The formation of galactic disks

An overview of Mo Mao & White 1998MNRAS 295 319

Introduction

● Disk formation in hierarchical cosmogonies● Two component theory: dark matter + gas (White

& Rees 1978) ● Extended massive halos are necessary to form

large observed spiral galaxies (Fall & Efstathiou 1980)

● Abundance of dark matter halos as function of mass and redshift (Press & Schechter 1974)

Index

● Models for disks in hierarchical cosmogonies● Non-self gravitating disks in isothermal spheres

● Self-gravitating disks in halos with realistic profiles

● The systematic properties of disks● Rotation curves

● Disk Instability

● Disk scale lengths and formation times

● Disk surface densities

● Tully Fisher relation

● High-redshift disks and damped Ly alpha systems● The effect of a central bulge

α

Basic assumptions

● The mass of a disk is some fixed fraction of that of the halo in which is embedded

● The angular momentum of the disk is also a fixed fraction of that of its halo

● The disk is a thin centrifugally supported structure with an exponential surface density profile

● Only dynamically stable systems correspond to real galaxy disks

Non self-gravitating disks in isothermal spheres

● Density profile given by an isothermal sphere

● Radius given by the spherical collapse model

● Energy given by the virial theorem

● Temporal evolution

Non self-gravitating disks in isothermal spheres

● Disk mass is a fixed fraction of the dark matter halo

● Disks are in centrifugal balance and have an exponential surface density profile

● Disk mass, disk scale length and central surface density related through

● Disk total angular momentum

● Disk angular momentum is a fixed fraction of the dark matter halo

● Using Dark matter halo dimensionless spin parameter

● Disk scale length

● Disk central surface density

Non self-gravitating disks in isothermal spheres

● At a given circular velocity, disk are less massive, smaller, and have higher surface density with

redshift, since H(z) increases with time

● At a given redshift, disks are larger and less compact in halos with larger dimensionless spin

parameter, because they contract less before coming to centrifugal equilibrium

● Numerical simulations (Warren et al. 1992) show that the distribution of dimensionless spin parameter

can be approximated by a log normal function with parameters 0.05 and 0.5.

● Almost no dependence with dark halo mass M or power spectrum P(k)

Applications

● An upper limit for the disk to halo mass ratio M_d/M=m_d is given by the universal baryon fraction

f_B=Omega_B/Omega_0=0.019 h^{-2}/0.3~0.1, but if the efficiency of disk formation is low then

m_d/f_B<1

● Assuming that the specific angular momentum of the disk and the halo are the same, J/M=J_d/M_d=1,

then j_d=m_d (Fall & Efstathiou 1980)

● For the Milky way, M_disk~6x10^10 solar masses, scale length R_d~3.5 kpc and rotational velocity

Vc~220 km/s at the solar radius R_sun~8 kpc

● Equations for M_d and R_d give constrains on H(z)/H_0, then is possible to infer the formation

redshift z_form < 4

● Disk mass equation can be cast into a Tully Fisher relation assuming a mass-to-light ratio M_d/L_d,

and using the zero-point of the Tully-Fisher relation z_form<3.

Self-gravitating disks in halos with realistic profiles

● Density profile of a dark matter halo in cosmological numerical simulations (Navarro Frenk & White

1997) given by for halos in equilibrium, for all halo masses and

independent of the cosmological model. The scale radius is r_s and delta_0 is a characteristic

overdensity.

● Mass given by where x=r/r200 is the normalized radius

and c=r200/r_s is the concentration parameter

● There is a relation between the concentration parameter and the characteristic overdensity

● Energy given by the virial theorem

Adiabatic Contraction

● If gravitational effects of the disk were negligible, its rotation curve would simply follow the circular

velocity curve of the unperturbed halo Vc^2=G M(r)/r

● Disk formation alters the rotation curve not only through the direct gravitational effects of the disk,

but also through the contraction it induces in the inner regions of the dark halo

● The halo responds adiabatically to the slow assembly of the disk, and it remains spherical as it

contracts: the angular momentum of individual dark matter particles is the conserved

● The final mass is the sum of the dark matter mass inside the initial radius and the mass contributed by

the disk with

● The angular momentum is

Adiabatic Contraction

● Using the dimensionless spin parameter definition we obtain

where and

● Comparing these equations with the isothermal sphere case we notice 2 differences: the factor f_c due

to the different energy resulting from the different density profile and f_R due to both the different

density profile and to the gravitational effects of the disk

● It is useful to have a fitting formula for

● For a given set of parameters V200, c, lambda, m_d and j_d these set of equations must be solved by

iteration to yield the scale length R_d and the rotation curve where

Rotation Curves

● Predicted rotation curves reach a maximum at

with

● Rotation curve shape depends on c, md and

lambda; amplitude depends V200,r200 or M.

● Example for Md=5x10^10 solar masses, other

masses scale like Md^1/3

● Rotation curves as peaked as that of the lower

left panel are not observed

Disk instability

● Not all parameter combination give physically

realizable disks.

● If disk self-gravity is dominant then they are

dynamically unstable to the formation of a bar

● Efstathiou, Lake & Negroponte (1982) find

criterion for bar instability

● In this model

disk are stable

if

Disk scale lengths and formation time

● No attempt to follow the actual formation and

evolution of disks

● Measure Vc at 3Rd

● Sample of present day nearby spiral galaxies

(Courteau 1996, 1997; Broeils & Courteau

1997)

● Solid line for stable disks with md=0.05

● Short-dashed for md=0.025

● Long-dashed for md=0.1

Disk surface density

● Distribution of disk central surface density for

redshift z=0 for different values of disk mass

fraction m_d.

● Solid squares are observational data

● Halo mass from Press-Schechter and spin

parameter from lognormal distribution

obtained for numerical simulations

Tully-Fisher relation

● Tully-Fisher relation for stable disks at z=0.

● Giovanelli et al. (1997) derive for 555 spiral

galaxies in 24 clusters (dashed lines)

● Assume a mass-to-light ratio gamma=1.7 h

● Slope close to 3, consistent with Giovanelli I-

band data; but B-band slope is 2.5

Conclusions

● Study the population of galactic disks expected in hierarchical clustering models

● Simple model for the formation of disk galaxies

● Many observed properties of spiral explained

● All galaxies halos have the NFW universal density profile

● All disks have masses and angular momenta which are a fixed fraction of their halos Md/M<=0.05

and Jd/J~Md/M

● These results are not produce in a natural way in numerical simulations, Jd/J<<Md/M since gas loses

angular momentum to the dark matter during galaxy assembly and disks are too small.

Tully-Fisher relation

● The scatter about the mean Tully Fisher

relation as a function of the dimensionless spin

parameter

Tully-Fisher relation

● Mass to light ratio required to reproduce the

observed zero-point of the Tully Fisher relation

as given by Giovanelli et al. (1997)

High redshift disks and damped Lyman alpha systems

● Logarithm of the predicted abundance of

damped Lyman alpha systems at redshift z=2.5

as function of V_I, the rotational velocity of

the least massive disks allowed to contribute.

High redshift disks and damped Lyman alpha systems

High redshift disks and damped Lyman alpha systems

High redshift disks and damped Lyman alpha systems

The effect of a central bulge

The effect of a central bulge