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Parker (1958) suggested that the corona could not remain in static equilibrium
but must be continually expanding since the interstellar pressure
cannot contain a static corona. The continual expansion is called the
solar wind. The existence of a solar wind had been known from comet
observations but the properties predicted by Parker were confirmed by
the satellites Lunik III and Venus I in 1959 and by Mariner II in the
early 1960s.
The main assumptions of Parker's model are that the outflow is
steady, spherically symmetric and isothermal. It is
straightforward to relax the isothermal assumption and consider an
adiabatic or polytropic atmosphere. The basic steady equations are

(6.7) 

(6.8) 

(6.9) 
and

(6.10) 
The velocity is taken as purely radial so that
and gravitational acceleration obeys the inverse square
law,
. In spherical coordinates and
assuming a steady flow,
(6.7) gives

(6.11) 
and the raidial component of (6.8) becomes

(6.12) 
Defining the isothermal sound speed as
, the
gas law gives

(6.13) 
and substituting (6.11) and (6.13) into (6.12) we get
Then dividing by we obtain
Now we can use (6.11) to express and in
terms of as
Expanding the pressure gradient term and rearranging gives the final
radial equation of motion as

(6.14) 
where
. is important
since and is a critical point of
(6.14). If the velocity of the plasma reaches the sound speed
then the radius must either equal or else the velocity gradient
becomes infinite. Another way of saying this is to say that if then either or and if then
either
or .
(6.14) can be integrated to give a transendental equation
for the velocity in terms of the radius as

(6.15) 
where is a constant. The solutions for different values of are
shown in Figure 6.1.
Figure 6.1:
The solar wind velocity, , as a function of the radius,
for various values of the constant . The five different
classes of solution are indicated.

In Figure 6.1 five distinct types of solution are indicated.
Solution I is doubled valued and so is unphysical. It is not possible
for the plasma to leave the solar surface with a velocity below the
sound speed, reach a maximum radius below and then turn round
and return to the Sun with a supersonic speed. Solution II is also
double valued but it never even starts from the solar surface and it
is also unphysical. Solution III starts with a velocity greater than
the sound speed but such a fast steady outflow is not observed.
Hence, this solution must also be neglected. The only two types of
solution that are physically reasonable at this stage are IV and V.
Solution V is the particular case where the plasma leaves the
solar surface with a particular speed and passes through the critical
point (also called the sonic point) at and . Solution IV always remains below the sound speed and is called
the solar breeze solution. For solution V we choose the constant
so that and and this requires .Is
it possible to decide between solutions
IV and V?
Example 6.2.1
The behaviour of solution V for large
can be obtained quite
easily. From Figure
6.1, we may assume that
so
that (
6.15) is approximated by
Hence, from (
6.11) the density is given by
Thus,
tends to zero as
tends to infinity. Since the plasma
is isothermal the pressure also tends to zero. This means that the
solution can eventually match onto the interstellar plasma at large
distances from the Sun. Thus, solution IV is a physically realistic
model of the solar wind. It predicts that the plasma will be
supersonic beyond the critical point.
Example 6.2.2
The behaviour of solution IV for large
is quite different. Again
using Figure
6.1, we see that
tends to zero as
tends
to infinity. Thus, (
6.15) may now be approximated by
The mass continuity equation gives the density as
Since the density tends to a constant value so will the pressure.
Thus, the solar breeze solution is unphysical since it cannot be
contained by the extremely small interstellar pressure.
Thus, the Parker solar wind model is given by solution IV. The plasma
starts at the solar surface with a small velocity that increases
towards the critical point. At the critical point the speed reaches
the sound speed. Then the flow becomes (and remains) supersonic
while the gas pressure decreases.
Example 6.2.3
We may calculate the critical radius,
. The radius of the Sun
is
m. Assuming a typical coronal
temperature of
K the sound speed is
The critial radius is
To put this into context, the radius of the Earth's orbit is
. Thus, the solar wind is highly
supersonic by the time it reaches the Earth. To calculate the actual
wind speed from Parker's model we set and solve for .
Hence,
This may be solved using the NewtonRaphson method to give
Observations at give the quiet solar wind as
Thus, Parker's solar wind model gives quite a good estimation of the
velocity.
There are three points to note
 As an alternative energy equation it is possible to try a
polytrope of the form
 The real solar wind does not come from the whole of the solar
surface but only from the coronal hole regions where the magnetic
field is open. This is only about 20 percent of the whole surface.
The strong magnetic field of the other 80 percent is closed and
effectively holds in the hot corona.
 The real solar wind probably ends in a shock called the
heliopause at about .
Next: About this document ...
Up: The Solar Wind
Previous: Introduction
Prof. Alan Hood
20010111