The Solar Wind

The solar wind is a high-speed particle stream continuously blown out from the Sun into interplanetary space. It extends far beyond the orbit of the Earth, and terminates in a shock front, called the heliopause, where it interfaces with the weakly ionized interstellar medium. The heliopause is predicted to lie between 110 and 160 AU (1 Astronomical Unit is $1.5\times 10^{11}$m) from the centre of the Sun. Voyager 1 is expected to pass through the heliopause sometime in the next decade: hopefully, it will still be functional at that time!

In the vicinity of the Earth, (i.e., at about 1 AU from the Sun) the solar wind velocity typically ranges between 300 and 1400 ${\rm km}\,{\rm s}^{-1}$. The average value is approximately $500\,{\rm km}\,{\rm s}^{-1}$, which corresponds to about a 4 day time of flight from the Sun. Note that the solar wind is both super-sonic and super-Alfvénic.

The solar wind is predominately composed of protons and electrons.

Amazingly enough, the solar wind was predicted theoretically by Eugine Parker a number of years before its existence was confirmed using satellite data. Parker's prediction of a super-sonic outflow of gas from the Sun is a fascinating scientific detective story, as well as a wonderful application of plasma physics.

The solar wind originates from the solar corona. The solar corona is a hot, tenuous plasma surrounding the Sun, with characteristic temperatures and particle densities of about $10^6$K and $10^{14}\,{\rm m}^{-3}$, respectively. Note that the corona is far hotter than the solar atmosphere, or photosphere. In fact, the temperature of the photosphere is only about $6000$K. It is thought that the corona is heated by Alfvén waves emanating from the photosphere. The solar corona is most easily observed during a total solar eclipse, when it is visible as a white filamentary region immediately surrounding the Sun.

Let us start, following Chapman, by attempting to construct a model for a static solar corona. The equation of hydrostatic equilibrium for the corona takes the form

$$\frac{dp}{dr} = - \rho\,\frac{G\,M_\odot}{r^2},$$ (731)

where $G= 6.67\times 10^{-11}\,{\rm m}^{3}\,{\rm s}^{-2}\,{\rm kg}^{-1}$ is the gravitational constant, and $M_\odot=2\times 10^{30}\,{\rm kg}$ is the solar mass. The plasma density is written

$$\rho\simeq n\,m_p,$$ (732)

where $n$ is the number density of protons. If both protons and electrons are assumed to possess a common temperature, $T(r)$, then the coronal pressure is given by

$$p = 2\,n\,T.$$ (733)

The thermal conductivity of the corona is dominated by the electron thermal conductivity, and takes the form [see Eqs. (260) and (280)]

$$\kappa = \kappa_0\,T^{5/2},$$ (734)

where $\kappa_0$ is a relatively weak function of density and temperature. For typical coronal conditions this conductivity is extremely high: i.e., it is about twenty times the thermal conductivity of copper at room temperature. The coronal heat flux density is written

$${\bf q} = - \kappa\,\nabla T.$$ (735)

For a static corona, in the absence of energy sources or sinks, we require

$$\nabla\!\cdot\!{\bf q} = 0.$$ (736)

Assuming spherical symmetry, this expression reduces to

$$\frac{1}{r^2}\frac{d}{dr}\!\left(r^2\,\kappa_0\,T^{5/2}\,\frac{dT}{dr}\right) =0.$$ (737)

Adopting the sensible boundary condition that the coronal temperature must tend to zero at large distances from the Sun, we obtain

$$T(r) = T(a)\left(\frac{a}{r}\right)^{2/7}.$$ (738)

The reference level $r=a$ is conveniently taken to be the base of the corona, where $a\sim 7\times 10^5\,{\rm km}$, $n\sim 2\times 10^{14}\,{\rm m}^{-3}$, and $T\sim 2\times 10^{6}$K.

Equations (731), (732), (733), and (738) can be combined and integrated to give

$$p(r) = p(a) \exp\left\{\frac{7}{5}\,\frac{G\,M_\odot\,m_p}{2\,T(a)\,a} \left[\left(\frac{a}{r}\right)^{5/7}-1\right]\right\}.$$ (739)

Note that as $r\rightarrow\infty$ the coronal pressure tends towards a finite constant value:

$$p(\infty) = p(a)\,\exp\left\{-\frac{7}{5}\,\frac{G\,M_\odot\,m_p}{2\,T(a)\,a} \right\}.$$ (740)

There is, of course, nothing at large distances from the Sun which could contain such a pressure (the pressure of the interstellar medium is negligibly small). Thus, we conclude, with Parker, that the static coronal model is unphysical.

Since we have just demonstrated that a static model of the solar corona is unsatisfactory, let us now attempt to construct a dynamic model in which material flows outward from the Sun.