Angles and actions

1. Paul McMillan Angles and actions So why isn’t everyone using them?

2. In 3D we can only find them analytically for the isochrone potential* And 3D harmonic oscillator Can find them with 1D numerical integrals for any spherical potential or the Stäckelpotential (de Zeeuw 1985), but not in general. What’s the catch? * Contains Keplerpotential as special case.

3. I’m going to go through: What they are. How to find them for a very simple case Approximations for more general potentials. So, problem 1, how do we get them?* *Problem 2 is “How can we use them?”, and we’ll get to that.

4. What are angles and actions? (a little reminder) As usual, we have coordinates and momenta, generically labelled (q,p). Take special momenta J to be integrals of motion (i.e. constant), with conjugate coordinate θ. We know: Hamilton’s equations, J constant Therefore clearly H = H(J). So the other one of Hamilton’s equations is Actions tell you what orbit you’re on, angles tell you where on that orbit

5. Want to convert between this:

6. And this:

7. What are these actions? For a path where the ith component of θ increases by 2π while J stays constant (labelled γi): Because J and θ are canonical coordinates, this actually means* that for any canonical coordsp, q So, for example, consider a loop orbit in an axisymmetric potential. What’s Jφ? pφ= Rvφ= Lz, and clearly γi is a path φ:0->2π And JR? Look at the surface of section… *stated without proof. See Binney & Tremaine Appendix D

8. What are these actions? Clearly the nature of the potential & orbits changes nature of the actions Box orbits & loop orbits clearly differ – something like angular momentum will be one action for loops, not for boxes. In the case of loop orbits in an axisymmetric potential JR – extent of radial oscillation (0 is circular or shell, -> ∞ is -> escape) Jz – extent of vertical oscillation (0 is in plane, -> ∞ is -> escape (again)) Jφ – angular momentum about symmetry axis (can take either sign). N.B. labels are pretty arbitrary – JR could equally be called Jr,Juetc.

9. Geography of action space Description in actions & angles encourages one to think of orbits more generally than “the path followed by an object in the potential” Given enough time, an orbit will (generally) densely fill a volume in space We can think of an orbital torus, labelled J, as a solid object with defined velocities at each physical point Notice: Density -> ∞ at orbit edges (because velocity in some direction -> 0).

10. A couple of useful rules of thumb 1. A useful sense of the velocities of the orbit: <vi2> ≈ ΩiJi For example, consider a circular orbit: Jφ = R vφ and Ωφ = vφ/R So ΩφJφ= vφ2 2. Surfaces of constant energy in action space are approximately triangles

11. Example: Harmonic oscillator Nice simple potential, force in x/y direction only depends on x/y We can think of this in terms of a generating function S, such that Hamilton-Jacobi equation, which we split by saying that S = Sx(x,J) + Sy(y,J) Both sides must be functions of J only - K2(J) - otherwise we could change x or y at fixed J (and thus fixed E) and only one side would change. We therefore have separate equations to solve in x & y (Binney & Tremaine 2008 §3.5.1)

12. Example: Harmonic oscillator (continued) So we can put together two pieces of information to yield Jx or Jy as 1D integrals In this case we can do these integrals analytically (BT08) to yield And similar for Jy, so in fact the Hamiltonian can be written as (Frequencies are ωx & ωyas you would expect) We therefore know S(x,J), and can find θ

13. Separability Notice that what we did there was possible because we can separate out the relevant equations into x & y components. As you can imagine, we can do similar for spherically symmetric potentials, using r, θ, φ coordinates. In that case we get an analytic solution for the isochrone potential Non-spherically symmetric potentials? The only other available situations where these equations separate are the Stäckel potentials where they are separable in confocal ellipsoidal coords

14. Stäckel potentials One can go into exceptional depths in the discussion of Stäckelpotentials (if you don’t believe me, see de Zeeuw 1985) N.B. I do not want or expect you to grasp the details of this. Just get the rough idea. The description is far from rigorous. Restricting ourselves to the axisymmetric case, we can use an ellipsoidal coordinate system in the R-z plane Lines of constant v – hyperbola Lines of constant u – ellipses Solid line – orbits in Galactic (not Stäckel) potential – note they are ~bounded by lines of constant u,v

15. Stäckel potentials The requirement that these equations separate forces us to potentials of the form This gives us u & pu only v & pv only This implies, again, that both the right and left hand sides must be constant for a given orbit. For convenience, this is 2Δ2 I3 This then gives us E, Lz and I3 as integrals of motion (known analytically from u, v, pu, pv and the potential). We can convert these to actions with 1D integrals

16. Torus modelling Basic idea: 1. Take null tori from the isochrone 2. Warp them to fit the potential of interest 3. They are now tori in the new potential Job done.

17. isochrone

18. A null torus at constant H is an orbit What do you mean the torus is “null”? Null in the sense that Poincaré’s invariant is zero on its surface: “Invariant” – in this case under canonical transform*, including that to get actions, angles So, clearly, if J,θare canonical coords, the PI is zero U =0, clearly, for J=const *stated without proof. Again, see Binney & Tremaine Appendix D

19. A null torus at constant H is an orbit So if we have a surface J=const at const H, clearly we have, as required: Equally, since there are tori at neighbouring J for which H is also independent of θ: So J, θbehave exactly* as we need them to – this is an orbit. *I haven’t shown that θ is 2π periodic, but this can be done by rescaling

20. So we can do the following transformation: Actions and angles in toy potential (isochrone) Hamiltonian in true potential, ½|v|2 + φtrue(x) But while surfaces of constant JT are null, they are not at constant Htrue (unless it’s the same isochrone potential, in which case why are you doing this?), so they’re not orbits in the true potential. By itself, this is useless. But it is a stepping stone. To do anything useful we’ll need another step…

21. This step has to fulfil a number of criteria if this is going to work 1. It needs to ensure the tori are still null 2. It needs to ensure that everything is 2π-periodic wrt both θ and θT 3. It needs to ensure that in the end, Htrue is constant for a given J (*) * In practice, of course, we accept a numerical approximation that it is almost constant ** ** In some cases (near resonances) a torus with Htrueconst doesn’t exist for a given J. In that case we can think of tori fit close to constant Htrue as defining a new Hamiltonian close to the real one.

22. Preserving nullness Recall that Poincaré’s invariant is constant as you switch between canonical coordinates So we just to ensure that we’re transforming between JT, θT and new canonical coordinates J, θ Reminder: We have canonical coordinates (q,p) and we want to know the transform into new canonical coordinates (Q,P). A generating function S(q,P) allows us to do this with

23. So we have two equations: Clearly the 2π-periodicity in θT must be maintained, so I.e. S must be periodic in θT(plus a linear term – derivative still periodic) Write as Fourier series First term ensures θ also 2π-periodic; n ≠ 0 ensures θ = 0 where θT = 0 (McGill & Binney 1990)

24. S So, using S we’ve ensured that surfaces of constant J still have all the properties of an orbital torus, and given ourselves the ability to manipulate them. All we need to do is manipulate them such that H is constant for a surface at constant J.

25. Means that (from definition of S) Finding these is possible in practice on an orbit by orbit basis. So for a given value J’, we find the various Sn(J’) such that H is constant Note that this won’t constrain the relation of θ to θT directly, we need to find the various dSn/dJi|J’ separately (see later).

26. Periodicity gives constraints, so we have 1D example - actions only

27. The 2D case (current state of the art) The code you’ll be working with is for loop orbits in a static axisymmeticpotential (symmetric about z=0). Lz is one of the actions, so the problem reduces to a 2D one in the R-z plane. We have further constraints on the generating function from the requirement that J is real, and from symmetry with t and z. Where all vectors are 2D (R & z components) nR ≥ 0 nz even n ≠ (0,0)

28. Scheme (ignoring angles): GM, b, Lz,iso Sn 1. Pick J. 2. Guess isochroneparameters & take Sn=0 for all n. 3. At this value J, take a 2D grid of points in θT 4. Minimise the sum over this grid Derivative wrt parameters known via chain rule. T T

29. What about the angles? We’re only halfway there. The surface J=const is now a orbital torus. We know the range in x the orbit covers and what v it has at each point Known for given J BUT, we don’t know θ. We can’t follow an orbit, or know the density of an orbital torus any point. Not known (yet) Most obvious possibility: find Sn(J) and Sn(J+ε) In practice this is too noisy, we need another approach…

30. What about the angles? I would not claim we have a perfect scheme, however, best thus far: Exploit the fact that we already know x,v for the whole orbit, and know Known K*M K 1 n Unknown If we integrate an orbit (in the usual way) from x,vknown on the torus, then we have a set of simultaneous equations, each equation at a given t. In practice - easier to cover all θ if we start from many initial x,v K starting points, and take M points from each integrated orbit

31. So we integrate a spaced out set of orbits, and find θT along each. Solve the simultaneous equations to find θ, which should act like this: Toy angles True angles

32. Torus modelling summary We can find the complete orbit associated with a given J, but we have to go through intermediate steps: isochrone S and its derivatives wrt J Fit such that J=const is an orbital torus (H=const) first Then find dSn/dJ on this torus What if I can’t fit torus into H in this way?

33. Sometimes life’s a bit harder For example, consider an orbit with JR = 0 Generally this orbit will look like that on the right, single line in Rz-plane, not at r=const In isochrone (or any spherical potential) JRT = 0 is r=const, so our orbit would require JRT≠ 0 Solid: Orbit Dotted: constant r Because of periodicity, can’t have JRT > 0 without also having (impossible) JRT< 0 elsewhere

34. What’s the fix? The fix comes from recalling that the relevant integrals stay constant when change between canonical coordinates This means that we can apply a “point transform” that alters what we mean by the coordinates, such that the line r=const is mapped to the relevant Jr=0 orbit (worked out by integrating an orbit in the true potential) (Kaasalainen & Binney 1994) Note also that these point transforms are needed to deal with the minor orbit families (granddaughter orbits) (Kaasalainen 1995) Point transform

35. What do we want? Torus modelling takes a value J and outputs the relevant orbitx,v If you want to know about orbit with another J (or in another Φ) you have to start again from scratch. What if we know x,v& want to know J? Guess J, see if you’re close, iterate towards truth. (McMillan & Binney 2008) Slow, not obvious how to iterate towards solution, not guaranteed to get to a solution at all

36. Finding J given x,v An exciting (though to date untested) possibility is interpolation between tori We describe each torus with the parameters of the toy Φ and values Sn. Interpolate these parameters between a grid of tori, and we know the point x,vassociated with arbitrary J,θwhich we can hunt within. This has been used in a carefully controlled way to interpolate “over” a resonance in J space (the resonance can then be described using perturbation theory). Not tested for this work (I have some concerns).

37. Finding J given x,v: other approximations Given this problem, it’s very useful to have some approximations conveniently available. Two that I want to talk about. 1. Adiabatic approximation (decouple motion in R & z) 2. Stäckel approximation

38. 1. Adiabatic approximation Decouple radial & vertical motion Referred to as the “adiabatic approximation” because we assume that the vertical oscillations are much faster than the radial ones, so Jz calculated in this -> potential is conserved Also calculate radial action assuming radial motion in this effective potential Therefore given R & vRwe calculate JR as 1D integral, and given R, z & vz calculate Jz as another 1D integral. Detail – we can improve this term (Binney & McMillan 2011, Schönrich & Binney 2012)

39. It works OK – model evaluated with tori (black) and AA (blue/red) U Velocity distributions at Solar radius in the plane (left) and 1.5 kpc above the plane (right) U V V W W (blue – AA as quoted; red – with centrifugal term correction)

40. Comparing integrated orbit to AA Integrated orbit in Miyamoto-Nagai potential (curved line) Outlines: Prediction from AA (straight solid line) Tweaked AA (dotted lines) N.B. Values of Jz & Ez found from corner points of integrated orbits shown. Jzmuch closer to being conserved. But note an asymmetry in the integrated orbit – the two paths that pass through a given point don’t always attack it at the same angle I.e. not this (except at z=0) Instead this, which AA can’t reflect

41. This problem is reflected in surfaces of section at a given R (dR/dt>0) – note symmetry of AA This also means that the tilt of the velocity ellipsoid will always be zero for any model assessed by AA Dots – real, curve - AA Tilt in model at R=R0 (assessed with tori) Tilt in MW from RAVE data (Burnett thesis)

42. 2. Stäckelapproximation Recall that in a Stäckel potential, equations of motion are separable in ellipsoidal coordinates u,v – the velocity ellipsoid will have a tilt if we use a Stäckel potential to approximate. As suggested by the fact that real orbits are ~ bounded by constu,v, this is a reasonable approx to real potentials Two recent works with two different approaches to using this. Stäckel fitting – Sanders 2012 Stäckel “fudge” – Binney 2012

43. 2. Stäckelapproximation - fitting The simpler version to understand Integrate orbit in true potential, then fit a Stäckel potential Φs over relevant volume. Note that we do this fit individually for each orbit – we don’t attempt to do it for the whole potential at once. Recall that once we have this, we have 3 integrals of motion (E, Lz, I3) in Φs, and can find J in Φs with 1D integrals. Since Φs is close to the true Φ, these are close to the true actions. (Sanders 2012)

44. 2. Stäckelapproximation - fudge (Not my name for it – Binney calls it that) The idea here is broadly similar – the potential relevant for disc orbits is similar to a Stäckel potential. Here we pick the ellipsoidal coordinates (i.e. the value of Δ) in advance. Recall that the potential can be written So if our potential is ~ of Stäckel form, then we have Nearly independent of v Nearly independent of u

45. Using these we end up with equations for pu which ~ only depend on u, and similar for v So, again, we can do 1D integrals that get out the actions Only now we’re approximating that motion is separable in these ellipsoidal coords, not R, z – this is a better approximation Stäckel Also, this technique can be used to produce a table that you can interpolate in, making the process very quick Actions determined for various points on the same orbit – should be constant. AA

46. Summary The joys of actions & angles are available in reasonably sensible Galactic potentials But only through approximations Torus modelling is the most rigorous and complete approximation scheme, but it has the limitation that it takes you from J,θ to x,v and not the other way round (for now?). Alternative methods that use simpler assumptions (to ~trivially separable equations) can take you from x,v to J,θ

47. Uses

48. Any occasion on which you wish to describe regular orbits

49. Some examples of where they already have been used Modelling of discs Distribution function for discs has been suggested - limiting your freedom to treat (e.g.) vR & vφ independently. Modelling of stars trapped by resonances – explaining part of the local velocity distribution Using assumption that f(x,v) = f(J) to constrain Galactic potential Radial migration of stars – change in Jφ with tiny change in JR (Sellwood & Binney 2002) Halo NFW/Einasto profile of simulated CDM halos may be explained in terms of conserving average actions (Pontzen & Governato 2013) Structure of tidal streams (which do not lie on orbits).

50. Jeans’ theorum Any steady state solution distribution function f(x,v) in a given potential depends on x,v only through the integrals of motion (in this case actions). Makes sense if you think in terms of orbits: J doesn’t change. θ increases at rate independent of θ, Ω(J), so a uniform distribution stays uniform. Binney & Tremaine 2008 §4.2