Mean longitude is the

ecliptic longitude The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System b ...

at which an

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 ...

ing body could be found if its orbit were

circular Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circula ...

and free of perturbations. While nominally a simple longitude, in practice the mean longitude does not correspond to any one physical angle.

Definition

* Define a reference direction, ♈︎, along the

ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic again ...

. Typically, this is the direction of the March

equinox A solar equinox is a moment in time when the Sun crosses the Earth's equator, which is to say, appears directly above the equator, rather than north or south of the equator. On the day of the equinox, the Sun appears to rise "due east" and se ...

. At this point, ecliptic longitude is 0°. * The body's orbit is generally

inclined Incline, inclined, inclining, or inclination may refer to: *Grade (slope), the tilt, steepness, or angle from horizontal of a topographic feature (hillside, meadow, etc.) or constructed element (road, railway, field, etc.) *Slope, the tilt, steepn ...

to the ecliptic, therefore define the angular distance from ♈︎ to the place where the orbit crosses the ecliptic from south to north as the '' longitude of the ascending node'', ''Ω''. * Define the angular distance along the plane of the orbit from the

ascending node An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes. Planes of reference Common planes of refere ...

to the

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

as the '' argument of the pericenter'', ''ω''. * Define the '' mean anomaly'', ''M'', as the angular distance from the pericenter which the body would have if it moved in a circular orbit, in the same orbital period as the actual body in its elliptical orbit. From these definitions, the ''mean longitude'', ''l'', is the angular distance the body would have from the reference direction if it moved with uniform speed, :''l'' = ''Ω'' + ''ω'' + ''M'', measured along the ecliptic from ♈︎ to the ascending node, then up along the plane of the body's orbit to its mean position.

Discussion

Mean longitude, like mean anomaly, does not measure an angle between any physical objects. It is simply a convenient uniform measure of how far around its orbit a body has progressed since passing the reference direction. While mean longitude measures a mean position and assumes constant speed,

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 ...

measures the actual longitude and assumes the body has moved with its actual speed, which varies around its

elliptical orbit In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it ...

. The difference between the two is known as the

equation of the center In two-body, Keplerian orbital mechanics, the equation of the center is the angular difference between the actual position of a body in its elliptical orbit and the position it would occupy if its motion were uniform, in a circular orbit of the ...

.Meeus, Jean (1991). p. 222

Formulae

From the above definitions, define the '' longitude of the pericenter'' :''ϖ'' = ''Ω'' + ''ω''. Then mean longitude is also :''l'' = ''ϖ'' + ''M''. Another form often seen is the ''mean longitude at epoch'', ''ε''. This is simply the mean longitude at a reference time ''t''₀, known as the

epoch In chronology and periodization, an epoch or reference epoch is an instant in time chosen as the origin of a particular calendar era. The "epoch" serves as a reference point from which time is measured. The moment of epoch is usually decided by ...

. Mean longitude can then be expressed, :''l'' = ''ε'' + ''n''(''t'' − ''t''₀), or :''l'' = ''ε'' + ''nt'', since ''t'' = 0 at the epoch ''t''₀. where ''n'' is the '' mean angular motion'' and ''t'' is any arbitrary time. In some sets of

orbital elements Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same ...

, ''ε'' is one of the six elements.

References

{{DEFAULTSORT:Mean Longitude Orbits

Definition

Discussion

Formulae

See also

References