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Angular Momentum

Angular Momentum

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Angular Momentum

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  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

  2. Disk Size J / M Spin parameter  ~ RV Conservation of specific angular momentum  . / ~ virial ~ const J M R V R V  disk R  disk ~ virial R J/M R V

  3. Tidal-Torque Theory (TTT) Peebles 1976 White 1984

  4. N-body simulation of Halo Formation

  5. N-body simulation of Halo Formation

  6. Origin of Angular Momentum Tidal Torque Theory (TTT): q Peebles 1976 White 1984  proto-galaxy perturber Result:  J t T I  i ijk jl lk 2   3 Tidal: Inertia: 3  T I a q q d q      ij 0 0 ij i j q q   i j

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

  8. Tidal-Torque Theory       angular momentum in Eulerian patch comoving coordinates  3 ( ) ( , [ ) ( ) ( )] [ (  ) ( )] L t r t r t R t v t V t d r  Eulerian      cm cm       3 3     ( ) ( ) ( ) 1 [   ( , )] [ ( ) ( )] L t t a t x t x t X t x d x        / / ( ) 1 x r a v a x t     cm const. in m.d.   displacement from Lagrangian q to Eulerian x laminar flow      ( , ) ( , ) q x x q t q S q t      1 3 3 1 [ ( , )] ( , ) 1 (   )  x q t J q  t d x d q      acobian S   3 0 3 ( ) [( ) ( ( , ) )] ( , ) L t a q q S q t S q t d q   average over q in Γ    0 Lagrangian Zel’dovich approximation //       2 ( , ) ( ) ( ) ( ) ( , ) /[ 4 ( ) ( ) (   t )] S q t D t q q q t G t a t D t S S      grav  D    2 3 0 3 ( ) ( ) ( ) ( ) ( ) L t a t t a q q q d q      0 2 in a flat universe in EdS / 3 2 a D D      2nd-order Taylor expansion of potential about qcm=0 2 1        ( ) ) 0 ( q q q q q q q      i i j 2 q q q      i i j 0 q  0 q  D   2 ( ) ( ) ( ) L t a t t D I  i ijk jl lk  2 Deformation tensor Inertia tensor antisymmetric tensor     3 0 3 I a q q d q D 0    ijk lk l k jl  q  q  j l 0 q q   cm

  9. Tidal-Torque Theory D   2 ( ) ( ) ( ) L t a t t T I  2   i ijk jl lk D    antisymmetric jl   3 0 3 I Inertia tensor a q q d q  q  q 0  j l ijk lk l k 0 q q    cm Deformation tensor  Tidal tensor = Shear tensor / 3 T D D Only the trace-less part contributes   ij ij ii  ij / 3 I I  Quadrupolar Inertia ij ii ij q 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

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

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

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

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

  14. Alignment of I and T: Protohalos and Filaments

  15. Alignment of I and T: Protohalos and Filaments

  16. Stages in Halo Formation

  17. Spin axis and Large-Scale Structure TTT:  2  ( ) J I I   x yy zz y z    2  ( ) J I I   y xx zz x z    2  ( ) J I I   z xx yy x y   I I I   xx yy zz 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.

  18. Disk-Pancake Alignment in the Local Supercluster

  19. Halo Spin Parameter Fall & Efstathiou 1980 Barnes & Efstathiou 1984 Steinmetz et al. 1994-… Bullock et al. 2001b

  20. Halo Spin Parameter / 1 2 J E Peebles 76: dimensionless   / 5 2 GM 3 J / M Bullock et al. 2001 same for isothermal sphere   4 RV 3 1 GM    2 2 2 2 2 E M V    2 2 R D  TTT:  2 2 2 0 / 1 2 / 5 3 ~ ~ J a MR a M  0 ~ a 2 / 3 2 ~ D t a     2 2 1 1 ~ when : 1 ~ ~ ~   D D a J determined at turnaround    0  3 0 comoving ~ / ~ R M M 0 2 1 / 5 3 ~ / ~  E M Physical R a M M  3 1 3 ~ ~  R a M λ is constant, independent of a or M simulations: λ~0.05

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

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

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

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

  25. Universal Distribution of J inside Halos Bullock et al. 2001b  j  ( ) 1 M j M    j      1 0  / ( ) 2 ' ( ) ln( 1 ) 1  j J M j b VR b vir        j j  max 0  1 0 Two parameter family: spin parameter λ and shape parameter μ P(μ) μ-1 μ-1 λ

  26. 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

  27. Distribution of J in space Toy model: J by minor mergers l ( ) m l 2 l dM  2  r Tidal radius ( ) t t 3 M r r       M l r dr )] t  [ ( ( ) M r m l r r    t Assume m and j are deposited locally in a shell r [ ( )] d rV r dm   2 4 ( ) ( ) ( ) ( ) r r j r m r rV r   dr j dr , ( ) M r m l M M r      NFW halo j(r) /jmax j(r) /jmax s=1.3±0.3 M(<r) /Mv M(<r) /Mv

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

  29. 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

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

  31. Gas versus Dark Matter Navarro, Steinmetz

  32. Flat gaseous disk vs spheroidal DM halo

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

  34. Disk Formation MW size (SPH, Governato)

  35. Disk Formation – quiet history

  36. Disk Formation - mergers

  37. Disk Size J / M Spin parameter  ~ RV Conservation of specific angular momentum  . / ~ virial ~ const J M R V R V  disk R  disk ~ virial R J/M R V

  38. Disk Profile from the Halo J Distribution ( ) ( ) Mgas j f M j 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    / 1] 2 ( ) [ ( ) j r Vr GM r r In disk:   ( ) ( ) M j m r   halo disk  ( ) j  r j    ( ) ( ) m r f M j r j ( ) 1 M j M     max d v halo vir ( ) j j r j j  0 0 Assume isothermal sphere No adiabatic contraction ( ) ( ) M r j r rV r rV     vir r    1 2 r ' ( )  r R b    ( ) m r f M r max r  d v d v r r  /( ) 1  max r  d d  f M  r ( ) v d  r   d 2) 2 ( r r r d

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

  40. The Angular-Momentum Problem Navarro & Steinmetz

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

  42. The spin catastrophe observed disk j Simulated SPH Steinmetz, Navarro, et al.

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

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

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

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

  47. Model success: j distribution in halos simulations model

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