celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...

, the longitude of the periapsis, also called longitude of the pericenter, of an orbiting body is the

longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...

(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 Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Ea ...

were zero. It is usually denoted '' 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. 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 In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...

. 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. However, the angle ω is less ambiguously known as the

argument of periapsis The argument of periapsis (also called argument of perifocus or argument of pericenter), symbolized as ''ω'', is one of the orbital elements of an orbiting body. Parametrically, ''ω'' is the angle from the body's ascending node to its periap ...

Calculation from state vectors

''ϖ'' is the sum of the longitude of ascending node Ω (measured on ecliptic plane) and the

''ω'' (measured on orbital plane): :

\varpi = \Omega + \omega

which are derived from the

orbital state vectors In astrodynamics and celestial dynamics, the orbital state vectors (sometimes state vectors) of an orbit are Cartesian vectors of position (\mathbf) and velocity (\mathbf) that together with their time (epoch) (t) uniquely determine the trajector ...

Derivation of ecliptic longitude and latitude of perihelion for inclined orbits

Define the following: : 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: : : : The right ascension α and declination δ of the direction of perihelion are: : : If A < 0, add 180° to α to obtain the correct quadrant. The ecliptic longitude ϖ and latitude b of perihelion are: : : 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)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 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

External links

Determination of the Earth's Orbital Parameters
Past and future longitude of perihelion for Earth. {{orbits Orbits