Angular MomentumTidal-Torque Theory

Halo spin

Angular-momentum distribution within halos

Gas Condensation and Disk Formation

The AM Problem(s)

Thin disk, thick disk, bulge

Lecture

Disk Size

RVMJ /~λ

VRVRMJconst diskvirial ~~/. λ=Conservation of specificangular momentum

J/M

λ~virial

disk

RR

Spin parameter

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):qr

Γ

proto-galaxy perturber

Peebles 1976 White 1984

∫Γ= qdqqaI jiij33

00ρ

lkjlijki ITtJ ε∝

jiij qqT

∂∂

∂−=

φ2Tidal: Inertia:

Result:

Tidal-Torque Theory

HaloProto-halo:

a Lagrangian patch _

_

Tidal-Torque TheoryrdtVtvtRtrtrtL cmcm∫ −×−= Eulerian3)]()([)]()([),()(

γρ

rrrrrrangular momentum in Eulerian patch

1)(// −≡≡≡ txavarx ρρδ&rrrr xdxtXtxtxtattL cm33 )]()([)],(1[)()()( &r

rrrr×−+= ∫γ δρ

comoving coordinates

const. in m.d.

),(),( tqSqtqxxq rrrrrrr

−=→

qdxdtqJtqx acobian331 )1(),()],([1 =+→=+ − δδ

∫Γ ×−+−= Lagrangian33

00 ),()]),(()[()( qdtqSStqSqqatL &r

ρ

displacement fromLagrangian q to Eulerian xlaminar flow

average over q in _

SStDtatGtqqqtDtqS &rr

//)]()()(4/[),()()()(),( 2grav →=∇−= ρπϕφφ

qdqqqatDtatL ∫Γ ∇×−−=33

002 )()()()()( ϕρ&

r

Zel’dovichapproximation

qqqqqqq

qq

q jiqji

iqi

rrrr

rr

−≡∂∂

∂+

∂

∂+≈

== 0

2

021)0()( φφφφ2nd-order Taylor expansion

of potential about qcm=0

tDDa ∝∝ 2/32 &in a flat universe in EdS

0

2

==∂∂

∂−≡

cmqqlj

jl qqD φ

lkjlijki IDtDtatL ε)()()(2 &=

ijkεqdqqaI kllk33

00 ∫Γ≡ ρDeformationtensor

Inertiatensor

antisymmetrictensor

Tidal-Torque Theory

Tidal tensor = Shear tensor 3/ijiiijij DDT δ−≡

0

2

==∂∂

∂−≡

cmqqlj

jl qqD φlkjlijki ITtDtatL ε)()()(

2 &=

ijkεqdqqaI kllk33

00 ∫Γ≡ ρDeformation tensor Inertia tensor antisymmetric

Quadrupolar Inertia 3/ijiiij II δ−Only the trace-less part contributes

qr

Γ

perturber

L by gravitational coupling ofQuadrupole moment of _ withTidal field from neighboring fluctuations°Ṫ and I must be misaligned.

L∝t till~turnaround

TTT vs Simulations (Porciani, Dekel & Hoffman 2002)

Alignment of T and I:Spin originates from theresidual misalignment.

° ̇Small spin !

TTT vs. Simulations: Amplitude Growth RatePorciani, Dekel & Hoffman 02

Amplitude Direction

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

TTT predicts the spin amplitudeto within a factor of ~2,

but it is not a very reliablepredictor 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

The spin direction is correlated with the intermediateprincipal axis of the Iij tensor at turnaround.

In a large-scale pancake: the spin axis should tend tolie in the plane.

)(

)(

)(

2

2

2

yyxxz

zzxxy

zzyyx

IIyx

J

IIzx

J

IIzy

J

−∂∂

∂=

−∂∂

∂=

−∂∂

∂=

φ

φ

φTTT:

zzyyxx III >>

Spin axis and Large-Scale Structure

Disk-Pancake Alignment in theLocal Supercluster

Halo Spin ParameterFall & Efstathiou 1980

Barnes & Efstathiou 1984

Steinmetz et al. 1994-…

Bullock et al. 2001b

Halo Spin Parameter

2/5

2/1

GMEJ

≡λPeebles 76: dimensionless

RVMJ /

43

≡λ

2222 221

23

σσσ === VRGMME

Bullock et al. 2001 same for isothermal sphere

simulations: _~0.05

3/52/1200

22 ~~ MaMRDaJ φ∇&

110

22 ~~:1~when~ −−∇→∇ aDD φδφδ

2/32 ~~ atDa &

MMR ~/~comoving 030 ρ

3/512 ~/~ MaRME −MaMR 313 ~~Physical −ρ

TTT:

J determined at turnaround

_ is constant, independent of a or M

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 MergerBurkert & D’onghia 04

quiet halos with norecent major merger

J

time

_

J Distribution inside Halos

Bullock et al. 2001b

Universal Distribution of J inside Halos

1)(0

>+

=< µµjjjMjM vir

Bullock et al. 2001b

1)1ln()('2)(/1

10

0max −−−≡==−

= −µµµλµµ

bVRbjMJjj

Two parameter family:spin parameter _ and shape parameter _

P(_)

_-1

_

_-1

Distribution of J with radius:a power-law profile

s=1.3°æ0.3

j(r) /jmax

M(

Distribution of J in spaceToy model: J by minor mergers

−=drdMrrM

rl

llm tt

t )(2)( 32

)()]([ rMrlmrM t ∝→∝α

Tidal radius rl

)()]([)()()(4 2 rrVdrdm

drrrVdrmrjrr +=ρπ

rMMjlmrM ∝∝→∝∝ )(,

Assume m and j are deposited locally in a shell r

s=1.3°æ0.3

j(r) /jmax

M(

Formation of StellarDisks and Spheroids

inside DM HalosWhite & Rees 1978

Fall & Efstathiou 1980

Mo, Mao & White

Galaxy Types: Disks and Spheroids

• The morphology of a galaxy is a transient feature dictatedby 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

merger

hhalos

accretion

disk

Galaxy Formation in halosradiative cooling

cold

hot

cold gas → young stars

spheroid

→ old stars

Gas versus Dark MatterNavarro, Steinmetz

Flat gaseous disk vs spheroidal DM halo

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

Disk Size

RVMJ /~λ

VRVRMJconst diskvirial ~~/. λ=Conservation of specificangular momentum

J/M

λ~virial

disk

RR

Spin parameter

R

V

Disk Profile from the Halo J Distribution)()( jMfjM gas

Disk Profile: Shape Problem

r/Rvir

_d(r) [Md/Rv2]

r/Rvir

_d(r) [Md/Rv2]

Bullock et al. 2001b

The Angular-Momentum ProblemNavarro & Steinmetz

The Spin Catastrophe

observations simulations

jj

Navarro & Steinmetz et al.

The spin catastrophe

observed

SimulatedSPH

diskj

Steinmetz, Navarro, et al.

BBShalodisk

van den Bosch, Burkert & Swaters 2002

Observed j distribution in dwarfs

P(j/jtot )

j/jtot

Low fbaryons≈0.03

Missing low j

High λbaryons≈0.07

+DM halo

gas cooling

satellite

Over-cooling → spin catastrophe

dynamical friction

tidal stripping

Feedback cansave the day

Maller & Dekel 02

Merger history

t

Orbital-merger model:

Binney & Tremaine

Orbit parameters

and random orientation

Add orbital angular momentum in merger history

Succes of orbital-merger model

model simulationsMaller, Dekel & Somerville 2002

model

simulations

Model success: j distribution in halos

model

simulations

Low/high-j from minor/major mergers

Low-j from minor mergers

High-j from major mergersJ

Supernova Feedback: VSN (Dekel & Silk 86; Dekel &Woo 03)

Energy fed to the ISM during the “adiabatic” phase:

ff* tMM ≈∗&

)( ffrad*radSN ttMtME ∝≈ ∗&νε

KTT 51 10~atfor −∝Λ01.0≈

oMMV10

critcrit 103km/s100 ×≈→≈→ ∗

Energy required for blowout:2

gasSN VME ≈

Feedback in satellite halos

blow out →jb>jDMDM

hot gas

hot gas

fbvir VV >

2/fbvir VV <

fbvir VV ≤

jb

Model vs Data (Maller & Dekel 02)

spin parameter

BBSdata

model dwarfsVvir=60

baryon fractionmodel dwarfs

bright

BBSdata

BBS data: 14 dwarfs, van den Bosch, Burkert & Swaters 02

One free parameter in model: Vfeedback≅ 90 km s-1

J-distribution within galaxies

modeldata

disk

DM halo

BBS: van den Bosch, Burkert & Swaters 2002

Summary: feedback effect on spin

In big satellites (merging to big galaxies)heating → gas expansion Rb~RDM→ tidal stripping together → λbar~ λDM

In small satellites (merging to dwarfs)gas blowout → fbar downblowout of low j gas → λbar > λDM

Thin Disk and Thick DiskNavarro & Steinmetz

Dynamical Components of a Simulated galaxy

Orbital Circularity Orbital Circularity Abadi et al 03Abadi et al 03

Dynamical components of a simulated galaxyDynamical components of a simulated galaxy

non-rotatingspheroid thick disk thin disk

Formation of Thick DiskStellar satellite merging with disk: edge-on

Formation of Thick DiskStellar satellite merging with disk: face-on