1. 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?

2. 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

3. 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

4. What do we want?
Torus modelling takes a value J and outputs the relevant orbit x,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

5. 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,v associated with arbitrary J,θ which 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).

6. 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

7. 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 & vR we 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)

8. It works OK – model evaluated with tori (black) and AA (blue/red)
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)

9. 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.
Jz much 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

10. 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)

11. 2. Stäckel approximation
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 const u,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

12. 2. Stäckel approximation - 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)

13. 2. Stäckel approximation - 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

14. 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

15. 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

16. Uses

17. Any occasion on which you wish to describe regular orbits
The jist – Angular momentum, Radial motion, vertical motion.

18. 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).

19. Makes sense if you think in terms of orbits:
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

20. What’s a sensible disc distribution function?
Some basic requirements:
Need f(J) -> 0 as each Ji -> ∞
Want something that ~forms a double exponential disc (constant scaleheight)
Note that stars with a non-zero Jz are at their maximum vz when they go through z=0
Therefore, a phase-mixed population all at a given Jz will have density minimum at z=0
So we need f(J) maximum as Jz -> 0
Binney 2010, 2012

21. When in doubt, start separable

22. When in doubt, start separable
~Radial density profile (R ~ Jφ/vc)

23. When in doubt, start separable
~Radial density profile (R ~ Jφ/vc)
~Vertical distribution at Jφ

24. When in doubt, start separable
~Radial density profile (R ~ Jφ/vc)
~Vertical distribution at Jφ
~vR,vφ distribution at Jφ

25. When in doubt, start separable, and from a Gaussian
Common to approximate local velocity distribution as

26. When in doubt, start separable, and from a Gaussian
Or, better:
(Shu 1969, see also Dehnen 1999)
(Spitzer 1942)

27. When in doubt, start separable, and from a Gaussian
Or, better:
(Shu 1969, see also Dehnen 1999)
(Spitzer 1942)

28. When in doubt, start separable, and from a Gaussian
Or, better:
(Shu 1969, see also Dehnen 1999)
(Spitzer 1942)
In 1D case, ΩzJz = <vz2>, so possibly: ?

29. When in doubt, start separable, and from a Gaussian?
The problem is that as Ji -> ∞, Ωi -> 0, and ΩiJi -> 0, so fi -> const
Instead – use epicycle frequencies κ(Jφ), ν(Jφ)

30. What’s σ?
Note that σ is not a velocity dispersion. It is, however, related, and has the same units.
For an isothermal sheet, the velocity dispersion required to give const. scale height is proportional to Σ½.
Therefore it is common to assume that velocity dispersion in real galaxies (in both R & z) is too. So we will take

31. Putting it together

32. Putting it together

33. Putting it together
Accounts for change of variable J, and factors of 2π
Prevents Jφ, -Jφ symmetry – give net rotation
~Exponential disc in R

34. What can we do with that? vφ local vR local ρ(z) local vφ(z) local
Add many together, with varying σ(stellar age), can fit to local kinematics and density profile vφ
local vR
local
(Binney 2010, using adiabatic approximation)
ρ(z)
local
vφ(z)
local
Also vz local (not pictured)

35. Works quite well, but…
Would be improved by a systematic shift in vφ

36. Works quite well, but… Would be improved by a systematic shift in vφ
MAPs?
Would be improved by a systematic shift in vφ
That’s the effect of correcting an incorrect peculiar Solar velocity, which it turns out is what was being used (Dehnen & Binney 1998 was wrong, see also McMillan & Binney 2010; Schönrich, Binney & Dehnen 2010)