Determining the orbit from five relative apparent positions

Abstract

The article describes a new method for determining the orbit of a satellite of a distant body by five measured apparent relative positions. Objects can be components of a visual binary star. The method is based on old ideas proposed back in the 19th century. In 1883, T.N. Thiele theoretically deduced and proposed an interesting method for determining an orbit from three apparent relative positions for a given sector (areal) velocity of the body in apparent orbit. In this paper, this method is supplemented with analytical formulas for determining the sectoral velocity of a satellite of a star or planet in apparent orbit in five positions. Five relative positions are needed to accurately determine the sector velocity of a body in a visible orbit. Further, the parameters of the spatial orbit are determined by three positions. The semi-major axis of the orbit and the mean motion are determined independently. This makes it possible to determine the total mass of the system under study. The proposed method is first tested on a model example, and then its adequacy is shown for the problem of determining the orbit of a visual binary star ZZ Tau based on real observations. In this example, the method is tested, its adequacy is shown. We also consider the application of the method for determining the preliminary orbit of an asteroid satellite with the aim of subsequent refinement of the orbit based on collection of observations over a large time interval. The reliability of the method has yet to be tested on specific problems. For convenience, an online service has been developed on the Internet allowing to calculate the orbit from the reader’s observations.