Mean anomaly

Formulae

The mean anomaly M can be computed from the eccentric anomaly E and the eccentricity e with Kepler's equation:

$${\displaystyle M=E-e\sin E.}$$

Mean anomaly is also frequently seen as

$${\displaystyle M=M_0+n\left(t-t_0\right),}$$

where M₀ is the mean anomaly at the epoch t₀, which may or may not coincide with τ, 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 ϖ as the longitude of the pericenter, the angular distance of the pericenter from a reference direction. Define 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

$${\displaystyle M=\ell -\varpi .}$$

Mean angular motion can also be expressed,

$${\displaystyle n=\sqrt{\frac{\mu }{a^3}},}$$

where μ is the gravitational parameter, which varies with the masses of the objects, and a is the semi-major axis of the orbit. Mean anomaly can then be expanded,

$${\displaystyle M=\sqrt{\frac{\mu }{a^3}}\left(t-\tau \right),}$$

and here mean anomaly represents uniform angular motion on a circle of radius a.

Mean anomaly can be calculated from the eccentricity and the true anomaly v by finding the eccentric anomaly and then using Kepler's equation. This gives, in radians: $${\displaystyle M=\mathrm{atan2}\left(\sqrt{1-e^2}\sin \nu ,e+\cos \nu \right)-e\frac{\sqrt{1-e^2}\sin \nu }{1+e\cos \nu }}$$ where atan2(y, x) is the angle from the x-axis of the ray from (0, 0) to (x, 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: $${\displaystyle M=\nu +2\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\left[\frac{1}{n}+\sqrt{1-e^2}\right]{\beta }^n\sin n\nu }$$

with ${\displaystyle \beta =\frac{1-\sqrt{1-e^2}}{e}}$

$${\displaystyle M=\nu -2e\sin \nu +\left(\frac{3}{4}{e}^2+\frac{1}{8}{e}^4\right)\sin 2\nu -\frac{1}{3}{e}^3\sin 3\nu +\frac{5}{32}{e}^4\sin 4\nu +\mathcal{O}\left(e^5\right)}$$

A similar formula gives the true anomaly directly in terms of the mean anomaly:

$${\displaystyle \nu =M+\left(2e-\frac{1}{4}{e}^3\right)\sin M+\frac{5}{4}{e}^2\sin 2M+\frac{13}{12}{e}^3\sin 3M+\mathcal{O}\left(e^4\right)}$$

A general formulation of the above equation can be written as the equation of the center:

$${\displaystyle \nu =M+2\sum _{s=1}^{\mathrm{\infty }}\frac{1}{s}\left[J_s(se)+\sum _{p=1}^{\mathrm{\infty }}{\beta }^p(J_{s-p}(se)+J_{s+p}(se))\right]\sin (sM)}$$