Parker Model of Solar Wind

By symmetry, we expect a purely radial, steady-state, coronal outflow. The radial equation of motion of the corona [which is a modified version of Equation (8.2)] takes the form (Richardson 2019)

$$\rho\,u\,\frac{du}{dr} = -\frac{dp}{dr} - \rho\,\frac{G\,M_\odot}{r^{2}},$$ (8.61)

where $u$ is the radial expansion speed. The continuity equation [which is equivalent to Equation (8.1)] reduces to (Richardson 2019)

$$\frac{1}{r^{2}}\frac{d(r^{2}\,\rho\,u)}{dr} = 0.$$ (8.62)

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

$$p = \frac{2\,\rho\,T}{m_p},$$ (8.63)

where $T$ is a constant. More realistic equations of state complicate the analysis, but do not significantly modify any of the physics results (Priest 1984).

Equation (8.62) can be integrated to give

$$r^{2}\,\rho\,u = I,$$ (8.64)

where $I$ is a constant. The previous expression simply states that the mass flux per unit solid angle, which takes the value $I$, is independent of the radius, $r$. Equations (8.61), (8.63), and (8.64) can be combined to give

$$\frac{1}{u} \,\frac{du}{dr}\left(u^2 - \frac{2\,T}{m_p}\right) = \frac{4\,T}{m_p\,r} - \frac{G\,M_\odot}{r^{2}}.$$ (8.65)

Let us restrict our attention to coronal temperatures that satisfy

$$T < T_0 \equiv \frac{G\,M_\odot\,m_p}{4\,a},$$ (8.66)

where $a$ is the radius of the base of the corona. For typical coronal parameters, $T_0\simeq 5.8\times 10^6$ K, which is certainly greater than the temperature of the corona at $r=a$. For $T<T_0$, the right-hand side of Equation (8.65) is negative for $a<r<r_c$, where

$$\frac{r_c}{a} = \frac{T_0}{T},$$ (8.67)

and positive for $r_c<r<\infty$. The right-hand side of (8.65) is zero at $r=r_c$, 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 (8.65) can be zero at the critical radius. Either

$$u^2(r_c) = u_c^{2} \equiv \frac{2\,T}{m_p},$$ (8.68)

or

$$\frac{du(r_c)}{dr} = 0.$$ (8.69)

Note that $u_c$ is the coronal sound speed.

As is easily demonstrated, if Equation (8.68) is satisfied then $du/dr$ has the same sign for all $r$, and $u(r)$ is either a monotonically increasing, or a monotonically decreasing, function of $r$. On the other hand, if Equation (8.69) is satisfied then $u^2-u_c^{2}$ has the same sign for all $r$, and $u(r)$ has an extremum close to $r=r_c$. The flow is either super-sonic for all $r$, or sub-sonic for all $r$. These possibilities lead to the existence of four classes of solutions to Equation (8.65), with the following properties:

1. $u(r)$ is sub-sonic throughout the domain $a<r<\infty$. $u(r)$ increases with $r$, attains a maximum value around $r=r_c$, and then decreases with $r$.
2. a unique solution for which $u(r)$ increases monotonically with $r$, and $u(r_c) = u_c$.
3. a unique solution for which $u(r)$ decreases monotonically with $r$, and $u(r_c) = u_c$.
4. $u(r)$ is super-sonic throughout the domain $a<r<\infty$. $u(r)$ decreases with $r$, attains a minimum value around $r=r_c$, and then increases with $r$.

These four classes of solutions are illustrated in Figure 8.2.

Figure 8.2: The four classes of Parker outflow solutions for the solar wind.

Each of the classes of solutions described previously fits a different set of boundary conditions at $r=a$ and $r\rightarrow\infty$. 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 because 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 $r=a$, because they both predict sub-sonic flow in this region. However, the Class 1 and Class 2 solutions behave quite differently as $r\rightarrow\infty$, and the physical acceptability of these two classes hinges on this difference.

Equation (8.65) can be rearranged to give

$$\frac{du^2}{dr}\left(1-\frac{u_c^{2}}{u^2}\right) = \frac{4\,u_c^{2}}{r} \left(1-\frac{r_c}{r}\right),$$ (8.70)

where use has been made of Equations (8.66) and (8.67). The previous expression can be integrated to give

$$\left(\frac{u}{u_c}\right)^2 -\ln\left(\frac{u}{u_c}\right)^2 = 4\,\ln\left(\frac{r}{r_c}\right) + 4\,\frac{r_c}{r} + C,$$ (8.71)

where $C$ is a constant of integration.

Let us consider the behavior of Class 1 solutions in the limit $r\rightarrow\infty$. It is clear from Figure 8.2 that, for Class 1 solutions, $u/u_c$ is less than unity and monotonically decreasing as $r\rightarrow\infty$. In the large-$r$ limit, Equation (8.71) reduces to

$$\ln\left(\frac{u}{u_c}\right) \simeq -2\,\ln\left(\frac{r}{r_c}\right),$$ (8.72)

so that

$$u\propto \frac{1}{r^{2}}.$$ (8.73)

It follows from Equation (8.64) that the coronal density, $\rho$, approaches a finite, constant value, $\rho_\infty$, as $r\rightarrow\infty$. Thus, the Class 1 solutions yield a finite pressure,

$$p_\infty= \frac{2\,\rho_\infty\,T}{m_p},$$ (8.74)

at large $r$, which cannot be matched to the much smaller pressure of the interstellar medium. Obviously, Class 1 solutions are unphysical.

Let us consider the behavior of the Class 2 solution in the limit $r\rightarrow\infty$. It is clear from Figure 8.2 that, for the Class 2 solution, $u/u_c$ is greater than unity and monotonically increasing as $r\rightarrow\infty$. In the large-$r$ limit, Equation (8.71) reduces to

$$\left(\frac{u}{u_c}\right)^2 \simeq 4\,\ln \left(\frac{r}{r_c}\right),$$ (8.75)

so that

$$u \simeq 2\,u_c\left[\ln\left(\frac{r}{r_c}\right)\right]^{1/2}.$$ (8.76)

It follows from Equation (8.64) that $\rho\rightarrow 0$ as $r\rightarrow\infty$. Thus, the Class 2 solution yields $p\rightarrow 0$ at large $r$, and can, therefore, be matched to the low pressure interstellar medium.

We conclude that the only solution to Equation (8.65) that is consistent with the physical boundary conditions at $r=a$ and $r\rightarrow\infty$ 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 (8.71) can be rewritten

$$\left(\frac{u^2}{u_c^{2}}-1\right)-\ln\left(\frac{u}{u_c}\right)^2 = 4\,\ln\left(\frac{r}{r_c}\right) + 4\left(\frac{r_c}{r}-1\right),$$ (8.77)

where the constant $C$ is determined by demanding that $u=u_c$ when $r=r_c$. Note that both $u_c$ and $r_c$ can be evaluated in terms of the coronal temperature, $T$, via Equations (8.67) and (8.68). Figure 8.3 shows $u(r)$ calculated from Equation (8.77) for various values of the coronal temperature. It can be seen that plausible values of $T$ (i.e., $T\sim 1$– $2\times 10^6$ K) yield expansion speeds of several hundreds of kilometers per second at 1 AU, which accords well with satellite observations. The critical surface where the solar wind makes the transition from sub-sonic to super-sonic flow is predicted to lie a few solar radii away from the Sun (i.e., $r_c\sim 5\,R_\odot$, where $R_\odot$ is the solar radius). Unfortunately, the Parker model's prediction for the density of the solar wind at the Earth is significantly too high compared to satellite observations. Consequently, there have been many further developments of this model. In particular, the unrealistic assumption that the solar wind plasma is isothermal has been relaxed, and two-fluid effects have been incorporated into the analysis (Priest 1984).

Figure 8.3: Parker outflow solutions for the solar wind. Each curve is labelled by the corresponding coronal temperature in degrees kelvin. The vertical dashed line indicates the mean radius of the Earth's orbit.