= Monatomic Bondi accretion
{title2=$\gamma=5/3,\quad \dot M_{\max}=\pi G^2M^2\rho_\infty c_\infty^{-3}$}
For isentropic spherical accretion with this <adiabatic index>, define $\lambda=\dot M c_\infty^3/(4\pi G^2M^2\rho_\infty)$ and $x=rc_\infty^2/(GM)$. The steady-flow equation is $\sqrt\lambda[\mathcal M^{3/2}/2+3\mathcal M^{-1/2}/2]=1+3x/2$. The bracket has minimum two at Mach one. Reaching arbitrarily small positive radii therefore requires $\lambda\le1/4$. For smaller rates the outer-rest branch remains subsonic; at the maximal rate it approaches Mach one only as $r\to0$, not at a finite <sonic point>. Supersonic outer branches fail the specified resting finite-density condition, and larger rates terminate at a positive minimum radius. An additional inner boundary prescription is needed to select a unique rate among the globally defined subsonic solutions.
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