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| In [[celestial mechanics]], '''mean anomaly''' is a [[parameter]] relating position and time for a body moving in a [[Kepler orbit]]. It is based on the fact that equal areas are swept in equal intervals of time by a line joining the focus and the orbiting body ([[Kepler's laws of planetary motion#Second law|Kepler's second law]]).
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| The mean anomaly increases uniformly from 0 to <math>2\pi</math> radians during each orbit. However, it is not an angle. Due to Kepler's second law, the mean anomaly is proportional to the area swept by the [[focus (geometry)|focus]]-to-body line since the last [[periapsis]].
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| The mean anomaly is usually denoted by the letter <math>M</math>, and is given by the formula:
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| :<math> M = n \, t = \sqrt{\frac{ G( M_\star \! + \!m ) } {a^3}} \,t </math>
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| where ''n'' is the [[mean motion]], ''a'' is the length of the orbit's [[semi-major axis]], <math>M_\star</math> and ''m'' are the orbiting masses, and ''G'' is the [[gravitational constant]].
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| The mean anomaly is the time since the last [[periapsis]] multiplied by the [[mean motion]], and the mean motion is <math>2\pi</math> divided by the [[orbital period|duration of a full orbit]].
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| The mean anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the [[eccentric anomaly]] and the [[true anomaly]]. 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) <math>\sqrt{\frac{ G( M_\star \! + \!m ) } {a^3}} \,\delta t</math> where <math>\delta t</math> represents the time difference. The other anomalies can hence be calculated.
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| ==Formulas==
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| The mean anomaly ''M'' can be computed from the [[eccentric anomaly]] ''E'' and the [[Eccentricity (mathematics)|eccentricity]] ''e'' with [[Kepler's Equation]]:
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| :<math>M = E - e \cdot \sin E</math>
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| To find the position of the object in an elliptic Kepler orbit at a given time ''t'', the mean anomaly is found by multiplying the time and the mean motion, then it is used to find the eccentric anomaly by solving Kepler's equation.
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| It is also frequently seen:
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| :<math>M = M_0 + nt</math>,
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| Again ''n'' is the mean motion. However, ''t'', in this instance, is the ''time since epoch'', which is how much time has passed since the measurement of ''M''<sub>0</sub> was taken. The value ''M''<sub>0</sub> denotes the ''mean anomaly at epoch'', which is the mean anomaly at the time the measurement was taken.
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| ==See also==
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| * [[Kepler orbit]]
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| * [[Ellipse]]
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| * [[Eccentric anomaly]]
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| * [[True anomaly]]
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| ==References==
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| * Murray, C. D. & Dermott, S. F. 1999, ''Solar System Dynamics'', Cambridge University Press, Cambridge.
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| * Plummer, H.C., 1960, ''An Introductory treatise on Dynamical Astronomy'', Dover Publications, New York. (Reprint of the 1918 Cambridge University Press edition.)
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| {{orbits}}
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| [[Category:Orbits]]
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| [[ru:Элементы орбиты#Аномалии]]
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