To catch a spacecraft ahead of you in orbit, don't hit the accelerator
Two spacecraft share a circular orbit, one trailing the other. The obvious move is for the rear craft to fire its engine and speed forward. Do that, and it climbs to a higher orbit, travels more slowly and falls further behind. Orbital rendezvous runs on rules that feel backwards.
Orbital mechanics, also called astrodynamics, applies Newton's laws of motion and gravity to rockets and satellites. Its rules of thumb explain the paradox. Lower orbits are faster, because gravity pulls harder closer in. Push forward on a circular orbit and you stretch it into an ellipse whose high point lies half an orbit away, with a longer period. Push backward and you dip toward a low point half an orbit ahead, and your lap gets shorter. So the chasing craft actually brakes, drops lower, gains ground, then climbs back. Real dockings take a series of carefully timed burns across several orbits, often hours or even days.
Another quirk: a single short burn can never move you from one circular orbit to another. Whatever point you fired at, you will pass through again every lap, while the rest of the path changes. And if no engine fires, an orbit's size and shape stay fixed, drag and other bodies aside.
The history runs through some famous names. Kepler published his first laws of planetary motion in 1609, and Newton's Principia of 1687 offered a way to work out an orbit from three sightings, which Edmond Halley used on comets, including the one named after him. In 1801 Carl Friedrich Gauss used three observations to pin down the six numbers describing the orbit of Ceres, letting astronomers find it again after it slipped from view. Descendants of such methods now run inside GPS receivers.
Samuel Herrick began shaping astrodynamics as its own discipline in the 1930s, encouraged by the rocket pioneer Robert Goddard, who believed space navigation would one day be needed. In the 1960s those techniques met powerful computers, and people could fly to the Moon and back. Newton's framework still does most of the work, with general relativity called in near strong gravity, such as orbits close to the Sun.
Source: Orbital mechanics
