projected semi-major axis

There are two projected semi-major axes of orbits commonly encountered in astronomy: the projection of the orbit's semi-major axis on the plane of the sky, e.g., when viewing a visual binary or direct imaging an extra-solar planet, and projection on the line of sight, when working out the orbit from radial velocity measurements over time or from timing variations of a pulsar. It is of note that given the geometry of rotation in three dimensions, the projection of the orbit's semi-major axis is not merely half the (apparent) longest distance between any two points on the projected orbit. However, that value and the projected semi-major axis each represent a lower-bound to the orbit's semi-major axis. The projected semi-major axis can be worked out if one of the projected foci is evident, one of which is very near the more massive body if the mass ratio is large, which makes a much more massive body's location a useful approximation. (Note that the foci of the projected ellipse are not the foci of the actual elliptical orbit.) There are formulae to determine or estimate the projected semi-major axis from sufficient observation data under some circumstances. Also, a distribution of semi-major axes can be statistically derived from the observed projected semi-major axes of a large number of bodies.