Abstract
The calculation of rocket trajectories is most often performed using purely numerical methods that account for all relevant parameters and provide the required results. There is a complementary need for analytical methods that make more explicit the effect of the various rocket and atmospheric parameters of the trajectory and can be used as test cases with unlimited accuracy. The available analytical methods take into account (i) variable rocket mass due to propellant consumption. The present paper includes four new analytical methods taking into account besides (i) also (ii) nonlinear aerodynamic forces proportional to the square of the velocity and (iii) exponential dependence of the mass density with altitude for an isothermal atmospheric layer. The four new methods can be used in “hybrid analytical-numerical” approach in which: (i) the atmosphere is divided into isothermal rather than homogeneous layers for greater physical fidelity; and (ii) in each layer, an exact analytical solution of the equations of motion with greater mathematical accuracy than a numerical approximation is used. This should allow a more accurate calculation of rocket trajectories while discretizing the atmosphere into a smaller number of layers. The paper therefore concentrates on four analytical methods of calculation of rocket trajectories in an isothermal atmospheric layers using new exact solutions of the equations of motion beyond those currently available in the literature. The four methods are developed first for the simpler case of a vertical climb and will be subsequently extended to the practically more relevant case of a gravity turn.
Highlights
Four new methods of analytical calculation of rocket trajectories are presented
The present paper aims at including the continuous dependence of aerodynamic forces on altitude through the mass density, and chooses as a starting point the dynamical trajectory equations (Section 2), using an earth-fixed Cartesian frame (Section 2.3) rather than the orthogonal frame moving along the trajectory (Section 2.1); in the latter case, altitude can be eliminated (Section 2.2) by introducing the radius of the curvature of the trajectory
The four methods I to IV of calculation of rocket trajectories have been applied to a single isothermal atmospheric layer, and use “analytical” solutions of the equations of motion; their accuracy would be improved by dividing the atmosphere into layers as is currently done in the “numerical”
Summary
Four new methods of analytical calculation of rocket trajectories are presented. The analytical methods have at least three advantages. Using the moving frame (b) and taking as variables the velocity and flight path angle, the acceleration components along and across the flight path are nonlinear; in this case, it is straightforward to eliminate time from the trajectory by dividing the longitudinal by the transversal acceleration, leading to a single differential equation for the path This method was originally applied [6] to calculate the ballistic trajectory of a projectile of constant mass in a uniform gravity field, with drag proportional to a power of the velocity and opposite to it; the extension of this solution to include the effects of thrust and variable mass has been researched extensively [7,8]. The case of free flight without thrust post burnout is considered by a fourth method IV, extending the known solution for constant atmospheric mass density [3], to exponential variation of altitude; this method leads to a series expansion with the terms of all orders calculated explicitly (Section 5.3)
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