On Selecting the Correct Root of Angles-Only Initial Orbit Determination Equations of Lagrange, Laplace, and Gauss

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. In this paper, a very simple method of determining the correct root from two or three non-spurious roots is presented, which doesn’t utilize any a priori knowledge and/or additional observations of the object. 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.