Torus modelling summary

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?

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

Because of periodicity, can’t have JRT > 0 without also having (impossible) JRT < 0 elsewhere

Solid: Orbit DoSed: constant r

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

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

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).

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

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)

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

V

W

Velocity distributions at Solar radius in the plane (left) and 1.5 kpc above the plane (right)

(blue – AA as quoted; red – with centrifugal term correction)

U

V

W

Comparing integrated orbit to AA Integrated orbit in Miyamoto-‐‑Nagai potential (curved line) Outlines: Prediction from AA (straight solid line) Tweaked AA (doSed 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 aSack it at the same angle I.e. not this (except at z=0)

Instead this, which AA can’t reflect

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

Tilt in model at R=R0 (assessed with tori)

Dots – real, curve -‐‑ AA

Tilt in MW from RAVE data (BurneS thesis)

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. 1.  Stäckel fiSing – Sanders 2012 2.  Stäckel “fudge” – Binney 2012

2. Stäckel approximation -‐‑ fiSing 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 aSempt 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)

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 wriSen So if our potential is ~ of Stäckel form, then we have Nearly independent of v Nearly independent of u

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 beSer approximation

Actions determined for various points on the same orbit – should be constant.

Stäckel AA

Also, this technique can be used to produce a table that you can interpolate in, making the process very quick

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

Any occasion on which you wish to describe regular orbits

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 (Ponwen & Governato 2013) Structure of tidal streams (which do not lie on 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

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

When in doubt, start separable

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


~Vertical distribution at Jφ


~Vertical distribution at Jφ

~vR,vφ distribution at Jφ

When in doubt, start separable, and from a Gaussian

Common to approximate local velocity distribution as


Or, beSer:

(Shu 1969, see also Dehnen 1999)

(Spiwer 1942)


Or, beSer:


(Spiwer 1942)


Or, beSer:


(Spiwer 1942)

In 1D case, ΩzJz = , so possibly:

?


? The problem is that as Ji -‐‑> ∞, Ωi -‐‑> 0, and ΩiJi -‐‑> 0, so fi -‐‑> const Instead – use epicycle frequencies κ(Jφ), ν(Jφ)

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

PuSing it together

PuSing it together

PuSing it together

Accounts for change of variable J, and factors of 2π

Prevents Jφ, -‐‑Jφ symmetry – give net rotation ~Exponential disc in R

What can we do with that? 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)

Works quite well, but…

Would be improved by a systematic shift in vφ

Works quite well, but…

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)