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, to ...

, 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 letter l ...

(measured from the point of the vernal equinox) at which the

periapsis An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary (astronomy), primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two ...

(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 Plane of reference, reference plane and the orbital plane or Axis of rotation, axis of direction of the orbiting object ...

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 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 ...

ω. The longitude of periapsis is a compound angle, with part of it being measured in the

plane of reference In celestial mechanics, the plane of reference (or reference plane) is the plane used to define orbital elements (positions). The two main orbital elements that are measured with respect to the plane of reference are the inclination and the longi ...

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 a p ...

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

mean longitude Mean longitude is the ecliptic longitude at which an orbiting body could be found if its orbit were circular and free of perturbations. While nominally a simple longitude, in practice the mean longitude does not correspond to any one physical angle ...

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.

Calculation from state vectors

''ϖ'' is the sum of the

longitude of ascending node The longitude of the ascending node (☊ or Ω) is one of the orbital elements used to specify the orbit of an object in space. It is the angle from a specified reference direction, called the '' origin of longitude'', to the direction of the asc ...

Ω (measured on ecliptic plane) and the argument of periapsis ''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