# Milking the spherical cow - on aspherical dynamics in spherical coordinates

Pontzen, Andrew; Read, Justin; Teyssier, Romain; Governato, Fabio; Gualandris, Alessia; Roth, Nina; Devriendt, Julien (2015). Milking the spherical cow - on aspherical dynamics in spherical coordinates. Monthly Notices of the Royal Astronomical Society, 451(2):1366-1379.

## Abstract

Galaxies and the dark matter haloes that host them are not spherically symmetric, yet spherical symmetry is a helpful simplifying approximation for idealized calculations and analysis of observational data. The assumption leads to an exact conservation of angular momentum for every particle, making the dynamics unrealistic. But how much does that inaccuracy matter in practice for analyses of stellar distribution functions, collisionless relaxation, or dark matter core-creation? We provide a general answer to this question for a wide class of aspherical systems; specifically, we consider distribution functions that are maximally stable', i.e. that do not evolve at first order when external potentials (which arise from baryons, large-scale tidal fields or infalling substructure) are applied. We show that a spherically symmetric analysis of such systems gives rise to the false conclusion that the density of particles in phase space is ergodic (a function of energy alone). Using this idea we are able to demonstrate that: (a) observational analyses that falsely assume spherical symmetry are made more accurate by imposing a strong prior preference for near-isotropic velocity dispersions in the centre of spheroids; (b) numerical simulations that use an idealized spherically symmetric setup can yield misleading results and should be avoided where possible; and (c) triaxial dark matter haloes (formed in collisionless cosmological simulations) nearly attain our maximally stable limit, but their evolution freezes out before reaching it.

## Abstract

Galaxies and the dark matter haloes that host them are not spherically symmetric, yet spherical symmetry is a helpful simplifying approximation for idealized calculations and analysis of observational data. The assumption leads to an exact conservation of angular momentum for every particle, making the dynamics unrealistic. But how much does that inaccuracy matter in practice for analyses of stellar distribution functions, collisionless relaxation, or dark matter core-creation? We provide a general answer to this question for a wide class of aspherical systems; specifically, we consider distribution functions that are maximally stable', i.e. that do not evolve at first order when external potentials (which arise from baryons, large-scale tidal fields or infalling substructure) are applied. We show that a spherically symmetric analysis of such systems gives rise to the false conclusion that the density of particles in phase space is ergodic (a function of energy alone). Using this idea we are able to demonstrate that: (a) observational analyses that falsely assume spherical symmetry are made more accurate by imposing a strong prior preference for near-isotropic velocity dispersions in the centre of spheroids; (b) numerical simulations that use an idealized spherically symmetric setup can yield misleading results and should be avoided where possible; and (c) triaxial dark matter haloes (formed in collisionless cosmological simulations) nearly attain our maximally stable limit, but their evolution freezes out before reaching it.

## Statistics

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16 citations in Web of Science®
11 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute for Computational Science 530 Physics English August 2015 22 Feb 2016 14:24 08 Dec 2017 18:35 Oxford University Press 0035-8711 This article has been accepted for publication in Monthly Notices of the Royal Astronomical Society © 2015 The Authors Published by Oxford University Press on behalf of Royal Astronomical Society. All rights reserved. Publisher DOI. An embargo period may apply. https://doi.org/10.1093/mnras/stv1032 arXiv:1502.07356v1