= Solution
Insert the pressure-negligible velocity law into the steady spherical mass flux. With $x=r/R_e$,
$$
\rho(r)=\rho_e\frac{R_e^2v_e}{r^2v(r)}=\rho_ex^{-3/2}.
$$
An <adiabatic equation of state> $p\rho^{-\gamma}=\text{constant}$ and the <ideal gas law> give
$$
\boxed{p(r)=p_ex^{-3\gamma/2},\qquad
T(r)=T_ex^{-3(\gamma-1)/2}.}
$$
For a monatomic gas with $\gamma=5/3$,
$$
\boxed{p\propto r^{-5/2},\qquad T\propto r^{-1}.}
$$
The normalization $T_e$, and hence $p_e=n_ek_BT_e$, 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.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2015/paper-1-ballistic-wind.png]
{title=Normalized velocity, density, temperature and pressure of a marginal-escape wind}
{description=Original power-law calculation for a monatomic gas in the pressure-negligible momentum approximation. Finite enthalpy requires a coupled wind solution.}
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