Introduction

Geomagnetic storms were first noticed in the 19th Century during which the Earth's magnetic field suddenly increased by about $10^{-3}$G and was followed by a slow decay. Normally these geomagnetic storms occured one or two days after a large solar flare. In addition, it was noticed that flares and geomagnetic activity seemed to have the same 11 year periodicity. Thus, the implication was that there was an electrical conncetion between the Earth and the Sun.

This suggestion of a connection between the Earth and the Sun led to the idea of an extended corona and in 1957 Chapman (1957) presented his model of a hydrostatic corona. The basic idea was to consider an energy equation with only conduction and hydrostatic pressure balance. Thus, the energy equation, in spherical coordinates, reduces to

$${d\over dr}\left (r^{2}\kappa {dT\over dr} \right ) = 0.$$ (6.1)

The conductivity, $\kappa$, in an ionised hydrogen plasma is given by

$$\kappa = \kappa_{0}T^{5/2},$$ (6.2)

where $\kappa_{0}$ is a constant. Notice how the conductivity depends on temperature. This means that thermal conduction is more efficient at smoothing out temperature variations at higher temperature values. Thus, integrating (6.1) gives

$$r^{2}\kappa_{0}T^{5/2}{dT\over dr} = C,$$ (6.3)

where $C$ is a constant. This may be rearranged into the form

$${2\over 7}{d\over dr}\left ( T^{7/2} \right ) = {C \over \kappa_{0} r^{2}},$$

and integrating gives

$$T^{7/2} = - {C\over \kappa_{0} r} + D.$$ (6.4)

The constants $C$ and $D$ are determined by the boundary conditions on the temperature, namely

$$T = T_{0} \hbox{ at }r = R_{\circ}, \qquad T \rightarrow 0 \hbox{ as } r \rightarrow \infty.$$

Hence, (6.4) becomes ($D=0$ and $C = - T_{0}^{7/2}\kappa_{0}R_{\circ}$)

$$T = T_{0}\left ({R_{\circ}\over r}\right )^{2/7}.$$ (6.5)

Notice that for a ``surface'', coronal temperature of $10^{6}$K the temperature of the corona at the Earth, $1 AU = 214R_{\circ}$, is about $10^{5}$K. This is actually a little low but does not seem too bad. The main problem with this model appears when hydrostatic pressure balance is considered. Thus,

$${dp\over dr} = - {GM_{\circ}\rho \over r^{2}}$$

and with the gas law $p = \rho { R}T$ and $T$ given by (6.5) we obtain

$$p = p_{0}\hbox{exp}\left \{{7 G M_{\circ}\rho_{0}\over 5 ... ...eft ({R_{\circ}\over r}\right)^{5/7} - 1 \right ] \right \},$$ (6.6)

where $p_{0}$ and $\rho_{0}$ are the values of the pressure and density at the solar surface. However, it is clear from (6.6) that the pressure tends to a constant value as $r$ tends to infinity. This does not make physical sense. As one tends to large distances away from the Sun the pressure should continue to drop down to the value of the interstellar pressure which is approximately $10^{-15}p_{0}$. In addition, the density tends to infinity for large values of $r$. Thus, the hydrostatic model is not physical.