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AY202a
Galaxies & Dynamics
Lecture 22:
Galaxy Evolution
Population Evolution
Tinsley ‘68, SSB ’73, JPH ’77
and the death of the Hubble/Sandage cosmology
program …
The Flux from a galaxy at time t and in band i is
n-1
(n-j)Δt
FG(i,t) =   [ Ψ(k,j) ∫
j
k
(n-j-1)Δt
FK(i,t’) dt’ ]
where FK(i,t’) is the flux of star of type k in
bandpass i at age t’ and ∫ F is the integrated flux in
the jth timestep
This is piecewise, time-weighted summation in an AgeFlux table for stars
Ψ(k,j) is the birth rate function of stars of type k in the
jth time step
Ψ(k,j)  Ψ(m,t) = m–α e–βt
The Initial Mass Function (IMF) is often parameterized as
m–α
with α = 2.35 as the Salpeter slope
The Star Formation Rate (SFR) is often parameterized as
an exponential in time R(t) = A e–βt
or as R(t) = m0/τ e–(t/τ) τ = 1 Bruzual “C” Model
 = 1 - e– (1 Gyr / τ)
Gaseous line + Continuum Emission
Starbursts
a.k.a. Composite galaxes --- its all in a name
How Fast They Change!
A 25 Myr Burst on an
old Spiral ΔL = x2
Spectrum
vs
Type
Kennicutt ‘92
Hβ as a diagnostic
Bursts
Young
Old α = 0.35
α = 3.35
Age
C Model vs time
μ = 0.7 model vs time
B&C = Bruzual & Charlot
GW = Worthey
BBCFN = Bertelli et al.
From Charlot, Worthy &
Bressan ‘96
Comparison to
Observations
Charlot, Worthy &
Bressan ‘96
Early Tracks
from I. Iben
X =0.708
Y = 0.272
Z = 0.020
Modern tracks from
Maeder & Meynet
(and there are others!)
Predicted Evolution
depends on [Fe/H] and
various assumptions re
opacity, mixing,
reaction rates, etc.
! There is not yet
agreement on these!
Star Formation Rates - redux
What drives SFR?
Schmidt ’59 SFR  ρg  dρg / dt
which leads to ρg(t)  ρ0 e -t/r
An exponentially declining SFR which is the
justification for the Tinsley, SSB, JPH, BC models
More general form has dρg/dt  ρgn
Best fitting current law (Kennicutt ’98)
SFR = (2.5 ±0.7)x10-4 (gas/ 1 M pc-2)1.4±0.5
M yr-1 kpc-2 (disk averaged SFR)
Relative Star Formation Rate RP/<R>
Present Rate
∫ Average Rate dt
b = RP/<R>
RP/<R> =
Im
0.2
Sc
0.4
Sb
B-V
Sa
0.8
1.0
SFR(z) = SFR(t)
Star Formation
history of the
Universe
Integrated Rate
ρv(z,λ) = comoving
luminosity
density at λ
Madau, Pozzetti & Dickinson ‘98
Dust?
GOODS
Blue = observed
Red = corrected to 0.2L*
Giavalisco
et al. ’04
Red = observed
Blue = corrected for dust
SF Histories
can be
complex
even in
simple
appearing
systems!
Smecker-Hane et
al.
Carina Dwarf
Population Synthesis Models
Depend on
IMF -- shape, slope , upper & lower mass
limits of integration
SFR -- detailed history
[Fe/H] – affects stellar colors, evolutionary
history
Y -- Helium content affects age, etc.
Gas -- Chemistry, density, distribution, infall etc.
Dust – related to [Fe/H], etc.
Non-thermal activity – presence of an AGN
“You can get anything you want at Alice’s Restaurant”
A. Guthrie
Galaxy Formation & Dynamical Evolution
Simple formation picture  Gravitational Instability
Start with Euler’s Equations and Newton
ρ/t + ·( ρ v) = 0
continuity
v/t + (v ·) v + 1/ρp + φ = 0 energy
2φ = 4πG ρ
potential
Static solution ρ0 = const P0 = const v = 0
Apply linear
ρ = ρ0 + ρ1
perturbation p = p0 + p1
analysis
v = v0 + v1
φ = φ0 + φ1
Now need an equation of state to relate pressure and
energy density.
Assume adiabatic (no spatial variation in Entropy)
Define the sound speed
VS2 ≡ (p/ρ)adiabatic
Then
p1
VS2 =
ρ1
We can write the perturbed Euler equations
(substituting for p) as
ρ1/t + ρ0 · v1 = 0
v1/t + (VS2/ρ0) ρ1 + φ = 0
2φ1 = 4πGρ1
Which can be combined as
ρ2/t2 -VS2 2ρ1 = 4πGρ0ρ1
which is a second order DE with solutions
ρ1(r,k) = δ(r,t)ρ0 = A e[-i k·r + i ωt] ρ0
Where ω and k satisfy the dispersion relation
ω2 = VS2 κ2 - 4πGρ0 ;
κ≡|k|
if ω is imaginary, then  exponentially growing modes
and for k less than some value, ω is imaginary and
modes grow or decay exponentially
define kJ = (4πGρ0 / VS2) ≡ Jeans wave number
with a dynamical timescale τdyn = (4πGρ0) -1/2
Jeans mass is the total mass inside a sphere of radius
λJ/2 = π/kJ
5/3
3
π
V
S
MJ = (4π/3) (π/ kJ)3 ρ0 =
3/2 1/2
6
Masses > MJ are unstable and will collapse
G
ρ
The Jeans problem can also be solved in an expanding
universe (c.f. Bonnor 1957, MNRAS 117, 104)
Characteristic formation times
Galaxy Spheroids
z ~ 20
AGN
z > 10
Dark Halos
z~ 5
Rich Clusters
z ~ 1-2
Spiral Disks
z~1
Superclusters, walls, voids z ~ 1
Details depend on Ω and the cause of structure formation
Rule of thumb from Peebles
Rich clusters have δρ/ρ ~ 100 inside ra
1 + zf ~ 2.5 Ω -1/3
& Globular Cluster systems
1 + zf ~ 8 h-2/3 Ω -1/3
so for our favorite numbers of h = 0.7
and ΩM = 0.25
GC formation is at z ~ 16
Time to go hunting in the dark ages!
Biased Galaxy Formation
Two themes
(1) By the mid 1980’s we know that
ξ(r), gal ~ ¼ ξ(r), rich clusters
so the amplitude of clustering for clusters is much
larger (20x) than that of galaxies. Clustering also
appears to be a function of galaxy mass
(2) ΩM from galaxy clustering etc. is only 0.25 not
1.00000…
So how could galaxies, etc. form efficiently?
N.Kaiser (’84) solved this by introducing the
idea of biased galaxy formation
1/f noise fluctuation spectrum
Galaxies form
Cut for
formation
δρ
Mean
cluster
And galaxies will cluster more than the underlying
dark matter.
If b is the linear biasing factor, then
ξ(r)Galaxies = b2 ξ(r)Dark matter
Coles &
Luchin ‘98
and
(δρ/ρ)Baryons = b (δρ/ρ)DM
and b2 = σ82(galaxies) / σ82(mass)
where σ8 is the variance in 8 Mpc spheres
(roughly where ξ(r)gal  1)
Real bias need not be linear, can be a function of
environment, etc. Current values ~ 1 to 1.5
Press Schechter Formalism
Galaxies and larger structures should be build up by
heirarchical clustering --- what happens after
fluctuations grow enough to form bound objects?
P&S assumed that the amplitudes of the fluctuations
could be described by a Gaussian distribution
p(Δ) =
1
(2π)1/2 σ(M)
exp -[
Δ2
2 σ2(M)
]
where Δ = δρ/ρ is the density contrast associated
with perturbations of mass M
The mean of the distribution is zero, but the variance
σ2(M), the mean squared fluctuation, is finite.
If only those fluctuations with Δ > ΔC collapse, the
fraction is
F(M) =
∞
1
(2π)½ σ(M)
Δ
exp-[
c
Δ2
2σ2(M)
] dΔ = ½ [1 – Φ(tc)]
where tc = Δc/ √2 σ and Φ(x) is the probability integral
defined by
Φ(x) = (2/√π) 
x
0
2
–t
e
dt
We can then relate the mean square density
perturbation to the power spectrum
σ2(M) = < Δ2> = A M –(3+n)/3
and we can write tc in terms of the mass
Δc
tc =
=
√2 σ(M)
Δc
√2 A½
M(3+n)/6 = (M/M*)(3+n)/6
with M* as a reference mass (2A/Δc2) 3/(3+n)
With some effort, we can write the mass function as
N(M) =
<ρ> γ
√π
M γ/2
M γ
( ) exp [ -( ) ]
2
M
M*
M*
Now
Press-Schechter
vs simulations
Extended PS
(includes nonsphericity)
Its surprising that
it works at all!
Dynamical Evolution
Galaxy shapes affected by dynamical interactions with
other galaxies (& satellites)
Galaxy luminosities will change with accretion &
mergers
SFR will be affected by interactions
Mergers – the simple model
Rate P = π R2 <vrel> N t
P = probability of a merger in time t
R = impact parameter N = density vrel = relative velocities
Roughly
N h-3
rc h
P = 2x10-4( 0.05 Mpc-3)( 20 kpc
vrel
)2 ( 300 km/s)
1/H0
a small number, but we see a lot in clusters
N ~ 103 – 104 N field
V rel ~ 3-5 V rel field
The problem was worked first by Spitzer & Baade in
the ’50’s, then Ostriker & Tremaine, Toomre2 and
others in the ’70’s
Mergers occur
depending on the
Energy and
Angular
Momentum of
the interaction
Results from n-body simulations:
(1) Cross sections for merging are enhanced if
angular momenta of the galaxies are aligned
(prograde) and reduced of antialigned (retrograde)
(2) Merger remnants will have both higher central
surface density and larger envelopes --- peaks and
puffs
(3) Head on collisions  prolate galaxies along the
line of centers, off center collisions  oblate
galaxies
An additional effect is Dynamical Friction (Chandrasekhar ’60)
A satellite galaxy, Ms, moving though a background of stars of
density ρ with dispersion σ and of velocity v is dragged by
tidal forces
wake formed
& exerts a
negative pull
(Schombert)
dv/dt = -4πG2 MS ρ v-2 [φ(x) – xφ’(x)] lnΛ
where
φ = error function
x = √2 v/σ
Λ = rmax/rmin (maximum & minimum
impact parameters)
usually rmin = max (rS, GMS/v2)
If you apply this to typical galaxy clustering distributions, on
average a large E galaxy has eaten about ½ its current
mass. Giant E’s in clusters are a special case.
Ostriker & Hausman ’78
Simulations for 1st ranked
galaxies (BCG’s)
1. Galaxies get brighter with
time due to cannibalism (L)
2. Galaxies get bigger with
time (β)
3. Galaxies get bluer with
time by eating lower L,
thus lower [Fe/H] galaxies
L
Core radius
Profile
Eisenstein et al…. BAO
We present the large-scale correlation function measured from a spectroscopic sample
of 46,748 luminous red galaxies from the Sloan Digital Sky Survey. The survey
region covers 0.72 h-3 Gpc3 over 3816 deg2 and 0.16<z<0.47, making it the best
sample yet for the study of large-scale structure. We find a well-detected peak in the
correlation function at 100 h-1 Mpc separation that is an excellent match to the
predicted shape and location of the imprint of the recombination-epoch acoustic
oscillations on the low-redshift clustering of matter. This detection demonstrates the
linear growth of structure by gravitational instability between z~1000 and the
present and confirms a firm prediction of the standard cosmological theory. The
acoustic peak provides a standard ruler by which we can measure the ratio of the
distances to z=0.35 and z=1089 to 4% fractional accuracy and the absolute distance
to z=0.35 to 5% accuracy. From the overall shape of the correlation function, we
measure the matter density Ωmh2 to 8% and find agreement with the value from
cosmic microwave background (CMB) anisotropies. Independent of the constraints
provided by the CMB acoustic scale, we find Ωm=0.273+/ 0.025+0.123
(1+w0)+0.137ΩK. Including the CMB acoustic scale, we find that the spatial
curvature is ΩK=-0.010+/-0.009 if the dark energy is a cosmological constant.
More generally, our results provide a measurement of cosmological distance, and
hence an argument for dark energy, based on a geometric method with the same
simple physics as the microwave background anisotropies. The standard
cosmological model convincingly passes these new and robust tests of its
fundamental
Likelihood contours of CDM models as a function of Ω mh2 and DV(0.35). The
likelihood has been taken to be proportional to exp(- χ 2/2), and contours
corresponding to 1 through 5 σ for a two-dimensional Gaussian have been
plotted. The one-dimensional marginalized values are Ω mh2 = 0.130 ± 0.010 and
DV(0.35) = 1370 ± 64 Mpc.
w = -1, non zero curvature
Solid lines constant curvature
Dashed lines constant Ωm
w =-1, non-zero curvature
Dashed lines are constant H0