Virtual singularities in the artificial satellite theory

Abstract

When the coordinate system used in perturbation theory presents a geometrical singularity and when the perturbation technique fails to take account of this, the solution developed may present singularities which are no longer easily explained by purely geometrical means. These singularities have been calledvirtual singularities by Deprit and Rom (1970). We propose to demonstrate that virtual singularities can in general be avoided by the use of Lie transforms. In general, it is sufficient to recognize that the original Hamiltonian function presents the d'Alembert characteristic with respect to pairs of action-angle variables and that the averaging operations preserve this characteristic. We then apply this criterion to the artificial satellite theory (for small to moderate eccentricity) showing that all of three possible virtual singularities can be avoided at the same time. A new set of elliptic elements, well suited to the problem at hand, is introduced.