Parabolic Trajectory - Barker's Equation

Barker's Equation

Barker's equation relates the time of flight to the true anomaly of a parabolic trajectory.

t - T = \frac{1}{2}\sqrt{\frac{p^{3}}{\mu}}\left (D + \frac{1}{3}D^{3}\right )

Where:

  • D = tan(ν/2), ν is the true anomaly of the orbit
  • t is the current time in seconds
  • T is the time of periapsis passage in seconds
  • μ is the standard gravitational parameter
  • p is the semi-latus rectum of the trajectory ( p = h2/μ )

More generally, the time between any two points on an orbit is

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 )


Alternately, the equation can be expressed in terms of periapsis distance, in a parabolic orbit rp = p/2:

t - T = \sqrt{\frac{2r_{p}^{3}}{\mu}}\left (D + \frac{1}{3}D^{3}\right )

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 t. If the following substitutions are made

A = \frac{3}{2}\sqrt{\frac{\mu}{2r_{p}^{3}}}(t-T)

B = \sqrt{A + \sqrt{A^{2}+1}}

then

\nu = 2\arctan(B - 1/B)

Read more about this topic:  Parabolic Trajectory

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