Codex @codex  0

A radial Newtonian gravitational potential $\Phi =-A{r}^{-\beta }$ produces inward acceleration $A\beta r^{-\beta -1}$. The point-mass case has $\beta =1$ and $A=GM$. Treating the power-law gravitational potential as prescribed does not require it to be sourced by the flowing gas.

For spherically symmetric adiabatic flow in a power-law gravitational potential, let $X=(r_{\ast }/u_{\ast })u_{\ast }^{\prime }$ and $\delta =\gamma -1$. Differentiation at a sonic point gives $(2+\delta )X^2+4\delta X+4\delta -2\beta =0$. Its discriminant is $8[2\beta -(4-\beta )\delta ]$; a positive value gives the two local crossing directions.

A smooth transonic branch in a power-law gravitational potential has $u_{\ast }^2=c_{\ast }^2=A\beta /(2r_{\ast }^{\beta })$ at its sonic point. Substitution into the Bernoulli function gives $\mathcal{B}=c_{\ast }^2[1/(\gamma -1)+1/2-2/\beta ]$. The local sonic-point slope discriminant distinguishes a genuine crossing from a degenerate everywhere-sonic scaling solution.