# Talk:Eccentric anomaly

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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).

## Contents

### 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. at0 23:57, 29 Mar 2005 (UTC)

This sentense in meaningless:

"The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit"

since an angle is defined from two directions. An angle cannot be defined from one direction and a point. —Preceding unsigned comment added by 62.180.229.222 (talk) 06:21, 5 November 2007 (UTC)

## Correct placement of the material

The eccentric/true anomalies logically belongs to the article Ellipse, these are two different ways of assigning a parameter to an ellipse in general, not only to an elliptic Kepler orbit. True anomaly is the then equally applicable to a parabola and a hyperbola. This as opposed to mean anomaly that only is relevant for an elliptic Kepler orbit, this is just a parameter transformation for the analytic solution of the differential equation of the two body problem. It has no geometrical significance for an ellipse as a geometric shape!

I plan to update the Ellipse article such that the proper definitinons of true and eccentric anomalies are included. I think the eccentric/true anomaly articles then should be reduced to pointers to Ellipse.

The picture of this article would be fine if just the word "orbit" is removed!

Stamcose (talk) 15:28, 22 November 2008 (UTC)

I have now updated the Ellipse article and the material of this article (except "mean anomaly") is now in the the Ellipse article. The convertion between "eccentric anomaly" and "mean anomaly" is in Kepler orbit where the mathematical analysis is carried through for which the "mean anomaly" is meaningful as a mathematical transformation of variable!

Correponding transformation

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

could be OK in a Mean anomaly article but not here! And the computation M => E is nowadays done with the Newton-Raphson algorithm, not as described here!

Stamcose (talk) 15:56, 23 November 2008 (UTC)

## Undid restore of content

I undid the last edit by Lasunncty which restored the Oct. 30 version. I agree with Stamcose that the content should be applicable to ellipses in general and not just orbits. However, I also think the current state of the article needs cleanup (for grammar, or just reduce to complete redirect/disambiguation). In any case I do not think a poorer version of the same content on the ellipse page should be in this article.

I myself was about to edit the article because I found it too confusing with only reference to periapsis etc. and not the semimajor axis, and the attempts to dance around using the focus (orbited body) as the frame of reference rather than the ellipse center, then I saw the Talk and history and saw the content was really elsewhere.

If there is any turmoil over this, I just ask the interested parties to work it out on the talk page. --Walt (talk) 17:28, 22 January 2009 (UTC)

## The figure is too complicated

I believe that the figure has too many details to be the definitive figure... It may be used to advocate some points later on, but the definitive figure should more like that of True Anomaly (same figure may apply here...) —Preceding unsigned comment added by 132.69.148.150 (talk) 14:31, 7 July 2010 (UTC)