Professor James Binney FRS

Abstract:

The challenge presented by computing actions for eccentric orbits in axisymmetric potentials is discussed. In the limit of vanishing angular momentum about the potential’s symmetry axis, there is a clean distinction between box and loop orbits. We show that this distinction persists into the regime of non-zero angular momentum. In the case of a Stäckel potential, there is a critical value (Formula presented) of the third integral (Formula presented) below which (Formula presented) does not contribute to the centrifugal barrier. An orbit is of box or loop type according as its value of (Formula presented) is smaller or greater than (Formula presented). We give algorithms for determining (Formula presented) and the critical action (Formula presented) below which orbits in any given potential are boxes. It is hard to compute the actions and especially the frequencies of orbits that have (Formula presented) using the Stäckel Fudge. A modification of the Fudge that alleviates the problem is described.

Abstract:

Galaxy models comprising several components (including dark matter) that are bound by the self-consistently generated gravitational field are readily constructed from distribution functions (DFs) that are analytic functions of the action integrals (Formula presented). We explain why such models have unphysical velocity distributions unless the DFs of hot components satisfy certain conditions as (Formula presented). We show how DFs for both isotropic and radially biased spherical systems can be constructed with specified (Formula presented). We show how to construct DFs for flattened systems with significant velocity anisotropy. Construction of self-consistent models rather than populations that are confined by an external potential leads to the conclusion that radially-biased spherical systems are generically unstable to quadrupolar perturbations. Chaos is likely key to maintenance of these constraints during adiabatic disc growth. If the DFs of dark haloes are radially biased, as simulations of cosmic clustering suggest, then models presented here suggest that dark haloes should be significantly oblate.

Abstract:

We outline the rationale for holding a Discussion Meeting thus titled at the Royal Society, London during 15 and 16 April 2024, and summarize what we learnt there. This article is part of the discussion meeting issue ‘Challenging the standard cosmological model’.