= Solution
A <stellar wind> removes <mass> and carries <energy> and <angular momentum> from a star into its surroundings. Hot gas can escape through thermal <pressure>; radiation transfers <momentum> through continuum or spectral-line <absorption>, and radiation acting on dust can drive surrounding gas. Magnetic stresses can extract rotational <energy> and <angular momentum>. These mechanisms have different dependencies on heating, opacity, stellar <luminosity> and field geometry.
For a steady spherical isothermal pressure-driven model, <mass> conservation gives $\dot M=4\pi r^2\rho u$. With $p=a^2\rho$ and constant <isothermal sound speed> $a$, the radial <Euler equation> becomes the <Parker wind equation>
$$
\left(u-\frac{a^2}{u}\right)\frac{du}{dr}=\frac{2a^2}{r}-\frac{GM}{r^2}.
$$
A smooth <transonic flow> solution crosses $u=a$ at $r_s=GM/(2a^2)$. Its inner subsonic atmosphere connects to a supersonic outflow; continuing heat input is needed to keep that expanding wind isothermal. Other boundary <pressures> can select a breeze rather than this escaping branch, and the mass-loss rate needs boundary/heating information. A polytropic outflow instead converts finite <enthalpy> into escape <energy> and has a different terminal budget.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2012/paper-1-pressure-driven-wind.png]
{title=Pressure-driven isothermal wind: a regular sonic crossing selects the transonic branch}
The <solar wind> supplies a nearby <plasma> example. Spectral-line <Doppler shifts> and profiles diagnose more distant winds; blueshifted <absorption> against stellar light indicates material moving toward the observer. Inference of a mass-loss rate additionally needs <density>, <ionization>, geometry and <velocity> information, rather than a line shift alone.
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