Dynamics of the Milky Way

Transcription

1 Jason Sanders (Oxford IoA) with James Binney Dynamics of the Milky Way Tidal streams & Extended Distribution Functions for the Galactic Disc Whistle-stop tour of a PhD

2 Where is the dark matter? We can currently only measure dark matter through its gravitational influence on Visible matter Photons emitted by visible matter (e.g lensing) Cannot measure accelerations only positions and velocities so must use statistics I will describe two tools useful for tackling this problem Extended Distribution Functions 2 Tidal Stream Modelling

3 Extended Distribution Functions Tidal Stream Modelling Local measurements of dark matter e.g. Kuijken & Gilmore 1989; Moni Bidin et al. 2012b; Garbari et al. 2012; Bovy et al. 2012a; Zhang et al. 2013; Piffl et al Global measurements of dark matter e.g. Koposov et al. (2010) Both use dynamical models to provide complementary constraints f(x,v) & Φ(x) 3

4 Action-angle modelling Convenient to use actions (J) & angles (θ) Canonical coordinates Simple equations of motion Adiabatic invariants When considering steady-state models, we must satisfy Can use the actions as arguments of f 4

5 Action-angle modelling Convenient to use actions (J) & angles (θ) Canonical coordinates J z Simple equations of motion Adiabatic invariants When considering steady-state models, we must satisfy J R Axisymmetric Can use the actions as arguments of f 5

6 Distribution Functions for the Galactic disc Model Φ(x) Multicomponent Galactic potential f(x,v) = f(j) Quasi-isothermal Data li, bi, l.o.s velocities, proper motions, parallaxes & abundances ([Fe/ H], [X/Fe]) Binney (2010, 2012) Is f(x,v) consistent with the data? Computing loglikelihood Geneva- Copenhagen survey, SEGUE, RAVE, Gaia- ESO, APOGEE, GALAH, Gaia 6 Sanders & Binney (2014, submitted)

7 Extended DFs Using chemical information Analytic metallicity with age Radial migration SFR (exponential decay) & IMF (Kroupa) Isochrones (BaSTI, Pietrinferni et al. 2004) 7

8 Why bother? Selection Functions Different chemical populations must live in same potential Bovy & Rix (2013) 8

9 Geneva-Copenhagen Survey Spectroscopic survey of ~10000 stars with Hipparcos parallaxes Predict SEGUE G dwarf data (in paper), RAVE data (WIP) Gilmore & Reid (1983) 9

10 Grillmair & Dionatos (2006) Stream modelling Result of tidal stripping from Milky Way satellites Not steady state, not phase-mixed But again we can use angle-action coordinates (Tremaine, 1999, Helmi & White, 1999, Eyre & Binney, 2011) 10

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12 Stream model in angle-frequency space 12

13 Finding potential Sanders,

14 Estimating angle-action variables (details ) A few special cases are analytic Kepler, harmonic oscillator, isochrone Most general potential for separable Hamilton-Jacobi equation = Stäckel potentials Stäckel potentials do not provide adequate global fits to observed potentials Want methods for finding actions in general potentials 14

15 Non-convergent procedures Convergent/iterative procedures Stäckel fitting Fit Stäckel potential to local region a given orbit probes and estimate actions in this potential (Sanders 2012) Torus machine Construct torus of given actions by finding generating function from toy Hamiltonians Designed for (J,Θ) to (x,v) (McMillan & Binney 2008) Stäckel Fudge Use equations for actions in a Stäckel potential assume potential is locally separable in ellipsoidal coordinates (Binney, 2012, [axisymmetric] and Sanders & Binney, 2014, submitted [triaxial]) Adiabatic approximation Assume potential is separable in polar coordinates (Schönrich & Binney, 2012) 15 Generating Function Construct generating function from time samples from orbit integration Designed for (x,v) to (J,Θ) (Sanders & Binney 2014) Iterative torus construction Iteratively find (J,Θ) from non-convergent procedure and construct torus with these actions J

16 Stäckel fitting Fit Stäckel potential to local region a given orbit probes and estimate actions in this potential (Sanders 2012) de Zeeuw & Lynden-Bell (1985) Axisymmetric multicomponent Galactic potential from McMillan (2011) 16

17 Non-convergent procedures Convergent/iterative procedures Stäckel fitting Fit Stäckel potential to local region a given orbit probes and estimate actions in this potential (Sanders 2012) Torus machine Construct torus of given actions by finding generating function from toy Hamiltonians Designed for (J,Θ) to (x,v) (McMillan & Binney 2008) Stäckel Fudge Use equations for actions in a Stäckel potential assume potential is locally separable in ellipsoidal coordinates (Binney, 2012, [axisymmetric] and Sanders & Binney, 2014, submitted [triaxial]) Adiabatic approximation Assume potential is separable in polar coordinates (Schönrich & Binney, 2012) 17 Generating Function Construct generating function from time samples from orbit integration Designed for (x,v) to (J,Θ) (Sanders & Binney 2014) Iterative torus construction Iteratively find (J,Θ) from non-convergent procedure and construct torus with these actions J

18 Stäckel Fudge Use equations for actions in a Stäckel potential assume potential is locally separable in ellipsoidal coordinates (Binney, 2012, [axisymmetric] and Sanders & Binney, 2014, submitted [triaxial]) 18

19 Non-convergent procedures Convergent/iterative procedures Stäckel fitting Fit Stäckel potential to local region a given orbit probes and estimate actions in this potential (Sanders 2012) Torus machine Construct torus of given actions by finding generating function from toy Hamiltonians Designed for (J,Θ) to (x,v) (McMillan & Binney 2008) Stäckel Fudge Use equations for actions in a Stäckel potential assume potential is locally separable in ellipsoidal coordinates (Binney, 2012, [axisymmetric] and Sanders & Binney, 2014, submitted [triaxial]) Adiabatic approximation Assume potential is separable in polar coordinates (Schönrich & Binney, 2012) 19 Generating Function Construct generating function from time samples from orbit integration, designed for (x,v) to (J,Θ) (Sanders & Binney 2014) Iterative torus construction Iteratively find (J,Θ) from non-convergent procedure and construct torus with these actions J

20 Generating Function Construct generating function from time samples from orbit integration, designed for (x,v) to (J,Θ) (Sanders & Binney 2014) Have expressions for toy angle-actions at each time sample Solve for true angle-actions 20

21 Conclusions Two complementary dynamical modelling tools vital for measuring where the dark matter in the Galaxy is. Extended DF allow us to exploit the rich chemical information that is becoming available from large spectroscopic surveys Stream modelling angle-frequency formalism is powerful Both rely on angle-action calculation there are various schemes to do this. 21