Editing Orbital elements (section)

=== Size and shape describing parameters ===
Two parameters are required to describe the size and the shape of an orbit. Generally any two of these values can be used to calculate any other (as described below), so the choice of which to use is one of preference and the particular use case.

* [[Eccentricity (orbit)|Eccentricity]] (''{{mvar|e}}'') — shape of the ellipse, describing how much it deviates from a perfect a circle. An eccentricity of zero describes a perfect circle, values less than 1 describe an ellipse, values greater than 1 describe a hyperbolic trajectory, and a value of exactly 1 describes a parabola.<ref name=":0" />
* [[Semi-major axis]] (''{{mvar|a}}'') — half the distance between the [[Apsis|apoapsis and periapsis]] (long axis of the ellipse). This value is positive for elliptical orbits, infinity for parabolic trajectories, and negative for hyperbolic trajectories, which can hinder its usability when working with different types of trajectories.<ref name=":02">{{Cite book |last=Vallado |first=David A. |title=Fundamentals of astrodynamics and applications |date=2022 |publisher=Microcosm Press |isbn=978-1-881883-18-0 |edition=4th |series=Space technology library |location=Torrance, CA |pages=41–112}}</ref>
* [[Semi-major and semi-minor axes|Semi-minor axis]] (''{{mvar|b}}'') — half the short axis of the ellipse. This value shares the same limitations as with the semi-major axis.
* [[Conic section#Conic parameters|Semi-parameter]] (''{{Mvar|p}}'') — the width of the orbit at the primary focus (at a [[true anomaly]] of ''{{Mvar|π/2}}'' or 90°). This value is useful for its use in the [[orbit equation]], which can return the distance from the central body given ''{{Mvar|p}}'' and the true anomaly for any type of orbit or trajectory. This value is also commonly referred to as the semi-latus rectum, and given the symbol ''{{Mvar|ℓ}}''. Additionally, this value will always be defined and positive unlike the semi-major and semi-minor axes.<ref name=":02" />
* [[Apsis|Apoapsis]] ({{math|{{var|r}}{{sub|a}}}}) — The farthest point in the orbit from the central body (at a true anomaly of ''{{Mvar|π}}'' or 180°). This quantity is undefined (or infinity) for parabolic and hyperbolic trajectories, as they continue moving away from the central body forever. This value is sometimes given the symbol ''{{Mvar|Q}}.''<ref name=":0" />
* [[Apsis|Periapsis]] ({{math|{{var|r}}{{sub|p}}}}) — The closest point in the orbit from the central body (at a true anomaly of 0). Unlike with apoapsis, this quantity is defined for all orbit types. This value is sometimes given the symbol ''{{Mvar|q}}.''<ref name=":0" />

For perfectly circular orbits, there are no points on the orbit that can be described as either the apoapsis or periapsis, as they all have the same distance from the central body. Additionally it is common to see the affix for apoapsis and periapsis changed depending on the central body (e.g. apogee and perigee for orbits of the [[Earth]], and aphelion and perihelion for orbits of the [[Sun]]).

Other parameters can also be used to describe the size and shape of an orbit such as the [[Eccentricity (mathematics)|linear eccentricity]], [[flattening]], and [[focal parameter]], but the use of these is limited.

==== Relations between elements ====
{{Further|Conic section|Apsis|Semi-major and semi-minor axes}}
This section contains the common relations between these orbital elements, but more relations can be derived through manipulations of one or more of these equations. The variable names used here are consistent with the ones described above.

Eccentricity can be found using the semi-minor and semi-major axes like so:{{Indent|5}}<math>e=\sqrt{1-\frac{b^2}{a^2}}</math>  when <math>a>0</math>, <math>e=\sqrt{1+\frac{b^2}{a^2}}</math> when <math>a<0</math>

Eccentricity can also be found using the apoapsis and periapsis through this relation:{{Indent|5}}<math>e=\frac{r_{a}-r_{p}}{r_{a}+r_{p}}</math>

The semi-major axis can be found using the fact that the line that connects the apoapsis to the center of the conic, and from the center to the periapsis both combined span the length of the conic, and thus the major axis. This is then divided by 2 to get the semi-major axis.{{Indent|5}}<math>a =\frac{r_{p}+r_{a}}{2}</math>

The semi-minor axis can be found using the semi-major axis and eccentricity through the following relations. Two formula are needed to avoid taking the [[square root]] of a negative number.{{Indent|5}}<math>b=a\sqrt{1-e^{2}}</math> when <math>e<1</math>, <math>b=a\sqrt{e^{2}-1}</math> when <math>e>1</math>

The semi-parameter can be found using the semi-major axis and eccentricity like so:{{Indent|5}}<math>p=a\left(1-e^{2}\right)</math>

Apoapsis can be found using the following equation, which is a form of the [[orbit equation]] solved for <math>\nu=\pi</math>.{{Indent|5}}<math>r_{a}=\frac{p}{1-e}</math> , when <math>e<1</math>

Periapsis can be found using the following equation, which, as with the equation for apoapsis, is a form of the [[orbit equation]] instead solved for <math>\nu=0</math>.{{Indent|5}}<math>r_{p}=\frac{p}{1+e}</math>