Eccentric Anomaly
The angle obtained by drawing the auxiliary circle of an ellipse with center O and focus F, and drawing a line perpendicular to the semimajor axis and intersecting it at A. The angle E is then defined as illustrated above. Then for an ellipse with eccentricity e,
| AF=OF-AO=ae-acosE. |
(1)
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But the distance AF is also given in terms of the distance from the focus r=FP and the supplement of the angle from the semimajor axis v by
| AF=rcos(pi-v)=-rcosv. |
(2)
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Equating these two expressions gives
| [画像: r=(a(cosE-e))/(cosv), ] |
(3)
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which can be solved for cosv to obtain
| [画像: cosv=(a(cosE-e))/r. ] |
(4)
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To get E in terms of r, plug (◇) into the equation of the ellipse
| [画像: r=(a(1-e^2))/(1+ecosv). ] |
(5)
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Rearranging,
| r(1+ecosv)=a(1-e^2) |
(6)
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and plugging in (◇) then gives
Solving for r gives
| r=a(1-ecosE), |
(9)
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so differentiating yields the result
| r^.=aeE^.sinE. |
(10)
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The eccentric anomaly is a very useful concept in orbital mechanics, where it is related to the so-called mean anomaly M by Kepler's equation
| M=E-esinE. |
(11)
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M can also be interpreted as the area of the shaded region in the above figure (Finch 2003).
See also
Eccentricity, Ellipse, Kepler's EquationExplore with Wolfram|Alpha
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References
Danby, J. M. Fundamentals of Celestial Mechanics, 2nd ed., rev. ed. Richmond, VA: Willmann-Bell, 1988.Finch, S. R. "Laplace Limit Constant." §4.8 Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 266-268, 2003.Montenbruck, O. and Pfleger, T. Astronomy on the Personal Computer, 4th ed. Berlin: Springer-Verlag, p. 62, 2000.Referenced on Wolfram|Alpha
Eccentric AnomalyCite this as:
Weisstein, Eric W. "Eccentric Anomaly." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EccentricAnomaly.html