The formation of galactic disks
An overview of Mo Mao & White 1998
MNRAS 295 319
Introduction
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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
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Models for disks in hierarchical cosmogonies
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Non-self gravitating disks in isothermal spheres
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Self-gravitating disks in halos with realistic profiles
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The systematic properties of disks
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Rotation curves
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Disk Instability
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Disk scale lengths and formation times
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Disk surface densities
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Tully Fisher relation
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α
High-redshift disks and damped Ly alpha systems
The effect of a central bulge
Basic assumptions
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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
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Density profile given by an isothermal sphere
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Radius given by the spherical collapse model
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Energy given by the virial theorem
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Temporal evolution
Non self-gravitating disks in
isothermal spheres
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Disk mass is a fixed fraction of the dark matter halo
Disks are in centrifugal balance and have an exponential surface density profile
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Disk mass, disk scale length and central surface density related through
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Disk total angular momentum
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Disk angular momentum is a fixed fraction of the dark matter halo
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Using Dark matter halo dimensionless spin parameter
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Disk scale length
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Disk central surface density
Non self-gravitating disks in
isothermal spheres
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At a given circular velocity, disk are less massive, smaller, and have higher surface density with
redshift, since H(z) increases with time
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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
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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.
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Almost no dependence with dark halo mass M or power spectrum P(k)
Applications
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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
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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)
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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
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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
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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
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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.
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Mass given by
where x=r/r200 is the normalized radius
and c=r200/r_s is the concentration parameter
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There is a relation between the concentration parameter and the characteristic overdensity
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Energy given by the virial theorem
Adiabatic Contraction
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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
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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
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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
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The final mass is the sum of the dark matter mass inside the initial radius and the mass contributed by
the disk
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The angular momentum is
with
Adiabatic Contraction
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Using the dimensionless spin parameter definition we obtain
where
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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
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It is useful to have a fitting formula for
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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
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Predicted rotation curves reach a maximum
at
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with
Rotation curve shape depends on c, md and
lambda; amplitude depends V200,r200 or M.
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Example for Md=5x10^10 solar masses, other
masses scale like Md^1/3
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Rotation curves as peaked as that of the lower
left panel are not observed
Disk instability
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Not all parameter combination give physically
realizable disks.
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If disk self-gravity is dominant then they are
dynamically unstable to the formation of a bar
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Efstathiou, Lake & Negroponte (1982) find
criterion for bar instability
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In this
model
disk are stable if
Disk scale lengths and formation
time
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No attempt to follow the actual formation and
evolution of disks
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Measure Vc at 3Rd
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Sample of present day nearby spiral galaxies
(Courteau 1996, 1997; Broeils & Courteau
1997)
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Solid line for stable disks with md=0.05
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Short-dashed for md=0.025
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Long-dashed for md=0.1
Disk surface density
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Distribution of disk central surface density for
redshift z=0 for different values of disk mass
fraction m_d.
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Solid squares are observational data
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Halo mass from Press-Schechter and spin
parameter from lognormal distribution
obtained for numerical simulations
Tully-Fisher relation
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Tully-Fisher relation for stable disks at z=0.
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Giovanelli et al. (1997) derive for 555 spiral
galaxies in 24 clusters (dashed
lines)
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Assume a mass-to-light ratio gamma=1.7
h
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Slope close to 3, consistent with Giovanelli Iband data; but B-band slope is 2.5
Conclusions
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Study the population of galactic disks expected in hierarchical clustering models
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Simple model for the formation of disk galaxies
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Many observed properties of spiral explained
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All galaxies halos have the NFW universal density profile
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All disks have masses and angular momenta which are a fixed fraction of their halos Md/M<=0.05
and Jd/J~Md/M
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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
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The scatter about the mean Tully Fisher
relation as a function of the dimensionless spin
parameter
Tully-Fisher relation
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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
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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