Analysis of the suitability of analytical, semi-analytical, and numerical approaches for important orbit propagation tasks

Astrodynamies encompasses phenomena on diverse and disparate time scales. That solar electromagnetic atmospheric density proxies are developed every few hours demonstrates phenomena on that scale. In the most simple two body Newtonian formulation that includes only inverse square gravitation, there are equilibrium solutions that persist forever. Tidal effects are diurnal. Collisions last only milliseconds. Explosion debris migrates over weeks or months. The equations of astrodynamies initial value problem in the analytical, semi-analytical, and numerical formulation of astrodynamies exhibit stiffness. We emphasize the well-known mathematical fact that explicit numerical methods can create numerical stiffness where there was no physical stiffness and can produce reasonable but erroneous outcomes. Implicit methods should always be used even though more computational operations might be required.