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'''Mean longitude''' is the [[longitude]] at which an [[Orbits|orbiting]] body could be found if its orbit were [[circular orbit|circular]], and free of [[perturbation (astronomy)|perturbations]], and if its [[inclination]] were zero.
'''Mean longitude''' is the [[Ecliptic coordinate system#Spherical coordinates|ecliptic longitude]] at which an [[orbit]]ing body could be found if its orbit were [[circular orbit|circular]] and free of [[perturbation (astronomy)|perturbations]]. While nominally a simple longitude, in practice the mean longitude does not correspond to any one physical angle.<ref name=Meeus1>
{{cite book
| last = Meeus
| first = Jean
| title = Astronomical Algorithms
| url = https://archive.org/details/astronomicalalgo00meeu_597
| url-access = limited
| publisher = Willmann-Bell, Inc., Richmond, VA
| year = 1991
|isbn=0-943396-35-2
|pages=[https://archive.org/details/astronomicalalgo00meeu_597/page/n201 197]–198}}
⚫
</ref>


==Description==
==Definition==
[[File:Orbit1-mean.png|thumb|400px|right|An orbiting body's ''mean longitude'' is calculated {{nowrap|''L'' {{=}} ''Ω'' + ''ω'' + ''M''}}, where ''Ω'' is the [[longitude of the ascending node]], ''ω'' is the [[argument of periapsis|argument of the pericenter]] and ''M'' is the ''[[mean anomaly]]'', the body's angular distance from the [[apsis|pericenter]] as if it moved with constant speed rather than with the [[Orbital speed|variable speed]] of an [[Elliptic orbit|elliptical orbit]]. Its ''[[true longitude]]'' is calculated similarly, {{nowrap|''l'' {{=}} ''Ω'' + ''ω'' + ''ν''}}, where ''ν'' is the [[true anomaly]].]]
Both the mean longitude and the [[true longitude]] of the body in the orbit described above would change at a constant rate over time.<ref name=Moulton_1970>
* Define a reference direction, ♈︎, along the [[ecliptic]]. Typically, this is the direction of the March [[equinox]]. At this point, ecliptic longitude is 0°.
Multon, F. R. (1970).
* The body's orbit is generally [[Orbital inclination|inclined]] 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''.
''An Introduction to Celestial Mechanics'', 2nd ed.,
* Define the angular distance along the plane of the orbit from the [[ascending node]] to the [[Apsis|pericenter]] as the ''[[Argument of periapsis|argument of the pericenter]]'', ''ω''.
p.&nbsp;182&ndash;183.
* 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.
New York, NY: Dover.
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.<ref name=Smart1>
{{cite book
| last = Smart
| first = W. M.
| title = Textbook on Spherical Astronomy
| publisher = Cambridge University Press, Cambridge
| year = 1977
|edition=sixth
|isbn=0-521-29180-1
|page=122}}
</ref>
</ref>
<ref name=Roy_1978>
Roy, A. E. (1978).
''Orbital Motion'',
p.&nbsp;174, (ISBN 0-470-99251-4).
New York, NY: John Wiley & Sons.
⚫
</ref>
<ref name=Brouwer_Clemence_1961>
Brouwer, D., & Clemence, G. M. (1961).
''Methods of Celestial Mechanics'',
p.&nbsp;45.
New York, NY: Academic Press.
</ref>
But real orbits are [[Orbital eccentricity|eccentric]] and so depart from circularity, at least slightly, even if presumed to be free from any perturbations. These unperturbed eccentric orbits are called [[Keplerian orbit|Keplerian ellipse]]s, and in them the progress of the orbiting body in true longitude does not change at a constant rate over time. So the mean longitude is an abstracted quantity for [[Keplerian orbits]], still proportional to the time, but now only indirectly related to the position of the orbiting body: the difference between the mean longitude and the true longitude is usually called the [[equation of the center]]. In such an elliptical orbit, the only times when the mean longitude is equal to the true longitude are the times when the orbiting body passes through [[periapsis]] (or pericenter) and [[apoapsis]] (or apocenter).


Sometimes the value defined in this way is called the "mean mean longitude", and the term "mean longitude" is used for a value that does have short-term variations (such as over a synodic month or a year in the case of the moon) but does not include the correction due to the difference between true anomaly and mean anomaly.<ref name=Simon>{{cite journal|display-authors=etal |last1=Jean-Louis Simon |title=Numerical expressions for precession formulae and mean elements for the Moon and the planets |journal=Astronomy and Astrophysics |date=1994 |volume=282 |page=663 |url=https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?db_key=AST&bibcode=1994A%26A...282..663S&letter=0&classic=YES&defaultprint=YES&whole_paper=YES&page=663&epage=663&send=Send+PDF&filetype=.pdf |bibcode=1994A&A...282..663S}}</ref><ref>{{cite web |title=Comprendre - Glossaire |url=https://promenade.imcce.fr/fr/pages2/221.html#longmoymoy |website=Promenade dans le système solaire |publisher=The FP7 ESPaCE Program |access-date=26 March 2024}}</ref>
In an orbit that is undergoing perturbations, an [[osculating orbit]] together with its (elliptical) osculating elements can still be defined for any point in time along the actual orbit. For each successive set of osculating elements, a mean longitude can be defined, as in the unperturbed case. But here, the changes in mean longitude over time will not only be those due to some constant rate over time; there will also be superimposed perturbations (and the rate itself is also perturbed). A set of mean elements can still be defined for such an orbit, after abstracting the perturbational variations with time. The term 'mean longitude' was already used for the unperturbed and osculating cases, and the corresponding mean longitude member in a set of mean elements, after abstraction of the periodic variations, is sometimes therefore called the 'mean mean longitude'. To arrive at a true longitude from a mean mean longitude, the perturbational terms must be applied as well as the equation of the center.
Also, sometimes the mean longitude (or mean mean longitude) is considered to be a slowly varying function, modeled with a [[Maclaurin series]], rather than a simple linear function of time.<ref name=Simon/>


==Calculation==
==Discussion==
The mean longitude <math>L\,</math> can be calculated as follows:&nbsp;<ref name=Roy_1978 />


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]] measures the actual longitude and assumes the body has moved with its [[Orbital speed|actual speed]], which varies around its [[Elliptic orbit|elliptical orbit]]. The difference between the two is known as the [[equation of the center]].<ref>Meeus, Jean (1991). p. 222</ref>
:<math>L=M + \varpi = M + \Omega + \omega\,</math>


==Formulae==
where:
⚫
From the above definitions, define the ''[[longitude of the periapsis|longitude of the pericenter]]''
*<math>M\, </math> is orbit's [[mean anomaly]],
:''ϖ'' = ''Ω'' + ''ω''.
⚫
*<math>\varpi \equiv \Omega + \omega\,</math> is [[longitude of the periapsis|longitude of the orbit's periapsis]],
Then mean longitude is also<ref name=Meeus1/>
*<math>\Omega\,</math> is the [[longitude of the ascending node]] and
:''L'' = ''ϖ'' + ''M''.
*<math>\omega\,</math> is the [[argument of periapsis]].

Another form often seen is the ''mean longitude at epoch'', ''ε''. This is simply the mean longitude at a reference time ''t''<sub>0</sub>, known as the [[Epoch (astronomy)|epoch]]. Mean longitude can then be expressed,<ref name=Smart1/>
:''L'' = ''ε'' + ''n''(''t'' − ''t''<sub>0</sub>), or
:''L'' = ''ε'' + ''nt'', since ''t'' = 0 at the epoch ''t''<sub>0</sub>.
where ''n'' is the ''[[mean motion|mean angular motion]]'' and ''t'' is any arbitrary time. In some sets of [[orbital elements]], ''ε'' is one of the six elements.<ref name=Smart1/>

==See also==
* [[Mean motion]]
* [[Orbital elements]]
* [[True longitude]]


==References==
==References==
{{reflist}}
<references />


{{Orbits}}
{{Orbits}}
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[[zh:平黃經]]

Latest revision as of 20:50, 23 October 2024

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.[1]

Definition

[edit]
An orbiting body's mean longitude is calculated L = Ω + ω + M, where Ω is the longitude of the ascending node, ω is the argument of the pericenter and M is the mean anomaly, the body's angular distance from the pericenter as if it moved with constant speed rather than with the variable speed of an elliptical orbit. Its true longitude is calculated similarly, l = Ω + ω + ν, where ν is the true anomaly.
  • Define a reference direction, ♈︎, along the ecliptic. Typically, this is the direction of the March equinox. At this point, ecliptic longitude is 0°.
  • The body's orbit is generally inclined 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 to the pericenter 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.[2]

Sometimes the value defined in this way is called the "mean mean longitude", and the term "mean longitude" is used for a value that does have short-term variations (such as over a synodic month or a year in the case of the moon) but does not include the correction due to the difference between true anomaly and mean anomaly.[3][4] Also, sometimes the mean longitude (or mean mean longitude) is considered to be a slowly varying function, modeled with a Maclaurin series, rather than a simple linear function of time.[3]

Discussion

[edit]

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 measures the actual longitude and assumes the body has moved with its actual speed, which varies around its elliptical orbit. The difference between the two is known as the equation of the center.[5]

Formulae

[edit]

From the above definitions, define the longitude of the pericenter

ϖ = Ω + ω.

Then mean longitude is also[1]

L = ϖ + M.

Another form often seen is the mean longitude at epoch, ε. This is simply the mean longitude at a reference time t0, known as the epoch. Mean longitude can then be expressed,[2]

L = ε + n(t − t0), or
L = ε + nt, since t = 0 at the epoch t0.

where n is the mean angular motion and t is any arbitrary time. In some sets of orbital elements, ε is one of the six elements.[2]

See also

[edit]

References

[edit]
  1. ^ a b Meeus, Jean (1991). Astronomical Algorithms. Willmann-Bell, Inc., Richmond, VA. pp. 197–198. ISBN 0-943396-35-2.
  2. ^ a b c Smart, W. M. (1977). Textbook on Spherical Astronomy (sixth ed.). Cambridge University Press, Cambridge. p. 122. ISBN 0-521-29180-1.
  3. ^ a b Jean-Louis Simon; et al. (1994). "Numerical expressions for precession formulae and mean elements for the Moon and the planets" (PDF). Astronomy and Astrophysics. 282: 663. Bibcode:1994A&A...282..663S.
  4. ^ "Comprendre - Glossaire". Promenade dans le système solaire. The FP7 ESPaCE Program. Retrieved 26 March 2024.
  5. ^ Meeus, Jean (1991). p. 222
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