eccentric anomaly
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In
orbital mechanics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of ...
, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
Kepler orbit Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
. The eccentric anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the
true anomaly In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main foc ...
and the mean anomaly.


Graphical representation

Consider the ellipse with equation given by: :\frac + \frac = 1, where ''a'' is the ''semi-major'' axis and ''b'' is the ''semi-minor'' axis. For a point on the ellipse, ''P'' = ''P''(''x'', ''y''), representing the position of an orbiting body in an elliptical orbit, the eccentric anomaly is the angle ''E'' in the figure. The eccentric anomaly ''E'' 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 hypotenuse ''a'' (equal to the ''semi-major'' axis of the ellipse), and opposite side (perpendicular to the ''major'' axis and touching the point ''P′'' on the auxiliary circle of radius ''a'') that passes through the point ''P''. The eccentric anomaly is measured in the same direction as the true anomaly, shown in the figure as ''f''. The eccentric anomaly ''E'' in terms of these coordinates is given by: :\cos E = \frac , and :\sin E = \frac The second equation is established using the relationship :\left(\frac\right)^2 = 1 - \cos^2 E = \sin^2 E, which implies that . The equation 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 length ''y'' as the distance from ''P'' to the ''major'' axis, and its hypotenuse ''b'' equal to the ''semi-minor'' axis of the ellipse.


Formulas


Radius and eccentric anomaly

The
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
''e'' is defined as: :e=\sqrt \ . From
Pythagoras's theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
applied to the triangle with ''r'' (a distance ''FP'') as hypotenuse: :\begin r^2 &= b^2 \sin^2E + (ae - a\cos E)^2 \\ &= a^2\left(1 - e^2\right)\left(1 - \cos^2 E\right) + a^2 \left(e^2 - 2e\cos E + \cos^2 E\right) \\ &= a^2 - 2a^2 e\cos E + a^2 e^2 \cos^2 E \\ &= a^2 \left(1 - e\cos E\right)^2 \\ \end Thus, the radius (distance from the focus to point ''P'') is related to the eccentric anomaly by the formula :r = a \left(1 - e \cos\right) \ . With this result the eccentric anomaly can be determined from the true anomaly as shown next.


From the true anomaly

The ''
true anomaly In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main foc ...
'' is the angle labeled \theta in the figure, located at the focus of the ellipse. It is sometimes represented by or . The true anomaly and the eccentric anomaly are related as follows. Using the formula for above, the sine and cosine of are found in terms of  : :\begin \cos E &= \frac = \frac = e + (1 - e \cos E) \cos f \\ \Rightarrow \cos E &= \frac \\ \sin E &= \sqrt = \frac ~. \end Hence, :\tan E = \frac = \frac ~. Angle is therefore the adjacent angle of a right triangle with hypotenuse \; 1 + e \cos f \;, adjacent side \; e + \cos f \;, and opposite side \;\sqrt \, \sin f \;. Also, :\tan\frac = \sqrt \,\tan\frac Substituting   as found above into the expression for , the radial distance from the focal point to the point , can be found in terms of the true anomaly as well: :r = \frac = \frac\, where :\, p \equiv a \left(\, 1 - e^2 \,\right) is called ''"the semi-latus rectum"'' in classical geometry.


From the mean anomaly

The eccentric anomaly ''E'' is related to the mean anomaly ''M'' by
Kepler's equation In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was first derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his '' Epi ...
: :M = E - e \sin E This equation does not have a
closed-form solution In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
for ''E'' given ''M''. It is usually solved by
numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
, e.g. the
Newton–Raphson method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-va ...
. It may be expressed in a Fourier series as :E = M + 2\sum_^ \frac\sin(n M) where J_(x) is the
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrar ...
of the first kind.


See also

*
Eccentricity vector In celestial mechanics, the eccentricity vector of a Kepler orbit is the dimensionless vector with direction pointing from apoapsis to periapsis and with magnitude equal to the orbit's scalar eccentricity. For Kepler orbits the eccentricity vec ...
* Orbital eccentricity


Notes and references


Sources

* 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 Orbits de:Exzentrische Anomalie