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The Alfvén radius characterizes where a magnetized stellar wind crosses its Alfvén surface. For a roughly radial wind, $v_{\mathrm{wind}}\simeq B/\sqrt{{\mu }_0\rho }$ and ${\dot{M}}_w\simeq 4\pi R_A^2\rho v_{\mathrm{wind}}$. The wind's angular-momentum lever arm is of order $R_A$: its torque is $\dot{J}\sim -{\dot{M}}_w\Omega R_A^2$, up to geometry factors. The precise wind invariant uses the cylindrical radius of the surface crossing, rather than necessarily its spherical radius.

Astrophysical fluid dynamics models gases and plasmas in stars, accretion discs, stellar winds and other astronomical flows. Its distinctive ingredients can include self-gravity, rotation, compressibility, radiative cooling and magnetic fields. ^[ref]

A useful scale comparison is the Mach number $\mathrm{Ma}=U/c_s$, with flow speed $U$ and sound speed $c_s$. Large compressions can produce shock waves; self-gravity can instead amplify density disturbances through Jeans instability. Conducting plasma adds magnetohydrodynamics. A continuum approximation must be justified: a very tenuous medium can require a kinetic description instead of a simple collisional fluid.

If an escaping stellar wind carries specific orbital angular momentum $\lambda J/M$ with constant $\lambda$, then $\dot{J}/J=\lambda \dot{M}/M$ and $J\propto M^{\lambda }$. Combining this with circular-binary orbital angular momentum gives the displayed period invariant. It is a different prescription from donor-wind angular-momentum loss, whose specific angular momentum generally varies with mass ratio.

A detached binary has both components lying inside their respective Roche lobes, so neither currently transfers matter by Roche-lobe overflow. A detached system may still exchange matter through a stellar wind.

A donor star is the component of a binary star that supplies matter to its companion. In Roche-lobe overflow, it fills its Roche lobe and gas passes near the inner Lagrange point. In wind mass transfer in a binary star, some gas lost in a stellar wind is captured by the other component without the donor necessarily filling its Roche lobe.

The donor's response to losing mass determines whether transfer weakens or strengthens. Compare its stellar radius response exponent with the Roche-lobe radius response exponent: if mass loss increases the relative overfill, transfer can run away. The donor need not currently be the more massive member, because binary mass-ratio reversal can occur after substantial transfer.

An isotropic escaping stellar wind has the donor star's mean specific orbital angular momentum. With ${\dot{M}}_d<0$, retained fraction $f$, negligible spin and no additional wind torque, $\dot{J}=(1-f){\dot{M}}_d{a}_d^2\Omega$. The retained matter redistributes angular momentum internally. Unlike isotropic re-emission from a binary star, the escaping material originates at the donor rather than the accretor. With no retention this reduces to Jeans-mode mass loss.

In a circular binary star let the donor star lose mass ${\dot{M}}_1<0$, with constant retained fraction $f$ and ${\dot{M}}_2=-f{\dot{M}}_1$. A donor star stellar wind escaping with the donor star's orbital specific angular momentum has $\dot{J}/J=(1-f){\dot{M}}_1{M}_2/[M_1(M_1+M_2)]$. Differentiating $J=G^{2/3}(P_{\mathrm{orb}}/2\pi {)}^{1/3}{M}_1{M}_2(M_1+M_2{)}^{-1/3}$ and using this loss law givesFor constant $f$ this integrates to $P_{\mathrm{orb}}{M}_1^{3f}{M}_2^3(M_1+M_2{)}^2=\mathrm{constant}$. If $f$ varies, the differential relation remains valid but the first factor must be replaced by $\exp (3\int fd\log M_1)$; treating a time-dependent exponent as a fixed power introduces an extra $\dot{f}\log M_1$ term.

D. R. Gies co-authored a spectroscopy study connecting stellar wind signatures with the X-ray behavior of Cygnus X-1. The team monitored H-alpha emission and separated components associated with different moving gas structures.

Comparing those line profiles with X-ray output supplied evidence about the supply of gas to an accreting black hole. The paper interpreted changes in the donor wind as a possible driver of changes in accretion rate and radiation state.^[ref]

A galactic wind is a large-scale flow carrying gas, energy and momentum away from a galaxy. Stellar winds and supernovae can maintain a hot, pressurized component that expands out of a star-forming region; an active galactic nucleus can supply another driving mechanism. Cooler entrained gas can coexist with the hot component.

Observations must distinguish the wind from unrelated point sources and foreground emission. In M82, ROSAT measurements separated extended soft X-ray emission from compact sources and traced emission away from the galactic plane. The emitting gas need not be the freely expanding wind itself: shocked clouds can dominate the detected radiation.^[ref]

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.

A high-mass X-ray binary has a massive donor star supplying an accreting neutron star or black hole. A strong stellar wind often feeds it, although Roche-lobe overflow and donor equatorial outflows also occur. The short lifetime of the donor associates these systems with young stellar populations.

Ian Stevens is an astrophysicist whose research includes massive stars and stellar winds. He coauthored an analysis of X-ray observations of the galactic wind in the starburst galaxy M82.

The team compared diffuse emission with models of a freely expanding wind and of clouds heated by shocks. The observations favoured shocked clouds over the simple free-wind model, showing how radiation can identify which component of an outflow is being measured.^[ref]

Institutional profile: research.birmingham.ac.uk/en/persons/ian-stevens/.

The interstellar medium is the gas, interstellar dust, magnetic field and cosmic rays between stars in a galaxy. It is a changing reservoir: molecular clouds supply star formation, while stellar winds and supernova remnants return material and stir or heat it. Different regions occupy different temperature and ionization states, so no single observation measures the whole medium. For example, the neutral-hydrogen 21-centimeter line probes neutral atomic hydrogen, whereas X-rays reveal much hotter plasma. Even neutral atomic gas can occupy cold and warm phases: the heating and cooling balance in the interstellar medium permits both over a restricted range of pressure.^[ref]

Loss of matter from a star or other astronomical system, for example through a stellar wind or explosive ejection. Mass loss rate conventions must distinguish the positive ejected rate $|\dot{M}|$ from the negative change of the retained mass, $\dot{M}$. In a binary star, the fate of an orbit also depends on the specific angular momentum carried away.

The mass loss rate is the positive rate at which a body loses mass. For a steady spherical stellar wind, the continuity equation gives $\dot{M}=4\pi r^2\rho u_r$, constant across spheres enclosing the source. It is positive for outward radial flow; an accretion convention often uses a separate positive inward rate.

In nonconservative binary evolution, matter can escape the system and/or external processes change its orbital angular momentum. Escaping overflow, a stellar wind and a common envelope can invalidate the fixed-total-mass and period-product invariants of conservative mass transfer.

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 equationA 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.

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.

An Alfvén wave is a transverse disturbance restored by magnetic tension. In a uniform conducting fluid with background ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$, a small transverse displacement ${\mathbf{\xi }}_{\perp }(z,t)$ bends the magnetic field lines. Magnetic flux freezing gives ${\mathbf{b}}_{\perp }=B_0{\partial }_z{\mathbf{\xi }}_{\perp }$, and the linear transverse Lorentz force density is $(B_0/{\mu }_0){\partial }_z{\mathbf{b}}_{\perp }$. Thus the Linear Alfvén wave equations reduce toFor this shear polarization, $\delta \rho =\delta p=0$ at linear order. The disturbance transports energy along the background magnetic field; a single traveling wave has equal perturbation magnetic energy and kinetic energy, with ${\mathbf{b}}_{\perp }=\mp \sqrt{{\mu }_0\rho }{\mathbf{u}}_{\perp }$ for propagation along $\pm {\mathbf{B}}_0$. These properties distinguish it from compressive magnetosonic waves.

The ideal magnetohydrodynamics approximation requires strong electrical coupling and wavelengths large compared with microscopic scales. Finite electrical resistivity and viscosity dissipate the wave's energy. In stellar winds and other astrophysical plasmas, Alfvén waves carry disturbances and can contribute to heating when dissipated. The Lundquist liquid-metal Alfvén wave experiment provides a laboratory counterpart: a conducting liquid in a strong applied magnetic field transmits a transverse magnetic disturbance, making the field's mechanical restoring action measurable.

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 equationA 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.

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.

In the stated gravity-dominated approximation, neglect the pressure force in the radial momentum equation. A steady outward stellar wind then obeysAt $R_e=R_{\mathrm{esc}}$, the escape velocity is $v_e=\sqrt{2G{M}_{\ast }/R_e}$, so $C=0$. The ballistic marginal-escape stellar wind hasThe outward flow decelerates and approaches zero speed at infinity; it is exactly at the escape threshold in this pressure-negligible model. An initial speed above that threshold gives a nonzero asymptotic speed.

Adiabaticity alone would instead conserve the full Bernoulli equation $v^2/2+h-G{M}_{\ast }/r=B$. If finite pressure is retained and $v_e$ equals the mechanical escape speed, then $B=h_e$, not zero. With $p=K{\rho }^{\gamma }$ and conservation of mass, the more general implicit solution isIt needs $h_e$ and $\gamma$ in addition to the escape-speed datum. The boxed explicit power law is the leading gravity-dominated result, not an exact solution for an arbitrary warm adiabatic wind.