= Solution
A <stellar wind> carries matter away from a <star>, altering its <mass>, <angular momentum> and surroundings. Hot coronal gas can escape through pressure-driven expansion; radiation transfers <momentum> to gas through continuum or <spectral line> absorption, or to dust that then drags gas. A <magnetic field> can transfer rotational <energy> and <angular momentum> to an outflow. These mechanisms need not give the same terminal speed or <mass loss rate>.
For a steady spherical pressure-driven model, <conservation of mass> gives $\dot M=4\pi r^2\rho u$. Combining the radial <Euler equation> with an <isothermal equation of state> $p=\rho a^2$ gives 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 branch> passes through $u=a$ at $r_s=GM/(2a^2)$, connecting a subsonic inner atmosphere to an escaping supersonic wind. The <isothermal Parker wind> requires continuing heat input as the gas expands; a <polytropic stellar wind> instead tracks a changing <temperature> and must satisfy its own terminal-energy condition. A <subsonic isothermal breeze> generally tends to a nonzero external <pressure>, so outward velocity alone does not select the escaping solution.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2014/paper-1-parker-wind.png]
{title=Transonic isothermal Parker wind and a pressure-confined breeze}
The <solar wind> supplies a nearby example of an outward <plasma> flow carrying a <magnetic field>. Profiles of <spectral lines>, especially absorption shifted by the <Doppler effect> together with emission, diagnose more distant <stellar winds>. The critical-point condition and boundary heating constrain $\dot M$; it is not determined by an escape-speed estimate alone.
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