= Polytropic accretion in a mixed inverse-power potential
{title2=$m c_\infty^2R_s^2-\frac{2m-3}{4}\mu R_s-\frac{2m-1}{2}\lambda=0$}
For $p=K\rho^{1+1/m}$, the <barotropic enthalpy function> is $mc_s^2$. 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.
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