The geometry of the Roche coordinates

Abstract

The exact geometry of the Roche curvilinear coordinates (ξ, η, ζ) in which ξ corresponds to the zero-velocity surfaces is investigated numerically in the plane, as well as in the spatial, case for various values of the mass-ratio between the two point-masses (m ₁,m ₂) constituting a binary system.

The geometry of zero-velocity surfaces specified by ξ-values at the Lagrangian points are first discussed by taking their intersections with various planes parallel to thexy-, xz- andyz-planes. The intersection of the zero-velocity surface specified by the ξ-value at the Lagrangian equilateral-triangle pointsL _4,5 with the planex=1/2 discloses two invariable curves passing through the pointsL _4,5 and situated symmetrically with respect to thexy-plane whose form is independent of the mass-ratio.

The geometry of the remaining two coordinates (η, ζ) orthogonal to the zero-velocity surfaces is investigated in thexy- andxz-planes from extensive numerical integrations of differential equations generated from the orthogonality relations among the coordinates. The curves η(x, y)=constant in thexy-plane are found to be separated into three families by definite envelopes acting as boundaries whose forms depend upon the mass-ratio only: the inner η-constant curves associated with the masspointm ₁, the inner η-constant curves associated with the mass-pointm ₂ and the outer η-constant curves. All the η-constant curves in thexy-plane coalesce at either of the Lagrangian equilateraltriangle pointsL _4,5, except for a limiting case coincident with thex-axis. The curves ζ(x, z)=constant in thexz-plane are also separated by definite envelopes depending upon the mass-ratio into different families: the inner ζ-constant curves associated with the mass-pointm ₁, the inner ζ-constant curves associated with the mass-pointm ₂ and the outer ζ-constant curves on both sides out of the envelopes. For larger values ofz, the curves ζ=constant tend asymptotically to the line perpendicular to thex-axis and passing through the centre of mass of the system, except for a limiting case coincident with thex-axis. The geometrical aspects of the envelopes for the curves η(x, y)=constant in thexy-plane and the curves ζ(x, z)=constant in thexz-plane are also discussed independently.

In the three-dimensional space, the Roche coordinates can be conveniently defined in such a way as to correspond to the polar coordinates in the immediate neighbourhood of the origin, and to the cylindrical coordinates at great distances. From numerical integrations of simultaneous differential equations generating spatial curves orthogonal to the zero-velocity surfaces, the surfaces η(x, y, z)=constant and the surfaces ζ(x, y, z)=constant are constructed as groups of such spatial curves with common values of some parameters specifying the respective surfaces.