Two-Dimensional Model for Estimating the Efficiencyof Angular Measurements in Elliptic Orbits

Orbits that are unfavorable for the observer, whose plane passes near the orbital plane, are considered. Replacing the observer’s motion with an approximately equivalent motion in the orbital plane leads to a simple two-dimensional model to measure the motion. The Rao–Cramer bounds for the estimates of the orbital parameters are determined. The observation elevation angles are related to true anomalies and orbital parameters by nonlinear equations. Differentiating the equations with respect to the orbit’s parameters gives the Jacobi matrix of single angular measurements, and then the Fisher information matrix, which is the base for the accuracy analysis. The most difficult part of determining the Fisher matrix is calculating the derivatives of the true anomalies with respect to the orbit’s parameters; this part is based on the numerical solution of a differential equation expressing Kepler’s second law. Examples of calculating the boundaries of the accuracy of estimates in a wide range of practical velocities and angles of incidence are given. The results show that it is possible to obtain an accuracy that is of practical interest in some parameters from the angular measurements.