Voyager and Helios measurements show the solar wind proton density falling off as…

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Fluid and magnetohydrodynamic description · Laboratory, industrial and space plasmas

Plasma Physics, Ch From particles to fluids: the distribution function, its moments and the two-fluid model

Problem

Voyager and Helios measurements show the solar wind proton density falling off as

n_p \propto r^{-2}

with heliocentric distance

r

(the flow speed is very nearly constant, so this follows from mass conservation through the expanding flux tube). At

r_1 = 1\,\mathrm{AU}

the measured proton temperature is

T_1 = 1.0\times10^5\,\mathrm{K}

; at

r_2 = 4\,\mathrm{AU}

the measured proton temperature is

T_2 = 3.0\times10^4\,\mathrm{K}

. The proton mean free path in the solar wind is of order the distance to the Sun or greater. ((a)) State the value of

\gamma_s

appropriate to treating the expanding proton gas with the adiabatic closure, sharing the compression energy isotropically among three velocity components, and say what physical idealisation this closure assumes about the heat flux

\mathbf{q}_s

. ((b)) Using that closure with

p\propto n^{\gamma_s}

and the measured density law

n_p\propto r^{-2}

, predict the proton temperature at

r_2 = 4\,\mathrm{AU}

given

T_1

r_1 = 1\,\mathrm{AU}

. ((c)) Compare the prediction with the measured

T_2

and explain, using the mean free path and the idea of a heat flux carried by free-streaming particles, why an exact match is not expected.

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