Near a target satellite, relative motion is governed by Hill/Clohessy-Wiltshire equations: a chaser slightly closer to Earth orbits faster, and one slightly further out orbits slower, creating characteristic curved relative-approach trajectories rather than straight lines.
ẍ − 3n²x − 2nẏ = Fx/m (radial)
ÿ + 2nẋ = Fy/m (along-track)
capture when |r| < r_capture and |v_rel| < v_safe
- Radial thrust — a burn toward/away from Earth, which the Coriolis-like coupling turns into along-track drift.
- Tangential thrust — a burn along the orbit direction, directly closing or opening along-track distance.
- Approach damping — extra braking thrust proportional to closing speed, standing in for an automated proximity-operations controller.
- Robotic arm reach — extends the chaser's manipulator once close enough to grapple the satellite's damaged panel.
This is the real physics behind missions like the Hubble servicing flights and in-orbit satellite-repair demonstrators (e.g. Northrop Grumman's Mission Extension Vehicle) — over-thrusting radially causes an unexpected sideways drift that catches many new orbital-mechanics students off guard.