Editing Orbital elements (section)

=== Epoch describing elements ===
Two elements are needed to describe the position of the body around its orbit, and the time at which this occurs. If this time is defined to be at a point where the specific position variable is a designated constant (usually zero), then it does not need to be specified.

* [[Epoch]] ({{math|{{var|t}}{{sub|0}}}}) — time at which one of the below elements is defined. Alternatively this is the point in time where the orbital elements were measured. Sometimes the epoch time is considered as part of the reference frame and is not listed as a distinct element.
* [[Time of Periapsis Passage|Time of periapsis passage]] (''{{math|{{var|T}}{{sub|0}}}}'') — time at which the orbiting body is at periapsis. This is also when the mean anomaly and true anomaly (and others) are zero, so they do not need to be defined. This value is not defined for circular orbits, as they do not have a uniquely defined point of periapsis.
* [[Mean anomaly]] at epoch ({{math|{{var|M}}{{sub|0}}}}) — mean anomaly at the epoch time. Mean anomaly is a mathematically convenient angle that increases linearly with time as if the orbit were perfectly circular. Zero is defined as being at periapsis, and one period spans 2''{{pi}}'' radians. The rate at which the mean anomaly increases is equal to the mean motion ''{{Mvar|n}}''. Because this angle is relative to periapsis, it is not defined for circular orbits.
* [[Mean longitude]] at epoch ({{math|{{var|L}}{{sub|0}}}}) — mean longitude at the epoch time. Mean longitude is similar to mean anomaly, in that it increases linearly with time and does not represent the real angular displacement. Unlike with mean anomaly, mean longitude is defined relative to the vernal point, which means it is defined for circular orbits.
* [[Eccentric anomaly]] at epoch ({{math|{{var|E}}{{sub|0}}}}) — the eccentric anomaly at the epoch time. Eccentric anomaly is defined at the angular displacement along the auxiliary circle of the ellipse (circle tangent to the ellipse both at apses). This value takes into account the varying speed of objects in elliptical orbits, but does not account for the elliptical shape of the orbit. As such, it still does not correspond to the real angular displacement of the orbiting body. Like with mean anomaly and true anomaly, the eccentric anomaly is measured relative to periapsis, and is not defined for circular orbits. The eccentric anomaly is also not defined for parabolic and hyperbolic trajectories, and instead the parabolic anomaly or hyperbolic anomaly are used.<ref name=":02" />
* [[True anomaly]] at epoch (<math>\nu_0</math>) — angle that represents the real angular displacement of the orbiting body at the epoch time, taking into account the varying speed and elliptical shape of an orbit. Like with mean anomaly, true anomaly is measured relative to periapsis, and thus it has the same limitations with circular orbits.
* [[True longitude]] at epoch ({{math|{{var|l}}{{sub|0}}}}) — the angular displacement of the orbiting body at the epoch time. Unlike with the true anomaly, the true longitude is measured relative to the vernal point, so it can be defined for circular orbits.
* [[Mean argument of latitude]] ({{math|{{var|u}}{{sub|M0}}}}) at epoch — the angular displacement of the orbiting body at the epoch time. Mean argument of latitude is similar to the mean anomaly and mean longitude, but instead it is measured relative to the ascending node. This means while it is well defined for circular orbits, it is not for equatorial orbits.<ref name=":02" />
* [[Argument of latitude]] at epoch ({{math|{{var|u}}{{sub|0}}}}) — the angular displacement of the orbiting body at the epoch time. This angle is measured relative to the ascending node, so while it is defined for circular orbits, it is not defined for equatorial orbits.
These elements are also used to describe the position of an object in its orbit in a more general context, and are not limited to describing the state at an epoch time.

==== 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>