Composite Stellar Populations 1

Stellar Objects: Composite Stellar Populations 1

Composite Stellar Populations

1 Deﬁnition and problems

Deﬁnition and Problems

A Composite Stellar Population (CSP) is a collection of stars formed at diﬀerent time and with diﬀerent initial chemical compositions. Clearly, the properties of a CSP depends on the star formation rate (SFR) — the amount (total stellar mass) formed as a function of time — and the initial chemical composition (or Age Metallicity Relation - AMR) of the stars. The SFR and AMR together are called the star formation history (SFH).

If the SFH is known, together with the IMF, the CSP of a system (e.g., a galaxy or its bulge) is then determined. But in general, the SFH is not known a priori, depending the gas accretion and removal from this system as well as the star formation eﬃciency and the stellar mass ejection back to the ISM (stellar feedback). None of these processes are well understood. The stellar feedback itself can in principle be inferred from the stellar evolution modeling, although how it aﬀects the other processes is still unclear. Therefore, a study of the CSP draws on results of stellar evolution and on ideas of galaxy formation; thus it stands at a boundary between these two major areas of astronomical research. Here we discuss two simple examples of such studies.

2 One-zone instantaneous recycling model

2.1 Major assumptions

• Primary metals such as O, Ne, and S (in contrast to N, which largely < results from secondary production), produced by massive stars (t ∼ 50 Myr), are returned to the ISM with a yield of p, which is independent of Z. O, Ne, and S, among other elements, are ”primary” products of stellar nucleosynthesis, whose yields depend relatively little on the abundances in the progenitor star. Stellar Objects: Composite Stellar Populations 2

• The returned metals are mixed in the galaxy thoroughly and rapidly. Heavy elements are dispersed by flows propelled by supernova explo- sions and stellar winds, as well as differential rotation, large scale flows induced by bars, etc. The timescale of the mixing is probably much shorter than that of the star formation. (The star formation in the solar neighborhood is fairly steady over the past 8-10 Gyr.)

• The mass Ms in the remnants of massive stars and in low-mass stars remains locked within them throughout the galaxy’s lifetime.

• The gas inﬂow to (or outﬂow from) a galaxy has a rate that is propor- tional to the star formation rate, i.e., δMf = νδMs.

2.2 Model Construction

From mass conservation,

δMs + δMg = νδMs, (1) therefore, δM δM = − g . (2) s 1 − ν

The mass Mh of heavy elements in the gas can then be expressed as

p − Z + νZ δM = pδM − ZδM + νZ δM = − f δM , (3) h s s f s 1 − ν g where the metallicity is deﬁned as M Z = h . (4) Mg

The increase of the metal abundance of the gas is then

δMhMg − δMgMh p + νZf − νZ δMg δZ = 2 = − . (5) Mg 1 − ν Mg Stellar Objects: Composite Stellar Populations 3

A couple of simple cases:

• Inﬂow (ν > 0): Assuming Zf is independent of Z and is equal to Z(t = 0),

p M (t) ν/(1−ν) Z(t)= 1 − g . (6) ν Mg(t = 0)

< • Closed-box or outﬂow (ν ∼ 0): Assuming Zf is equal to Z(t),

p M (t = 0) Z(t)= Z(t = 0)+ ln g . (7) 1+ |ν| n Mg(t) o

Clearly, the outﬂow only reduces the eﬀective yield of metals returned to the gas. When ν = 0, we recover the close-box model as described in §10.1 of the text book.

In this case, the total mass of stars formed before time t

Ms(< Z)= Mg(0) − Mg(t)= Mg(0) 1 − exp{−[Z − Z(0)]/p} . (8)

2.3 Model applications

• The simple closed-box model gives a good ﬁt to the metallicity distri- butions of G and K giant stars of the Galactic bulge.

−2 • Near the Sun, the Galactic disk contains 30 − 40M⊙ pc of stars, −2 −2 together with about 13M⊙ pc in gas, for a total of ∼ 50M⊙ pc . The average abundance in the gas is about Z(now) ≈ 0.7Z⊙ Assuming Z(t =0)=0 and ν = 0, we have

Z(now) = pln(50/13). (9)

So p ≈ 0.5Z⊙.

• The Sun is more metal rich than the local disk gas → incomplete mixing and/or inﬂow of metal-poor gas. Stellar Objects: Composite Stellar Populations 4

Events involving massive stars are concentrated in young star clusters, leading to the potential for localized enrichment. Infall of primordial (or not so primordial) gas from the environment of a galaxy may involve accretion of gas clouds or dwarf galaxies, possibly leading to localized depressions of heavy element abundances. • The solar neighborhood G-dwarf problem: the closed-box model over- estimates of metal poor stars (Z < Z⊙/4)) if Z(t = 0) = 0:

M (< Z⊙/4) 1 − exp[−Z⊙/4p] s = ≈ 0.52, (10) Ms(< 0.7Z⊙) 1 − exp[−0.7Z⊙/p] compared to the observed fraction of ≈ 0.25 for a sample of G dwarf stars in the solar neighborhood. → pre-enrichment of the gas and/or late infall of metal-poor gas . • Globular clusters contain little gas and a lot of stars that all have the same metal-poor composition. → single generation of star and effective ejection of the rest of gas. • The high O/Fe ratio in the halo and thick disk. This suggests a brief formation period, early in the history of the Galaxy. The thin disk has lower O/Fe and a metallicity distribution consistent with prompt initial enrichment (perhaps from the thick disk) combined with infall of extragalactic gas. • Characteristic abundances increase with galactic luminosity. This trend embraces irregulars as well as spirals. This trend may involve the escape of nucleosynthesis products in winds from galaxies with low escape velocities. Indeed, effective yields for low surface brightness galaxies as a class fall below those for normal galaxies. • Galaxies in the Virgo cluster core, with marked H I deficiencies, are more metal rich than spirals with normal H I content located in the periphery of the cluster and in the field. This may result from the curtailment of metal-poor infall onto galaxies in the cluster core, where they are immersed in the hot cluster medium. In contrast, infall continues onto spirals in the cluster periphery and in the field, restraining the increase in abundances with time. Stellar Objects: Composite Stellar Populations 5

The evolution of galaxies is a complicated issues and is not well understood. Our Galaxy and other nearby galaxies provides the necessary test-beds for understanding various important physical, chemical, and dynamical processes, which are essential for us to interpret observations of more distant galaxies.

3 Star formation history based on CMD ﬁts

Statistical comparison of observed and synthetic CMDs can be used to de- termine the SFH and AMR, plus other parameters such as extinction and distance to a CSP.

Basic assumptions: 1) stellar models are sufficiently accurate as the function of mass, age, and metallicity and available for all the populations presented in the observed CSP; 2) the prescribed IMF is realistic and independent of age and metallicity; and 3) data quality is sufficiently good with errors measured or modeled. In addition, the distance modulus and extinction of the observed CSP are often assumed a priori, which could in principle be determined via the same fitting procedure.

The observed CMD is divided onto a grid with N cells of speciﬁed or variable sizes in both the magnitude and color directions; each cell contains No(i) counts, where i = 1, ..., N. The morphology of the CMD can be used to provide useful limiting ranges of the age t and metallicity Z of the CSP.

The model synthetic CMD of the CSP can be expressed as a linear combina- tion of a set of elementary stellar populations. Each elementary population j is assumed to have a uniform distribution of age and metallicity within inter- vals ∆t and ∆Z. The sizes of ∆t and ∆Z, centered around n discrete values of age t and m metallicities, are determined by the limited counting statistics and photometric errors of the observed CMD as well as the uncertainties in j the theoretic modeling. The CMD of such an elementary population Ne (i) can be most conveniently generated using MC techniques.

The synthetic CMD can be expressed as

j Ns(i) = ΣjajNe (i)+ f(i) (11) Stellar Objects: Composite Stellar Populations 6

where j runs from 1 to n × m and the weight aj is to be determined, while f(i) is the number of interlopers (foreground and background stars) in each of the CMD cell i and is typically estimated with the CMD of an observation oﬀ the CSP.

Other effects one needs to consider include unresolved binaries and blending, which broaden the CMD. One way to account for the binary effect, for ex- ample, is to randomly assign a fixed fraction of stars with companions. For such a selected star with a mass of Mpr, the mass of the companion Mcomp is drawn according to

Mcomp = [ran(1 − qc)+ qc]Mpr (12) where qc is the minimum value of the ratio Mcomp/Mpr and is usually assumed to be 0.7 and ran is a random number with a ﬂat distribution between 0 and 1. So Mcomp has a value between 0.7Mpr and Mpr. The magnitude in a generic −0.4MA(1) −0.4MA(2) band A is then MA = −2.5log[10 + 10 ], where MA(1) and MA(2) are the magnitudes of the system components. Software codes are available for generating the synthetic CMDs, incorporat- ing stellar evolution and bolometric correction libraries and providing colors and magnitudes of stars as a function of age, mass, and metallicity. A variety of inputs for the IMF, SFH, AMR, and binarity are allowed.

The comparison of the observed and synthetic CMDs is done using some merit function, e.g.,

2 2 [No(i) − Ns(i)] χ = Σi . (13) No(i) with the index i running from 1 to N. The merit function may be minimized for each assumed value of distance and extinction, in addition to giving aj.

The weights aj provide the best-ﬁt model for the SFR and AMR. Stellar Objects: Composite Stellar Populations 7

4 Review

Key concepts: CSP, SFR, AMR, SFH, stellar chemical yield

In what kinds of systems may the one-zone instantaneous recycling model be applied?

What does the “instantaneous recycling” mean?

What are the potential limitations of the model?

Why does the O/Fe abundance ratio tend to decrease with stellar age in a CSP?

What is the chemical abundance evidence for the escape of nucleosynthesis products from galaxies?

What evidence is available for gas infall into galaxies?

What might be the limitations of the CMD-ﬁtting method discussed to infer the SFH?

What are the basic steps in the method?

How can one set the ranges of the stellar age and metallicity in the modeling of an observed CMD?