An analytic trajectory algorithm including earth oblateness
Abstract
A closedform solution is presented for the trajectory of a ballistic missile over an oblate earth to first order in the oblateness. Except for the possibility of a subset of trajectories with eccentricity near unity, this solution is valid for any nonzero value of eccentricity. The solution is derived through the application of classical perturbation theory to the equations of motion from which the missile's trajectory is expressed as a perturbation from a nominal Keplerian ellipse. This analysis is quite similar to that presented by Roberson, but extends his work somewhat by giving a solution to the time equation, a firstorder differential equation relating time to the true anomaly. The result is an augmented Kepler's equation which, analogous to the wellknown Kepler equation, must be solved iteratively for the final value of eccentric anomaly. The analytic solution is compared to a direct numerical integration of the equations of motion to substantiate its correctness.
 Publication:

Journal of the Astronautical Sciences
 Pub Date:
 March 1977
 Bibcode:
 1977JAnSc..25...35M
 Keywords:

 Ballistic Trajectories;
 Missile Trajectories;
 Trajectory Analysis;
 Algorithms;
 Differential Equations;
 Equations Of Motion;
 Iterative Solution;
 Numerical Integration;
 Oblate Spheroids;
 Perturbation Theory;
 Astrodynamics