Subsonic isothermal breeze ($u\propto r^{-2},\rho \to {\rho }_{\infty }>0$)

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.