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# Apsis

(Redirected from Apoapsis)
This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms). In architecture, apsis is a synonym for apse; Apogee is also the name of a video game publisher.

In astronomy, an apsis (plural apsides "ap-si-deez") is the point of greatest or least distance of the elliptical orbit of a celestial body from its center of attraction (the center of mass of the system).

The point of closest approach is called the periapsis and the point of farthest approach is the apoapsis. A straight line drawn through the periapsis and apoapsis is the line of apsides. This is the major axis of the ellipse, the line through the longest part of the ellipse.

Related terms are used to identify the body being orbited. The most common are perigee and apogee, referring to earth orbits, and perihelion and aphelion, referring to orbits around the sun.
We have:

• Periapsis: maximum speed $v_\mathrm{per} = \sqrt{ \frac{(1+e)\mu}{(1-e)a} } \,$  at minimum distance $r_\mathrm{per}=(1-e)a\!\,$ (periapsis distance)
• Apoapsis: minimum speed $v_\mathrm{ap} = \sqrt{ \frac{(1-e)\mu}{(1+e)a} } \,$  at maximum distance $r_\mathrm{ap}=(1+e)a\!\,$ (apoapsis distance)

where one easily verifies

$h = \sqrt{(1-e^2)\mu a}$
$\epsilon=-\frac{\mu}{2a}$

(each the same for both points, like they are for the whole orbit, in accordance with Kepler's laws of planetary motion (conservation of angular momentum) and the conservation of energy)

where:

Properties:

$e=\frac{r_\mathrm{ap}-r_\mathrm{per}}{r_\mathrm{ap}+r_\mathrm{per}}=1-\frac{2}{\frac{r_\mathrm{ap}}{r_\mathrm{per}}+1}=\frac{2}{\frac{r_\mathrm{per}}{r_\mathrm{ab}}+1}-1$

Note that for conversion from heights above the surface to distances, the radius of the central body has to be added, and conversely.

The arithmetic mean of the two distances is the semi-major axis $a\!\,$. The geometric mean of the two distances is the semi-minor axis $b\!\,$.

The geometric mean of the two speeds is $\sqrt{-2\epsilon}$, the speed corresponding to a kinetic energy which, at any position of the orbit, added to the existing kinetic energy, would allow the orbiting body to escape (the square root of the sum of the squares of the two speeds is the local escape velocity).

## Terminology

Various related terms are used for other celestial objects. 'Perigee', 'perihelion' and 'periastron' and corresponding 'apo-' forms are very frequently used in the astronomical literature, while '-lune', '-selene', '-cytherion', '-areion' and '-jove' are very occasionally used; other terms are never used in practice, with 'perigee' being quite commonly used as a generic 'closest approach to planet' term instead of specifically applying to the Earth.

Body Closest approach Farthest approach
Star Periastron Apastron
Black hole Perimelasma Apomelasma
Sun Perihelion Aphelion (1)
Mercury Perihermion Aphermion (1)
Venus Pericytherion Apocytherion
Earth Perigee Apogee
Moon Periselene/Pericynthion/Perilune Aposelene/Apocynthion/Apolune
Mars Periareion Apoareion
Jupiter Perizene/Perijove Apozene/Apojove
Saturn Perikrone/Perisaturnium Apokrone/Aposaturnium
Uranus Periuranion Apuranion
Neptune Periposeidion Apoposeidion