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

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 Stäckel potential (de Zeeuw 1985), but not in general. What’s the catch? * Contains Kepler potential as special case.

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

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

Want to convert between this:

And this:

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

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 J R – extent of radial oscillation (0 is circular or shell, -> ∞ is -> escape) J z – extent of vertical oscillation (0 is in plane, -> ∞ is -> escape (again)) J φ – angular momentum about symmetry axis (can take either sign). N.B. labels are pretty arbitrary – J R could equally be called J r, J u etc.

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).

A couple of useful rules of thumb 1. A useful sense of the velocities of the orbit: ≈ Ω i J i For example, consider a circular orbit: J φ = R v φ and Ω φ = v φ /R So Ω φ J φ = v φ 2 2. Surfaces of constant energy in action space are approximately triangles

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 = S x (x,J) + S y (y,J) Both sides must be functions of J only - K 2 (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)

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

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

Stäckel potentials One can go into exceptional depths in the discussion of Stäckel potentials (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

Stäckel potentials The requirement that these equations separate forces us to potentials of the form This gives us u & p u only v & p v only This implies, again, that both the right and left hand sides must be constant for a given orbit. For convenience, this is 2Δ 2 I 3 This then gives us E, L z and I 3 as integrals of motion (known analytically from u, v, p u, p v and the potential). We can convert these to actions with 1D integrals

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. Torus modelling

isochrone

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,θ are canonical coords, the PI is zero =0, clearly, for J=const *stated without proof. Again, see Binney & Tremaine Appendix D U

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, θ behave 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

So we can do the following transformation: But while surfaces of constant J T are null, they are not at constant H true (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… Actions and angles in toy potential (isochrone)Hamiltonian in true potential, ½|v| 2 + φ true (x)

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 θ T 3. It needs to ensure that in the end, H true 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 H true const doesn’t exist for a given J. In that case we can think of tori fit close to constant H true as defining a new Hamiltonian close to the real one.

Recall that Poincaré’s invariant is constant as you switch between canonical coordinates 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 Preserving nullness So we just to ensure that we’re transforming between J T, θ T and new canonical coordinates J, θ

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

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.

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

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

The code you’ll be working with is for loop orbits in a static axisymmetic potential (symmetric about z=0). L z 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. The 2D case (current state of the art) Where all vectors are 2D (R & z components) n R ≥ 0 n z evenn ≠ (0,0)

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

Most obvious possibility: find S n (J) and S n (J+ε) In practice this is too noisy, we need another approach… 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)

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 What about the angles? If we integrate an orbit (in the usual way) from x,v known 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 Known Unknown K1n K*M

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

Fit such that J=const is an orbital torus first Then find dS n /dJ on this torus What if I can’t fit torus into H in this way? 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