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Abstract:

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 $I_{3\rm crit}(E)$ of the third integral $I_3$ below which $I_3$ does not contribute to the centrifugal barrier. An orbit is of box or loop type according as its value of $I_3$ is smaller or greater than $I_{3\rm crit}$. We give algorithms for determining $I_{3\rm crit}(E)$ and the critical action $J_{z\rm crit}$ below which orbits in any given potential are boxes. It is hard to compute the actions and especially the frequencies of orbits that have $J_z\simeq J_{z\rm crit}$ 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 . We explain why such models have unphysical velocity distributions unless the DFs of hot components satisfy certain conditions as . We show how DFs for both isotropic and radially biased spherical systems can be constructed with specified . 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:

Abstract Code is presented that computes and exploits orbital tori for any axisymmetric gravitational potential. The code is a development of the agama software package for action-based galaxy modelling and can be downloaded as the agama b code library. Although coded in C++, most of its functions can be accessed from Python. We add to the package functions that facilitate confronting models with data, which involve sky coordinates, lines of sight, distances, extinction, etc. The new torus generator can produce tori for both highly eccentric and nearly circular orbits that lie beyond the range of the earlier torus-mapping code. Tori can be created by interpolation between tori at very low cost. Tori are fundamentally devices for computing ordinary phase-space coordinates from angle-action coordinates, but agama b includes an action finder that returns angle-action coordinates from any given phase-space location. This action finder yields the torus through the given point, so it includes the functionality of an orbit integrator. The action finder is more accurate and reliable but computationally more costly than the widely used Stäckel Fudge. We show how agama b can be used to generate sophisticated but cheap models of tidal streams and use it to analyse data for the GD1 stream. With the most recently published distances to the stream, energy and angular momentum imply that the end that must be leading is trailing, but extremely small changes to the distances rectify the problem.

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’.