Past exam of the physics 1 paper of the University of Oxford / 2014 / 11 / c

For a fixed-speed orbital launch-direction error, the initial radius and speed are correct, so the specific orbital energy is still $\epsilon =-GM/(2R)$ and the new semi-major axis is still $a=R$. Resolve the launch velocity into radial and tangential components, taking $\alpha$ as the in-plane deviation from the intended tangent:For a nondegenerate orbit $\cos \alpha \ne 0$, the orbital eccentricity follows from $j^2=GMa(1-e^2)$:The intended circular orbit has radius $R$ everywhere, so the largest radial distance between the orbit curves occurs at either apsis:This is the geometric separation of the two orbit curves, not the distance between two satellites compared at equal elapsed times. The latter also depends on orbital phase. A purely radial launch is a degenerate zero-angular-momentum trajectory through the central body and requires separate treatment of the collision.