Error propagation in the numerical solutions of the differential equations of orbital mechanics

The relationship between the eigen values of the linearized differential equations of orbital mechanics and the stability characteristics of numerical methods is presented. It is shown that the Cowell, Encke, and Encke formulation with an independent variable related to the eccentric anomaly all have a real positive eigen value when linearized about the initial conditions. The real positive eigen value causes an amplification of the error of the solution when used in conjunction with a numerical integration method. In contrast an element formulation has zero eigen values and is numerically stable.