The Two-Point Boundary-Value Problem for Rocket Trajectories

Abstract

The two dimensional gravity turn problem is addressed allowing for the effects of variable rocket mass due to propellant consumption, thrust and thrust vector angle, lift and drag forces at an angle-of-attack and atmospheric mass density varying with altitude; Coriolis and centrifugal forces are neglected. Three distinct analytical solutions are obtained for constant: propellant flow rate, thrust, thrust vector angle, angle-of-attack and acceleration of gravity; the lift and drag are assumed to be proportional to the square of velocity, and the mass density is assumed to decrease exponentially with altitude. The method III uses power series of time for the horizontal (downrange) and vertical (altitude) coordinates; the method II replaces the altitude as variable by the atmospheric mass density and method I by its inverse. Thus the three solutions have distinct properties, e.g., I and III converge best close to lift-off and II close to burn-out. The three solutions: I, II, III, can be applied in isolation (or matched in combination) to the single-point boundary-value problem (SPBVP) of finding the trajectory with given initial conditions at launch. They can also be used as pairs in six distinct ways (I + II, I + III, II + III or reverse orders) to solve the two-point boundary-value problem (TPBVP), viz.: from given conditions at launch achieve one (not more) specified condition at burn-out, e.g., ã desired horizontal velocity for payload release. Each of the six distinct combinations of methods of addressing the TPBVP shares three features: (i) it can determine if there is a solution, viz. if the rocket has enough performance to reach the desired burn-out condition; (ii) if the desired burn-out condition is achievable it can calculate the complete trajectory from launch to burn-out; (iii) it can determine the range of achievable burn-out conditions, e.g., the minimum and maximum possible horizontal velocity at burn-out for given initial conditions at launch.