 Polytropic accretion in a mixed inverse-power potential

For $p=K{\rho }^{1+1/m}$, the barotropic enthalpy function is $m{c}_s^2$. In $\Phi =-\lambda /R^2-\mu /R$, a regular sonic point obeys $c_{s,s}^2=\lambda /R_s^2+\mu /(2R_s)$, and its Bernoulli integral yields the displayed quadratic. For $K,\lambda ,\mu >0$ and the usual positive polytropic index, its unique positive root exists precisely when $m>1/2$. The mass accretion rate is $4\pi R_s^2{\rho }_{\infty }{c}_{s,s}(c_{s,s}^2/c_{\infty }^2{)}^m$. If indices with $0<1+1/m<1$ are admitted, $m<-1$ also permits a positive sonic root; negative compressibility indices do not.