= Apsidal invariant on an unbound central-force branch
{title2=$\psi_{\rm peri}=\psi-\operatorname{Arg}(u-u_c-iu'/\nu)/\nu$}
For $u=u_c[1+e\cos\nu(\psi-\psi_0)]$ with $\nu>0$ and $e\geq1$, the positive-radius scattering branch has $|\nu(\psi-\psi_0)|<\arccos(-1/e)\leq\pi$. 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.
Back to article page
