Parker wind

A Parker wind is a steady, spherically symmetric outflow accelerated from a gravitating body through a sonic point. Smoothness selects the transonic solution from the family of formal steady solutions.

The steady isothermal Parker wind differential equation also has positive solutions describing subsonic flow at every radius. Write their first integral as $w-\ln w=4\ln x+4/x+C$, where $w=u^2/c^2$ and $x=r/r_s$. On the small-$w$ branch at large radius, $w\sim e^{-C}{x}^{-4}$, so $u\sim c{e}^{-C/2}{x}^{-2}$. Constant spherical mass flow rate then gives nonzero limiting mass density and hence finite external pressure. This distinguishes a pressure-confined breeze from an escaping wind into vanishing pressure. Specifying outward motion alone does not select the supersonic asymptotic branch.

For unattenuated isotropic radiation from a point source, the radiation pressure force on each absorbing particle is $\sigma L/(4\pi c{r}^2)$. Dividing by its mass $\bar{m}$ changes the effective inverse-square inward acceleration to $[GM-\sigma L/(4\pi \bar{m}c)]/r^2$. The wind equation therefore places the maximum of a subsonic isothermal breeze at the displayed radius, provided the bracket is positive. If it is nonpositive, the positive-temperature subsonic outflow decreases in speed throughout its domain and has no interior speed maximum. A cross section per particle requires that particle's mass; an opacity per unit mass instead absorbs this conversion.

A steady spherical subsonic isothermal breeze with isothermal sound speed $a$ obeys $(u-a^2/u)u^{\prime }=2a^2/r-GM/r^2$. Since $0<u<a$, its finite-radius speed maximum lies at $r_M=GM/(2a^2)$. Spherical mass conservation then gives the displayed density, where ${\mathcal{M}}_M=u(r_M)/a\in (0,1)$ is the peak Mach number. This free parameter is selected by a boundary condition, not by $T$, $M$ and $\dot{M}$ alone. Therefore ${\rho }_M>\dot{M}{a}^3/(\pi G^2{M}^2)$, while a peak Mach number of order unity gives that expression as a characteristic scale.

A direct demonstration of the free parameter comes from the first integral $u^2/2-a^2\ln (u/a)=2a^2\ln (r/r_M)+GM/r+C$. Its additive constant remains free for a breeze; imposing the regular transonic branch would be a different condition.

A steady spherical Parker wind with constant isothermal sound speed $c$ and negligible wind self-gravity obeys $4\pi r^2\rho u=\dot{M}$ and $(u-c^2/u)u^{\prime }=2c^2/r-GM/r^2$. A regular sonic point has $u=c$ at $r_s=GM/(2c^2)$. With $x=r/r_s$ and $w=u^2/c^2$, integration gives the displayed equation for the smooth accelerating transonic branch. Its slope at the sonic point is $u^{\prime }(r_s)=c/r_s$. Maintaining constant gas temperature during expansion generally requires heat input, so this model is distinct from an adiabatic wind with $\gamma >1$.

On the escaping supersonic isothermal Parker wind branch, $w=u^2/c^2\to \infty$. Its integrated equation gives $w=4\ln x+\ln w-3+4/x$, so $w/(4\ln x)\to 1$ and the title result follows. A further iteration yields $w=4\ln x+\ln (4\ln x)-3+o(1)$. The leading logarithmic growth holds for any supersonic branch with a fixed finite integration constant, but does not describe a subsonic isothermal breeze. The unbounded speed is a formal consequence of indefinite isothermal heating in a prescribed central gravitational potential.

A polytropic Parker wind is isentropic with $p=K{\rho }^{\gamma }$. Its sound speed changes with density, and a wind reaching infinity with positive terminal kinetic energy requires $1<\gamma <5/3$.

For radial speed $u$, sound speed $c_s$, and central mass $M$, steady spherical mass and momentum conservation giveAt a regular sonic point, $u=c_s$ and $r=GM/(2c_s^2)$ simultaneously.