Optimum Geometric Distribution of Stellar Observations in Astronomical Positioning

Abstract

Astronomical positioning can determine the astronomical longitude and latitude of a station based on observations of celestial bodies. The accuracy of astronomical positioning depends on the observation instruments, surveying environment, and geometric distribution of stellar observations. For astronomical theodolite, existing theories support that the best positioning accuracy can only be obtained with uniformly distributed stars on an equal-altitude circle. In fact, inhomogeneous distribution can also obtain the best positioning accuracy. To illustrate this problem, a quantitative index to determine the optimum geometric distribution of stellar observations by introducing the mean resultant length of vectors in directional statistics is proposed, providing a theoretical basis for studying high-precision astronomical positioning and automated star selection algorithms. Multiple optimum distributions of observations with constant numbers of stars during simulation experiments using the equal-altitude method have been demonstrated. Regarding the approximate equal-altitude method, the inhomogeneous distribution of stars guided by the conclusion described in this paper was also optimum. It could achieve the same accuracy as the uniform distribution. These results can aid in overcoming the limitations of the existing uniform distribution theory for astronomical positioning and provide a theoretical basis for achieving high-precision and high-efficiency astronomical positioning by inhomogeneous distribution stellar observations, especially when only a small number of stars can be observed.