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 Spin parameter Conservation of

specific angular momentum ~ J /M RV const. J / M ~ RvirialV ~ RdiskV Rdisk ~ Rvirial J/M

R V Tidal-Torque Theory (TTT) Peebles 1976 White 1984 N-body simulation of Halo Formation N-body simulation of Halo Formation Origin of Angular Momentum Tidal Torque Theory (TTT):

Peebles 1976 White 1984 q proto-galaxy Result: J i t ijk T jl I lk 2 Tidal: Tij q q i j

3 3 Inertia: I ij 0 a0 qi q j d q perturber Tidal-Torque Theory Proto-halo: Halo a Lagrangian patch

Tidal-Torque Theory L (t ) angular momentum in Eulerian patch comoving coordinates Eulerian

(r , t ) [r (t ) Rcm (t )] [v (t ) Vcm (t )] d 3r 3 L (t ) (t ) a (t ) [1 ( x , t )] [ x (t ) X cm (t )] x d 3 x

x r / a v ax / (t ) 1 const. in m.d. displacement from q x x (q , t ) q S (q , t ) Lagrangian q to 1

3 3 Eulerian x laminar flow 1 [ x(q, t )] J acobian (q, t ) (1 ) d x d q L (t ) 0 a03 Lagrangian Zeldovich approximatio n

[( q q ) ( S (q, t ) S )] S (q, t ) d 3 q average over q in 2 S ( q, t ) D(t ) (q ) (q ) grav (q, t ) /[4G (t ) a (t ) D(t )] S // S L (t ) a 2 (t ) D (t ) 0 a03 (q q ) (q) d 3q in a flat a 2 D D 3 / 2 t in EdS universe

2 nd 2 -order Taylor expansion (q ) (0) q 1 qi q j q q q i qi q0 2 qi q j of potential about qcm=0 q 0 Li (t ) a 2 (t ) D (t ) ijk D jl I lk 2

Deformatio D jl n tensor q j ql q qcm 0 Inertia I a 3 0 0 tensor lk q q d

l k 3 q antisymmetri c tensor ijk

Tidal-Torque Theory Li (t ) a 2 (t ) D (t ) ijk T jl I lk 2 D jl q j ql q qcm 0 I lk 0 a03 ql qk d 3 q ijk

antisymmetri c Deformation tensor Inertia tensor Tidal tensor = Shear tensor Quadrupolar Inertia Tij Dij Dii ij / 3 Only the trace-less part I ij I ii ij / 3 contributes L by gravitational coupling of

Quadrupole moment of with Tidal field from neighboring fluctuations T and I must be misaligned. Lt till ~turnaround q 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 Amplitude

Direction 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. Alignment of I and T: Protohalos and Filaments

Alignment of I and T: Protohalos and Filaments Stages in Halo Formation Spin axis and Large-Scale Structure TTT: 2 Jx ( I yy I zz ) yz 2 Jy ( I xx I zz )

xz 2 Jz ( I xx I yy ) xy I xx I yy I 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 Peebles 76: dimensionless

Bullock et al. 2001 TTT: JE 1/ 2 GM 5 / 2 3 J /M 4 RV

same for isothermal sphere 3 1 GM E M 2 2 V 2 2 2 2 2 R J ~ a 2 D 20 MR02 ~ a1/ 2 M 5 / 3 a 2 D ~ t ~ a 3 / 2 J determined at turnaround ~ D2

when ~ 1 : 20 ~ D 1 ~ a 1 comoving R03 ~ M / 0 ~ M E ~ M 2 / R ~ a 1M 5 / 3 Physical R 3 ~ 1M ~ a 3 M is constant, independent of a or M simulations: ~0.05 Distribution of Halo Spins

<> ~ 0.04 ln ~ 0.5 Spin vs Mass, Concentration, History distribution is universal correlated with ac, anti-correlated with C Spin Jump in a Major Merger

Burkert & Donghia 04 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 M ( j ) M vir j j0 j 1 jmax j0 1

J / M j0b( ) 2VR ' b( ) ln(1 1 ) 1 Two parameter family: spin parameter and shape parameter P( ) -1 -1

Distribution of J with radius: a power-law profile j(r)~Ms s j(r) /jmax s=1.30.3 M( M (

r ) r lt2 r3 dr M r

r m[lt (r )] M (r ) Assume m and j are deposited locally in a shell r d [rV (r )] dm rV (r ) dr dr j(M ) M r

4 r 2 (r ) j (r ) m(r ) M r, ml NFW halo j(r) /jmax j(r) /jmax s=1.30.3 M( M( 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

Flat gaseous disk vs spheroidal DM halo Disk/Bulge Formation (gas only) Steinmetz) (Navarro, Disk Formation MW size (SPH, Governato) Disk Formation quiet history

Disk Formation - mergers Disk Size Spin parameter Conservation of specific angular momentum ~ J /M RV

const. J / M ~ RvirialV ~ RdiskV Rdisk ~ Rvirial J/M R V 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 M halo ( j ) M vir Assume isothermal sphere No adiabatic contraction j j0 j M gas ( j ) f M ( j ) In disk:

j (r ) Vr [GM (r )r ]1/ 2 M halo ( j ) mdisk (r ) 1 M r md (r ) f M v j (r ) jmax j (r ) rV (r ) rVvir

r md (r ) f M v rd r d (r ) j (r ) j0 j (r ) r rmax f M v rd 2 r (rd r ) 2

rd 2 ' Rv b 1 ( ) rmax rd /( 1) Disk Profile: Shape Problem Bullock et al. 2001b d(r) [Md/Rv2] d(r) r/Rvir

[Md/Rv2] r/Rvir The Angular-Momentum Problem Navarro & Steinmetz The Spin Catastrophe Navarro & Steinmetz et al. observation s

j simulation s j The spin catastrophe observe d jdisk Simulated SPH

.Steinmetz, Navarro, et al Observed j distribution in dwarfs disk halo BBS Low fbaryons0.03 Missing low j High baryons0.07

P(j/jtot ) j/jtot van den Bosch, Burkert & Swaters 2002 Over-cooling spin catastrophe Maller & Dekel 02 satellite tidal stripping +

DM halo dynamical friction gas cooling 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 model

Maller, Dekel & Somerville 2002 simulations Model success: j distribution in halos simulations model Low/high-j from minor/major mergers High-j from simulations model

Low-j from minor mergers major mergers J Feedback in satellite halos Vvir Vfb hot gas j DM Vvir Vfb / 2 DM blow out jb>jDM hot gas j =j b DM

Vvir Vfb Model vs Data (Maller & Dekel 02) BBS data: 14 dwarfs, van den Bosch, Burkert & Swaters 02 baryon fraction model dwarfs BBS

data spin parameter bright BBS data model dwarfs Vvir=60 One free parameter in model: Vfeedback 90 km

-1 J-distribution within galaxies DM halo data model disk BBS: van den Bosch, Burkert & Swaters 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 Thin Disk and Thick Disk Navarro & Steinmetz Dynamical Components of a Simulated galaxy Dynamical

Dynamical components components of of aa simulated simulated galaxy galaxy non-rotating spheroid thick disk thin disk

Orbital OrbitalCircularity Circularity Abadi Abadiet etal al03 03