Rendezvous: catching the space station

In December 1965 Gemini 6 and Gemini 7 flew within 30 centimetres of each other, 300 km above the Earth. It took two years of theory and practice to get there: the first attempts had shown that flying straight at your target, as a pilot would, does not work in orbit.

In orbit, to catch up you drop down, not speed up.

Look

Everything in orbit follows one rule: lower is faster. A lower orbit is shorter and the spacecraft moves faster along it, so it gains on anything higher up.

1. Launch into a lower orbit, behind the target. The capsule is timed to start some way behind the station and slightly below it.
2. Phasing. Lap after lap, the lower capsule gains a little. Here, 20 km below the station, it gains 1.6 degrees per orbit.
3. Two small burns, a mini Hohmann transfer, lift the capsule into the station’s orbit just behind it.
4. Final approach. The last few hundred metres are flown slowly, straight along the flight direction, by computers watched by the crew. At contact the closing speed is a few centimetres per second.

Understand

Orbital period: T = 2π√(r³/μ). With μ = 398,600 km³/s²:

• Capsule at 400 km (r = 6,778 km): 92.6 minutes.
• Station at 420 km (r = 6,798 km): 93.0 minutes.

Each capsule orbit, the station covers only 92.6/93.0 of a lap: the capsule gains 360° × (1 − 92.6/93.0) ≈ 1.6°, about 190 km. Closing a 20° gap takes about 13 orbits, roughly 20 hours.

The transfer from 400 to 420 km costs about 5.6 m/s for each of the two burns: tiny compared with the 7.7 km/s of orbital speed.

Master

Near the target, rendezvous engineers use the Clohessy-Wiltshire (Hill) equations: linearised motion in a frame that orbits with the target, with x radial, y along-track and n the target’s mean motion:

ẍ − 3n²x − 2nẏ = 0,   ÿ + 2nẋ = 0,   z̈ + n²z = 0

They explain the strange behaviour: a push forwards (+ẏ) makes the chaser rise (x grows) and then drift backwards. Approach paths are designed along the y axis (V-bar, from behind) or the x axis (R-bar, from below, where the natural motion helps brake), with safety rules so that a failed engine never leads to a collision.