THE METALLICITY DISTRIBUTION FUNCTION AND CHEMICAL EVOLUTION
References: Binney & Merrifield, Sections 5.2, 5.3, 10.7.2.
Also helpful are Carney in Star Clusters, Section 5.1, and McWilliam's
(1997, ARAA, 35, 503) nice review.
CHEMICAL EVOLUTION IN A CLOSED BOX
The Metallicity Distribution Function is a hallmark of a
For example, we know that the halo is characterized by
a mean [Fe/H] ~ -1.6 while the MDF of the disk is centered
closer to solar metallicity.
The variable MDF between populations is an important component in
our understanding of the evolution of galaxies like the Milky Way.
Under the assumption that metallicity increases with
time, we often depend on metallicity as a relative chronometer.
With something as complicated as the chemical evolution of a stellar population, let alone an
entire Galaxy, it is customary to start with
a simple model, then see how it needs to be modified to explain observations.
A simple model to explain the MDF in a population is one that pertains
to a closed-box.
In a closed-box, no material enters or leaves the zone of study.
As time progresses stars form from the ISM gas and
massive stars return hydrogen, helium and metals to the ISM.
We assume that turbulent motions of the gas keep it well
stirred and homogeneously mixed through time.
The supply of gas gets consumed with time as some fraction
gets locked up terminally into stellar remnants.
We typically start by assuming the instantaneous recycling
approximation: No time delay between formation of a generation
of stars and ejection of heavy metals into the ISM by that
generation's massive stars.
Generally ok when the major contributor is Type II
SNe from massive stars.
Not a good assumption when interested in the
products of Type Ia SNe or AGB stars, for example.
(Corrections for the time delay in this case can be added
to the overall theory.)
Under these assumptions, we find that the mean metallicity of the
system is driven by the yield of heavy metals put into the
ISM by each generation compared to the mass locked up
(As in the definition of Z in the Sun) let Z (t ) describe the metallicity of the
gas with time, which is the mass of heavy metals divided
by the total mass of the gas:
Z = h / g
(for the Sun, Z = 0.02).
That is, the fraction of the gas mass that is in heavy elements is
For simplicity, let's assume that the initial metallicity
is zero, we start with all gas and no stars, but the mass
in gas and stars are related by conservation of mass at all times.
The yield, p, is the relative increase in the heavy
elements produced for the amount of mass locked up in the combination of
(long-lived stars + stellar remnants), so that we can describe the mass of metals
produced in this generation of stars as
is the mass of
[long-lived + remnant] stars that remain after the massive stars of the generation
have died off.
(We assume that the difference between the total mass converted into
stars and the total mass in long-lived stars plus remnants is negligible --
i.e., that the fractional mass in massive stars [less their remnants] is small).
But there is also some withdrawal of metals from the system
and locked into stars,
Z δs, so
that the net change in the heavy element content of the gas is the
amount of metals produced by a star formation episode less the amount of metals that
get put into the stars made (remember, we're assuming instantaneous recycling):
δh = p δs - Z δs = (p - Z ) δs .
The metallicity of the system then changes by an amount (given by partial
differentiation of the
definition of Z = h / g given above)
δZ = δ (h / g ) =
(δh / g) - (h / g2)δg =
(δh - Z δg ) / g
but inserting the prior equation for
and using that δs =
and the yield is given by the differential enrichment, dZ,
with respect to the fraction of mass turned into stars
ds/g , or equivalently, by the differential enrichment, dZ,
with respect to gas mass reduction
-dg / g :
Then we integrate the relation for dZ above to obtain the metallicity at some time, t,
(assuming the yield is always the same):
In principle, if we are following chemical evolution from a pristine
initial state, then Z(0) = 0.
If we recall that at time t = 0 all of the mass was initially
in gas, and that mass is conserved, then
s(t ), so that:
Either of the preceding two equations suggest that the observed metallicity of
the gas in the system is directly related to the logarithm
of the fraction of the mass still remaining in gaseous form (defined as
Z = p ln (1 / μ) =
- p ln (μ)
The above analysis gave a relationship between the metallicity of the
gas and the fraction of mass in gas, but we can also predict
the metallicity distribution of the stars.
By the above equation relating Z(t) to
g , the mass of stars that have less than
metallicity Z (t ) is:
s [< Z(t )] =
s (t ) =
- g (t ) =
g (0)[1 - e-Z( t )/p ].
Therefore, at a time t the fraction of stars with
Z < Z (t ) is given by
where Z / p = ln (1 / μ)
is from above.
The above equation is important for generating the cumulative stellar metallicity
distribution function (i.e. how many stars have a metallicity less than
some value), which can be converted to a differential form (as shown in
the next figure).
Thus, this leads us to expect to see stars in a population
distributed over a wide range of
metallicity with a characteristic Metallicity Distribution Function
shape that resembles the following:
An example of an MDF produced from a closed box model and assuming a particular value
for the yield, from Carney in Globular
Clusters, SAAS-FEE Advanced Course No. 28. Stars
in the left hand part of the distribution were formed
when the system was mostly gas, whereas the metal-rich stars
were formed near the end of the gas conversion into stars.
Several things to note:
Initially, before many supernovae have gone off,
the probability of finding a star of any metallicity,
including low, is small.
As more gas is converted into stars, the
system becomes more enriched and the probability of finding
a more enriched star increases.
When more than half of the gas has been converted
into stars, the number of stars of higher metallicity
declines (by definition).
As the gas gets used up, fewer and fewer
increasingly enriched stars can be made.
The final MDF has a long tail to low metallicities.
The median metallicity of the stars created in the
closed box system will be a small fraction of the present
metallicity of the gas Z(t ):
This can be see, e.g., by inserting a small
value for μ, e.g.,
~0.1 (something like the fraction of mass in gas in
the solar neighborhood), into above equation.
We get that more than half of the stars in the
system have less than ~Z (t )/4.
As may be seen from the equation
Z / p = ln(1 / μ)
actual value of this median metallicity is driven
by the yield, p, and this is a function of the IMF.
Note that none of the above analysitcal analysis made any explicit reference to a specific time or timescale,
except for t = 0 and t = NOW.
That is, the resulting MDF is independent of the timescales for
star formation, and only relies on the amount of gas consumed, on whatever timescale that is accomplished.
IMPLICATIONS FOR GALACTIC STELLAR POPULATIONS
The Closed Box Scenario is a basic prescription for chemical evolution, and a starting,
simple model against which to test hypotheses.
There are several properties of the Galactic stellar populations
that show that a closed box model is not appropriate for them individually:
The Mean Metallicity Problem
Note the following result that can be derived from the above analysis:
(obtained by integrating [ln(μ) dμ] from 1 to μ and dividing by integral of
[dμ] in same range)
which approaches unity as μ goes to zero.
This means that the mean metallicity of a system, Z,
approaches the yield, p, as the gas is used up.
Now the yield is presumably related to the shape of the IMF (because it depends on the
number of high mass stars created), whereas we have seen that
the shape of the IMF seems to be fairly universal under a variety of
Then why don't the halo, IPII and disk populations -- and for that
matter, dSph galaxies -- all of which have μ nearly
or equal to zero, have exactly the same MDF?
For example, the mean [Fe/H] of dSphs and the halo are about -1.6 (see
figure), the IPII
is about -0.7, and the disk is near 0.
A halo field star MDF from
Carney (2001, in Star Clusters, SAAS-FEE Advanced Course
No. 28, Springer Verlag.
The obvious differences in the MDFs of the different
stellar populations points to the fact that a "closed-box" treatment
is not appropriate for each of these populations (considered
Clearly the differences can be explained if the halo
lost gas and therefore was not able to reach the same
net metal production as it would had it retained gas (see below
on where the halo gas may have gone).
Similarly, for dSph galaxies, gas must have been lost during
The G-dwarf Problem
In the above calculation for the solar neighborhood,
where μ ~ 0.1, we find that about half of the stars in the
solar neighborhood should have about 1/4 the metallicity of
the most metal rich local stars (say [Fe/H] ~ +0.2, as may be
seen in the figure below).
Thus we expect about 1/2 of the local stars to have metallicity
[Fe/H] < -0.4 or so.
Instead, we have far fewer than this:
Disk star MDFs from various authors (from
Chiappini et al. 1997).
As suggested in the above disk MDF, the median nearby metallicity
is [Fe/H] ~ -0.1.
The local MDF actually finds only a few percent of stars in the
solar neighborhood with metallicity less than the nominal closed-box
predicted median value of [Fe/H] ~ -0.4.
This difference between the observed and predicted MDFs among
local disk stars --
"the paucity of metal-poor stars in the
solar neighborhood with respect to that predicted by the closed box model"
-- is commonly referred to as the
(Gratton 1999, Chiappini et al. 1997), but obviously
it can apply to stars of other types (e.g., Mould 1982).
Think why it is the case that G dwarfs would be
the natural stellar type first used to study the problem and
see it reliably.
The classical G dwarf problem shown by way of cumulative distribution
functions (note that the normalization on the abscissa is with respect
to a star with 2 Zsun ). The solid line is the observed
data for stars in the solar neighborhood. The dashed line is the prediction
for a closed-box model.
A number of solutions to the G-dwarf problem have been proposed.
A variable yield, p with Z, which is the
same as saying that the IMF changes with metallicity.
For example, if we had bimodal star formation
where the early, metal-poor IMF only made higher mass stars,
then the G-dwarf problem could be solved within a
Unfortunately, this model runs into problems accounting
for metallicity gradients in disks, which require the
opposite trend for the IMF.
Thuan et al. (1975) experimented with variable
yields (IMFs) with time and found that the G-dwarf
problem is solved only by so strongly biasing the
early IMF against low mass stars that in order to get them
later on they must be formed at a rate so quick that
all of the gas gets used up in a few Gyr.
Infall of gas onto the disk.
We shall explore the accreting box model below.
A larger "closed-box" needs to be taken into account.
That is, the disk gas started out pre-enriched.
For example, the "closed box" may pertain to
all of the disk -- thin and thick -- and maybe even include
I.e., the "closed box" may have to be enlarged to
be the Milky Way itself.
LEAKY BOX MODEL
Both of the problems stated above can be reasonably accommodated
with modifications to the closed box scenario that allows outflows
and inflows of gas.
Let's take the latter example first, a leaky box model
mass is driven out from the system, say from supernovae.
Assume that the rate at which supernovae drive gas out of the system
is a function of the star formation rate:
where t is the total mass of
stars and gas in the system and η is a constant.
This makes some sense since star formation leads to the
creation of stars of all masses, and the highest mass fraction
We assume that we are in a low density environment (like in the
early extended Milky Way or an SZ fragment?)
where the escape energy can be imparted to the gas.
If we integrate this differential from 0 to time t we obtain
(assuming no stars initially):
The mass of gas in the box at any time is the total mass in the box at that time,
t , which has been reduced as just
above, less the stars. So:
As before, we have the change in metallicity given by the yield times
the differential production of gas turned to stars (remember, we
assume the gas is instantaneously recycled/enriched, even if some gas
then gets blown out), and substituting we have:
Unlike before, however, the increase in heavy element abundances, dZ
relative to mass locked up in stars ds / g, is not
equivalent to the differential enrichment, dZ, with respect to gas mass
reduction (-dg / g) because some gas is being lost from the system:
The net effect, compared to the closed box model, is to make the
effective yield smaller by a factor (1 + η).
The net effect is a reduced enrichment progression:
We obtain an equation for the number of stars less than a
certain metallicity in this case similar to that obtained before but
with a reduced, effective yield:
If s (Z=0) = 0, then t(0) = g(0), and we integrate the equation
for dZ to get:
Note extra (1 + η) factors...
For example, when the gas is all used up, we have created
a mass of stars =
t / (1 + η),
and we get a mean metallicity of
If the system loses no gas, η = 0 and
we reduce back to the original situation.
At the other extreme, if supernovae are efficient in
removing gas, η >> 1 and the mean abundance
becomes p / η.
Thus, the effective yield declines at the ease with which the
metal-enriched gas can leave the system.
This model has been applied to the halo:
If p ~ 0.02 (which is the solar abundance, where we
have assumed μ small and
η ~ 0 for the disk), then you find that for the
same p you need a halo η of ten or more
to get down to a halo mean metallicity.
The metallicity distribution for the halo can be reasonably accommodated
with a leaky box model that has an effective yield of:
peff = p / (1 + η) = 5.7 x 10-4
From Binney and Merrifield Figure 10.39.
A variant of this model is where the mass is not lost in simple proportion
to the star formation rate, but rather the "wind" is very powerful with respect to
the escape velocity of the system so that all of the gas is suddenly cleared
out in one event.
In this case you would have a terminated MDF.
From Carney, in Star Clusters.
Such a model might apply to something like a globular cluster
or a dwarf spheroidal if they formed slowly enough to let some
self-enrichment occur before formation of the catastrophic gas clearing.
Binney & Merrifield discuss how one can relate the mean enrichment
level with the velocity dispersion of the system, and use this to
explain the metallicity trends in elliptical galaxies, for example:
Here σZ is a parameter to
be obtained by observations.
The point is that as the stellar velocity dispersion,
σ* goes up, the mass of the
system is higher and more of the gas is retained.
Where Did the Halo Gas Go?
We see that a leaky box model can explain the MDF of the halo. Where did all of that gas
lost by the halo end up?
Interestingly, the mass of bulge ~ estimated mass lost from
halo (~2-3 x 1010 solar masses)
Moreover, the angular momentum distribution of the halo and bulge are more similar than
are the halo and disk populations:
Wyse and Gilmore (1992) figure from Carney's contribution to Star Clusters.
These points suggest that the bulge, NOT the disk, may have
received most of the relatively low angular momentum
gas from the halo (so the halo+disk would not form a "closed-box", but
But the bulge is now strongly enriched (mean [Fe/H] ~ -0.25),
having even super-solar metallicity stars.
because the gas is trapped deep in potential well -- transformation
of gas to stars more complete?
The above figure does suggest, however, that the G dwarf problem in the disk is not
solved by including effects from the halo (infall of gas or pre-enrichment by the halo).
ACCRETING BOX MODEL
The above diversion suggests that the halo gas didn't fall into the disk.
Nevertheless, one solution to the G-dwarf problem is the accretion of gas into
the disk (for now let's not question from whence it comes...).
Note that this gas need not even be pre-enriched --
if it happens to settle in at just the correct rate, will yield the same
result: the number of metal-poor stars can be diluted
by the creation of a large number of more metal-rich stars.
In the normal closed box model, the reason that the MDF winnows out at the
high Z end is that by the time enrichment progresses to very high levels,
there is simply not that much gas left to put into stars -- you just run out of gas.
In the accreting box model, we remove this problem by letting enrichment
proceed, but we don't let the box run out of gas.
Imagine a system that has progressed through chemical enrichment
by, say, 90% gas consumption.
At this point roughly half the stars will have less than 1/4 the
current Z level.
A "trick" is to relatively slowly introduce gas.
Imagine a small incremental addition,
of primordial gas fed in at
a steady state rate equivalent to the rate at which gas is being turned
into stars in the box.
At the same time that δ gets locked
into stars at current metallicity Z, a mass
p δ of freshly
made heavy elements is returned to the system at metallicity
If we keep feeding the system at the rate stars are forming,
the net metallicity of the gas will eventually reach Z = p
(i.e., we keep resupplying gas to replace that which is being used to churn
out heavy elements at the same nominal yield, p > Z[t] ).
By introducing a constant (but not huge) supply of gas, we
are able to make more stars out of the ever higher metallicity gas,
and also have enough supply of gas to keep making lots of stars to contribute
more yield to continue to enrich the gas.
After a sufficiently long time, when the gas has enriched to the level of
the yield, we produce stars of metallicity p, and these dilute out by
sheer numbers the
initially made, lower metallicity stars.
Formulated mathematically, we still have the equations from the closed box
situation giving the rate of metal production and enrichment as:
δh = p δs - Z δs = (p-Z ) δs .
δZ = δ (h / g ) = (δh - Z δg ) / g
But now the total mass is varying (for now, let's do general math, not assuming the rate
of the gas infall is tuned to the rate stars are made):
Eliminating δh and
δs we get (i.e., combining
the last three equations):
The last term on the right is what we got from the closed box model, but now
we have the extra (p-Z) δt /
g term that involves the effect of adding the
new mass in gas.
This can be turned into the differential equation:
Binney & Merrifield show you how to solve this with a change of variables
and obtain for the case described above of a constant gas mass (i.e.
δg = 0)
and initial gas metallicity of zero:
(g = constant)
One can see, as in the verbal description above, Z --> p when
δg (i.e. when lots
of gas is eventually added over time).
In this scheme, where g is being held constant,
the number of stars more metal poor than Z is
g ; we can
insert the above equation to find:
(g = constant)
For the solar neighborhood, where only 10% of the mass is in gas (i.e.,
t (t0 ) /
g = 10), we find that the
actual yield must then be about the current enrichment level p ~ Z.
This is actually pretty high!
Since the mean Z in the solar neighborhood is about solar, we then find:
(g = constant)
which is a number close to that observed and so we see we have a
reasonable solution to the G dwarf problem, albeit by invoking a very
We saw one solution to the G dwarf problem is the accreting box model, where
there is continuous infall of matter, though it suggests a very high yield.
Another way the problem could be resolved is by the thin disk inheriting heavy elements
already created in the halo and thick disk.
Recall that we found in the closed box model:
If we just used this equation (i.e., Z = -p ln(μ)) for the disk
(μ=0.1), we would
obtain an effective yield for the disk of p = 0.009, which is about double
the yield one actually observes in star forming galaxies like the Magellanic Clouds.
Since we believe in a fairly universal IMF, the differences in yield
Therefore, let us see what happens when we include pre-enrichment.
In this scenario, when we integrate to account for the disk, we do not
assume Z (0) = 0 (which gives the equation just above),
but rather some pre-enriched value
Z (0) = Zi , a metallicity
below which no disk stars will have:
The thick disk metallicity is about [Fe/H]=-0.7 (a quarter solar = 0.005), and the current
Z is about 0.02; if we use these values for Zi we then
get p = 0.0065, or more similar to the Magellanic Clouds.
Thus, we can account for differences in metallicity in different star forming environments
while still retaining the features of the closed-box scenario if we just deduce different
For example, in the Magellanic Clouds we would use
Zi = 0, whereas in the Milky Way disk
we would pre-enrich to, say, a quarter solar.
The Milky Way disk model works, then, as long as pre-enrichment by a
thick disk takes place.
Resolution of the classical G dwarf problem by way of pre-enrichment (dotted
(Note that the normalization on the abscissa is with respect
to a star with 2 Zsun ).
IP II, THICK DISK ABUNDANCES
If the thick disk is the "pre-enrichment" phase of the the
thin disk, its MDF should resemble the "missing part" of the closed box model
for the thin disk MDF.
As we have said, it has an apparent peak [Fe/H] ~ -0.7
It also has the very metal poor tail, [Fe/H] < -2.0 -- "metal-weak thick
disk" (Norris 1986, Morrison et al. 1990, Majewski 1992, Beers &
Sommer-Larsen 1995, Morrison & Martin 1998)
There is a small vertical abundance gradient within the thick disk:
d[Fe/H] / dz <~ -0.10 dex/kpc (~ small), but things become more
metal rich with the transition into the thin disk.
Metallicity gradient for the disk by Yoss et al. (1987).
Note, all of this is rather hampered by the usual separation problems:
Thin disk: σ [Fe/H] large,
Halo: σ [Fe/H] large,
References: Yoss et al. 1987, Majewski 1992, Sandage & Fouts 1987,
Yoshii et al. 1987.
It is difficult at present to know how properly to isolate thick disk and
halo stars, which have overlapping MDFs, and the transition from thin to
thick disk MDF is, of course, vague.
A promising avenue to do this seems to be by taking advantage of chemical abundance
patterns -- i.e., looking at how multiple elements track one another (e.g.,
alpha-elements versus iron -- see below).
CASE STUDY: THE DISK AGE-METALLICITY RELATION and the BREAKDOWN OF THE
In the "well-mixed" assumption, the mean abundance of the ISM rises
steadily with time, and all stars formed at time t should have
abundance Z (t ).
That is, there should be a well-defined Age-Metallicity Relation
This is not what is observed for disk stars:
No real AMR seen for disk stars
(Marsakov et al. 1990, Edvardsson et al. 1993) or open clusters
At any age - wide abundance spread (Geisler et al. 1992)
Overall mean increase only few 0.1 dex over large age range.
This suggests that the ISM of the disk has not, in fact, been
chemically well mixed over its lifetime.
Of course, stars are visiting the solar neighborhood from other
mean radii, depending on the ellipticity of orbits.
The overall age-metallicity relation in the Milky Way, showing the bulk
properties and approximate ranges of the different Galactic components. The obvious
trends of metallicity, [Fe/H], to increase with time and spatial concentration
toward the Galactic plane and central region suggest that the components of the
Galaxy formed and evolved in a coherent and continuous process. However, the
dispersion of the relation is everywhere so large that for some components,
different independent origins and evolutionary connections are possible.
Figure and caption from Buser 2000, Science, 287, 5450, 69.
But, there are radial abundance gradients found in the disk: -0.091 ± 0.014 dex/kpc
This is a common feature of disk galaxies.
One also finds that the gas fraction tends to be lower in the
center of the galaxy than in the outer parts.
This is roughly consistent with expectations from the closed box
Z = p ln (1 / μ) =
- p ln (μ)
Suggests that over the history of the disk, radius played
more important role in determining chemistry rather than age.
CASE STUDY: BREAKDOWN OF THE INSTANTANEOUS RECYCLING APPROXIMATION and
An important (and useful) example of how the instantaneous recycling approximation
breaks down is exhibited by the trend of [O/Fe] ratios with [Fe/H].
An early plot of [O/Fe]-[Fe/H] from the late 1980s.
A 2018 version of the same plot using APOGEE data (figure courtesy of Chris Hayes). Note how
the APOGEE data show at least three distinct groups of stars in this plane.
As proposed by Tinsley (1979), Type Ia supernovae (resulting from mass
accretion in a binary system onto a C-O white dwarf) are in fact the primary
producers of iron in the Galaxy.
On the other hand oxygen is primarily produced by Type II supernovae, which
come from high mass stars.
The same holds true for other α elements (nuclei that are combinations
of α nuclei): O, Ne, Mg, Si, S, A, Ca, (and Ti, Cr).
(Note that these are called primary nuclides because they can be made
in a star that starts out with only H and He. Secondary nuclides are those
that are created using a primary nuclide formed in an earlier stellar generation.)
(Top Row:) APOGEE DR14 (from 2018) plots of three other alpha elements, Mg, Si, and Ca.
(Bottom Row:) The same plot but now for the "integrated alpha" lines in the APOGEE survey.
Note how the various alpha elements show the same overall character in these representations, but with
some variations in detail.
(Figures courtesy of Chris Hayes).
The figures above shows that there is a sudden change in the [alpha/Fe]
ratios of Milky Way stars around [Fe/H] ~ -1.
Apparently, there is a transition in the source of heavy elements contributed
to the ISM (from Type II to Type Ia supernovae) at the
point where the galaxy has enriched to about [Fe/H] = 1.
This is an interesting feature, because 5 solar mass stars
are the likely main contributor of Type Ia SN, and these have a
main sequence lifetime of ~1 Gyr.
Thus, the above plot suggests that it took
approximately 1 Gyr for the Galaxy to enrich to
[Fe/H] = -1.
If this metallicity is characteristic of the IPII,
we see an interesting timescale imprinted in the
Note that the Delay Time Distribution of Type Ia supernovae is still
debated, as is the primary contributing progenitors -- e.g., from Single Degenerate (SD, meaning
one white dwarf and a red giant in a common envelope with accretion onto the WD) or
Double Degenerate (DD, two white dwarf stars merging).
Here is a plot showing a compilation of results from different assessments of the
Delay Time Distribution, using the rates of observed supernovae Ia in other galaxies
as a guide.
Time Delay Distribution from Type Ia supernovae, from Howell (2011, Nature Communications 2, Article 350). Here is the original caption (see Howell for
Adpted from Maoz, Sharon, and Gal-Yam. All solid circles except the first are delay times derived from cluster SN rates, assuming formation at z=3. The first solid circle results from a constraint on the observed iron-to-stellar mass ratio in clusters. Blue stars are derived from SN rates and galaxy stellar populations in the Lick observatory supernova survey. Green triangles (including an arrow denoting the 95% confidence upper limit) are constraints from SN remnants in the Magellanic Clouds. Red squares are based on SN Ia candidates in E/S0 galaxies at z=0.4-1.2. Curves are power laws t-1.1 (solid), and t-1.3 (dashed), constrained to pass through the last point. The prediction from DD mergers is t-1. Reproduced by permission of the AAS. "
Nevertheless, whatever the Time Delay Distribution, clearly oxygen better satisfies the "instantaneous recycling"
approximation than does iron.
But this particular breakdown of the instantaneous recycling approximation
is a very useful one, since the "knee" in the [O/Fe] to [Fe/H] distribution
can be associated with a specific time.
The position of the knee thus provides information on the star formation
rate in a system.
If the SFR is high, then the gas will be able to reach a higher [Fe/H]
before the first SN Ia occur, and the knee will occur at a higher
The fraction of stars at [Fe/H] less than the knee gives
information on the star formation timescale.
From A. McWilliam (1997, ARAA 35, 503).
So, for example, what can we infer about the relative star formation
histories of the Milky Way, the LMC and the Sgr dSp based on the following plots:
Typical distribution for [O/Fe] to [Fe/H] for Galactic and
LMC stars from Smith et al. (2001).
From APOGEE data: Red points are Sgr dSph stars compared to the rest from the MW (by Chris Hayes).
Note that the IMF of the stellar system can also be inferred from
the above diagram, since theoretical predictions of the SN II element yield
shows that [α/Fe] increases with increasing progenitor mass
(Woosley & Weaver 1995).
If a system has an IMF skewed to high mass stars, then the
[α/Fe] ratio will be higher.
From A. McWilliam (1997, ARAA 35, 503).
In fact, Si, S, Ca, Ti, Cr, while Type II SN products, are
not as heavily weighted towards most massive progenitors.
From A. McWilliam (1997, ARAA 35, 503).
Thus, comparison of these elements to, e.g, O or Mg, gives
information about relative numbers of massive stars of different masses
in the IMF (and potentially map the history of the IMF shape).
Note this example for stars in the Galactic bulge, which compares
[(Mg+Ti)/Fe] to [(Si+Ca)/Fe] abundances to the [α/Fe] sequence
for the disk.
The enhanced Mg/Fe suggests that the bulge may have been enriched by
relatively more 35 solar mass supernovae.
From A. McWilliam (1997, ARAA 35, 503).
Another important comparison is the Milky Way trends of [α/Fe] to
[Fe/H] compared to the dSph trends:
The traditional view that has developed is that the [α/Fe]
ration in dSphs are diminished with respect to Milky Way stars
(particularly in the range of [Fe/H] for halo and thick disk stars), as
e.g., in this plot:
Colored dots representing halo (cyan), thick disk (green) and thin disk
stars (red) from the Milky Way (from Venn et al. 2004)
compared to stars from various Milky Way dSph
galaxies (from Shetrone et al. 2003, Geisler et al. 2004). The small
black points represent extremely retrograde stars and the small blue
dots have "high energy" in the Toomre diagram; both of these components
of the halo are suspected to be accreted.
The implication that has been drawn from such comparisons is
that dSphs therefore cannot be the primary contributors of stars
to the halo/thick disk.
This has been a longstanding problem for the "dSphs as Searle & Zinn
However, it is becoming apparent that this view was shortsighted.
As people determine the [α/Fe] ratios for the most metal poor stars
in dSphs, they are starting to see [α/Fe] enhancements comparable
to those seen in halo stars.
An expanded abundance distribution in the Sculptor dSph galaxy to the most
metal poor Sculptor stars from Geisler et al.
(2005, ApJ, 129, 1428) and Shetrone et al. (2003, AJ, 125, 684).
When one considers that it is these oldest, most metal-poor stars
that are the most likely to be contributed by a dSph to the halo
(i.e., the population in a dSph longest subject to tidal forces and
stripping and also typically the population at the largest radius
in a dSph) then it makes sense that the halo most closely resembles
these oldest dSph stars (Majewski et al. 2002, ASPC, 285, 199).
This point is especially obvious (again) by looking at the evolution
of the MDF in the Sgr tidal tails (Chou et al., ApJ, submitted):
The fact that stars contributed to a halo by a dSph may look nothing like
the stars currently in the dSph is evident by the vast difference in the
MDF seen in the Sgr core (top panel) ad in the Sgr tidal tails (lower panels).
Finally, that many of the [Fe/H] < -1 stars in dSphs are
not α-enhanced tells us what about the relative enrichment history of
dwarf spheroidal galaxies compared to the Milky Way?
Does this make sense in light of the information we discussed above?
Chemical Evolution of the Disk
Notice the relative timescales suggested for the formation of the disk components
of the Milky Way implied by the Venn et al. figure of [α/Fe]-[Fe/H] above.
Notice that the thick disk stars mainly populate the point after the
contribution of Type Ia supernovae.
This suggests that the bulk of the gas out of which thick disk stars formed
already had contributions from Type Ia SN and therefore must have formed
from fairly enriched gas more than ~1 Gyr after the first star formation.
Notice also the smooth progression of abundance patterns between thick
and thin disk stars.
Does this imply that these two components share a common evolution (i.e., are
two extremes of the same structural component?
What does the Hodge population box for the thick disk look like?
Combine disk AMR, n[Fe/H], SFR(t)
Combine IPII age, n[Fe/H]
IPII shown as distinct burst ("D") at 12 Gyr
Just one of a series of disk bursts?
Or distinct in other way?
Broad [Fe/H] distribution at all ages
Results from the APOGEE project have made provided major new insights into the evolution of the disk, many of which are so
new that the models have not yet been able to definitively made to explain the results.
By working in the infrared, where extinction of the disk is minimized,
APOGEE can provide maps of the chemical distributions of stars across
all parts of the Milky Way, and allowing the ability to study the variations in chemistry with position,
enabling reconstruction of the chemical history of different stellar populations.
Some of the discoveries include:
Clear signatures of metallicity gradients across the disk.
The gradient of metallicity across the disk. From Hayden et al. 2014, AJ, 147, 116.
Clear evidence of age variations across the disk, which strongly indicate a steady and
consistent "inside-out" growth of the disk.
New abilities to measure the ages of individual stars through the
methods of asteroseismology, gyrochronology, and C/N abundance ratios. Age is perhaps the most important
parameter for any investigation of evolutionary processes, but until recently we could generally
only age date clusters of stars reliably. These new abilities to measure ages for individual
field stars across the Galaxy is radically advancing our ability to explore this history of our
Galaxy's stellar populations.
Video showing the distribution of stellar ages measured from C/N abundance ratios (calibrated by
stars having ages from asteroseimology) in giant stars as
measured by the APOGEE survey. Colored dots show the 70,000 APOGEE RGB stars by age. As the video
shows, the oldest stars are found in the Galactic center, while younger stars are seen farther out in the
disk. This suggests that the Galaxy grows from the inside-out. Data and video from Ness et al. (2016).
A clear bifurcation of the disk chemical evolution sequence into "high alpha" and "low alpha"
sequences that overlap at highest metallicities.
Left: the [alpha/Fe] vs. [Fe/H] diagram for our sample of APOGEE RC stars with S/N > 150 (3954 stars).
The bimodality in [alpha/Fe] at low [Fe/H] is clearly visible and extends over ~0.6dex in metallicity.
The error bar in the lower left corner shows representative precisions of 0.05dex in [Fe/H] and 0.027dex in [alpha/Fe].
Right: a schematic of this [alpha/Fe] vs. [Fe/H] diagram showing the main features: the low-alpha group (red),
high-alpha group (blue), and an intermediate-alpha "valley."
The hashed red/blue shows the overlap region between the low- and high-alpha stars. The black dots are RC stars with S/N > 150.
From Nidever et al. (2014).
Nidever et al. interpret this pattern as "requiring ... a mix of two or more populations with distinct enrichment histories."
The high alpha sequence can be modeled as a leaky box model, and with the "star formation
efficiency (SFE)" defining the star formation rate as SFR=(SFE)*Mgas, as follows:
Fiducial GCE model for the high-alpha sequence with SFE=~4.5 × 10-10 yr-1.
In the figures below,
the labeled open circles indicate the time in Gyr, to show the change in abundance of each model as a
function of time.
Here is how it looks when one varies the SFE and the η (mass loss) parameter:
Tracks in [alpha/Fe] vs. [Fe/H] for GCE models with varying star-formation efficiency (SFE, top) and outflow rate
(=η×SFR, middle), as marked in the legend. The labeled open circles indicate the abundance of each model
at the given time in Gyr. The APOGEE-RC stars (S/N > 150) are shown as filled black points. The heavy black curve
indicates the fiducial model that best fits the observed high-alpha sequence. (Bottom)
The star-formation history of the GCE models normalized by the SFR of the fiducial model at t= 2 Gyr.
Nidever et al. conclude that the high alpha sequence is easily modeled
with simple Galactic chemical evolution models and an
"average" SFE of ~4.5 X 10-10 per year (which is similar to the nearly constant
value found in the molecular-gas-dominated regions of nearby spirals).
Note that this high alpha sequence is surprisingly constant in shape throughout the Galaxy
(see figure showing the abundance patterns across the Galaxy below) .
"This result suggests that the early evolution of the Milky Way disk was characterized by stars that shared a similar star-formation history and were formed in a well-mixed, turbulent, and molecular- dominated ISM with a gas consumption timescale (SFE-1) of ~2 Gyr."
However, the low alpha sequence is a bit more difficult to understand.
Clearly we need a distinct, second population of stars, at least in the outer Galactic
One way to potentially explain the low-alpha sequence is with the accretion of "pristine" gas
about 8 Gyr ago. This makes the end of the high-alpha sequence ``jump back" to the low metallicity
end of the low-alpha sequence suddenly.
This scenario may be difficult to match to age distributions of stars in the disk.
Below shows an alternative of two other GCE models with parameters chosen to approximately reproduce the observed locus of the low-# stars with a single chemical evolutionary sequence. This can sort of be done...
but how to explain the sudden turn on?
Two GCE models intended to reproduce the low-alpha sequence in the [alpha/Fe] vs. [Fe/H] plane. The blue line shows a model with an outflow rate of 2.5×SFR and SFE = 2 × 10-10 yr-1, which reaches the density peak of the low-alpha sequence. The red line shows a model with a time-varying outflow rate (η = 3 for t < 4 Gyr and η = 1.5 for t > 4 Gyr) that runs through the whole low-alpha sequence. The labeled open circles indicate the abundance of each model at the given time in Gyr. The APOGEE-RC stars are shown as filled black points.
Clear variations in the chemical abundance patterns across the disk.
APOGEE results for
the [α/Fe] vs. [M/H] diagram for 70,000 red giant stars across the Galactic disk, broken up in
zones of radius and distance from the Galactic plane. From Hayden et al. (2015).
The effects of orbital blurring (due to the secular evolution in stellar orbits to become more elliptical
over time) and churning from
radial migration of stars likely plays an important role in the present distributions
of stars in the Galaxy, which then effects the local age-metallicity relations, and other chemodynamical distribution
Even the distribution functions for individual elements (here, iron) for different parts
of the Galaxy hold important clues to post-star formation mixing of stars from place to place.
In this figure showing the Metallicity Distribution Function across the Galaxy, we see
the variation in the distribution shape, which Hayden et al. (2015) interpret as having to do
with radial migration of stars to different radii in the disk. From Hayden et al. (2015).
A model showing the difference between the current stellar radius (green stripe = solar neighborhood) and birth radii
(black dashed line for all stars, but colored lines for stars of different ages).
From Minchev et al. (2018, A&A, submitted).
Based on a technique where local stars of a given age are tracked to their birth radius under the assumption of the
evolution of the Galactic radial metallicity gradient over time, solar neighborhood stars here are coded by their
age (left), birth radii (center) and orbital guiding radii (right). The two rows are two different stellar samples.
From Minchev et al. (2018, A&A, submitted).
Neutron Capture Elements (and Instantaneous Recycling Reprise)
Just briefly (due to lack of time -- but see Section 5.2 in Binney and Merrifield for
Normal nucleosynthesis in high mass stars ends with the production of iron, which has the
highest binding energy of all atoms.
Nuclei of elements beyond the "iron peak" cannot be efficiently produced by charged-particle
interactions because of the large Coulomb repulsion between nuclei.
Instead, these are synthesized by successive neutron captures onto iron peak nuclei (neutrons
not affected by Coulomb repulsion),
followed by β decays.
This can happen along two paths that produce two characteristic
abundance patterns depending on ambient neutron density:
s-process (slow neutron capture) occurs at relatively low neutron densities
(~106-1011 cm-3) and so when the neutron captures occur on a
timescale (100-105 years) long enough that after each neutron capture, the
product nucleus has time to decay if it is unstable, and all potential
β decays have time to occur.
s-process element paths.
From Pagel's Nucleosynthesis and Chemical Evolution of Galaxies.
r-process (rapid neutron capture) occurs in environments with an intense
neutron flux (densities >1020 cm-3)
so that the captures happen on timescales short (0.01 to 10 seconds)
compared to β decay.
This produces more neutron-rich isotopes, as seen below.
From Pagel's Nucleosynthesis and Chemical Evolution of Galaxies.
Another version of the
chart of the nuclides, from Sneden & Cowan (2003, Science, 299, 5603, 70).
At a given proton (atomic) number,
isotopes toward the left are proton rich, and those to the right are the
neutron-rich ones that are the subject of this article. The stable nuclides
are marked by black boxes; n-capture in s-process synthesis occurs near these
nuclei close to the "valley of β-stability." The jagged diagonal black line
represents the limit of experimentally determined properties of nuclei, and the
magenta line represents the r-process "path." Vertical and horizontal black
lines represent closed neutron or proton shells, sometimes referred to as
"magic numbers." Color shading denotes the different (log) time scales for
In a steady flow of neutrons, the abundance of each isotope is inversely proportional to
its neutron capture cross-section.
The closed neutron shells with 50, 82, and 126 neutrons have small
neutron capture cross-sections, leading to abundance peaks near these
points (see inset to figure above).
From Pagel's Nucleosynthesis and Chemical Evolution of Galaxies.
Similarly, even numbered nuclei have smaller cross-sections than odd
nuclei, leading to an odd-even effect in the abundances of elements.
The s-process is thought primarily to occur during thermal pulses in the intershell convection
none (between the He- and H-burning shells) in low mass (1-3 solar mass) AGB stars.
In these pulses, neutrons are released by α capture onto 13C --
These neutrons are captured by pre-existing iron seed nuclei already in the
atmosphere of the star from previous generations of stars.
Obviously, then, this process does not begin to enrich the ISM until
1-3 solar mass stars from a second generation of stars evolve off the main
sequence (after several billion years).
The site of the r-process is still debated, but thought to be associated with core collapse
supernovae or neutron star mergers.
As you may suspect, investigation of these heavy nuclei provides a much more detailed
picture of the star formation history of a galaxy.
For example, the s-process element barium shows a steady increase in [Ba/Fe]
with increasing [Fe/H], showing the delay in Ba production from the lower mass stars
having main sequence lifetimes of several Gyr.
[Ba/Fe] trends with [Fe/H]. The horizontal scale stretches from [Fe/H]=-1.1 to +0.3.
From Edvardsson et al. 1993, A&A, 275, 101.
On the other hand, europium, a pure r-process element, shows a [Eu/Fe]
trend with [Fe/H] that looks very much like what is seen for [α/Fe].
This is consistent with the notion that both r-process and α elements
are made in Type II supernovae.
From McWilliam, 1997, ARAA, 35, 503.
In the end I hope it is evident why there is so much excitement in the use of detailed
chemical abundance analyses in interpreting the evolution of galaxies, since information
on the IMF, SF timescales, star forming sites, etc. might be gleaned from this type