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The chaser's path in the target's local frame: along-track (V-bar) horizontal, radial (R-bar) vertical, target at the origin. A radial offset drives a secular along-track drift; ẏ₀ = −2n·x₀ closes the loop.
drift—n—
Relative motion of a chaser near a target in a circular orbit, from the Clohessy–Wiltshire equations. First-cut: closed-form LVLH trajectory.
The chaser's path in the target's local frame: along-track (V-bar) horizontal, radial (R-bar) vertical, target at the origin. A radial offset drives a secular along-track drift; ẏ₀ = −2n·x₀ closes the loop.
Rendezvous is analyzed not in inertial space but in a frame riding with the target: the local-vertical/local-horizontal (LVLH) frame, with axes along the radial (“R-bar”), along-track (“V-bar”), and cross-track directions. For a chaser close to a target on a near-circular orbit, the linearized equations of relative motion are the Clohessy–Wiltshire equations, published by W. H. Clohessy and R. S. Wiltshire in 1960 — themselves a form of the equations G. W. Hill derived for the Moon’s motion in 1878.
Their closed-form solution captures the behavior that makes docking counterintuitive. A purely radial nudge does not move you radially — it opens a secular drift along-track. A small along-track push at the same altitude leaves you on the same orbit (a co-orbit) with no relative motion at all. Only a specific pairing of offset and velocity, ẏ₀ = −2n·x₀, closes the relative orbit into a bounded ellipse that returns to its start. Approaches are therefore flown deliberately along V-bar or R-bar with a sequence of small, timed burns, not by pointing at the target and thrusting.
The CW model is exact only for a circular target orbit and small separations; the Tschauner–Hempel equations extend it to eccentric targets. It nonetheless underlies the terminal phase of essentially every crewed and cargo docking, from Gemini and Apollo to the ISS.