Transcript "Actions and Angles", Lecture Notes for the Gaia School Mexico

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
Hyades
In reality it’s far from smooth
Many, many attempts to explain
these overdensities with reference
to interaction with bars or spirals
Hercules
Inner & Outer Lindblad resonances
There’s a perturbation rotating with pattern speed Ωp –
what are the resonances?
If the perturbation is m-fold symmetric, then a star feels a
force that has frequency |m(Ωp - Ωφ)|, so resonances are of
the form
For l=0, Ωp = Ωφ so clearly this is co-rotation
For l=+1, Ωp > Ωφ, so star is outside co-rotation (OLR)
For l=-1, Ωp < Ωφ, so star is inside co-rotation (ILR)
N.B. the more usual form replaces ΩR with κ, which is in the limit of small JR
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
Hyades
Hercules
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 Ωp shifts left or
right on this plot)
Hyades sticks out like a
sore thumb.
real
q-iso
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 Ωp shifts left or
right on this plot)
Hyades sticks out like a
sore thumb.
real
q-iso
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.
before
What about angles? We expect a
similar trap – stars librate about a
combination of the angles
after
It’s clear that the Hyades stars are not
evenly distributed in angle
If you reverse the sign
of vR and vz, J is
unchanged, so if the
Hyades was just
associated with an f(J),
then it would look the
same for vR>0 and vR<0
It doesn’t.
The problem – selection effects
These stars are all in the Solar
neighbourhood (within ~100pc).
The range in θφ is small and it is
highly correlated with θR
θφ is ~ φ of guiding centre
(epicycle approx)
If star is at apo/pericentre,
θφ ≈ φ ≈ 0
The difference between θφ and φ
increases towards θR = ±π/2, by
an amount that depends on JR
So if you simply look
for an overdensity in
lθR+mθφ…
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, vlos, 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. vR, vz), 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 nd bins, and very few stars per
bin.
http://astrowiki.ph.surrey.ac.uk/dokuwiki/doku.php?id=start
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 nα stars with observed quantities uα. 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 vlos with some precision (or not at all…)
So integral reduces to one down the line-of-sight [i.e. P(l,b) = δ(l-lα)δ(b-bα)] . Other
terms in P(u’|uα, σα) straightforward.
P(u’|Model) is trickier.
P(u’|Model)
We have:
DF
f(x,v) = f(J)
Luminosity function F(M)
Selection effects
S(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(x,v,M|Model)
Normalisation
Change of variables
So we have nα+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
Integral over actions.
Replace with Monte-Carlo
Looks ugly (change of variable again), but can
be found given known J, l, b ,s’
“A” can be found from the
same set of Jk 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)
What’s gone wrong?
When we change potential, the orbit library changes.
We can keep the values of Jk 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
Tori Jk used to evaluate Likelihood
Observation (with uncertainties)
What’s gone wrong?
When we change potential, the orbit library changes.
We can keep the values of Jk 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
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
What’s gone wrong?
When we change potential, the orbit library changes.
We can keep the values of Jk 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
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?
More tori in the library used to evaluate the integral?
It would take ~ 109, so no.
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
That integral again
At it’s simplest, for each star we sample a set of points u’k,α from the
pdf P(u’|uα,σα). We then evaluate this integral as
Since we know J(x,v), we know f(x,v). Recall:
Normalisation, A, is all that’s
left to work out
r6 cos b
Normalising P(u’|Model)
Normalisation
constant
Rewrite as
Monte Carlo
integral
Sampling density
Note that L is proportional to ANα – 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 xk, vk at
which you evaluate A in all of the Φ
Do all of this, and it works!
J(x,v)
Again, error bars are numerical uncertainties
Compare to previous:
Torus
library
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)
Streams
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…
(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 J0 and being on a orbit at
angle θ0(t) = θ0(0)+Ω0t
How do the values J, θ of the stars in the
stream relate to J0, θ0?
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 Ω0+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.
ΔΩ?
We know from our original definitions that
So, expanding to first
order in ΔJ around Ω0
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 ê1, ê2, ê3 & related eigenvalues λ1, λ2, λ3.
If λ1 >> λ2, λ3 then Δθ will come close to being aligned with
ê1, with a much smaller spread in the ê2, ê3 directions
That is roughly what is found for realistic Galactic
potentials. Typical values for λ1/λ2 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 ê1 and Ω0 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
Position
Blue line – from
Hessian
Angle
Black line –
progenitor orbit
ϕ ≈ 1.5°
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  ΔJ t) is difficult to use,
because we need to know ΔJ, 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 θΦ,i, ΩΦ,i for each star i
Fit straight lines to e.g. θR against θφ and ΩR against Ωφ
Compare gradients of these lines
True potential minimises difference of these gradients