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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. Disk Size. J /. M. Spin parameter. . ~. RV. Conservation of specific angular momentum. . /.
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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
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
Tidal-Torque Theory (TTT) Peebles 1976 White 1984
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
Tidal-Torque Theory Proto-halo: a Lagrangian patch Γ Halo Γ
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
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
TTT vs Simulations (Porciani, Dekel & Hoffman 2002) Alignment of T and I: Spin originates from the residual misalignment. → Small spin !
TTT vs. Simulations: Amplitude Growth Rate Porciani, Dekel & Hoffman 02 Direction Amplitude
TTT vs Simulations: Scatter (Porciani, Dekel & Hoffman 2002)
TTT predicts the spin amplitude to within a factor of ~2, but it is not a very reliable predictor of spin direction.
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.
Disk-Pancake Alignment in the Local Supercluster
Halo Spin Parameter Fall & Efstathiou 1980 Barnes & Efstathiou 1984 Steinmetz et al. 1994-… Bullock et al. 2001b
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
Distribution of Halo Spins <λ> ~ 0.04 Δlnλ ~ 0.5
Spin vs Mass, Concentration, History λ distribution is universal λ correlated with ac, anti-correlated with C
Spin Jump in a Major Merger Burkert & D’onghia 04 λ quiet halos with no recent major merger J time
J Distribution inside Halos Bullock et al. 2001b
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
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
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
Formation of Stellar Disks and Spheroids inside DM Halos White & Rees 1978 Fall & Efstathiou 1980 Mo, Mao & White
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
Galaxy Formation in halos radiative cooling cold hot spheroid merger disk accretion hhalos cold gas young stars old stars
Gas versus Dark Matter Navarro, Steinmetz
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
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
Disk Profile: Shape Problem Bullock et al. 2001b Σd(r) [Md/Rv2] Σd(r) [Md/Rv2] r/Rvir r/Rvir
The Angular-Momentum Problem Navarro & Steinmetz
The Spin Catastrophe Navarro & Steinmetz et al. observations simulations j j
The spin catastrophe observed disk j Simulated SPH Steinmetz, Navarro, et al.
Observed j distribution in dwarfs Low fbaryons0.03 Missing low j High baryons0.07 halo disk BBS P(j/jtot ) j/jtot van den Bosch, Burkert & Swaters 2002
Over-cooling spin catastrophe Maller & Dekel 02 satellite tidal stripping + DM halo gas cooling dynamical friction Feedback can save the day
Orbital-merger model: Add orbital angular momentum in merger history Merger history Orbit parameters Binney & Tremaine t and random orientation
Succes of orbital-merger model simulations model Maller, Dekel & Somerville 2002
Model success: j distribution in halos simulations model
Low/high-j from minor/major mergers High-j from major mergers J simulations model Low-j from minor mergers