Editing Mean anomaly (section)

==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>