Inorbital mechanics, theeccentric anomaly is anangular parameter that defines the position of a body that is moving along anellipticKepler orbit, the angle measured at the center of the ellipse between the orbit's periapsis and the current position. The eccentric anomaly is one of three angular parameters ("anomalies") that can be used to define a position along an orbit, the other two being thetrue anomaly and themean anomaly.

Consider the ellipse with equation given by:

${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,}$

wherea is thesemi-major axis andb is thesemi-minor axis.

For a point on the ellipse,${\displaystyle P=P(x,y)}$, representing the position of an orbiting body in an elliptical orbit, the eccentric anomaly is the angleE in the figure. The eccentric anomalyE is one of the angles of a right triangle with one vertex at the center of the ellipse, its adjacent side lying on the major axis, having hypotenusea (equal to the semi-major axis of the ellipse), and opposite side (perpendicular to the major axis and touching the pointP' on the auxiliary circle of radiusa) that passes through the pointP. The eccentric anomaly is measured in the same direction as the true anomaly, shown in the figure as${\displaystyle \theta }$. The eccentric anomalyE in terms of these coordinates is given by:^[1]

${\displaystyle \cos E=\frac{x}{a},}$

and

${\displaystyle \sin E=\frac{y}{b}}$

The second equation is established using the relationship

${\displaystyle {\left(\frac{y}{b}\right)}^2=1-{\cos }^{2}E={\sin }^{2}E}$,

which implies that${\displaystyle \sin E=\pm \frac{y}{b}}$. The equation${\displaystyle \sin E=-\frac{y}{b}}$ is immediately able to be ruled out since it traverses the ellipse in the wrong direction. It can also be noted that the second equation can be viewed as coming from a similar triangle with its opposite side having the same lengthy as the distance fromP to the major axis, and its hypotenuseb equal to the semi-minor axis of the ellipse.

Theeccentricitye is defined as:

${\displaystyle e=\sqrt{1-{\left(\frac{b}{a}\right)}^2}.}$

FromPythagoras's theorem applied to the triangle withr (a distance${\displaystyle FP}$) as hypotenuse:

Thus, the radius (distance from the focus to pointP) is related to the eccentric anomaly by the formula

${\displaystyle r=a\left(1-e\cos E\right).}$

With this result the eccentric anomaly can be determined from the true anomaly as shown next.

Thetrue anomaly is the angle labeled${\displaystyle \theta }$ in the figure, located at the focus of the ellipse. It is sometimes represented byf orν. The true anomaly and the eccentric anomaly are related as follows.^[2]

Using the formula forr above, the sine and cosine ofE are found in terms off :

Hence,

${\displaystyle \tan E=\frac{\sin E}{\cos E}=\frac{\sqrt{1-e^2}\sin f}{e+\cos f}.}$

where the correct quadrant forE is given by the signs of numerator and denominator, so thatE can be most easily found using anatan2 function.

AngleE is therefore the adjacent angle of a right triangle with hypotenuse${\displaystyle 1+e\cos f,}$ adjacent side${\displaystyle e+\cos f,}$ and opposite side${\displaystyle \sqrt{1-e^2}\sin f.}$

Also,

${\displaystyle \tan \frac{f}{2}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}}$

Substituting${\displaystyle \cos E}$ as found above into the expression forr, the radial distance from the focal point to the pointP, can be found in terms of the true anomaly as well:^[2]

${\displaystyle r=\frac{a\left(1-e^2\right)}{1+e\cos f}=\frac{p}{1+e\cos f}}$

where

${\displaystyle p\equiv a\left(1-e^2\right)}$

is called thesemi-latus rectum in classical geometry.

The eccentric anomalyE is related to themean anomalyM byKepler's equation:^[3]

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

This equation does not have aclosed-form solution forE givenM. It is usually solved bynumerical methods, e.g. theNewton–Raphson method. It may be expressed in aFourier series as

${\displaystyle E=M+2\sum _{n=1}^{\mathrm{\infty }}\frac{J_n(ne)}{n}\sin (nM)}$

where${\displaystyle J_n(x)}$ is theBessel function of the first kind.

• Eccentricity vector
• Orbital eccentricity
• Universal variable formulation

• Murray, Carl D.; & Dermott, Stanley F. (1999);Solar System Dynamics, Cambridge University Press, Cambridge, GB
• Plummer, Henry C. K. (1960);An Introductory Treatise on Dynamical Astronomy, Dover Publications, New York, NY (Reprint of the 1918 Cambridge University Press edition)

• Orbits

• Articles with short description
• Short description is different from Wikidata