On selecting the correct root of angles-only initial orbit determination equations of Lagrange, Laplace, and Gauss

Abstract

This paper is concerned with a classical yet still mystifying problem regarding multiple roots of the angles-only initial orbit determination (IOD) polynomial equations of Lagrange, Laplace, and Gauss of the form: f(x) = x8 + ax6 + bx3 + c = 0 where a,c < 0. A possibility of multiple non-spurious roots of this 8th-order polynomial equation with b > 0 has been extensively treated in the celestial mechanics literature. However, the literature on applied astrodynamics has not treated this multiple-root issue in detail, and not many specific numerical examples with multiple roots are available in the literature. Recently, Gim Der has claimed that the 200-year-old, angles-only IOD riddle associated with the discovery and tracking of asteroid Ceres has been finally solved by using a new angles-only IOD algorithm that doesn't utilize any a priori knowledge and/or additional observations of the object. In this paper, a very simple method of determining the correct root from two or three non-spurious roots is presented. The proposed method exploits a simple approximate polynomial equation of the form: g(x) = x8 + ax6 = 0. An approximate polynomial equation, either g(x) = x8 + c = 0 or g(x) = x8 + ax6 = x6(x2 + a) = 0, can also be used for quickly estimating an initial guess of the correct root.