Editing Orbital elements (section)

==== 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>