Astronomy & Celestial Mechanics
the sky, computed
Concept

Proximity operations — where it appears

Manoeuvring within a few kilometres of another spacecraft. The velocities involved are centimetres a second, the dynamics are dominated by Coriolis terms in a rotating frame, and an intuition trained on translation is reliably wrong about them.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Also named here as the clohessy–wiltshire equations — the same set of essays touches all of them, so they are one junction rather than several.

A burn along the track moves the chaser 8330 m backwards. Three 0.5 m/s impulses from rest alongside a target in a 400 km circular orbit, followed for 2 revolutions in the frame riding on the target. Along-track distance runs across the page with the direction of travel to the left, and radial distance up. The prograde burn ends 8330 metres behind after one revolution — exactly 6πΔv/n, and it is behind rather than ahead because the burn raised the orbit and a higher orbit takes longer. The retrograde burn ends 8330 metres ahead by the same arithmetic with the sign reversed. The radial burn is the third case and the strange one: it opens a closed loop and returns exactly to where it started after a revolution, having gone nowhere at a cost of 0.5 m/s. That is not a curiosity but the basis of the R-bar approach, in which a vehicle closes on a station from below along a path that costs nothing to abandon.

A burn that moves the wrong way

In the frame riding on an orbiting target, a thrust along the direction of travel leaves a chaser eight kilometres behind after one lap, a radial thrust returns it exactly to where it started, and every free relative orbit is the same ellipse — twice as long along the track as it is across.

spaceflight · Rendezvous The relative orbit when the target's own is not circular. Free relative motion of a chaser about a target, over one revolution, for four eccentricities of the target's orbit — drawn by propagating two exact Keplerian orbits and differencing them in the target's own rotating frame rather than by any linearised solution. The chaser differs from the target by 0.0035 in eccentricity, so the two semi-major axes are identical. Along-track distance runs across the page. At e = 0 the path is the familiar closed ellipse, 2.00 times as long along the track as across it — recovered here from an exact propagation, which is the check that the calculation is right. Raise the target's eccentricity and the path stops being an ellipse and starts breathing: the separation varies by a factor of 2.00 over one revolution at e = 0 and 2.26 at e = 0.7, because the frame itself is now turning at a rate that changes through the orbit and the chaser is swept around it fastest at periapsis. Every one of them still closes exactly, and that is the one statement of the circular theory that survives — two orbits of equal semi-major axis have equal periods and cannot drift apart, whatever else differs between them.

The relative orbit that breathes

Every result in the close-approach frame — the two-to-one ellipse, the burn that moves the wrong way, the eight kilometres behind — assumes the target's own orbit is circular. Around an eccentric target none of the geometry survives. One statement does, and it is about energy rather than shape.

spaceflight · Rendezvous Two ranges that look identical until something is spent. The bearing to a target, as a camera measures it, for two relative orbits that differ only in size — one 2.5 times further away than the other in every coordinate at every instant. Before the manoeuvre the two histories are identical to machine precision, and that is not a numerical accident but a property of the equations: the linearised relative dynamics are homogeneous, so scaling a whole trajectory gives another trajectory, and a direction is unchanged by scaling. An angles-only tracker is therefore exactly blind to range, and no amount of observing improves it. At 42 per cent of the way through, both vehicles fire the same 0.05 m/s impulse — and the histories separate, by up to 2.3 degrees, because an impulse measured in metres per second is the one quantity in the problem that does not scale with the trajectory. The manoeuvre is what makes range observable, which is why an autonomous vehicle navigating on a camera alone flies a deliberate zig-zag rather than a straight approach, and why the manoeuvre is chosen for what it reveals as much as for where it goes.

A camera that cannot see how far

The equations that make a close approach tractable are linear and homogeneous, so scaling an entire relative trajectory gives another valid one with every bearing identical. A vehicle navigating on angles alone is therefore not merely imprecise about range — it is exactly blind to it, and the only thing that restores the scale is spending propellant.

spaceflight · Rendezvous

Named alongside it

The objects these essays reach for when they reach for this one.

The Clohessy–Wiltshire equationsImpulseRelative motionSafety ellipseAngles only navigationApproach ellipseDegeneracyDrift rateEccentricityFormation flyingKalman filterLine of sight

All concepts