View source for Longitude of periapsis

[[File:Orbit1.svg|thumb|right|ϖ = Ω + ω in separate planes.]]
In [[celestial mechanics]], the '''longitude of the periapsis''', also called '''longitude of the pericenter''', of an orbiting body is the [[longitude]] (measured from the point of the vernal equinox) at which the [[periapsis]] (closest approach to the central body) would occur if the body's orbit [[inclination]] were zero. It is usually denoted ''[[Pi (letter)#Variant pi|ϖ]]''.

For the motion of a planet around the Sun, this position is called '''longitude of perihelion''' ϖ, which is the sum of the longitude of the ascending node Ω, and the [[argument of perihelion]] ω.<ref>{{cite book |title=Explanatory Supplement to the Astronomical Almanac |chapter=Chapter 8: Orbital Ephemerides of the Sun, Moon, and Planets |chapter-url=ftp://ssd.jpl.nasa.gov/pub/eph/planets/ioms/ExplSupplChap8.pdf |page=26 |editor-last1=Urban |editor-first1=Sean E. |editor-last2=Seidelmann |editor-first2=P. Kenneth |publisher=University Science Books}}</ref><ref>{{cite journal |last1=Simon |first1=J. L. |last2=Bretagnon |first2=P. |last3=Chapront |first3=J. |last4=Chapront-Touzé |first4=M. |last5=Francou |first5=G. |last6=Laskar |first6=J. |display-authors=1 |year=1994 |journal=Astronomy and Astrophysics |title=Numerical expressions for precession formulae and mean elements for the Moon and the planets |volume=282 |pages=663–683,672|bibcode = 1994A&A...282..663S }}</ref>

The longitude of periapsis is a compound angle, with part of it being measured in the [[plane of reference]] and the rest being measured in the plane of the [[orbit]]. Likewise, any angle derived from the longitude of periapsis (e.g., [[mean longitude]] and [[true longitude]]) will also be compound.

Sometimes, the term ''longitude of periapsis'' is used to refer to ''ω'', the angle between the ascending node and the periapsis. That usage of the term is especially common in discussions of binary stars and exoplanets.<ref>{{cite book|author=Robert Grant Aitken|title=The Binary Stars|url=https://archive.org/details/binarystars00aitkgoog|year=1918|publisher=D.C. McMurtrie|series=Semicentennial Publications of the University of California|page=[https://archive.org/details/binarystars00aitkgoog/page/n228 201]}}</ref><ref>[http://ad.usno.navy.mil/wds/orb6/format.html "Format"] {{Webarchive|url=https://web.archive.org/web/20090225210208/http://ad.usno.navy.mil/wds/orb6/format.html |date=2009-02-25 }} in [http://ad.usno.navy.mil/wds/orb6.html ''Sixth Catalog of Orbits of Visual Binary Stars''] {{Webarchive|url=https://web.archive.org/web/20090412084731/http://ad.usno.navy.mil/wds/orb6.html |date=2009-04-12 }}, William I. Hartkopf & Brian D. Mason, U.S. Naval Observatory, Washington, D.C. Accessed on 10 January 2018.</ref> However, the angle ω is less ambiguously known as the [[argument of periapsis]].

==Calculation from state vectors==
''ϖ'' is the sum of the [[longitude of ascending node]] Ω (measured on ecliptic plane) and the [[argument of periapsis]] ''ω'' (measured on orbital plane):
<math display="block">\varpi = \Omega + \omega</math>

which are derived from the [[orbital state vectors]].

==Derivation of ecliptic longitude and latitude of perihelion for inclined orbits==
Define the following:
{{unbulleted list | style = padding-left: 1.5em
| i, inclination
| ω, argument of perihelion
| Ω, longitude of ascending node
| ε, obliquity of the ecliptic (for the standard equinox of 2000.0, use 23.43929111°)
}}

Then:
{{unbulleted list | style = padding-left: 1.5em
| {{larger|{{math|1=A = cos ω cos Ω – sin ω sin Ω cos i}}}}
| {{larger|{{math|1=B = cos ε (cos ω sin Ω + sin ω cos Ω cos i) – sin ε sin ω sin i}}}}
| {{larger|{{math|1=C = sin ε (cos ω sin Ω + sin ω cos Ω cos i) + cos ε sin ω sin i}}}}
}}

The right ascension α and declination δ of the direction of perihelion are:
{{block indent | em = 1.5 | text = {{larger|{{math|1=tan α = {{big|{{sfrac|B|A}}}}}}}}}}
{{block indent | em = 1.5 | text = {{larger|{{math|1=sin δ = C}}}}}}

If A < 0, add 180° to α to obtain the correct quadrant.

The ecliptic longitude ϖ and latitude b of perihelion are:
{{block indent | em = 1.5 | text = {{larger|{{math|1=tan ϖ = {{larger|{{sfrac|sin α cos ε + tan δ sin ε|cos α}}}}}}}}}}
{{block indent | em = 1.5 | text = {{larger|{{math|1=sin b = sin δ cos ε – cos δ sin ε sin α}}}}}}

If cos(α) < 0, add 180° to ϖ to obtain the correct quadrant.

As an example, using the most up-to-date numbers from Brown (2017)<ref>Brown, Michael E. (2017) “Planet Nine: where are you? (part 1)” The Search for Planet Nine. http://www.findplanetnine.com/2017/09/planet-nine-where-are-you-part-1.html</ref> for the hypothetical [[Planet Nine]] with i = 30°, ω = 136.92°, and Ω = 94°, then α = 237.38°, δ = +0.41° and ϖ = 235.00°, b = +19.97° (Brown actually provides i, Ω, and ϖ, from which ω was computed).

==References==
{{reflist}}

==External links==
* [https://data.giss.nasa.gov/modelE/ar5plots/srorbpar.html Determination of the Earth's Orbital Parameters ] Past and future longitude of perihelion for Earth.

{{orbits}}

[[Category:Orbits]]