Solar Wind

The solar wind is a high-speed particle stream continuously blown out from the Sun into interplanetary space (Priest 1984). 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, which is the mean Earth-Sun distance, is $1.5\times 10^{11}$ m) from the center of the Sun (Suess 1990). The Voyager 1 spacecraft is inferred to have crossed the heliopause in August of 2012 (Webber and McDonald 2013).

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 ${\rm km}\,{\rm s}^{-1}$ (Priest 1984). 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, and is predominately composed of protons and electrons.

The solar wind was predicted theoretically by Eugine Parker (Parker 1958) a number of years before its existence was confirmed by means of satellite data (Neugebauer and Snyder 1966). Parker's prediction of a super-sonic outflow of gas from the Sun is a fascinating application of plasma physics.

The solar wind originates from the solar corona, which 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 (Priest 1984). The corona is actually 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 (Priest 1984). 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 (Chapman 1957), by attempting to construct a model for a static solar corona. The equation of hydrostatic equilibrium for the corona takes the form

$$\frac{dp}{dr} = - \rho\,\frac{G\,M_\odot}{r^{2}},$$ (8.51)

where $G= 6.67\times 10^{-11}\,{\rm m}^{3}\,{\rm s}^{-2}\,{\rm kg}^{-1}$ is the gravitational constant, $M_\odot=2\times 10^{30}\,{\rm kg}$ the solar mass (Yoder 1995), and $r$ the radial distance from the center of the Sun. The plasma density is written

$$\rho\simeq n\,m_p,$$ (8.52)

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

$$p = 2\,n\,T.$$ (8.53)

The thermal conductivity of the corona is dominated by the electron thermal conductivity, and takes the form [see Equations (4.70) and (4.89)]

$$\kappa = \kappa_0\,T^{5/2},$$ (8.54)

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

$${\bf q} = - \kappa\,\nabla T.$$ (8.55)

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

$$\nabla\cdot{\bf q} = 0.$$ (8.56)

Assuming spherical symmetry, this expression reduces to (Richardson 2019)

$$\frac{1}{r^{2}}\frac{d}{dr}\!\left(r^{2}\,\kappa_0\,T^{5/2}\,\frac{dT}{dr}\right) =0.$$ (8.57)

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

$$T(r) = T(a)\left(\frac{a}{r}\right)^{2/7}.$$ (8.58)

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 (Priest 1984).

Equations (8.51), (8.52), (8.53), and (8.58) can be combined and integrated to give

$$p(r) = p(a) \exp\left\{\frac{7}{5}\,\frac{G\,M_\odot\,m_p}{2\,T(a)\,a} \left[\left(\frac{a}{r}\right)^{5/7}-1\right]\right\}.$$ (8.59)

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

$$p(\infty) = p(a)\,\exp\left[-\frac{7}{5}\,\frac{G\,M_\odot\,m_p}{2\,T(a)\,a} \right] = p(a)\,\exp\left[-\frac{14}{5}\,\frac{T_0}{T(a)}\right],$$ (8.60)

where $T_0$ is defined in Equation (8.66). There is, of course, nothing at large distances from the Sun that could contain such a pressure (the pressure of the interstellar medium is negligibly small). Thus, we conclude, following Parker, that the static coronal model is unphysical.

We have just demonstrated that a static model of the solar corona is unsatisfactory. Let us, instead, attempt to construct a dynamic model in which material flows outward from the Sun.