Editing Orbital elements (section)

==== Relations between elements ====
This section contains the common relations between the set of orbital elements described above, 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. These formulae also hold true for conversions between these elements in general.

Epoch can be found given the time of periapsis passage, the mean anomaly at epoch, and mean motion like so:{{Indent|5}}<math>t_{0}=T_{0}+\frac{M_{0}}{n}</math>

Time of periapsis passage can be found from the epoch, mean anomaly at epoch, and mean motion by re-arranging the previous equation like so:{{Indent|5}}<math>T_0=t_{0}-\frac{M_{0}}{n}</math>

Mean anomaly can be found from the eccentric anomaly and eccentricity using Kepler's equation like so:{{Indent|5}}<math>M=E-e\sin E</math>

Mean longitude can be found using the mean anomaly at epoch and the longitude of periapsis.{{Indent|5}}<math>L=M+\varpi</math> or <math>L=M+\omega+\Omega</math>

Eccentric anomaly can be found with the mean anomaly and eccentricity using [[Kepler's equation]] through various means, such as iterative calculations or numerical solutions (for some values of {{Mvar|e}}). Kepler's equation is given as{{Indent|5}}<math>E=M+e\sin E</math>,

and can be solved through a [[root-finding algorithm]] (usually [[Newton's Method]]) like so:{{Indent|5}}<math>E_{n+1} = E_{n} + \frac{ M-E_{n} + e \sin(E_{n})}{ 1 - e \cos(E_{n})}</math>

Typically a starting guess of either <math>M</math>, <math>M-e</math>, <math>M+e</math>, or <math>M+e\sin M</math> are used.<ref name=":02" /><ref>{{Cite web |last=Standish |first=E. Myles |last2=Williams |first2=James G. |date= |title=Approximate Positions of the Planets |url=https://ssd.jpl.nasa.gov/planets/approx_pos.html |access-date=20 February 2025 |website=NASA Solar System Dynamics}}</ref> This iteration can be repeated until a desired level of tolerance is reached.

True anomaly can be found from the eccentric anomaly and  through the following relations. The quadrant of the solution can be resolved using an [[Atan2|atan2(y,x)]] function.<ref name=":02" />{{Indent|5}}<math>\sin\nu = \frac{\sqrt{1-e^{2}}\sin E}{1-e\cos\left(E\right)}, \cos\nu =\frac{\cos E-e}{1-e\cos E}</math>

True longitude can be found using the true anomaly and longitude of periapsis through the following relation:{{Indent|5}}<math>l=\nu+\varpi</math> or <math>l=\nu+\omega+\Omega</math>

Mean argument of latitude can be calculated using the mean anomaly and argument of periapsis like so:{{Indent|5}}<math>u_{M}=\Omega+M</math>

Argument of latitude can be found using the true anomaly and argument of periapsis like so:{{Indent|5}}<math>u=\nu+\Omega</math>