(c/a) ~ 0.6
(b/a) ~ 0.8
NFW ✓
For Milky Way mass halos…
Slide 8
Slide 8 text
what about
REAL LIFE
Slide 9
Slide 9 text
Weak lensing
Constraint on mean
projected halo ellipticity
van Uitert et al. (2012)
eh
⇡ 0.4 ± 0.25
(c/a)
proj
= 0.6 ± 0.25
Slide 10
Slide 10 text
Diego et al. (2014)
(c/a) ~ 0.55
Strong lensing
Slide 11
Slide 11 text
No content
Slide 12
Slide 12 text
Sag.
Orphan
Pal 5
Slide 13
Slide 13 text
Tidal streams are sensitive to internal progenitor
dynamics and the potential in which they form
Slide 14
Slide 14 text
Results from the Sagittarius stream:
Newberg et al. 2007
Law & Majewski 2010
Ibata et al. 2013
Deg & Widrow 2013
Ibata et al. 2001
spherical
Johnston et al. 2005
Vera-Ciro & Helmi 2013
oblate
spherical
triaxial
maybe still spherical
oblate → triaxial
triaxial
Slide 15
Slide 15 text
How do streams form?
Slide 16
Slide 16 text
rtide
⇠ f
✓
m
Menc
◆1/3
R
f ⇠ O(1)
m2 << m1
R
Slide 17
Slide 17 text
potential center
L1
L2
Slide 18
Slide 18 text
No content
Slide 19
Slide 19 text
Tidal disruption is simple
Slide 20
Slide 20 text
How should we fit
stream data?
With a
generative model
Slide 21
Slide 21 text
progenitor
Lagrange
points
progenitor orbit
potential center
Streakline
Küpper et al. (2012)
Slide 22
Slide 22 text
t = 0
Streakline
Küpper et al. (2012)
Slide 23
Slide 23 text
t = 1
Streakline
Küpper et al. (2012)
Slide 24
Slide 24 text
t = 2
Streakline
Küpper et al. (2012)
Slide 25
Slide 25 text
t = 3
Streakline
Küpper et al. (2012)
Slide 26
Slide 26 text
model stream
observed stars
Streakline
Slide 27
Slide 27 text
Streakline
Slide 28
Slide 28 text
Rewinder
Slide 29
Slide 29 text
t = 0
Price-Whelan et al. (2014)
Rewinder
Slide 30
Slide 30 text
Rewinder
Price-Whelan et al. (2014)
t = -1
evaluate
likelihood
Slide 31
Slide 31 text
Rewinder
Price-Whelan et al. (2014)
t = -2
evaluate
likelihood
Slide 32
Slide 32 text
Rewinder
Price-Whelan et al. (2014)
t = -3
Slide 33
Slide 33 text
ith star
orbit
progenitor
orbit
potential
unbinding
time
velocity
dispersion
~tidal
radius
leading/
trailing
Price-Whelan et al. (2014)
p(wi
| wp, ✓p, ⌧i, )
progenitor
internals
{N(ri
| rp
± rtide
ˆ
rp, r)|⌧i
N(vi
| vp, v)|⌧i
Slide 34
Slide 34 text
⌧ub
K
unbinding time
leading/trailing tail
M mass today
any parametrization
per star
progenitor
potential
marginalize out
Rewinder
model parameters
Slide 35
Slide 35 text
No content
Slide 36
Slide 36 text
No content
Slide 37
Slide 37 text
⌧ub
K
unbinding time
leading/trailing tail
M mass today
any parametrization
per star
progenitor
potential
(l, b, d, µl, µb, vr)
(l, b, d, µl, µb, vr)
marginalize out
WE’RE
HOSED
Rewinder
Price-Whelan et al. (2014)
Slide 38
Slide 38 text
Nparams
/ 6Nstars
Slide 39
Slide 39 text
50 −100 −50 0 50
X [kpc]
−100 −50 0 50
X [kpc]
x
z
y
Potential:
Miyamoto-Nagai disk
+
Hernquist spheroid
+
Triaxial, log. halo
(q1, qz, , vh, rh)
Price-Whelan et al. (2014)
Slide 40
Slide 40 text
−100 −50 0 50
X [kpc]
−100 −50 0
X [kpc]
x
“Data”:
Price-Whelan et al. (2014)
8 “RR Lyrae” stars
Gaia velocity errors
2% distance errors
+ Progenitor, same errors
Price-Whelan et al. (2014)
8 RR Lyrae stars
2% distance errors
No proper motions
Slide 43
Slide 43 text
Price-Whelan et al. (2014)
Recover unobserved
proper motion for stars
Slide 44
Slide 44 text
Next
Marginalize true phase-space positions of the stars
Marginal likelihood has fixed dimensionality set by
potential params., progenitor params
Price-Whelan et al. (in prep.)
Slide 45
Slide 45 text
Nparams
/ 6Nstars
Good:
Bad:
- test particle orbits (no N-body)
- arbitrary potentials
- observational uncertainties / missing data
- less sensitive to observational biases
Slide 46
Slide 46 text
Tidal streams are sensitive to the
Milky Way potential at large scales
When modeling streams, must use a
probabilistic model, e.g. Rewinder
SMHASH will measure precise distances
to stars in Sgr & Orphan streams
Slide 47
Slide 47 text
Chaos and tidal
streams
2)
Slide 48
Slide 48 text
Mean density of streams evolves faster relative to
spherical/oblate
Chaos may become important, mixing can reduce
density in streams
In triaxial potentials:
Slide 49
Slide 49 text
For regular orbits:
˙
J =
@H
@✓
= 0
˙
✓ =
@H
@J
= ⌦
(
x
,
v
) ! (
✓
,
J
)
Slide 50
Slide 50 text
Angle-action coords.
Slide 51
Slide 51 text
Triaxial / 3 dof = 6D phase space = 3D torus
embedded in
Regular orbit:
Slide 52
Slide 52 text
x
y
px
py
Slide 53
Slide 53 text
θ1
θ2
J1
J2
Slide 54
Slide 54 text
⌦2/⌦1
θ1
θ2
J1
J2
Slide 55
Slide 55 text
x
=
X
k
ak e
i!kt
!k = nk ⌦1 + mk ⌦2 + lk ⌦3
⌦i = Fi(J1, J2, J3)
Slide 56
Slide 56 text
NAFF
FFT orbit
Convolve with Hanning filter
Find and subtract strongest component
[repeat]
Laskar (1990)
Valluri & Merritt (1998)
(Numerical Approximation of Fundamental Frequencies)
Slide 57
Slide 57 text
1
2
⌦1
⌦2
Frequency Diffusion
Chaotic
orbit
Regular
orbit
Slide 58
Slide 58 text
Regular Chaotic
Frequency Diffusion
Price-Whelan et al. (2015)
Slide 59
Slide 59 text
Regular Chaotic
Slide 60
Slide 60 text
Frequency diffusion rate measures the
magnitude of chaos for a given orbit
x
z
Price-Whelan et al. (2015)
short axis
tubes
Intermediate
axis
long axis
tubes
box
Slide 65
Slide 65 text
How does chaos effect
ensembles of orbits?
Slide 66
Slide 66 text
Triaxial / 3 dof = 6D phase space = 3D torus
Regular orbit:
Triaxial / 3 dof = 6D phase space = 5D energy surface
Chaotic orbit:
Slide 67
Slide 67 text
Chaotically mixed ➞ uniform on energy surface
⇢E0
(x) /
Z
d3v (H E0)
/
p
E0 (
x
)
Price-Whelan et al. (2015)
(projected into configuration space)
Slide 68
Slide 68 text
Globular cluster-like “debris”
Ensembles
E
⇡ 0.5% E0
Start at pericenter, evolve for 25 periods
Slide 69
Slide 69 text
“Distance” to fully-mixed state
DKL(
t
) =
Z
d
x ⇢
(
x, t
) ln ⇢
(
x, t
)
⇢E0
⇡
N
X
i
ln ⇢
(
xi, tj)
⇢E0
Price-Whelan et al. (2015)
Slide 70
Slide 70 text
Price-Whelan et al. (2015)
DKL
Time
Regular
Chaotic
}
Slide 71
Slide 71 text
Price-Whelan et al. (2015)
Slide 72
Slide 72 text
x
z
Price-Whelan et al. (2015)
Slide 73
Slide 73 text
What does chaos do to the
observability of streams?
Slide 74
Slide 74 text
x
z
Slide 75
Slide 75 text
No content
Slide 76
Slide 76 text
Price-Whelan et al. (2015)
Regular Chaotic
Slide 77
Slide 77 text
Chaos is generic to triaxial potentials
Frequency diffusion seems to be predictive of
observables, e.g., stream morphology
More realistic potentials will be time-
dependent, lumpy, multi-component, all of
which may enhance chaos
The mere existence of thin streams around the
Milky Way places constraints on the potential