What is the solar wind speed as function of distance from the Sun?

Question

Someone must have calculated at least a "zero-order approximation" of the behavior of solar wind speed, density, and pressure as a function of distance from the Sun, but the heck if I can find it. Anyone know of some good references for this?

Answer 1

A fairly complete discussion is found in Ryden's online coursework. The relevant chapter is online here.

An interesting takeaway is that the velocity increases as it leaves the sun, and then flattens out. It's pretty much flat by the time it reaches Earth, and from then on it's simply a density going down with r^2.

Answer 2

Someone must have calculated at least a "zero-order approximation" of the behavior of solar wind speed, density, and pressure as a function of distance from the Sun, but the heck if I can find it. Anyone know of some good references for this?

The theory of the solar wind evolution as a function of radial distance from the sun has been analytically calculated to at least first order for over 30 years [e.g., Schwartz and Marsch, 1983]. While these approximations are okay, it has been known for nearly as long that the evolution of some parameters like electron temperature (e.g., see https://physics.stackexchange.com/a/218643/59023 for definition in a kinetic gas) depend upon the speed of the solar wind in which said temperature is measured [e.g., Pilipp et al., 1990].

Some parameters seem to follow what one would naively expect from simple expansion models while others like the electron heat flux does not follow the expectation of a collisionless expansion along the quasi-static magnetic field [e.g., Scime et al., 1994]. Also, the ion temperature change with radial distance has long been known to decrease much less rapidly than simple adiabatic expansion would predict [e.g., Richardson et al., 1995]. Note that the temperatures are not isotropic and the particle are not in thermal or thermodynamic equilibrium [e.g., Wilson et al., 2018].

There is also the so called double-adiabatic relations, that assume no collisions and no heat fluxes, which predict the following relationships between parameters: $$ \begin{align} T_{\perp} & \propto B \tag{0a} \\ T_{\parallel} & \propto \left( \frac{n}{B} \right)^{2} \tag{0b} \end{align} $$ where $n$ is the number density, $B$ is the quasi-static magnetic field magnitude, $T_{j}$ is the j^th component of the temperature, and $\parallel$($\perp$) is parallel(perpendicular) to the local quasi-static magnetic field vector. For a truly adiabatic expansion, the density and temperature would follow: $$ \begin{align} n\left( r \right) & \propto \left( \frac{r}{R_{o}} \right)^{-2} n_{o} \tag{1a} \\ T\left( r \right) & \propto \left( \frac{r}{R_{o}} \right)^{-4/3} T_{o} \tag{1b} \end{align} $$ where $r$ is radial distance from the sun, $R_{o}$ is some reference distance, and $n_{o}$($T_{o}$) is a typical number density(temperature) at $R_{o}$. Note that in the absence of collisions or heat flux or heating/cooling, one would recover the double-adiabatic relations in Equations 0a and 0b with the resultant relationships being $T_{\perp} \propto r^{-2}$ and $T_{\parallel}$ ~ constant. This is not observed, i.e., neither temperature component is constant and neither decreases anywhere near as quickly as $r^{-2}$ [e.g., Stverak et al., 2015].

The relationship in Equation 1a is nearly true because it has been measured in multiple studies that the total particle flux, $n \ V \ r^{2}$ ($V$ is bulk flow speed), is a constant with radial distance. Since it is also well known from measurements that the solar wind speed is roughly constant with radial distance, after some threshold distance from the solar surface, the constancy of the total particle flux implies that $n \propto r^{-2}$ [e.g., Schwartz and Marsch, 1983, and references therein].

If magnetic flux is conserved, then we can argue under spherical symmetry that: $$ \begin{align} B_{r}\left( r \right) & \propto \left( \frac{R_{o}}{r} \right)^{2} B_{o} \tag{2a} \\ B_{\phi}\left( r \right) & \propto - \frac{R_{o}^{2}}{r} B_{o} \tag{2b} \end{align} $$ where $B_{o}$ is the typical magnetic field magnitude at $R_{o}$ and the subscript $r$($\phi$) corresponds to the radial(azimuthal) direction. This is known as the Parker spiral approximation, after Eugene Parker.

References

• W.G. Pilipp, et al., "Large-scale variations of thermal electron parameters in the solar wind between 0.3 and 1 AU," J. Geophys. Res. 95(A5), pp. 6305-6329, doi:10.1029/JA095iA05p06305, 1990.
• J.D. Richardson, et al., "Radial evolution of the solar wind from IMP 8 to Voyager 2," Geophys. Res. Lett. 22(4), pp. 325-328, doi:10.1029/94GL03273, 1995.
• E.E. Scime, et al., "Regulation of the solar wind electron heat flux from 1 to 5 AU: Ulysses observations," J. Geophys. Res. 99(A12), pp. 23,401-23,410, doi:10.1029/94JA02068, 1994.
• S.J. Schwartz and E. Marsch "The radial evolution of a single solar wind plasma parcel," J. Geophys. Res. 88(A12), pp. 9919-9932, doi:10.1029/JA088iA12p09919, 1983.
• S. Stverak, et al., "Electron energetics in the expanding solar wind via Helios observations," J. Geophys. Res. 120(10), pp. 8177-8193, doi:10.1002/2015JA021368, 2015.
• L.B. Wilson III, et al., "The Statistical Properties of Solar Wind Temperature Parameters Near 1 au," Astrophys. J. Suppl. 236(2), pp. 41, doi:10.3847/1538-4365/aab71c, 2018.