Relative-motion frame. The target (station) is on a circular orbit of radius r0 with mean motion n = √(μ/r03). The LVLH (Hill) frame rotates with it: x points radially outward, y along the direction of flight (the V-bar), z along the orbit normal. For small separations ρ ≪ r0 the relative motion of a chaser obeys the Clohessy–Wiltshire (Hill) equations (Hill 1878; Clohessy & Wiltshire 1960):
x″ − 2n y′ − 3n² x = ax y″ + 2n x′ = ay z″ + n² z = az
With a = 0 they have a closed form (the state-transition matrix) in τ = nt: x(t) = (4 − 3cosτ)x0 + (sinτ/n)x′0 + (2/n)(1 − cosτ)y′0; y(t) = 6(sinτ − τ)x0 + y0 − (2/n)(1 − cosτ)x′0 + (4 sinτ − 3τ)y′0/n; z(t) = z0cosτ + (z′0/n)sinτ. This is what the dashed “CW prediction” follows.
Why a prograde burn drops you behind. A burn along +y raises the orbit on the far side, so the chaser's semi-major axis grows. A higher orbit has a longer period, so it slowly loses angle to the station. In CW terms the guiding-centre (mean) radial offset is xc = 4x0 + 2y′0/n = δa and the secular along-track drift is y′drift = −(3/2) n xc, i.e. Δy = −3π δa per orbit. To catch up you retrograde-burn, drop to a lower and faster orbit, and later raise it again.
Football orbit. If xc = 0 (y′0 = −2n x0, x′0 = 0) there is no drift and the chaser traces a closed 2:1 ellipse, x = x0cosτ, y = y0 − 2x0sinτ: the along-track extent is exactly twice the radial extent.
Two-impulse rendezvous. Write r(T) = Φrr r0 + Φrv v0+. The first burn must give v0+ = Φrv-1(rf − Φrr r0); the second, at arrival, removes the velocity v(T)− = Φvr r0 + Φvv v0+. The in-plane determinant is (8 − 8cosτ − 3τ sinτ)/n², so the cost spikes where it passes through zero (e.g. whole orbits), and the cross-track part is singular when T is a multiple of half an orbit.
V-bar and R-bar. Holding a straight line along the V-bar at speed v needs a continuous radial thrust of 2nv (cancelling the Coriolis term), so a pure impulsive straight approach does not exist and real vehicles use short “hops” between hold points. On the R-bar the gravity gradient 3n²x points away from the station for a chaser below it and Coriolis couples in, so a chaser that stops thrusting drifts off instead of coasting in. This passive safety is why R-bar approaches have been used for capture by robotic arms. The autopilot here adds the CW feed-forward terms so both lines are straight; the Δv it spends is reported. Switch on “Cut thrust” to see the passive behaviour.
Corridor and contact speed. The drawn cone (±10°), the flagged closing-speed limit 0.10 + 0.005 · range (m/s) and the autopilot's schedule v = 0.04 + 0.003 · range (m/s) are illustrative; public docking practice for visiting vehicles has contact speeds of order 0.1 m/s or less, and the page flags a contact above 0.10 m/s. They are not the requirements of any particular vehicle.
Accuracy limits. CW linearises the gravity gradient about the target's circular orbit. The neglected terms are of relative size ρ/r0, so the position error grows like ρ2/r0 (error/r0 ∝ (ρ/r0)2) and builds up over several orbits; experiment 1 measures the exponent. Keep the plan short and re-plan from the measured state, as flight software does.
Model and assumptions. The truth model integrates the chaser with RK4 (1 s step) in an inertial frame under point-mass gravity (μ = 3.986004×1014 m3/s2, RE = 6371 km); the target is exactly circular. No J2, drag, eccentricity, attitude dynamics, thruster pulsing or sensor noise. Burns are instantaneous; the autopilot is continuous thrust capped at 0.05 m/s². The Sun is held fixed in this view (no eclipses) and the Earth only turns at the orbital rate. The station and capsule are generic illustrative models without any real vehicle's markings.
A physics teaching model for the principles of orbital rendezvous. All numbers are illustrative and are not mission data or flight procedures.