Apsidal invariant on an unbound central-force branch (${\psi }_{\mathrm{peri}}=\psi -\mathrm{Arg}(u-u_c-i{u}^{\prime }/\nu )/\nu$)

For $u=u_c[1+e\cos \nu (\psi -{\psi }_0)]$ with $\nu >0$ and $e\ge 1$, the positive-radius scattering branch has $|\nu (\psi -{\psi }_0)|<\arccos (-1/e)\le \pi$. Thus the principal complex argument in the displayed expression is continuous throughout that branch, unlike at an apocentre of a bound orbit. Its derivative is $\nu$, so the angle modulo $2\pi$, or the associated unit apsidal vector in the orbital plane, is a stationary integral of motion fixing the periapsis direction. Together with energy and three components of specific angular momentum, it gives five regular scattering labels even for irrational $\nu$. Periodicity is therefore necessary for five orbit labels in recurrent bound motion, not in general for escaping trajectories.