Estimating the Potential Accuracy of Some Orbital Parameters from Angular Measurements: a 2D model

Abstract

The problem of estimating some parameters of a cosmic body orbits from angular measurements is considered. Contrary to the existing opinion that angular measurements are unsuitable for obtaining accuracies that are of practical interest, it is shown that, if the orbit is such that the impact place is in the vicinity of the observer, then such parameters as the impact point and time can be determined with quite acceptable accuracy. A fairly simple means for evaluating the accuracy of the "motion-measurement" process by reducing it to an analysis in a plane is described. The analysis is based on several principles. A family of orbits unfavorable to the observer is singled out, and an analysis is performed on this family. These are orbits whose plane touches the observer's Earth parallel, with the touching point being the intersection point of the observer and cosmic body trajectories. The article substantiates a simple approximate plane model, in which the observer’s true motion is replaced without changing its velocity by motion in the orbit plane. Information about the impact point is contained in the observations when the height decreases, which makes it possible to consider the gravity acceleration of as a constant in the descending mode. The motion model is simplified compared with the Keplerian motion: a simple motion scheme is obtained in the polar coordinate system. As a result, it becomes unnecessary to solve the differential equation representing the change in acceleration. A measurement formula that takes into account the Earth curvature and the observer latitude is derived, and the parameters determining the analyzed scheme are added: the distance to the impact point, incidence angle and impact moment, and horizontal velocity. The model accuracy is analyzed using the Fisher matrix. As a result, an approximate problem on a plane with four parameters is obtained instead of a problem in 3D space with seven parameters. The difficulties with the accuracy of estimating the displacement and impact moment in the model remain, and they are analyzed. The accuracy is estimated using the concept of Fisher information and the multidimensional Rao-Kramer information inequality instead of analyzing the processing algorithm.