Angular Momentumin Halos & Galaxies

Physics 463, Spring 07 Lecture 11

angular momentum

• Tidal torque theory

• Halo spin

• The angular momentum distribution in halos

• Gas condensation & Disk formation

• The AM problems

• gas AM vs dark matter AM

how do galaxies get their spin?

M33

the spin parameterDisk Size

RV

MJ /~!

VRVRMJconstdiskvirial

~~/. !=Conservation of specific

angular momentum

J/M

!~virial

disk

R

R

Spin parameter

R

V

tidal torque theoryOrigin of Angular Momentum

Tidal Torque Theory (TTT):

qr

!

proto-galaxy perturber

Peebles 1976 White 1984

!"= qdqqaI jiij33

00#

lkjlijki ITtJ !"

ji

ijqq

T!!

!"=

#2Tidal: Inertia:

Result:

angular momentum in protohalos grows linearly with time.

Tidal-Torque Theory

Halo

Proto-halo:

a Lagrangian patch _

_

Tidal-Torque TheoryrdtVtvtRtrtrtL

cmcm! "#"=Eulerian

3)]()([)]()([),()($

%rrrrrr

angular momentum in Eulerian patch

1)(// !""" txavarx ##$&rrrr xdxtXtxtxtattLcm

33 )]()([)],(1[)()()( &rrrrr

!"+= #$ %&comoving coordinates

const. in m.d.

),(),( tqSqtqxxqrrrrrrr

!="

qdxdtqJtqx acobian

331 )1(),()],([1 =+!=+" ##

!" #$+$=Lagrangian

33

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

%

displacement from

Lagrangian q to Eulerian x

laminar flow

average over q in _

SStDtatGtqqqtDtqS&rr

//)]()()(4/[),()()()(),( 2

grav !="#= $%&''

qdqqqatDtatL !" #$%%= 33

00

2 )()()()()( &'&r

Zel’dovich

approximation

qqqqqqq

qq

qji

qji

i

qi

rrrr

rr

!"##

#+

#

#+$

== 0

2

02

1)0()(

%%%%2nd-order Taylor expansion

of potential about qcm=0

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

0

2

==!!

!"#

cmqqlj

jlqq

D$

lkjlijki IDtDtatL !)()()( 2 &=

ijk!qdqqaI kllk

33

00 !"# $Deformation

tensor

Inertia

tensor

antisymmetric

tensor

angular momentum comes from gravitational coupling of the quadrupole moment of the protohalo’s mass distribution with the

tidal field. torque depends on the misalignment.

Dekel

continuity eq. implies

for small fluctuations, the mapping q--x is reversible

Stages in Halo Formation

first collapse along major axis of inertia & tidal fieldfilament breaks into clumps. clumps merge.

Porciani, Dekel & Hoffman 02

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

Amplitude Direction

conclusion: predicts spin amplitude within a factor of two; only useful for average properties; not a good predictor of spin direction

TTT vs Simulations (Porciani, Dekel & Hoffman 2002)

Alignment of T and I:

Spin originates from the

residual misalignment.

° ̇Small spin !

halo spin parameters

• typical values are 0.02-0.11

Barnes & Efstathiou 87, Ryden 88, Warren et al 92, Steinmetz & Bartelman 95, Cole & Lacey 96, Gardner 01, Bullock et al 01, Maccio et al 06

λ ≡

JE1/2

GM5/2λ

′

=J

√

2MV R

Bullock et al 01Peebles 76

approximately: rotational support in units of the virial velocity dispersion

distribution of the spin parameter

doesn’t depend on M, z, cosmology Bullock et al 01

λ′

=J

√

2MV R

lognormal distribution

angular momentum growth through mergers

Wechsler 01; Vitvitska et al 02

angular momentum growth through mergers

0

1

2

3

4

5

6

7

!f/!i

P(!

f/!

i)

1.00

Mf/Mi<1.1 1.1<Mf/Mi<1.25 1.25<Mf/Mi

!i<0.025

0.025<!i<0.055

0.055<!i

!f/!i

P(!

f/!

i)

1.23

!f/!i

P(!

f/!

i)

1.98

0

1

2

3

4

5

6

!f/!i

P(!

f/!

i)

0.98

!f/!i

P(!

f/!

i)

0.95

!f/!i

P(!

f/!

i)

1.14

0.1 1.0

0

1

2

3

4

5

6

!f/!i

P(!

f/!

i)

0.97

0.1 1.0

!f/!i

P(!

f/!

i)

0.89

0.1 1.0 10.0

!f/!i

P(!

f/!

i)

0.84

0.0

0.5

1.0

1.5

2.0

2.5

3.0

!f/!i

P(!

f/!

i)

1.25

!f/!i

P(!

f/!

i)

2.56

0.1 1.0

0.0

0.5

1.0

1.5

2.0

2.5

!f/!i

P(!

f/!

i)

1.28

0.1 1.0 10.0!f/!i

P(!

f/!

i)

0.94

all major mergers !i<0.025

0.025<!i<0.055 0.055<!i

basic picture: growth of spin in halos is a random walkmuch of spin-up comes from a sequence of minor mergers

Wechsler 01; Vitvitska et al 02

angular momentum growth through mergersSpin Jump in a Major Merger

Burkert & D’onghia 04

quiet halos with no

recent major merger

J

time

_

angular momentum profiles

M(< j) = Mv

µj

j0 + j

high spin halos have J more evenly distributed

a two-parameter family: spin & shape

Bullock et al 01

model

simulations

Low/high-j from minor/major mergers

Low-j from minor mergers

High-j from major mergers

J

Maller, Dekel & Somerville 02

radial profile of j

~>10% have significant misalignment between shells Bullock et al 01

merger

hhalos

accretion

disk

Galaxy Formation in halos

radiative cooling

cold

hot

cold gas ! young stars

spheroid

! old stars

basic picture

morphology is a transient feature of galaxies set by its merging history

disks are formed during quiet periods

classic disk formation picture

• gas initial well mixed in a smoothly rotating halo

• angular momentum exists due to tidal torques

• falls in to form an angular momentum-supported exponential disk

• assumes that the specific angular momentum of the disks are the same as their host halos

Fall & Efstathiou 1980, Blumenthal et al 1986, Mo, Mao & White 1998

classic disk formation picture

• gas initially well mixed in a smoothly rotating halo

• mass of the disk is a fixed fraction of the halo mass

• angular momentum is a fixed fraction of the halo AM

• disk is thin with an exponential surface density

• only dynamically stable systems can be disks.

• falls in to form an angular momentum-supported disk

• gives an estimate for the sizes & rotation curves of disks

Mo, Mao & White 1998

Disk Profile from the Halo J Distribution

)()( jMfjM gas <=<Assume the gas follows the halo j distribution

2/1])([)( rrGMVrrj ==

)()( rmjM diskhalo !<

Assume conservation of j

during infall from halo to disk.

In disk: lower j at lower r

In disk:

max

0

)()(

)()( jrj

rjj

rjMfrm vd <

+= µ1)(

0

>+

=< µµ

jj

jMjM virhalo

virrVrrVrjrM ==!" )()(

max)( rrrr

rMfrm

d

vd <+

= µ

2)(2)(

rrr

rMfr

d

dvd

+=!

"

µ

)1/(

)('2

max

1

!=

= !

µ

µ"

d

vd

rr

bRr

Assume isothermal sphere

No adiabatic contraction

Dekel

merger

hhalos

accretion

disk

Galaxy Formation in halos

radiative cooling

cold

hot

cold gas ! young stars

spheroid

! old stars

basic picture

That’s all well and good. But does it have anything to do with galaxies?

a perpetual problem to form realistic disk galaxies in cosmological simulations

Orbital Circularity Orbital Circularity

Abadi et al 03Abadi et al 03

Dynamical components of a simulated galaxyDynamical components of a simulated galaxy

non-rotating

spheroid thick disk thin disk

basic picture, that the various components are set by the merging history is probably right.

The Spin Catastrophe

observations simulations

j

j

Navarro & Steinmetz et al.

a perpetual problem to form realistic disk galaxies in cosmological simulations

The spin catastrophe

observed

Simulated

SPH

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 "baryons!0.07

angular momentum problems

• the observed spin component of galaxies is comparable or larger to that of dm halos, but cooling of the baryons should make it smaller

• baryons in observed dwarfs seem to lack the low-j and high-j tails of the distribution of angular momentum for dark halos

are DM & galaxy J really the same?

not likely. see van den Bosch et al 02,

Wise & Abel

although the two components experience the same torques and the same merging processes, they undergo very different relaxation, heating, etc.

dm: violent relaxation; gas: shock heating + feedback

momentum distribution given by

pðlÞ ¼ !lðl$ 1Þð!l þ l$ 1Þ2

; ð14Þ

which corresponds to the universal profile of equation (9).Here

! ¼ 1ffiffiffi2

p"gas

1$ l 1$ ðl$ 1Þ ln l

l$ 1

" #$ %& '; ð15Þ

where "gas is the spin parameter of the gas in the halo underconsideration (listed in Table 1). We adopt l ¼ 1:25, whichcorresponds to the median value of all halos analyzed byB01.

Whereas the disks forming out of the AMDs of the gasanalyzed here have surface brightness profiles that are veryclose to exponential (corresponding to a straight line in thepanels on the left), this is not the case for disks forming out

of the universal AMD. The largest difference between thetwo surface brightness profiles is apparent at small radii.This is due to the bulge-formation scenario included in onecase but not the other. In fact, as shown by van den Bosch(2001), if a simple bulge formation scenario is included, theuniversal AMD also yields near-exponential disks. Oneproblem that is not solved by turning low angular momen-tum material into a bulge, however, is the fact that the uni-versal AMD yields disks with a clear truncation radius thatoccurs at too high surface density (van den Bosch 2001).However, the disks forming out of the AMDs of the gasanalyzed here continue as exponentials to much lower sur-face brightness, suggesting that the cell averaging adoptedby B01 underestimated the true jmax.

Thus, the angular momentum distributions of the gas inprotogalaxies seem consistent with the observed surfacedensity distributions of disk galaxies if the material withnegative specific angular momentum is assumed to make abulge component. Nevertheless, several problems remain.

Fig. 9.—Results at z ¼ 0 for the high-resolution simulation of cluster S0 described in x 3.3. The upper left panel plots pðlvÞ for both the dark matter (solidline) and the gas (dashed line). As for the halos presented in Fig. 5, the dark matter and gas have virtually indistinguishable distributions of specific angularmomentum. The upper right panel compares pðlÞ (solid line) with pðlvÞ for the gas (cf. Fig. 6). Note that even when considering the streaming motions of thegas, there is a large fraction of gas mass with negative specific angular momentum (see also Table 1). The lower two panels plot "ðRÞ and D#ðRÞ for halo S0,where the solid (dashed) lines correspond to the dark matter (gas). A comparison with Figs. 7 and 8 shows that halo S0 behaves similar as the halos analyzed inx 3.2.

32 VAN DEN BOSCH ET AL. Vol. 576

remember, standard model assumes specific angular momentum conservation

!"

#$%&'(()*+,

!"

!-+$.*/$)

01*/2*(+

3451!'(()*+,&&&&&&&&67*+&'1*%*%

8*9$)&621*77*+,

j-rich mass is stripped off,compact baryonic

component survives and loses j as it

sinks to the center

Maller & Dekel 02

0.00 0.02 0.04 0.06 0.08!"

0

10

20

30

40

P(!")

d!"

!"0 = 0.005

#! = 0.51

!"0 = 0.036

#! = 0.53

if cooling is the problem, maybe heating is the solution

Supernova Feedback: VSN (Dekel & Silk 86; Dekel &

Woo 03)

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

ff*tMM !"

&

)( ffrad*radSN ttMtME !" #&$%

KTT51

10~atfor!

"#

01.0!

oMMV

10

critcrit103km/s100 !"#"# $

Energy required for blowout:

2gasSN VME !

if cooling is the problem, maybe heating is the solution...

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

similar model by van den Bosch, Abel & Hernquist 02

Feedback in satellite halos

blow out !jb>jDMDM

hot gas

hot gas

fbvirVV >

2/fbvirVV <

fbvirVV !

jb<jDM

jb=jDM

in small satellites heating can blow out gas, low j preferentially

• new results from Wise & Abel: feedback from SF & SNe @ z ~ 15 drastically affect the spin distribution of the baryons.

• clearly, a full understanding of angular momentum in galaxies is still in its infancy.

• this understanding is crucial to understand the sizes and shapes of galaxies