= Density at the maximum speed of an isothermal breeze
{title2=$r_M=GM/(2a^2),\quad\rho_M=\dot M a^3/(\pi G^2M^2\mathcal M_M)$}
A steady spherical <subsonic isothermal breeze> with <isothermal sound speed> $a$ obeys $(u-a^2/u)u'=2a^2/r-GM/r^2$. 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 $\rho_M>\dot M a^3/(\pi G^2M^2)$, 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.
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