Space Trajectories. Flat Earth Solutions

Space trajectory problems can rarely be solved analytically. Assuming a flat Earth with constant gravity, with no atmosphere or with a simplified atmosphere model, the equations of motion are greatly simplified, enabling explicit solutions to be found in some cases. These solutions are useful for rapidly giving orders of magnitude of performance in the pre-project phase, or for initializing numerical resolution with more realistic models. This article describes the resolution of four classic launch trajectory problems on flat Earth, followed by a method for moving on to the solution on round Earth.

The first problem deals with maximum-range ballistic firing at constant acceleration. The solution on a flat Earth with no atmosphere shows that the optimum thrust direction is constant. The thrust attitude is obtained by solving an equation with one unknown that depends solely on the level of acceleration.

The second problem deals with injection into orbit at constant acceleration. The solution for a flat Earth with no atmosphere shows that the direction of thrust follows a linear tangent law. The problem is reduced to an equation with one unknown, enabling us to evaluate the thrust required according to the level of acceleration and the target altitude.

The third problem deals with the vertical arc of maximum apogee, assuming variable thrust. This is Goddard's problem, whose solution may involve a non-maximum variable thrust arc in the presence of aerodynamic braking. This singular arc is determined analytically as a function of the rocket's drag.

The fourth problem deals with in-orbit injection, assuming variable thrust. The optimal solution on a flat Earth with no atmosphere consists of a single arc with maximum thrust. The optimal control problem is reduced to a system with two unknowns, easily solved by a firing method.

The final part deals with the transition from flat Earth and constant gravity to round Earth and variable gravity. The transformation involves a change of coordinates and a continuation method that gradually introduces the Earth's curvature and variable gravity.

The four problems are independent. This article presents the modeling, the analytical solution of the optimal control problem and an illustrative numerical application for each one.