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 '' 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. Likewise, any angle derived from the longitude of periapsis (e.g., mean longitude and

true longitude In celestial mechanics true longitude is the ecliptic longitude at which an orbiting body could actually be found if its inclination were zero. Together with the inclination and the ascending node, the true longitude can tell us the precise directi ...

) 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 periapsi ...

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 traject ...

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