Gauss meets Newton again: How to make Gauss orbit determination from two position vectors more efficient and robust with Newton–Raphson iterations

Abstract

Gauss’s method for determining the six classical idealized Keplerian two-body orbital parameters from two position vectors and two times is elegantly formulated. It first reduces the three-dimensional velocity determination problem to two equation in two variables x, y. These two equations take the form: (i) x = F( y; l, m), (ii) y = G( x; l). Here l and m are parameters that are functions of the two position vectors and the corresponding time span between them. Gauss’s approach was to form a functional iteration ( y-iteration) by substituting (i) into (ii) An alternate second functional iteration ( x-iteration) is formed by substituting (ii) into (i). In either case a further reduction of dimension from two to one is obtained. The method first appeared in the early-19th century and is ubiquitous in the astronomy and astro-dynamics literature. However, it could not exploited until the mid 20th century advent of powerful radars. These can provide initial elliptical orbital positional observations of such natural space objects as the moon, comets, asteroids, and meteors as well as man-made objects such as satellites and ballistic missile in the mid-course phase of their trajectories. There are, however, three basic problems with the classical approach: (1) it breaks down when the angular difference in true anomaly between the two position vectors grows larger than π/4, (2) the classical first guess, y 0 = 1, may not lie in the convergence region of the method’s functional iteration, and (3) the functional iteration, when it works, is slowly linearly convergent. In this paper we show that applying a Newton–Raphson iteration to the fixed-point equation that defines the functional iteration, in y when the angular spread is sufficiently small, or in x when the spread is large, together with trivial first guesses solves all three problems at once. Moreover, we prove a theorem on a simple necessary and sufficient condition for the existence of unique solutions and put forth a conjecture on the concavity of the two fixed-point equation graphs. This Newton–Raphson iteration approach extends the range of algorithm applicability to a true anomaly angular spread close to π, a basic singularity of the Gauss method.