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GALAXIES 626
The Milky Way II.
Chemical evolution:
Chemical evolution
Observation of spiral and irregular galaxies show that the
fraction of heavy elements varies with the fraction of the
total mass which is in the form of gas:
heavy
element
abundance
gas fraction
What we need to model galactic chemical evolution
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Initial conditions (open or closed­box; chemical composition of the gas)
Birthrate versus mass SFRxIMF
Stellar yields (how elements are produced and inserted into the ISM)
Gas flows (infall, outflow, radial flow)
Gas Infall at the present time
Summary of Nucleosynthesis
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During the Big Bang light elements are formed, ●
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Spallation process in the ISM produces 6Li, Be and B Supernovae II produce alpha­elements (O, Ne, Mg, Si, S, Ca), some Fe
Summary of Nucleosynthesis
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Type Ia SNe produce mainly Fe and Fe­
peak elements plus some traces of elements from C to S
Type Ia SN nucleosynthesis
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A Chandrasekhar mass (1.44 Msun) explodes by C­
deflagration
C­deflagration produces 0.6 Msun of Fe plus traces of other elements from C to Si
Type II SNe
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Type II SNe arise from the core collapse of massive stars (M=8­40 Msun) and produce mainly alpha­elements (O, Mg, Si, Ca...) and some Fe
Stellar Yields
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We call stellar yield the newly produced and ejected mass of a given chemical element by a star of mass m
Stellar yields depend upon the mass and the chemical composition of the parent star
Stellar Yields
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Low and intermediate mass stars (0.8­8 Msun): produce He, N, C They die as C­O white dwarfs, when single, and can die as Type Ia SNe when binaries
Massive stars (M>8­10 Msun): produce mainly alpha­elements, some Fe, light and explode as core­collapse SNe
Primary and Secondary elements
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We define a primary element as an element produced directly from H and He
A typical primary element is carbon or oxygen which originate from the 3­ alpha reaction
We define a secondary element an element produced starting from metals already present in the star at birth (e.g. Nitrogen produced in the CNO cycle)
Simple Model and Secondary Elements
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The solution of the Simple model of chemical evolution for a secondary element Xs formed from a seed element Z
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Xs is proportional to Z^(2)
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Xs/Z goes like Z
Primary versus secondary
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Figure from Pettini et al. (2002)
Small dots are extragalactic HII regions
Red triangles are Damped Lyman­alpha systems (DLA)
Dashed lines mark the solution of the simple model for a primary and a secondary element
Stellar Yields
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Oxygen yields from massive stars
Different studies agree on O yields
Oxygen increases continuously with stellar mass from 10 to 40 Msun
Not clear what happens for M>40 Msun
Stellar Yields
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Yields for Fe in massive stars Stellar Yields
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Mg yields from massive stars
Big differences among different studies
Mg yields are too low to reproduce the Mg abundances in stars
The returned fraction
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We define returned fraction the amount of mass ejected into the ISM by an entire stellar generation
Instantaneous recycling approximation (IRA) is assumed , namely stellar lifetimes of stars with M> 1 Msun are neglected and all more massive stars evolve instantaneously
The Yield per stellar generation
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The yield per stellar generation of a single chemical element, can be defined as Where p_im is the stellar yield and the instantaneous recycling approximation has been assumed
Instantaneous Recycling Approximation
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The I.R.A. states that all the stars with masses < 1 Msun live forever but also that the stars with masses > 1 Msun die instantaneously (poor approximation)
I.R.A. affects mainly the chemical elements produced on long timescales (e.g. N and Fe)
POSSIBLE MODELS
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a) Start from a gas cloud already present at t=0 (monolithic model). No flows allowed (closed­
box) OR
b) Assume that the gas accumulates either quickly or slowly and the system undergoes inflows and outflows (open model)
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c) We assume that the gas at t=o is primordial (no metals)
OR
d) We assume that the gas at t=o is pre­enriched by Pop III stars
Simplest model is the one-zone model.
Consider the history of an annulus of a spiral galaxy
at some radius R. Make several simplifying assumptions:
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No material (gas, stars) enters or leaves the annulus
Initially the annulus contains only gas, with no heavy
elements (i.e. just hydrogen, helium)
As stars are formed, massive stars explode
`instantaneously’ as supernovae, returning
enriched gas to the ISM
Turbulent motions keep the gas well mixed, so
it has a single well-defined composition
How does the metal fraction of the gas evolve with time?
Let mass of interstellar gas in annulus be Mg
Mass of heavy elements in gas Mh
Define metallicity:
Zº
Mh
Mg
Suppose the mass of stars at this time is Ms
Imagine forming new stars, with mass ∆’Ms
Of these:
• Stars with mass M > 8 Solar masses explode rapidly
as supernovae, returning metals to the ISM
• Lower mass stars, with mass ∆Ms, remain
The mass of heavy elements produced by this episode of
star formation is p∆Ms, defining the yield p.
Total change in the mass of heavy elements due to
star formation is then:
DM h = pDM s − ZDM s = p− Z  DM s
existing metals locked up in
low mass stars
Corresponding change in metallicity is:
DZ =D
 
Mh
Mg
=
DM h
Mg
−
Mh
M 2g
DM g =
By conservation of mass:
DM s =­ DM g
1
Mg
 DM
h
−ZDM g

Combining the two previous equations get:
DZ =­ p
DM g
Mg
Can write this as a differential equation:
dM g
dZ
=­
Mg
p
If p is a constant (ie does not vary between subsequent
generations of stars), then this integrates immediately
to give:
Z  t  =­ p ln
[ ]
Mg t 
Mg 0 
Gas mass over
total mass
(using assumption that Z(t=0) was zero). Relation of this
form is very roughly what is observed in some galaxies.
Simple Model by Element
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If we assume that Xi is the abundance by mass of an element i, we have:
where Abundance ratios
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Under the assumption of the instantaneous recycling approximation the ratio of two abundances is equal to the ratio of the corresponding yields:
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When the I.R.A. is relaxed it is no longer true that the ratio between the abundances of two different elements is equal to the ratio of the corresponding yield because of the time dependent effects
The G-dwarf problem
A basic requiremenf for a model is that it
predict the relative numbers of low mass stars
with different metallicities. Does the simple model work?
All stars with metallicity < Z must have been formed at
time t < t1. Mass of such stars is:
[
]
M s ¿ Z  t 1  = M s  t 1 = M g  0 − M g  t 1 
¿ Mg
 0  [ 1−e
 / p ]
−Z t
1
Apply this analysis to the Solar neighborhood.
Ratio Mg(t) / Mg(0) is the ratio of the surface density of
gas to the total surface density of stars + gas near the Sun.
Approximately 0.1. Metallicity of this gas is Z ~ 0.02.
Derive p = 0.009 using previous equation.
Using this estimate of p, what mass of stars should have
metallicity Z < 0.25 Solar value (ie Z < 0.005)?
[
M s [ Z 0 . 005 ] =M g  0  1−e−0 . 005/ 0 . 009
]
Find that roughly half of the stars should be this metal poor.
Observations suggest that only 2% of F and G stars near
the Sun have this low a metallicity! G-dwarf problem.
What’s wrong with the model?
• p is not a constant (required trend seems unlikely)
• Region near the Sun is not a closed box - gas
enters and / or leaves
• Initial metallicity was non-zero
G­dwarf distribution Could radial flows in the disk do this?
Stars near the Sun
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Stars have random velocities w.r.t. LSR
Spiral structure increases random velocities over time
Hipparcos data
Spiral structure
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Local Standard of Rest (LSR) on circular(?) orbit around GC
Shifts stars radially
Sun has probably moved out ~2kpc
N­body simulation
Pollution
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Older stars have fewer heavy elements
Radial migration leads to big spread in [Fe/H] at given age
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Sun more metal­rich than local gas ­ even now
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Sun probably formed ~2 kpc nearer centre
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not nearly large enough effect however
Change the overall assumptions?
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The Simple Model of galactic chemical evolution assumes....
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One­zone, closed ­box model (no infall or outflow)
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IMF constant in time
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Instantaneous recycling approximation
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Instantaneous mixing approximation
The outflow law
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The rate of gas loss from a galaxy through a galactic wind can be expressed as:
Simple model with outflow
The Infall law
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The infall rate can simply be constant in space and time
Or described by an exponential law:
Simple model with infall
Complete Equations (including time dependence)
Definitions of variables
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dGi/dt is the rate of time variation of the gas fraction in the form of an element i
Xi(t) is the abundance by mass of a given element i
Qmi is a term containing all the information about stellar evolution and nucleosynthesis
Definition of variables
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A =0.05­0.09 is the fraction in the IMF of binary systems of that particular type to give rise to Type Ia SNe. B=1­A
Tau_m is the lifetime of a star of mass m
f(mu) is the distribution function of the mass ratio in binary systems
A(t) and W(t) are the accretion and outflow rate, respectively
The formation of the Milky Way
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Eggen, Lynden­Bell & Sandage (1962) suggested a rapid collapse lasting 300 Myr for the formation of the Galaxy
Searle & Zinn (1978) proposed a central collapse but also that the outer halo formed by merging of large fragments taking place over a timescale > 1Gyr
Different approaches in modelling the MW
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Serial approach: halo, thick and thin disk form as a continuous process Parallel approach: the different galactic component evolve at different rates but they are inter­connected Different approaches in modelling the MW
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Two­infall approach: halo, thick and thin disk form out of different infall episodes
Focus on this...
A scenario for the formation of the Galaxy
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One two­infall model assumes two main episodes of gas accretion
During the first one the halo and bulge formed, the second gave rise to the disk
Recipe for this two­infall model
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SFR­ Kennicutt’s law with a dependence on the surface gas density (exponent k=1.5) plus a dependence on the total surface mass density (feedback). Threshold of 7 solar masses per pc squared
IMF, Scalo (1986) normalized over a mass range of 0.1­100 solar masses
Exponential infall law with different timescales for inner halo (1­2 Gyr) and disk (inside­out formation with 7 Gyr at the S.N.)
Solar Vicinity
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We study first the solar vicinity, namely the local ring at 8 kpc from the galactic center
Then we study the properties of the entire disk from 4 to 22 Kpc Stellar Lifetimes
The star formation rate (threshold effects)
Stellar abundances
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[X/Fe]= log(X/Fe)_star­log(X/Fe)_sun is the abundance of an element X relative to iron and to the Sun
The most recent accurate solar abundances are from Asplund et al. (2005)
Previous abundances from Anders & Grevesse (1989) and Grevesse & Sauval (1998)
The main difference is in the O abundance, now lower
Predicted SN rates
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Type II SN rate (blue) follows the SFR
Type Ia SN rate (red) increases smoothly (small peak at 1 Gyr)
Time­delay model
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Blue line= only Type II SNe to produce Fe
Red line= only Type Ia SNe to produce Fe
Black line: Type II SNe produce 1/3 of Fe and Type Ia SNe produce 2/3 of Fe
Detailed predictions of the two­
infall model
Different timescales for disk formation
G­dwarf distribution Ok for this SFR history
Many constraints....
Abundance Gradients
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The abundances of heavy elements decrease with galactocentric distance in the disk
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Gradients of different elements are slightly different (depend on their nucleosynthesis and timescales of production)
How does the gradient form?
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If one assumes the disk to form inside­out, namely that first collapses the gas which forms the inner parts and then the gas which forms the outer parts
Namely, if one assumes a timescale for the formation of the disk increasing with galactocentric distance, the gradients are well reproduced
Abundance gradients
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Predicted and observed abundance gradients ●
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Data from HII regions, PNe and B stars, red dot is the Sun
The gradients steepen with time (from blue to red)
The Galactic Bulge
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A model for the Bulge (green line, Ballero et al. 2006)
Yields from Francois et al. (04), SF efficiency of 20 Gyr^(­1), timescale of accretion 0.1 Gyr
Data from Zoccali et al. 06, Fulbright et al. 06, Origlia &Rich (04, 05)
The Galactic Bulge
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Metallicity Distribution of Bulge stars, data from Zoccali et al. (2003) and Fulbright et al. (2006) (dot­dashed)
Models from Ballero et al. 06, with different SF eff.
The Galactic Bulge
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Models with different IMF
The best IMF for the Bulge is flatter than in the S.N: and flatter than Salpeter
Best IMF: x=0.95 for M> 1 solar mass and x=0.33 below
Conclusions on the Bulge
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The best model for the Bulge suggests that it formed by means of a strong starburst
The efficiency of SF was 20 times higher than in the rest of the Galaxy
The IMF was very flat, as it is suggested for starbursts
The timescale for the Bulge formation was 0.1 Gyr and not longer than 0.5 Gyr
Conclusions on the Milky Way
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The Disk at the solar ring formed on a time scale not shorter than 7 Gyr
The whole Disk formed inside­out with timescales of the order of 2 Gyr in the inner regions and 10 Gyr in the outer regions
The inner halo formed on a timescale not longer than 2 Gyr Gradients from Cepheids are flatter at large Rg than gradients from other indicators
GALAXIES 626
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