Bi-elliptic transfer

In astronautics and aerospace engineering, the bi-elliptic transfer is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations, require less delta-v than a Hohmann transfer maneuver. The bi-elliptic transfer consists of two half-elliptic orbits. From the initial orbit, a first burn expends delta-v to boost the spacecraft into the first transfer orbit with an apoapsis at some point away from the central body. At this point a second burn sends the spacecraft into the second elliptical orbit with periapsis at the radius of the final desired orbit, where a third burn is performed, injecting the spacecraft into the desired orbit. While they require one more engine burn than a Hohmann transfer and generally require a greater travel time, some bi-elliptic transfers require a lower amount of total delta-v than a Hohmann transfer when the ratio of final to initial semi-major axis is 11.94 or greater, depending on the intermediate semi-major axis chosen. The idea of the bi-elliptical transfer trajectory was first published by Ary Sternfeld in 1934. The three required changes in velocity can be obtained directly from the vis-viva equation where is the speed of an orbiting body, is the standard gravitational parameter of the primary body, is the distance of the orbiting body from the primary, i.e., the radius, is the semi-major axis of the body's orbit. In what follows, is the radius of the initial circular orbit, is the radius of the final circular orbit, is the common apoapsis radius of the two transfer ellipses and is a free parameter of the maneuver, and are the semimajor axes of the two elliptical transfer orbits, which are given by and Starting from the initial circular orbit with radius (dark blue circle in the figure to the right), a prograde burn (mark 1 in the figure) puts the spacecraft on the first elliptical transfer orbit (aqua half-ellipse).