Stellar wind

A stellar wind is a continuous outflow of gas from a star. Radiation, gas pressure, waves, or magnetic fields can supply its acceleration, and the resulting mass loss can determine a star's final remnant.

An episode of particularly strong mass loss near the end of Asymptotic giant branch evolution, capable of removing the remaining envelope rapidly compared with its prior evolution. Radiation acting on dust can transfer momentum to the gas; pulsations help lift material into cooler dust-forming layers. Envelope removal terminates sustained hydrogen-burning shell and helium-burning shell activity.

A magnetized stellar wind extracts angular momentum through matter transport and Maxwell stress tensor. The Alfvén radius supplies an effective lever arm larger than the stellar radius, with geometry factors absorbed into an averaged lever arm. Dynamo action couples the torque to rotation by supplying the magnetic field. For a simplified radial wind of fixed speed, $R_A^2$ scales as $B_{\ast }^2{R}_{\ast }^4/({\mu }_0{\dot{M}}_w{v}_w)$, so unsaturated $B_{\ast }\propto \Omega$ gives a torque proportional to $-{\Omega }^3$. Magnetic topology and wind thermodynamics can alter this scaling.

If the moment of inertia I changes slowly and wind-driven magnetic braking of a solar-type star has torque $-K{\Omega }^3$, then ${\Omega }^{-2}(t)={\Omega }^{-2}(t_0)+2K(t-t_0)/I$. At sufficiently late times within this unsaturated approximation, rotation scales as $t^{-1/2}$. This empirical main-sequence trend is not a universal law for saturated rapid rotators, changing stellar structure or every magnetic geometry.

Using an axisymmetric hydrodynamic mass flux function, set $s=s(\psi )$, $\ell =R{u}_{\varphi }=\ell (\psi )$ and $\epsilon =\epsilon (\psi )$. The azimuthal vorticity is ${\omega }_{\varphi }=-R\nabla \cdot (\nabla \psi /(\rho R^2))$. The meridional projection of the Crocco form of the unsteady Euler equation gives the displayed equation. The mass density also satisfies the Bernoulli relation $\epsilon =|\nabla \psi {|}^2/(2{\rho }^2{R}^2)+{\ell }^2/(2R^2)+w+\Phi$, so an equation of state is part of closing the system.

A steady spherical polytropic flow escaping a point-mass potential conserves the outward mass loss rate and the Bernoulli function $C=u^2/2+c_s^2/(\gamma -1)-GM/r$. For positive terminal speed, a finite-radius smooth sonic point requires $C>0$ and $1<\gamma <5/3$. Its sound speed obeys $c_{s,c}^2=2C(\gamma -1)/(5-3\gamma )$, and evaluating the conserved flux there selects the transonic mass loss rate. Other parameter limits need separate treatment.

Put $y=\rho r^{4/(\gamma +1)}$ and $\alpha =4(\gamma -1)/(\gamma +1)$. The polytropic stellar wind equations become $F(r)=C{r}^{\alpha }+GM{r}^{\alpha -1}=G(y)=\gamma K{y}^{\gamma -1}/(\gamma -1)+{\dot{M}}^2/(32{\pi }^2{y}^2)$. At a regular sonic point both derivatives vanish. Matching their minima allows the two positive-$y$ branches to join with finite slope; unequal minima either leave a forbidden radial interval or separate the subsonic and supersonic branches. This reduces a singular differential-equation crossing to a geometric minimum comparison.

In an optically thick steady spherical stellar wind of constant mean molecular weight ${\mu }_{\mathrm{mol}}$, let $v$ be the outward speed and $a_g^2=k_{B}T/({\mu }_{\mathrm{mol}}{m}_u)$ the isothermal sound speed squared. Mass conservation gives ${\rho }^{\prime }/\rho =-2/r-v^{\prime }/v$, where a prime denotes differentiation with respect to radius $r$. Use $P_{\mathrm{gas}}=\rho a_g^2$ and radiative diffusion to write $P_{\mathrm{rad}}^{\prime }/\rho =-g\Gamma$, where $g=G{M}_r/r^2$ and $\Gamma =L_r/L_{\mathrm{Edd}}$ uses the local Eddington luminosity $L_{\mathrm{Edd}}=4\pi cG{M}_r/\kappa$ and opacity $\kappa$. Substituting into the momentum equation yields the displayed wind equation. Smooth sonic point crossing requires both its denominator and numerator to vanish. The radiation-pressure gradient already supplies radiative acceleration; adding that force separately would count it twice.

For an optically thick diffusion-controlled wind, put $\beta =P_{\mathrm{gas}}/(P_{\mathrm{gas}}+P_{\mathrm{rad}})$. Diffusion gives $(\log T{)}^{\prime }=-g\Gamma \beta /[4a_g^2(1-\beta )]$. At $\Gamma =1$, the numerator of the gas and radiation pressure stellar wind equation is $2a_g^2/r+g\beta /[4(1-\beta )]>0$ for finite-temperature gas and $0<\beta <1$. A subsonic branch then decelerates and cannot pass a regular sonic point. This statement is specific to subsonically launched winds in this transport model, not to every supersonic solution or every form of stellar-wind driving.