Editing Orbital elements (section)

== Common sets of elements ==

=== Classical Keplerian elements {{anchor|Keplerian}} ===
While in theory, any set of elements that meets the requirements above can be used to describe an orbit, in practice, certain sets are much more common than others.

The most common elements used to describe the size and shape of the orbit are the semi-major axis ({{Mvar|a}}), and the eccentricity ({{Mvar|e}}). Sometimes the semi-parameter ({{Mvar|p}}) is used instead of {{Mvar|a}}, as the semi-major axis is infinite for parabolic trajectories, and thus cannot be used.<ref name=":02" /><ref name=":0">{{Cite web |last=Weber |first=Bryan |title=Orbital Mechanics |url=https://orbital-mechanics.space/intro.html |access-date=21 February 2025 |website=Orbital Mechanics |at="Orbital Nomenclature", and "Classical Orbital Elements"}}</ref>

It is common to specify the period ({{Mvar|P}}) or mean motion ({{Mvar|n}}) instead of the semi-major axis in Keplerian element sets, as each can be computed from the other provided the [[standard gravitational parameter]] (<math>\mu</math>) is known for the central body though the relations above.

For the epoch, the epoch time ({{Mvar|t}}) along with the [[mean anomaly]] ({{math|''M''<sub>0</sub>}}), [[mean longitude]] ({{math|''L''<sub>0</sub>}}), [[true anomaly]] (<math>\nu_0</math>) or (rarely) the [[eccentric anomaly]] ({{math|''E''<sub>0</sub>}}) are often used. The time of periapsis passage ({{math|''T''<sub>0</sub>}}) is also sometimes used for this purpose.<ref name=":0" />

It is also quite common to see either the mean anomaly or the mean longitude expressed directly, without either {{math|''M''<sub>0</sub>}} or {{math|''L''<sub>0</sub>}} as intermediary steps, as a [[Linear function (calculus)|linear function]] of time. This method of expression will consolidate the mean motion as the slope of this linear equation. An example of this is provided below:{{Indent|5}}<math>M(t)=M_{0}+n(t-t_{0})</math>

=== Elements by body type ===
The choice of elements can differ depending on the type of astronomical body. The eccentricity (''{{mvar|e}}'') and either the semi-major axis (''{{mvar|a}}'') or the distance of periapsis (''{{mvar|q}}'') are used to specify the shape and size of an orbit. The longitude of the ascending node ({{math|Ω}}) the inclination (''{{mvar|i}}'') and the argument of periapsis (''{{mvar|ω}}'') or the [[longitude of periapsis]] (''{{mvar|ϖ}}'') specify the orientation of the orbit in its plane. Either the Mean longitude at epoch ({{math|''L''<sub>0</sub>}}) the mean anomaly at epoch ({{math|''M''<sub>0</sub>}}) or the time of periapsis passage ({{math|''T''<sub>0</sub>}}) are used to specify a known point in the orbit. The choices made depend whether the vernal equinox or the node are used as the primary reference.<ref name="Green2">{{cite book |last=Green |first=Robin M. |title=Spherical Astronomy |date=1985 |publisher=Cambridge University Press |isbn=978-0-521-23988-2}}</ref><ref name="Danby2">{{cite book |last=Danby |first=J. M. A. |title=Fundamentals of Celestial Mechanics |date=1962 |publisher=Willmann-Bell |isbn=978-0-943396-20-0}}</ref>
{| class="wikitable"
|+Sets of orbital elements
!Object
!Elements used
|-
|Major planet
|{{math|''e'', ''a'', [[inclination|''i'']], [[ascending node|Ω]], [[longitude of periapsis|''ϖ'']], [[mean longitude|''L''<sub>0</sub>]]}}
|-
|Comet
|{{math|''e'', [[periapsis|''q'']], ''i'', Ω, [[argument of periapsis|''ω'']], ''T''<sub>0</sub>}}
|-
|Asteroid
|{{math|''e'', ''a'', ''i'', Ω, ''ω'', [[mean anomaly|''M''<sub>0</sub>]]}}
|}

=== Two-line elements ===<!--This section is linked from [[Epoch (astronomy)]]-->
{{Main|Two-line element set}}

Orbital elements can be encoded as text in a number of formats. The most common of them is the [[NASA]] / [[NORAD]] '''"two-line elements"''' (TLE) format,<ref name="Kelso_FAQ2">{{cite web |last=Kelso |first=T.S. |title=FAQs: Two-line element set format |url=http://celestrak.com/columns/v04n03/ |url-status=live |archive-url=https://web.archive.org/web/20160326061740/http://celestrak.com/columns/v04n03/ |archive-date=26 March 2016 |access-date=15 June 2016 |website=celestrak.com |series=CelesTrak}}</ref> originally designed for use with 80 column punched cards, but still in use because it is the most common format, and 80-character ASCII records can be handled efficiently by modern databases.

The two-line element format lists the eccentricity ({{Mvar|e}}), inclination ({{Mvar|i}}), longitude of the ascending node ({{Math|Ω}}), argument of periapsis (''{{mvar|ω}}''), mean motion (''{{Mvar|n}}''), epoch ({{math|{{var|t}}{{sub|0}}}}), and mean anomaly at epoch ({{math|{{var|M}}{{sub|0}}}}).<ref name="Kelso_FAQ2" /><ref name=":02" /> Since the format is primarily meant for orbits of the Earth, the standard gravitational parameter (''{{Mvar|μ}}''), can be assumed and used to calculate the semi-major axis with the mean motion via the relations above.

Depending on the application and object orbit, the data derived from TLEs older than 30 days can become unreliable. Orbital positions can be calculated from TLEs through [[Simplified perturbations models|simplified perturbation models]]  ([[SGP4]] / [[SDP4]] / SGP8 / SDP8).<ref>{{cite book |title=Explanatory Supplement to the Astronomical Almanac |publisher=University Science Books |year=1992 |editor-last=Seidelmann |editor-first=K.P. |edition=1st |place=Mill Valley, CA}}</ref>

Example of a two-line element:<ref>{{cite web |title=SORCE |url=http://www.heavens-above.com/orbitdisplay.asp?lat=0&lng=0&alt=0&loc=Unspecified&TZ=CET&satid=27651 |archive-url=https://web.archive.org/web/20070927203806/http://www.heavens-above.com/orbitdisplay.asp?lat=0&lng=0&alt=0&loc=Unspecified&TZ=CET&satid=27651 |archive-date=2007-09-27 |website=Heavens-Above.com |series=orbit data}}</ref><pre>
1 27651U 03004A   07083.49636287  .00000119  00000-0  30706-4 0  2692
2 27651 039.9951 132.2059 0025931 073.4582 286.9047 14.81909376225249
</pre>

=== Delaunay variables ===
The Delaunay orbital elements were introduced by [[Charles-Eugène Delaunay]] during his study of the motion of the [[Moon]].<ref name="Aubin-20142">{{cite book |last=Aubin |first=David |title=Biographical Encyclopedia of Astronomers |publisher=Springer New York |year=2014 |isbn=978-1-4419-9916-0 |place=New York City |pages=548–549 |chapter=Delaunay, Charles-Eugène |doi=10.1007/978-1-4419-9917-7_347}}</ref> Commonly called ''Delaunay variables'', they are a set of [[canonical variables]], which are [[action-angle coordinates]]. The angles are simple sums of some of the Keplerian angles, and are often referred to with different symbols than other in applications like so:

* the [[mean longitude]]: <math>\ell = L = M + \omega + \Omega</math>,
* the [[longitude of periapsis]]: <math>g = \varpi = \omega + \Omega</math>, and
* the [[longitude of the ascending node]]: <math>h = \Omega</math>

along with their respective [[Conjugate momentum|conjugate momenta]], ''{{mvar|L}}'', ''{{mvar|G}}'', and ''{{mvar|H}}''.<ref name="Shevchenko-20172">{{cite book |last=Shevchenko |first=Ivan |title=The Lidov–Kozai effect: applications in exoplanet research and dynamical astronomy |publisher=Springer |year=2017 |isbn=978-3-319-43522-0 |publication-place=Cham}}</ref> The momenta ''{{mvar|L}}'', ''{{mvar|G}}'', and ''{{mvar|H}}'' are the [[Action-angle coordinates|''action'' variables]] and are more elaborate combinations of the Keplerian elements ''{{mvar|a}}'', ''{{mvar|e}}'', and ''{{mvar|i}}''.

Delaunay variables are used to simplify perturbative calculations in celestial mechanics, for example while investigating the [[Kozai–Lidov oscillations]] in hierarchical triple systems.<ref name="Shevchenko-20172" /> The advantage of the Delaunay variables is that they remain well defined and non-singular (except for ''{{mvar|h}}'', which can be tolerated) even for circular and equatorial orbits.