Orbital rendezvous

Why must a spacecraft slow down in order to catch up with another craft ahead of it in the same orbit?

On a motorway, catching the car ahead is not a puzzle: press the accelerator. In June 1965, Jim McDivitt tried the same thing in orbit, closing on the spent booster stage that had launched Gemini 4. He pointed at it, thrusted toward it — and watched it drift away. The attempt was abandoned having burned a substantial share of the mission's fuel. He was not doing it wrong. He was doing what works everywhere except in orbit. Why should adding speed put you further behind?

Start by noticing what "speed" even means up there. A circular orbit is not a choice of speed at a place; it is a balance. At a given radius there is exactly one speed at which falling and moving forward cancel into a circle. Faster and you no longer curve down as fast as the ground curves away — you rise. Slower and you sink. Radius and speed are locked together, not independent.

So follow the burn. You fire your engine along your direction of travel. Where does that energy go? You are briefly moving faster than a circle at that radius allows, so you climb. And climbing costs — the extra kinetic energy is spent buying altitude. By the time you have coasted round to the far side of your orbit, you are higher, and moving slower than you were before the burn.

Pause on that, because it is the strange part. You added energy and ended up slower. There is no contradiction: your total orbital energy did rise. It is simply that in orbit, energy buys height, and height is paid for in speed. A satellite in a low orbit races along at nearly eight kilometres per second; one far out at geostationary distance ambles at about three.

Now bring in the thing we actually care about, which is not speed but angular rate — how quickly you sweep around the planet, because that is what closes or opens a gap. Kepler's third law ties the orbital period to the semi-major axis: the period grows as that distance to the power of three-halves. Raise your orbit and your lap takes longer. Two effects, both against you: you are slower, and your track is longer.

So what does the prograde burn actually accomplish? It puts you in a bigger, lazier orbit. Your target, still in the original one, keeps sweeping round at the old rate — and pulls ahead. Speeding up made you fall behind. And what if you do the opposite? Fire retrograde, and you drop into a smaller orbit with a shorter period. You come round faster, and every lap you gain a slice of angle on the craft above. To catch up, you go down.

How much? Take the Space Station's neighbourhood, about 400 kilometres up, one lap in roughly 92 minutes. Drop two kilometres and the period shortens by around two and a half seconds. That sounds trivial, but it repeats: each lap closes something like nineteen kilometres of the gap along the track. Rendezvous is therefore not a chase but a phasing problem — sit in a slightly lower orbit for as many revolutions as it takes, then raise yourself back up to match, arriving alongside because you have arranged to be in the same place at the same time.

Three refinements connect those steps.

The one most worth stating plainly: nothing here is unusual physics. There is no special "orbital force" reversing things. The counterintuitive result falls straight out of Newton with one substitution — in orbit, thrust does not primarily change your speed, it changes your orbit, and the orbit then dictates your speed and your period.

Second, the reversal is about the long game, not the last few metres. Thrust toward a target does move you toward it immediately. The trouble is that the orbital effect accumulates while the direct effect does not, so over a few minutes the direct push wins and over an orbit the orbital consequence buries it. That is why the final approach — inside a few hundred metres, over a few minutes — can be flown by pointing and nudging, using small translations along carefully chosen approach lines, while everything before it must be flown as orbit changes. Crews think in the rotating frame of the target, where the standard tool is the Clohessy-Wiltshire equations, and in that frame the odd behaviour reappears as a rule: push forward and you drift backward and up.

Third, a burn's effect shows up half a revolution away. Firing prograde raises the point opposite you, leaving your current altitude untouched. A rendezvous is therefore a sequence of burns at chosen points, each shaping the orbit somewhere else — which is why these operations are planned in advance rather than flown by eye.

• How a Hohmann transfer picks the cheapest possible route between two circular orbits.
• Why the same logic governs interplanetary launch windows, and what a gravity assist adds to it.

Every term the collection defines is gathered in the glossary.