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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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## 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

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