Of course, discs aren’t actually axisymmetric This has consequences for the velocity distribution seen locally A df made from quasi-isothermals produces a smooth velocity distribution in the solar neighbourhood

Of course, discs aren’t actually axisymmetric This has consequences for the velocity distribution seen locally A df made from quasi-isothermals produces a smooth velocity distribution in the solar neighbourhood In reality it’s far from smooth

Of course, discs aren’t actually axisymmetric This has consequences for the velocity distribution seen locally A df made from quasi-isothermals produces a smooth velocity distribution in the solar neighbourhood In reality it’s far from smooth Hercules Hyades Many, many attempts to explain these overdensities with reference to interaction with bars or spirals

Inner & Outer Lindblad resonances There’s a perturbation rotating with pattern speed Ω p – what are the resonances? If the perturbation is m-fold symmetric, then a star feels a force that has frequency |m(Ω p - Ω φ )|, so resonances are of the form For l=0, Ω p = Ω φ so clearly this is co-rotation For l=+1, Ω p > Ω φ, so star is outside co-rotation (OLR) For l=-1, Ω p < Ω φ, so star is inside co-rotation (ILR) N.B. the more usual form replaces Ω R with κ, which is in the limit of small J R

Lindblad resonances in the Solar neighbourhood Hercules has been associated with an OLR with the bar (Dehnen 2000, Antoja et al 2013) I’m going to focus on the Hyades, and look at this in angles & actions Hercules Hyades

Lindblad resonances in the Solar neighbourhood Hercules has been associated with an OLR with the bar (Dehnen 2000, Antoja et al 2013) I’m going to focus on the Hyades, and look at this in angles & actions Actions first. Dotted lines show possible I/OLRs (changing Ω p shifts left or right on this plot) Hyades sticks out like a sore thumb. q-iso real

q-iso real Lindblad resonances in the Solar neighbourhood Hercules has been associated with an OLR with the bar (Dehnen 2000, Antoja et al 2013) I’m going to focus on the Hyades, and look at this in angles & actions Actions first. Dotted lines show possible I/OLRs (changing Ω p shifts left or right on this plot) Hyades sticks out like a sore thumb.

This sort of thing is seen in simulations (Sellwood 2012) In carefully controlled simulations Sellwood found trapping of particles (at an ILR) that looks rather like the Hyades in action space. What about angles? We expect a similar trap – stars librate about a combination of the angles before after

It’s clear that the Hyades stars are not evenly distributed in angle If you reverse the sign of v R and v z, J is unchanged, so if the Hyades was just associated with an f(J), then it would look the same for v R >0 and v R <0 It doesn’t.

The problem – selection effects These stars are all in the Solar neighbourhood (within ~100pc). The range in θ φ is small and it is highly correlated with θ R θ φ is ~ φ of guiding centre (epicycle approx) If star is at apo/pericentre, θ φ ≈ φ ≈ 0 The difference between θ φ and φ increases towards θ R = ±π/2, by an amount that depends on J R

So if you simply look for an overdensity in lθ R +mθ φ … Or if you try to build models f(J,θ), using tori, of each type to fit…

But further away…

Finding the gravitational potential A very standard way of finding Φ is to say that the system must be in equilibrium. Therefore df is f(integrals) Choose a form for f(integrals), and fit to data in a given Φ. Repeat for many Φ. Best Φ wins! E.g. Schwartzchild’s method. Integrate many orbits in a given potential until they’re “well sampled” – effectively f(J) Weight each orbit individually ( * ) and find combination of non-negative weights that best fits your data. * Or in “bundles” of similar orbits

Milky Way data is not like external Galaxy data It’s much, much richer. Typically l, b, distance (with some accuracy, in parallax or distance modulus), μ, v los, magnitude, colour(s), [Fe/H], [α/Fe], other chemicals… These are often very different from quantities we can associate with a Galactic model (e.g. v R, v z ), which can cause intense confusion with uncertainties (e.g. what do you do if the measured parallax is negative?) The high dimensionality makes binning impractical (if you have n bins in each of the d dimensions, you have n d bins, and very few stars per bin.

It is (moderately) well known that in the limits of perfect data, Schwarzschild modelling fails for two reasons. Both are due on the fact that there’s a one-to-one relationship between data points and orbits. 1)DF is a set of individually weighted delta functions in J. Can build a perfect model in any ( * ) Φ by putting an orbit through each data point. It’s no better than any other model – no Φ information. 2)If you place orbits independently of the data, none will go through any data points. Using tori rather than integrated orbits? 1)DF is smooth f(J). Not all Φ equally likely (at the cost of possible biases). 2)Torus description gives all possible (x,v) reached by orbit with actions J, not just those reached (and stored) within integration time. Won’t be good enough for exact data, but Milky Way-esque?

What does this calculation look like? Set of n α stars with observed quantities u α. Maximise: Do as an integral over observable space The observables are very precise l,b, pretty precise magnitude (will stick to one for simplicity), and μ, ϖ and v los with some precision (or not at all…) So integral reduces to one down the line-of-sight [i.e. P(l,b) = δ(l-l α )δ(b-b α )]. Other terms in P(u’|u α, σ α ) straightforward. P(u’|Model) is trickier.

We have: DF f(x,v) = f(J) Luminosity function F(M) Selection effectsS(u’) With M absolute magnitude. If a star with true observables u’ exists, there is a probability S(u’) that it enters the catalogue. P(u’|Model) NormalisationChange of variables P(x,v,M|Model)

So we have n α +1 integrals to perform Big benefit of action-angle variables: Tori are δ-functions in J, and this is a high dimensional integral, so perform as Monte-Carlo integral. Common example is area of a shape More generally, can find integrals with “importance sampling”. Sample points from known pdf p, find integral of f

Divide up the integral Looks ugly (change of variable again), but can be found given known J, l, b,s’ Integral over actions. Replace with Monte-Carlo “A” can be found from the same set of J k used to do this integral

So we can do the integrals to find Where our model is f(J) in a potential Φ. In each potential, maximise L by varying f(J), the true potential should have the highest value of L… Nope

What’s gone wrong? Numerical noise associated with evaluating the likelihood this way has overwhelmed the signal (Error bars purely numerical)

When we change potential, the orbit library changes. We can keep the values of J k the same, but they will correspond to different x,v. So, since the data are so precise, it is easy for observations to “slip between the cracks” in one case but not the other What’s gone wrong? Tori J k used to evaluate Likelihood Observation (with uncertainties)

When we change potential, the orbit library changes. We can keep the values of J k the same, but they will correspond to different x,v. So, since the data are so precise, it is easy for observations to “slip between the cracks” in one case but not the other What’s gone wrong? If we change potential, the tori shift in x,v and the observation doesn’t, so the calculated likelihood can fall to zero, just because our observation no longer hits any of our tori

When we change potential, the orbit library changes. We can keep the values of J k the same, but they will correspond to different x,v. So, since the data are so precise, it is easy for observations to “slip between the cracks” in one case but not the other What’s gone wrong? Even in less extreme cases, the number of tori that go through the observation changes, which causes numerical noise that ruins the calculation

How can we fix this? The schematic diagrams suggest that what we need to do is hold the points where we evaluate the integral fixed w.r.t. the observations. To do that we need to evaluate J(x,v). This is why I went on about the need for approximations that can do this. Need to slightly reassess how we do this integral More tori in the library used to evaluate the integral? It would take ~ 10 9, so no.

That integral again At it’s simplest, for each star we sample a set of points u’ k,α from the pdf P(u’|u α,σ α ). We then evaluate this integral as Since we know J(x,v), we know f(x,v). Recall: r 6 cos b Normalisation, A, is all that’s left to work out

Normalising P(u’|Model) Normalisation constant Rewrite as Monte Carlo integral Sampling density Note that L is proportional to A Nα – so this matters a lot We care about the ratio of likelihoods in different Φ, so we care about the ratio of the values A in these potentials. So, once again, a key element of doing this right is fixing the values x k, v k at which you evaluate A in all of the Φ

Do all of this, and it works! Again, error bars are numerical uncertainties J(x,v ) Torus library Compare to previous:

Real data Analysis of almost exactly this kind has been done for stars observed by Segue Divide up the observed stars by [α/Fe], [Fe/H] and fit each one separately Result: vertical force as a function of R (Bovy & Rix 2013)

More and more streams are being found around the Milky Way, associated with disrupted clusters or satellite galaxies. It is very common to try to use them to learn about the MW potential. It is tempting to imagine that these streams lie very close to an individual orbit, and try to find the orbit under that approximation… Streams (This section – Eyre & Binney 2011; Sanders & Binney 2013, both papers)

Except… The objects in the stream all started from ~the same point. The fact they’re not on the same orbit is why there’s a stream! The progenitor can be thought of as having actions J 0 and being on a orbit at angle θ 0 (t) = θ 0 (0)+Ω 0 t How do the values J, θ of the stars in the stream relate to J 0, θ 0 ?

Let’s assume we have stars stripped from a satellite and then evolving freely in the Galactic potential. Their actions remain constant after they’re stripped, differing* from those of the progenitor by ΔJ, and with frequencies Ω 0 +ΔΩ. So, the difference in angle is And the 2nd term (the spread at disruption) is negligible (again, because otherwise it wouldn’t be a stream) So what’s ΔΩ? *Note that fitting a single orbit assumes ΔJ -> 0. That approximation is ok by itself. It’s the approximation about the angle distribution that causes trouble.

We know from our original definitions that ΔΩ? So, expanding to first order in ΔJ around Ω 0 So we have Single orbit approximation

What do we require (per this formalism) for a thin stream to form? The RHS is matrix × vector, and D has orthogonal eigenvectors ê 1, ê 2, ê 3 & related eigenvalues λ 1, λ 2, λ 3. If λ 1 >> λ 2, λ 3 then Δθ will come close to being aligned with ê 1, with a much smaller spread in the ê 2, ê 3 directions That is roughly what is found for realistic Galactic potentials. Typical values for λ 1 /λ 2 are ~ 6 to 40 No one really seems to know why.

So, does this actually matter? Orbit fitting doesn’t claim to provide any information about t, so all we care about is: Are ê 1 and Ω 0 parallel? Consider angle ϕ (0 if parallel) ϕ depends on both the potential and J. For the Kepler potential, ϕ=0°, always. For the isochrone potential, ϕ~1-3° For more realistic potentials, ϕ can range up to ~40° (seems to depend loosely on how far from spherical Φ is in the vicinity of the orbit)

So, does this actually matter? Here’s a fairly typical stream near apocentre in an isochrone potential Blue line – from Hessian Black line – progenitor orbit ϕ ≈ 1.5° Position Angle Φ Even with a small value of ϕ, the error in the potential is ~20% Sanders & Binney showed that similar (or larger) effects can be expected for real Milky Way streams

So what can we do? In practice, the full analysis (Δθ ≈ D  Δ J t) is difficult to use, because we need to know Δ J, and how it varies with t (which is not the same for all stars). Simpler is just Both Δθ and ΔΩ depend on the potential. The correct potential is one in which Δθ and ΔΩ are parallel vectors for all stars.

A simple(ish) algorithm Pick a trial potential Φ Find θ Φ,i, Ω Φ,i for each star i Fit straight lines to e.g. θ R against θ φ and Ω R against Ω φ Compare gradients of these lines True potential minimises difference of these gradients