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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
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