 Solution

Insert the pressure-negligible velocity law into the steady spherical mass flux. With $x=r/R_e$,An adiabatic equation of state $p{\rho }^{-\gamma }=\text{constant}$ and the ideal gas law giveFor a monatomic gas with $\gamma =5/3$,The normalization $T_e$, and hence $p_e=n_e{k}_B{T}_e$, was not specified and remains a boundary parameter. If it is large enough for the pressure gradient to affect acceleration, replace the ballistic velocity by the full adiabatic Bernoulli equation solution. The general mass-flux and thermodynamic relations still hold, but these particular radius powers then need not hold.