Eccentric anomaly

The eccentric anomaly of point P is the angle E. The center of the ellipse is point C, and the focus is point F.

In celestial mechanics, eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit.

For a point P orbiting in an ellipse, the eccentric anomaly is the angle E in the figure to the right. It is determined by drawing a line perpendicular to the major axis of the ellipse through the point P and locating its intercept P′ with the auxiliary circle, a circle of radius a\,\! (the semi-major axis of the ellipse) that enscribes the entire ellipse. This intersection P′ is called the corresponding point to P. The radius of the auxiliary circle passing through the corresponding point makes an angle E with the major axis.[1]

The eccentric anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the true anomaly and the mean anomaly.

Coordinates

If the center of coordinates is taken as the center of the ellipse, the coordinates of a point P(x, y) on the ellipse satisfy the equation

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

where a and b are the semi-major and semi-minor axes determining the length (2a) and width (2b) of the ellipse.

The eccentric anomaly E in terms of these coordinates is given by:

\cos E = \frac{x}{a}\quad \mathrm {and} \quad \sin E = \frac{y}{b} \ .

The above equations can be established by drawing the auxiliary circle of radius a enclosing the elliptical path and the minor auxiliary circle of radius b inscribed within the path.[1] The first equation is established by the definition of E. By extending a vertical line through point P to the auxiliary circle, a right triangle is formed with base that is the x-coordinate of P, and hypotenuse a, establishing the first equation. The second equation is established using the minor auxiliary circle. A horizontal line through P intersects this minor auxiliary circle of radius b, establishing another right triangle with altitude y and hypotenuse b. Labeling the adjacent angle E′:

\sin E' = \frac{y}{b} \ .

It is next established that E′ = E. From the equation for the ellipse and the Pythagorean trigonometric identity:

\frac{y}{b} = \sqrt { 1 - \left( \frac {x}{a}\right) ^2 } = \sqrt { 1 - \cos^2 E } = \sin E \ ,

establishing E′ = E.

Formulas

Radius and eccentric anomaly

The eccentricity e is defined as:

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

From Pythagoras' theorem applied to the triangle with r as hypotenuse:

\begin{align} r^2 &= b^2 \sin^2E + (ae-a\cos E)^2 \\ &=a^2(1-e^2)(1-\cos^2 E)+a^2 (e^2 -2e \cos E +\cos^2 E)\\ &=a^2 -2a^2e \cos E +a^2e^2 \cos^2 E \\ &=a^2 (1-e \cos E )^2\\ \end{align}

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

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

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

From the true anomaly

The true anomaly is the angle labeled f in the figure, located at the focus of the ellipse; it is often referred to as θ as in the calculations below. The true anomaly and the eccentric anomaly are related as follows.[2]

Using the formula for r above, the sine and cosine of E are found in terms of θ:

\cos E = \frac{x}{a} = \frac{ae +r \cos \theta}{a} = e+ (1-e \cos E) \cos \theta \ \to \cos E = \frac{ e + \cos \theta }{1 + e \cos \theta }
\sin E = \sqrt{1 - \cos^2 E} = \frac{ \sqrt{1 - e^2} \, \sin \theta }{1 + e \cos \theta } \ .

Hence,

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

Angle E is therefore the adjacent angle of a right triangle with hypotenuse 1 + e cosθ, adjacent side e + cosθ, and opposite side √(1-e2) sinθ.

Also,

\tan \frac{\theta}{2} = \sqrt{\frac{1+e}{1-e}} \cdot \tan \frac{E}{2}

Substituting cosE as found above into the expression for r, the radial distance from the focal point to the point P, can be found in terms of the true anomaly as well:[2]

r = \frac{a \left( 1-e^2\right)}{1+e\cos \theta} \ .

From the mean anomaly

The eccentric anomaly E is related to the mean anomaly M by Kepler's equation:[3]

M = E - e \cdot \sin E

This equation does not have a closed-form solution for E given M. It is usually solved by numerical methods, e.g. Newton-Raphson method.

In-line references and notes

  1. ↑ 1.0 1.1 George Albert Wentworth (1914). "The ellipse §126". Elements of analytic geometry (2nd ed.). Ginn & Co. p. 141.
  2. ↑ 2.0 2.1 James Bao-yen Tsui (2000). Fundamentals of global positioning system receivers: a software approach (3rd ed.). John Wiley & Sons. p. 48. ISBN 0-471-38154-3.
  3. ↑ Michel Capderou (2005). "Definition of the mean anomaly, Eq. 1.68". Satellites: orbits and missions. Springer. p. 21. ISBN 2-287-21317-1.

Background references

See also