= Inner-boundary selection of isothermal Bondi flow
{title2=$\lambda\le e^{3/2}/4$}
For steady <Isothermal Bondi accretion> from gas at rest at infinity, define $\lambda=\dot M c_s^3/(4\pi G^2M^2\rho_\infty)$, $x=r/r_s$ and $y=u/c_s>0$, where $r_s=GM/(2c_s^2)$. The <Bernoulli function> and <mass conservation> give $y^2/2-\ln y=2\ln x+2/x-\ln(4\lambda)$. The left side has minimum $1/2$ at $y=1$, and the radius-dependent part on the right has minimum two at $x=1$. Existence over all positive radii therefore requires the displayed bound. Equality permits the smooth accelerating-inward <transonic branch> and gives the usual <isothermal Bondi accretion rate>. Smaller positive rates have an everywhere-subsonic reservoir-connected branch, and zero rate gives a hydrostatic atmosphere. An absorbing inner boundary inside the sonic radius selects the critical inflow; steady isothermal equations and the outer density alone do not specify that inner condition.
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