Triaxial cosmological haloes and the disc of satellites

We construct simple triaxial generalizations of Navarro-Frenk-White haloes. The models have elementary gravitational potentials, together with a density that is cusped like 1/r at small radii and falls off like 1/r(3) at large radii. The ellipticity varies with radius in a manner that can be tailored to the user's specification. The closed periodic orbits in the planes perpendicular to the short and long axes of the model are well described by epicyclic theory, and can be used as building blocks for long-lived discs. As an application, we carry out the simulations of thin discs of satellites in triaxial dark halo potentials. This is motivated by the recent claims of an extended, thin disc of satellites around the M31 galaxy with a vertical rms scatter of similar to 12 kpc and a radial extent of similar to 300 kpc. We show that a thin satellite disc can persist over cosmological times if and only if it lies in the planes perpendicular to the long or short axis of a triaxial halo, or in the equatorial or polar planes of a spheroidal halo. In any other orientation, then the disc thickness doubles on similar to 5 Gyr time-scales and so must have been born with an implausibly small vertical scaleheight.