Debris Capture Rendezvous — Clohessy-Wiltshire Proximity Ops

Scenario

Debris tumble rate 6.0°/s

Time scale 1.0×

Thrust (ΔV per burn 0.05 m/s)

Approaching target…

Telemetry

Range to debris

50.0 m

Closing speed

0.00 m/s

ΔV used

0.00 m/s

Mission time

0 s

How it works

The chaser's motion relative to the tumbling debris is governed by the Clohessy-Wiltshire (Hill's) equations — the standard linearised model used for real orbital rendezvous and proximity operations, expressed in a frame centred on the debris with R = radial (away from Earth), V = along-track, N = cross-track:

ẍ − 2n ẏ − 3n²x = a_R
ÿ + 2n ẋ         = a_V
z̈ + n²z          = a_N

n = mean motion of the reference orbit (≈ 0.0011 rad/s at 500 km altitude)

Notice the coupling: a burn along V (+V/−V) changes the radial drift too, and a purely radial burn (+R/−R) induces along-track drift via the Coriolis-like 2nẋ / 2nẏ terms — this is why rendezvous trajectories curve instead of going straight at the target, and why real chaser spacecraft (e.g. ATV, Dragon, Astroscale's ELSA-d) fly curved football-shaped approach ellipses rather than straight lines.

+V / −V, +R / −R, +N / −N — fire small 0.05 m/s impulses along each axis; every burn costs propellant, tracked as total ΔV.
Debris tumble rate — sets how fast the target rotates; a robotic-arm or net capture becomes physically viable once you are inside the 5 m capture sphere and your closing speed drops under 0.2 m/s — tumble rate does not block capture here but represents the added difficulty (and torque) an arm would have to absorb on contact.
Capture — triggers automatically once range and closing speed both cross their thresholds; running out of the budget without capturing ends the mission.

Real-world relevance: this is the same relative-motion model that ESA's ClearSpace-1 and Astroscale's ADRAS-J missions plan their final-approach burns around when closing on non-cooperative, tumbling debris.

Chaser spacecraft Debris (frame origin) Relative trajectory

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Debris Capture Rendezvous — Clohessy-Wiltshire Proximity Operations