Editing Parabolic trajectory (section)

==Barker's equation==

Barker's equation relates the time of flight <math>t</math> to the true anomaly <math>\nu</math> of a parabolic trajectory:<ref>
{{cite book
 | last1 = Bate
 | first1 = Roger
 | last2 = Mueller
 | first2 = Donald
 | last3 = White
 | first3 = Jerry
 | title = Fundamentals of Astrodynamics
 | url = https://archive.org/details/fundamentalsofas00bate
 | url-access = registration
 | publisher = Dover Publications, Inc., New York
 | year = 1971
 | isbn = 0-486-60061-0
}} p 188</ref>
:<math>t - T = \frac{1}{2} \sqrt{\frac{p^3}{\mu}} \left(D + \frac{1}{3} D^3 \right)</math>

where:
*<math>D = \tan \frac{\nu}{2}</math> is an auxiliary variable
*<math>T</math> is the time of periapsis passage
*<math>\mu</math> is the standard gravitational parameter
*<math>p</math> is the [[conic section#Features|semi-latus rectum]] of the trajectory (<math>p = h^2/\mu</math> )

More generally, the time (epoch) between any two points on an orbit is 
:<math>
  t_f - t_0 = \frac{1}{2} \sqrt{\frac{p^3}{\mu}} \left(D_f + \frac{1}{3} D_f^3 - D_0 - \frac{1}{3} D_0^3\right)
</math>

Alternately, the equation can be expressed in terms of periapsis distance, in a parabolic orbit <math>r_p = p/2</math>:
:<math>t - T = \sqrt{\frac{2 r_p^3}{\mu}} \left(D + \frac{1}{3} D^3\right)</math>

Unlike [[Kepler's equation]], which is used to solve for true anomalies in elliptical and hyperbolic trajectories, the true anomaly in Barker's equation can be solved directly for <math>t</math>. If the following substitutions are made
:<math>\begin{align}
  A &= \frac{3}{2} \sqrt{\frac{\mu}{2r_p^3}} (t - T) \\[3pt]
  B &= \sqrt[3]{A + \sqrt{A^{2}+1}}
\end{align}</math>

then
: <math>\nu = 2\arctan\left(B - \frac{1}{B}\right)</math>

With hyperbolic functions the solution can be also expressed as:<ref>{{cite journal
 | last1 = Zechmeister
 | first1 = Mathias
 | title = Solving Kepler's equation with CORDIC double iterations
 | journal = MNRAS
 | date = 2020
 | volume= 500
 | issue = 1
 | pages = 109–117
 | doi = 10.1093/mnras/staa2441
 | doi-access = free
 | arxiv = 2008.02894
 | bibcode = 2021MNRAS.500..109Z
}} Eq.(40) and Appendix C.</ref>

: <math>\nu = 2\arctan\left(2\sinh\frac{\mathrm{arcsinh} \frac{3M}{2}}{3}\right)</math>

where

: <math> M = \sqrt{\frac{\mu}{2r_p^3}} (t - T)</math>