Editing Apsis (section)

==Mathematical formulae==

These [[formula]]e characterize the pericenter and apocenter of an orbit:
; Pericenter: Maximum speed, <math display="inline">v_\text{per} = \sqrt{ \frac{(1 + e)\mu}{(1 - e)a} } \,</math>, at minimum (pericenter) distance, <math display="inline">r_\text{per} = (1 - e)a</math>.
; Apocenter: Minimum speed, <math display="inline"> v_\text{ap} = \sqrt{\frac{(1 - e)\mu}{(1 + e)a} } \,</math>, at maximum (apocenter) distance, <math display="inline">r_\text{ap} = (1 + e)a</math>.

While, in accordance with [[Kepler's laws of planetary motion]] (based on the conservation of [[angular momentum]]) and the conservation of energy, these two quantities are constant for a given orbit:
; [[Specific relative angular momentum]]: <math>h = \sqrt{\left(1 - e^2\right)\mu a}</math>
; [[Specific orbital energy]]: <math>\varepsilon = -\frac{\mu}{2a}</math>

where:
* <math display="inline">r_\text{ap}</math> is the distance from the apocenter to the primary focus
* <math display="inline">r_\text{per}</math> is the distance from the pericenter to the primary focus
* ''a'' is the [[semi-major axis]]:
*: <math>a = \frac{r_\text{per} + r_\text{ap}}{2}</math>
* ''μ'' is the [[standard gravitational parameter]]
* ''e'' is the [[orbital eccentricity|eccentricity]], defined as
*: <math>e = \frac{r_\text{ap} - r_\text{per}}{r_\text{ap} + r_\text{per}} = 1 - \frac{2}{\frac{r_\text{ap}}{r_\text{per}} + 1}</math>

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

The [[arithmetic mean]] of the two limiting distances is the length of the semi-major axis ''a''. The [[geometric mean]] of the two distances is the length of the [[semi-minor axis]] ''b''.

The geometric mean of the two limiting speeds is
:<math>\sqrt{-2\varepsilon} = \sqrt{\frac{\mu}{a}}</math>

which is the speed of a body in a circular orbit whose radius is <math>a</math>.