Roche Equipotentials Research Articles

On the integrability of the Roche coordinates

The aim of the present paper is to prove that the system of partial differential equations, which define a set of curvilinear coordinates ξ, η, ζ that are orthogonal to the Roche equipotentials ξ(r, θ, ϕ) incorporating the effects of both rotationaland tidal distortion, does not admit of any formal integrals; and can be solved only numerically in an asymptotic manner. This fact is related with analytic properties of the problem of three bodies, in which ξ represents the potential.

Gravity-darkening in eclipsing binaries

A quantitative analysis of the ellipticity effect in close binary systems of the W Ursae Maioris type with spectroscopically known mass-ratios discloses that the photometric ellipticities of these variables are, in general, more than twice as large as the dynamical ellipticities (due to the geometrical distortion alone) computed for contact models of centrally condensed stars. If, moreover, this excess of the photometric over dynamical ellipticity is attributed to the phenomenon of gravity-darkening over distorted surfaces of the constituent components, this darkening must be present to a considerablylarger degree than that predicted by the theory of radiative transfer — a result difficult to reconcile with the existence of extensive sub-surface convection zones in late-type Main-Sequence stars. If, on the other hand, gravity-darkening is present in W UMa-type systems in the amount predicted by the theory (let alone in the presence of sub-surface convection), the only effective way of reconciling the theory with the observations would be to assume that the mean fractional radii of their constituent components are appreciablylarger than those appropriate for contact models in which the two components just fill the largestclosed Roche equipotentials capable of containing their mass. In other words, it would appear that the W UMa-type stars considered in this paper may constitute a single dumb-bell figure rather than two distinct components; and that the observed variations of light, colour or radial velocity are invoked by the axial rotation (and consequent variable cross-section) or this dumb-bell figure rather than to the contribution of individual stars.