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ASTR 5610, Majewski [SPRING 2018]. Lecture Notes

ASTR 5610 (Majewski) Lecture Notes


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.

The Metallicity Distribution Function is a hallmark of a stellar population.

The variable MDF between populations is an important component in our understanding of the evolution of galaxies like the Milky Way.

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.

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 in stars:


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:
  1. 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 conditions.

    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 separately).
    • 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 their evolution.

  2. 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 "G-dwarf problem" (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 closed-box context.
        • 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 the halo.

          I.e., the "closed box" may have to be enlarged to be the Milky Way itself.


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 where some 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.

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 d s / g, is not equivalent to the differential enrichment, dZ, with respect to gas mass reduction (-d g / 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:

This model has been applied to the halo:

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.

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?

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).


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...).

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 δt >> 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 s(<Z) = t(Z) - 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.

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 high yield.


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.

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) = Z i , 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 Z i 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 Z i .

Resolution of the classical G dwarf problem by way of pre-enrichment (dotted lines). (Note that the normalization on the abscissa is with respect to a star with 2 Zsun ).


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.


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 ).

This is not what is observed for disk stars:

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


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.

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.

What does the Hodge population box for the thick disk look like?

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:

Neutron Capture Elements (and Instantaneous Recycling Reprise)

Just briefly (due to lack of time -- but see Section 5.2 in Binney and Merrifield for more description).

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:

In a steady flow of neutrons, the abundance of each isotope is inversely proportional to its neutron capture cross-section.

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.

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.

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 of information.

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All material copyright © 2003,2006,2008,2010,2012,2018 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 551 and Astronomy 5610 at the University of Virginia.