The fundamental work of Carl Frederic Gauss [Gauss 1809] introduced the theory of the normal distribution of the observation errors and the method of least square to solve for orbits; the two concepts are related by a theorem, by which the probability distribution of the least square fit in the space of orbital elements is also normal -Gaussian, as we say today- in the linear approximation. Gauss developed this theory (and more) to solve the problems of orbit determination and recovery for the very first asteroids, (1) Ceres and (2) Pallas. For a modern discussion of the transformation of a Gaussian probability distribution under a coordinate change see e.g. [Jazwinski 1970], Chap. 2, Section 6.
These methods have been used by the astronomers ever since; with the space age the linearised Gaussian theory become the standard tool of astrodynamics [Cappellari et al., 1976]. The orbit determination of a spacecraft is, however, different in that the tracking schedule is built in the mission analysis, thus the number of observations is designed to be very large; this legitimates, by the central limit theorem, the hypothesis of a Gaussian distribution of the observation errors (provided some biases are also estimated). Moreover, the linearity hypothesis is applicable because a spacecraft is always tracked with the purpose of keeping the orbit uncertainty within a small region, where the linear approximation is very good; if nonlinear error propagation is needed, then somebody in the mission team has done a very dangerous mistake.
For asteroid and comet orbit determination, on the contrary, both the Gaussian error distribution and the linearity of the error propagation are often not applicable. Real observations are the most precious, and scarce, resource, thus their numbers are in most cases to small to reliably use statistical arguments. Especially for observations performed in the recent years, the accidental astrometric reduction errors are less important than the star catalogue errors, known to have regional biases; thus the error distribution is in fact not Gaussian. If an asteroid is lost, by definition the uncertainty of its position on the celestial sphere is large, and the linearity hypothesis is not applicable, not even as an approximation.
The goal of this paper is to remove both hypotheses, the Gaussian error distribution and the applicability of the linear approximation. The problem raised by the nonlinear effects has already been stated, and some solutions proposed, in [Muinonen and Bowell 1993]; we shall comment later on the need for algorithms more efficient than the ones they propose. Nonlinearity is discussed in Sections 3, 4 and 5; this Section has the purpose of restating the classical linear theory in a way dependent from the size of the observation errors, but independent from the shape of their distribution.
It is possible to describe the domain of possible solutions in terms of an optimization problem only. To this purpose, we shall use the fact that the orbit determination is obtained by minimizing a target function Q, which is essentially the sum of squares of the residuals (observed minus computed position on the celestial sphere). In this framework, the uncertainty of the solution arises from the fact that the minimum point of Q is as good a solution as other points in the parameter space such that the residuals have essentially the same size. None of the results of this Section is new, but they are presented with an interpretation different from the one available in the textbooks.