Minimum separation of phase-shifted sphere oscillations (${\ell }_{\min }={\ell }_0-2\delta |\sin (\varphi /2)|$)

The centre separation for equal-amplitude oscillations has minimum ${\ell }_0-2\delta |\sin (\varphi /2)|$. A bound ${\ell }_0\ge \delta$ alone cannot guarantee nonoverlap. An asymptotic mobility calculation needs this minimum much larger than the sphere radii.