PYTHON · NUMPY

Relative motion: Clohessy-Wiltshire

A chaser sits near a target on a circular orbit with a small along-track offset and no relative velocity. What happens?

Propagate a chaser's motion near a target on a circular orbit, and discover the two behaviours that make rendezvous non-obvious.

01Challenge

Near a target on a circular orbit, relative motion follows the linear Clohessy-Wiltshire equations. Implement the state-transition matrix, then discover two surprises: an along-track offset holds station, but a radial offset drifts along-track.

Predict firstA chaser sits near a target on a circular orbit with a small along-track offset and no relative velocity. What happens?

02Model

Close to a target, you do not reason in absolute orbits; you reason in the target's local frame, radial (out) and along-track. The Clohessy-Wiltshire equations linearise the relative motion there, and their solution is a state-transition matrix: give it a relative position and velocity now and it hands you the state at any later time. Two of its predictions are counter-intuitive and matter for rendezvous. Put a chaser directly ahead of the target with zero relative velocity and it just stays there, because it is on the same orbit with the same period. But offset it radially, higher or lower, and even with zero relative velocity it drifts along-track, because a different altitude means a different period. Rendezvous has to account for both.

03Guided practice

1. Mean motion. n = sqrt(mu / a^3) sets the orbital rate; a = R_earth + altitude.

2. The CW blocks. With s=sin(nt), c=cos(nt), the planar state-transition blocks map (r0, v0) to (r, v). They are standard; the cell gives them.

3. The two behaviours. Propagate an along-track offset (it holds) and a radial offset (it drifts). Read both.

04Feedback

PASS WHEN PASS once both lines print: the along-track case stays near (0, 120), and the radial case shows a large along-track number.

If your run fails, check:

If the along-track case drifts, check the CW blocks: rr[0][1] should be 0 and rr[1][1] should be 1.
Use radians: n*t is an angle, and sin/cos take radians.

05Retrieve & space

Why does a purely along-track offset hold station? (same orbit, same period, so the separation is constant).
Why does a radial offset drift along-track? (a different altitude means a different orbital period).
What object does the CW state-transition matrix act on? (the relative position-and-velocity state in the target's local frame).

06Mastery & project

You can propagate relative motion with Clohessy-Wiltshire and explain the along-track fixed point and the radial drift.

Before planning any close approach, propagate the passive relative motion first: it tells you which way the chaser drifts if you do nothing.

GO FURTHER · THE LIVE PLAYGROUND

On-orbit capture (live) ↗ · The rendezvous this relative-motion model underlies.

← Composing the budget Two-impulse rendezvous →

PYTHON · NUMPY

Relative motion: Clohessy-Wiltshire

Implement CW propagation and confirm the along-track fixed point and the radial along-track drift.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you