## A Geometric Derivation and a Solution of Barkers Equation for the Time of Flight Along Parabolic Trajectories

### A. Pathan (2005), *JBIS*, **58**, 82-89

**Refcode**: 2005.58.82

**Keywords**: Parabolic trajectory, Barker's Equation, Geometric derivation, Time of flight

**Abstract:**Historically the geometric derivations of the time of flight along the conic section trajectories were developed over three centuries ago with the exception of the parabolic trajectory. Here a geometric derivation of the time of flight along a parabolic trajectory usually referred to as Barker's Equation is presented. Expressed in terms of radii only, the determination of the time of flight along a parabolic trajectory for lunar and planetary flights is shown to be very convenient, especially when using canonical units. Given the true anomaly and the periapsis distance, the time of flight in a given central force field is readily determined from Barker's Equation. The inverse problem, that of finding the true anomaly given the time of flight and the periapsis distance, particularly for short times of flight presents significant difficulties. These difficulties involve the preservation of the numerical accuracy of the solution of the cubic equation for the true anomaly . Here a solution for Barker's Equation, valid for all flight times is presented.

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