useful coordinates for modelling dynamical systems
The halo of the Galaxy is rich in substructure. This substructure is the result of The Milky Way accreting material from smaller satellite galaxies. Debris from the satellites is tidally stripped by the Milky Way, forming long tidal streams of stars. These structures are of interest as they reveal something about the gravitational potential of the Galaxy, and hence the underlying dark matter distribution.
A typical orbit in the Milky Way disc (left). Complex orbital paths can be represented more simply as straight line paths over a torus (right). The action variables characterise the torus.
To construct models of galaxies and tidal streams, I use dynamical quantities called the action integrals. Each star in the Galaxy feels the gravitational pull of all the other stars, the gas and the dark matter causing it to follow an orbit.
In the movie, there is an example of such an orbit. Although this path looks complicated, it can be simplified by transforming to action coordinates. These are constants of the motion and characterise a torus on which the star moves. The position on the torus is then characterised by the angle coordinates which simply increase linearly in time (depicted in the second movie).
With these quantities we can build up dynamical models. However, although the actions are conceptually simple, in practice they are difficult to calculate. I have developed approximate schemes for their computation and published code that performs these calculations on github.
We review the available methods for estimating actions, angles and frequencies of orbits in both axisymmetric and triaxial potentials. The methods are separated into two classes. Unless an orbit has been trapped by a resonance, convergent, or iterative, methods are able to recover the actions to arbitrarily high accuracy given sufficient computing time. Faster non-convergent methods rely on the potential being sufficiently close to a separable potential, and the accuracy of the action estimate cannot be improved through further computation. We critically compare the accuracy of the methods and the required computation time for a range of orbits in an axisymmetric multicomponent Galactic potential. We introduce a new method for estimating actions that builds on the adiabatic approximation of Schönrich & Binney and discuss the accuracy required for the actions, angles and frequencies using suitable distribution functions for the thin and thick discs, the stellar halo and a star stream. We conclude that for studies of the disc and smooth halo component of the Milky Way, the most suitable compromise between speed and accuracy is the Stäckel Fudge, whilst when studying streams the non- convergent methods do not offer sufficient accuracy and the most suitable method is computing the actions from an orbit integration via a generating function. All the software used in this study can be downloaded from https://github.com/jls713/tact.
@article{2016MNRAS.457.2107S,author={{Sanders}, Jason L. and {Binney}, James},title={{A review of action estimation methods for galactic dynamics}},journal={\mnras},keywords={methods: numerical, galaxies: kinematics and dynamics, Astrophysics - Astrophysics of Galaxies},year={2016},month=apr,volume={457},number={2},pages={2107-2121},doi={10.1093/mnras/stw106},archiveprefix={arXiv},eprint={1511.08213},primaryclass={astro-ph.GA},adsurl={https://ui.adsabs.harvard.edu/abs/2016MNRAS.457.2107S},adsnote={Provided by the SAO/NASA Astrophysics Data System}}
2015
MNRAS
A fast algorithm for estimating actions in triaxial potentials
We present an approach to approximating rapidly the actions in a general triaxial potential. The method is an extension of the axisymmetric approach presented by Binney, and operates by assuming that the true potential is locally sufficiently close to some Stäckel potential. The choice of Stäckel potential and associated ellipsoidal coordinates is tailored to each individual input phase-space point. We investigate the accuracy of the method when computing actions in a triaxial Navarro-Frenk-White potential. The speed of the algorithm comes at the expense of large errors in the actions, particularly for the box orbits. However, we show that the method can be used to recover the observables of triaxial systems from given distribution functions to sufficient accuracy for the Jeans equations to be satisfied. Consequently, such models could be used to build models of external galaxies as well as triaxial components of our own Galaxy. When more accurate actions are required, this procedure can be combined with torus mapping to produce a fast convergent scheme for action estimation.
@article{2015MNRAS.447.2479S,author={{Sanders}, Jason L. and {Binney}, James},title={{A fast algorithm for estimating actions in triaxial potentials}},journal={\mnras},keywords={methods: numerical, Galaxy: kinematics and dynamics, galaxies: kinematics and dynamics, Astrophysics - Astrophysics of Galaxies},year={2015},month=mar,volume={447},number={3},pages={2479-2496},doi={10.1093/mnras/stu2598},archiveprefix={arXiv},eprint={1412.2093},primaryclass={astro-ph.GA},adsurl={https://ui.adsabs.harvard.edu/abs/2015MNRAS.447.2479S},adsnote={Provided by the SAO/NASA Astrophysics Data System}}
2014
MNRAS
Actions, angles and frequencies for numerically integrated orbits
We present a method for extracting actions, angles and frequencies from an orbit’s time series. The method recovers the generating function that maps an analytic phase-space torus to the torus to which the orbit is confined by simultaneously solving the constraints provided by each time step. We test the method by recovering the actions and frequencies of tori in a triaxial Stäckel potential, and use it to investigate the structure of orbits in a triaxial potential that has been fitted to our Galaxy’s Sagittarius stream. The method promises to be useful for analysing N-body simulations. It also takes a step towards constructing distribution functions for the triaxial components of our Galaxy, such as the bar and dark halo.
@article{2014MNRAS.441.3284S,author={{Sanders}, Jason L. and {Binney}, James},title={{Actions, angles and frequencies for numerically integrated orbits}},journal={\mnras},keywords={methods: numerical, Galaxy: kinematics and dynamics, galaxies: kinematics and dynamics, Astrophysics - Astrophysics of Galaxies},year={2014},month=jul,volume={441},number={4},pages={3284-3295},doi={10.1093/mnras/stu796},archiveprefix={arXiv},eprint={1401.3600},primaryclass={astro-ph.GA},adsurl={https://ui.adsabs.harvard.edu/abs/2014MNRAS.441.3284S},adsnote={Provided by the SAO/NASA Astrophysics Data System}}
2012
MNRAS
Angle-action estimation in a general axisymmetric potential
The usefulness of angle-action variables in galaxy dynamics is well known, but their use is limited due to the difficulty of their calculation in realistic galaxy potentials. Here we present a method for estimating angle-action variables in a realistic Milky Way axisymmetric potential by locally fitting a Stäckel potential over the region an orbit probes. The quality of the method is assessed by comparison with other known methods for estimating angle-action variables of a range of disc and halo-type orbits. We conclude by projecting the Geneva- Copenhagen survey into angle-action space.
@article{2012MNRAS.426..128S,author={{Sanders}, Jason},title={{Angle-action estimation in a general axisymmetric potential}},journal={\mnras},keywords={methods: numerical, Galaxy: kinematics and dynamics, solar neighbourhood, Galaxy: structure, galaxies: kinematics and dynamics, Astrophysics - Astrophysics of Galaxies},year={2012},month=oct,volume={426},number={1},pages={128-139},doi={10.1111/j.1365-2966.2012.21698.x},archiveprefix={arXiv},eprint={1208.2813},primaryclass={astro-ph.GA},adsurl={https://ui.adsabs.harvard.edu/abs/2012MNRAS.426..128S},adsnote={Provided by the SAO/NASA Astrophysics Data System}}
