The most fuel-efficient orbital maneuver
A Hohmann transfer is an elliptical orbit connecting two circular orbits of different altitudes. Named after Walter Hohmann (1925), it requires exactly two impulsive burns: one at the low orbit to accelerate into the transfer ellipse, and one at the high orbit to circularize. This simplicity and efficiency make Hohmann transfers the standard method for space missions, from near-Earth satellites to interplanetary probes.
The transfer orbit is tangent to both circular orbits, meaning the spacecraft's closest point (periapsis) touches the low orbit and its farthest point (apoapsis) touches the high orbit. The fuel cost (delta-v) depends on the mass ratio and specific impulse of the rocket engine, but for any two given orbits, the Hohmann transfer minimizes total delta-v compared to other continuous-thrust or impulsive strategies.
Why two burns, not one
A single instantaneous velocity change cannot change both the periapsis and apoapsis simultaneously if starting from a circular orbit. A single burn at the low orbit would create an elliptical orbit with periapsis at the starting altitude and apoapsis elsewhere; the destination altitude would not be reached. Only a second burn at apoapsis raises the periapsis, circularizing the destination orbit.
For transfer to a lower orbit (e.g., from geostationary to low Earth orbit), the maneuver reverses: one retrograde burn to enter the transfer ellipse, then another retrograde burn at the low orbit to circularize. Total delta-v is symmetrical either way.