. 
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Transition from a supersonic to a subsonic solar wind

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