Eccentric anomaly

The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipse's circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. In the diagram below, it is E (the angle zcx). ==Calculation== In astrodynamics eccentric anomaly ''E'' can be calculated as follows: :

E=\arccos {{1-\left [ \mathbf{r} \right ] / a} \over e}

where: *

\mathbf{r}\,\!

is the orbiting body's orbital position vector (segment ''sp''), *

a\,\!

is the orbit's semi-major axis (segment ''cz''), and *

e\,\!

is the orbit's orbital eccentricity. The relation between ''E'' and ''M'', the mean anomaly, is: :

M = E - e \cdot \sin{E}.\,\!

For small values of

e

(

e < 0.6627434

) this equation can be solved iteratively, starting from

E_0 = M

and using the relation

E_{i+1} = M + e\,\sin E_i

. The first few terms of the expansion in power series of

e

are: *

E_1 = M + e\,\sin M

E_2 = M + e\,\sin M + \frac{1}{2} e^2 \sin 2M

E_3 = M + e\,\sin M + \frac{1}{2} e^2 \sin 2M 
              + \frac{1}{8} e^3 (3\sin 3M - \sin M)

. For references on details of this derivation, as well as other more efficient methods of solution, see Murray and Dermott (1999, p.35). For a derivation of the limiting value of

e

see Plummer (1960, section 46). The relation between ''E'' and ''T'', the true anomaly, is: :

\cos{T} = {{\cos{E} - e} \over {1 - e \cdot \cos{E}}}

or equivalently :

\tan{T \over 2} = \sqrt{{{1+e} \over {1-e}}} \tan{E \over 2}.\,

The relations between the radius (position vector magnitude) and the anomalies are: :

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

and :

r = a{(1 - e^2) \over (1 + e \cdot \cos{T})}.\,\!

==See also== * Kepler's laws of planetary motion * Mean anomaly * True anomaly ==References== * Murray, C. D. & Dermott, S. F. 1999, Solar System Dynamics, Cambridge University Press, Cambridge. * Plummer, H.C., 1960, An Introductory treatise on Dynamical Astronomy, Dover Publications, New York. (Reprint of the 1918 Cambridge University Press edition.) Astrodynamics Celestial mechanics

Eccentric anomaly

I think the calculation of

E

from

M

is not correct. Could someone point out if there is any error in the following. If noone replies in a while I will replace the expression in the page. If

M = E - e\,\sin E

, the iteration relation would read

E_{i+1} = M + e\,\sin E_i

. Then, by taking

E_0 = M

and using expansions of trigonometric functions, we obtain: *

E_1 = M + e\,\sin M

E_2 = M + e\,\sin M + \frac{1}{2} e^2 \sin 2M

E_3 = M + e\,\sin M + \frac{1}{2} e^2 \sin 2M 
              + \frac{1}{8} e^3 (3\sin 3M - \sin M)

Also it should be noted that this series expansion fails for

e>0.6627434

(Murray and Dermott 1999, p.35). ===Reference=== * Murray, C. D. & Dermott, S. F. 1999, Solar System Dynamics, Cambridge University Press, Cambridge. I have now added the information above to the page. User:AtoUser talk:Ato 23:57, 29 Mar 2005 (UTC)

See other meanings of words starting from letter:

E

EA | EB | EC | ED | EF | EG | EH | EI | EJ | EK | EL | EM | EN | EO | EP | ER | ES | ET | EU | EW | EX | EY | EZ |

Words begining with Eccentric_anomaly:

Eccentric_anomaly
Eccentric_anomaly