Editing True anomaly (section)

===From the mean anomaly===
The true anomaly can be calculated directly from the [[mean anomaly]] <math>M</math> via a [[Fourier expansion]]:<ref name="Battin 1999 p. 212">{{cite book | last=Battin | first=R.H. | title=An Introduction to the Mathematics and Methods of Astrodynamics | publisher=American Institute of Aeronautics & Astronautics | series=AIAA Education Series | year=1999 | isbn=978-1-60086-026-3 | url=https://books.google.com/books?id=OjH7aVhiGdcC&pg=PA212 | access-date=2022-08-02 | page=212 (Eq. (5.32))}}</ref>

:<math>\nu = M + 2 \sum_{k=1}^{\infty}\frac{1}{k} \left[ \sum_{n=-\infty}^{\infty} J_n(-ke)\beta^{|k+n|} \right] \sin{kM}</math>
with [[Bessel functions]] <math>J_n</math> and parameter <math>\beta = \frac{1-\sqrt{1-e^2}}{e}</math>.

Omitting all terms of order <math>e^4</math> or higher (indicated by <math>\operatorname{\mathcal{O}}\left(e^4\right)</math>), it can be written as<ref name="Battin 1999 p. 212"/><ref name="Smart p. ">{{cite book | last=Smart | first=W. M. | title=Textbook on Spherical Astronomy | year=1977 | url=https://wangsajaya.files.wordpress.com/2015/02/textbook-on-spherical-astronomy-smart-6ed-1977.pdf | bibcode=1977tsa..book.....S | page=120 (Eq. (87))}}</ref><ref>{{cite book |last=Roy |first=A.E. |title=Orbital Motion |url=https://forum.fh-aachen.org/cms/index.php?attachment%2F9683-orbital-motion-fourth-edition-pdf%2F#page=78&zoom=100,0,0 |year=2005 |location=Bristol, UK; Philadelphia, PA |publisher=Institute of Physics (IoP) |edition=4 |page=78 (Eq. (4.65)) |isbn=0750310154 |bibcode=2005ormo.book.....R |access-date=2020-08-29 |archive-date=2021-05-15 |archive-url=https://web.archive.org/web/20210515142200/https://forum.fh-aachen.org/cms/index.php?attachment%2F9683-orbital-motion-fourth-edition-pdf%2F#page=78&zoom=100,0,0 |url-status=dead }}</ref>

:<math>\nu = M + \left(2e - \frac{1}{4} e^3\right) \sin{M} + \frac{5}{4} e^2 \sin{2M} + \frac{13}{12} e^3 \sin{3M} + \operatorname{\mathcal{O}}\left(e^4\right).</math>

Note that for reasons of accuracy this approximation is usually limited to orbits where the eccentricity <math>e</math> is small.

The expression <math>\nu - M</math> is known as the [[equation of the center]], where more details about the expansion are given.