View source for Mean anomaly

{{Short description|Specifies the orbit of an object in space}}
[[File:Mean anomaly diagram.png|thumb|upright=1.2|Area swept out per unit time {{colorbox|lightgrey}}{{nbsp}}by an object in an [[elliptic orbit|elliptical orbit]], and {{colorbox|pink}}{{nbsp}}by an imaginary object in a [[circular orbit]] (with the same orbital period). Both sweep out equal areas in equal times, but the angular rate of sweep varies for the elliptical orbit and is constant for the circular orbit. Shown are '''mean anomaly''' and [[true anomaly]] for two units of time. (Note that for visual simplicity, a non-overlapping circular orbit is diagrammed, thus this circular orbit with same orbital period is not shown in true scale with this elliptical orbit: for scale to be true for the two orbits of equal period, these orbits must intersect.)|alt=]]
{{Astrodynamics}}
In [[celestial mechanics]], the '''mean anomaly''' is the fraction of an [[elliptic orbit|elliptical orbit's]] period that has elapsed since the orbiting body passed [[apsis|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 [[apsis|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.<ref>
{{cite book 
 | last = Montenbruck
 | first = Oliver
 | title = Practical Ephemeris Calculations
 | url = https://archive.org/details/practicalephemer00mont
 | url-access = limited
 | publisher = [[Springer-Verlag]]
 | year = 1989
 | isbn = 0-387-50704-3
 | page = [https://archive.org/details/practicalephemer00mont/page/n52 44]
}}</ref><ref>
{{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 
 | page = [https://archive.org/details/astronomicalalgo00meeu_597/page/n186 182]
}}</ref>

==Definition==
Define {{mvar|T}} as the time required for a particular body to complete one orbit. In time {{mvar|T}}, the [[radius vector]] sweeps out 2{{pi}} radians, or 360°. The average rate of sweep, {{mvar|n}}, is then 

<math display="block">n = \frac{\,2\,\pi\,}{T} = \frac{\,360^\circ\,}{T}~,</math> 

which is called the ''[[mean motion|mean angular motion]]'' of the body, with dimensions of radians per unit time or degrees per unit time.

Define {{mvar|&tau;}} as the time at which the body is at the pericenter. From the above definitions, a new quantity, {{mvar|M}}, the ''mean anomaly'' can be defined 

<math display="block">M = n\,(t - \tau) ~,</math> 

which gives an angular distance from the pericenter at arbitrary time {{mvar|t}}<ref>
{{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 = 113
}}</ref> with dimensions of radians or degrees.

Because the rate of increase, {{mvar|n}}, is a constant average, the mean anomaly increases uniformly (linearly) from 0 to 2{{pi}} radians or 0° to 360° during each orbit. It is equal to 0 when the body is at the pericenter, {{pi}} radians (180°) at the [[apsis|apocenter]], and 2{{pi}} radians (360°) after one complete revolution.<ref>Meeus (1991), p. 183</ref> If the mean anomaly is known at any given instant, it can be calculated at any later (or prior) instant by simply adding (or subtracting) {{mvar|n⋅δt}} where {{mvar|δt}} represents the small time difference.

Mean anomaly does not measure an angle between any physical objects (except at pericenter or apocenter, or for a circular orbit). It is simply a convenient uniform measure of how far around its orbit a body has progressed since pericenter. The mean anomaly is one of three angular parameters (known historically as "anomalies") that define a position along an orbit, the other two being the [[eccentric anomaly]] and the [[true anomaly]].

===Mean anomaly at epoch===
The ''mean anomaly at epoch'', {{mvar|M}}{{sub|0}}, is defined as the instantaneous mean anomaly at a given [[Epoch (astronomy)|epoch]], {{mvar|t}}{{sub|0}}. This value is sometimes provided with other orbital elements to enable calculations of the object's past and future positions along the orbit. The epoch for which {{mvar|M}}{{sub|0}} is defined is often determined by convention in a given field or discipline. For example, planetary ephemerides often define {{mvar|M}}{{sub|0}} for the epoch [[J2000]], while for earth orbiting objects described by a [[two-line element set]] the epoch is specified as a date in the first line.<ref>{{Cite web |title=Space-Track.org |url=https://www.space-track.org/documentation#/tle |access-date=2024-08-19 |website=www.space-track.org}}</ref>

==Formulae==
The mean anomaly {{mvar|M}} can be computed from the [[eccentric anomaly]] {{mvar|E}} and the [[orbital eccentricity|eccentricity]] {{mvar|e}} with [[Kepler's equation]]:

<math display="block">M = E - e \,\sin E ~.</math>

Mean anomaly is also frequently seen as

<math display="block">M = M_0 + n\left(t - t_0\right) ~,</math>

where {{mvar|M}}{{sub|0}} is the mean anomaly at the epoch {{mvar|t}}{{sub|0}}, which may or may not coincide with {{mvar|&tau;}}, the time of pericenter passage. The classical method of finding the position of an object in an elliptical orbit from a set of orbital elements is to calculate the mean anomaly by this equation, and then to solve Kepler's equation for the eccentric anomaly.

Define {{mvar|ϖ}} as the ''[[longitude of the periapsis|longitude of the pericenter]]'', the angular distance of the pericenter from a reference direction. Define {{mvar|ℓ}} as the ''[[mean longitude]]'', the angular distance of the body from the same reference direction, assuming it moves with uniform angular motion as with the mean anomaly. Thus mean anomaly is also<ref>Smart (1977), p. 122</ref>
:<math display="block">M = \ell - \varpi~.</math>

[[Mean motion|Mean angular motion]] can also be expressed,

<math display="block">n = \sqrt{\frac{\mu}{\;a^3 \,} \,}~,</math>

where {{mvar|μ}} is the [[Standard gravitational parameter|gravitational parameter]], which varies with the masses of the objects, and {{mvar|a}} is the [[semi-major axis]] of the orbit. Mean anomaly can then be expanded,

<math display="block">M = \sqrt{\frac{\mu}{\; a^3 \,}\,}\,\left(t - \tau\right)~,</math>

and here mean anomaly represents uniform angular motion on a circle of radius {{mvar|a}}.<ref>
{{cite book 
 | last = Vallado  | first = David A.
 | year = 2001
 | title = Fundamentals of Astrodynamics and Applications
 | edition = 2nd
 | pages = 53–54
 | publisher = Microcosm Press  | location = El Segundo, California
 | isbn = 1-881883-12-4
}}</ref>

Mean anomaly can be calculated from the eccentricity and the [[true anomaly]] {{mvar|v}} by finding the eccentric anomaly and then using Kepler's equation. This gives, in radians:
<math display="block">M = \operatorname{atan2}\left( \sqrt{1 - e^2} \sin \nu ,  e + \cos \nu \right)-e \frac{\sqrt{1 - e^2} \sin \nu }{1 + e \cos \nu}</math>
where [[atan2]](y, x) is the angle from the x-axis of the ray from (0,&nbsp;0) to (x,&nbsp;y), having the same sign as y.

For parabolic and hyperbolic trajectories the mean anomaly is not defined, because they don't have a period. But in those cases, as with elliptical orbits, the area swept out by a chord between the attractor and the object following the trajectory increases linearly with time. For the hyperbolic case, there is a formula similar to the above giving the elapsed time as a function of the angle (the true anomaly in the elliptic case), as explained in the article [[Kepler orbit]]. For the parabolic case there is a different formula, the limiting case for either the elliptic or the hyperbolic case as the distance between the foci goes to infinity – see [[Parabolic trajectory#Barker's equation]].

Mean anomaly can also be expressed as a [[series expansion]]:<ref>
{{cite book 
 | last = Smart  | first = W. M.
 | year = 1953
 | title = Celestial Mechanics
 | publisher = Longmans, Green, and Co.  | place = London, UK
 | page = 38
}}</ref>
<math display="block">M = \nu +2\sum_{n=1}^{\infty}(-1)^n \left[\frac{1}{n} +\sqrt{1-e^2} \right]\beta^{n}\sin{n\nu}</math>

with <math>\beta = \frac{1-\sqrt{1-e^2}}{e}</math>

<math display="block">M = \nu - 2\,e \sin \nu + \left( \frac{3}{4}e^2 + \frac{1}{8}e^4 \right)\sin 2\nu - \frac{1}{3} e^3 \sin 3\nu + \frac{5}{32} e^4 \sin 4\nu + \operatorname{\mathcal{O}}\left(e^5\right)</math>

A similar formula gives the true anomaly directly in terms of the mean anomaly:<ref>
{{cite book
 |last=Roy |first=A. E.
 |year=1988
 |title=Orbital Motion
 |edition=1st
 |publisher=A. Hilger |place=Bristol, UK; Philadelphia, Pennsylvania
 |isbn=0852743602
}}
</ref>

<math display="block">\nu = M + \left( 2\,e - \frac{1}{4} e^3 \right) \sin M + \frac{5}{4} e^2 \sin 2M + \frac{13}{12} e^3 \sin 3M + \operatorname{\mathcal{O}}\left(e^4\right)</math>

A general formulation of the above equation can be written as the [[equation of the center]]: <ref>{{Cite book| last=Brouwer|first=Dirk |title=Methods of celestial mechanics| publisher=Elsevier| year=1961| pages=e.g. 77}}</ref>

<math display="block"> \nu = M +2 \sum_{s=1}^{\infty} \frac{1}{s} \left[ J_{s}(se) +\sum_{p=1}^{\infty} \beta^{p}\big(J_{s-p}(se) +J_{s+p}(se) \big)\right]\sin(sM) </math>

==See also==
* [[Kepler's laws of planetary motion]]
* [[Mean longitude]]
* [[Mean motion]]
* [[Orbital elements]]

==References==
{{reflist|25em}}

==External links==
* [http://asa.usno.navy.mil/SecM/Glossary.html Glossary entry ''anomaly, mean''] {{Webarchive|url=https://web.archive.org/web/20171223151613/http://asa.usno.navy.mil/SecM/Glossary.html |date=2017-12-23 }} at the US Naval Observatory's [http://asa.usno.navy.mil/index.html ''Astronomical Almanac Online''] {{Webarchive|url=https://web.archive.org/web/20150420225915/http://asa.usno.navy.mil/index.html |date=2015-04-20 }}

{{orbits}}

[[Category:Orbits]]