1. Lecture Angular Momentum
Tidal-Torque Theory
Halo spin
Angular-momentum distribution within halos
Gas Condensation and Disk Formation
The AM Problem(s)
Thin disk, thick disk, bulge

3. Disk Size Spin parameter Conservation of specific angular momentum J/M

4. Tidal-Torque Theory (TTT)
Peebles White 1984

5. N-body simulation of Halo Formation

6. N-body simulation of Halo Formation

7. Origin of Angular Momentum
Tidal Torque Theory (TTT):
Peebles White 1984
proto-galaxy
perturber
Tidal:
Inertia:
Result:

8. Tidal-Torque Theory
Proto-halo:
a Lagrangian patch Γ Γ
Halo

9. Tidal-Torque Theory angular momentum in Eulerian patch
comoving coordinates
const. in m.d.
displacement from Lagrangian q to Eulerian x
laminar flow
average over q in Γ
Zel’dovich approximation
in a flat universe
in EdS
2nd-order Taylor expansion of potential about qcm=0
Deformation tensor
Inertia tensor
antisymmetric tensor

10. Tidal-Torque Theory
Deformation tensor
Inertia tensor
antisymmetric
Tidal tensor = Shear tensor
Only the trace-less part contributes
Quadrupolar Inertia
L by gravitational coupling of Quadrupole moment of Γ with Tidal field from neighboring fluctuations →T and I must be misaligned.
L∝t till ~turnaround
perturber

11. TTT vs Simulations (Porciani, Dekel & Hoffman 2002)
Alignment of T and I: Spin originates from the residual misalignment.
→ Small spin !

12. TTT vs. Simulations: Amplitude Growth Rate Porciani, Dekel & Hoffman 02
Direction

13. TTT vs Simulations: Scatter (Porciani, Dekel & Hoffman 2002)

14. TTT predicts the spin amplitude to within a factor of ~2,
but it is not a very reliable predictor of spin direction.

15. Alignment of I and T: Protohalos and Filaments

16. Alignment of I and T: Protohalos and Filaments

17. Stages in Halo Formation

18. Spin axis and Large-Scale Structure
TTT:
The spin direction is correlated with the intermediate principal axis of the Iij tensor at turnaround.
In a large-scale pancake: the spin axis should tend to lie in the plane.

19. Disk-Pancake Alignment in the Local Supercluster

20. Halo Spin Parameter Fall & Efstathiou 1980 Barnes & Efstathiou 1984
Steinmetz et al …
Bullock et al. 2001b

21. λ is constant, independent of a or M
Halo Spin Parameter
Peebles 76: dimensionless
Bullock et al. 2001
same for isothermal sphere
TTT:
J determined at turnaround
λ is constant, independent of a or M
simulations: λ~0.05

22. Distribution of Halo Spins
<λ> ~ 0.04
Δlnλ ~ 0.5

23. Spin vs Mass, Concentration, History
λ distribution is universal
λ correlated with ac, anti-correlated with C

24. Spin Jump in a Major Merger
Burkert & D’onghia 04 λ
quiet halos with no recent major merger J
time

25. J Distribution inside Halos
Bullock et al. 2001b

26. Universal Distribution of J inside Halos
Bullock et al. 2001b
Two parameter family: spin parameter λ and shape parameter μ
P(μ)
μ-1 λ
μ-1

27. Distribution of J with radius: a power-law profile
j(r)~Ms s
j(r) /jmax
s=1.3±0.3
M(<r) /Mv
Mvir

28. Distribution of J in space
Toy model: J by minor mergers r
Tidal radius
Assume m and j are deposited locally in a shell r
M(<r) /Mv
j(r) /jmax
NFW halo
s=1.3±0.3
j(r) /jmax
M(<r) /Mv

29. Formation of Stellar Disks and Spheroids inside DM Halos
White & Rees 1978
Fall & Efstathiou 1980
Mo, Mao & White

31. Galaxy Types: Disks and Spheroids
The morphology of a galaxy is a transient feature dictated by the mass accretion history of its dark matter halo
most stars form in disks; spheroids result from subsequent mergers
disks result from smooth gas accretion; oldest disk stars are often used to date the last major merger event

32. Galaxy Formation in halos
radiative cooling
cold
hot
cold gas  young stars
merger
hhalos
accretion
disk
spheroid
 old stars

33. Gas versus Dark Matter
Navarro, Steinmetz

34. Flat gaseous disk vs spheroidal DM halo

35. Disk/Bulge Formation (gas only) (Navarro, Steinmetz)

36. Disk Formation MW size (SPH, Governato)

37. Disk Formation – quiet history

38. Disk Formation - mergers

39. Disk Size Spin parameter Conservation of specific angular momentum J/M

40. Disk Profile from the Halo J Distribution
Assume the gas follows the halo j distribution
Assume conservation of j during infall from halo to disk.
In disk: lower j at lower r
In disk:
Assume isothermal sphere No adiabatic contraction

41. Disk Profile: Shape Problem
Bullock et al. 2001b
r/Rvir
Σd(r) [Md/Rv2]
Σd(r) [Md/Rv2]
r/Rvir

42. The Angular-Momentum Problem
Navarro & Steinmetz

43. Navarro & Steinmetz et al.
The Spin Catastrophe
Navarro & Steinmetz et al.
observations
simulations j j

44. The spin catastrophe
observed
Simulated
SPH
Steinmetz, Navarro, et al.

45. Observed j distribution in dwarfs
BBS
halo
disk
Observed j distribution in dwarfs
Low fbaryons0.03
Missing low j
High baryons0.07
P(j/jtot )
j/jtot
van den Bosch, Burkert & Swaters 2002

46. Over-cooling  spin catastrophe
Maller & Dekel 02
tidal stripping +
DM halo
gas cooling
satellite
dynamical friction
Feedback can save the day

47. Orbital-merger model:
Add orbital angular momentum in merger history
Merger history t
Binney & Tremaine
Orbit parameters
and random orientation

48. Succes of orbital-merger model
simulations
Maller, Dekel & Somerville 2002

49. Model success: j distribution in halos
simulations

50. Low/high-j from minor/major mergers
model
simulations
High-j from major mergers J
Low-j from minor mergers

51. Feedback in satellite halos
hot gas
jb<jDM DM
blow out jb>jDM
hot gas
jb=jDM

52. Model vs Data (Maller & Dekel 02)
BBS data: 14 dwarfs, van den Bosch, Burkert & Swaters 02
baryon fraction
spin parameter
model dwarfs
bright
BBS
data
BBS
data
model dwarfs
Vvir=60
One free parameter in model: Vfeedback 90 km s-1

53. J-distribution within galaxies
data
model
DM halo
disk
BBS: van den Bosch, Burkert & Swaters 2002

54. Summary: feedback effect on spin
In big satellites (merging to big galaxies)
heating  gas expansion Rb~RDM
 tidal stripping together  bar~ DM
In small satellites (merging to dwarfs)
gas blowout  fbar down
blowout of low j gas  bar > DM

55. Thin Disk and Thick Disk
Navarro & Steinmetz

56. Dynamical Components of a Simulated galaxy

57. Dynamical components of a simulated galaxy
non-rotating spheroid
thick disk
thin disk
Orbital Circularity
Abadi et al 03