where $X_{\rm i0}/X_{\rm j0}=y_{\rm i}/y_{\rm j}$. The value of $T$ could in
this case be found by substituting into Equation (4.16) the yield ratio (given
by nucleosynthesis theory) and the abundance ratio in the early Solar System
(obtained from the abundances of decay products in meteorites). The age of the
Galaxy, which is simply $T$ plus the age of the Sun, could then be determined.

Of course, the elements were probably synthesized continually during the
period $T$ from first star formation in the Galaxy until the Solar System
condensed, so Equation (4.16) is unrealistic. As first shown by Schramm &
Wasserburg (1970), one can define a useful age parameter, here denoted $T_{\rm
ij}$, analogously to the solution of (4.16) for $T$:

$T_{\rm ij}\equiv\frac{1}{\lambda_{\rm i}-\lambda_{\rm
j}}\ln\left[\frac{y_{\rm i}/y_{\rm j}}{X_{\rm i}(T)/X_{\rm j}(T)}\right],$
(4.17)

which can, in principle, be evaluated from meteoritic and nuclear data
independently of Galactic evolution. In the limit of long-lived elements,
($\lambda_{\rm i}T\ll 1,\>\lambda_{\rm j}T\ll 1$), $T_{\rm ij}$ is just the
mean age of elements in the gas, $T_{Z}$, at the time when the Solar System
formed (Tinsley, 1975c).

The relation between the mean age ($T_{Z}$) and the elapsed time ($T$) is
model-dependent, so estimates of $T_{\rm ij}$ for long-lived pairs of elements
do not give $T$ directly. Different possibilities include the following.

  1. 1.

If essentially all nucleosynthesis of the relevant elements took place in an
initial burst, then $T=T_{Z}$.

  2. 2.

The simple model for chemical evolution gives the intuitive result that the
mean age is half the elapsed time, $T_{Z}=T/2$; however, because this model is
discrepant with stellar metallicities, one cannot assume that $T$ is given
simply by $2T_{Z}$ in reality.

  3. 3.

In extreme infall models, the ISM and heavy elements in it have a mean age
${\sim}M_{\rm g}/f$ at all times greater than $M_{\rm g}/f$; so if chemical
evolution in the disk was strongly affected by infall before the Solar System
formed, the value of $T_{Z}$ obtained from meteoritic abundances may reflect
only the timescale for infall, independently of the age $T$.

  4. 4.

There are consistent models for the Solar neighborhood, with some infall
and/or some early enrichment, that have values of $T_{Z}\simeq T/2$ (as
emphasized by Hainebach & Schramm, 1977), but since not all plausible models
have this relation it cannot be used confidently (as emphasized by Tinsley,
1977c).

In summary, there is a large uncertainty in any age estimate of the Galaxy
derived from nucleochronology, except of course that the age of the Solar
System is a reliable lower limit!

The initial Solar System abundances of short-lived radioactive elements are
sensitive to rates of nucleosynthesis at times immediately preceding the
solidification of the meteorites. Their abundances suggest that the
nucleosynthesis of most elements ceased ${\sim}10^{8}\>\rm yr$ before the
solidification time, but some material was produced only ${\sim}10^{6}\>\rm
yr$ earlier. Interpretations of these timescales include passage of the pre-
Solar material through spiral arms at $10^{8}$-yr intervals; enrichment by
fresh supernova ejecta each $10^{6}\>\rm yr$, which could result from the
average supernova rate in the Solar neighborhood; a last-minute supernova that
triggered the formation of the Solar System; and locking of radioactive
elements, with their decay products, into grains long before the Solar System
formed. These possibilities are reviewed briefly by Podosek (1978), and
discussed in detail by several authors in a conference proceedings edited by
Gehrels (1978). As yet there is no consensus on the interpretation of short-
lived radioactivities in the early Solar System, but ultimately they should
provide valuable information on star formation and interstellar chemistry.

##  5 Chemical Evolution of Galaxies

For other galaxies, and for regions of our own outside the Solar neighborhood,
there is much less detailed information on abundance distributions, but there
are some striking trends that call for explanations involving the formation
and later dynamical evolution of galaxies. A few relevant properties have been
mentioned in Section 1.1, and now further details will be described and some
of the theoretical models reviewed. Other general reviews of this subject
include those by Audouze & Tinsley (1976), Pagel (1978b), and Trimble (1975).

###  5.1 Outline of Relevant Data

Abundances are very often found to decrease outward in galaxies: gradients
have been observed in the H ii regions of disks, in disk stars, and in the
stars of spheroidal systems including elliptical galaxies and the bulge-halo
components of spirals.

In the Galactic disk, H ii regions within a few kpc of the Sun have an average
gradient $d{\rm[O/H]}/dr\simeq-0.1\>\rm kpc^{-1}$ (where $r$ is the
galactocentric distance), while stars of intermediate age have a gradient
$d{\rm[Fe/H]}/dr\simeq-0.05\>\rm kpc^{-1}$; an open cluster about $10\>\rm
kpc$ from the Sun in the anticenter direction, of age only $\lesssim
10^{8}\>\rm yr$, is apparently as metal-poor as $\rm[Fe/H]<-1$ (Christian &
Janes, 1979). (Oxygen and iron abundances are quoted for the ISM and stars,
respectively, because these are the best observed elements). The uncertainties
are such that the apparent age dependence of the gradient may or may not be
real. These data are reviewed by Janes (1977), Mayor (1977), and Peimbert
(1977). H ii regions in external galaxies generally show gradients of a
similar magnitude (e.g. Smith, 1975; Shields & Searle, 1978; and references
therein). However, only a marginal gradient appears in the Large Magellanic
Cloud and none in the Small Cloud (Pagel et al., 1978).

The most obvious explanation for these gradients would be that the outer
regions of disks are less chemically evolved than the inner regions, in the
sense of having converted a smaller fraction of their gas to stars. The simple
model, for example, would predict a $Z$ gradient given by $Z=y\ln\ \mu^{-1}$
arising from a gradient in the gas fraction $\mu$ (Section 3.2.1). However,
the best studied galaxies (the Milky Way and M101) probably do not have a
sufficient gradient in $\mu$ for this explanation to suffice. Gordon & Burton
(1976) found that the combined surface densities of atomic and molecular
hydrogen lead to a nearly constant gas fraction ($\mu\sim 0.05$) at $R>4\>\rm
kpc$ in the Galaxy, and Shields & Searle (1978) noted a similar problem in
M101. The amount of ISM interior to the Sun could be overestimated, since the
$\rm H_{2}$ density is derived from observations of CO molecules on the
assumption of a constant abundance ratio $\rm CO/H_{2}$; if in fact $\rm
CO/H_{2}$ increases inward because of a $\rm C/H$ abundance gradient, then
there is less $\rm H_{2}$ than had been thought at small radii (Peimbert,
1977). With this correction, the Galaxy probably has some gradient in $\mu$,
which could account in part for the interstellar composition gradient. Of
course, the simple model is known to be invalid in the Solar neighborhood, so
we do not expect the formula $Z=y\ln\ \mu^{-1}$ to explain gradients in
detail. Other ways of generating gradients will be mentioned in Section 5.5.

The Galactic halo stars also have a metallicity gradient. Studies of
individual stars in globular clusters show that a spread of metallicities of
$\rm[Fe/H]\sim-2$ to $0$ occurs in the innermost $5\>\rm kpc$ (measured from
the Galactic center, at any angle to the disk), while the upper limit drops to
${\sim}{-1}$ at greater radii (Cowley et al., 1978; Searle & Zinn, 1978). It
is not clear whether a systematic decline in iron and/or CNO abundances
persists further out (McClure, 1979; Kraft, 1978).

Many elliptical and S0 galaxies have gradients of integrated colors and line
strengths in their spectra that are best interpreted as composition gradients.
The same quantities also vary systematically with the absolute magnitudes of E
and S0 galaxies, as illustrated in Figure 10, indicating that the brighter
galaxies are more metal-rich. A thorough review of this subject is given by
Faber (1977). The analysis and calibration of abundance effects in the
integrated light of a galaxy are much more complicated than for individual
stars, because line strengths are strongly affected by the mixture of stellar
temperatures and luminosities, and because the whole population in the HR
diagram is shifted by effects of metallicity on the interiors and atmospheres
of stars (Section 2.4.5). Until recently, it was not clear whether all
elements heavier than helium enter the composition trends in E and S0
galaxies, or whether a few with strong spectral features (N, Mg, Na), are
mainly responsible. Cohen (1978) has now made a detailed observational study
of some lines of Ca, Na, Mg, and Fe, together with approximate theoretical
predictions of how their strengths should vary with the composition of a
population of stars; she finds no evidence against the abundances of all of
these elements varying in the same proportions.

Figure 10: A color–magnitude diagram for elliptical and S0 galaxies in several
clusters of galaxies (Visvanathan & Sandage, 1977). The color $(u-V)_{\rm c}$
is corrected for reddening, redshift, and aperture effects; magnitudes are
adjusted to the distance of the Virgo cluster using redshift ratios.
(_Crosses_ denote Virgo galaxies not used by Visvanathan & Sandage, 1977 in
their linear fit to the data). The _straight lines_ are a linear fit, with
$\pm 2\sigma$ boundaries. Despite the excellent linear fit over a large range
of magnitudes, the brightest few points for the Virgo cluster alone (_filled
circles_) and for other clusters (_open circles_) show a tendency to level off
in color.

The color–magnitude relation for elliptical galaxies is linear over a wide
range of metallicities (Figure 10), which suggests a power-law relation
between metallicity and luminosity. A tentative calibration, subject to
revision when both the data and the theoretical color–metallicity relation are
more secure, is

$Z_{\rm s}\propto L_{\rm B}^{0.3}$ (5.1)

(Tinsley, 1978b), where $Z_{\rm s}$ is the average metallicity of stars in an
elliptical galaxy of blue luminosity $L_{\rm B}$. This relation was derived by
from population models differing only in metallicity, so that the stars in all
cases had the same IMF and age (as outlined in Section 6.1.2). Such models
also predict that the mass-to-luminosity ratio should depend on metallicity,
yielding a relation $M_{\rm s}/L_{\rm B}\propto L_{\rm B}^{0.13}$. A relation
very similar to this has been obtained observationally by Schechter & Gunn
(1979) for the cores of elliptical galaxies. Equation (5.1) therefore
corresponds to a tentative metallicity–mass relation,

$Z_{\rm s}\propto M_{\rm s}^{0.25},$ (5.2)

where $M_{\rm s}$ is the mass of stars (not any extended hidden halo
material), and the main uncertainties in the exponent are due to the
color–magnitude data and to the color–metallicity calibration. There is some
evidence that the color–magnitude relation levels off at the magnitudes of the
brightest cluster galaxies, as can be seen, for example, from the brightest
few points in Figure 10.

Figure 11: (a) Star formation rates and (b) metallicities of newly formed
stars (i.e., $Z$ of the gas), at several radii in a collapse model for the
formation of a spherical galaxy (Larson, 1974a). The radius in pc is marked on
each curve, and the three ticks indicate the times at which star formation is
$10\%$, $50\%$, and $90\%$ complete (relative to the final mass of stars) at
that radius.

Heavy elements have been detected in intergalactic gas, and since these
elements almost certainly come from galactic stars they are relevant to the
chemical evolution of galaxies. A feature due to iron has been observed in the
diffuse X-ray emission spectra of several rich clusters of galaxies; the
interpretation is that these clusters contain hot gas (${\sim}10^{8}\>\rm K$),
emitting by thermal Bremsstrahlung, with approximately Solar abundances of
iron. The mass of intergalactic gas inferred from the data is model-dependent
(e.g. Bahcall & Sarazin, 1977; Perrenod, 1978), and is roughly between 1 and
30 times the total luminosity of cluster galaxies, in solar units. Now the
mass of stars in galaxies is ${\sim}10$ times their total luminosity, in solar
units101010This mass is not to be confused with the virial masses of clusters,
which are ${\sim}100$ times the total luminosity and which provide the main
evidence for hidden non-stellar matter in association with galaxies., and
their average metallicity is approximately Solar, so the intergalactic medium
(IGM) in the rich clusters apparently contains about the same total mass of
iron as do the stars themselves. These observations suggest that galaxies
sometimes lose enough metal-rich gas to affect their own chemical evolution
substantially.

Another striking observation of metal-rich IGM is absorption due to Ca and Mg
in the spectrum of a quasar (3C 232) lying $1.9^{\prime}$ on the sky from a
foreground spiral galaxy (NGC 3067); the absorption redshift matches that of
the galaxy, and neutral hydrogen absorption has been detected at the same
redshift in the radio spectrum of the quasar (Haschick & Burke, 1975). The
line strengths and positions of the objects imply that there is gas with
roughly Solar abundances at least $17\>\rm kpc$ from the center of the galaxy
(Boksenberg & Sargent, 1978).

###  5.2 Abundance Gradients in Spheroidal Systems

Most stars in elliptical galaxies and in the bulge-halo components of spirals
were probably formed within a few times $10^{9}\>\rm yr$ at a time
${\sim}10^{10}\>\rm yr$ ago. The abundance gradients in these systems
therefore reflect processes occurring during the time of star formation in a
protogalaxy; several such processes have been suggested as possible causes of
gradients.

####  5.2.1 Dissipative collapse

The most extensive models exploring the effects of star formation during the
collapse of an initially gaseous protogalaxy are those reviewed by Larson
(1976a). In the spheroidal components of model disk galaxies, and in model
ellipticals, star formation becomes increasingly concentrated toward the
center as the density builds up. This effect is illustrated in Figure 11 (a),
which shows the SFR as a function of time at several radii in a spherical
collapse model. Stars formed at a given radius remain in orbits with little
net inward motion, but the gas sinks further in because it is dissipative
(i.e., its kinetic energy of radial motion is partly lost via collisionally
induced radiation). Thus the metals ejected by evolving stars are carried
inward by the gas, and an abundance gradient develops in the gas. As stars
continue to form, their compositions reflect this gaseous abundance gradient.
Figure 11 (b) shows the evolution of metallicity of newly formed stars (i.e.,
$Z$ of the gas) at several radii in a spherical model, and the rapid
development of a gradient is clear. The same process of dissipation produces a
central concentration in the gas density, which leads to a condensed nucleus
of stars.

If there were no dissipation, the stars and gas would collapse together and
the metals would not be concentrated inward. Thus in the outer parts of some
of these models, where the protogalactic density is too low for dissipation to
be effective, no stellar abundance gradient appears. The possible lack of a
gradient in metallicities of globular clusters beyond ${\sim}10\>\rm kpc$ from
the Galactic center has therefore been interpreted as showing that the
collapse of the Galaxy began with the stars and gas in free-fall; conversely,
the gradient at smaller radii is interpreted as showing the effects of
dissipation at a later stage of the collapse (Larson, 1977a).

####  5.2.2 A gradient in the IMF

Aside from any effects of gas flows, negative metallicity gradients could be
produced by gradients in the IMF that led to a yield decreasing outward. Since
the yield (Equation 3.12) depends on the relative numbers of metal-producing
stars, possibilities would be a steeper slope for massive stars or more low-
mass “inert” stars, at larger radii. In the latter case, the stars that
survive to the present would still have a radial gradient in their mass
function, with an interesting consequence: most of the luminosity of an old
population of stars comes from dwarfs near the MS turnoff and evolving giants,
while most of the mass is in less massive objects that contribute little
light; thus the $M/L$ ratio increases with the proportion of very low-mass
stars, and the postulated gradient in the IMF would lead to an outward
increase of $M/L$. Such a trend is indeed observed, although it is sometimes
ascribed to an extended “halo” of non-stellar condensed objects that formed
too soon to affect chemical evolution (Section 2.2.2). van den Bergh (1975)
has suggested that the IMF tends to have more massive stars in regions of high
density, and this view of the origin of metallicity gradients is part of his
evidence. The hypothesis of a gradient in the IMF in spheroidal systems has no
convincing theoretical basis, and the trends it would explain can arise in
other ways, but nevertheless systematic variations in the IMF could be as
important as they are hard to verify.

####  5.2.3 Finite stellar lifetimes

The timescale for star formation in a protogalaxy could be comparable to the
lifetimes of some metal-producing stars, in which case stars formed early
would be relatively unenriched. Thus if the outermost stars formed before most
of the central ones, there would be a negative metallicity gradient. In the
models in Section 5.2.1 (Larson, 1976a) it is assumed that all metals are
produced by stars with lifetimes $<3\times 10^{7}\>\rm yr$, so this effect is
negligible; but iron production by Type I supernovae could in fact be
significant on longer timescales (Section 2.4.4). What timescales are
relevant? The minimal collapse time for a protogalaxy is the free-fall time,

$t_{\rm ff}=1.7\times 10^{6}\left(\frac{M}{10^{11}\>\rm
M_{\odot}}\right)^{-\frac{1}{2}}\left(\frac{R}{1\>\rm
kpc}\right)^{\frac{3}{2}}\>\rm yr,$ (5.3)

where $M$ and $R$ are the mass and radius. For example, a galaxy with
$M=2\times 10^{11}\>\rm M_{\odot}$ and $R=15\>\rm kpc$ has $t_{\rm ff}=7\times
10^{7}\>\rm yr$; and a protogalaxy of the same mass collapsing from $R=50\>\rm
kpc$ has $t_{\rm ff}=4\times 10^{8}\>\rm yr$. Much longer timescales for star
formation are possible if the dissipation is slow, and the collapse time of
the system can be much longer if its boundary initially expands with the
Universe (Gunn & Gott, 1972). At least the outer parts of large galaxies could
therefore be metal-poor partly because of the finite lifetimes of metal-
producing stars.

A potential test is to look for variations in relative abundances. For
example, if oxygen comes only from very massive stars but iron comes partly
from stars of intermediate mass (Section 2.4; Section 4.4.1), then iron should
be more deficient than oxygen in the outermost stars. The hypothesis of a
gradient in the IMF of massive stars would predict the opposite trend in
relative abundances. Current data do not detect any gradients in relative
abundances, but oxygen itself has not been studied and nor have the faint
outer regions of elliptical galaxies.

It is quite possible that all of the processes discussed above were effective
in producing abundance gradients in spheroidal systems, so clear choices among
the theories are not to be expected.

###  5.3 The Metallicity–Mass Relation for Elliptical Galaxies

The correlation between metallicity and mass (color and luminosity) of
elliptical galaxies has been explained in several ways, of which two will be
reviewed here. These each involve dynamical effects during galaxy formation,
resulting in less complete conversion of the protogalactic gas to stars, and
so to a smaller final mean stellar metallicity, in smaller systems. One could,
of course, invoke differences in the IMFs of galaxies as a function of their
mass, but there is no independent evidence for a trend of the required form.

####  5.3.1 Supernova-driven winds

Star formation and chemical enrichment are cut off in a protogalaxy if the
residual gas is lost, and a possible loss mechanism is a galactic wind
energized by supernova explosions. Galactic winds were first analyzed for a
steady-state case by Johnson & Axford (1971) and Mathews & Baker (1971);
similar analyses have been made for nuclear bulges of spirals by Faber &
Gallagher (1976) and for bulge-disk systems by Bregman (1978). A galaxy
sustains a steady-state wind if the supernova rate divided by the rate of
supply of gas (from evolving stars) gives the gas enough energy to escape from
the galactic potential well. For protogalaxies, we are interested not in the
steady state, but in conditions for the initiation of a wind that can remove
essentially all of the residual gas. Larson (1974b) discussed possible effects
of supernovae in heating the gas, and adopted a simple approximation as the
condition for its removal: the residual gas is assumed to be lost suddenly
when the total heat input from supernovae has provided the gas with the escape
energy, assuming uniform conditions throughout the protogalaxy. This
approximation is plausible enough to suggest how the loss condition scales
with the relevant parameters, but there are unavoidably large uncertainties in
the astrophysics involved so the results are not very secure. The scaling can
be derived as follows.

Let $E$ be the thermal energy imparted to the ISM by supernovae when a unit
mass of stars is formed; $E$ is proportional to the fraction of stars that
become supernovae, to the mean kinetic energy of material ejected in a
supernova explosion, and to the efficiency with which this energy is
transferred to the ISM as heat. (The last factor is the most uncertain). As an
approximation, let $E$ be treated as a constant, despite finite stellar
lifetimes and complicated effects of the clumpiness of the ISM, its chemical
composition, etc. Let us consider a spherical protogalaxy of mass $M$ that has
formed a mass $M_{\rm s}$ of stars and has residual gas mass $M_{\rm
g}=M-M_{\rm s}$. The condition for gas to escape can be written

$\rm Potential\>energy\>of\>gas=Energy\>from\>supernovae,$

i.e,

$K\frac{GMM_{\rm g}}{R}=EM_{\rm s},$ (5.4)

where $K$ depends on the density distribution in the galaxy and will be
assumed constant as another simplification. Large elliptical galaxies are
observed to be more tightly bound than small ones, so a greater fraction of
their gas must be converted to stars before the condition (5.4) is satisfied;
therefore, their surviving stars have a greater mean metallicity. Other
consequences of this scenario are that the more massive galaxies collapse more
extensively before star formation is cut off, so they are predicted to have
more condensed nuclei and steeper metallicity gradients than smaller galaxies
(Section 5.2.1). These trends are observed, lending support to this type of
origin for the increase of metallicity with mass.

The form of the metallicity–mass relation can be accounted for using the same
approximate model. Let the initial mass–radius relation for protogalaxies have
the form

$M\propto R^{\alpha},$ (5.5)

so Equation (5.4) can be written

$M^{1-\frac{1}{\alpha}}\left(M-M_{\rm s}\right)\propto M_{\rm s}.$ (5.6)

Asymptotic equations for the mean metallicity of stars can be derived from
very general considerations: the mass of metals synthesized and ejected is
$yM_{\rm s}$, so at early stages of evolution when $M\simeq M_{\rm g}$, we
have approximately

$Z_{\rm s}\propto\frac{yM_{\rm s}}{M_{\rm g}}\simeq\frac{yM_{\rm
s}}{M},\>\>\left(Z_{\rm s}\ll y\right).$ (5.7)

At late stages, Equation (3.21) predicts that in all cases where mass is
conserved,

$Z_{\rm s}\rightarrow y,\>\>{\rm as}\>\>M_{\rm s}\rightarrow M.$ (5.8)

The results from numerical collapse models verify these relations in cases of
interest here. Substituting Equation (5.7) into (5.6), we find the stellar
metallicity–mass relation,

$Z_{\rm s}\propto M_{\rm s}^{\frac{\alpha-1}{2\alpha-1}},\>\>\left(Z_{\rm
s}\ll y\right).$ (5.9)

The tentative empirical relation (5.2) is obtained if $\alpha=1.5$, which
agrees fairly well with the observed mass–radius relation for elliptical
galaxies if one considers how the stellar system must swell (to conserve
energy) when the gas is lost (Tinsley, 1978b). According to Equation (5.8),
the power-law relation between $Z_{\rm s}$ and $M_{\rm s}$ must level off at
large masses, with $Z_{\rm s}\rightarrow y$ in the limit when essentially all
the original material is converted to stars; this behavior agrees with the
levelling of the color–magnitude relation at the magnitudes of the brightest
cluster galaxies.

The critical parameter $E$ can plausibly have a value that would give the
right scale for the $Z_{\rm s}$–$M_{\rm s}$ relation (Larson, 1974b), but its
value is very uncertain so the success of this theory must be considered
tentative. The interaction between supernovae and the ISM could, in fact, be
so weak as to drive a wind in only the very smallest protogalaxies.

####  5.3.2 Bursts of star formation in merging subsystems

Since the largest elliptical galaxies are the most metal-rich, a natural
hypothesis is that chemical enrichment accompanied the _growth_ of galaxies by
successive mergers among small subsystems. As noted in Section 1.2, gaseous
protogalaxies probably consisted of many dense lumps, so it is only a change
of viewpoint to consider these as merging subsystems rather than as a
collapsing unit. Moreover, extrapolations backward in time from the incidence
of strongly interacting galaxies in sight today suggest that collisions and
coalescence were common processes in the early lives of galaxies (Toomre,
1977; Vorontsov-Velyaminov, 1977). A property of colliding galaxies most
relevant to chemical evolution is that they often appear to be undergoing
intense bursts of star formation induced by the dynamical disturbance (Section
7.2), so it is reasonable to assume that star formation was caused in the past
by coalescence of subsystems in a protogalaxy. A qualitative model of chemical
enrichment by this process has been proposed by Tinsley & Larson (1979):
elliptical galaxies form by a hierarchical sequence of mergers among
subsystems, starting from small unenriched gas clouds; a burst of star
formation occurs at each merger, so at each stage of growth the fraction of
the total mass in stars increases and the mean metallicities of stars and gas
increase. In this picture, the final mass of an elliptical galaxy is
determined by the total mass of the small pieces initially present in its
surroundings. When these have all been mopped up, efficient star formation
stops. Any residual gas may get swept away if the system is moving through an
ambient IGM, or possibly blown out in a wind; if it remains bound to the
system, it could settle to a disk and leave the “elliptical galaxy” as the
central bulge of a spiral.

The resulting $Z_{\rm s}$–$M_{\rm s}$ relation depends on the _efficiency_ of
star formation as a function of the mass of the system (i.e., the system that
has been built after a given number of mergers), where efficiency is defined
as the mass of stars formed (in a given burst) per unit mass of gas. An
approximately power-law relation between $Z_{\rm s}$ and $M_{\rm s}$ can be
obtained only if the efficiency increases with the total mass of the system,
i.e., with successive mergers. For example, a relation

${\rm Efficiency\>of\>star\>formation\propto(Total\>mass)}^{p},$ (5.10)

where $p$ is a constant, leads to

$Z_{\rm s}\propto M_{\rm s}^{\frac{p}{1+p}},\>\>\left(Z_{\rm s}\ll y\right),$
(5.11)

with the usual limit $Z_{\rm s}\rightarrow y$ when all the gas is consumed.
The relation (5.10) can be justified qualitatively by considerations of gas
compression during collisions and mergers of subsystems. To reproduce the
tentative empirical relation (5.2), Equation (5.11) needs $p=1/3$, which is
consistent with the compression arguments. Equation (5.11) results from (5.10)
independently of such details as the mass distribution of merging pieces, and
it can be understood as follows: Equation (5.7) is true in any models with
mass conservation (including here conservation of the total mass of merging
pieces), while Equation (5.10) gives, dimensionally,

$\frac{M_{\rm s}}{M_{\rm g}}\propto M^{p},$

so that

$M_{\rm s}\propto M^{1+p}\>\>{\rm when}\>M_{\rm g}\simeq M\>\>\left(M_{\rm
s}\ll M\right);$ (5.12)

Equation (5.11) then follows from (5.7) and (5.12). The power law is again
predicted to level off, with $Z_{\rm s}\rightarrow y$ at high masses,
according to Equation (5.8).

As a theory for the origin of the metallicity–mass relation, this model has
the advantage of invoking processes that can be studied in nearby interacting
galaxies, but it remains to be seen whether the structural properties of
elliptical galaxies are fully consistent with its dynamical implications.

####  5.3.3 Mergers of stellar systems

The color–magnitude (metallicity–mass) relation for elliptical galaxies is
apparently affected in a way that has nothing to do with chemical evolution:
central cluster galaxies accrete their neighbors, by the process of dynamical
friction. There is no star formation during these mergers, because the
galaxies involved are ellipticals or S0s with almost no gas. Thus the growth
in luminosity is not accompanied by chemical enrichment, and it can make the
growing system bluer because the surrounding galaxies that it accretes are
generally smaller than the central giant. Galactic cannibalism by dynamical
friction was first proposed by Ostriker & Tremaine (1975), and later papers
(e.g. Hausman & Ostriker, 1978, and references therein) have developed its
implications for cosmological tests, the origin of core–halo structure of cD
galaxies, the luminosity function of galaxies in clusters, and the
color–magnitude relation itself. The process obviously tends to make the
color–magnitude relation turn over toward bluer colors at the bright end. This
effect has been proposed as a test for the occurrence of cannibalism in
clusters, but the results are not unambiguous because there is an intrinsic
upper limit, $Z_{\rm s}\rightarrow y$, to the average stellar metallicity in
the models discussed above, that leads to a flattening of the relation anyway.
Strong evidence that galaxies in the centers of clusters _do_ merge with each
other is given by the lumpy appearance of the central supergiant (cD) members
of some clusters; the lumps are interpreted as recently swallowed galaxies,
and the timescale for them to merge into a smooth system is generally
$<10^{9}\>\rm yr$ (Gunn, 1977).

###  5.4 The Intergalactic Medium and Gas Lost from Galaxies

Loss of interstellar gas from galaxies can both affect their own evolution, as
discussed for example in Section 5.3 above, and be a significant source of
metal-enriched IGM.

####  5.4.1 Loss of metals from galaxies

The mass of metals lost from an elliptical galaxy can be estimated by the
following argument, which is independent of the method of gas loss. The mass
of metals ever made by stars in the galaxy is ${\sim}yM_{\rm s}$ (by the
definition of the yield, Equation 3.12), and the mass of metals presently in
its stars is $Z_{\rm s}M_{\rm s}$, so the mass lost to the IGM at some stage
is ${\sim}(y-Z_{\rm s})M_{\rm s}$. This reasoning was used by Larson &
Dinerstein (1975) to predict a substantial metal-rich IGM in clusters of
galaxies, and a number of models with similar results have been advanced since
the iron X-ray line was discovered. An essentially model-independent estimate
can be made as follows.

Let $\phi(M_{\rm s})$ be the mass function of elliptical galaxies in a
cluster. Then the total mass of metals they have supplied to the ISM is

$M_{Z1}=\int\left[y-Z_{\rm s}\left(M_{\rm s}\right)\right]M_{\rm
s}\phi\left(M_{\rm s}\right)\ dM_{\rm s},$ (5.13)

where $Z_{\rm s}(M_{\rm s})$ is a function derivable from the color–magnitude
relation. In practice, $M_{\rm s}$ is expressed in terms of luminosity, and
$\phi(M_{\rm s})$ is obtained from the luminosity function. The value of $y$
should be taken as the maximum $Z_{\rm s}$ of an elliptical galaxy, which is
hard to obtain since the extensive outer regions that are probably metal-poor
are seldom observed; setting $y$ equal to the mean metallicity of local stars
(a little under $\rm Z_{\odot}$) is equivalent if elliptical galaxies have the
local IMF. In a calculation equivalent to the one just outlined, Tinsley &
Larson (1979) found that a cluster of elliptical galaxies would contain a mass
${\sim}(2-5)\>\rm M_{\odot}Z_{\odot}$ of intergalactic metals per solar
luminosity.

This is a very significant quantity of metals. For example, about $1/3$ of the
luminosity of the Coma cluster is due to its elliptical galaxies, so they
would provide a mass ${\sim}1\>\rm M_{\odot}Z_{\odot}$ of metals per solar
luminosity of the cluster, corresponding to ${\sim}0.1\>\rm Z_{\odot}$ of
metals per unit mass of galaxies (the ordinary stellar mass). If the bulges of
spiral and S0 galaxies also lost their metals due to the IGM, this estimate
could be doubled, but some of those metals may be in the disks (cf. Section
4.3.1). Iron can be considered as a representative metal in this calculation,
so we predict ${\sim}1\times\rm(Fe/H)_{\odot}\times M_{\odot}$ of iron in the
IGM per solar luminosity of the cluster. The actual mass of iron is quite
uncertain, and could be equal to the predicted amount.

####  5.4.2 Overall gas loss from galaxies

The total mass of gas lost from elliptical galaxies is a less predictable
quantity, depending on gas flows within the galaxies and gain or loss of gas
during the time of star formation. Nevertheless, some estimates are
interesting.

In order of magnitude, almost any model will predict a mean metallicity of the
lost gas that exceeds the mean metallicity of stars, since the gas has the
composition of the last and most metal-rich stars formed; i.e., $Z_{\rm
i}\gtrsim Z_{\rm s}$. The mass of gas lost is therefore $M_{\rm gi}=M_{\rm
Zi}/Z_{\rm i}\lesssim M_{\rm Zi}/Z_{\rm s}$. With $Z_{\rm s}\lesssim y\sim\rm
Z_{\odot}$ for the mean of a typical cluster of galaxies, we therefore expect
that $M_{\rm gi}\sim M_{\rm Zi}/\rm Z_{\odot}$, very roughly. This implies
that the elliptical galaxies in the Coma cluster have ejected ${\sim}1\>\rm
M_{\odot}$ of gas per solar luminosity of the cluster, which is at the lower
end of the range of estimates of the cluster gas content, from X-ray data
(Section 5.1). Various specific calculations using the models discussed in
Sections 5.3.1 and 5.3.2 lead to similar results within a factor ${\sim}3$. It
is therefore uncertain whether gas loss from ellipticals could be the entire
source of gas in a cluster like Coma, or whether some of the IGM is simply
primordial material that was never in a galaxy. (In the latter case, the
intergalactic metals could nevertheless have been supplied by galaxies).

An interesting comment on the origin of the cluster IGM has been made by
Ostriker (1977b): the distribution of morphological types of galaxies in
clusters like Coma differs from the field distribution in having a much
smaller fraction of spirals, many more S0s, and somewhat more ellipticals
(Oemler, 1974). If one “corrects” the cluster galaxies by adding disk matter
until the overall ratio (disks)/(elliptical galaxies + bulges) equals that of
the field, the mass of extra disk matter required is ${\sim}50\%$ of the
(ordinary) mass of galaxies in the cluster. This in turn is a significant
fraction of the mass of cluster IGM. Ostriker (1977b) therefore proposes that
some of the IGM is material that would have been made into disks in a less
dense environment, but instead was swept up into a hot ambient IGM. This idea
ties in with several scenarios for the formation of disk galaxies, in which
the disk is made from diffuse gas that is accreted after the formation of a
spheroidal component. For example, in the model of Ostriker & Thuan (1975), a
significant fraction of the disk is shed by halo stars; and in the picture of
Tinsley & Larson (1979), the disk forms from diffuse gas after denser pieces
have merged to make the bulge.

####  5.4.3 Ejection from evolving stars in elliptical galaxies

The stellar population in elliptical galaxies (and S0 galaxies and the
bulge–halo components of spirals) is predominantly very old, and the light of
these galaxies is dominated by red giants. It is almost certain that such
stars lose a few tenths of a solar mass, between the MS turnoff and the end of
their lives, to die as white dwarfs of typically $0.7\>\rm M_{\odot}$ (Section
2.4.1). This mass has been included in the total mass loss considered above,
but it is interesting to calculate also the present _rate_ of mass loss by
stars in elliptical galaxies.

For an analytical estimate, let us assume that all the stars in the system
formed at the same time, $t=0$. Let $M_{0}$ be the mass of stars formed, and
let $\phi(m)$ be the IMF; the mass of stars formed in the mass interval
$(m,\>m+dm)$ is therefore

$n(m)\ dm=M_{0}\phi(m)\ dm,$ (5.14)

by Equation (2.1). Now imagine these stars peeling off the MS and dying soon
afterward as they reach their lifetimes $\tau_{\rm m}$. The number of stars
dying per unit time is clearly

$D(t)=n\left(m_{\rm t}\right)\left|\frac{dm}{d\tau_{\rm m}}\right|_{\tau_{\rm
m}=t},$ (5.15)

where $m_{\rm t}$ is the turnoff mass ($\tau_{\rm m}=t$). The stellar
mass–lifetime relation can be approximated by a power law,

$\frac{m}{m_{1}}=\left(\frac{\tau_{\rm m}}{\tau_{1}}\right)^{-\theta},$ (5.16)

where $\tau_{1}$ is the lifetime of a fiducial mass $m_{1}$ and $\theta\simeq
0.25$ in the mass range of interest ($m_{\rm t}\sim 1\>\rm M_{\odot}$). It is
convenient to use a power-law IMF, Equation (2.3), normalized to
$\phi(m_{1})\equiv\phi_{1}$; masses in only the small range ${\sim}0.5-1\>\rm
M_{\odot}$ are relevant to the following calculation, so this IMF may be a
reasonable approximation even if a single power law would not apply to all
masses. The ejection rate can be obtained by multiplying $D(t)$ by $(m_{\rm
t}-w_{\rm m})$, the mass lost per star with remnant mass $w_{\rm m}$, with the
result

$E(t)=M_{0}\phi_{1}\theta\frac{m_{1}}{\tau_{1}}\left(m_{\rm t}-w_{\rm
m}\right)\left(\frac{t}{\tau_{1}}\right)^{-1+\theta x}.$ (5.17)

Since $(m_{\rm t}-w_{\rm m})$ changes slowly with time and $\theta x$ is
probably only a few tenths, this expression shows that the ejection rate
$E(t)$ varies approximately as $t^{-1}$. In Section 6.2.4, an analytical
expression is derived for the luminosity of stars in this model, and it is
shown that Equation (5.17) leads to a ratio of ejection rate to integrated
blue luminosity of the population,

$\frac{E}{L_{\rm B}}\sim 0.02\>\rm M_{\odot}\ L_{B\odot}^{-1}\ Gyr^{-1}$
(5.18)

at a present time ${\sim}10^{10}\>\rm yr$.

Most elliptical galaxies have a mass of neutral hydrogen that is less than 0.1
times their luminosity, in solar units, and many better studied ellipticals
have less neutral hydrogen than 0.01 times their luminosity (Knapp et al.,
1978). Faber & Gallagher (1976) argue that significant amounts of ISM cannot
be hiding in elliptical galaxies in ionized or molecular form. Thus the
ejection rate given by Equation (5.18) would provide more than the observed
amount of gas in a few Gyr, or even in less than $1\>\rm Gyr$. Possible fates
for this gas have been thoroughly discussed by Faber & Gallagher (1976); they
note that star formation at the rate in Equation (5.18) would be detectable
(unless only low-mass stars form), so they conclude that the gas ejected from
stars is being continually lost from the galaxies. On the other hand, Oemler &
Tinsley (1979) argue that star formation at the rate required to use up this
gas could have escaped detection in most ellipticals, and could account for
their supernova rate.

###  5.5 Abundance Gradients in Disks

Abundance gradients in disk stars and gas cannot be fully accounted for by
gradients in the gas fraction (Section 5.1), so it is of interest to see
whether dynamical processes analogous to those discussed in Section 5.2, for
spheroidal systems, could be responsible. A gradient in the IMF could again be
invoked, but this mechanism will not be discussed further.

Figure 12: (a) Star formation rates at several radii in the equatorial plane
of a collapse model for the formation of a disk galaxy (Larson, 1976c). The
radius in pc is marked on each curve, and the three ticks indicate the times
at which star formation is $10\%$, $50\%$, and $90\%$ complete (relative to
the final mass of stars) at that radius. (b) Metal abundances in the gas
(relative to the yield) in the equatorial plane of the same model (Tinsley &
Larson, 1978). In this Figure, the radii are given in kpc, ticks have the same
meaning as before, and open circles denote the time of maximum gas density at
each radius.

####  5.5.1 Effects of infall

The idea that disks of galaxies form by accretion, incorporating metals from
the young halo, has been suggested by dynamical models and supported by the
metallicities of stars in the Solar neighborhood (Section 4.3). The properties
of the accretion process that affect chemical evolution are the metallicity of
infalling gas ($Z_{\rm f}$) and the ratio of SFR to infall rate ($\psi/f$). If
there is a radial gradient in these quantities, there must be a corresponding
metallicity gradient in the disk. In particular, the metallicity of the gas at
any time tends to a value given by Equation (4.6), $Z\rightarrow ky+Z_{\rm
f}$, where $k=(1-R)\psi/f$. Thus the disk gas has about this metallicity at
the present time, in regions where infall is at all effective, i.e., where $k$
is not so large that $Z$ takes too long to approach its asymptotic value; the
stars in turn reflect the metallicity of the gas at the time when they formed.

In the dynamical models studied by Tinsley & Larson (1978), the value of
$\psi/f$ decreases outward in the disk at all times, because star formation is
less efficient at low densities; $Z_{\rm f}$ is negligible at late times at
all radii, but it has a significant negative gradient at early stages when
metals from the young halo (central bulge) were most important near the
center. Figure 12 (a) illustrates the SFR versus time at several radii in the
equatorial plane of one of these models, and Figure 12 (b) gives the
corresponding metallicities in the gas. At small radii, most stars form early
from metal-rich infalling gas, while the outer regions experience star
formation on a much longer timescale and are still relatively metal-poor. The
gas at the present time thus has a negative metallicity gradient due mainly to
the gradient in $\psi/f$, while the gradient in stellar abundances is due
partly to the early gradient in $Z_{\rm f}$. The sizes of the model gradients
are comparable to observed values, so this very schematic model may possibly
be showing some effects that occur in real disk galaxies.

Generalizing these results, we can conclude as in Section 4.3 that the
chemical properties of disks could plausibly be strongly affected by gas flows
that constitute the formation of the disk itself.

####  5.5.2 Effects of radial gas flows

Radial inflow of gas in disks possibly occurs as a result of transfer of
angular momentum by viscosity, loss of angular momentum from the gas to spiral
or bar-like density waves, and other mechanisms (e.g. Kalnajs, 1978; Ostriker,
1977a). These processes are rather speculative, since the inflow in many
models could be a numerical artifact, but it is interesting to see how
chemical evolution could be affected by flow velocities of a plausible
magnitude.

The metals are concentrated inward by this process, as by gaseous dissipation
in a collapsing spheroidal system (Section 5.2.1). Let us consider an annulus
of a galaxy between radii $r$ and $r+\delta r$, measured in the disk. The
chemical evolution of this annulus can be studied in the instantaneous
recycling approximation, using Equations (3.17) and (3.19). Let $M_{\rm g}$
and $\psi$ in those equations be replaced by $2\pi rM_{\rm g}\delta r$ and
$2\pi r\psi\delta r$, respectively, where $M_{\rm g}$ and $\psi$ now denote
the corresponding surface densities; let $f$ be replaced by the net rate of
inflow into the annulus, i.e., $F(r)-F(r+\delta r)=-(\partial F/\partial
r)\delta r$, where $F$ is a flow rate (in $\rm M_{\odot}\ yr^{-1}$) with a
positive sign for outward motion; and let $Z_{\rm f}f$ be replaced by the net
rate of inflow of metals, which is $Z(r)F(r)-Z(r+\delta r)F(r+\delta
r)=-Z(\partial F/\partial r)\delta r-(\partial Z/\partial r)F\delta r$. The
equations then reduce to

$\frac{\partial M_{\rm g}}{\partial t}=-(1-R)\psi-\frac{1}{2\pi
r}\frac{\partial F}{\partial r},$ (5.19)

and

$M_{\rm g}\frac{\partial Z}{\partial t}=y(1-R)\psi-\frac{1}{2\pi
r}\frac{\partial Z}{\partial r}F.$ (5.20)

Equation (5.20) shows that the radial flow is consistent with a steady-state
abundance gradient,

$\frac{\partial(Z/y)}{\partial r}\sim 2\pi r\frac{(1-R)\psi}{F},$ (5.21)

which is negative if the flow is inward. The flow causes $Z$ to change on a
timescale

$\tau_{\rm F}\sim\frac{2\pi r^{2}M_{\rm g}}{\lvert F\rvert}.$ (5.22)

$F$ can be expressed in terms of the flow velocity $v$, where $\lvert
F\rvert=$ (mass of gas in the annulus)/(time for gas to flow across $\delta
r$) $=2\pi rM_{\rm g}\lvert v\rvert$. The timescale for radial flow to be
effective is thus

$\tau_{\rm F}\sim\frac{r}{\lvert v\rvert},$ (5.23)

and the corresponding gradient can be written

$\frac{\partial(Z/y)}{\partial\ \ln(r)}\sim\frac{\tau_{\rm F}}{\tau_{*}},$
(5.24)

where $\tau_{*}\equiv M_{\rm g}/(1-R)\psi$ is the timescale for star formation
to use up the gas. These relations show that rapid inflow, with a timescale
less than that for star formation, quickly obliterates any radial metallicity
gradient, while slow inflow can lead to a significant one.

Substituting values of $\psi$, $M_{\rm g}$, and $r$ for the Solar
neighborhood, it is found that the interstellar abundance gradient (Section
5.1) is consistent with inflow at a few $\rm km\ s^{-1}$, carrying a flux
${\sim}1\>\rm M_{\odot}\ yr^{-1}$; the timescale for the gradient to change is
a few Gyr. There is no strong evidence for the occurrence of systematic gas
flows of this magnitude in the Galaxy, but nor can they be ruled out.

Sanders (1977) has suggested that the deep minimum in the surface density of
gas in the Galaxy, in an annulus between $0.6$ and $4\>\rm kpc$, could be due
to inflow into the central $600\>\rm pc$, where the total quantity of ISM is
enough to fill the depleted annulus. If so, inflow could perhaps be fueling
the strong star formation at the Galactic center.

##  6 Approaches to Photometric Evolution

Evolution of stars in galaxies affects not only their chemical compositions,
but also their integrated luminosities, colors, and spectra. Photometric and
chemical evolution can be studied separately, because they depend largely on
complementary properties of a galaxy: the colors at a given time are governed
strongly by the current rate of star formation relative to its past average
value, whereas the chemical composition depends mainly on the integrated past
star formation relative to the gas content and on the ratio of the SFR to gas
flow rates. In studying photometric evolution, we can ignore the effects of
ISM, except in correcting colors for any reddening or gaseous emission, and we
can avoid assumptions relating the SFR to the gas supply. Of course, a
complete understanding of the properties of a galaxy would include the
relations among its history of star formation, gas content and gas flows, and
chemical composition (cf. Figure 1), but more can be learned by tackling
pieces of the puzzle separately first.

###  6.1 Aims and Methods

Models for the stellar population of a galaxy address three related questions:
what types of stars are present, what history of star formation produced them,
and what was the population and its photometric properties in the past? The
answers to these questions have many applications, such as interpreting in
terms of star formation the correlations between photometric and morphological
properties of galaxies, and predicting changes on cosmological timescales.

Methods of constructing population models can be divided into three
categories: “population synthesis” with no explicit evolution, evolutionary
models, and analytical approximations.

####  6.1.1 Population synthesis

This approach is to find the “best” mixture of stars to match the colors of
the galaxy under study. The inferred distribution of stars in the HR diagram
then contains information on their past formation rate and the IMF, and it is
often possible to judge the mean chemical composition and even to detect minor
components of high or low metallicity. The procedure is to observe the colors
of the galaxy and a variety of nearby stars, generally including narrow-band
photoelectric colors and indices giving the strengths of spectral features
that are sensitive to stellar temperature, luminosity, or composition. Then
synthetic colors are computed for various mixtures of stars and compared with
the galaxy colors. The search for an optimal mixture can be made in many ways,
ranging from trial-and-error to elaborate computer algorithms; the method
generally used, quadratic programming, was introduced to the field by Faber
(1972). Because of observational errors, and because the available nearby
stars do not include all types in the galaxy under study, a perfect fit is
seldom found, and the solution that (formally) minimizes the errors is not
necessarily the most plausible. The choice of a “best” synthetic population
must therefore be based on imposed astrophysical constraints; these include
such requirements as a smooth MS luminosity function, and a distribution of
subgiants and giants that could plausibly arise from evolution off the MS. The
lack of an objectively defined best fit means that the final solution depends
strongly on the imposed constraints, as emphasized by Williams (1976).

There are often several astrophysically acceptable synthetic populations that
match the galaxy colors equally well but correspond to significantly different
histories of star formation. An example of such ambiguity appears in models
for elliptical galaxies, reviewed by Faber (1977). All studies agree that
_most_ of the light of these galaxies, from blue through infrared wavelengths,
comes from an old stellar population with a distribution in the HR diagram
like an old open cluster plus an extended giant branch (cf. Figure 5).
However, such models almost always fail by a few percent to account for the
light around $3500~{}$\mathrm{\text{Å}}$$ (e.g. $U-B$ is predicted to be too
red by ${\sim}0.1\>\rm mag$), so they are lacking some hot stellar component
that is present in the real galaxies. To date it has been impossible to
determine whether the hot stars are a few upper-main-sequence stars, implying
ongoing star formation, or a minor population of old objects such as
horizontal-branch stars or blue stragglers (which can be seen in the
color–magnitude diagram for the old cluster M67, Figure 5). Obviously, it
would be very interesting to know if typical elliptical galaxies really are
still making stars at a slow residual rate! For the central bulge of M31,
which is optically indistinguishable from an elliptical galaxy, there are
broadband colors down to $1550~{}$\mathrm{\text{Å}}$$, but even these data
have not resolved the ambiguity (Wu et al., 1980). A new avenue has been
opened by a demonstration that the integrated light of stars in a nuclear
bulge region of our own galaxy matches exactly the integrated light of
comparable regions of spirals and ellipticals (Whitford, 1978). The brighter
stars are individually observable in the Galactic bulge, so a star-by-star
synthesis of their contribution to the light is possible. Perhaps in the end
this approach will tell whether an old population alone can account for the
ultraviolet light.

Another general problem is that even the best defined regions in the HR
diagram cannot be interpreted uniquely in terms of a past history of star
formation. The models are insensitive to many details of the IMF and SFR, for
two basic reasons:

  1. 1.

the integrated light of galaxies is dominated by regions of the HR diagram
that depend theoretically on rather few parameters of star formation; and

  2. 2.

some types of stars, such as red giants, may have evolved from a wide range of
MS masses (Section 2.4.2), so they cannot be traced uniquely back to an
initial mass and time of star formation.

The second of these problems is avoided in the evolutionary method described
next, but the first remains and will be discussed below.

####  6.1.2 Evolutionary models

This approach relies primarily on stellar evolution theory to suggest
allowable populations, as follows. Theoretical tracks (or isochrones) of stars
in the HR diagram are used to compute the stellar population that would arise,
at a given age, from a given SFR and IMF, with a given chemical composition;
the integrated colors are then calculated, using observed colors of stars in
appropriate parts of the HR diagram, and the results are compared with the
colors of the galaxy under study. The aim is to derive from a series of models
the SFR, IMF, age, and composition(s) that best match the galaxy, and thereby
to learn not only about its present stellar population but also about its past
history and past photometric properties.

In practice, stellar evolution is not well enough understood for fully
theoretical models to be reliable. The main problems are related to late
stages of evolution, including particularly the giant branches in old stellar
populations, whose effects on models for elliptical galaxies are reviewed by
Faber (1977). These problems are alleviated by using statistical studies of
nearby giants to provide semi-empirical evolutionary tracks (Section 2.4.1),
and by allowing the most uncertain types of stars to be present in numbers
that are treated as adjustable parameters. This method thus closely resembles
some non-evolutionary population syntheses in which the constraints are chosen
to represent stellar evolutionary tracks (O’Connell, 1976).

The evolutionary approach has several advantages. The best established aspects
of stellar evolution theory are incorporated, so the resulting population is a
_possible_ one as far as can be determined. Uncertainties cannot be formally
calculated, but from trials with a variety of assumptions one can estimate
subjectively the allowable range of parameters. Often this range is small
enough to lead to useful conclusions about the past history of star formation,
and to predictions of photometric changes of cosmological interest. For
example, it is possible to determine the slope of the IMF in elliptical
galaxies closely enough to be sure that their integrated luminosity declines
with increasing age (Section 6.2).

Uncertainties in the conclusions from this method arise partly from
uncertainties in stellar evolution, and partly from the intrinsic
insensitivity of integrated colors to many parameters of interest – a problem
found earlier with population syntheses. Two parts of the HR diagram tend to
dominate the integrated light, as illustrated spectroscopically by the work
of, e.g., Morgan & Mayall (1957) and Morgan & Osterbrock (1969): B stars on
the upper main sequence, and late G through early M giants. These dominant
regions are extended out to O stars in ultraviolet light and to late M giants
in the infrared. If young stars are absent, low-mass giants dominate at visual
and longer wavelengths, so the colors depend much more on stellar evolution
than on the IMF or past SFR; at shorter wavelengths, however, turnoff stars
are seen so the colors give some information on the age of the system. If
young stars are present, the light at short wavelengths is dominated by OB
stars, whose relative numbers depend on the IMF and whose total numbers
(relative to red stars) depend on the ratio of the present SFR to its
integrated past value. Stars with lifetimes from a few times $10^{8}\>\rm yr$
to just below the age of the system (usually A and F stars) contribute
relatively little light, so there is little information on either their part
of the IMF or the detailed time-dependence of the SFR. In Section 7.1, models
will be discussed that illustrate the dominance of the upper main sequence
and/or low-mass giants, depending on the SFR.

Programs for constructing evolutionary models have been described by Tinsley
(1968, 1972a, 1978b), Searle et al. (1973), Tinsley & Gunn (1976a), and Larson
& Tinsley (1978). The mechanical details are far less troublesome than the
input “data” representing stellar tracks, and it is easy to obtain numerical
accuracy far exceeding the astrophysical certainty of the calculations. There
are two types of this technique.

  1. 1.

The first method is to supply the computer with evolutionary tracks in the HR
diagram for stars with a series of discrete masses, or with isochrones for a
series of discrete ages; separate stellar data are used for each chemical
composition of interest. Then, for a given IMF and SFR, the calculation yields
the numbers of stars on a large grid of points in the HR diagram, as a
function of the age of the system.

  2. 2.

The second method uses the first type of program once only for each IMF and
composition, to give the integrated colors at a series of ages of a model
whose SFR consists of a single initial burst. These are then regarded as the
colors of “generations” of stars with a given age (and IMF and composition).

A model with any prescribed SFR can then be treated, at each age, as the sum
of such generations in proportions given by the SFR. The number of generations
whose properties must be combined to obtain the integrated colors of any model
is much smaller than the number of points in the HR diagram that are referred
to directly in the first method, so the second approach is more economical. In
either method, it is clearly possible to add arbitrary numbers of stars of
undetermined evolutionary status, in the spirit of population synthesis.

While models with Solar metallicity can rely on nearby stars to provide colors
and semi-empirical evolutionary tracks, there is no such convenient sample for
other compositions. In making models for non-Solar metallicities, it is often
most convenient to change the “standard” models differentially, rather than
starting from scratch with tracks and colors for each set of stars. Faber
(1973) first used the metallicity effects discussed in Section 2.4.5 to
estimate differential changes in the integrated colors of elliptical galaxies,
as a function of metallicity, and her methods have been adapted by others
subsequently. Recent results for elliptical galaxies have been cited in
Section 5.1. The calculations of metallicity effects in integrated light are
still much less secure than one would like, and there is a need for more basic
work on stellar evolution and atmospheres at non-Solar compositions, including
non-Solar abundance ratios among elements heavier than helium (Faber, 1977).

####  6.1.3 Analytical approximations

Some of the results from evolutionary models can be understood qualitatively
using analytical approximations. These have proved particularly tractable for
models in which all the stars form in a single initial burst, which is a first
approximation to the population in elliptical galaxies. Such models will be
considered next.

###  6.2 Evolution of a Single Generation of Stars

Many numerical models designed to match detailed photometry of elliptical
galaxies have shown that nearly all the light at visual and longer wavelengths
can be accounted for by a very old population, with a turnoff near the Sun’s
position on the MS. The metallicities of the dominant stars appear to be
within a factor of two of Solar in wide-aperture observations of giant
ellipticals, although their centers may be more metal-rich and small
ellipticals are metal-poor (Section 5.1). Reviews by van den Bergh (1975) and
Faber (1977) cover the history and recent status of this subject, and a few
subsequent developments have been referred to in Section 6.1. The implications
of a predominantly very old population for the evolution of elliptical
galaxies are best understood using analytical approximations.

####  6.2.1 Content and luminosity

Let us consider a single generation of stars, formed with total mass $M_{0}$
in a short burst (as in Section 5.4.3), with a fixed chemical composition near
Solar. The population evolves by peeling off the MS, as can be visualized from
Figures 2 and 3.

The IMF will be taken to be a power law, normalized to
$\phi(m_{1})\equiv\phi_{1}$, where $m_{1}$ is the turnoff mass at a fiducial
time $\tau_{1}$. The power-law approximation need only hold over a small mass
interval, since the light at present comes almost entirely from stars between
$0.4\>\rm M_{\odot}$ and turnoff, and the turnoff mass at ages of interest,
${\sim}5-20\>\rm Gyr$, lies in the small range ${\sim}0.9-1.2\>\rm M_{\odot}$.

At a time $t$ after star formation, the MS stars present have masses from the
lower limit at formation, $m_{\rm L}$, up to the turnoff mass $m_{\rm t}$,
which is given by substituting $\tau_{\rm m}=t$ in Equation (5.16). Thus the
number of dwarfs with masses in the interval $(m,\ m+dm)$ is, by Equation
(2.3),

$\medsize n_{\rm d}(m)\ dm=M_{0}\phi(m)\
dm=M_{0}\phi_{1}\left(\frac{m}{m_{1}}\right)^{-(1+x)}\ dm,\>\>m_{\rm L}\leq
m\leq m_{\rm t}.$ (6.1)

Stars slightly more massive than $m_{\rm t}$ are present as giants, and their
total number is the number of stars that were on the MS with lifetimes between
$t$ and $t-\tau_{\rm g}$, where $\tau_{\rm g}$ is the duration of post-MS
evolution for masses ${\sim}m_{\rm t}$. (The term “giants” is used loosely
here to mean all post-MS stars; the analysis can easily be modified to refer
to any portion of post-MS evolution). The number of giants is therefore

$n_{\rm g}(t)=M_{0}\phi(m_{\rm t})\left|\frac{dm}{d\tau_{\rm
m}}\right|_{\tau_{\rm m}=t};\>\>\tau_{\rm
g}=M_{0}\phi_{1}\theta\frac{m_{1}}{\tau_{1}}\tau_{\rm
g}\left(\frac{t}{\tau_{1}}\right)^{-1+\theta x}.$ (6.2)

The luminosity of individual dwarfs in the mass range of interest can be
approximated by a power law,

$\ell_{\rm d}(m)=\ell_{1}\left(\frac{m}{m_{1}}\right)^{\alpha},$ (6.3)

where $\alpha\simeq 5$. For giants, an average luminosity $\ell_{\rm g}$ is
defined so that the product $\ell_{\rm g}\tau_{\rm g}$ gives correctly the
integrated light output during post-MS evolution. The values of $\ell_{1}$,
$\alpha$, and $\ell_{\rm g}$ of course depend on the wavelength interval of
interest, and so do the results below relating to luminosities. (For
bolometric light, the product $\ell_{\rm g}\tau_{\rm g}$ is proportional to
the amount of nuclear energy used, but it has no such interpretation in
restricted wavelength bands).

The integrated luminosities and masses of dwarfs and giants can now be derived
from Equations (6.1) – (6.3) and Equation (5.16). It will be assumed in the
integrals that $m_{\rm L}\ll m_{1}$. The total mass of dwarfs at time $t$
depends critically on whether the slope of the IMF $(x)$ is less than or
greater than 1:

$\displaystyle\frac{M_{0}\phi_{1}m_{1}^{2}}{x-1}\left(\frac{m_{\rm
L}}{m_{1}}\right)^{-x+1},\>\>\>\,x>1,$ (6.4a) $\displaystyle
M_{0}\phi_{1}m_{1}^{2}\ \ln\left(\frac{m_{\rm t}}{m_{\rm
L}}\right),\>\>\>\>\>\;x=1,$ (6.4b)
$\displaystyle\frac{M_{0}\phi_{1}m_{1}^{2}}{1-x}\left(\frac{t}{\tau_{1}}\right)^{-\theta(1-x)},\>\>x<1.$
(6.4c)

Giants have a total mass ${\sim}m_{\rm t}n_{\rm g}(t)$, and one can quickly
verify that the mass ratio of giants to dwarfs is greatest in the case $x<1$,
and is at most ${\sim}\tau_{\rm g}/t\sim 0.1$; the contribution of giants to
the total mass will therefore be neglected. The integrated luminosity of
dwarfs is

$L_{\rm d}(t)=\int_{m_{\rm L}}^{m_{\rm t}}\ell_{\rm d}(m)n_{\rm d}(m)\
dm=\frac{M_{0}\phi_{1}m_{1}\ell_{1}}{\alpha-x}\left(\frac{t}{\tau_{1}}\right)^{-\theta(\alpha-x)},$
(6.5)

on the assumption $x<\alpha$, which is justified below. Finally, the
integrated luminosity of giants is

$L_{\rm g}(t)=\ell_{\rm g}n_{\rm
g}(t)=M_{0}\phi_{1}\theta\frac{m_{1}}{\tau_{1}}\ell_{\rm g}\tau_{\rm
g}\left(\frac{t}{\tau_{1}}\right)^{-1+\theta x}.$ (6.6)

The above relations will be used to derive some interesting properties of this
single generation of stars.

####  6.2.2 Remnants of dead stars

There may be a significant dark mass in the form of remnants of stars
initially above $m_{\rm t}$, especially if the IMF has a fairly shallow slope
so these stars were relatively numerous. Although it is probably a very poor
approximation to extrapolate the IMF to high masses with the slope $x$ used
near $1\>\rm M_{\odot}$, the equations will be written to show how the
contributions of remnants can be estimated in the simplest cases. (These
results can easily be modified to allow for a variable slope). In this
approximation, it will be assumed that all remnants have the same mass $w$,
and that all stars above $m_{\rm t}$ are dead. Then the total mass of remnants
is $w$ times the number of stars formed with masses between $m_{\rm t}$ and
the upper limit $m_{\rm U}$:

$M_{\rm w}(t)=w\int_{m_{\rm t}}^{m_{\rm U}}M_{0}\phi(m)\
dm=\frac{M_{0}\phi_{1}m_{1}w}{x}\left(\frac{t}{\tau_{1}}\right)^{\theta x},$
(6.7)

assuming $m_{\rm U}\ll m_{\rm t}$ and $x>0$. The relative mass of remnants is
potentially greatest if $x<1$, and then Equation (6.4c) shows that $M_{\rm
w}/M_{\rm d}\sim w/m_{\rm t}$, which could be close to unity. This result is
obviously strongly dependent on the assumption of a single power law for the
whole IMF, which would exaggerate the mass of remnants if, for example,
elliptical galaxies have a curved IMF like the function in the Solar
neighborhood (Figure 4). It may be concluded that dead remnants could possibly
affect the total mass by a factor ${\sim}2$, which cannot be predicted with
any confidence from constraints on the slope of the IMF at turnoff.

####  6.2.3 The ratio of giants to dwarfs in the light

Some spectral features in the integrated light of elliptical galaxies depend
sensitively on the relative amounts of light contributed by giant and dwarf
stars at the feature wavelength. Examples are an iron hydride band at
$0.99\>\rm\upmu m$, known as the Wing-Ford band, which Whitford (1977) has
found to be extremely strong in late dwarfs but weak in late giants; and a
carbon monoxide band at $2.2\>\rm\upmu m$, studied especially by Frogel et al.
(1978, and earlier papers cited therein), which has the opposite behavior,
being much stronger in late giants than in late dwarfs. Since the light of
elliptical galaxies at those wavelengths must be dominated by late-type stars,
the galaxies should show a weak FeH band and a strong CO band if giants
outshine dwarfs, and vice versa. As the following analysis shows, the relative
luminosities of giants and dwarfs give important information on the slope of
the IMF, which in turn affects many other properties of elliptical galaxies
including the rate of evolution of total luminosity; it is the significance of
this effect for cosmological tests (Section 6.2.6) that has motivated much of
the analysis of spectral features.

Equations (6.5) and (6.6) together give an approximate expression for the
relative luminosities of giants and dwarfs:

$G(t)\equiv\frac{L_{\rm g}(t)}{L_{\rm d}(t)}=\theta(\alpha-x)\frac{\ell_{\rm
g}\tau_{\rm
g}}{\ell_{1}\tau_{1}}\left(\frac{t}{\tau_{1}}\right)^{\theta\alpha-1}.$ (6.8)

An alternative expression is obtained by substituting Equations (5.16) and
(6.3):

$G(t)=\theta(\alpha-x)\frac{\ell_{\rm g}\tau_{\rm g}}{\ell_{\rm d}\left(m_{\rm
t}\right)t}.$ (6.9)

The term $\ell_{\rm g}\tau_{\rm g}$ is the amount of energy radiated (at a
given wavelength) by a star of approximately turnoff mass after it leaves the
MS, while $\ell_{\rm d}\left(m_{\rm t}\right)t$ is the energy radiated during
MS evolution. Thus Equation (6.9) says that the value of $G$ in bolometric
light is, in order of magnitude, equal to the ratio of nuclear fuel consumed
after leaving the MS to that consumed on the MS; since stars near $1\>\rm
M_{\odot}$ burn the hydrogen in only $10\%$ of their mass while on the MS but
in $70\%$ before they die (Section 2.4.1), this fuel ratio is ${\sim}6$. This
high value is the underlying reason why giants can outshine dwarfs in the
integrated light of a galaxy, despite their very short lifetimes. Giants tend
to be especially dominant at long wavelengths, because most of the energy from
the giant branch as a whole comes from red giants.

The fuel burning ratio is not the only factor affecting $G$, however. The term
$(\alpha-x)$ in Equation (6.9) introduces a dependence on $x$, the slope of
the IMF. A larger value of $x$ reduces the contribution of giants simply by
reducing the number of stars in the mass range of giants (just above turnoff)
relative to those still on the MS. The dependence of $G$ on $x$ is of great
practical importance, since it allows spectroscopic criteria to set
constraints on $x$. The work of Whitford (1977) and Frogel et al. (1978) shows
that the red–infrared light of elliptical galaxies is strongly dominated by
giants, to an extent that $x$ must be less than 2, and possibly less than 1.
These constraints are consistent with the IMF in the Solar neighborhood, which
has $x<1$ in the relevant mass range (Figure 4 and Equation 2.9).

The infrared spectra of elliptical galaxies set constraints not only on the
IMF but also on the relative numbers of M giants of different spectral types
that populate the giant branch. As noted in Section 6.1.2, these numbers are
not firmly predicted by stellar evolution theory, so studies of galaxy spectra
can add to an understanding of late stages in the lives of low-mass stars.
This application of galaxy models is discussed by Faber (1977), Tinsley
(1978b), and references therein.

####  6.2.4 The stellar mass loss rate relative to luminosity

An expression for the rate of mass loss from stars has been derived in Section
5.4.3, but Equation (5.17) is not in a useful form for comparing with
observable quantities. It is possible to obtain a useful equation for the
ejection rate per unit integrated luminosity, because both quantities scale
with the populations of stars near turnoff.

From Equations (6.5) and (6.8), the total luminosity can be written

$L(t)=\left[1+G(t)\right]L_{\rm
d}(t)=\frac{M_{0}\phi_{1}m_{1}\ell_{1}}{\alpha-x}(1+G)\left(\frac{t}{\tau_{1}}\right)^{-\theta(\alpha-x)}.$
(6.10)

Then, with Equation (5.17), the ratio of ejection rate to luminosity is

$\frac{E(t)}{L(t)}=\frac{\theta(\alpha-x)}{\ell_{1}\tau_{1}}\frac{m_{\rm
t}-w_{\rm m}}{1+G}\left(\frac{t}{\tau_{1}}\right)^{\theta\alpha-1},$ (6.11)

which shows that the ratio depends only slowly on time. A more useful relation
for finding the present ratio is given by substituting Equation (5.16) and
(6.3) to eliminate $\ell_{1}$ and $\tau_{1}$, with the result

$\frac{E(t)}{L(t)}=\theta(\alpha-x)\frac{m_{\rm t}-w_{\rm
m}}{1+G}\frac{1}{\ell_{\rm d}\left(m_{\rm t}\right)t}.$ (6.12)

This ratio can be estimated for present-day ellipticals as follows. From
spectroscopic studies in _blue_ light, $G\simeq 1$ (the value of $G$ is
greater in red or bolometric light); and approximate values of the other
quantities are $\alpha\simeq 5$, $\theta\simeq 0.25$, $m_{\rm t}\simeq 1\>\rm
M_{\odot}$, $w_{\rm m}\simeq 0.7\>\rm M_{\odot}$, $\ell_{\rm d}\simeq 1\>\rm
L_{\rm B\odot}$, $t\simeq 10\>\rm Gyr$, $x\simeq 1$. The result from Equation
(6.12) is then $E/L_{\rm B}\simeq 0.015\>\rm M_{\odot}\ L_{\rm
B\odot}Gyr^{-1}$, of which the significance was discussed in Section 5.4.3.

####  6.2.5 The mass-to-luminosity ratio

An analytical estimate of $M_{\rm s}/L$ can be made using the mass of stars
$M_{\rm s}\simeq M_{\rm d}(t)$ (neglecting the small contribution of giants
and the very uncertain contribution of dead remnants), and the total
luminosity $L(t)$. From Equations (6.4a) – (6.4c), it is clear that the result
depends strongly on whether $x\lessgtr 1$. Moreover, it depends critically on
the assumption that $x$ is constant down to $m_{\rm L}$, since the least
massive stars (or sub-stellar objects) can be numerous enough to dominate the
mass while contributing negligibly to the light. If $x<1$, the result from
Equations (6.4c) and (6.10) is

$\frac{M_{\rm s}}{L}=\frac{\alpha-x}{1-x}\frac{1}{1+G}\frac{m_{\rm
t}}{\ell_{\rm d}\left(m_{\rm t}\right)},\>\>x<1,$ (6.13)

which is proportional to the mass-to-luminosity ratio of turnoff stars. If
$x>1$ (or if $x$ increases from a value below 1 at turnoff to above 1 at
smaller masses) Equation (6.4a) shows that $M_{\rm s}/L$ increases in
proportion to $m_{\rm L}^{-(x-1)}$, so it is sensitive to a quantity that
cannot be determined photometrically.

In all cases, photometric data (star counts, population syntheses,
spectroscopic estimates of $x$) yield only a _lower limit_ to the true mass-
to-luminosity ratio $(M/L)$ of a galaxy, since any amount of mass could be
present in hidden form. When the masses of galaxies are determined
dynamically, the empirical $M/L$ values often increase to such large values in
the outer regions that a large amount of hidden mass must indeed be present
(e.g. Spinrad et al., 1978). For this reason, values of $M/L$ determined from
population syntheses and equivalent methods are sometimes called “photometric
$M/L$ ratios” to distinguish them from the ratios of actual mass (defined
dynamically) to luminosity.

####  6.2.6 Evolution of luminosity and the Hubble diagram

Figure 13: Schematic Hubble diagram showing how both the deceleration
parameter ($q_{0}$) and evolution of galaxies affect the departure from
linearity. _Lines_ are schematic “theoretical” curves for two values of
$q_{0}$, _dots_ are hypothetical data points, and _arrows_ indicate
qualitatively how they should be corrected if the net effect of evolution is
to make distant galaxies intrinsically brighter than nearby ones. More
precisely, evolution (in this sense) at the rate of a few percent of a
galaxy’s luminosity per Gyr makes the true value of $q_{0}$ smaller by about
unity than the value inferred from the uncorrected data points.

In one of the classic cosmological tests, the Hubble diagram, logarithmic
redshifts of galaxies are plotted against against their apparent magnitudes,
as illustrated schematically in Figure 13. For a sample with a well-defined
mean absolute magnitude, this diagram can be regarded heuristically as a plot
of “recession velocity” versus “distance”. At small redshifts, the regression
line is linear with a slope corresponding to Hubble’s Law, $\rm
redshift\propto distance$. At large redshifts, the deviation from linearity
measures the change in the ratio “velocity” / “distance” with distance itself;
since the lookback time (the light-travel time) increases with distance, the
curvature of the Hubble diagram thus gives a measure of the past expansion
rate of the Universe, and in particular of its deceleration. The deceleration
parameter $q_{0}$ can take only positive values in the simplest cosmological
models of General Relativity, the Friedmann models, and $1/2$ is a critical
value: if $q_{0}>1/2$, the deceleration is large enough for the expansion
eventually to be reversed, but if $0<q_{0}\leq 1/2$, the Universe will expand
forever; if in fact $q_{0}$ is negative, indicating that the expansion is
accelerating, more complicated cosmological models are required. Evolution of
galaxies enters the picture because the lookback times sampled must be many
Gyr for the deceleration to be detectable; the galaxies then had significantly
different luminosities, so the “distance” parameter, apparent magnitude,
cannot be estimated on the assumption of a constant absolute magnitude. The
departure of the Hubble diagram from linearity is very sensitive to evolution:
if the luminosities of elliptical galaxies grow fainter at a few percent per
Gyr, for example, the apparent value of $q_{0}$ (inferred from the shape of
the Hubble diagram) exceeds its true value by several tenths. This problem has
been discussed by Humason et al. (1956), Sandage (1961b, c), Gunn & Oke
(1975), and Tinsley (1972b, 1977b). For an approximate estimate of the
evolutionary correction to $q_{0}$, the above analytical equations can be
used.

From Equation (6.10), we have

$\frac{d(\ln\ L)}{d(\ln\ t)}=-\theta(\alpha-x)+\frac{t}{1+G}\frac{dG}{dt},$
(6.14)

and Equation (6.8) can be used to evaluate $dG/dt$. The term $\ell_{\rm
g}\tau_{\rm g}$ in the expression for $G(t)$ depends only slowly on time,
because giant branch evolution depends only weakly on mass in the relevant
range, so only the explicit time-dependence need be considered and that term
gives $(t/G)(dG/dt)=\theta\alpha-1$. Substituting in Equation (6.14), we have

$\frac{d(\ln\ L)}{d(\ln\ t)}=-\theta(\alpha-x)+\frac{G}{1+G}(\theta\alpha-1).$
(6.15)

The second term in Equation (6.15) is not very important, since
$(\theta\alpha-1)$ is a few tenths and $G/(1+G)$ lies between 0 and 1. The
main term is therefore simply $(-\theta\alpha+\theta x)$, which can be written

$\frac{d(\ln\ L)}{d(\ln\ t)}\simeq-1.3+0.3x.$ (6.16)

Essentially the same result is obtained for the evolution of luminosity in
numerical population models. The examples in Figure 14 show the predicted
dependence on the IMF: the rate at which $M_{V}$ gets dimmer is slower in
models with a larger value of $x$. Since giants supply most of the light, this
behavior is mainly because, when $x$ is large, the giant branch is fed by a
more richly populated main sequence as time goes on.

Figure 14: Evolution of colors and magnitudes of single-generation models for
the stellar population in elliptical galaxies (Tinsley & Gunn, 1976a). Curves
are for three values of the slope of the IMF: _solid lines_ , $x=2$; _dashes_
, $x=1$; _dots_ , $x=0$. Note that if $x$ is small, colors evolve slowly but
magnitudes evolve quickly.

In the Hubble diagram, evolution means that departures from linearity are due
not only to $q_{0}$ but also to systematic changes in the absolute magnitudes
of galaxies (Figure 13). If the curvature is interpreted without regard to
evolution, the result is an apparent value of $q_{0}$ that differs from the
true value by

$\Delta q_{0}\equiv{\rm apparent\ value-true\ value}\simeq-1.5\frac{d(\ln\
L)}{d(\ln\ t)}$ (6.17)

(e.g. Tinsley, 1977b). A first-order estimate, from Equation (6.16), is
therefore

$\Delta q_{0}\simeq 2.0-0.4x.$ (6.18)

The slope of the IMF, $x$, emerges as the critical parameter. As discussed in
Section 6.2.3, spectroscopic studies indicate that $x<2$, and possibly $x<1$.
In the first case, $|\Delta q_{0}|\gtrsim 1$, and in the second case, $|\Delta
q_{0}|\gtrsim 1.5$. In either case, the correction for evolution is big enough
to make a qualitative difference to the type of cosmology inferred.

Current estimates of the apparent value of $q_{0}$ range from ${\sim}0$ (Gunn
& Oke, 1975) to ${\sim}1.5$ (Kristian et al., 1978), the differences being due
to unknown sampling and observational effects. Downward corrections of order
unity are clearly important in determining whether the true value of $q_{0}$
is greater than $1/2$, less than $1/2$, or even negative.

A value of $x\gtrsim 5$ would be needed to make stellar evolution negligible
in the Hubble diagram, but such a steep IMF would make the late dwarfs
dominate the infrared light to an extent that is precluded by the giant-
dominated spectra of elliptical galaxies. A possible loophole is the
following. Equation (6.16) depends on the value of $x$ for stars near turnoff,
while the infrared spectra depend on the ratio of giants to dwarfs of types K
and M; if the IMF were to turn over sharply between ${\sim}0.5$ and $1\>\rm
M_{\odot}$ (i.e., having far fewer less massive stars that the turnoff slope
would predict), one could have both a steep slope at turnoff and a very small
contribution from dwarfs to the infrared light. This idea is, of course,
completely ad hoc, since the IMF in the Solar neighborhood has $x<2$ for all
masses $<10\>\rm M_{\odot}$, and does not cut off above ${\sim}0.2\>\rm
M_{\odot}$. It is therefore most reasonable to conclude that elliptical
galaxies have giant-dominated spectra because the IMF has a fairly shallow
slope at turnoff; if so, their luminosity evolves fast enough to make the
apparent value of $q_{0}$ exceed its true value by 1 or more.

However, this is not all we need to know to unravel the Hubble diagram. The
galaxies used for this test are the central cluster giants, which are believed
to grow secularly by cannibalizing their neighbors (Section 5.3.3). This
process could plausibly lead to a growth rate in the total stellar population
of several percent per Gyr, with a corresponding increase of luminosity in
opposition to the effect just discussed. The dynamical effects cannot yet be
calculated accurately enough for a correction to be applied to the Hubble
diagram, so this test does not yet give a usefully accurate value of $q_{0}$.
The situation is reviewed by Tinsley (1977b).

####  6.2.7 Evolution of colors

Predictions of color evolution are of interest because they can be tested by
observations of distant elliptical galaxies whose ages are several Gyr younger
than nearby galaxies.

The colors of a single-generation population become redder with age, if the
main course of stellar evolution is peeling off the MS at turnoff and
following the red giant branch. The main contribution to the color change is
the redward evolution of the turnoff, since giant evolution is insensitive to
turnoff mass in the range of interest. Consequently, colors evolve faster if
the light is less giant-dominated, i.e. if $x$ is larger, in contrast to the
integrated luminosities just discussed. This behavior is illustrated in Figure
14.

Qualitatively different behavior is predicted if the stars can lose enough
mass to become blue horizontal-branch stars, instead of the red “clump” giants
that normally represent the core helium burning stage of metal-rich low-mass
stars (Section 2.4.1). It has been suggested that such stars lose mass at a
variety of rates, some becoming late red giants and others becoming blue.
Numerical models for galaxy populations in which mass loss occurs
stochastically on the red giant branch have been studied by Ciardullo &
Demarque (1978). Because evolution to a blue position in the HR diagram occurs
only if the star has a small mass of envelope left, the fraction of giants
becoming blue increases as the turnoff mass decreases. The upshot is that the
integrated colors of the model galaxies evolve blueward after ${\sim}8\>\rm
Gyr$.

Observations of distant elliptical galaxies are ambiguous on this point, as
reviewed by Spinrad (1977). Some of the most distant central cluster galaxies
known, with redshifts ${\sim}0.6$, have intrinsic colors that are bluer than
those of nearby ellipticals, but the distant galaxies were selected on the
basis of strong radio emission so they may be atypical. If they are typical,
the color change is about that expected according to the type of models that
evolve monotonically toward redder colors (e.g. Figure 14); the lookback time
sampled is ${\sim}4-7\>\rm Gyr$, depending on the cosmological model. Another
sample of central cluster galaxies with redshifts up to nearly $0.5$ has no
systematic dependence of color on redshift that can be disentangled from the
intrinsic scatter (Wilkinson & Oke, 1978).

Dramatic color differences between nearby and distant galaxy populations in
clusters have been discovered by Butcher & Oemler (1978a, b). Nearby clusters,
i.e. those with lookback times $<1\>\rm Gyr$, have galaxy populations that are
strongly correlated with the cluster morphology: loose, irregular clusters
have a large fraction of spiral galaxies, and centrally concentrated, regular
clusters have very few spirals and mainly S0 and elliptical galaxies; the
brighter galaxies in regular clusters are correspondingly all red. However, in
two regular clusters with lookback times ${\sim}5\>\rm Gyr$, the bright
galaxies are found to have a wide range of colors, including many that are as
blue as late-type spiral galaxies. On the assumption that the distant regular
clusters represent younger versions of the nearby ones, these very blue
galaxies must evolve (in a few Gyr) into red S0s or ellipticals. The color
change observed is many times greater than any predictions based on the
evolution of single-generation populations, so it is concluded that those
galaxies were actively forming stars just a few Gyr ago. Presumably, they are
mainly the precursors of S0 galaxies seen in nearby clusters, in which star
formation is undetectable.

##  7 Colors and Star Formation Rates

The stellar populations in most galaxies are far more complicated than those
in ellipticals, because young stars are important contributors to the light.
The time-dependence of the SFR is therefore an important parameter in addition
to the three quantities (age, IMF, and metallicity) used to characterize old
populations, and the latter quantities could also be changing in time.
Moreover, the colors of spiral and irregular galaxies are often affected by
internal reddening and gaseous emission lines. Despite the complications
presented by these galaxies, it is especially interesting to try to understand
their photometric properties in terms of histories of star formation.
Applications of such studies include explaining correlations between form and
photometric properties, finding what physical conditions are conducive to star
formation, and searching for young galaxies.

Models for galaxies with ongoing star formation are usually numerical;
analytical approximations are cumbersome except in the simplest case of a
constant SFR (Tinsley, 1973). This Section will consider models that study
only a few simple properties, mainly just UBV colors. Although more can be
learned from spectroscopic details, the UBV system has the advantage of an
extensive and homogeneous compilation of galaxy colors in the Second Reference
Catalogue of Bright Galaxies (de Vaucouleurs et al., 1976; to be referred to
as RC2).

###  7.1 UBV Colors of Normal Galaxies

The UBV colors of a sample of morphologically normal galaxies are shown in
Figure 15; the crosses are all elliptical and S0 galaxies, and the dots are a
variety of morphological types, which we consider first. The colors of these
galaxies form such a narrow distribution in the two-color diagram that it is
tempting to look for one dominant parameter that could vary among galaxies and
lead to a one-dimensional set of colors. Because the appearance and spectra of
galaxies suggest a progression of star-forming activity, ranging from very
active in late-type irregulars to negligible in ellipticals, it is natural to
suggest that the color sequence is due to different proportions of young and
old stars. Population syntheses and evolutionary models have confirmed this
view, and their conclusions can be summarized (with some oversimplification)
in a “standard scenario” for galaxy evolution: normal galaxies have the same
IMF and mean stellar metallicities, and they are of the same age, but they
differ in the time-dependence of their SFRs; in particular, the latest
(bluest) types of galaxies form stars on a long timescale, while the earliest
(reddest) ceased star formation long ago. This hypothesis is obviously
inaccurate in detail, but it provides a useful starting point. It will be used
to construct a series of “standard” galaxy models, whose colors will be
compared with observations, and then the effects of factors other than the SFR
will be considered in turn.

####  7.1.1 “Standard” models

Figure 15: Two-color diagram for morphologically normal galaxies and globular
clusters. _Filled circles_ : galaxies from the _Hubble Atlas_ (Sandage,
1961a), excluding peculiars and those with galactic latitudes
$|b|<20^{\circ}$, with corrected colors from the RC2; the _error cross_ is for
this sample, and the _solid line_ is its mean locus estimated by eye (Larson &
Tinsley, 1978). _Crosses_ : E and S0 galaxies in the Virgo cluster, with
colors from Sandage (1972), corrected for reddening according to the RC2
formulae. _Open circles_ : galactic globular clusters (excluding those with
$E_{B-V}>0.05$), with colors from Harris & Racine (1979), corrected for
reddening according to the RC2 formulae. Figure 16: Theoretical two-color
diagram for galaxies with monotonic SFRs, Solar metallicity, and the local IMF
(Section 7.1). _Heavy line_ : “standard” models, i.e. those of age $10\>\rm
Gyr$, with SFRs ranging from constant at the top to a single initial burst at
the bottom. _Light solid lines_ : models differing from the standard set only
in age, as indicated. _Dashes_ : models differing from the standard set only
in having an IMF with a constant slope, as marked. _Dash-dot line_ : models
differing from the preceding set with $x=1$ only in having an upper stellar
mass limit $m_{\rm U}=10\>\rm M_{\odot}$, whereas all other models shown have
$m_{\rm U}=30\>\rm M_{\odot}$. _Arrows_ : approximate estimates of the effect
on colors of blue and red galaxies, respectively, of altering the metallicity
by the factor indicated.

The methods discussed in Section 6.1.2 have been used by various authors to
construct models corresponding to the standard scenario; the (typical) results
shown here are from Larson & Tinsley (1978). Let us consider models with the
IMF of the Solar neighborhood, Solar (or old-disk) metallicity, and an age of
$10\>\rm Gyr$; the exact choice of these standard parameters is not critical,
as shown below. The models have monotonically decreasing SFRs ($\psi$),
ranging from constant to a single initial burst lasting $10^{7}\>\rm yr$.
Different series of models have different monotonic functions $\psi(t)$
between these extremes, such as exponential functions, negative powers of
time, and combinations of the constant and single-burst models as two
components in different proportions. The colors of these series in the UBV
diagram all lie very near the locus indicated by a single heavy line in Figure
16: the model with a constant SFR is at the top of this line, that with an
initial burst is at the bottom, and the form of the curve in between is
essentially the same for all series with various functional forms for
$\psi(t)$. The theoretical locus for these standard models is very close to
the observed mean locus for galaxies of different types (line in Figure 15),
so the standard scenario is at least superficially consistent. Two conclusions
can be stated.

  1. 1.

_The UBV colors of normal galaxies can in general be accounted for by models
with the same age, metallicity, and IMF_ ; most of the observed colors lie in
the range predicted for monotonically decreasing SFRs, within the
observational errors. A further conclusion is that late-type galaxies are not
necessarily young, even though their appearance and blue-light spectra are
dominated by short-lived OB stars; the integrated colors are instead
consistent with an underlying population of stars with ages up to many
billions of years. These conclusions have been stressed in the context of
evolutionary models by Tinsley (1968), Searle et al. (1973), Larson & Tinsley
(1974, 1978), and Huchra (1977). Caveats and deviations from the norm are
discussed below.

  2. 2.

_Models with monotonically declining SFRs (and with the same age, metallicity,
and IMF) define a one-parameter sequence in the ( $U-B$, $B-V$) plane_.
Inspection of the models shows that the parameter is the ratio of the present
SFR to its average value over the lifetime of the system; or equivalently the
SFR per unit mass of stars ($\psi_{1}/M_{\rm s}$ or
$\psi_{1}/\overline{\psi}_{1}t_{1}$); or equivalently the inverse of these
quantities, a _timescale for star formation_
$T_{1}\equiv\overline{\psi}_{1}t_{1}/\psi_{1}$, in the notation of Section
2.2. Galaxies of the latest morphological types are the bluest objects in
Figure 15, and these evidently have the longest timescales for star formation,
while the earliest types, which are the reddest, have the shortest timescales;
this point will be discussed further in Section 7.1.6.

The one-parameter sequence shows that UBV colors for a given value of $T_{1}$
are almost independent of the functional form of $\psi(t)$, as long as it is
monotonic. An unfortunate consequence of this result is that the UBV colors of
a galaxy (if near the mean locus in Figure 15) cannot give any more
information about the SFR than the quantity $T_{1}$. In practice, they give
less information, because of ambiguities due to possible variations of
metallicity, etc., as shown below.

The sensitivity of colors to the single parameter $T_{1}$ is due to the
dominance of low-mass giants and/or young OB stars in the light of galaxies,
as discussed in Section 6.1.2. In effect, the contribution of low-mass giants
is proportional to the number of long-lived stars ever formed, and the
contribution of upper-main-sequence stars is proportional to the present SFR,
so their ratio is proportional to $T_{1}$. The integrated colors are
insensitive to details of the past SFR because A – F dwarfs with intermediate
lifetimes contribute relatively little light, and because the nature of the
giant branch changes little over a wide range of MS lifetimes for the
precursor stars. For the same reasons, it is difficult to extract
significantly more information about the history of star formation in a galaxy
from more detailed photometry than from UBV colors.

We next consider some possible problems with the simple one-parameter
scenario.

####  7.1.2 Possible effects of errors

Three systematic discrepancies between the models and data can be seen on
comparing Figures 16 and 15: the heavy theoretical line lies about $0.05\>\rm
mag$ about the empirical mean locus, some galaxies are bluer than the bluest
model, and some are redder than the reddest model. The systematic offset is no
more than could be due to errors in the stellar evolution tracks, judged from
series of models based on alternative tracks. If this offset is corrected ad
hoc by moving the heavy theoretical line downward, there are still some bluer
and redder galaxies than predicted. The differences seem to be too big to
ascribe to uncertainties in the stellar evolution used, and they cannot be
corrected by redefining the “standard” age or metallicity, since an
improvement at the red end would leave more discrepant galaxies at the blue
end, and vice versa. Nor can the “standard” IMF be changed, since if the IMF
is universal it must be the same as the local function. Therefore, not all of
the discrepancies between the heavy line and the data are due to theoretical
errors within the framework of the standard scenario.

Although the mean error bar shown for the data in Figure 15 is small, some of
the colors may have significantly larger errors due to uncertainties in the
reduction. The colors plotted were corrected in the RC2 on a statistical basis
for Galactic and internal reddening, so excessively red and blue galaxies
could result from inappropriate corrections in a few cases. A reddening vector
(from the RC2) is shown in Figure 15, and it indicates that galaxies away from
the ends of the distribution could not be moved far from the mean locus except
by extremely large over- or underestimates of their reddening, because the
vector happens to lie almost parallel to the mean locus itself.

Emission lines can affect the colors of late-type galaxies, but the estimates
made by Huchra (1977) indicate that morphologically normal galaxies are
unlikely to have strong enough gaseous emission for this to be important.

In summary, it seems likely that some normal galaxies have colors that are too
red or too blue to be accounted for by the standard scenario. Two questions
arise: How can the discrepant galaxies be accounted for? And could normal
galaxies have significant variations in age, IMF, or metallicity that do not
show up on the UBV plot?

####  7.1.3 Variations in age

Light lines in Figure 16 indicate the effects of allowing ages between 5 and
$20\>\rm Gyr$. The loci for different ages overlap, so most of the galaxies in
Figure 15 could have any ages in this range. The extreme colors, however, do
depend on age, and the bluest and reddest data points could be accounted for
if the ages of galaxies vary by a factor ${\sim}4$. We shall see that this is
not the only possible explanation of those data points, since metallicity
effects are probably important.

####  7.1.4 Variations in metallicity

Effects of different stellar metallicities ($Z_{\rm s}$) can be estimated as
outlined in Section 6.1.2, and some approximate results are indicated in
Figure 16; the slope of the vector for red galaxies is empirical, but the
slope for blue galaxies and the length of each vector are uncertain by factors
${\sim}2$.

As discussed in Section 5.1, the sequence of colors for E–S0 galaxies (crosses
in Figure 15) is regarded as one of metallicity. This sequence closely
overlaps the locus of galaxies with different SFRs and ages, so UBV colors
alone cannot unambiguously give the SFR parameter ($T_{1}$), age, and $Z_{\rm
s}$ for a population of stars. The reddest points in Figure 15 are giant
elliptical galaxies, which almost certainly have a mean $Z_{\rm s}$ greater
than Solar (in the aperture used for the colors); if the galaxies have some
residual star formation, it is undetected to date, but it could affect the
colors enough to change the estimated $Z_{\rm s}$ and/or age somewhat.

The bluest points in Figure 15 are small late-type galaxies. It is known from
studies of the gaseous emission lines in some such galaxies that they can be
significantly metal-poor (e.g. a factor of 4 in the Small Magellanic Cloud;
Pagel et al., 1978), so a low $Z_{\rm s}$ could help to make these points very
blue. The effects of abundance changes on the colors of blue galaxies are too
uncertain to say whether another effect, such as a somewhat younger age, is
also required to account for their being bluer than the standard sequence.

####  7.1.5 Variations in the IMF

To show possible effects of variations in the IMF, Figure 16 includes the loci
of UBV colors of models differing from the standard sequence only in their
IMF. Two of the variants have constant slopes, $x=1$ and $x=2$, while the
third has $x=1$ and an upper limit $m_{\rm U}=10\>\rm M_{\odot}$ compared to
$30\>\rm M_{\odot}$ in all other cases111111The models in Figures 16 and 18
use an IMF slightly different from Equation (2.9): the upper limit is $30\>\rm
M_{\odot}$ (except as stated), and the slope is $x=1.3$ ($\phi\propto
m^{-2.3}$) for all $m>2\>\rm M_{\odot}$. The UBV colors would be little
affected if Equation (2.9) itself, with $m_{\rm U}$ taking any value $\geq
30\>\rm M_{\odot}$, were used (cf. Huchra, 1977).. For blue galaxies, the
local IMF gives colors between those for $x=1$ and $x=2$. In general, the
colors are redder with a larger value of $x$ or a smaller value of $m_{\rm
U}$, since there are relatively few upper-MS stars. Comparisons with Figure 15
show that the variants illustrated are about the largest deviations from the
local IMF that one could have without predicting a greater color spread than
is observed.

This conclusion applies only to the bluer galaxies, and only to stars $\gtrsim
1\>\rm M_{\odot}$ that contribute significantly to their light. It is clear
from Figure 16 that the UBV colors of redder galaxies ($B-V\gtrsim 0.8$) are
very insensitive to the variations of IMF considered. Additional information
on the IMF, derived from spectroscopic studies and $M/L$, was discussed in
Section 2.2.2. Possible departures from the local IMF discussed there are a
lack of stars above $10\>\rm M_{\odot}$ in some early-type spirals, and
ubiquitous variations in the fraction of very low-mass objects. In elliptical
galaxies, the IMF between ${\sim}0.4$ and $1\>\rm M_{\odot}$ cannot be very
much steeper than the local function (Section 6.2.3).

####  7.1.6 Relevance to the formation and structure of normal galaxies

To summarize the preceding discussion, every aspect of the standard scenario
has been shown to have its weaknesses: the metallicity, IMF, and age are known
to vary from one galaxy to another and/or could vary significantly without
affecting the locus of normal UBV colors. Nevertheless, it is true that the
main parameter causing the progression of colors of morphologically normal
galaxies is the timescale for star formation. The average UBV colors of
galaxies of different Hubble types lie along the middle of the distribution in
Figure 15, with the latest types at the top and the earliest at the bottom (de
Vaucouleurs, 1977). Thus _there is a strong correlation between the structure
of galaxies and their timescales for star formation_. Star formation seems to
be most efficient in the galaxies with the highest bulge-to-disk ratios, and,
among the spirals, it is most efficient in those with the tightly wound spiral
arms.

An obvious question is whether the shape of a galaxy is a consequence of its
efficiency of star formation, or whether, conversely, the timescale for star
formation is determined by the structure. Both effects are believed to be
present. On one hand, more efficient early star formation leads to a galaxy
with a greater bulge-to-disk ratio (Section 1.2): the formation of a
spheroidal component requires that the stars in this component formed on a
timescale less than that for gaseous dissipation in the protogalaxy. On the
other hand, the present structure of a galaxy governs its large-scale
dynamics, which in turn has important effects on star formation (Section
2.3.3); for example, a more prominent bulge implies a greater surface density,
which can lead to stronger gas compression and so to more efficient star
formation in the disk. The papers cited in Section 1.2 and Section 2.3.3 give
many more detailed discussions, and further references, on the origin of the
Hubble sequence of galaxy types and the numerous properties of galaxies that
correlate with their forms.

S0 galaxies, which are disk galaxies without spiral structure, are an
inscrutable class. Like elliptical galaxies, they have colors consistent with
no ongoing star formation (or possibly a little, showing at short wavelengths;
Section 6.1.1). There are currently two types of theory on the origin of S0
galaxies: in the first, they are former spirals that have no more star
formation because their ISM was lost, either in a collision with another
galaxy or by ram-pressure sweeping due to motion through an ambient IGM; in
the second type of theory, S0 galaxies had intrinsically very efficient star
formation in their disks at early stages. The second type of theory is
preferred by some authors because S0 galaxies in isolation and in dense
clusters have essentially the same low contents of neutral hydrogen (Faber &
Gallagher, 1976) and the same distributions of colors (Sandage & Visvanathan,
1978), and because there are some structural differences between S0s and
spirals (Burstein, 1978). Nevertheless, the first picture is supported
circumstantially by the high proportion of S0 galaxies in clusters, especially
in their central regions, and especially in clusters with hot IGM (Oemler,
1974; Melnick & Sargent, 1977). Arguments based on colors are not decisive,
because models in which star formation stopped long ago have similar present
colors to those in which it stopped only a few Gyr ago (Biermann & Tinsley,
1975). Whatever mechanism cuts off star formation, the very blue galaxy
content of distant, regular clusters strongly suggests that many S0 galaxies
were actively making stars only ${\sim}4\>\rm Gyr$ ago (Section 6.2.7). It is
especially interesting that some nearby clusters contain “anemic” spirals,
with weak spiral structure and a subnormal neutral hydrogen content, that have
been interpreted as disk systems at a stage of evolution between normal
spirals and stripped S0s (van den Bergh, 1976b).

###  7.2 Colors of Peculiar Galaxies

Galaxies with morphological peculiarities have peculiar colors too, as
illustrated in Figure 17, which is a UBV diagram for systems in the _Atlas of
Peculiar Galaxies_ (Arp, 1966). The width of their color distribution is in
striking contrast to the narrow locus of normal galaxies (Figure 15), and on
closer inspection the width turns out to be due almost entirely to interacting
galaxies, which are shown as crosses in Figure 17 (Larson & Tinsley, 1978).
The implication is that dynamical disturbances have led to an unusual star
formation history, so a study of these galaxies and their colors might shed
some light on the process of star formation in general.

Figure 17: Two-color diagram for galaxies in the _Atlas of Peculiar Galaxies_
(Arp, 1966), excluding those with galactic latitudes $|b|<20^{\circ}$, with
corrected colors from the RC2 and other sources cited by Larson & Tinsley
(1978). _Crosses_ denote interacting systems. _Open circles_ are two Type I
Seyfert galaxies, whose colors may be affected by non-thermal emission.

####  7.2.1 Bursts of star formation and blue colors

If the SFR in a galaxy does not decrease monotonically, colors very different
from those of the standard models in Section 7.1 can be obtained. The idea of
star formation in “flashes” or “bursts” was introduced by Searle & Sargent
(1972) (see also Searle et al., 1973) to explain the very blue colors of some
dwarf irregular galaxies, and it has appeared in other contexts including
elliptical galaxies with patches of star formation (van den Bergh, 1975) and
the peculiar galaxies discussed here.

The effects of a burst of star formation on a formerly red galaxy are
illustrated in Figure 18, where the heavy curve is the locus of standard
models aged $10\>\rm Gyr$, from Figure 16. The colors of younger galaxies are
shown in two extreme cases: the dotted line is a model with a constant SFR,
evolving through the ages shown (in Gyr), and the heavy dashed line (on the
left) is a model whose star formation stopped at $10^{7}\>\rm yr$. The latter
curve can be regarded as the evolution of a cluster of stars formed in a
period of $10^{7}\>\rm yr$, or equivalently as the colors resulting from stars
formed in a burst lasting $10^{7}\>\rm yr$. The light solid lines are the loci
of models made of two components:

  1. 1.

a red galaxy aged $10\>\rm Gyr$, with no ongoing star formation; and

  2. 2.

stars formed in a burst of duration $10^{7}\>\rm yr$, seen at ages
$10^{7}\>\rm yr$ (upper line) and $10^{8}\>\rm yr$ (lower light solid line).

Figure 18: Theoretical two-color diagram showing the colors of young galaxies
(_dotted and heavy dashed lines_) and an old red galaxy with bursts of star
formation of various strengths and ages (_light solid and dashed lines_).
Details of these lines and their labels are explained in Section 7.2.1. The
_heavy solid line_ is the locus of standard old models, from Figure 16.

The numbers along the upper curve are a burst strength parameter, defined as
the mass ratio of stars formed in the burst to stars in the old red galaxy.
Finally, the light dashed lines represent the evolution of a composite system,
from age $10^{7}\>\rm yr$ when the star formation in the burst stops; these
are lines of constant burst strength, and they cross the lower light solid
line when the age is $10^{8}\>\rm yr$. (The heavy dashed line is the limiting
case of a burst of infinite strength, i.e., without any underlying old stars).
It can be seen that a burst of star formation in a red galaxy gives colors
initially above the normal UBV locus, since the young stars cause an
“ultraviolet excess”; as the burst ages, the colors evolve across the normal
locus after nearly $10^{8}\>\rm yr$; then they fall below the normal line and
eventually become imperceptibly different from the colors of an undisturbed
old galaxy.

Evidently the colors of peculiar galaxies in Figure 17 can be explained by
bursts of star formation of various strength and ages in galaxies with various
initial colors. Using a series of models like those in Figure 18, Larson &
Tinsley (1978) found that most of the Arp (1966) galaxies can be accounted for
with bursts of strength less than $5\%$ and duration $<2\times 10^{7}\>\rm
yr$; a few more deviant colors are probably due to observational scatter,
internal reddening, non-thermal continuum emission (in two Type I Seyfert
galaxies in the sample), and strong gaseous emission lines, whose effects on
colors are discussed by Huchra (1977). The strongest bursts of star formation
are inferred for galaxies in distorted close pairs, often with bridges or
tails, and in apparently single systems with tails and filamentary streamers.
Dynamical models for colliding galaxies predict features like these in cases
of strong tidal deformation and recent mergers (Toomre, 1977), so it appears
that violent dynamical interactions lead to star formation. For this reason,
it has been suggested that the stars in elliptical galaxies could have formed
in bursts due to collisions and mergers among protogalactic subsystems
(Section 5.3.2).

Figure 18 shows that the colors of models with bursts of strength $10\%$ and
infinity are very similar. The differences are less than the observational
uncertainties for many faint peculiar galaxies, and the theoretical
uncertainties in stellar evolution, metallicity effects, etc. In other words,
it is not possible to tell from UBV colors alone whether a galaxy is really
young or has $90\%$ of its mass in old stars! Some of the galaxies near the
upper left in Figure 17 are very chaotic in appearance, and are tempting
candidates for truly young galaxies, but in all cases the colors are
inconclusive and the appearance could be due to a violent collision or
irregularly distributed star formation in an old galaxy.

Colors at longer wavelengths, such as $V-K$, are much more sensitive than
$B-V$ to the presence of some old stars that would distinguish between a truly
young system and one with a burst strength ${\sim}10\%$ (Struck-Marcell &
Tinsley, 1978). Galaxies with very active star formation are often dusty, so
red colors could be due to reddening rather than to age; on the other hand,
very blue values of $V-K$ would indicate a lack of red giants, and much
stronger limits could be put on the mass of any underlying old component. As
yet, there are no known nearby galaxies in which the dominant young stellar
component could not be masking a significant mass of old stars.

####  7.2.2 Highly reddened galaxies

Some regions of galaxies that are suspected of having intense star formation
are extremely dusty, and they show thermal infrared (IR) emission that is
interpreted as re-radiation of starlight by the dust. Example of such regions
include the centers of M82 and NGC 253, and the dust band around NGC 5128
(e.g. Kleinmann, 1977; Telesco, 1978). Star formation is indicated by early-
type spectra and blue colors in unobscured patches (e.g. van den Bergh, 1971,
1978), emission from interstellar molecules (e.g. Whiteoak, 1978), and the
lack of more plausible explanations for the IR emission (e.g. Kleinmann,
1977). In NGC 5128, the dust band has an IR luminosity of a few times
$10^{10}\>\rm L_{\odot}$ (Telesco, 1978), which rivals the visual luminosity
of the entire elliptical galaxy.

If the IR luminosity is assumed to represent the bolometric luminosity of
buried stars, an SFR can be estimated (Struck-Marcell & Tinsley, 1978): models
like those of Section 7.1 show that any system with a mass-to-luminosity ratio
$M_{\rm s}/L_{\rm bol}<0.5$ is so dominated by young stars that $L_{\rm bol}$
is almost directly proportional to the SFR; with the local IMF, the relation
is

$\psi\simeq(0.1-0.4)\frac{L_{\rm bol}}{\rm L_{\odot}}\>\rm M_{\odot}\
Gyr^{-1}.$ (7.1)

An upper limit to the time for which star formation could have continued at
this rate is approximately $M_{\rm s}/\psi$, so the limiting timescale
$\tau_{\rm s}$ depends only on $M_{\rm s}/L_{\rm bol}$, according to the
relation

$\tau_{\rm s}\equiv\frac{M_{\rm s}}{\psi}\sim(3-10)\frac{M_{\rm s}/L_{\rm
bol}}{\rm M_{\odot}/L_{\odot}}\>\rm Gyr.$ (7.2)

Some galactic nuclei have such strong IR emission that $M_{\rm s}/L_{\rm bol}$
is only a few hundredths, so the timescale is only a few times $10^{8}\>\rm
yr$. The dust-band region of NGC 5128 is also making stars at a prodigious
rate: given a luminosity ${\sim}10^{10}\>\rm L_{\odot}$, Equation (7.1) leads
to an SFR of about $2\>\rm M_{\odot}\ yr^{-1}$, so a respectable disk of stars
could be built in just a few times the dynamical timescale of the system. van
den Bergh (1975) has suggested that NGC 5128 could evolve into an early-type
spiral seen edge-on, like the Sombrero galaxy M104.

Since most of the bolometric light comes from massive stars, the above ratios
of SFR to $L_{\rm bol}$ could be overestimates if low-mass stars are not
forming. For example, in the models discussed, stars above $10\>\rm M_{\odot}$
contribute $90\%$ of $L_{\rm bol}$, but stars below $1\>\rm M_{\odot}$ account
for $80\%$ of the mass formed into stars. There is no reason to suspect a lack
of low-mass stars, however, especially since low-mass protostars (T Tauri
stars) are associated with dark clouds in the Milky Way.

These very dusty galaxies with intense star formation lead again to the
question of what truly young galaxies would look like. In the absence of dust
they would be very blue, like the young models in Figure 18, but it seems that
the regions of galaxies with the most intense bursts are the very reddened
ones just described. This question is important in the context of primeval
galaxies at large redshifts, i.e. the early stages of most normal galaxies
that now have ages ${\sim}10\>\rm Gyr$. Starting with the ideas of Partridge &
Peebles (1967), most models for primeval galaxies have assumed that hot young
stars would be visible and lead to detectable luminosities (despite the great
distances) during a brilliant early burst of star formation (e.g. Meier, 1976;
Kaufman & Thuan, 1977; Sunyaev et al., 1978). A model for a very dusty
primeval galaxy has been studied by Kaufman (1976), and Sunyaev et al. (1978)
considered the possibility of substantial radiation from dust. If the very
dusty nearby galaxies are the best analogs of truly primeval systems, as
suggested by Larson (1976b), their prospects for detection at optical
wavelengths are dim. Observational searches have so far produced null results.

Another factor making primeval galaxies hard to detect could be that most
galaxies have such a long timescale for star formation that there is not a
very bright phase at an early peak SFR (Tinsley, 1978a; Tinsley & Larson,
1979). One argument is that most spiral galaxies have colors that correspond
to rather long timescales for star formation (Section 7.1), precluding a
significant early burst with a corresponding peak in luminosity. Moreover,
even elliptical galaxies could form their stars over rather long time
intervals if star formation occurs during a series of mergers among subsystems
(Section 5.3.2). According to these ideas, late-type galaxies could be fainter
in the past than they are now, and early-type galaxies could experience their
brightest evolutionary stages only a few Gyr ago. Galaxies in interestingly
early evolutionary stages have indeed already been found at moderate
redshifts: distant clusters have excess numbers of blue galaxies (Section
6.2.7; Butcher & Oemler, 1978a), and counts in the field show a large excess
of blue galaxies at faint apparent magnitudes (Kron, 1978). Further studies of
these phenomena will surely shed light on the ways in which stars and galaxies
have formed during cosmological time.

##  8 Conclusion

Returning to the outline of galactic evolution in Figure 1, one can see how
much remains to be learned before the jigsaw puzzle will be complete enough
for a clear picture to emerge. Essentially every aspect of the subject needs
further observational and theoretical study, so galactic evolution will long
be a fertile field for research.

## Acknowledgements

I am grateful to Pierre Demarque, Richard B. Larson, Curtis Struck-Marcell,
and Bruce A. Twarog for their suggestions and help in the preparation of this
review. The work was supported in part by the National Science Foundation
(Grant AST77-23566) and the Alfred P. Sloan Foundation.

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