 Pericentre-speed qualification of Kepler orbit classes

At a genuine pericentre, the radial velocity vanishes and the radial acceleration is nonnegative. The radial equation therefore requires $v_p^2\ge GM/r_p$. In terms of the displayed factor k, a circle has k=1, a noncircular ellipse has $1<k<\sqrt{2}$, a parabola has $k=\sqrt{2}$ and a hyperbola has $k>\sqrt{2}$. If $0<k<1$ at a tangential turning point, that point is the apocentre, despite the negative orbital energy. Bound-energy classification alone does not establish which turning point is being specified.