where is the radial expansion speed. The continuity equation reduces to

In order to obtain a closed set of equations, we now need to adopt an equation of state for the corona, relating the pressure, , and the density, . For the sake of simplicity, we adopt the simplest conceivable equation of state, which corresponds to an

where is a constant. Note that more realistic equations of state complicate the analysis, but do not significantly modify any of the physics results.

Equation (742) can be integrated to give

Let us restrict our attention to coronal temperatures which satisfy

where is the radius of the base of the corona. For typical coronal parameters (see above), K, which is certainly greater than the temperature of the corona at . For , the right-hand side of Eq. (745) is negative for , where

and positive for . The right-hand side of (745) is zero at , implying that the left-hand side is also zero at this radius, which is usually termed the ``critical radius.'' There are two ways in which the left-hand side of (745) can be zero at the critical radius. Either

or

Note that is the coronal

As is easily demonstrated, if Eq. (748) is satisfied then has the
same sign for all , and is either a monotonically
increasing, or a monotonically decreasing, function of . On the other
hand, if Eq. (749) is satisfied then has the same
sign for all , and has an extremum close to . The flow
is either super-sonic for all , or sub-sonic for all . These
possibilities lead to the existence of *four* classes of solutions
to Eq. (745), with the
following properties:

- is sub-sonic throughout the domain . increases with , attains a maximum value around , and then decreases with .
- a unique solution for which increases monotonically with , and .
- a unique solution for which decreases monotonically with , and .
- is super-sonic throughout the domain . decreases with , attains a minimum value around , and then increases with .

Each of the classes of solutions described above fits a different
set of boundary conditions at and
. The
*physical* acceptability of these solutions depends on these
boundary conditions. For example, both Class 3 and Class 4 solutions can
be ruled out as plausible models for the solar corona since they predict
*super-sonic* flow at the base of the corona, which is not observed, and is
also not consistent with a static solar photosphere. Class 1 and Class 2 solutions
remain acceptable models for the solar corona on the basis of their
properties around , since they both predict sub-sonic flow in this region.
However, the Class 1 and Class 2 solutions behave quite differently
as
, and the physical acceptability of these two
classes hinges on this difference.

Equation (745) can be rearranged to give

(750) |

where is a constant of integration.

Let us consider the behaviour
of Class 1 solutions in the limit
. It is
clear from Fig. 17 that, for Class 1 solutions, is less than unity and monotonically
decreasing as
. In the large- limit, Eq. (751)
reduces to

(752) |

(753) |

(754) |

Let us consider the behaviour of the Class 2 solution
in the limit
. It is
clear from Fig. 17 that, for the Class 2 solution, is greater than unity and monotonically
increasing as
. In the large- limit,
Eq. (751) reduces to

(755) |

(756) |

We conclude that the only solution to Eq. (745) which is consistent
with physical boundary conditions at and
is
the Class 2 solution. This solution predicts that the
solar corona expands radially outward at
relatively modest, sub-sonic velocities close to the Sun,
and gradually accelerates
to super-sonic velocities as it moves further away from the Sun.
Parker termed this continuous, super-sonic expansion of the corona
the *solar wind*.

Equation (751) can be rewritten