True longitude

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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 direction from the central object at which the body would be located at a particular time.

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


The true longitude l can be calculated as follows: [1] [2] [3]

l = ν + ϖ


True anomaly angular parameter that defines the position of a body moving along a Keplerian orbit; angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse (the point around which the object orbits)

In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse.

Longitude of the periapsis longitude (measured from the point of the vernal equinox) at which the periapsis (closest approach to the central body) would occur if the bodys inclination were zero

In celestial mechanics, the longitude of the periapsis, also called longitude of the pericenter, of an orbiting body is the longitude at which the periapsis would occur if the body's orbit inclination were zero. It is usually denoted ϖ.

Argument of periapsis one of the orbital elements used to specify the orbit of an object in space

The argument of periapsis, 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 periapsis, measured in the direction of motion.

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Ecliptic apparent path of the Sun on the celestial sphere

The ecliptic is the mean plane of the apparent path in the Earth's sky that the Sun follows over the course of one year; it is the basis of the ecliptic coordinate system. This plane of reference is coplanar with Earth's orbit around the Sun. The ecliptic is not normally noticeable from Earth's surface because the planet's rotation carries the observer through the daily cycles of sunrise and sunset, which obscure the Sun's apparent motion against the background of stars during the year.

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Mean anomaly one of the orbital elements used to specify the orbit of an object in space

In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical two-body problem. It is the angular distance from the pericenter which a fictitious body would have if it moved in a circular orbit, with constant speed, in the same orbital period as the actual body in its elliptical orbit.

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.

A non-inclined orbit is an orbit coplanar with a plane of reference. The orbital inclination is 0° for prograde orbits, and π (180°) for retrograde ones. If the plane of reference is a massive spheroid body's equatorial plane, these orbits are called equatorial; if the plane of reference is the ecliptic plane, they are called ecliptic.

Orbital inclination change is an orbital maneuver aimed at changing the inclination of an orbiting body's orbit. This maneuver is also known as an orbital plane change as the plane of the orbit is tipped. This maneuver requires a change in the orbital velocity vector at the orbital nodes.

Circular orbit

A circular orbit is the orbit with a fixed distance around the barycenter, that is, in the shape of a circle.

In celestial mechanics, the plane of reference 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 longitude of the ascending node.

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Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities. Those Greek letters which have the same form as Latin letters are rarely used: capital A, B, E, Z, H, I, K, M, N, O, P, T, Y, X. Small ι, ο and υ are also rarely used, since they closely resemble the Latin letters i, o and u. Sometimes font variants of Greek letters are used as distinct symbols in mathematics, in particular for ε/ϵ and π/ϖ. The archaic letter digamma (Ϝ/ϝ/ϛ) is sometimes used.

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In celestial mechanics, the Kozai mechanism or Lidov–Kozai mechanism or Kozai–Lidov mechanism, also known as the Kozai, Lidov–Kozai or Kozai–Lidoveffect, oscillations, cycles or resonance, is a dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body under certain conditions, causing the orbit's argument of pericenter to oscillate about a constant value, which in turn leads to a periodic exchange between its eccentricity and inclination. The process occurs on timescales much longer than the orbital periods. It can drive an initially near-circular orbit to arbitrarily high eccentricity, and flip an initially moderately inclined orbit between a prograde and a retrograde motion.

The perifocal coordinate (PQW) system is a frame of reference for an orbit. The frame is centered at the focus of the orbit, i.e. the celestial body about which the orbit is centered. The unit vectors and lie in the plane of the orbit. is directed towards the periapsis of the orbit and has a true anomaly of 90 degrees past the periapsis. The third unit vector is the angular momentum vector and is directed orthogonal to the orbital plane such that:

In celestial mechanics, the argument of latitude is an angular parameter that defines the position of a body moving along a Kepler orbit. It is the angle between the ascending node and the body.

Nodal precession is the precession of the orbital plane of a satellite around the rotational axis of an astronomical body such as Earth. This precession is due to the non-spherical nature of a rotating body, which creates a non-uniform gravitational field. The following discussion relates to low Earth orbit of artificial satellites, which have no measurable effect on the motion of Earth. The nodal precession of more massive, natural satellites like the Moon is more complex.


  1. Multon, F. R. (1970). An Introduction to Celestial Mechanics (2nd ed.). New York, NY: Dover. pp. 182–183.
  2. Roy, A. E. (1978). Orbital Motion. New York, NY: John Wiley & Sons. p. 174. ISBN   0-470-99251-4.
  3. Brouwer, D.; Clemence, G. M. (1961). Methods of Celestial Mechanics. New York, NY: Academic Press. p. 45.