1.3. A brief history of the discovery of the solar wind

Until fairly recently, it was believed that the Sun was a perfect, unchanging sphere, and that the only source of interaction between the Sun and the other planets was the force of gravitation. This idea began to change with the invention of the telescope by Galileo in 1609 and his subsequent observation of sunspots on the surface of the Sun in 1610. However, even this discovery was not enough to persuade the scientists of the time and this was explained away as the possible transit of the inner planets across the solar disk.
In 1859, British astronomer Richard Carrington, while observing sunspots from a projected image of the Sun, noticed two patches of particularly intense light appear and fade within 5 minutes from the largest visible sunspot group \citep{carrington1859description}. Some time later, large disturbances in telegraph systems were observed, along with intense auroral events. To Carrington, this was suggestive of a connection between the event that he had observed on the Sun (now known as a solar flare) and the geomagnetic disturbances that followed.
It was suggested by Kristian Birkeland in 1908 that since there appeared to be a causal relation between solar activity and geomagnetic disturbances, it was possibly due to energetic particles (or ''corpuscular radiation") being emitted by the Sun at all times in all directions and the interaction of these particles with the Earth's magnetic field caused these geomaagnetic events \citep{birkeland1908norwegian}. However, this idea did not gain significant traction in the scientific community.
In the early 1950s, German physicist Ludwig Biermann, when analyzing tails of comets, would come to the same conclusion as Birkeland, thus providing more support to the idea of the existence of the solar corpuscular radiation \citep{Biermann51}. It was observed that comets have two types of tails: one gaseous, which pointed straight away from the Sun, and the other which curves away, made up of dust. The shape of the curved tail could be explained as being produced by the solar radiation pressure and gravity acting on the dust grains.
However, the gaseous tail raised some interesting questions, as they showed irregularities that accelerated away from the Sun. Biermann proposed that this can be described as the interaction of the constant solar corpuscular radiation with the comets' tails.

In 1957, Chapman reasoned that the high temperature of the corona implied a high conductivity of the coronal electrons, which in turn implied a small temperature gradient. This meant that there should be an extension of this high coronal temperature well out into interplanetary space. This argument can then be combined with the assumption of hydrostatic equilibrium, in order to deduce the radial dependence of the pressure and density of the solar corona. Hydrostatic equilibrium implies that the inward force acting on any fluid element due to gravity is balanced by the outward force due to the pressure gradient. This can be expressed as

\frac{d P}{d r} = - \frac{G M_{⊙} ρ}{r^{2}}

Solving this yields pressures of $10^{-6} $ N/m $^{2}$ as $r \to \infty$ , which cannot be supported by known estimates of the pressure of the local interstellar medium, which is of the order of $10^{- 13} - 10^{- 14}$ N/m $^{2}$ .
Thus, the solar corona, as described by Chapman's model, cannot blend into the interstellar background. Eugene Parker realized this problem, stating in his now famous paper, ''... we conclude that probably it is not possible for the solar corona, or, indeed, perhaps the atmosphere of any star, to be in complete hydrostatic equilibrium out to large distances." \citep

Instead of assuming a corona in hydrostatic equilibrium, Parker adopted a hydrodynamic approach, using the equations of mass, momentum and energy conservation:

\frac{1}{r^{2}} \frac{d}{d r} (r^{2} ρ u) = 0 $ $ $ $ \frac{1}{u} \frac{d u}{d r} = - \frac{d P}{d r} - \frac{G M_{⊙} ρ}{r^{2}}

\frac{1}{r^{2}} \frac{d}{d r} [r^{2} ρ u (\frac{1}{2} u^{2} + \frac{3}{2} \frac{P}{ρ})] = - \frac{1}{r^{2}} \frac{d}{d r} (r^{2} P u) - ρ u \frac{G M_{⊙}}{r^{2}} + S (r)

where $S (r)$ represents an energy source or sink.
To simplify the problem, Parker assumed a polytropic equation of state, effectively eliminating the energy equation from our system.

P = P_{0} {(\frac{ρ}{ρ_{0}})}^{α}

For the present discussion, we will consider the case $α = 1$ , which corresponds to an isothermal corona. Then, the pressure is given by
$P = 2 n k T$ and assuming that the corona is made up of a quasi-neutral plasma, we have $ρ = n (m_{p} + m_{e}) \approx n m$ . Substituting these expressions of pressure and mass density in the momentum equation gives us the following equation after a few lines of algebra:

\frac{1}{u} \frac{d u}{d r} (u^{2} - \frac{2 k T}{m}) = \frac{4 k T}{m r} - \frac{G M_{⊙}}{r^{2}}

We notice that there exists a value of $r$ for which the RHS is zero, which is termed the ''critical radius'', $r_{c}$ . This can happen when one or both of the following is true:

u^{2} (r_{c}) = \frac{2 k T}{m}, or \frac{1}{u} \frac{d u}{d r} |_{r = r_{c}} = 0

This leads us to 4 classes of solutions for $u (r)$ .

$u (r)$ increases monotonically from $r_{0}$ to $r_{c}$ , and then decreasing with increasing $r$ .
$u (r)$ increases monotonically from the base of the corona, at $r_{0}$ , with $u (r_{c}) = u_{c} = \sqrt{2 k T / m}$ .
$u (r)$ decreases monotonically from the base of the corona, at $r_{0}$ , with $u (r_{c}) = u_{c} = \sqrt{2 k T / m}$ .
$u (r)$ decreases from $r_{0}$ to $r_{c}$ , and then increases with increasing $r$ .

Although these solutions are all mathematically possible, they show very different behavior at the boundary conditions $r = 0$ and $r \to \infty$ .

\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{Figures/Parker_solution_SW.png}
\caption{Different classes of solutions to \eqref{eq: Parker_equation}. \textit{Figure reproduced from} \citep{Hundhausen1972book}}.
\label{fig:parker_SW}
\end

Classes 3 and 4 are not feasible solutions, since for these solutions, $u (r) > \sqrt{2 k T / m}$ near $r_{0}$ . Given a coronal temperature of $T \approx 10^{6} K$ , this implies $u (r) > 100$ km/s, which is too high for plasma motions deep in the corona.
Classes 1 and 2 both have $u (r)$ increasing with distance from $r_{0}$ to $r_{c}$ , but exhibit different behavior as $r \to \infty$ . It can be shown that Class 1 solutions give a finite pressure at large distances, which leads to the same problem as Chapman's solution, with the solar wind not being able to blend into the interstellar background.
The solution corresponding to Class 2 implies that $u (r)$ keeps on increasing as $r \to \infty$ , which yields a vanishing pressure at large distances. This solution corresponding to a continuous, supersonic flow of plasma was referred to by Parker as the 'solar wind'. However, this view was challenged by Chamberlain who discarded the idea of a isothermal corona, instead assuming an evaporative model, and proposed the idea of a subsonic 'solar breeze'.
It would not be until in situ observations became available that this debate would be settled once and for all. The first measurements confirming the existence of a continuous supersonic plasma flow were made by the Mariner 2 spacecraft \citep{NeugebauerSnyder66}, thus validating Parker's theory.