The Roche coordinates for the rotational problem

Abstract

Curvilinear coordinates in three dimensions associated with the Roche model distorted by centrifugal force alone constitute a Lamé family, of which one (ξ-) coordinate can be defined by equipotential surfaces which are known in closed algebraic form; the other (η) becomes identical with the meridional planes of the rotationally distorted Roche model; while the third (ζ) then follows from the requirements of orthogonality to the others. The explicit form of such coordinates in terms of the polar or cartesian systems has already been established by the author (Kopal, 1970) correctly to quantities of the first order in superficial distortion of the respective Roche model. In the present paper this latter restriction on accuracy will be removed, and expressions constructed for the ζ-coordinate in the form of infinite series which are exact and converge rapidly for any distortion below that which entails equatorial break-up.