Radial turning points for an inverse-square height well (${\dot{r}}^2=2\mathcal{E}+2g/r-(L^2+2g)/r^2$)

For the height profile $h(r)=r^{-2}-r^{-1}$, set $A=L^2+2g$ and use the planar small-slope model. Its effective potential per unit mass is $V_{\mathrm{eff}}=A/(2r^2)-g/r$. Release with negligible radial speed at $r_0$ gives $\mathcal{E}=A/(2r_0^2)-g/r_0$ and the displayed radial-speed law. A bounded trajectory requires $\mathcal{E}<0$, equivalently $r_0>A/(2g)$. The two positive turning radii are $r_0$ and $A{r}_0/(2g{r}_0-A)$; they coincide at the circular solution $r_0=A/g$. Zero energy is marginal escape, not bounded motion.