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ANGULAR MOMENTUMAND THE STRUCTURE OF DM HALOS
Chiara Tonini
Special guest: Andrea LapiDirector: Paolo Salucci
C.T., A. Lapi & P. Salucci (astro-ph/0603051, ApJ in press)
Plan of the talk
I - Halos in phase-space (microscopic)
NFW halo and its dynamical properties
Perturbing the halo with angular momentum
A new equilibrium configuration
II – Angular momentum transfer: a toy-model (macroscopic)
Dynamical friction and galaxy formation (El-Zant et al. 2001, 2004)
III – Discussion
Halos in phase-space
The microscopic state of the system is determined by the 6-D phase-space distribution function: probability density in space
and velocity vdvrfr 3),()(
Function of the integrals of motion
vfdvOfdO 33Macroscopic observables:
In case of isotropic systems: f(E)
d
d
d
d
df
08
1)(
20 2
1vE 0
Eddington’s inversion:
Binney & Tremaine 1987
Halos in phase-space: NFW
Standard theory of hierarchical clustering:
density profile: 2
2
3 )1(
)(
4)(
cxx
cgc
R
Mx
v
v
x
cxcgvx v
)1ln()()( 2
gravitational potential
2
2
1)(r
r anisotropy profile
MACROSCOPIC
Navarro, Frenk & White 1997
If the anisotropy profile is nontrivial, the symmetry of the halo is not described by a one-variable DF
)(2
22
2
0 Lr
Lff
a
TvrL 2-D tangential component of the internal, randomly-oriented motions of the DM
22 2 arL particle orbital energy
22 2 arLQ the particles are less bound, due to the increase of tangential motions
ar anisotropy radius
Halos in phase-space: NFW
01ar
Lokas & Mamon 2001
dr
r
dQ
dQf
Q
a
)(12
1)(
0 2
2
2
2
22/50
Cudderford 1991
Halos in phase-space: NFW
Halos in phase-space: a.m. perturbation
)(2
22
2
0 Lr
Lff
a
2L 22 LL E EE ar
new equilibrium state
dQdLLQfGdr
d
rdr
d arrQr
)/1/()(2
0
2
02
2 222
),(42
rearrangement of the halo particles in phase-space:
new density, anisotropy and potential
the halo must conserve energy and angular momentum after the perturbation
Halos in phase-space: a.m. perturbation
dQQQfrr
rr
a
2/1
0 0122
22/3
))(()2/3(
)1(
)/1(
2)2()(
density, integrated over
L^2:I
for small radii:
0 Q
dr
r
dQ
dQf
Q
a
)(12
1)(
0 2
2
2
2
22/50 energy part of the
DFII
PoissonIII
)2/()1(20 )( rr
)2/()21(2)( rr
0 1 r
2/1 const
NFW
Halos in phase-space: a.m. perturbation
2)/1/()(2
0 2222
2
0 02
222
)/1)(/()(2
)()(
2)( dL
rrrLQ
LLdQQf
rrL
arrQr
a
angular momentum transferred to the halo:
)2/()21(2)( rr
22
22
)(a
a
rr
rrr
)(r
)2/(32/10 ))(()( rrrrL
DF
)1,0( ar NFW
)21,21( ar CORELIKE
0 1 r
2/1 const
NFW
Halos in phase-space: a.m. perturbation
Angular momentum transfer: a toy-model
Is there any physical mechanism that can account for an angular momentum transfer to the halo?
modifying the angular momentum profile down to the inner regions
compatible with spiral galaxies (no mergings?)
involving baryons, where the discrepancy is present
affecting the microscopic state of the system
galaxy formation is the most promising scenario: the baryonic collapse and the dynamical friction
CM222 )/( rLvv r
)0(r )0(e )1/()1( eerr
r
r r
r
r Cfr
dvdr
MFvdvdr
dt
dE
)/(
/)/(
r
r r
r
r Cfr
dvdr
vMFLdvdr
dt
dL
)/(
)/()/(
22 /1/1
)]()([2)0(
rr
rrL
2)()0(
2vrE
20
3
22')'(
ln4v
vdvfMGF
v
Cf
Angular momentum transfer: a toy-model
Recipe for galaxy formation:
0.16 fractional mass of baryons, in self-gravitating clouds
uniformly distributed between 0 and the virial radius
Maxwellian velocity distribution 22 2/1)( xx vr 0xv
Monte Carlo
Angular momentum transfer: a toy-model
power-law CC MM )( ]1010[: 25 CM]20[:
collapse time from 0 to 2 Gyr
El-Zant et al. 2004
Angular momentum transfer: a toy-model
the angular momentum profile produced by dynamical friction is compatible with that needed to perturb the halo DF and transform the halo equilibrium configuration from NFW to whatever…the tangential motions are enhanced, the symmetry between the velocity components is broken
Is this a general property of DM halos?
Discussion
)(fTotally isotropic systems: 222TyTxR vvv
),( 2Lf Tangentially-anisotropic systems:
222TyTxR vvv
),,( 2ZLLf 222
TyTxR vvv Spinning systems:
),()(),,( 210
2ZZ LLfQfLLQf odd in Lz vvT
DiscussionBaryons piling up in the center of the halo deepen the well: negligible1) feedback processes expel most of the baryons 2) energy transfer enhances halo expansion 3) after symmetry breaking, the isotropic enhancement of the sigma-components does not interfere
Cloud mass function can affect the galactic morphology through dynamical friction timescales:
DF efficiently deprives big clouds of all their angular momentum, theycollapse early in the very center of the halo feeding the spheroidal componentsmall clouds are slow in setting down, they retain a larger fraction of theirangular momentum and are more likely to end up in a rotating disk (gradual assembly, inside-out?)
Star formation in clouds could possibly disrupt them and offset the DF effect, but the timescales of SF are in general longer than that of the collapse (consistent with starburst regimes in the center of galaxies, cold flows of gas collapsing to the center)
Mo & Mao 2004
Simulations?
Dynamical friction with gas clumps is sub-grid, at least in galactic halos
Chung-Pei Ma & Michael Boylan-Kolchin 2004
Robertson et al. 2005
DM – DM gravitational scattering
disks and bulges can indeed be originated from the merging of gaseous progenitors in hydrodynamical simulations
Conclusions
An angular momentum perturbation in a NFW Dark Matter halo transforms the halo equilibrium configuration, leading to new density and anisotropy profiles:
injection of angular momentum flattening of the cusp
Dynamical friction in the early stages of galaxy formation can provide the halo with the necessary amount of angular momentum (not a unique plausible mechanism)
C.T., A. Lapi & P. Salucci (astro-ph/0603051, ApJ in press)