The Solar Wind
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This is a Demonstration of the Parker model (Parker 1958) for the solar wind. The plot shows the solar wind flux velocity as a function of the distance from the Sun. The temperature is assumed to be constant. You can change its value to see how it influences the resulting velocity. Distance is expressed in astronomical units (1 AU = 149 598 000 km). The blue dashed line indicates where the velocity equals the sound speed. The black dashed line shows the location of the Earth.
The solar wind is a stream of ionized particles, mainly hydrogen (therefore electrons and protons) and helium, ejected by the Sun's corona. These particles achieve a very high kinetic energy and are thus able to escape the Sun's gravity. E. Parker modeled the solar wind with a steady-state outflow. Parker's model assumes that the solar wind flux behaves like an ideal gas expanding isothermally into a vacuum. The pressure contribution of the magnetic field is neglected. The solar wind flux velocity as a function of radial distance is obtained from Euler's equation of motion and the equation of continuity, assuming spherical symmetry and steady state (with all time derivatives equal to zero).
v = (, 0, 0),
where = gas velocity, = radial distance, = pressure, = mass of the Sun, = gravitational constant, and = density.
From the ideal gas equation of state, , where = Boltzmann constant, = gas temperature, and = mass of the ionized particle . We consider the ionized hydrogen; therefore the pressure is given by the sum of the pressures of the protons and electrons, which are considered to be equal: . Thus, we obtain , where = mass of the proton. One can define:
where = sound speed in the plasma and = sonic radius (distance at which the solar wind velocity is equal to the sound speed). The point (, ) is called the "sonic point" (or "critical point").
Then, the differential equation can be reduced to the dimensionless form: (1).
Integration with Mathematica
The integral of equation (1) involves the Lambert -function, named ProductLog in Mathematica. The Lambert -function is defined as the inverse of the function . It is a multivalued function with an infinite number of solution branches, labeled by convention by an integer subscript: , for = 0, ±1, ±2, …. If is a real number, the only two branches that take on real values are and . When using DSolve with Mathematica, we find only the branch. We thus input the solution for manually. In this way, we obtain several possible solutions. Only one of these solutions is a monotonic increasing function, and thus physically consistent with the solar wind. Finally, we back substitute for and and plot the result for the temperature varying between 0.5×and 4× kelvin. The blue dashed vertical line passes through the sonic point (the point at which the velocity of the flow is equal to the sound velocity in plasma) and separates the subsonic and supersonic domains. Since the sound velocity changes as the temperature changes, the separation line is different for different isotherms. The black dashed line represents the Earth's orbit (r = 1 AU).
This Demonstration shows that, regardless of the temperature of the plasma, the solar wind particles achieve a supersonic velocity within a short distance. Even though it is very simple, Parker's model agrees with observations.
E. N. Parker, "Dynamics of the Interplanetary Gas and Magnetic Fields," Astrophysical Journal, 128, 1958 p. 664.