Robyn Sanderson
NSF Postdoctoral Fellow
Columbia University/NYU

Review  of  ac/on-­‐angle  variables
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Review  of  ac/on-­‐angle  variables
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

The  accreted  stellar  halo  is  clumpy  in  ac/on  space
View  in  Galac/c  coordinates
View  in  ac/on  space
(using  input  poten/al)
1
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015
0
3
2
6
5 n
*
= 497198
M = 2.7x10
12 b = 8 kpc
M
ü
4
5
L z
0
H kpc
2
Myr
1 L
5

Ac/ons  are  most  clustered  in  the  correct  poten/al
J r
= p
GM
2 E
1
2
⇣
L + p
L 2 + 4 GM b
⌘ Potential parameters
Observations
Both
M true
=2.7x10
12 M
⊙
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Ac/ons  are  most  clustered  in  the  correct  poten/al
J r
= p
GM
2 E
1
2
⇣
L + p
L 2 + 4 GM b
⌘ Potential parameters
Observations
Both
M true
=2.7x10
12 M
⊙
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

The  Kullback-­‐Leibler  divergence  measures  clustering
D
KL
( p || q ) =
Z ln
✓ p ( x )
◆ p ( x ) dx q ( x )
Clumpier distribution = larger relative entropy
D
KL
= 1.1
Smoother distribution = smaller relative entropy
D
KL
= 0.4
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Tes/ng  the  ac/on-­‐clustering  method
1. Make  a  mock  halo  in  a  known  poten/al  (start  with   accreted  component  only)
2. “Observe”  halo  with  Gaia
3. Find  the  poten/al  that  maximizes  the  KLD
4. Determine  confidence  intervals  (“error  bars”)
5. Compare  the  answer  to  input  values
6. Varia/ons:  incorrect/different  func/onal  forms,
+smooth  component,  different  error  models,  etc.
For  details,  see  Sanderson,  Helmi,  &  Hogg  2015,  ApJ,  801,  98
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Errors  blur,  but  do  not  destroy  the  informa/on
Without errors With Gaia (pre-launch) errors
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015
Sanderson, Helmi, & Hogg, 2015

Results  of  tests  with  a  mock   isochrone   halo
Log
10
conditional probability
Without  errors With  pre-­‐launch  Gaia  errors
67 , 95 , 99 % confidence d
GC equivalents
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Simulated  stellar  halo  in  cosmological  poten/al
15  thin  streams  chosen  by  hand  from  tagged  Aquarius  halo  Aq-­‐A
(Cooper+2010,  Helmi+2011)
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015
Sanderson,  Hartke,  &  Helmi,  in  prep

Best-­‐fit  halo  matches  present-­‐day  mass  profile
Model:  smooth,  spherical  NFW
*Membership  info  not  used  in  fit*
Sanderson,  Hartke,  &  Helmi,  in  prep
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Best-­‐fit  halo  matches  present-­‐day  mass  profile
Model:  smooth,  spherical  NFW
*Derived  from  (r
-­‐2
,  ρ
-­‐2
)   of  DM *
Sanderson,  Hartke,  &  Helmi,  in  prep
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Follow-­‐up  observa/ons
RVs
•
Northern  Hemisphere :  WEAVE  on  William
Herschel  Telescope  (INT/IAC,  La  Palma,  Canary
Islands)
•
Southern  Hemisphere :  4MOST  on  VISTA  telescope   in  Chile  (ESO,  Cerro  Paranal,  Chile)
•
4m  telescope  with  mul/ple-­‐fiber  instrument
•
~10 6  halo  stars  (per  survey)
•
RV  errors  <2  km  s -­‐1  to  V=20-­‐22
•
Adds  RVs  to  stars  with  Gaia  proper  mo/ons
•
•
20%  photometric  distances
Distances   to  2%,  RV  errors  5-­‐10  km  s
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015
-­‐1

Improved  distances  increase  informa/on,  but   decrease  sampling
KIII giants, Gaia parallaxes RRLe distances
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Observa/ons  with  larger  average  d
GC
do  bener
Sanderson,  in  prep
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Advantages
•
No  need  to  assign  membership  of  stars  in  streams
( membership  probability  is  an  output )
•
Robust  to  adiaba/c  /me-­‐evolu/on  of  host,   substructure,  some  chao/c  diffusion
•
Can  use  any  informa/ve  parameter  (i.e.  abundances)
Disadvantages
•
Needs  6D  phase  space  info
•
Not  a  forward  model  (no  way  to  treat  uncertain/es)
•
Derailed  by  large  (>50%)  smooth  component
•
Limited  to  integrable  poten/als
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Metallici/es  as  a  4 th  dimension
[Fe/H]  vs  L z
[Fe/H]  vs  L z
L  vs  L z
J r
vs  L z
Metallicities and spreads from empirical relations of
Kirby et al.
(2011), constant
M/L
Z/Z
⊙
might be better
(e.g. Leaman
2012)
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

Metallici/es  as  a  fourth  dimension
With  [Fe/H]  (4D) max  D
KL
=  2.27
N  =  30,860
∛ N  ~  30
∜ N  ~  13
Without  [Fe/H]  (3D) max  D
KL
=  2.34
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

The  takeaway
•
Tidal  streams  are  sensi/ve  to  the  poten/al  of  the
Galac/c  dark  maner  halo
•
Streams  form  clumps  in  ac/on  space  that  s/ll  look   clumpy  at  Gaia  precision  (δπ/π  <  0.2)
•
Maximizing  D
KL
of  ac/on  distribu/on   idenCfies  the   potenCal  parameters
•
Rela/on  between  D
KL   and  posterior  probability  lets   us  set  confidence  intervals
•
Expected  number  of  MW  streams  is  enough
•
Stream  membership  info   not  required
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015

The  future
•
Metallicity/abundances  as  extra  dimensions
•
More  realis/c  poten/als:  flanening,  triaxiality
•
Halo-­‐to-­‐halo  stochas/city
•
Uses  for  the  recovered  ac/on  space
R.E.  Sanderson  •  LG  Astrosta/s/cs  •  1  June  2015