To catch the spacecraft ahead of you, slow down
Point at the target, fire the engine, and you will end up further behind than when you started.
Two spacecraft share a circular orbit 400 km above the Earth. One is ten kilometres ahead of the other. The one behind would like to close the gap.
Every instinct says to point at it and burn. Doing so is the one manoeuvre guaranteed to make things worse.
Lower is faster
The speed needed to hold a circular orbit is v = √(GM/r). The radius is in the denominator, so a lower orbit is a faster one. This is not a small effect and it is not a technicality: it is the whole of orbital rendezvous.
Open Orbital Period and it reports 1.54 hours at its default. That works out to a radius of 6,770 km, about 399 km of altitude, which is where the International Space Station actually flies. The orbital speed there is 7.67 km/s.
Now the consequence. Period goes as r3/2, so dropping the orbit shortens the year as well as raising the speed. Drop by ten kilometres and the period falls by 12.28 seconds. Every orbit, the lower craft arrives back at the meeting point twelve seconds early, which along the track is a gain of 94 kilometres.
Why burning towards the target fails
Thrusting forwards adds energy. Added energy raises the orbit, and specifically it raises the far side of it, turning your circle into an ellipse whose low point is where you burned.
A bigger orbit has a longer period. So after one lap the target, still on the original circle, has come back to the meeting point before you do. You aimed at it, spent fuel, and fell behind.
To catch it you burn retrograde, against your motion. That lowers the opposite side of the orbit, shortens your period, and you begin gaining on the target every lap. When the gap has closed you burn prograde to circularise and match velocities.
The manoeuvre is counterintuitive in the exact sense: slowing down is how you catch up, and the reason is that in orbit the throttle does not control speed, it controls altitude, and altitude controls speed.
The same trade on an ellipse
Kepler Orbits makes the exchange visible on a single orbit. At its default eccentricity of 0.4 it reports 45.5 km/s at perihelion and 19.5 km/s at aphelion, a factor of 2.33 between the fastest and slowest points of one unchanging path.
Nothing is pushing the body faster near the star. It is trading height for speed the way a dropped ball does, and the ratio of the two speeds is exactly (1 + e)/(1 − e), which for e = 0.4 gives 1.4/0.6 = 2.33. The tool's two readouts are that fraction.
Kepler's second law, equal areas in equal times, is the same statement made geometrically, and it is conservation of angular momentum wearing seventeenth-century clothing.
Where the ladder ends
If lower is faster, the obvious question is how far the pattern runs. Escape Velocity gives the other end: 11.2 km/s from Earth's surface, which is exactly √2 times the circular speed at the same radius.
That factor of √2 is worth knowing, because it says the difference between staying and leaving is 41%, not some enormous multiple. Orbit is not most of the way to space in the altitude sense. It is most of the way in the speed sense, which is the point the rocket equation makes about why the fuel tank is the vehicle.
Everything above assumes two bodies and no atmosphere. Real rendezvous adds drag, which lowers orbits on its own and therefore quietly speeds spacecraft up, and station keeping exists largely to undo it.
References (1)
- phasing and rendezvous Vallado. Fundamentals of Astrodynamics and Applications, ch. 6.