Solutions to the compact debris flight equations

Abstract

Analytic, approximate analytic, and numerical solutions to the compact debris flight equations are presented. Analysis shows that, after release from rest, the slope of the particle trajectory adjusts from an initial slope of −Ω to a final slope of −Ω where Ω is the inverse of the Tachikawa number. For Ω<1 the trajectory steepens whereas for Ω>1 the path becomes less steep over the full flight distance. However, for all values of Ω the trajectory initially steepens before adjusting to its steady trajectory slope. The final steady straight line trajectory is shown to project back to a virtual release height that is calculated numerically and shown to be a function of Ω. Approximate analytical solutions for the flight distance required to achieve the final steady-state slope (−Ω) are presented and show that the transition height is a function of Ω for small values of Ω but is independent of Ω for larger values. The transition height is shown to be very large for a broad range of physically realistic conditions. Contour plots are presented that summarize the change in trajectory, horizontal flight distance, horizontal and vertical velocity, and kinetic energy as a function of vertical distance traveled and Ω.