Fixed-speed orbital launch-direction error ($a=R,e=|\sin \alpha |,\Delta r_{\max }=R|\sin \alpha |$)

Launching at the correct circular orbital speed but at in-plane angle $\alpha$ from the tangent leaves specific orbital energy $-GM/(2R)$, hence semi-major axis $R$. The specific angular momentum becomes $\sqrt{GMR}\cos \alpha$, so $j^2=GMa(1-e^2)$ gives $e=|\sin \alpha |$. The actual periapsis and apoapsis are $R(1\mp e)$, equally far from the intended circular orbit. The maximum geometric radial separation is therefore $Re$. This is not an equal-time separation between two moving satellites.