Department of Mathematics, University of Pisa, Italy
We consider initial orbit determination for asteroids and comets assuming small numbers observations and/or short observational time arcs as compared to the orbital period.
In the six--dimensional phase space of orbital elements, the orbital covariance matrix describes an uncertainty ellipsoid centered at the least--squares orbital elements. For a small number of observations, we cannot necessarily utilize the covariance matrix in determining, e.g., the uncertainties of the orbital elements. Nevertheless, the covariances always offer information about the orbit determination problem. Computing the eigenvalues and eigenvectors of the covariance matrix, we unfold the shape and orientation of the ellipsoid.
We have obtained intriguing first results for the lost asteroid (719) Albert and the newly--discovered comet C/1995 O1 Hale--Bopp: the uncertainty ellipsoids appear almost as a line of variation in the phase space (Muinonen 1996, MNRAS, in press). To gain more insight, we will solve the eigenproblem for a larger selection of asteroids making use of three observations only. Eigenorbits---orbital elements that lie on the line of variation---can prove useful in initial orbit determination, identification, and linkage in, e.g., the Spaceguard Survey for near--Earth asteroids and comets.