CLOSED-FORM TRAJECTORY SOLUTION FOR SHALLOW, HIGH-ALTITUDE ATMOSPHERIC FLIGHT

A closed-form approximate solution for shallow, high-altitude atmospheric flight, consistent with aerobraking passes is proposed. The solution includes expressions for velocity, flight-path angle, and altitude for lifting, high-speed atmospheric flight, which can be used to quickly evaluate trajectories. The complete derivation of the solution is presented. The solution is based on the assumptions of small flight-path angles and altitude rate changing linearly with respect to time. Results show a good match between the proposed approximate solution and numerical integration of the full equations of motion for a variety of trajectory parameters, including vacuum periapsis altitudes, initial flight-path angles and velocities, and vehicle aerodynamic coefficients. Larger, but bounded errors are present in predicted atmospheric exit velocities. Generally, results show that the predicted final velocity has a maximum error of approximately 0.6% in nominal conditions where the assumptions hold. Exit velocity errors are lower for trajectories that dissipate less energy during atmospheric flight. Finally, a Monte Carlo simulation is used to show how errors in altitude, flight-path angle, and velocity remain bounded in the presence of perturbations. Overall, results indicate that the proposed approximate solution can be used for first-order fast trajectory design for aerobraking and other grazing atmospheric trajectories.