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Hot topics on Galaxy Formation
and Evolution
2. The IMF of galaxies
Roberto Saglia
Max-Planck Institut
für extraterrestrische Physik
Garching, Germany
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2. The varying IMF of Galaxies
•  Introduction
•  What is the IMF and how to measure it
•  The integrated galaxy IMF
•  High-mass IMF constraints from emission: Ha
•  Low-Mass IMF constraints from absorption:
NIR indices
•  IMF constraints from mass measurements:
stellar dynamics and gravitational lensing.
The problem of dark matter.
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References
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Weidner 2004 & Kroupa, 2004, MNRAS, 350, 1503 (WK)
Weidner & Kroupa 2005, ApJ, 625, 754
Howersten & Glazebrook 2008, ApJ, 75, 163 (HG)
Van Dokkum & Conroy, 2010, Nature, 468, 940
Van Dokkum & Conroy, 2011 ApJ, 735, L13
Thomas et al. 2011, MNRAS, 415, 545
Cappellari et al., 2012, Nature, in press
Kroupa et al. 2012, Stellar Systems and Galactic Structure, Vol. V (K12)
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A more modern view
of the Hubble classification
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Kormendy & Bender, 2012, ApJS, 198, 2
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Milky Way (Sbc-galaxy) in different wavebands
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The color-magnitude diagram
of all Hipparcos stars
with relative distance error
< 0.1.
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The HRD of a Globular Cluster
The stars of a stellar cluster have the same age and the same
chemical composition
dependence only on mass
HST
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The HRDs of Open Clusters of Different Ages
Zero Age Main
Sequence (ZAMS)
high masses
Plejades
small masses
Turn-off point
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The Initial mass function (IMF)
IMF = distribution of stellar masses at the time of birth. Definition:
Φ(m) = number of stars formed per mass-interval [m,m+dm] and per total mass of formed
stars (Unit = 1/mass2)
mΦ(m) = mass of stars per mass-interval and per total mass
mΦ(m)dm = mass of stars within mass-interval and per total mass
resulting normalization:
∫ mΦ(m)dm = 1
Observational determination of IMF:
"   from the solar neighbourhood using star counts and taking into account the lifetime of
stars; requires assumptions concerning the star formation history in the solar neighbourhood: low-mass stars (M < 1MΘ) have not disappeared since their creation, while
O- and B-stars disappear within 107 yrs
"   from star counts in star forming regions with infrared photometry (dust extinction smaller
in IR)
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Note: sometimes it is claimed that the IMF is bimodal, i.e. high-mass and low-mass stars
form in separate regions, or, in the same region at different times.
Common analytical approximations:
Salpeter-IMF:
0.1 ≤
(APJ 121, 161 (1955))
m
≤ 100 : Φ(m)  0.17m−2.35
M
Miller-Scalo-IMF:
(see also figure on next page)
mass-fraction
0.1 ≤
m
≤ 0.5
M
0.93m−0.85
0.31
0.5 ≤
m
≤ 1.0
M
0.46m−1.85
0.31
1.0 ≤
m
≤ 3.16
M
0.46m−3.4
0.26
m
≤ 100 0.21m−0.27
M
0.12
3.16 ≤
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Φ(m) =
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Pagel: Nucleosynthesis and Chemical Evolution, Cambridge University Press 1997
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The IMF of Star Clusters
ξ (m) ∝ m−α
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α = 2.35 : Salpeter
Kroupa IMF
0.5M e
WK
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0.08M e
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A collection of IMFs
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IMFs and cumulative functions
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... has a constant slope
(to first approximation)
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K12
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Metallicity and density
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High mass slope α 3 :
flatter at high densities and low metallicities?
Marks et al. 2012, MNRAS, in press
Padova, March 2012
(modeling involved)
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Observed ranges of the IMF?
Marks et al. 2012, MNRAS, in press
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Gas ejection rate of all stars:
Basics of Galaxy Evolution
M = Ms + Mg
E (t ) =
t −τ m ( m )
Φ (m)dm
mass of stars dying at time t
m - wm : ejected mass
dM s
=Ψ−E
dt
⎧Ψ = star formation rate
⎨
⎩ E = gas ejection rate of all stars
Ψ t −τ m ( m ) Φ (m): birth rate at t-τ m =
death rate at time t, stars
of different generations are
involved
τ m : main-sequence lifetime at mass m
= −Ψ + E + f − e
Remnant mass:
Spectrum of a galaxy
f (λ , t ) = ∫ f m,Z (λ , t − t ')Ψ(t − t ')Φ(m, t ')dt ' dm
Luminosity : L(t ) = ∫ f (λ, t )dt
m
mt : turn-off mass at time t = lowest
⎧ f = rate on infalling gas
⎨
⎩e = rate on ejected gas
dt
∫ [m − w ] Ψ
mt
⎧ M = total mass
⎪
⎨ M s = mass in stars
⎪ M = mass in gas
⎩ g
dM
= f −e
dt
dM g
∞
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⎧⎪ wm = 0.11m + 0.45M  (m < 6.8M  )
⎨
(m ≥ 6.8M  )
⎪⎩ wm = 1.5M 
Mass-to-light ratio M / L(t )
see: Tinsley 1980 and corrections by Maeder 1992
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IMF Variations at low and high m
Steep IMF at low
masses:
Flat IMF at high
masses:
Lots of M-dwarfs
Lots of 'black'
remnants, i.e.
neutron stars
and black holes,
X-ray binaries
Old stellar populations have large
M/L
Extrapolated
'evidence'
for high
metallicities?!
Marks et al. 2012 MNRAS
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Old stellar populations have
large M/L.
Lots of SNII:
More a elements
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Basics of star formation
We first discuss under which conditions a gas cloud will collapse. We consider an infinite
uniform gas cloud in hydrostatic equilibrium with initial conditions: ρ = ρ0 = const.,

T = T0 = const and v 0 = 0.
The basic equations of hydrodynamics (without magnetic fields!) which govern the dynamics of the cloud are:
Continuity equation:
Equation of motion:
Poisson Equation:
Equation of state for an ideal gas:

∂ρ
+ ∇ ⋅ ( ρ v) = 0
∂t


∂v 
∇P
+ ( v ⋅ ∇)v = − ∇Φ
∂t
ρ
∇ 2Φ = 4π G ρ
P=
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k
ρT = Cs2 ρ
µ
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where Ф is the gravitational potential and Cs is the speed of sound.
We assume that we know an equilibrium solution to which we add a small perturbation:
ρ = ρ0 + ρ1
P = P0 + P1
Φ = Φ 0 + Φ1
  
v = v0 + v1
We furthermore assume a constant speed of sound (and temperature!, the gas is assumed
to be isothermal). Inserting the perturbed solution into the equation and neglecting all
second order terms, we obtain differential equations for the perturbations. These can be
combined to give a wave equation:
which is solved by:
⎛ 2 1 ∂ 2 4π G ρ0 ⎞
⎜ ∇ − C 2 ∂t 2 + C 2 ⎟ ρ1 = 0
⎝
⎠
s
s
 
ρ1 = Aexpi( k ⋅ r − ω t)

This represents a planar density wave with wave vector k and angular velocity ω fulfilling
the dispersion relation
2
2 2
ω = k Cs − 4π G ρ0
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This relation shows that ω is real only when k is larger than a critical value kj = (4πGρ0)1/2/
Cs, or when the wavelength is smaller than a critical wavelength
1/ 2
λj =
2π ⎛ π ⎞
=⎜
⎟
k j ⎝ G ρ0 ⎠
Cs
In that case the density perturbation is oscillating, but the amplitude will not grow. However,
if:
λ > λj
the frequency ω becomes imaginary, so that density perturbations will grow exponentially
with time. Thus the region of the density perturbation has to be large enough (and massive
enough) to start collapsing. This is known as the Jeans criterion. The collapse time can be
estimated via:
τ collapse ≈
1/ 2
λj
⎛ π ⎞
=⎜
⎟
Cs ⎝ G ρ 0 ⎠
We finally define the critical or Jeans mass as:
or, explicitely
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4
M j = volume ⋅ density = πλ 3j ρ
3
⎛ πk ⎞
Mj =⎜
⎟
⎝ Gµ ⎠
3/ 2
T 3/ 2 ρ −1/ 2
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⎛ T ⎞
M j = 1.2 ⋅105 M e ⎜ 2 ⎟
⎝ 10 K ⎠
3/ 2
⎛
⎞
ρ
⎜ −24
3 ⎟
10
g
/
cm
⎝
⎠
−1/ 2
µ −3/ 2
Therefore, for typical properties of the diffuse ISM, Mj is much larger than the mass
of a star. Giant molecular clouds, on the other hand, are well above their Jeans mass.
However, the temperature of the gas in molecular clouds unterestimates the internal energy
significantly. Most of the internal energy is in the relative motions of the cloudlets which is of
the order of up to 10km/s (see above). Therefore, giant molecular clouds do not seem to be
in a collapse phase yet but rather appear as virialized quasi-stationary systems. Magnetic
field pressure can further stabilize a cloud. (Side remark: if all molecular clouds in the Milky
Way would collapse in the time scale estimated 1/(Gρ)1/2, the star formation rate in the Milky
Way would have to be 200MΘ/yr, observed is only ≈ 1MΘ/yr).
Collisions between clouds, or tidal forces (e.g. from the Galactic spiral density wave) can
however finally trigger the collapse. Once the cloud is compressed and the density increases, the collapse can start. Also instabilities in the cloud itself may finally cause it to
collapse without external influence.
At lower metallicities, cooling is less efficient, therefore higher temperatures are present in
the clouds --> larger masses are needed for collapse --> a flatter IMF at high masses is
expected. At high densities, the crossing time is so short that protostars will 'coagulate'
tending to create again a flatter IMF at high masses.
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Fragmentation of collapsing clouds
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From the observations, we know that molecular clouds have complex substructure indicating that fragmentation takes place already in the pre-collapse phase. We now discuss the
conditions which determine the size of the fragments. For fragmentation to proceed, Mj
should not be constant during collapse but should decrease with increasing density.
For an isothermal collapse we found
M j ∝ ρ −1/ 2
(isothermal collapse)
which means Mj indeed decreases with rising density. The situation is opposite for an
adiabatic collapse, where T ~ P2/5, so from P ~ ρT for an ideal gas, or T ~ ρ2/3, we have
M j ∝ T 3/ 2 ρ −1/ 2 ∝ ρ 1/ 2
(adiabatic collapse)
In other words, Mj increases with density in an adiabatic collapse.
To ensure that the collapse is isothermal, the gravitational energy released during the collapse of the cloud should be radiated away in order not to increase the internal energy
of the gas. Therefore the collapse time τcollapse should be much larger than the thermal
adjustment timescale τadj:
τ collapse  τ adj
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As we have seen, the collapse time for a cloud(let) only depends on its density and is:
τ collapse ≈ ( G ρ )
−1/ 2
⎛
⎞
ρ
≈ 108 y ⎜ −24
3 ⎟
⎝ 10 g / cm ⎠
−1/ 2
On the other hand, the thermal adjustment timescale for interstellar clouds is found to be
of the order τadj ≈ 100y, which obviously fulfills the criterion for isothermal collapse. Thus
Mj ~ ρ−1/2 which in turn leads to continued fragmentation during collapse.
We now want to estimate the lower mass limit for fragmentation. For this purpose we
have to know when the collapse will change from isothermal to adiabatic.
We first calculate the energy radiated per second during the isothermal collapse. The
total gravitational energy released in the collapse is Eg = GM2/R during the timescale
τ ≈ (Gρ)−1/2, so that the luminosity is given by
Eg
1/ 2 ⎛ M ⎞
GM 2
1/ 2
3
Lg ≈
=
(G ρ ) = (3G / 4π ) ⎜ ⎟
τ
R
⎝ R⎠
5/ 2
This luminosity should be much smaller than that of a black body with the same temperature and volume. Because, if the cloud does emit black body radiation, it means that
it is optically thick, has reached an equilibrium configuration and does not loose energy
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efficiently anymore. The luminosity emitted in the black body case would be:
Lc = 4π R 2σ T 4
When Lc ≥ Lg the collapse becomes adiabatic because the released gravitational energy
cannot be radiated away. Obvioulsy, the condition Lc = Lg corresponds to a lower mass
limit for fragmentation:
(
Mj = π /9
9
) (
1/4
σG
3
) (
−1/2
R/µ
)
9/4
T 1/4 = 10−2 M T 1/4
where again R = (3/4π)1/3(M/ρ)1/3 and the equation for the Jeans mass given above have
been used. Because of the exponent 1/4 of the temperature, the temperature dependence
of this mass is small. For temperatures of 1000 K we obtain 0.05MΘ.
This shows why during the transition from diffuse HI clouds to molecular clouds fragmentation already takes place. It is this fragmentation which may actually slow down the further
collapse of the molecular cloud because the fragments can start to move as individual
almost ’collisionless particles’. This permits relative velocities of the cloudlets which are
larger than what corresponds to their internal thermal motions and helps to stabilize the
molecular cloud.
Interestingly, the lower mass limit which we obtain for the cloudlets is similar the lowest
masses which are observed for stars.
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The maximum star mass
Stars are born in Star Clusters
of a Star Cluster and only there!
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Largest star mass
in a star cluster: m max
1=
mmax*
∫
ξ (m) dm
mmax
Mass of the star cluster:
M ecl =
mmax
∫
mξ (m) dm
mmin
Maximum stellar mass:
m max* (150M  ?)
Padova, March 2012
K12
Hot topics on galaxy formation and evolution 2
The mass distribution of Star
Clusters
Page 28
−β
P ( M ecl ) ∝ M ecl
β ≈ 2 − 2.35
Hunter et al. 2003,
AJ, 126, 1836
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The SFR- M ecl ,max relation
Mass of a star formation event:
M ecl ,max
∫
M TOT =
M ecl P ( M ecl )dM ecl
M ecl ,min
SFR = M TOT / δ t
100
10
dt=1 Myr
→ M ecl ,max = f ( SFR )
Dotted: b=2
Dashed: b=2.4
M ecl =
mmax
∫
mξ (m)dm
mmin
→ mmax = g ( SFR )
WK
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The composite IMF of galaxies
Page 30
All stars in galaxies are made in star clusters:
ξ IGIMF (m, t ) = ∫
t
∫
M ecl ,max ( SFR ( t '))
0 M ecl ,min
ξ (m ≤ mmax ( M ecl ))P( M ecl )dM ecl dt ' / t
100 Myr burst
14 low level bursts
Low constant SF
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The slope(s) of the IGIMF
WK
Padova, March 2012
Salpeter
Page 31
K12
Hot topics on galaxy formation and evolution 2
IMF constraints from emission
Page 32
Star forming galaxies have
ionized gas that produces
emission lines:
The Orion Nebula in optical light, from the Hubble Space Telescope. The bar at the lower
left is a wall of gas viewed edge-on. This gas glows because it is ionized by the four hot
”Trapezium stars” just to the left of center.
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Star formation rate indicators:
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Diagnostic plots
Ionization photons can be produced by young stars in regions of star formation or by central
non-thermal sources, i.e. AGNs. Diagnostic plots allow to disentagle the two cases:
q: ionization parameter
z: metallicity
Kewley et al. 2001,ApJ, 556, 12
The lines show the line ratios predicted when the ionization source is a
star bursting population for different metallicities.
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The photoionizing continuum of AGNs has a significant fraction of its energy in
the X-rays. These photons can penetrate deeply into the neutral regions and
produce large partly ionized zones with H+/H~0.2-0.4 that do not exist in
HII regions.
Ions of atoms having ionization potentials similar to H (O, S, N…) will also be
present. The hot free electrons produced by X-ray photoionization are available
to excite and increase the strenght of lines produced by collisional excitation,
like OIII, SII, NII. Therefore we expect stronger intensities of the [OIII] 5007 line
relative to the Hb, and stronger intensities of the [SII] or [NII] lines with respect
to Ha than in star forming regions.
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The equivalent width
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I cont − I λ
w = ∫ Rλ d λ = ∫
dλ
line
line
I cont
The equivalent width w evidently has the unit
of a wavelength. Geometrically, the product
of continuum flux times equivalent width covers
the same area in a spectrum as the absorption
or emission line does.
The equivalent width increases with increasing number of ions in the corresponding quantum
state. In the optically thin case, the increase is linear. For higher densities of ions, the
absorption line starts to saturate and the equivalent width almost stops increasing (at which
density this happens depends on the Doppler broadening).
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The Kennicutt
method
Page 37
The equivalent width of Hα vs B − V
color can be used to constrain the initial
mass function of stars in spirals (figure
from Kennicutt et al. ApJ No 272, p54,
1983) The three evolutionary models
shown in the figure use different IMFs
(from top to bottom):
Ф ~ m−2
Ф ~ m−2.35
Ф ~ Miller-Scalo IMF
Wλ ;
or:
Wλ ;
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Hα flux
continuum flux
present SFR
luminosity of old stars
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How to measure the IMF slope:
the Kennicutt method
Page 38
The Ha equivalent width is sensitive to the ratio of the number of
massive stars (driving the emission line strenght) to the number of
giant stars (driving the continuum). When combined with a color it can
constrain the slope of the IMF.
log Hα EW
Age
Metallicity
α = Γ +1
t=1.1 decl.
Z=0.025
t=1.5 inc.
0.005
HG08
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g-r
Page 39
SDSS
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SDSS Data Products
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•  Images of 1/4 of the full sky (10000 square-degrees) in 5 wavelength bands
•  photometry of 100 million objects
•  spectra of > 1 million objects
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Hoversten &
Glazebrook 2008
The SDSS
Sample
Page 41
130602 galaxies with
good lines S/N, Ha representing SF and not
AGN. Colors corrected
for dust using Balmer
decrement and K-corrected
to z=0.1.
χi2 (Γ, Z , t , SFH ) =
(ci − c(Γ, Z , t , SFH )) / σ c2i ) 2 +
( wi − w(Γ, Z , t , SFH )) / σ w2i ) 2
Dust vector :
Balmer decrement f = Hα / H β
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Page 42
A variable
IMF slope?
α −1
SMC
MW
Salpeter
Many alternative
explanations tested
and rejected:
Malquist bias, redshift,
aperture, extiction,
f ratio, bursty star
formation histories.
HG08
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Constraints on the
IMF from
absorption lines
Page 43
S
⎛ ⎞
I ⎜ A ⎟ = Δ − ∫ dλ
C
⎝ ⎠
Δ
0
C
S
Δ
GC NGC 6626
Worthey et al. 1994 ApJS 94, 687
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Simple Stellar Population models
Page 44
1 Age, 1 Metallicity, Initial Mass Function
IMF ∝ M −α ,α ≈ −2.35
E-AGB 5-6
HB 4-5
SGB 2-3
RGB 3-4
dMS 0-1
MS-TO 1-2
f c*, j =
Fc*, j
∑F
L. Greggio
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*
c, j
j
Hot topics on galaxy formation and evolution 2
Simple Stellar Population models
Page 45
1 Age, 1 Metallicity, Initial Mass Function
E-AGB 5-6
HB 4-5
SGB 2-3
RGB 3-4
dMS 0-1
MS-TO 1-2
f c*, j
*
⎛
⎞
I
j
*
*
=
; Fl , j = Fc , j ⎜1 − ⎟
*
⎜ Δ⎟
∑ Fc, j
⎝
⎠
Fc*, j
j
IMF ∝ M −α ,α ≈ −2.35
*
j
Fitting Function: I (Te , g , Z )
⎛ ∑F ⎞
⎛ F⎞
⎜
⎟
*
l, j
I SSP = Δ ⎜1 − l ⎟ = Δ ⎜1 − j * ⎟ =
⎝ Fi ⎠
⎜ ∑ Fc , j ⎟
j
⎝
⎠
* *
⎛
⎞
F
c, j I j
*
*
⎜ ∑ Fc , j − Fc , j +
⎟
Δ ⎟
j
⎜
=Δ
*
⎜
⎟
F
∑
c, j
⎜⎜
⎟⎟
j
⎝
⎠
= ∑ I *j × f c*, j
j
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Page 46
In the
optical:
M-dwarfs
not
important
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Page 47
NIR indices
in stars
Strong features
present only in
M dwarfs
9900
10000
o
A
M dwarfs flux (more)
important in the NIR
A steep IMF will
have lots of M dwarfs
8300
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8500
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Page 48
NIR
indices
Van Dokkum & Conroy
2010, Nature, 468, 940
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Constraining the IMF
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NaI and FeH Wing-Ford
indices in GCs
Page 50
M31 globular
clusters have
'normal' IMF
Van Dokkum &
Conroy, 2011
ApJ 375, L13
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Comparing GCs and Es
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IMF constraints from mass
measurements
Page 52
•  Compute the luminosity of a galaxy L
•  Compute the M/L of the stellar population that best reproduces the colors and/or
spectrum/line indices of a galaxy for a given IMF
•  Measure the mass of a galaxy using:
stellar dynamics or gravitational lensing M
•  Compute the ratio M/L
•  Compare the two values, taking into account the amount of dark matter and the
mass of central black holes
Padova, March 2012
Hot topics on galaxy formation and evolution 2
Gas ejection rate of all stars:
Basics of Galaxy Evolution
M = Ms + Mg
E (t ) =
t −τ m ( m )
Φ (m)dm
mass of stars dying at time t
m - wm : ejected mass
dM s
=Ψ−E
dt
⎧Ψ = star formation rate
⎨
⎩ E = gas ejection rate of all stars
Ψ t −τ m ( m ) Φ (m): birth rate at t-τ m =
death rate at time t, stars
of different generations are
involved
τ m : main-sequence lifetime at mass m
= −Ψ + E + f − e
Remnant mass:
Spectrum of a galaxy
f (λ , t ) = ∫ f m,Z (λ , t − t ')Ψ(t − t ')Φ(m, t ')dt ' dm
Luminosity : L(t ) = ∫ f (λ, t )dt
m
mt : turn-off mass at time t = lowest
⎧ f = rate on infalling gas
⎨
⎩e = rate on ejected gas
dt
∫ [m − w ] Ψ
mt
⎧ M = total mass
⎪
⎨ M s = mass in stars
⎪ M = mass in gas
⎩ g
dM
= f −e
dt
dM g
∞
Page 53
⎧⎪ wm = 0.11m + 0.45M  (m < 6.8M  )
⎨
(m ≥ 6.8M  )
⎪⎩ wm = 1.5M 
Mass-to-light ratio M / L(t )
see: Tinsley 1980 and corrections by Maeder 1992
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The M/L ratio of stellar populations
For main sequence stars:
(0.1M  < M < 100M  )
v  Mass–luminosity relation
L∝M4
for M > 0.6M 
L∝M2
for M < 0.6M 
:
:
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Hot topics on galaxy formation and evolution 2
M/L from SSPs
Page 55
If you now the (mean) age and
metallicity of a galaxy, Simple
Stellar Population Models can
predict their mass-to-light
ratio for a given IMF.
M / LKroupa = 0.56 M / LSalpeter
The problem is then to compute
the mean age and metallicity
of a galaxy.
Maraston 1998, 300, 872
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Page 56
The age-metallicity
degeneracy problem:
colors are hopelessly
degenerate in age and
metallicity !
absorption lines in the
blue can help: hotter
stars at MSTO have
stronger Balmer lines,
this allows to break the
degeneracy (see further
below)
(Worthey 1993)
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Page 57
Illustration of effects of age and metallicity on isochrones (Worthey 1993), both higher age
and higher metallicity make the integrated colors redder and cause stronger metal
absorption lines (spectra get more dominated by K, M giants)
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Ages and
metallicities
Page 58
Lick indices Mgb , < Fe >, H β
to break the age-metallicity
[Z / H ] = log(Z / H ) − log(Z / H )e
degeneracy
Ze = 0.02
Trager et al. 2000 AJ, 120, 165
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The Bulge of M31
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Page 59
Saglia et al. 2010, A&A, 509, 61
Hot topics on galaxy formation and evolution 2
The M/L of Coma galaxies
Page 60
M/L from
dynamics
Average of
Salpeter
SSP M/L
M / LKroupa =
0.56M / LSalpeter
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The IMF changes
with sigma
Page 61
Thomas et al. 2011, Wegner et al. 2012
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The density profiles
Page 62
If all 'stellar' mass in excess of Kroupa is
'moved' to the DM profiles, they all become
isothermal...
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Jeans Modeling
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Page 63
Hot topics on galaxy formation and evolution 2
Highest density ellipticals
Page 64
4000 densest ellipticals from SDSS: dark matter not important
Salpeter IMF needed to explain the dynamical masses
Dutton et al. 2012, MNRAS in press
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Page 65
Masses of galaxies from gravitational lensing
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Basics of Gravitational Lensing
Page 66
The light path from the source to the observer can then be broken up into three distinct
zones:
1.
Light travels from the source to a point close to the lens through unperturbed
spacetime, since b « Dd.
2.
Near the lens the light is deflected.
3.
Light travels to the observer through unperturbed spacetime, since b « Dds.
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Hot topics on galaxy formation and evolution 2
Page 67
In a naive Newtonian approximation one would derive:
α=
v z 1 dΦ
1 dΦ
= ∫
dt = 2 ∫
dl
c c 
dz
c dz
∗
*: acceleration in z direction; because the acceleration doesn’t depend on the energy of
the photons, gravitational lenses are achromatic.
This result differs only by a factor of two from the correct general relativistic result:
 2 
α = 2 ∫ ∇ ⊥ Φ dl
G.R.
c


where the deflection angle α , written as vector α perpendicular to the light propagation l, is
the integral of the potential gradient perpendicular to the light propagation.
For a point mass the potential can be written as:
Φ(l, z) =
Therefore:
−GM
(l 2 + z 2 )1/2
dΦ
+GMz
= 2
dz (l + z 2 )3/2
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
= (∇ ⊥ Φ)
Hot topics on galaxy formation and evolution 2
Page 68
After integration:
2
α= 2
c
∞
∞
∞
⎤
GMz
4GMz
dl
4GMz ⎡
l
dl
=
=
∫−∞ (l 2 + z 2 )3/ 2
c 2 ∫0 (l 2 + z 2 )3/ 2
c 2 ⎢⎣ z 2 (l 2 + z 2 )1/ 2 ⎥⎦ 0
Thus the deflection angle α for a light ray with impact parameter b = z near the point mass
M becomes:
α=
4GM 2 RS
=
c 2b
b
where RS = 2GM/c2 is the Schwarzschild radius of the mass M, i.e. the radius of the black
hole belonging to the mass M.
Therefore for the sun (MΘ ≈ 2 · 1033 g → RS ≈ 3.0 km) we get a deflection angle α at the
Radius of the sun (≈ 700000km) of:
α ,R  1.7''

In order to calculate the deflection angle α caused by an arbitrary mass distribution (e.g.
a galaxy cluster) we use the fact that the extent of the mass distribution is very small
compared to the distances between source, lens and observer:
Δl « Dds and Δl « Dd
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Page 69
Therefore, the mass distribution of the lens can be treated as if it were an infinitely thin
mass sheet perpendicular to the line-of-sight. The surface mass density is simply obtained
by projection.
The plane of the mass sheet is called the lens plane. The mass sheet is characterized by
its surface mass density


∑ (ξ ) = ∫ ρ (ξ , l ) dl
Δl

The deflection of a light ray passing the lens plane at ξ by a mass element


2
dm = ∑ (ξ ')d ξ ' at ξ ' is:
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Page 70
dα =
4Gdm
 
2
c |ξ −ξ ' |
To get the deflection caused by all mass elements, we have to integrate over the whole
surface. Doing this we must take into account that, e.g., the deflection caused by mass
elements lying on opposite sides of the light ray may cancel out. Therefore we must add
the deflection angles as vectors:
 



4G (ξ − ξ ')∑ (ξ ') 2
 
α (ξ ) = 2 ∫
d ξ'
2
c
|ξ −ξ ' |
Special case: For a spherical mass distribution the lensing problem can be reduced to one
dimension. The deflection angle then points toward the center of symmetry and we get:
α (ξ ) =
4GM (< ξ )
c 2ξ
where ξ is the distance from the lens center and M(< ξ) is the mass enclosed within radius
ξ,
ξ
M (< ξ ) = 2π ∫ ∑ (ξ ')ξ ' dξ '
0
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Page 71
Lensing Geometry and Lens Equation
deflection angle just computed
Important relations:
αˆ ⋅ Dds = α ⋅ Ds
θ ⋅ Ds = β ⋅ Ds + αˆ ⋅ Dds
Note: The distances D are angular
diameter distances.
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Page 72
Using the previous two equations one obtains the so called lens equation:
β = θ −α = θ −
Dds
αˆ
Ds
The lens equation relates the real position (angle) of the source (without a lens) with the
position of the lensed image.
Important note: only angular distances are needed for deriving the lens equation. In general,
i.e. over cosmological distances: Dds ≠ Ds − Dd.
Consider now a circularly symmetric lens with an arbitrary mass profile. Due to the rotational
symmetry of the lens system, a source, which lies exactly on the optical axis
(θ = α ↔ β = 0 ) is imaged as a ring. This ring is the so called Einstein ring:
β =0 a
θ =α =
Dds
D 4G M (< ξ )
⋅ αˆ = ds ⋅ 2 ⋅
Ds
Ds c
ξ
The radius of the Einstein ring can be calculated using the previous equation and ξ = Ddθ:
Dds 4G
θ =
⋅ 2 ⋅ M <θE
Ds Dd c
2
E
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Einstein radius
Hot topics on galaxy formation and evolution 2
The SLAC survey
Page 73
Bolton et al. 2006,
ApJ, 638, 703
Sloan Lens ACS Survey: search for spectra
having multiple nebular emission lines
at redshift higher than the target galaxy.
Measure mass inside the Einstein radius
z=0.5812
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@0.32
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Page 74
Comparison Lensing-Dynamics
rEin ≈ 0.5re
rEin ≈ 0.75re
Dynamical models
with dark matter match
lensing results well
the dynamical M/L
ratios are good.
Dynamical models
without dark matter
underestimate the lensing
mass
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Page 75
The SWELLS survey
The Sloan WFC Edge-on Late-type Lens Survey
Brewer et al, astro-ph/1201.1677
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Spirals have a Kroupa/
Chabrier IMF
Stellar mass fraction >1 if Salpeter
Hot topics on galaxy formation and evolution 2
Constraints on the IMF from the
IGM metallities
The light of the (old,
Page 76
massive) ellipticals of
galaxy clusters comes
from ~solar stars.
The oxigen and silicon
in the IGM came from
massive stars that
exploded as SNII.
→ the M O / LB or the M Si / LB
ratios probe the IMF slope.
They are compatible with a
Salpeter IMF.
Renzini 2005
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Conclusions 1.
Page 77
•  The IMF observed in our MW is Salpeter power-law at M>Msol
and turns to Kroupa/Chabrier at smaller stellar masses.
•  GCs at low metallicities and/or Ultra Compact galaxies at high densities
might have a steeper than Salpeter IMF at high stellar masses
•  Speculations that larger numbers of low mass stars than Kroupa/Chabrier
might be present in high metallicity systems
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Conclusions 2.
Page 78
•  If stars are only produced in star clusters, then the IMF of a
galaxy is the sum of the IMFs of all star clusters produced there
•  A star cluster will be populated up to a maximal star mass, that
depends on the total mass of the star cluster
•  The maximal total mass of a star cluster is set by the current star
formation rate
•  Galaxies with low star formation rate histories will have steeper
IMFs, because they will seldon manage to build high mass stars.
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Conclusions 3.
Page 79
•  The EW of Ha carries information on the IMF slope (at high mass)
•  The modeling of the Ha EW and colors of SLOAN galaxies
suggests that the IMF slope is a function of the galaxy luminosity:
dwarf galaxies have steeper IMFs than giant spirals (at high mass).
•  Modeling of the stellar dynamics of elliptical galaxies allows to
constrain the luminous and dark matter content
•  Stellar M/L ratio from age and metallicity derived from line indices
( M / L )dyn / ( M / L )* indicates a Kroupa IMF at low σ and a Salpeter
(or even steeper IMF at low masses) at high σ
•  NIR indices suggest steeper than Salpeter IMF (at low mass)
in high s Es
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Page 80
Yet to come
Constraints on the evolution of the IMF with time might come from the
analysis of the evolution of elliptical galaxies --> see final lecture on
size evolution.
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