Editing Eccentric anomaly (section)

==Formulas==

===Radius and eccentric anomaly===
The [[eccentricity (mathematics)|eccentricity]] ''e'' is defined as:

:<math>e=\sqrt{1 - \left(\frac{b}{a}\right)^2 } \ . </math>

From [[Pythagoras's theorem]] applied to the triangle with ''r'' (a distance ''FP'') as hypotenuse:
:<math>\begin{align}
  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{align} </math>

Thus, the radius (distance from the focus to point ''P'') is related to the eccentric anomaly by the formula
:<math>r = a \left(1 - e \cos{E}\right) \ .</math>

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 <math>\theta</math> in the figure, located at the focus of the ellipse. It is sometimes represented by {{mvar|f}} or {{mvar|v}}. The true anomaly and the eccentric anomaly are related as follows.<ref name=Tsui>{{cite book |first=James Bao-yen |last=Tsui |year=2000 |title=Fundamentals of Global Positioning System receivers: A software approach |edition=3rd |page=48 |publisher=[[John Wiley & Sons]] |isbn=0-471-38154-3 |url=https://books.google.com/books?id=jPRCxNDZqDQC&pg=PA48}}</ref>

Using the formula for {{mvar|r}} above, the sine and cosine of {{mvar|E}} are found in terms of {{mvar|f}}&nbsp;:
:<math>\begin{align}
              \cos E &= \frac{\,x\,}{a} = \frac{\, a e + r \cos f \,}{a} = e + (1 - e \cos E) \cos f \\
  \Rightarrow \cos E &= \frac{\, e + \cos f \,}{1 + e \cos f} \\
              \sin E &= \sqrt{\, 1 - \cos^2 E \;} = \frac{\, \sqrt{\, 1 - e^2 \;} \, \sin f \,}{ 1 + e\cos f } ~.
\end{align}</math>

Hence,
:<math>\tan E = \frac{\, \sin E \,}{\cos E} = \frac{\, \sqrt{\, 1 - e^2 \;} \, \sin f \,}{e + \cos f} ~.</math>

where the correct quadrant for {{mvar|E}} is given by the signs of numerator and denominator, so that {{mvar|E}} can be most easily found using an [[atan2]] function.

Angle {{mvar|E}} is therefore the adjacent angle of a right triangle with hypotenuse <math>\; 1 + e \cos f \;,</math> adjacent side <math>\; e + \cos f \;,</math> and opposite side <math>\;\sqrt{ \, 1 - e^2 \; } \, \sin f \;.</math>

Also,
:<math>\tan\frac{\, f \,}{2} = \sqrt{\frac{\, 1 + e \,}{1 - e}\,}  \,\tan\frac{\, E \,}{2}</math>

Substituting {{math|cos}}&nbsp;{{mvar|E}} as found above into the expression for {{mvar|r}}, the radial distance from the focal point to the point {{math|P}}, can be found in terms of the true anomaly as well:<ref name=Tsui/>
:<math>r = \frac{a \left(\, 1 - e^2 \,\right)}{\,  1 + e \cos f \, } =  \frac{p}{\,  1 + e \cos f \, }\,</math>
where
:<math>\, p \equiv a \left(\, 1 - e^2 \,\right) </math>
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]]:<ref name=Capderou>{{cite book |title=Satellites: orbits and missions |author=Michel Capderou |chapter=Definition of the mean anomaly, Eq. 1.68 |page=21 |chapter-url=https://books.google.com/books?id=BAihdjtLZXcC&pg=PA21 |isbn=2-287-21317-1 |year=2005 |publisher=Springer}}</ref>
:<math>M = E - e \sin E</math>

This equation does not have a [[closed-form solution]] for ''E'' given ''M''. It is usually solved by [[numerical methods]], e.g. the [[Newton's method|Newton–Raphson method]]. It may be expressed in a [[Fourier series]] as
:<math>E = M + 2\sum_{n=1}^{\infty } \frac{J_{n}(ne)}{n}\sin(n M)</math>
where <math>J_{n}(x)</math> is the [[Bessel function]] of the first kind.