# Evolution of Stars and Gas in Galaxies

Beatrice M. Tinsley
Yale University Observatory

(First published 1980, in Fundamentals of Cosmic Physics, Volume 5, pp.
287–388)

††pubyear: 1980††pagerange: Evolution of Stars and Gas in Galaxies–References

##  1 Overview

Essentially everything of astronomical interest is either part of a galaxy, or
from a galaxy, or otherwise relevant to the origin or evolution of galaxies.
Diverse examples are that the isotropic composition of meteorites provides
clues to the history of star formation billions of years ago, and cosmological
tests for the deceleration of the Universe are strongly affected by changes in
the luminosities of galaxies during the lookback time sampled. The aim of this
article is to review some of the vital connections that galaxy evolution makes
among many astronomical phenomena.

The evolution of galaxies can be broadly divided into three areas: dynamical
evolution, chemical evolution, and the evolution of photometric properties.
Chemical and photometric evolution are the main topics of this review, and
dynamical processes will be considered only where they directly affect the
others. The article is intended to be self-contained for readers with a
general knowledge of astronomy and astrophysics but with no special expertise
on galaxies. Because the background and literature relevant to the evolution
of stars and gas in galaxies are so extensive, complete coverage cannot be
attempted. Instead, comprehensive reviews and recent papers are emphasized in
the references; these include detailed works in closely related fields, such
as galaxy formation and nucleosynthesis, that are treated very briefly here.
The history of the subject has been discussed in several recent reviews (e.g.
Sandage et al., 1975; Spinrad & Peimbert, 1975; Trimble, 1975; van den Bergh,
1975), so here the emphasis will be on current ideas.

This first Section gives an overview of relevant properties of galaxies and
theoretical ideas on how they formed and evolved, followed by an outline of
the rest of the article.

###  1.1 Stellar Populations and Chemical Composition of Galaxies

Galaxies are made of stars and interstellar matter (ISM), with a variety of
properties that correlate remarkably with the forms of galaxies as they appear
on the sky. Introductory references to supplement the following brief outline
include the classic lectures of Baade (1963) and subsequent reviews by Morgan
& Osterbrock (1969), King (1971, 1977), van den Bergh (1975), and Sandage et
al. (1975). The _Hubble Atlas of Galaxies_ (Sandage, 1961a) is an essential
source book for photographs and descriptions of typical galaxies, while the
_Atlas of Peculiar Galaxies_ (Arp, 1966) illustrates a great variety of
unusual forms.

Most galaxies can be arranged in a natural sequence of morphological types,
the Hubble sequence. The “earliest” are elliptical (E) galaxies, with
apparently smooth distributions of yellow–red stars and seldom any signs of
internal structure (except that individual stars and globular clusters can be
resolved in the closest ones); and the “latest” are irregulars (Irr I) with no
symmetry and with disorganized patches of blue stars, hot gas, and dust
throughout111By a historical misfortune, the early-type galaxies are dominated
by late-type (red) stars, and the late-type galaxies are dominated by early-
type (blue) stars!. In between are the spirals, in order from Sa types with
large nuclear bulges and tightly wound arms to Sc types with small bulges and
open arms; the nuclear bulges closely resemble elliptical galaxies, while
spiral arms, or more often bits of spiral arms, are traced by patchy blue
stars and gas and by dark dust lanes, superposed on an underlying smoother
disk of redder stars. The galaxies classified as S0 have disks but lack spiral
structure and its associated young stars and gas; Hubble put them between
types E and Sa, but van den Bergh (1976b) pictures them as a sequence parallel
to the spirals, ordered by bulge-to-disk ratio. Another “dimension” of the
Hubble sequence is based on the presence or absence of a bar in the center,
from the ends of which the spiral arms seem to emerge; there are barred
spirals of types SBa with tight arms through SBc with loosely wound arms. The
Hubble sequence has been elaborated by de Vaucouleurs (1959) into a more
detailed system indicating intermediate forms and additional structural
features.

Although the sequence E–S0–Sa–Sb–Sc–Irr I was defined by the shapes of
galaxies, it is very much a sequence of stellar populations. The earliest
galaxies show no signs of hot young stars and (usually) have undetectably
little gas or dust, while the latest galaxies are undergoing active star
formation and have significant amounts of ISM. The Yerkes classification
system of galaxies (Morgan & Mayall, 1957; Morgan & Osterbrock, 1969)
emphasizes the correlation between forms and stellar populations of galaxies.
For spirals, this system is based simply on the central concentration of
light, and it is found that the highly concentrated galaxies have late-type
nuclear spectra (indicating a mixture of G, K, and M stars), while those with
little central concentration have early-type spectra (dominated by B stars)
even in the nucleus. The Yerkes types of galaxies are correlated loosely with
Hubble types, which is not surprising since the Hubble classification is based
partly on the amount of light concentrated into a nuclear bulge and partly on
the tightness of spiral structure. The increasing prevalence of blue, young
stars as one goes from elliptical through irregular galaxies is reflected in a
general progression from redder to bluer integrated colors.

From studies of the integrated spectra and colors of galaxies, it is concluded
that nearly all galaxies have some very old stars, and that their different
proportions of red and blue stars in the light are due mainly to different
current rates of star formation. Most (if not all) galaxies appear to be many
billions of years old, but whereas significant star formation ceased billions
of years ago in ellipticals, it has occurred at a roughly constant rate in the
latest types of galaxies. Theories of galaxy formation, and of star formation
within galaxies, are challenged to explain this close correlation between the
forms of galaxies and their histories of star formation.

Some galaxies fit nowhere into a tidy sequence of types, and they are
classified as simply “peculiar” or as one of the special types of peculiars
such as ring galaxies or Irr II (which are early-type galaxies in form but
strewn with dust, young stars etc.). Many of the peculiar systems are
interacting galaxies, distorted by tidal effects (Toomre & Toomre, 1972).
Sometimes the colors of peculiar galaxies are so blue that any underlying old
population is completely outshone by newly-formed stars, and sometimes they
are so red that most of the activity must be hidden in dust clouds. Although
these very peculiar galaxies are rare, they are of special interest since some
of them may be undergoing the kinds of violent changes and rapid star
formation that characterized normal galaxies in their youth (Larson, 1976b).

The chemical compositions of galaxies in general show much less diversity than
their stellar populations. Few instances are known of interstellar gas that is
deficient or overabundant in heavy elements by more than a factor of four
relative to the Sun, and although stellar metallicities range over more than
two orders of magnitude, they are usually within a factor of three of
Solar222The term “metals” in this field generally includes all elements
heavier than helium. True metals, especially iron, are most easily detected in
the spectra of stars, while the common non-metallic elements, especially
oxygen, are most accessible in the ISM. Another loose usage in this field is
that the word “element” frequently means a particular nuclide (e.g., the
“element” 13C).. Many of the differences among chemical compositions in
galaxies are systematic: metallicities of both stars and gas tend to decrease
outward from the centers of galaxies, and average metallicities tend to
increase with galaxy luminosity.

Models for chemical evolution of galaxies aim to account for their
compositions in terms of the production of elements by stars (mainly) and the
mixing of stellar ejecta with interstellar gas. Gas flows often play important
roles in chemical evolution, diluting the products of nucleosynthesis with
unenriched material from outside the galaxy, and carrying metals from one part
of the galaxy to another.

Many more details are known about the stellar population in the Solar vicinity
of our own galaxy than anywhere else, and some of these details contain clues
to large-scale processes of galaxy formation. For example, stars with high
space velocities that make them members of the halo333The word “halo” is used
in the astronomical literature in two ways: (i) its classical meaning, a
spheroidal population of ordinary stars, typified by globular clusters; and
(ii) an invisible spheroidal component that provides much more galactic mass
than can be ascribed to known stars and gas. The first meaning is implied
throughout this article, unless otherwise stated. population are metal-poor by
factors of ten or more relative to the Sun, while stars with disk motions have
metallicities almost entirely within a factor of three of Solar. Since the
classic paper of Eggen et al. (1962), this difference has been interpreted as
evidence that the halo stars formed first, before much chemical enrichment by
deaths of massive stars had taken place. Those authors also used the
kinematics of halo stars, and a model for the collapse of the Galaxy from
particular initial conditions, to infer a free-fall timescale, ${\sim}2\times
10^{8}\>\rm yr$, for the collapse and formation of halo stars. Ages that have
recently been determined for halo and disk stars now point toward a slower
collapse, however; globular clusters in the Galaxy appear to have a range of
ages from ${\sim}12-16\>\rm Gyr$, while very few disk stars near the Sun are
older than $10\>\rm Gyr$ (Demarque & McClure, 1977; Saio, 1977; Saio et al.,
1977; Twarog, 1980). The picture presented by these values is of a rather slow
collapse, and a very long timescale for star formation to get underway in the
outer disk of the Galaxy. A very young star cluster at least $10\>\rm kpc$
from the Sun in the Galactic anticenter direction has recently been assigned a
metallicity as low as those of halo stars (Christian & Janes, 1979), so it
seems that the outermost disk is still in a very early stage of chemical
evolution.

In general, composition differences among the old stars of galaxies serve as
frozen-in tracers of their early chemical evolution, while abundances in young
stars and the ISM indicate the progress of continuing enrichment. These
effects are closely tied to the history of star formation in various regions
of galaxies, which relates them to factors determining photometric properties.
It will be seen later in this review that the aspects of galactic evolution of
greatest importance for photometric evolution and for chemical evolution,
respectively, are to a large extent complementary, so the two fields of study
tend to yield different types of information about the evolution of galaxies.

###  1.2 Galaxy Formation and Evolution

The striking correlations between the forms and contents of galaxies, and
systematic changes of content with position in galaxies, point to the
importance of dynamical processes in chemical and photometric evolution. Many
theoretical ideas on the formation and dynamical evolution of galaxies have
been motivated by these correlations, and dynamical models in turn have led to
further understanding of such properties. A brief sketch of some current
theoretical views of galaxy formation and dynamical evolution will therefore
be given here. A fuller non-technical introduction to the field is given by
Larson (1977c), and more technical reviews are by Doroshkevich et al. (1978),
Gott (1977), Jones (1976), Larson (1976a), and Rees (1977).

In the conventional cosmological picture, the primeval gas emerging from the
Big Bang some $10-20\>\rm Gyr$ ago consisted almost entirely of hydrogen and
helium. There must have been large enough density perturbations in this gas
for some regions to be locally bound and to collapse despite the expansion of
the Universe, and the existence today of galaxies and bound clusters with a
$10^{10}$-fold range of masses shows that there was a very wide spread of
density perturbations. A favored mass scale is ${\sim}10^{6}\>\rm M_{\odot}$,
which is the Jeans mass in the Universe after (re-)combination of the primeval
plasma about $10^{6}\>\rm yr$ after the Big Bang. Maybe star formation first
began in lumps of this mass, which could themselves have been part of larger-
scale perturbations destined to collapse eventually into a massive galaxy.
Although models for galaxy formation often start from an idealized smooth
density distribution, gas in this state would be Jeans unstable on much
smaller scales than the protogalaxy, and one would expect star formation to
occur in regions of much higher density than average.

Independently of cosmological details, a protogalaxy can be pictured as a
lumpy gas cloud governed by self-gravity (and perhaps by perturbations due to
neighbors) rather than by the expansion of the Universe. If no stars formed,
the gaseous protogalaxy would collapse, collisions between gas clouds would
dissipate the energy parallel to the net angular momentum vector, and the
system would become a flat disk. (The process of “dissipation” in this context
usually implies that some of the kinetic energy of the gas is lost by
collisionally induced radiation). If, on the other hand, stars form in much
less than the collapse time, there would be no dissipation and a spheroidal
system of stars would be produced; subclustering due to the initially lumpy
gas distribution would be wiped out except in regions with about the mass and
density of globular clusters. A spheroidal system can also be formed in much
longer than the free-fall time if dissipation takes much longer, perhaps
because the gas is in clouds that collide infrequently.

In general, the formation of spheroidal stellar systems – elliptical galaxies
and the bulges of spirals – requires that stars form on a timescale less than
that of gaseous dissipation, while the formation of a disk requires the
opposite situation. The amount of disk possessed by a galaxy therefore depends
on the amount of gas that remained to collapse dissipatively, after efficient
star formation in the spheroidal component. This dynamical picture is
consistent with the great ages, ${\sim}10^{10}\>\rm yr$, inferred for stars in
spheroidal systems, and the contrasting range of ages (including ongoing star
formation) in disks.

In detail, the differences between spheroidal and disk systems are not quite
so clear-cut, because the formation of condensed nuclei in elliptical galaxies
and the central bulges of spirals probably involved some dissipation. The
enhanced metallicities of nuclear stars are thought to be due to an inward
concentration of metals, as gas enriched by massive stars further out
dissipated its energy and condensed toward the center.

S0 galaxies appear to be intermediate systems, having disks but no young
stars. One class of theories of their origin suggests that these properties
are intrinsic, in that star formation efficiently used up the gas after it had
collapsed to a disk. In another class of theories, S0 galaxies are former
spirals that are devoid of young stars because their ISM was swept away,
either in a collision with another galaxy or by an intergalactic wind due to
motion of the galaxy through intergalactic gas. The high frequency of S0
galaxies in dense clusters, where intergalactic gas has been detected,
suggests that at least some were formed in the latter way, but other S0
galaxies are quite isolated and may be of the “intrinsic” type.

Sweeping by collisions and intracluster gas are examples of interactions with
the environment that probably affect the evolution of many galaxies, including
apparently normal ones and very peculiar objects. Other effects include
mergers of galaxies in the centers of clusters and in close binaries, tidal
disruption in close passages, and bursts of star formation due to such violent
interactions among gas-rich galaxies. A spiral galaxy may also accrete gas
from its surroundings during most of its life, gradually forming a more
massive and more extended disk and acquiring new gas to fuel continued star
formation. Various aspects of interactions between galaxies and their
environment are reviewed by Ostriker (1977b), Saar & Einasto (1978), and
Toomre (1977).

###  1.3 Plan of this Review

From the preceding outline, it should be clear that attempts to understand the
evolution of stars and gas in galaxies inevitably get involved in very diverse
aspects of astronomical theory and observation. This is not a field in which
one can hope to develop a complete theory from a simple set of assumptions,
because many relevant data are unavailable or ambiguous, and because galactic
evolution depends on many complicated dynamical, atomic, and nuclear processes
which themselves are incompletely understood. Because a logical development is
inappropriate, this article treats different aspects of galactic evolution as
pieces of a jigsaw puzzle that may some day be put together.

Figure 1: A schematic outline of the links between various constituents of
galaxies and evolutionary processes. _Solid lines_ indicate the main factors
in chemical evolution, and _dashed lines_ are for photometric evolution.

Figure 1 is a very schematic diagram of relations among some of the most
important factors in galactic evolution. It serves to summarize parts of the
preceding outline, to suggest how a final jigsaw might be linked, and to show
how the topics discussed below are related to each other. Two main sets of
processes and constituents are indicated.

  1. 1.

Chemical evolution depends mainly on the areas connected by solid lines, which
indicate that stars form from the ISM and return some of their mass, including
new products of nucleosynthesis, to the ISM at death; the composition of the
ISM also depends on gas flows to and from the region under consideration.

  2. 2.

Photometric properties depend mainly on the areas on the areas connected by
dashed lines, indicating that the history of star formation and the evolution
of individual stars determine the population at any time.

Sections 3, 4, and 5 below are on chemical evolution and Sections 6 and 7 are
on the evolution of photometric properties. Since the formation and evolution
of stars are vital to all aspects of galactic evolution, these processes are
reviewed first in Section 2.

##  2 The Formation and Evolution of Stars

The evolution of galaxies depends critically on properties of stars as a
function of mass: their formation rates, evolution in the Hertzsprung–Russell
(HR) diagram, lifetimes, and the mass and composition of material returned to
the ISM. This Section reviews the properties of stars that are most relevant
to later discussion of galaxy evolution.

###  2.1 Basic Physical Properties of Stars

Table 1 lists for reference some properties of main sequence (MS) stars with
chemical compositions typical of the nearby disk population, i.e. with Solar
or slightly lower metallicities. Because stars evolve in the HR diagram while
still on the MS, this Table only applies to a particular stage of evolution
for each mass: stars with total MS lifetimes less than $10\>\rm Gyr$ (masses
$\gtrsim 1\>\rm M_{\odot}$) are listed with the properties they have in the
middle of this lifetime; less massive stars are listed at age $5\>\rm Gyr$ or
with mean empirical properties. The theoretical properties of MS stars of a
given mass and composition differ slightly in different series of
calculations, and they are sensitive to the helium and heavy-element
abundances, even within the range of values applicable to disk stars, so the
tabulated values should not be accepted too literally. This Table is intended
only to provide approximate relations between such quantities as spectral type
and MS lifetime, for quick reference.

Table 1: Representative data for main sequence stars. Quantities tabulated are typical values for stars of approximately Solar composition, from a variety of sources including mainly the following. Masses $m\leq 0.6\>\rm M_{\odot}$: data from Veeder (1974); $0.8\leq m\leq 10\>\rm M_{\odot}$: theoretical tracks of Mengel et al. (1979) and Alcock & Paczynski (1978); $m\geq 20\>\rm M_{\odot}$: theoretical tracks of Chiosi et al. (1978) with maximum mass loss rates. For $m\leq 0.6\>\rm M_{\odot}$, the mean empirical main sequence is given; for $0.8\leq m\leq 1.0\>\rm M_{\odot}$, points are for an age of $5\>\rm Gyr$; and for more massive stars, points are for the middle of the main sequence lifetime ($\tau_{\rm ms}/2$).  $m\>\rm\left(M_{\odot}\right)$ | $\tau_{\rm ms}$ | $\log\left(L/\rm L_{\odot}\right)$ | $M_{\rm V}$ | $\log\ T_{\rm eff}$ | $B-V$ | Sp
---|---|---|---|---|---|---
0.15 | – | -2.5 | 14.2 | 3.48 | 1.80 | M7
0.25 | – | -2.0 | 12.0 | 3.52 | 1.60 | M5
0.4 | – | -1.4 | 10.0 | 3.57 | 1.48 | M1
0.6 | – | -0.9 | 7.6 | 3.64 | 1.18 | K5
0.8 | 25 | -0.4 | 6.0 | 3.70 | 0.88 | K1
0.9 | 15 | -0.2 | 5.4 | 3.73 | 0.76 | G8
1.0 | 10 | 0.0 | 4.9 | 3.76 | 0.64 | G2
1.1 | 6.4 | 0.2 | 4.3 | 3.79 | 0.56 | F8
1.2 | 4.5 | 0.4 | 3.7 | 3.82 | 0.47 | F6
1.3 | 3.2 | 0.5 | 3.5 | 3.84 | 0.42 | F5
1.4 | 2.5 | 0.7 | 3.0 | 3.86 | 0.36 | F2
1.5 | 2.0 | 0.8 | 2.8 | 3.88 | 0.30 | F0
2 | 0.75 | 1.3 | 1.4 | 3.98 | 0.00 | A0
3 | 0.25 | 2.1 | -0.2 | 4.10 | -0.12 | B7
4 | 0.12 | 2.6 | -0.6 | 4.18 | -0.17 | B5
6 | 0.05 | 3.2 | -1.5 | 4.30 | -0.22 | B3
8 | 0.03 | 3.6 | -2.2 | 4.35 | -0.25 | B1
10 | 0.02 | 3.9 | -2.7 | 4.40 | -0.27 | B0.5
15 | 0.01 | 4.4 | -3.7 | 4.45 | -0.29 | B0.5
20 | 0.008 | 4.7 | -4.3 | 4.48 | -0.30 | B0
30 | 0.006 | 5.1 | -5.1 | 4.51 | -0.31 | O9.5
40 | 0.004 | 5.4 | -5.7 | 4.53 | -0.31 | O9
60 | 0.003 | 5.7 | -6.2 | 4.58 | -0.32 | O5

An overview of stellar evolution in the HR diagram is provided by Figures 2
and 3. Figure 2 is a theoretical HR diagram with the zero-age MS and
evolutionary tracks for stars of a few masses; the tracks are drawn lightly in
regions where stellar evolution is rapid (relative to neighboring points on
the same track) since few stars are observed at those stages. A given star
spends only about $10\%$ of its MS lifetime in stages of evolution beyond the
MS, so the supergiant and giant regions of the HR diagram contain a relatively
ephemeral population of stars. Much the same information is given empirically
in Figure 3, which is a reproduction of the composite color–magnitude diagram
for open clusters of Sandage & Eggen (1969). The clusters have too few stars
for the faster stages of evolution to be well represented; not only are there
gaps between the MS and the giant branch, but the latest giants are absent
from all but one or two clusters. Despite this problem, the diagram of Sandage
& Eggen (1969) gives a vivid picture of stellar evolution.

Figure 2: Theoretical HR diagram indicating main sequence lifetimes and
evolutionary tracks of stars with approximately Solar composition. The zero-
age main sequence (ZAMS) is shown, with a few stellar masses ($\rm M_{\odot}$)
and lifetimes (yr). Also shown are representative evolutionary tracks, shaded
heavily where a star evolves slowly relative to adjacent points. _Sources_ :
ZAMS and tracks for $m\leq 3\>\rm M_{\odot}$: Mengel et al. (1979), $Z=0.01$,
$Y=0.20$; ZAMS for $3<m\leq 10\>\rm M_{\odot}$: Alcock & Paczynski (1978),
$Z=0.01$, $Y=0.29$; track for $m=4.5\rm\>M_{\odot}$: Harris & Deupree (1976),
$Z=0.01$, $Y=0.28$; ZAMS and tracks for $m\geq 20\rm\>M_{\odot}$: Chiosi et
al. (1978), case with maximum mass loss rate, $Z=0.02$, $Y=0.28$. Figure 3:
Composite color–magnitude diagram for open clusters (Sandage & Eggen, 1969).
Ages at the right refer to the blue point for a given cluster. Modern
distances and age scales correspond to younger ages, e.g., $5\times
10^{9}\>\rm yr$ for NGC 188 (Demarque & McClure, 1977). This diagram is an
empirical counterpart of Figure 2.

Using these Figures, one can envisage the evolution of a single generation of
stars in a galaxy, starting with their populations on the zero-age MS. As time
passes, the top of the MS peels away and short-lived supergiants appear; the
MS turnoff evolves down to mid-B spectral type in $10^{8}\>\rm yr$, and by
$10^{9}\>\rm yr$ late A and later stars remain on the MS; between $10^{9}$ and
$10^{10}\>\rm yr$, the turnoff evolves from late A to early G, and there is a
population of evolving K and M giants. An elliptical galaxy, for example, that
had no star formation for the last many billions of years, would be expected
to consist only of K and M giants and the MS up to a late F or early G
turnoff. In a galaxy like our own, where star formation has presumably been
continuous for billions of years, the whole array of MS stars, turnoff, and
giant branches can be found.

###  2.2 The Initial Mass Function

Because the properties of stars depend strongly on their masses, the
distribution of stellar masses at birth is a function vital to the photometric
and chemical evolution of galaxies. Various definitions and notations for this
function are used in the literature, so care is needed to avoid confusion in
comparing different papers. The following notation is used here: the number of
stars formed in the mass interval ($m$, $m+dm$) and in the time interval ($t$,
$t+dt$) is

$\phi\left(m\right)\psi\left(t\right)dm\ dt,$ (2.1)

where $\psi\left(t\right)$ is the total mass of stars formed per unit time,
and $\phi\left(m\right)$, which may itself be a time-dependent function, is
therefore normalized so that

$\int_{0}^{\infty}m\phi\left(m\right)dm=1.$ (2.2)

The function $\psi\left(t\right)$ is called the star formation rate (SFR) and
$\phi\left(m\right)$ is called the initial mass function (IMF). It is often
useful to approximate $\phi\left(m\right)$ by a power law, at least over some
range of masses, and in these cases the notation used here is

$\phi\left(m\right)\propto m^{-\left(1+x\right)},$ (2.3)

where $x$ is called the slope of the IMF444Quantities equivalent to $x$, $-x$,
$\left(1+x\right)$, and $-\left(1+x\right)$ have been called “the slope of the
IMF” by different authors..

A comprehensive review of the IMF in the Solar neighborhood and elsewhere has
recently been written by Scalo (1978), so the following account includes only
those aspects that are most relevant to the rest of this article.

####  2.2.1 The local IMF

The IMF in the Solar neighborhood can be derived from counts of field stars,
using principles originally derived by Salpeter (1955), and generalized to
allow for a time-dependent SFR by Schmidt (1959, 1963), among others. A recent
thorough rediscussion of the data and methods is given by Miller & Scalo
(1979).

Stars are counted as a function of absolute magnitude, and the counts must
first be reduced to the present mass distribution of stars, $n(m)$, where
$n(m)\ dm$ will denote the present number of MS stars in the mass interval
($m$, $m+dm$). Several non-trivial problems arise at this stage, including the
following:

  1. 1.

The number of stars in each absolute magnitude interval must be corrected for
post-MS stars, which requires a knowledge of color or spectral type; these
corrections can be large – e.g., near $M_{\rm V}=+1$ there are about equal
numbers per unit volume of A dwarfs and K giants.

  2. 2.

Field stars of different masses have very different distributions
perpendicular to the Galactic plane, the scaleheight decreasing with
increasing mass. The function $n(m)$ obtained for stars in a unit volume
therefore differs systematically from the function for a unit column
perpendicular to the plane. Because one is normally interested in the whole
population of stars formed at all heights (or formed in the plane and
accelerated to more extended orbits), near the Sun’s galactocentric distance,
the star counts should be reduced to the _average number per square parsec_ at
this distance from the Galactic center.

  3. 3.

The transformation between magnitudes and masses of MS stars depends on their
chemical composition.

  4. 4.

Stars evolve in the HR diagram while on the MS, so there is not a unique
mass–luminosity relation even for a given composition.

The papers cited above discuss how these problems can be handled, and the
resulting uncertainties in $n(m)$.

Let us assume that we know the present mass distribution of MS stars per
square parsec in the Solar neighborhood, $n(m)$. This function can be used to
estimate the IMF and SFR as follows. Stars with lifetimes $\left(\tau_{\rm
m}\right)$ greater than the age of the Galaxy $\left(t_{1}\right)$ have
accumulated since star formation began ($t\equiv 0$), so Equation (2.1) gives
directly

$n(m)=\int_{0}^{t_{1}}\phi\left(m\right)\psi\left(t\right)dt,\>\>\tau_{\rm
m}\geq t_{1}.$ (2.4)

If the IMF is assumed constant, this expression simplifies to

$n(m)=\phi\left(m\right)\overline{\psi}_{1}t_{1},\>\>\tau_{\rm m}\geq t_{1},$
(2.5)

where $\overline{\psi}_{1}$ is the average past SFR; Equation (2.5) can be
used in any case if $\phi\left(m\right)$ is interpreted as the average past
IMF. The mass with $\tau_{\rm m}=t_{1}$ is called the present turnoff mass
($m_{1}$) since it defines the lowest MS turnoff point in the HR diagram for
local field stars. In practice, one must consider how the mass–lifetime
relation depends on metallicity, but to illustrate the principles here these
effects will be ignored; approximate round numbers are $m_{1}\simeq 1\>\rm
M_{\odot}$, $t_{1}\simeq 10\>\rm Gyr$.

The present population of stars with $\tau_{\rm m}<t_{1}$ includes only those
that were formed at times less than $\tau_{\rm m}$ ago, so from Equation (2.1)
their mass distribution is

$n(m)=\int_{t_{1}-\tau_{\rm
m}}^{t_{1}}\phi\left(m\right)\psi\left(t\right)dt,\>\>\tau_{\rm m}<t_{1}.$
(2.6)

If $\phi(m)$ is constant, it can be taken out of the integral, but $n(m)$
still depends on details of the past SFR. A simpler equation can be written
for stars with lifetimes shorter than the present timescale for changes in
$\psi\left(t\right)$:

$n(m)=\phi\left(m\right)\psi_{1}\tau_{\rm m},\>\>\tau_{\rm m}\ll t_{1},$ (2.7)

where $\psi_{1}$ is the present SFR. The function $n(m)$ is supposed to be
averaged over the patchy distribution of the youngest stars in the Solar
neighborhood, so Equation (2.7) is probably a good approximation for $m\gtrsim
2\>\rm M_{\odot}$, i.e., for stars with lifetimes $\lesssim 1\>\rm Gyr$ and MS
spectral types earlier than about A0; it would hold for all $m>m_{1}$ if the
SFR were constant.

From the empirical function $n(m)$, therefore, one can use Equation (2.5) to
derive the shape of the (average) IMF for stars below about $1\>\rm M_{\odot}$
and Equation (2.7) to derive the shape of the IMF for stars above about
$2\>\rm M_{\odot}$. These two pieces of the IMF are not determined with the
same multiplicative factors, since the quantities actually determined are
$\phi\left(m\right)\overline{\psi}_{1}t_{1}$ and $\phi\left(m\right)\psi_{1}$,
respectively. In particular, the relative values of the IMF for $1\>\rm
M_{\odot}$ and $2\>\rm M_{\odot}$ depend on the ratio

$T_{1}\equiv\frac{\overline{\psi}_{1}t_{1}}{\psi_{1}},$ (2.8)

which is evidently a timescale for star formation in the Solar neighborhood.

There is an intermediate mass interval, $1\lesssim m/\rm M_{\odot}\lesssim 2$,
for which Equation (2.6) cannot be simplified, so the shape of the IMF cannot
be derived independently of details of $\psi(t)$. This gap is usually filled
by assuming that the functions for higher and lower masses can be smoothly
interpolated, as illustrated, for example, by Schmidt (1963, Figure 1). In
this way, $\phi(m)$ for the intermediate mass range is determined without too
much ambiguity and constraints are set on the quantity $T_{1}$.

Figure 4: The initial mass function in the Solar neighborhood, shown in the
form $\log\left[\psi_{1}m\phi(m)\right]$ versus $\log\ m$, where $\phi(m)$ is
the IMF itself, normalized as in Equation (2.2), and $\psi_{1}$ is the present
star formation rate in $\rm M_{\odot}\ pc^{-2}\ Gyr^{-1}$. Masses are in solar
units. _Solid line and shaded area_ : range of functions derived by Miller &
Scalo (1979), allowing for uncertainties in the basic data and in the history
of the star formation rate. _Dashed line_ : power-law segments, given by
Equation (2.9); the slopes of these lines are $-x$ in the notation of Equation
(2.3). (The dashed line and the shaded area are discrepant for $-0.4<\log\
m<0.0$ because different stellar luminosity functions were used).

The IMF derived along the preceding lines by Miller & Scalo (1979) is
illustrated in Figure 4, where the quantity plotted (logarithmically) is
$m\phi(m)\psi_{1}$. The solid line is their analytical relation for the case
of a constant SFR and a Galactic age $t_{1}=12\>\rm Gyr$:
$m\phi(m)\psi_{1}=3.83\exp[-1.09(\log m+1.02)^{2}]\>\rm pc^{-2}\ Gyr^{-1}$
(with $m$ in solar units). The shaded area shows the range of uncertainty
allowed by their analytic fits to limiting cases, based on uncertainties in
the basic data and the limits for smoothly joining the low- and high-mass
ends, as explained above. Miller & Scalo (1979) stress that, if one is making
models for the Solar neighborhood, the choice of an IMF within the shaded area
should be made consistently with the adopted SFR function, for otherwise the
two functions together would not reproduce the observed star counts.

Power laws are very tractable analytically, so it is tempting to fit straight
lines to the IMF derived from the star counts. Salpeter (1955) originally
derived a famous slope $x=1.35$ for the local IMF, but subsequent star counts
and stellar lifetimes show that a single power law for the whole mass spectrum
is a poor approximation; serious systematic errors can be introduced in some
applications by ignoring the increase of slope with mass in the local IMF. The
four dashed straight lines in Figure 4 are intended as a compromise between
the data and analytical convenience. (The discrepancy with the shaded area
between $0.4$ and $1\>\rm M_{\odot}$ arises because the dashed line for those
masses is based on the luminosity function of Wielen (1974), which is flatter
than the function adopted by Miller & Scalo, 1979, Figure 1). The equations of
these four lines, which have slopes $-x$ in the logarithmic plot (see Equation
2.3), are:

$\displaystyle m\phi(m)\psi_{1}=$ $\displaystyle\ 1.00m^{-0.25},\>\>0.4<m\leq
1.0,$ (2.9) $\displaystyle\ 1.00m^{-1.0},\>\>1.0<m\leq 2.0,$ $\displaystyle\
1.23m^{-1.3},\>\>2.0<m\leq 10.0,$ $\displaystyle\ 12.3m^{-2.3},\>\>10.0<m\leq
50.0,$

in units of $\rm pc^{-2}\ Gyr^{-1}$, with masses in solar units. (It is
fortuitous that the coefficient 1.00 occurs in these equations). A slope is
not given for $m<0.4\>\rm M_{\odot}$ because M dwarfs contribute little to the
light of any galaxy and nothing directly to chemical evolution; their
contribution to the total mass can be included with the invisible objects
below $0.1\>\rm M_{\odot}$.

Integrating Equation (2.9) with respect to mass, we find that the present SFR
for masses $0.4-50\>\rm M_{\odot}$ is $3.0\>\rm M_{\odot}\ pc^{-2}\ Gyr^{-1}$.
The total SFR, $\psi_{1}$, could be derived if we knew the contribution from
objects below $0.4\>\rm M_{\odot}$, and $\psi_{1}$ in turn would allow the IMF
to be normalized as in Equation (2.2). As described in Section 2.3.1, Miller &
Scalo (1979) find $\psi_{1}$ in this way by defining as “stars” only objects
above $0.1\>\rm M_{\odot}$, for which there are star counts giving the shape
of the IMF. Alternatively, $\psi_{1}$ can be estimated indirectly (see Section
2.3.1). The values found are within a factor of ${\sim}2$ of $10\>\rm
M_{\odot}\ pc^{-2}\ Gyr^{-1}$, which is several times greater than the rate
for objects above $0.4\>\rm M_{\odot}$.

####  2.2.2 The IMF at other times and places

Several questions can be asked about the constancy of the IMF. Is the IMF the
same in regions other than the Solar neighborhood? Is it constant in time,
locally and elsewhere? If variations occur, do they involve the shape of the
IMF for stars above say $0.4$ or $1\>\rm M_{\odot}$, or do they involve only
the mass fraction in “inert” low-mass objects?

Star clusters in the Galaxy and the Magellanic Clouds have a variety of MS
luminosity functions indicating significant variations in the IMF on the scale
of clusters (Da Costa, 1977; Freeman, 1977; others reviewed by Scalo, 1978);
also, there are regions of the Milky Way where the only newborn stars seem to
be T Tauri stars, with masses $\lesssim 2\>\rm M_{\odot}$, and other regions
abounding in young OB stars. These observations, however, do not necessarily
imply variations on galaxy-wide scales in averages over many sites of star
formation. In fact, counts of field stars in nearby galaxies have not revealed
any significant deviations from the local IMF (Butcher, 1977; Hardy, 1977;
Lequeux, 1979). The clearest evidence for large-scale variations is the mass
function for nearby halo stars of Schmidt (1975); this function has $x\simeq
2$ for $0.25-0.75\>\rm M_{\odot}$, whereas local disk stars have $x<1$ in the
same mass range. There are also some early-type spiral galaxies with
widespread patches of young blue stars but a deficiency of H ii regions,
suggesting that the IMF is deficient in O stars; examples are the Sombrero
galaxy, M104 (van den Bergh, 1976a; Schweizer, 1978), and NGC 2841 (Kormendy,
1977).

The upper part of the IMF $\left(m>0.4\>\rm M_{\odot}\right)$ can be studied
indirectly by comparing colors of galaxies with those of models based on
various IMFs. As discussed in Section 7, the results are consistent with the
local IMF holding (on large scales) everywhere, but in some respects the test
is rather insensitive. For example, a galaxy like the Sombrero is so dominated
by old yellow–red stars that the _integrated_ colors at optical wavelengths
would be changed negligibly by the presence or absence of stars above $10\>\rm
M_{\odot}$ in the IMF. The red–infrared spectra of galaxies show that giant
stars, rather than late dwarfs, provide most of the light at these
wavelengths, and this result sets some constraints on the IMF for stars of
${\sim}0.4-1\>\rm M_{\odot}$: if the IMF were much steeper than the local
function (say if $x>2$), low-mass dwarfs would contribute much of the infrared
light, contrary to the spectroscopic information.

Because of this giant dominance in the infrared light, photometric
observations are very insensitive to the mass fraction in stars below
$0.4\>\rm M_{\odot}$, i.e. M dwarfs; these stars could be numerous enough to
dominate the mass of the system while contributing very little light! A handle
on the mass fraction in invisible stars is given by the mass-to-luminosity
($M/L$) ratio, which is crudely a ratio of very low-mass to more massive
stars. In the Solar neighborhood, the integrated luminosity of known stars
(per $\rm pc^{3}$), including giants, divided by the dynamically-estimated
density of mass, yields a ratio $M/L_{\rm B}\simeq 3\>\rm
M_{\odot}/L_{B\odot}$ (Faber & Gallagher, 1979). If the IMF and proportion of
invisible matter were invariant, all galaxies should have this $M/L$ ratio,
apart from a systematic increase by a factor of ${\sim}10$ from the bluest
galaxies to the reddest (because of the decreasing fraction of young blue
stars), and apart from a scatter ${\sim}10\%$ due to different mass fractions
of ISM. In fact, observed $M/L$ values show a _scatter_ by a factor of 10 or
more at a given color, and a tendency to increase from values of a few in the
central parts of galaxies to ${\sim}100$ in the outer regions (Faber &
Gallagher, 1979). It appears that different proportions of hidden matter exist
in different galaxies, and that the invisible mass fraction increases with
radius. If this matter condensed before ordinary star formation began in
galaxies, as in some pictures of the formation of heavy invisible halos (e.g.
White & Rees, 1978), then it would have no direct effect on chemical or
photometric evolution. (It would have indirect effects, because the motions of
gas and stars in the galaxy would be influenced by the hidden mass). However,
if there are variations in the low-mass part of the stellar IMF, chemical
evolution would be affected significantly, because variable proportions of
stellar matter would be returned to the ISM by evolving stars.

Certain chemical properties of galaxies can be explained by invoking
variations in the shape of the IMF for massive stars. For example, the extreme
paucity of metal-poor dwarfs in the Solar neighborhood could be due to an
initial burst of massive metal-producing stars; and variations in relative
abundances of heavy elements could be due to variable proportions of the stars
that synthesize such elements. But in all cases, other explanations not
involving variations in the IMF are available, as discussed in Sections 4 and
5.

In summary, significant variations in the IMF may be rare enough for the
assumption of a universal function (the local IMF) to be useful for many
contexts. However, it is equally relevant to ask how galactic evolution would
be affected by changes in both the form of the IMF for visible stars and the
relative weight of very low-mass objects.

###  2.3 Rates of Star Formation

The rate of star formation is one of the main factors in galactic evolution.
The shape of a galaxy depends on the timescale for star formation relative to
collapse and dissipation timescales, the colors and luminosity depend on the
age distribution of stars, and chemical evolution depends on the SFR relative
to the gas flow rates and the gas mass. Various ways of estimating SFRs in
galaxies are reviewed here.

####  2.3.1 The local SFR

There are several ways of estimating the present SFR in the Solar
neighborhood, which is the quantity in $\rm M_{\odot}\ pc^{-2}\ Gyr^{-1}$
denoted $\psi_{1}$ above. Stars above ${\sim}2\>\rm M_{\odot}$ are no problem,
because they are so short-lived that their numbers scale directly with
$\psi_{1}$ (Equation 2.7), but the ages of less massive stars are unknown, as
is the mass fraction contained in objects below $0.1\>\rm M_{\odot}$. Most of
the methods that have been suggested for overcoming these problems reduce in
principle to estimating the average past SFR ($\overline{\psi}_{1}$) and then
the ratio $\overline{\psi}_{1}/\psi_{1}$ 555Another approach in the literature
is to find the formation rate of relatively massive stars (above $2\>\rm
M_{\odot}$ or a greater limit), and to scale to the total SFR via an _assumed_
IMF. Obviously, the problems mentioned above are sidestepped rather than
solved, so the conclusions should be regarded as correspondingly uncertain..
The following two methods are typical.

  1. 1.

A dynamical estimate of the total surface density in the Solar neighborhood,
called the “Oort limit”, can be derived from motions of stars perpendicular to
the Galactic plane; its value is probably ${\sim}80-100\>\rm M_{\odot}\
pc^{-2}$ (Oort, 1965). Of the total surface density, ISM in various forms
accounts for $5-10\>\rm M_{\odot}\ pc^{-2}$ (Gordon & Burton, 1976), so stars
are presumed to contribute ${\sim}70-95\>\rm M_{\odot}\ pc^{-2}$ (neglecting
any possible non-stellar condensed objects). This is not the total mass of
stars ever formed, because some fraction $R$ has been ejected back to the ISM
by evolving and dying stars; $R$ depends on the IMF, whose normalization is
not known without $\psi_{1}$ itself, but all reasonable normalizations give
$R$ in the range ${\sim}0.1-0.3$ (as shown in Section 3.1). The mass of stars
ever formed is thus $10\%-30\%$ greater than the mass now present; the mean
past SFR is thus given by $75<\overline{\psi}_{1}t_{1}<125\>\rm M_{\odot}\
pc^{-2}$.

The next step is to use the ratio $T_{1}$ (Equation 2.8), derived by joining
smoothly the IMFs with different multiplicative factors for stars above
$2\>\rm M_{\odot}$ and below $1\>\rm M_{\odot}$, respectively. Plausible
values of $T_{1}$ lie between 5 and $20\>\rm Gyr$ (Tinsley, 1976; Miller &
Scalo, 1979), so we are left with the following range of values for $\psi_{1}$
itself: $1.5<\psi_{1}<25\>\rm M_{\odot}\ pc^{-2}\ Gyr^{-1}$. The uncertainties
at various stages of the derivation are mutually independent, and the $20\%$
uncertainty in $R$ is relatively unimportant. A more stringent upper limit
could be obtained if one could assume that the SFR has not increased since the
time the oldest disk stars formed, ${\sim}12\>\rm Gyr$ ago; then
$T_{1}>12\>\rm Gyr$ and $\psi_{1}<10\>\rm M_{\odot}\ pc^{-2}\ Gyr^{-1}$.
However, although the assumption of a monotonically decreasing SFR is common,
recent work on the age distribution of disk stars (Twarog, 1980) shows that
the SFR more likely _increased_ between 12 and $5\>\rm Gyr$ ago, and was
approximately constant in the last ${\sim}5\>\rm Gyr$. In this case, $T_{1}$
is probably between about 5 and $10\>\rm Gyr$, and the above limits on
$\overline{\psi}_{1}t_{1}$ lead to $8<\psi_{1}<25\>\rm M_{\odot}\ pc^{-2}\
Gyr^{-1}$. Values only ${\sim}10-20\%$ of these have usually been derived in
the literature, on the assumption that $\psi(t)$ has been a decreasing
function for at least $10\>\rm Gyr$.

  2. 2.

Instead of using the Oort limit to estimate the total mass of stars ever
formed, one can use star counts to obtain the existing population (and to
allow for the total mass of stars that have evolved away in the time since
they formed) down to ${\sim}0.1\>\rm M_{\odot}$, and neglect any less massive
objects. In this way, Miller & Scalo (1979) derive a range of values
$43<\overline{\psi}_{1}t_{1}<144\>\rm M_{\odot}\ pc^{-2}$. Combining these
values with appropriate limiting values of $T_{1}$, they find self-consistent
values in the range $3<\psi_{1}<7\>\rm M_{\odot}\ pc^{-2}\ Gyr^{-1}$.

Method (i) above gave larger values than those of Miller & Scalo (1979) for
two reasons: the estimated mass of stars now present above $0.1\>\rm
M_{\odot}$ falls short of estimates based on the Oort limit (although the
discrepancy may not be significant), and Miller & Scalo (1979) considered both
decreasing and increasing SFRs. Altogether, it is plausible to suggest that
$\psi_{1}$ lies within a factor of about two of $10\>\rm M_{\odot}\ pc^{-2}\
Gyr^{-1}$.

####  2.3.2 The SFR elsewhere

Estimates of the SFR in other places use various indicators of the numbers of
fairly massive stars, and convert these to a total SFR by means of an assumed
IMF, which is usually the local function (or an assumed power-law
approximation). Methods that have been used include the following.

  1. 1.

The brightest individual stars, supergiants, and occasionally upper-MS stars,
can be counted in nearby galaxies. Apart from complications due to differences
in chemical composition, the numbers of massive stars in given stages of
evolution should be directly proportional to the SFR, if the IMF is the same
everywhere; their numbers relative to counts of similar stars in the Solar
neighborhood thus scale fairly directly with the SFR. Lequeux (1979) has
applied this method to galaxies of the Local Group, using various types of
stars, and several authors (e.g. Searle et al., 1973; Larson & Tinsley, 1974)
have used such rare objects as supernovae in the same way for more distant
galaxies.

  2. 2.

If hot young stars are present in a region, they generally emit most of the
Lyman continuum photons that are available for ionizing interstellar hydrogen.
The flux of $\rm H\alpha$, $\rm H\beta$, or free–free radio emission is
therefore approximately proportional to the SFR. This method has been applied
to external galaxies by Cohen (1976) using integrated $\rm H\alpha$ equivalent
widths, and by Huchra (1977) using $\rm H\beta$; Smith et al. (1978) have
studied SFRs in the Galaxy using radio emission from H ii regions.

  3. 3.

The integrated colors of galaxies depend strongly on their relative
proportions of young and old stars; as a result, UBV colors in general provide
a rough estimate of the ratio (SFR) / (mass of old stars). This approach will
be discussed in detail in Section 7.

  4. 4.

Regions of star formation are often heavily obscured by dust, but even so the
infrared luminosity (starlight re-radiated by dust grains) gives a measure of
the SFR, as discussed in Section 7.2.2.

Most of these methods rely strongly on the assumption of a universal IMF,
since they sample only the very massive stars; integrated colors can reveal
large departures from an assumed IMF (Section 2.2.2), but even these are
completely blind to stars below ${\sim}0.5\>\rm M_{\odot}$. Even granted the
assumption of universality, important uncertainties arise in scaling from the
observations to a total SFR, because the statistics of very massive stars are
poor. Large systematic errors can arise from the use of a single power law for
the whole IMF (a common practice), as can be clearly seen from Figure 4.

Despite the uncertainties, the results of various studies have led to a
coherent picture. In the Galaxy, some $10\%$ of the current star formation is
occurring in the innermost $1\>\rm kpc$, and most of the remainder is
concentrated in a ring between 5 and $8\>\rm kpc$ from the center, which is
the site of most of the Galaxy’s giant molecular clouds, infrared emission,
and other signs of intense star formation. (There are several relevant reviews
in the symposium volume edited by Burton, 1979). Morphologically normal
galaxies show a systematic trend of SFR with type, which can be described as a
progression from a very short timescale for star formation in the earliest
types to very long in the latest. Some peculiar galaxies appear to be
undergoing intense bursts of star formation. These properties will be
discussed later in Section 7.

####  2.3.3 Factors affecting the SFR

An obvious question to ask is _why_ stars form at various rates in various
places. Factors affecting SFRs have been reviewed by Larson (1977b), and some
salient points will be mentioned here.

A popular assumption, following Schmidt (1959), is that the SFR varies as a
power $n$ of the gas density. However, attempts to determine $n$ empirically
run into two problems: $n$ appears to vary within and among galaxies,
indicating that the gas density is not the only relevant quantity; and the
very definition of “gas density” is ambiguous since it depends strongly on the
spatial resolution of the observations. Consequently, a power-law formula for
the SFR cannot be given the status of a physical law, but it can be regarded
as a useful parametrization in some circumstances (as in the later study of
Schmidt, 1963).

The physical environment needed for star formation is evidently cool,
gravitationally bound gas clouds, so the main question is: under what
conditions does interstellar gas clump into clouds? Larson (1977b) argues that
gas compression is the key requirement for star formation, and that both
large- and small-scale dynamical processes are important. On small scales,
interstellar shock and ionization fronts have been widely discussed as causes
of star formation. These processes can be self-sustaining, since supernovae
and H ii regions arising from recently-formed stars may induce further star
formation nearby (e.g. Elmegreen & Lada, 1977). Large-scale compression
mechanisms that may lead to star formation include gravitational settling of
gas into a thin layer, density waves of bar or spiral form, high-velocity
collisions between gas streams in young or interacting galaxies, and accretion
of intergalactic clouds. If star formation can be self-propagating over long
distances, differential rotation can spread the active regions out to produce
spiral structure (Gerola & Seiden, 1978). Large-scale processes are especially
relevant to the long-term evolution of galaxies, and since some of them
involve interactions with external matter, galaxies do not necessarily evolve
as independent systems. This point has been emphasized in a review by Saar &
Einasto (1978). Further discussion of star formation can be found in the
reviews cited, many references therein, and in a symposium volume edited by de
Jong et al. (1977).

Although some understanding has developed of the conditions under which stars
form, no formula for the SFR has emerged that can make a useful _prediction_
for any galaxy. Lynden-Bell (1977) has pointed out that a proper expression
for the SFR would contain so many unknown parameters as to be useless. He
wrote, “This rate $R$ (the SFR) probably depends on gas density $\rho_{\rm
g}$, gas sound speed $c_{\rm s}$, shock frequency $\omega_{\rm s}$, shock
strength $V_{\rm T}/c_{\rm s}$, gas rotation $\Omega$ and shearing rate $A$,
the magnetic field strength $\lvert\underaccent{\tilde}{B}\rvert$, the gas
metal abundance $Z$, and possibly the background star density $\rho_{*}$. Thus

$R=R\left(\rho_{\rm g},\ c_{\rm s},\ \omega_{\rm s},\ V_{\rm T}/c_{\rm s},\
\Omega,\ A,\ \lvert\underaccent{\tilde}{B}\rvert,\ Z,\ \rho_{*}\right).$

However, if we knew the true functional form of $R$ and offered it to a galaxy
builder he would probably tell us ‘Oh, go and jump in the lake, that’s far too
complicated’.”

The only feasible approach to galaxy building is to be less ambitious than to
want a physical formula for the SFR. Instead, schematic expressions, guided by
the above ideas on relevant factors, can be tested to see how galactic
evolution is affected by various parameters. Some examples already mentioned
just above and in Section 1 are that self-propagating star formation can lead
naturally to spiral structure, and that the final shape of a galaxy depends on
the ratios of the timescales for star formation, collapse, and gaseous
dissipation in the protogalaxy. Many examples of schematic expressions for the
SFR, in models for chemical and photometric evolution, will be considered
later.

###  2.4 Stellar Evolution Beyond the Main Sequence

Late stages of stellar evolution are important to galaxies because stars burn
much of their nuclear fuel after leaving the main sequence. The products of
this late fuel guzzling provide most of the chemical enrichment of galaxies,
and the energy liberated provides most of their integrated light. The
following outline of stellar evolution and nucleosynthesis emphasizes aspects
that will be relevant later in the discussion of galaxies, and necessarily
only skims the surface of these topics. Further details and references can be
found in the comprehensive review by Trimble (1975), and in other papers cited
below.

####  2.4.1 Stars near Solar mass

Giant stars of approximately Solar metallicity and mass are especially
interesting because they provide most of the light of elliptical and S0
galaxies and the nuclear bulges of spirals. The dominant old-disk population
of the Solar neighborhood provides a convenient sample of such stars for
detailed study; these may be somewhat younger than most stars in spheroidal
systems, but evolution on the giant branch is thought to be rather insensitive
to the initial stellar mass in the small range whose lifetimes are
${\sim}5-15\>\rm Gyr$.

Figure 5: Color–magnitude diagrams for the old open cluster M67 (Racine, 1971;
_filled circles_) and for red giants in old-disk groups (Eggen et al., 1962;
Tinsley & Gunn, 1976b; _open circles_); _crosses_ denote variable stars. The
M67 data have been reduce to $M_{\rm bol}$ versus $R-I$, as in Tinsley & Gunn
(1976b). This diagram illustrates the types of stars that apparently
contribute most of the light in old, metal-rich systems, such as giant
elliptical galaxies and the nuclear bulges of spirals. Because of the small
total cluster population, there are no late giants in M67, and because of
selection effects the visually brightest stars are over-represented in the
groups.

Figure 5 gives color–magnitude diagrams of the old open cluster M67 and a
sample of field giants, which are necessary for completeness because the
cluster is so poor that it contains none of the short-lived M giants.
Selection effects in the field sample mean that earlier giants are badly
under-represented relative to later types, and an unbiased luminosity function
would show roughly a factor of two decrease in the number of stars per
bolometric magnitude interval, for each magnitude increase in luminosity up
the giant branch. Some relevant aspects of the stellar evolution responsible
for this distribution of stars are as follows (e.g. Iben, 1974; Tinsley &
Gunn, 1976b):

  1. 1.

When the central $10\%$ of hydrogen has been converted to helium, the star
leaves the MS and its site of hydrogen burning moves from the center to a
shell around the helium core; as the shell is established, the star moves
along the subgiant branch (at $M_{\rm bol}\simeq 2.5$ in Figure 5) to the base
of the giant branch, at which point the helium core mass is ${\sim}0.2\>\rm
M_{\odot}$.

  2. 2.

During the first ascent of the giant branch to the so-called “tip” at $M_{\rm
bol}\simeq-3$, the helium core grows to ${\sim}0.45\>\rm M_{\odot}$. Although
this ascent occupies only about $10\%$ of the preceding lifetime, the energy
emitted, which is nearly proportional to the mass of hydrogen burned, is
greater.

  3. 3.

Helium ignition in the degenerate core – the “helium flash” – halts the ascent
of the giant branch, and the star very quickly drops to a lower luminosity,
$M_{\rm bol}\simeq 0.5$, where it remains for about $10^{8}\>\rm yr$ while the
core helium is burned to carbon (and a little oxygen). This stage appears as
the horizontal branch to the left of the giant branch in HR diagrams of
globular clusters, but most metal-rich stars form a clump of early K giants
superposed on the ascending giant branch; some metal-rich blue horizontal-
branch stars are found in the old-disk population, and these may represent a
minority that lost a large fraction of their envelopes at the helium flash
(e.g. Butler et al., 1976). Although the total energy emitted by core-helium-
burning stars is relatively small, blue horizontal-branch stars can make an
important contribution to the light of galaxies whose bluest stars otherwise
are at the turnoff.

  4. 4.

The star then ascends the giant branch for a second time (called the
asymptotic giant branch stage because it appears as an asymptote to the left
of the first giant branch in the HR diagrams of globular clusters), with two
shells burning hydrogen and helium respectively. The star can become brighter
and cooler than the first giant branch “tip”, as seen in Figure 5 but not in
all HR diagrams for globular clusters.

  5. 5.

The end of this stage of evolution is determined by mass loss, without which
the whole star would become a carbon-oxygen white dwarf. In fact, there is
much evidence that several tenths of a solar mass of envelope are lost before
the burning shells reach out to consume the whole initial mass of hydrogen.
For example, the average mass of white dwarfs is ${\sim}0.7\>\rm M_{\odot}$,
well below the turnoff mass (${\sim}1\>\rm M_{\odot}$) for field stars;
planetary nebulae, most of which belong to the old-disk population, represent
${\sim}0.2\>\rm M_{\odot}$ of ejected matter; and spectroscopic observations
of M giants show that they are losing mass in a wind. The existence of both
very late M giants and blue horizontal-branch stars in the old-disk population
suggests that mass loss occurs at a variety of rates among stars of a given
initial mass and composition. Since the average final core mass of solar-mass
stars must be the white dwarf mass, ${\sim}0.7\>\rm M_{\odot}$, the energy
liberated during the second ascent of the giant branch is comparable to that
liberated on the first ascent, with the difference that most of it now appears
in the near infrared from very cool stars.

Current stellar models do not predict post-MS evolution accurately enough to
be used directly in synthetic galaxies. The main uncertainties are the depth
of the surface convection zone, which determines the effective temperature at
a given luminosity, and mass loss; a related problem is the outward mixing of
products of nucleosynthesis, which are observed on the surfaces of many giants
but cannot be predicted in detail. Consequently, galaxy models must rely
heavily on empirical studies of giant populations. The implications of these
problems for understanding photometric properties of elliptical galaxies are
reviewed by Faber (1977). Mass loss and mixing also mean that even these low-
mass stars contribute significant amounts of newly synthesized elements to the
ISM, including for example carbon and nitrogen, which are often overabundant
at the surfaces of giants and in planetary nebulae. The isotopes 13C and 14N
are by-products of hydrogen burning by the CNO cycle, while 12C must be a
product of core helium burning (e.g. Trimble, 1975; Iben & Truran, 1978).

####  2.4.2 Stars of $1-4\>\rm M_{\odot}$

Figure 6: Composite HR diagrams for (mainly) post-main-sequence stars in
galactic clusters grouped by age (Harris, 1976). Ages ($\tau$ in yr) are
indicated. These diagrams illustrate the distribution of supergiants expected
in galaxies containing stars of the appropriate ages.

White dwarfs appear in open clusters where their progenitors must have been
stars with initial masses up to ${\sim}4\>\rm M_{\odot}$ (e.g. Tinsley,
1975a). The substantial mass loss required is in agreement with stellar models
that allow for winds (Fusi-Pecci & Renzini, 1976; Mengel, 1976); these models
suggest that stars up to some limit ${\sim}3-6\>\rm M_{\odot}$ lose so much
mass that they fizzle before the point of carbon ignition, which would
otherwise occur (possibly explosively) when the degenerate core reached
$1.4\>\rm M_{\odot}$. Therefore, stars with MS lifetimes in the whole range
from ${\sim}10^{8}$ to ${\sim}10^{10}\>\rm yr$, from mid-B to early G on the
MS, evolve internally similarly to the solar-mass stars discussed above, and
they make similar contributions to nucleosynthesis.

####  2.4.3 Stars above $8\>\rm M_{\odot}$

Stars initially more massive than ${\sim}8\>\rm M_{\odot}$, which are early B
and O stars on the MS, are believed to ignite carbon (quietly) in a non-
degenerate core and to proceed through many stages of nuclear burning. Models
acquire an onion-skin structure of successive elements, with unaltered
envelope material (mostly hydrogen) on the outside, then layers of helium,
carbon, oxygen, neon, magnesium, etc., with the ashes of each nuclear reaction
providing fuel for the next. These stars and their contributions to
nucleosynthesis are discussed by Arnett (1978), Weaver et al. (1978), and
Chiosi (1979), the last stressing the very important effects of mass loss.
Theoretical models predict that the stars undergo core collapse after the last
exoergic reaction, formation of iron-peak elements, and the event is expected
to send a shock wave through the star leading to explosive nucleosynthesis of
most of the less abundant elements. Because Type II supernovae are associated
with the most massive stars in galaxies, they are believed to represent these
explosive stellar deaths. The lower initial mass limit for stars with this
fate is somewhere in the range $6-10\>\rm M_{\odot}$, depending on the
structure of stellar cores at carbon ignition; it is sometimes suggested that
there is also an upper limit for such deaths, stars above the higher limit
collapsing to black holes that swallow most of the new elements (e.g. Wheeler,
1978a).

The evolution of massive stars in the HR diagram (as well as internally) must
be strongly affected by mass loss, since winds of ${\sim}10^{6}\>\rm
M_{\odot}\ yr^{-1}$ are inferred from the spectra of MS O stars and much
higher mass-loss rates are common in supergiants (Conti, 1978). These rates
mean that the stellar mass is changing significantly on an evolutionary
timescale. The evolutionary tracks of such stars in the HR diagram cannot be
predicted reliably (e.g. Chiosi et al., 1978; Stothers & Chin, 1978), so
models for galaxies with young stars are subject to some uncertainty.
Empirical illustrations of the evolutionary tracks of massive stars are
provided by compilations of HR diagrams of open clusters in age groups
(Harris, 1976), of which the youngest few are reproduced in Figure 6. The
general trends with age are clear, but there are obviously many uncertainties
due to the small numbers of supergiants, and a scatter that may be due partly
to the presence of clusters with different metallicities.

####  2.4.4 Stars of intermediate mass

There may be some stars of mid-B type on the MS, with masses in the interval
${\sim}4-8\>\rm M_{\odot}$, that neither die quietly as white dwarfs nor
follow the sequence of nuclear burning stages through core collapse and (Type
II) supernova explosion. Such stars would be characterized by carbon ignition
in a degenerate core, and it is possible that they do not exist if the upper
limit for white dwarf formation exceeds the lower limit for non-degeneracy. If
they do exist, their final evolution is very uncertain, since it is not known
whether degenerate carbon ignition would blow a star apart or lead quietly to
further burning; this problem is reviewed by Wheeler (1978a).

Regardless of their final fate, stars of about $4-8\>\rm M_{\odot}$666Values
of stellar masses will from now on be understood to be initial values, unless
otherwise stated. make important contributions to galactic enrichment in
elements that are produced during hydrogen and helium burning and are mixed up
to the surface. Overabundances of carbon, nitrogen, barium and other elements
are seen in the spectra of many giants in this mass range, and their
theoretical origin is discussed by Iben & Truran (1978).

The evolution of stars between 4 and $8\>\rm M_{\odot}$ in the HR diagram is
subject to the same uncertainties that beset more and less massive stars. The
groups of clusters in Harris (1976) again give a useful guide, but they are
rather sparse, as seen in Figure 6.

About half of all supernovae in spiral galaxies are Type I supernovae (SNI),
which, because of their sites in galaxies are associated with stars below
${\sim}6\>\rm M_{\odot}$. Their origin has been controversial for some time,
as reviewed by Tammann (1974, 1977), Tinsley (1975a, 1977a), and Wheeler
(1978a). This leaves a serious gap in our understanding of stellar deaths and
nucleosynthesis, especially since SNI appear to eject significant quantities
of iron (Kirshner & Oke, 1975). Alternative theories include the following.

  1. 1.

SNI are exploding white dwarfs. This has long been the standard theory,
because SNI occur in elliptical galaxies where only stars ${\sim}1\>\rm
M_{\odot}$ are conventionally believed to be dying. However, the white dwarf
theory conflicts with the large envelopes (of supergiant size and $\gtrsim
1\>\rm M_{\odot}$) that are invoked to explain the early spectra and light
curves of SNI, and with the clear association of SNI with a young stellar
population in spiral and irregular galaxies (Oemler & Tinsley, 1979).

  2. 2.

SNI are stars ${\sim}2\>\rm M_{\odot}$ that live for ten times longer than
normal by mixing completely on the MS. Wheeler (1978b) proposed this model to
be consistent with the envelope masses, populations in elliptical galaxies,
the extreme deficiency of hydrogen in SNI spectra, and some properties of the
Crab nebula. Its problems are that occasional complete mixing of MS stars must
be postulated ad hoc, and that often SNI are associated with a much younger
stellar population.

  3. 3.

SNI are stars of intermediate mass, ${\sim}4-6\>\rm M_{\odot}$. This was
suggested by Renzini (1976) on the grounds that these stars could blow away
their hydrogen-rich envelopes before dying, and by Oemler & Tinsley (1979) to
account for the association of SNI with ongoing star formation in spiral and
irregular galaxies. This hypothesis implies a rate of star formation in
elliptical galaxies that should be marginally detectable.

  4. 4.

There are two kinds of SNI, those from low-mass stars that appear in
elliptical galaxies, and those from more massive stars that account for the
frequency in spiral and irregular galaxies (Dallaporta, 1973). While this
hypothesis avoids the problems of the preceding ones, the properties of SNI
(while not exactly homogeneous) do not divide naturally into two classes
suggestive of very different stellar origins.

In any case, the production of iron by SNI and the red-giant nucleosynthesis
mentioned above show that stars with a very wide range of masses contribute to
galactic enrichment in various elements.

####  2.4.5 Effects of initial composition

Stars now forming and evolving in different regions of galaxies have at least
a ten-fold range of metallicities, and the first stars everywhere were
presumably very metal-poor. It is therefore important to ask how stellar
evolution is affected by initial composition.

The evolution of stars from ${\sim}0.7-1\>\rm M_{\odot}$ has been widely
studied as a function of metallicity ($Z$) and helium abundance ($Y$), for the
interpretation of HR diagrams of globular clusters (e.g. Iben, 1974; Mengel et
al., 1979; Sweigart & Gross, 1978). Much less attention has been paid to more
massive stars. In the largest systematic survey to date, stars up to $7\>\rm
M_{\odot}$ with a wide range of $Z$ and $Y$ have been followed to the base of
the giant branch (Mengel et al., 1979); Sweigart & Gross (1978) have followed
some of these up the giant branch. A few more massive models have been studied
by Trimble et al. (1973) and Harris & Deupree (1976), and Alcock & Paczynski
(1978) have followed the evolution of stars up to $10\>\rm M_{\odot}$, with
various $Z$, through core helium exhaustion. The results of these calculations
suggest that important systematic differences in nucleosynthesis may occur as
a result of initial $Z$ differences, but they do not go far enough to provide
quantitative estimates. Models for chemical evolution usually assume that the
production of primary elements – those that are made from hydrogen and helium
that were initially in the star – is a function of stellar mass alone, but
this assumption may be leading to systematic errors. (An exception is the
study undertaken by Arnett, 1971 of explosive nucleosynthesis of certain
neutron-rich primary elements which may depend systematically on the star’s
initial $Z$; Arnett, 1971 showed how such effects could be tested using
relative abundances of metals in metal-poor stars).

Some elements are believed to be synthesized in amounts that depend _directly_
on the initial composition of the star, because they are made not only from
hydrogen and helium but also from heavier elements that were present
initially. These are called secondary elements, and some examples are:

  1. 1.

13C and 14N are made in stellar envelopes as by-products of hydrogen burning
in the CNO cycle, mainly from 12C that was present initially777If any fresh
12C, a product of helium-burning in the star’s own core, gets mixed to the
hydrogen-burning region and processed in this way, the resulting 13C and 14N
would be called primary products..

  2. 2.

Barium is made from the addition of neutrons to iron nuclei, probably in the
intershell region of asymptotic-branch giants (e.g. Iben & Truran, 1978).

Because the structure of stars at different stages of evolution depends on
their initial composition, these and other elements may also be affected
_indirectly_ by initial composition.

Extensive calculations of stellar models lie ahead before the dependence of
both primary and secondary nucleosynthesis on composition is adequately
understood. Almost all calculations to date have used Solar relative
abundances of the heavy elements (all scaled with the single initial
metallicity parameter $Z$), and it will be important also to find out how
stellar evolution and nucleosynthesis depend on variations among these
relative abundances.

Another set of questions concerns changes in the HR diagram that would affect
the colors of galaxies with stars of different compositions. A population of
low $Z$ is in general bluer than normal for two reasons: the effective
temperature of a star of a given mass and evolutionary stage is (in most cases
of interest) higher at lower $Z$, because metal-poor stars tend to have
smaller radii; and most colors at a given effective temperature are bluer at
lower $Z$, because the reduced line-blanketing restores more flux at short
wavelengths than at long wavelengths. These effects are partially offset when
one considers populations of a given age, because the turnoff mass is lower at
lower $Z$, but this factor is relatively unimportant. Some implications of
these changes will be discussed in Section 5.1 and Section 7.1.4.

##  3 Aims and Methods of Chemical Evolution

Studies of chemical evolution aim to account for abundance distributions of
the elements, including the variation of stellar metallicities with age and
position in the Galaxy, abundance gradients in galaxies, variations in
relative abundances of elements heavier than helium, and related observations.
The main processes governing chemical evolution are indicated in Figure 1:
star formation, nucleosynthesis, mass loss from evolving and dying stars, and
gas flows. Because of the importance of dynamical processes, another
motivation for studying chemical evolution is that it leads to clues to the
dynamical evolution of galaxies, as examples in Sections 4 and 5 will show.
Details of the basic processes have been extensively reviewed elsewhere, so
the emphasis here will be on methods of modeling chemical evolution, and
selected applications. Reviews for further background material include Audouze
& Tinsley (1976), Faber (1977), Pagel (1978a, b), Trimble (1975), van den
Bergh (1975), and others cited in Section 2. There is no clearer introduction
to chemical evolution than the seminal paper by Schmidt (1963) on the subject.

###  3.1 Basic Assumptions and Equations

Models for chemical evolution follow, analytically or numerically, abundance
changes in the ISM of a region and the resulting abundance distributions in
stars. For most purposes, the region under study in a particular model can be
assumed to have ISM of uniform composition and to gain or lose mass through
gas flows only; complicated models for whole galaxies are generally divided
up, for computation, into zones with these properties. The basic equations are
then easy to derive.

The total mass $M$ of the system changes according to the net inflow rate $f$
of gas:

$\frac{dM}{dt}=f.$ (3.1)

($f$ is often called the infall or accretion rate). The mass of stars $M_{\rm
s}$ changes via star formation and mass loss from evolving stars:

$\frac{dM_{\rm s}}{dt}=\psi-E,$ (3.2)

where $\psi$ is the SFR and $E$ is the total ejection rate from stars of all
masses and ages. Similarly, the mass of ISM, called “gas” ($M_{\rm g}$),
changes through star formation, ejection, and net inflow into the region:

$\frac{dM_{\rm g}}{dt}=-\psi+E+f,$ (3.3)

Since $M=M_{\rm s}+M_{\rm g}$, the sum of Equations (3.2) and (3.3) is simply
(3.1).

The fraction of gas in the system is written

$\mu\equiv\frac{M_{\rm g}}{M},$ (3.4)

so the mass of stars is obviously

$M_{\rm s}=\left(1-\mu\right)M.$ (3.5)

The ejection rate can be written in terms of the IMF and SFR, Equation (2.1),
if one makes the usual approximation that each star undergoes its entire mass
loss after a well-defined lifetime. Let $w_{\rm m}$ be the remnant mass and
$\tau_{\rm m}$ the lifetime of a star of (initial) mass $m$; the star was
therefore formed at time ($t-\tau_{\rm m}$) if it dies at time $t$. Then the
ejection rate at time $t$ is

$E(t)=\int_{m_{\rm t}}^{\infty}\left(m-w_{\rm m}\right)\psi\left(t-\tau_{\rm
m}\right)\phi(m)\ dm,$ (3.6)

where $m_{\rm t}$ is the mass with $\tau_{\rm m}=t$ (the turnoff mass) and
$\phi(m)$ is the IMF at time ($t-\tau_{\rm m}$). The approximation of sudden
mass loss at the end of each star’s lifetime is usually a reasonable one,
since for most stars nearly all of the mass loss is thought to occur in a
final small fraction of their lives; mass loss on the MS is important for O
stars, however, so the approximation fails if timescales $<10^{7}\>\rm yr$ are
of interest.

For illustration, let us consider the evolution of a single “metal” abundance
parameter $Z$, where “metals” are one or all of the common primary elements
(e.g., C, O, Fe), and let us assume that their production is a function only
of stellar mass, not of initial composition. (The methods for calculating this
schematic $Z$ are readily adapted to more realistic cases). Let us also assume
for simplicity that material ejected from stars is mixed instantly throughout
the ISM in the region under study. Then the mass of metals in the gas, which
is $ZM_{\rm g}$, evolves via star formation (putting metals from the ISM into
stars), ejection, and gas flows, according to the equation

$\frac{dZM_{\rm g}}{dt}=-Z\psi+E_{Z}+Z_{\rm f}f,$ (3.7)

where $E_{Z}$ is the total ejection rate of metals from stars and $Z_{\rm f}$
is the mean metal abundance in infalling gas. $E_{Z}$ includes both newly
synthesized metals and those that were in the star from birth and re-ejected.
Let $p_{\rm zm}$ be the mass fraction of a star of mass $m$ that is converted
to metals and ejected; then the rate of ejection of new metals from stars at
time $t$ is

$\int_{m_{\rm t}}^{\infty}mp_{\rm zm}\psi\left(t-\tau_{\rm m}\right)\phi(m)\
dm.$ (3.8)

Unprocessed material occupies a mass ($m-w_{\rm m}-mp_{\rm zm}$) of the
ejected part of a star of mass $m$, and the metal abundance in this region is
$Z(t-\tau_{\rm m})$. Thus the total ejection rate of old and new metals is
obtained by adding the ejection rate from all masses of these unprocessed
metals to the rate (Equation 3.8):

$\medsize E_{Z}(t)=\int\limits_{m_{\rm t}}^{\infty}\left[\left(m-w_{\rm
m}-mp_{\rm zm}\right)Z\left(t-\tau_{\rm m}\right)+mp_{\rm
zm}\right]\psi\left(t-\tau_{\rm m}\right)\phi(m)\ dm.$ (3.9)

The mean metallicity of stars ever formed, denoted $Z_{\rm s}$, can be
obtained by writing an equation for the conservation of metals: a mass $Z_{\rm
s}M_{\rm s}$ of metals is stored in stars, a mass $ZM_{\rm g}$ is in the gas,
and their sum is the mass of new metals ever ejected, which is obtained by
integrating Equation (3.8) over time. Thus $Z_{\rm s}$ is given by

$Z_{\rm s}M_{\rm s}+ZM_{\rm g}=\int_{0}^{t}\int_{m_{\rm
t^{\prime}}}^{\infty}mp_{\rm zm}\psi\left(t^{\prime}-\tau_{\rm
m}\right)\phi(m)\ dt^{\prime}dm.$ (3.10)

It is convenient to define some integrals that depend on the IMF and
parameters of stellar evolution but not on $\psi(t)$. The _returned fraction_
$R$ is defined by the integral

$R\equiv\int_{m_{1}}^{\infty}\left(m-w_{\rm m}\right)\phi(m)\ dm,$ (3.11)

where $m_{1}$ is the present turnoff mass, and is usually taken to be $1\>\rm
M_{\odot}$ in evaluating $R$. Because the IMF is normalized as in Equation
(2.2), $R$ is the fraction of mass put into stars at a given time that is
thereafter returned to the ISM in the lifetime of a solar-mass star; for
example, if the stars in a galaxy were all formed in a single initial burst
with a total mass $M$, then ${\sim}10^{10}\>\rm yr$ later the mass in stellar
form would be $M(1-R)$. An estimate of $R$ can be obtained using the local IMF
as approximated in Equation (2.9), with a white dwarf mass of $0.7\>\rm
M_{\odot}$ as the remnant mass $m_{\rm w}$ for $m\leq 4\>\rm M_{\odot}$ and a
neutron star mass of $1.4\>\rm M_{\odot}$ for $m>4\>\rm M_{\odot}$; the result
is, $R=0.17\times(10\>\rm M_{\odot}\ pc^{-2}\ Gyr^{-1}/\psi_{1})$.

A second useful integral is the _yield_ $y$, which is the mass of new metals
ejected (eventually) when unit mass of matter is locked into stars. The yield
can be defined for any element of interest, and for the “metals” considered
above it is given by

$y\equiv\frac{1}{1-R}\int_{m_{1}}^{\infty}mp_{\rm zm}\phi(m)\ dm.$ (3.12)

Again, if a mass $M$ of stars were formed in a single initial burst, then
${\sim}10^{10}\>\rm yr$ later, the mass of new metals ejected would be
$y(1-R)$. Sometimes the quantities $p_{\rm zm}$ are called stellar yields, and
$y$ is called the net yield.

Although $R$ and $y$ are defined with a specific lower mass limit, their
values do not depend strongly on $m_{1}$, so they turn out to be useful
parameters in many situations. For example, $R$ has been used in Section 2.3.1
to relate the mass of stars now in the Solar neighborhood to the mass ever
formed, giving a reasonable estimate even though star formation did not occur
in a single burst; and the yield will prove to be a first approximation to the
abundance of metals resulting from a given IMF and stellar production
parameters (e.g., $p_{\rm zm}$), independently of the detailed history of the
system.

Many insights into chemical evolution can be obtained using analytical
approximations to the above equations, while for detailed modeling they can be
cast into forms suitable for numerical computation. These techniques will be
outlined in turn.

###  3.2 Analytical Approximations

The equations of chemical evolution are especially easy to handle if one makes
an approximation known as _instantaneous recycling_ : stars are divided into
two classes, those that live forever (masses less than some value $m_{1}$) and
those that die as soon as they are born ($m>m_{1}$). Thus the lower mass
limits in the integrals in Equations (3.6) – (3.10) are taken to be the time-
independent $m_{1}$, and the assumed instant deaths allow the arguments
($t-\tau_{\rm m}$) to be replaced by simply ($t$). (Note that the assumption
now made of death immediately after birth, so $\tau_{\rm m}\equiv 0$, is a
much stronger and less realistic assumption than the earlier one of sudden
mass loss, which simply allowed $\tau_{\rm m}$ to be well-defined for each
star). Equations (3.6) and (3.9) reduce at once to the much simpler relations,

$E(t)=R\psi(t),$ (3.13)
$E_{Z}(t)=RZ(t)\psi(t)+y(1-R)\left[1-Z(t)\right]\psi(t),$ (3.14)

in which all time-dependent quantities are evaluated at the current time $t$
only. Since $Z\ll 1$ in all situations of interest, Equation (3.14) can be
written more simply

$E_{Z}(t)=RZ(t)\psi(t)+y(1-R)\psi(t).$ (3.15)

Substituting in Equations (3.2), (3.3), and (3.7), we have now

$\frac{dM_{\rm s}}{dt}=(1-R)\psi,$ (3.16) $\frac{dM_{\rm
g}}{dt}=-(1-R)\psi+f,$ (3.17) $\frac{dZM_{\rm
g}}{dt}=-Z(1-R)\psi+y(1-R)\psi+Z_{\rm f}f.$ (3.18)

An equation for the metal abundance $Z$ of the gas, rather than the mass of
metals it contains, is obtained by substituting Equation (3.17) into (3.18),
using the identity $M_{\rm g}dZ/dt=d(ZM_{\rm g})/dt-ZdM_{\rm g}/dt$; the
result is

$M_{\rm g}\frac{dZ}{dt}=y(1-R)\psi+\left(Z_{\rm f}-Z\right)f.$ (3.19)

For simplicity, let us assume that the IMF is constant. (The methods developed
for a constant IMF can be adapted when necessary to more complicated cases).
The quantities $R$ and $y$ are therefore constants.

Equation (3.16) now gives an expression for the mass of stars at time $t$,

$M_{\rm s}(t)=(1-R)\int_{0}^{t}\psi\left(t^{\prime}\right)\
dt^{\prime}\equiv(1-R)\overline{\psi}t.$ (3.20)

The mean metallicity of stars can be obtained from Equation (3.10), using the
instantaneous recycling approximation and the assumption of a constant IMF:
the integrals with respect to time and mass separate, and are simply $M_{\rm
s}/(1-R)$ and $y(1-R)$ respectively, so the result is $Z_{\rm s}M_{\rm
s}+ZM_{\rm g}=yM_{\rm s}$, from which

$Z_{\rm s}=y-\frac{\mu}{1-\mu}Z.$ (3.21)

This result shows that $Z\rightarrow y$ as $\mu\rightarrow 0$, which simply
states that, when there is no gas, all the metals ever made and ejected
($yM_{\rm s}$) are incorporated into later generations of stars. An estimate
of the yield for the local IMF is therefore the mean metallicity of stars in
the Solar neighborhood (where the gas fraction is small, $\mu\sim 0.05$),
i.e., $y\simeq 0.8\>\rm Z_{\odot}$. The approximation $y\sim\>\rm Z_{\odot}$
will be useful below.

Further solutions to the above equations depend on the assumptions made about
gas flows, so two extreme cases will be considered for illustration.

####  3.2.1 A closed system, initially unenriched gas

In this case, we set $f=0$, and the initial values are $M_{\rm g0}=M$, $M_{\rm
s0}=0$, $Z_{0}=Z_{\rm s0}=0$. Obviously, $M=\rm constant$. Time can be
eliminated as an explicit variable if Equation (3.19) is divided by (3.17),
with the result

$M_{\rm g}\frac{dZ}{dM_{\rm g}}=-y,$ (3.22)

i.e.,

$Z=y\ln\left(\frac{M_{\rm g0}}{M_{\rm g}}\right)=y\ln\left(\frac{M}{M_{\rm
g}}\right)=y\ln\left(\frac{1}{\mu}\right).$ (3.23)

This solution is valid as long as $Z\ll 1$. If not, Equation (3.14) must be
used instead of (3.15); the result is then $Z=1-\mu^{y}$, which reduces to
(3.23) if $Z\ll 1$, and has the limit $Z\rightarrow 1$ as $\mu\rightarrow 0$.
(This formally correct limit is of course absurd, and is a consequence of
neglecting the effects of metallicity on stellar nucleosynthesis). Original
derivations of these results were given by Talbot & Arnett (1971), and
Equation (3.23) was derived independently by Searle & Sargent (1972). Through
Equations (3.21) and (3.23), $Z_{\rm s}$ can also be expressed as a function
of $\mu$.

####  3.2.2 A system with infall balanced by star formation

Here it is assumed that star formation just keeps up with the rate of infall
plus stellar gas loss:

$\psi=f+R\psi,$ (3.24)

so that, from Equation (3.17),

$M_{\rm g}={\rm constant}=M_{0}.$ (3.25)

The picture is that the total mass grows by infall, while star formation
maintains a constant gas mass. Other initial conditions are $M_{\rm s0}=0$ and
$Z_{0}=Z_{\rm s0}=0$, and the infalling gas will be assumed to be unenriched
so that $Z_{\rm f}=0$. To eliminate time in this case, we divide Equation
(3.17) by (3.1), with the result

$M_{\rm g}\frac{dZ}{dM}=y-Z.$ (3.26)

It is useful to solve this Equation in terms of a parameter $\nu$ that gives
the ratio of mass accreted to the initial mass:

$\nu\equiv\frac{M-M_{0}}{M_{0}}=\frac{M-M_{\rm g}}{M_{\rm
g}}=\frac{1}{\mu}-1.$ (3.27)

Then Equation (3.26) has the solution

$Z=y\left(1-e^{-\nu}\right),$ (3.28)

a result due originally to Larson (1972a). Equation (3.21) again gives $Z_{\rm
s}$. An important difference between this infall model and the preceding
closed model is that here $Z\rightarrow y$ as $\mu\rightarrow 0$; an
equilibrium is set up between the rate of infall of metal-free gas and the
rate of enrichment by evolving stars.

####  3.2.3 Generalities

Although the above two models are extremely schematic, their Equations (3.23)
and (3.28) for the “metal” abundance of the gas have some common properties
that are found in a wide variety of models for chemical evolution. The
following generalizations are approximately true even when instantaneous
recycling is a poor approximation, but they break down if the IMF is time-
dependent.

  1. 1.

_$Z$ is proportional to the net yield, $y$._ Thus any primary elements whose
stellar production parameters are independent of composition will have
abundances in proportion to their respective yields. The yields in turn can be
calculated from Equation (3.12), for a given IMF, independently of any model
for star formation or gas flows in the system. The utility of this result is
that theories of nucleosynthesis can be tested by comparing predicted relative
yields of various elements with their observed relative abundances.

  2. 2.

_$Z/y$ depends chiefly on current properties of the system_, and is
insensitive to its past history. This statement is exactly true in the above
models because of the assumption of instantaneous recycling (which led to the
cancellation of time as an explicit variable). In numerical models that allow
for finite stellar lifetimes, the same result holds approximately, and
relevant current properties include the gas fraction ($\mu$) and the ratio of
SFR to gas flow rate ($\psi/f$).

  3. 3.

_$Z/y$ depends rather weakly on model-dependent quantities_ such as $\mu$ and
$\psi/f$. Consequently, regions of galaxies with gas fractions differing by
orders of magnitude may have interstellar abundances differing by only small
factors. This prediction allows one to test whether theories of
nucleosynthesis give the right absolute amounts of elements, as well as the
right relative amounts; in particular, the yield of an element, calculated for
the local IMF, should be in order of magnitude equal to its abundance in the
Solar System or nearby stars.

###  3.3 Numerical Models

Chemical evolution is best studied numerically in cases where approximations
that make the analytical approach transparent (or possible) break down.
Situations in which instantaneous recycling is no longer a useful
approximation include the following:

  1. 1.

If the SFR is a strongly decreasing function of time, systematic errors result
from setting $\psi(t-\tau_{\rm m})=\psi(t)$; in particular, low-mass stars
formed early die later in much greater numbers (relative to massive stars)
than would be predicted on the basis of the current SFR.

  2. 2.

If one is interested in time-dependent abundance ratios arising from
nucleosynthesis in stars of different lifetimes, the effects would be entirely
lost by neglecting finite stellar lifetimes. Even if instantaneous recycling
can be assumed, the analytical approach becomes intractable in models with
time-dependent IMFs, or in considerations of radioactive elements with
lifetimes of billions of years, except in very special cases.

For numerical computation, equations such as (3.1) – (3.10) are expressed as
differences in finite steps of time and sums over a grid of stellar masses.
The programming required is little more than careful book-keeping, so the
effort in models for chemical evolution goes not into techniques but into the
astrophysical input. Many examples of numerical models will be found in the
references cited in Sections 4 and 5, in the discussion of particular
problems.

##  4 Chemical Evolution in the Solar Neighborhood

The extensive data available on chemical compositions of nearby stars and H ii
regions, and details of the isotopic composition of Solar System material,
have led to many efforts to build consistent models for chemical evolution in
the Solar neighborhood. In trying to explain the abundance distributions,
these models also shed light on dynamical processes that have affected the
local region of the Galaxy, and they set some constraints on theories of
nucleosynthesis.

The “neighborhood” must be carefully defined to include the range of
environments occupied by nearby stars and their predecessors in their motions
around the Galaxy, and to include stars that were formed near the plane but
now occupy a thicker disk. A small “local swimming hole” around the Sun (in
Baade’s words) will not do. Instead, the Sun’s “nucleogenetic pool” is
usefully defined as a region about a kpc wide in the Galactic plane (to
include most stellar epicyclic motions and periodic gas motions), extended
around the Galaxy at the Sun’s galactocentric distance (to include passages of
local material through spiral arms and related small-scale patchiness), and
extending about a kpc out of the plane (to include stars whose periodic
motions carry them far from the plane). This region is shaped like a
cylindrical shell, and quantities like the mass of stars, SFR, etc. are
expressed in units per square parsec, referring to _an average column at the
Solar galactocentric distance_. Spatial averages over this pool can be
interpreted as short-term time averages during the history of a given small
amount of material, and models generally consider explicitly only timescales
longer than passages between spiral arms, etc. (Some special studies, such as
the interpretation of elements with radioactive lifetimes
${\sim}10^{6}-10^{9}\>\rm yr$, do need to consider short-term changes).
Although the region has been defined to include most relevant periodic
motions, it cannot be treated as a closed system, because it is probably
affected by systematic radial gas flows in the disk and by infall from
outside, especially at early times.

Halo stars are sometimes treated as part of the “neighborhood” (in which case
its extent out of the plane must be many kpc), but more often the halo is
treated as a separate subsystem of the Galaxy, whose effects on the evolution
of the disk are perturbations from outside. Properties of halo stars are in
any case relevant to the Solar neighborhood, so they will be discussed along
with the properties of disk stars.

###  4.1 Outline of Relevant Data

Chemical abundances in the Solar neighborhood and in the Solar System have
been reviewed in detail by Trimble (1975) and several authors at a recent
conference at Yale University. An outline of the points most relevant to this
article will be given here. Because of systematic radial gradients in chemical
abundances in the Galaxy (Section 5.1), the following remarks apply only to
stars within a few kpc of the Sun’s galactocentric distance.

Figure 7: The relation between age and metallicity for disk stars in the Solar
neighborhood (Twarog, 1980). Boxes denote the range of ages referred to by
each point, and the $\pm 1\sigma$ spread in $\rm[Fe/H]$ for the age group.
Error bars attached to the points denote the standard error in the mean. Note
that the mean metallicity increases by a factor of less than four during the
lifetime of the disk.

Stars with halo kinematics are all metal-poor, with $\rm[Fe/H]\lesssim-1$,
whereas stars with disk motions nearly all have $\rm[Fe/H]\gtrsim-0.5$. (As
usual, abundance ratios in square brackets are logarithms of the ratios
relative to their Solar values). Since most halo stars are probably older than
most disk stars (Section 1.1), their composition differences indicate a
systematic increase of metallicity with time of formation. The disk stars
alone show a systematic increase of metallicity with time of formation,
illustrated in Figure 7, but the trend is only about a factor of three over
the whole lifetime of the disk. Open clusters show a very weak dependence of
metallicity on age, if any, because they are found at such a wide range of
galactocentric distances that the radial gradient in metallicity obscures the
time-dependence.

A powerful tool for chemical evolution is the metallicity distribution of all
stars ever formed, which is given by samples of G–M dwarfs since these stars
have remained on the MS since they formed. The distribution can be roughly
approximated by a log-normal function, with a mean $\langle\left[\rm
Fe/H\right]\rangle=-0.1$ and standard deviation $\sigma\left(\left[\rm
Fe/H\right]\right)=0.2$, allowing for observational errors (Pagel & Patchett,
1975; Pagel, 1978a; and references therein). Its striking characteristic is
the extreme paucity of metal-poor stars: only a few percent of stars in the
local cylinder (including halo stars) have $\rm[Fe/H]<-1$. The lack of metal-
poor disk stars proves to be one of the most powerful constraints on models
for the Solar neighborhood, as discussed below.

Stellar “metallicities” are commonly denoted $\rm[Fe/H]$ because most
photometric indices of the abundance of heavy elements are sensitive to
blanketing by numerous iron lines. Detailed abundance analyses show that other
elements heavier than helium usually occur in fixed proportions, so a single
metallicity index is often adequate. However, the exceptions to this
uniformity are often of great interest. Certain abundance variations in
_evolved_ stars are believed to show products of the stars’ own
nucleosynthesis that are mixed to the surface and lost in a stellar wind
(Section 2.4) – direct observational evidence for enrichment of the ISM. Some
elements also show relative abundance variations in _unevolved_ stars, and in
these cases the anomalies must have been present in the ISM from which the
star formed.

Besides these interesting abundance variations, there are others that should
be ignored or “corrected” for in the context of chemical evolution. Extraneous
effects include depletion of metals in the diffuse interstellar gas due to
their incorporation into grains; anomalous interstellar abundance ratios,
especially between isotopes of the same element, caused by chemical
fractionation888This is a case of real chemistry, not in the special sense of
“chemical” evolution of galaxies. Other terminology has been suggested to
avoid the term “chemical evolution” now that real chemistry of the ISM is
becoming an important topic in astronomy, but the alternative suggestions have
their own problems. For example, “nuclear evolution” can be confused with the
evolution of the nuclei (centers) of galaxies, and it obscures the great
importance of gas flows, in addition to nucleosynthesis, in determining the
evolution of chemical abundances in galaxies.; and large abundance variations
of various elements on the surfaces of Ap and related stars, produced by
differential diffusion of elements in the stellar atmospheres (Trimble, 1975).
Unfortunately, it is not always clear whether a particular composition anomaly
is due to some such extraneous effect, to nucleosynthesis in the star
observed, or to significant composition changes in the ISM.

###  4.2 The “G-Dwarf Problem”

The metallicity distribution of long-lived disk dwarfs leads to a problem
originally discovered by van den Bergh (1962) and Schmidt (1963): there are
far fewer metal-poor stars than one predicts using the most straightforward
assumptions. This contradiction is called the “G-dwarf problem”, although a
stellar metallicity distribution has now been confirmed for M dwarfs (Mould,
1978). The straightforward assumptions involved are the following:

  1. 1.

the Solar neighborhood can be modeled as a closed system;

  2. 2.

it started as $100\%$ metal-free gas;

  3. 3.

the IMF is constant; and

  4. 4.

the gas is chemically homogenous at any time.

Any model based on these assumptions is called a _“simple” model_. It is now
recognized that assumptions (i) and (ii) are almost certainly false and that
more realistic alternatives lead to a natural solution of the G-dwarf problem;
assumptions (iii) and (iv) are probably also unrealistic, but there is less
evidence for believing that they are violated sufficiently to solve the
problem entirely. Although the G-dwarf problem is no longer regarded as a
difficulty in chemical evolution, it will be reviewed here because of its
earlier importance in motivating many models. Moreover, a review of this
problem serves to emphasize that simple models cannot be used consistently in
any arguments relating to chemical evolution in the Solar neighborhood – the
G-dwarf metallicities are so much at variance with such models that they must
ignore very important effects, and could lead to highly misleading results in
other contexts. This conclusion also leads to suspicion of simple models for
other regions of galaxies.

The problem itself can be derived analytically, following Pagel & Patchett
(1975), using the instantaneous recycling approximation and the results for a
simple model in Section 3.2.1. Let us consider the cumulative metallicity
distribution of stars ever formed (which is represented observationally by
dwarfs of mid-G type and later). From Equation (3.5), the fraction of all
stars that had been made while the gas fraction was $\geq\mu$ is

$\frac{M_{\rm s}}{M_{\rm s1}}=\frac{1-\mu}{1-\mu_{1}},$ (4.1)

where subscripts 1 denote present values. From Equation (3.23), these stars
were formed while the metallicity of the gas was $\leq y\ln\ \mu^{-1}$. The
fraction of stars with metallicities $\leq Z$, denoted $S(Z)$, is therefore

$S(Z)=\frac{M_{\rm s}}{M_{\rm s1}}=\frac{1-\exp(-Z/y)}{1-\mu_{1}}.$ (4.2)

It is useful to eliminate the yield using the relation $y=Z_{1}/(\ln\
\mu_{1}^{-1})$ (again Equation 3.23), so Equation (4.2) can finally be written
in terms of the metallicity ratio $Z/Z_{1}$ and the present gas fraction
$\mu_{1}$:

$S(Z)=\frac{1-\mu_{1}^{Z/Z_{1}}}{1-\mu_{1}}.$ (4.3)

This distribution is compared in Figure 8 with a log-normal approximation to
the data. (The model illustrated has $\mu_{1}=0.05$, and the shape of the
curve is insensitive to $\mu_{1}$ within the range of likely values).
Obviously, there are far fewer metal-poor stars in the Solar neighborhood than
predicted by this formula. The assumptions defining the simple model allow
only two significant quantities to define a specific model: the IMF and the
SFR. However, the IMF affects only the yield, which does not appear in
Equation (4.3) because metallicities have been scaled to the present value;
the SFR was eliminated in the derivation of Equation (3.23), and numerical
models show that very nearly the same result is obtained if the approximation
of instantaneous recycling is dropped. Thus the failure of the “simple” class
of models is independent of the particular IMF or SFR used. It may seem at
first sight that one could eliminate the problem by making the metals rapidly
using a large SFR at early times; however, since there is initially unenriched
gas and a constant IMF, a fixed quota of metal-poor G dwarfs must form along
with the massive stars that produce metals – no matter how rapidly it all
happens.

Figure 8: Cumulative stellar metallicity distribution functions. $S(Z)$ is the
fraction of stars that have metallicities $\leq Z$, with a maximum value
$Z_{1}$. _Solid line_ : log-normal representation of the data for stars in the
Solar neighborhood; $Z_{1}$ is taken as $2\>\rm Z_{\odot}$ since more metal-
rich stars are very rare; $\langle\log\ Z/\rm Z_{\odot}\rangle=-0.1$, and
$\sigma(\log\ Z/\rm Z_{\odot})=0.15$, allowing for observational errors (e.g.
Pagel, 1978a). _Long dashes_ : the “simple” model for chemical evolution
(Section 4.2). _Short dashes_ : an extreme infall model (Section 4.2.1).
_Dots_ : a model with a finite initial metallicity (Section 4.2.2).

Attempts to solve the G-dwarf problem have motivated numerous alternative
models for the Solar neighborhood, in which one or more of the four
straightforward assumptions are dropped. The following outline considers each
assumption in turn, and some types of model will be developed from a different
viewpoint in Section 4.3. Further details and references, on these and other
models, can be found in reviews by Audouze & Tinsley (1976), Pagel & Patchett
(1975), Tinsley (1974) and Trimble (1975).

####  4.2.1 Infall

It is unlikely that the region called the “Solar neighborhood” has evolved as
a closed system. In particular, the disk could have grown gradually by
accretion, on a timescale similar to that for star formation, as proposed by
Larson (1972b) and Sciama (1972). The effect on the stellar metallicity
distribution is then as follows (Larson, 1972a). Suppose the disk consists
initially of unenriched gas, as in the simple case, but with a very small mass
(surface density). The first stars formed are metal-poor, but their numbers
are proportional only to the initial disk mass; star formation substantially
enriches the gas while the disk is still far below its final mass. Then
further star formation continually enriches the new gas that is accreted, so
ultimately the only very metal-poor stars are the few initial ones.

The model in Section 3.2.2 illustrates this behavior analytically. According
to Equation (3.28), $Z$ remains essentially constant at the yield value after
the disk mass has increased to several times its initial value. The stellar
metallicity distribution, $S(Z)$, can be derived analytically for this model
(Tinsley, 1975b), and the result is a function strongly peaked at $Z=y$, as
shown in Figure 8. Here we have a problem opposite to that of the simple
model: there are too few stars in the model with $Z<Z_{1}$! The problem is
alleviated if the width of the observed distribution is due more to errors
than has been thought, but in any case this particular model is very
schematic, showing effects of infall in a most extreme way, and more realistic
models with infall (Section 4.3) predict metallicity distributions
intermediate between the extreme and the simple cases.

####  4.2.2 Pre-enrichment of the disk gas

If the gas that formed the disk had a significant initial metal abundance, the
number of more metal-poor disk stars would obviously be zero. This idea was
introduced by Truran & Cameron (1971), who postulated a pregalactic burst of
massive stars. Ostriker & Thuan (1975) considered the effects of massive stars
in the young halo, whose ejecta enriched the gas that was condensing into a
disk; the likely importance of this effect will be reviewed below (Section
4.3.1).

The effects of pre-enrichment can be seen analytically by simply adding a
finite initial metal abundance $Z_{0}$ to the solution of Equation (3.22).
Then (3.23) becomes $Z=Z_{0}+y\ln\ \mu^{-1}$, and the exponent $Z/Z_{1}$ in
(4.3) is replaced by $(Z-Z_{0})/(Z_{1}-Z_{0})$. Figure 8 shows the resulting
stellar metallicity distribution in the case $Z_{0}=0.2Z_{1}$, and there is
clearly no excess of metal-poor stars. A more realistic picture will be
discussed in Section 4.3.

####  4.2.3 Variable IMF

The original solution to the G-dwarf problem, proposed by Schmidt (1963), was
to postulate an IMF richer in massive stars at early times; the number of
long-lived stars formed before the gas reaches a given metallicity is
therefore reduced. (An analytical solution illustrating this effect is given
by Pagel & Patchett, 1975). Many alternative models with variable IMFs have
been considered subsequently, but they all suffer from being ad hoc: although
arguments can be offered for massive stars forming more easily at low
metallicity, they are not compelling, and there is no independent way of
verifying suggested changes in the IMF in the distant past.

####  4.2.4 Metal-enhanced star formation

Talbot & Arnett (1973) suggested that there are always large chemical
inhomogeneities in the ISM, and that stars form preferentially in regions with
above-average metallicity. Talbot (1974) has developed this idea into a
physical model, in which the enhanced cooling by heavy elements favors star
formation in metal-rich regions. For this process alone to solve the G-dwarf
problem, it would require both very large chemical inhomogeneities and very
inefficient star formation in metal-poor regions; more plausibly, metal-
enhanced star formation leads only to some reduction in the proportion of
metal-poor stars.

In summary, none of the four assumptions underlying the simple model is likely
to be strictly true, and departures from any or all of them could alter the
predicted metallicity distribution in the required sense of including fewer
metal-poor stars.

###  4.3 Effects of Galaxy Formation

In retrospect, the failure of the simple model is not surprising, because the
Solar neighborhood was treated as a closed box, whereas gas flows during the
formation of the Galaxy would affect chemical evolution strongly in two ways:

  1. 1.

some pre-enrichment of the initial disk gas by massive stars in the young
halo, and

  2. 2.

later accretion of metal-poor gas.

The “formation” of the Galaxy could in the latter sense be continuing today.

####  4.3.1 Metals from the young halo

The amount of initial enrichment by massive halo stars can be estimated from
the present mass of long-lived stars in the halo. Let $y_{\rm h}$ be the yield
during the time of halo formation. Then, by definition of the yield, a mass
$M_{\rm h}$ of surviving halo stars implies that a mass ${\sim}y_{\rm h}M_{\rm
h}$ of metals was made and ejected at an early time; of these metals, a mass
$Z_{\rm h}M_{\rm h}$ remains locked in halo stars (where $Z_{\rm h}$ is their
mean metallicity), and the remaining $(y_{\rm h}-Z_{\rm h})M_{\rm h}$ either
fell to the disk or was lost from the Galaxy. Ostriker & Thuan (1975) suggest
on geometrical grounds that about half of those metals were retained by the
disk. Now the halo yield, $y_{\rm h}$, may differ from the disk yield because
the IMF for low-mass stars appears to differ between the two regions (Section
2.2.2), but for a very rough estimate we can consider $y_{\rm h}\sim{\rm
Z_{\odot}}\gg Z_{\rm h}$. The mass of metals falling on the disk is therefore
${\sim}0.5\ {\rm Z_{\odot}}M_{\rm h}$, and the initial metallicity of the disk
gas is this quantity divided by the initial disk mass. The appropriate value
of $M_{\rm h}$ is not the mass of halo stars in a local cylinder, but the mass
(mostly further out) from which ejected gas would fall to this part of the
disk; since the surface density of halo stars is locally only a few percent of
the disk value, and it decreases with increasing radius (Schmidt, 1975),
$M_{\rm h}$ is probably only a small fraction of the present mass of the disk
in the local cylinder. However, the initial disk mass may also have been
small. Altogether, it is possible that metals captured from the young halo
could have been enough to have enriched the initial disk gas significantly. A
value of $Z_{0}/\rm Z_{\odot}\sim 0.2$ would completely solve the G-dwarf
problem, as illustrated above.

Early enrichment by halo stars appears as an important effect in the dynamical
models for the formation of disk galaxies of Larson (1976c), whose chemical
evolution has been studied further by Tinsley & Larson (1978). At model radii
corresponding to the Sun’s position in the Galaxy, $Z$ reaches about a third
of its final value before stars with disk kinematics begin to form, and the
initial disk mass in these models is about $10\%$ of the final value. As a
result, the fraction of local stars with $Z<\rm Z_{\odot}/3$ is no greater
than observed.

It may be concluded that the disk in the Solar neighborhood plausibly started
with a small fraction of its present mass, in the form of gas that had been
significantly enriched by massive stars in the young halo.

####  4.3.2 Later metal-poor infall

In the dynamical models just mentioned, the outer disk grows on a timescale of
billions of years by accretion of metal-poor gas; star formation occurs on a
similar timescale, so the metallicity tends to reach a roughly constant value
near the yield, as in the schematic infall model of Section 3.2.2.

A significant additional factor, first suggested by Lynden-Bell (1975), is
that the ratio of SFR to infall rate ($\psi/f$) is unlikely to remain strictly
constant in a realistic picture; its value is more likely to increase as the
surrounding gas is used up, as indeed occurs in the dynamical models. An
asymptotic relation between $\psi/f$ and $Z$ can be derived using the
Equations of Section 3.2 with the instantaneous recycling approximation. Let

$(1-R)\psi=kf,$ (4.4)

where $k$ is a constant, so Equation (3.19) becomes

$M_{\rm g}=\frac{dZ}{dt}=\left(ky+Z_{\rm f}-Z\right)f.$ (4.5)

According to this Equation, there is an asymptotic value of $Z$,

$Z\rightarrow ky+Z_{\rm f},$ (4.6)

so the metal abundance of the gas tends toward a value $ky$ if $Z_{\rm f}=0$.
Thus if $\psi/f$ increases, so does $Z$, and there is steady enrichment rather
than the constant $Z$ obtained in Section 3.2.2. Examples of this behavior are
seen in the numerical models of Tinsley & Larson (1978), in particular in the
regions between ${\sim}7$ and $15\>\rm kpc$ which have general properties
resembling the Solar neighborhood. The slow increase of $Z$ leads to a less
peaked stellar metallicity distribution than that in the “extreme” infall
model shown in Figure 8.

####  4.3.3 Timescales for chemical evolution

The original evidence for infall of extragalactic gas was the explanation of
the high-velocity clouds of neutral hydrogen (Oort, 1970), which required an
infall rate comparable to the SFR. Further evidence is given by the timescale
for star formation to consume the ISM in the Solar neighborhood, as pointed
out by Larson (1972b): the local surface density of ISM is ${\sim}8\>\rm
M_{\odot}\ pc^{-2}$ while the SFR is ${\sim}10\>\rm M_{\odot}\ pc^{-2}\
Gyr^{-1}$ (Section 2.3.1), so the ISM is being used up on a timescale

$\tau_{*}\equiv\frac{M_{\rm g}}{(1-R)\psi}\sim 1\>\rm Gyr.$ (4.7)

If there is not a source of new gas, flowing in at about the SFR, we must be
living very near the end of active star formation in this part of the disk.
Various constraints related to interstellar pressure balance and Galactic
X-ray emission allow an infall rate up to ${\sim}1\>\rm M_{\odot}\ pc^{-2}\
Gyr^{-1}$ in the Solar neighborhood (Cox & Smith, 1976). This is enough to
extend the lifetime of the gas against total consumption by only ${\sim}10\%$,
but all of the above estimates are rather uncertain, and it is possible that
$f\sim\psi$ in the Solar neighborhood, so the gas is continually replenished.

A related timescale is that for chemical enrichment, which can be written

$\tau_{\rm Z}\equiv\frac{Z}{\left|dZ/dt\right|}=\frac{ZM_{\rm
g}}{\left|y(1-R)\psi+\left(Z_{\rm f}-Z\right)f\right|},$ (4.8)

using Equation (3.19). If infall were negligible, this Equation would give
$\tau_{\rm Z}=\tau_{*}(Z/y)\simeq\tau_{*}\sim 1\>\rm Gyr$; we would then
expect to see $Z$ increasing on a timescale of $1\>\rm Gyr$ now, which is not
consistent with the data in Figure 7. However, if gaseous inflow is occurring
at a rate comparable to the SFR, $\tau_{\rm Z}$ could be much longer.

Other processes than infall have been suggested to account for the slow
enrichment rate in the Solar neighborhood. For example, various authors
beginning with Truran & Cameron (1971) have suggested a time-dependent IMF
with a greater yield in the past; thus at present $Z\gg y$, so $\tau_{\rm
Z}\gg\tau_{*}$.

Whatever such additional processes may affect the evolution and distribution
of metal abundance, it seems almost inevitable that gas flows related to
galaxy formation have a strong influence on chemical evolution. The dynamical
models discussed above were designed to reproduce the structural properties of
galaxies (Larson, 1976c), so it is significant that they predict both pre-
enrichment of the disk gas and later metal-poor infall, which thus appear to
be the most natural solutions to the G-dwarf and timescale problems of the
Solar neighborhood. From these results, it is to be expected that the chemical
evolution of other regions of galaxies is affected by gas flows; examples will
be reviewed in Section 5.

###  4.4 Relative Abundances of the Elements

Relative abundances of elements heavier than helium provide information on
both nucleosynthesis and galactic evolution, in a variety of ways that are
discussed extensively in the literature (see reviews by Audouze & Tinsley,
1976, Pagel, 1978a, and Trimble, 1975). Here the emphasis will be on points
relevant to galactic evolution.

####  4.4.1 Primary elements from different stars

Stars of different masses eject new primary elements in different proportions,
so if the ISM is not well mixed we expect to see some variations in relative
abundances on the surfaces of unevolved stars. (Variations would arise
similarly from any dependence of primary nucleosynthesis on the initial
compositions of stars, Section 2.4.5). Until recently, most reported
variations were within the uncertainties of measurement and analysis, but some
trends are now emerging.

The relative abundances of carbon, oxygen, and iron (for which the data are
reviewed by Pagel, 1978a) are an interesting example. All unevolved stars with
reliable carbon abundances have $\rm[C/Fe]=0$ within the errors, i.e. a Solar
abundance ratio999This statement is tentative, since very few metal-poor stars
have quantitative carbon abundances and some globular clusters show
preliminary evidence for carbon excesses and/or deficiencies (McClure, 1979).,
but oxygen is systematically overabundant in metal-poor stars, a typical value
being $\rm[O/Fe]=0.5$ in stars with $\rm[Fe/H]<-1$. Since these stars have
halo kinematics, it appears that the yield of oxygen was relatively high
during the early life of the Galaxy. Oxygen is the most abundant element after
H and He, so it would be more appropriate to say that the yield of iron (and
perhaps carbon) was relatively low.

One possible explanation is that the IMF was different, favoring massive
stars, which have a higher oxygen/carbon ratio in their ejecta according to
current stellar models (e.g. Arnett, 1978). Another possibility is that
significant fractions of iron and carbon are ejected by stars with such long
lives that they die too late to enrich the halo stars. This idea agrees
qualitatively with the mass dependence of stellar nucleosynthesis outlined in
Section 2.4: carbon stars (and planetary nebulae) with initial masses from
${\sim}1-8\>\rm M_{\odot}$ indicate that carbon production takes place partly
in very long-lived stars; Type I supernovae, with iron-enriched envelopes,
show that iron is ejected at least partly by stars of $\lesssim 6\>\rm
M_{\odot}$; but there is no evidence for oxygen production outside the massive
stars ($\gtrsim 10\>\rm M_{\odot}$) that are theoretically predicted to be its
main source. Models showing the effects of nucleosynthesis in stars with such
a wide range of lifetimes have been studied by Tinsley (1979). It is found
that systematic variations in relative abundances arise if star formation
occurs on timescales less than the lifetime of the least massive star
producing significant amounts of an element of interest. For example, iron
produced by stars of intermediate mass (${\sim}4-6\>\rm M_{\odot}$) is
underabundant relative to oxygen if stars form on a timescale
${\sim}10^{8}\>\rm yr$. Even if the whole time for halo formation is several
billion years, as indicated by the age spread of globular clusters (Section
1.1), such variations will appear in halo stars if they were formed in a
series of short bursts. A very inhomogeneous protogalaxy, with stars forming
in dense lumps, has been suggested on dynamical grounds (Section 1.2), so it
is interesting to note how chemical abundances could be affected.

####  4.4.2 Secondary elements

Secondary elements are those synthesized from “seed” elements heavier than
helium that were already present in the star at birth; some examples have been
mentioned in Section 2.4.5. The stellar yields of these elements increase with
metallicity, so their abundances are expected to increase more rapidly (and to
be relatively lower in metal-poor stars) than those of primary elements. As
reviewed by Pagel (1978a), this prediction is borne out qualitatively by
relative underabundances of nitrogen and barium in metal-poor stars, but the
data are in quantitative disagreement with theoretical models.

Figure 9: Ratios of primary to secondary abundances versus metallicity. _Data
points_ (and shaded area with many points) are from sources cited by Tinsley
(1979), and _dashed lines_ correspond to a simple scenario for secondary
nucleosynthesis (Section 4.4.2). The nitrogen data are all for dwarfs; barium
data include giants, but these do not give a different picture from the dwarfs
alone, so there is probably no significant contamination by the stars’ own
nucleosynthesis.

The canonical prediction for secondary abundances is that they increase as the
square of primary abundances, e.g. $\rm Ba/H\propto(Fe/H)^{2}$ or $\rm
Ba/Fe\propto Fe/H$. This prediction arises, as follows, from the conventional
assumption that a star of given mass produces and ejects an amount of
secondary element in direct proportion to the star’s initial primary seed
abundance. Let us consider a secondary element “$s$”, with primary seed $Z$,
and let $p_{\rm sm}$ be the mass of new $s$ ejected by a star of mass $m$ with
initial metallicity $\rm Z_{\odot}$; with another initial metallicity $Z$, the
amount of new $s$ ejected is $p_{\rm sm}(Z/\rm Z_{\odot})$. Thus the
expression like (3.8) for the rate of ejection of new $s$ is

$\int_{m_{\rm t}}^{\infty}mp_{\rm sm}\frac{Z\left(t-\tau_{\rm m}\right)}{\rm
Z_{\odot}}\psi\left(t-\tau_{\rm m}\right)\phi(m)\ dm.$ (4.9)

The instantaneous recycling approximation to this expression can be written
$y_{\rm s}(Z/{\rm Z_{\odot}})(1-R)\psi$, where $Z$ and $\psi$ are current
values. The equation analogous to (3.19) for the abundance $X_{\rm s}$ of $s$
is therefore

$M_{\rm g}\frac{dX_{\rm s}}{dt}=y_{\rm s}\left(\frac{Z}{\rm
Z_{\odot}}\right)(1-R)\psi+\left(X_{\rm sf}-X_{\rm s}\right)f.$ (4.10)

In the simple model of Section 3.2.1, the equation analogous to (3.22) is

$M_{\rm g}\frac{dX_{\rm s}}{dM_{\rm g}}=-y_{\rm s}\left(\frac{Z}{\rm
Z_{\odot}}\right).$ (4.11)

Now $Z$ is given by Equation (3.23), so (4.11) can be solved for $X_{\rm s}$,
with the result

$X_{\rm s}=\frac{1}{2}\left(\frac{y_{\rm s}}{y\rm Z_{\odot}}\right)Z^{2},$
(4.12)

which is the canonical prediction $X_{\rm s}\propto Z^{2}$ mentioned above.
Since Solar abundances are near the mean disk values, they must also be
approximately the yield values, so the prediction is equivalent to $X_{\rm
s}/Z\propto Z$ with a zero-point near the Solar values.

Stellar nitrogen and barium abundances are plotted in Figure 9, where the
canonical relation would correspond in each case to the straight line at
$45^{\circ}$ through about the Solar values (zeros in the logarithmic
notation). It can be seen that the nitrogen abundances scatter greatly about
the predicted relation, mainly exceeding the predicted value in metal-poor
stars; and barium abundances are almost all greater than predicted, no matter
where one places this zero-point among the data for stars near Solar
composition.

What is wrong? There is a logical flaw in making predictions with the simple
model because it is known to be untenable on account of the stellar
metallicity distributions (Section 4.2); furthermore, instantaneous recycling
may be a bad approximation for secondary elements that are made in red giants
down to initial masses ${\sim}1\>\rm M_{\odot}$. Plausible modifications to
the above theory lead to no better agreement, however, as the following
examples show (Tinsley, 1979).

  1. 1.

Numerical calculations allowing for delayed ejection by long-lived stars make
the situation worse, because they predict even smaller secondary abundances in
the most metal-poor stars.

  2. 2.

Infall makes the situation even worse, because it predicts a steeper rise than
$X_{\rm s}/Z\propto Z^{2}$ at the metal-rich end.

The latter result can be derived simply in the extreme infall model of Section
3.2.2. Use of Equation (4.10), with $X_{\rm sf}=0$ and (3.28) for $Z$, leads
to a result analogous to (3.28),

$X_{\rm s}=\frac{y_{\rm s}y}{\rm Z_{\odot}}\left(1-e^{-\nu}-\nu
e^{-\nu}\right).$ (4.13)

This formula reduces to Equation (4.12) in the limit of low abundances, but
its asymptotic limit, $X_{\rm s}\rightarrow y_{\rm s}y/\rm Z_{\odot}$, is
twice the value obtained by extrapolating (4.12) to the asymptotic value
$Z\rightarrow y$. Thus a plot of $X_{\rm s}/Z$ versus $Z$ for this model would
have a slope of $45^{\circ}$ at first, but would become steeper at the top,
obviously in disagreement with the data in Figure 9.

From these results, it appears that the problem in Figure 9 is not due to a
simplistic choice of model for chemical evolution, but to wrong assumptions
about nucleosynthesis. For nitrogen, it has often been suggested that a
primary source of nucleosynthesis could contribute significantly (e.g. Truran
& Cameron, 1971), and this component of the nitrogen abundance would make the
ratio $\rm N/Fe$ more nearly constant. A possible scenario for primary
nitrogen production is that some carbon produced by helium burning in stars of
intermediate mass gets mixed into the hydrogen-burning shell region and there
converted to nitrogen; the models in Iben (1976) illustrate this possibility.
Nitrogen abundances in H ii regions, in our own and other galaxies, show a
scatter similar to the stellar data, and primary production has again been
proposed as an explanation (e.g. Smith, 1975; Edmunds & Pagel, 1978).

For barium there is no obvious solution to the discrepancy in Figure 9, since
a primary source is unlikely in terms of nuclear physics. The calculations of
stellar structure and nucleosynthesis used to predict barium production are so
complicated that results for Solar initial metallicities cannot be
extrapolated to others (Iben & Truran, 1978); it is therefore likely that the
conventional assumption, that production scales directly with the initial $\rm
Fe/H$ of a star of given mass, is wrong.

Other aspects of secondary nucleosynthesis relevant to galactic evolution are
discussed in the references cited at the beginning of Section 4.4 and in the
proceedings of a meeting on CNO isotopes in astrophysics (Audouze, 1977).

####  4.4.3 Radioactive elements

The field of nucleochronology (or cosmochronology or nucleo-cosmochronology)
uses meteoritic abundances of the decay products of certain radioactive
elements to derive information on the age distribution of elements in the
early Solar System. This information is relevant to the rates of star
formation and nucleosynthesis before the Sun formed, and to the age of the
Galaxy. Nucleochronology has been reviewed recently by Fowler (1979), Schramm
(1974), and Trimble (1975), and its relevance to galactic evolution is
discussed further by Reeves (1972), Reeves & Johns (1976), and in other papers
referenced below. Some general principles will be outlined here.

Let us consider the abundance $X_{\rm i}$ of a radioactive nuclide with decay
constant $\lambda_{\rm i}$ and net yield $y_{\rm i}$. The equations of
chemical evolution must be modified to allow for decay in the ISM and in
stellar envelopes; the equation analogous to (3.7) is

$\frac{dX_{\rm i}M_{\rm g}}{dt}=-X_{\rm i}\psi+E_{\rm i}+X_{\rm
if}f-\lambda_{\rm i}X_{\rm i}M_{\rm g},$ (4.14)

where $E_{\rm i}$ is given, analogously to Equation (3.9), by

$\medsize E_{\rm i}(t)=\int\limits_{m_{\rm t}}^{\infty}\left[\left(m-w_{\rm
m}-mp_{\rm im}\right)X_{\rm i}\left(t-\tau_{\rm m}\right)e^{-\lambda_{\rm
i}\tau_{\rm m}}+mp_{\rm im}\right]\psi\left(t-\tau_{\rm m}\right)\phi(m)\ dm.$
(4.15)

The decay timescale, $\lambda_{\rm i}^{-1}$, is of course independent of the
system’s timescales for star formation and infall, so straightforward
analytical solutions for $X_{\rm i}$ are possible only in special cases, even
if instantaneous recycling is used. In fact, the long-lived radioactive
elements are useful just _because_ their abundances are sensitive to
evolutionary details (such as the past SFR and the age) that cancel in the
instantaneous recycling approximation and hardly affect other abundance
parameters in detailed models.

The simplest situation would be if all the elements formed in an initial
burst. Then their decay in the ISM and stellar envelopes would lead to
abundances at time $t$ given by $X_{\rm i}(t)=X_{\rm i0}\exp(-\lambda_{\rm
i}t)$, where $X_{\rm i0}$ is the initial abundance and is proportional to the
amount of element $i$ made, i.e. to its yield $y_{\rm i}$. In particular, at
the time $T$ when the Solar System formed, two such elements would have an
abundance ratio

$\frac{X_{\rm i}(T)}{X_{\rm j}(T)}=\frac{X_{\rm i0}}{X_{\rm
j0}}\exp\left[-\left(\lambda_{\rm i}-\lambda_{\rm j}\right)T\right],$ (4.16)