Sonic-point slope discriminant
= Sonic-point slope discriminant
For <spherically symmetric adiabatic flow> in a <power-law gravitational potential>, let $X=(r_*/u_*)u_*'$ 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.