galaxy age = 2/3 1 / H0 - 109 yr,
where the last term, which is about 5-15% the first term, depending
on the correct value of the Hubble constant H0,
represents the estimate of the time required to form galaxies.
Clearly we hope to have several methods to independently determine
the age of the Galaxy. There are in fact four reliable methods:
The ages of the oldest clusters of stars (globular
clusters).
The ages of the oldest white-dwarf stars (cooling times).
The ages of galactic solar-type stars in
conjunction with the astronomically observed concentration of
radioactive thorium in their surfaces.
Radioactivity age of the elements in the solar system.
This article will detail only the ``nuclear cosmochronologies,''
numbers 3 and 4. However, for the sake of completeness, the results at
present of the first two methods would seem to be:
The oldest globular clusters appear to be
15-17 x 109 years old, although ages outside this range
cannot strictly be excluded at the present time.
The oldest white dwarfs, which are also the faintest white
dwarfs, appear to be 9-11 x 109 years old, although ages
outside this range may not be strictly excluded. This is a
relatively new method having good future prospects.
The reader should be warned that the two ages above need not
agree. In the first place, the globular clusters are distant objects
that are probably as old as the Galaxy itself, whereas the very faint
white dwarfs can be studied only in the galactic disk, near the
position of the Sun, which could be a considerably later aspect of
galactic structure than the early phase that formed globular
clusters. In the second place, both age estimates involve formidable
observing problems for astronomers, plus formidable theoretical
problems concerning the physical structure of the stars studied.
For comparison's sake the nuclear cosmochronologies currently
seem to yield similar ages:
The inferred extent of thorium decay in the oldest
solar-type
stars in the neighborhood of the Sun appears to indicate that
they are 9-14 x 109 years old.
The radioactive dating techniques suggest that the
chemical elements in the solar system are 10-15 x 109 years old.
These last two ages also apply to the formation of the galactic disk
structure near the Sun rather than to the Galaxy itself. From even
this simple survey it can be seen that no unique age can be ascribed
to the Galaxy; but neither is there any glaring or irresolvable
conflict comparable to the controversy over the Hubble constant
itself.
RADIOACTIVITY AGES
Use of radioactivity to date events has a venerable history. It has
measured, for example, ages of terrestrial fossils, ages of
terrestrial and lunar rocks, the age of the Earth itself and of
meteorites that fall onto the Earth, and, of special interest here,
the age of the chemical elements. This last application is conceived
for a universe that began with only hydrogen and helium and that has
fused the heavier elements from them as a by-product of the
thermonuclear evolution of the interiors of stars. If the heavy
elements were synthesized within stars, it becomes sensible to ask
when that occurred.
The property of radioactivity that enables these applications is
that all radioactive decays are quantum mechanical transitions, all of
which occur probabilistically. For each radioactive nucleus there
exists a transition probability , which defines the chance that the
nucleus will radioactively decay to its daughter nucleus during the
next small time step. Unlike things of direct human experience, that
probability is independent of how long the radioactive nucleus has
already existed. That is, radioactivity as a process has no
memory. Formally, one writes that the probability for decay in the
next infinitesimal time interval dt is equal to the product dt,
independent of the age of the radioactive nucleus. But each
radioactive nucleus has a different transition probability .
A direct consequence is that if a large number N (t) of radioactive
nuclei exist at some time, the number of them that will decay in the
next interval dt is given by the product dN = -N (t) dt, where the
minus sign means a ``reduction in the number.'' From techniques of
integral calculus follows the famous law of exponential decay,
N (t) = N (t0) e- (t - t0),
where t0 is the initial time when the number was N
(t0). One says that
if there exist no sources of production of additional radioactive
nuclei, the number of each declines exponentially. The famous
concept, half-life, can be defined as the time required for an
exponentially declining population to be halved. Its value
t1/2 = ln 2 / . The value of each half-life is determined by
measurement of a population of identical radioactive nuclei. It is
intellectually stimulating that a fundamental quantum process without
memory can be used as a clock by counting the population. There
is no way of predicting when a single radioactive nucleus will decay.
The first application to nuclear cosmochronology was advanced in
1929 by Lord Rutherford (Ernest Rutherford), famous pioneer of nuclear
radioactivity. Shortly after the determination that uranium has two
naturally occurring radioactive isotopes, 235U and
238U, having
different half-lives t1/2 (235) = 0.704 Gyr and
t1/2 (238) = 4.47 Gyr
(where Gyr = 109 yr), and having the lighter shorter-lived
isotope less
abundant In the ratio 235U / 238U = 0.00725,
Rutherford suggested that
they had initially been produced in equal abundance but that time had
caused a much greater decline of the 235U population. Application of
this argument gives an age of 6 Gyr for uranium, about 1.5 Gyr older
than the Earth itself; however, the argument is based on incorrect
assumptions about the nature of the astrophysical system. Nonetheless,
it conveys the history and spirit of radioactivity ages. In a nutshell
Rutherford's observation was that radioactive nuclei could not always
have existed. He also pioneered the idea of daughter chronologies - that
the age can he calculated from the ratio of abundances of the
radioactive parent to that of the stable daughter to which it decays.
In geochronological applications it is necessary to know independently
something of the initial abundances within the object under study (a
rock, an antler, the Earth, etc.), because it is the formation time of
that object that is sought. In nuclear cosmochronology one seeks the
age of the elements themselves. The independent information needed
for that problem is knowledge of what the natural abundances would
have been were the parents stable instead of radioactive. The ratio of
the abundance of 238U, for example, to the abundance it would
have had were it stable is called the remainder, r
(238U). Values of the
remainders must be determined from nucleosynthesis theory, independent
of the chronology! This determination consists of calculating the
production rate of a radioactive nucleus relative to those of stable
nuclei. Those remainders, along with a similar determination from
nucleosynthesis theory of what the daughter abundances would have been
were their parents stable, contain the age information. The
logistical process can be summarized schematically:
(natural abundances) + (nucleosynthesis theory) -> (remainders),
(remainders) + (astrophysical model) -> (age of elements).
The second of these two equations notes explicitly that the age
calculated is model dependent. General astrophysical theory must
provide a model setting for the production of new nuclei and for their
incorporation into the matter destined eventually for our solar
system, for it is the abundances in the solar system that are the
basic data.
The useful radioactive nuclei are given in
Table 1 in decreasing
order of half-life. Also given there are the stable daughters to which
they decay and recent evaluations from nucleosynthesis theory of
relevant production ratios. Cosmoradiogenic chronologies, those based
on daughter abundances, list the remainder at solar birth of the parent.
Table 1. Useful Radioactive Nuclei
|
Nucleus
| Half-life
| Daughter
| Production Ratio
|
|
87Rb
| 48.0 Gyr
| 87Sr
| r(87) = 0.92-0.07+0.05
|
187Re
| 42.8 Gyr
| 187Os
| r(187) = 0.90-0.03+0.01
232Th
| 14.1 Gyr
| 208Pb
| p(232) / p(238) = 1.65 ± 0.15 r(232)
unmeasurable
|
238U
| 4.47 Gyr
| 206Pb
| r(238) very uncertain
|
235U
| 0.704 Gyr
| 207Pb
| p(235) / p(238) = 1.34 ± 0.2 r(235) =
0.12 ± 0.05
|
244Pu
| 82 Myr
| (232Th)
| 244Pu / 238U = 0.007
|
129I
| 16 Myr
| 129Xe
| 129I / 129Xe = 1.5 x 10-4
|
107Pd
| 6.5 Myr
| 107Ag
| 107Pd / 107Ag = 3 x 10-5
|
53Mn
| 3.7 Myr
| 53Cr
| 53Mn / 53Cr = 3 x 10-4
|
26Al
| 0.75 Myr
| 26Mg
| 26Al / 26Mg = 4 x 10-5
|
| |
The first five table entries occur naturally in the Earth and are
the useful ones for determining galactic age. The last five entries
are ``extinct radioactivities,'' detectable only by their radiogenic
daughters in meteorites, and which yield interesting constraints on
the nucleosynthesis rate shortly prior to solar formation. For the
extinct radioactivities the entry gives the abundance ratio inferred
to have existed in the early solar system. This information is more
useful for the general astrophysics of solar system formation than for
the age of the elements, for which their half-lives are too short.
Certain features of chemical evolution in the solar neighborhood
require special emphasis within the framework of cosmochronology. First,
the presolar nucleosynthesis of the elements is now believed to be
distributed in time, rather than occurring at one point in time as
Rutherford initially assumed. This requires that the presolar gas
concentration of a radioactive parent be governed by a production term
as well as die decay rate term
dN / dt = p (t) - N.
Second, these N (t) are not total interstellar abundances, which
depend on the variable and unknown total mass of interstellar gas
MG (t), but are rather concentrations within solar
matter (normally
defined per 106 Si atoms). Thus NA (t) =
XA (t) in usual astronomical
nomenclature. Third, the foregoing requires that p (t) be not
actually the galactic production rate, but technically the birthdate
spectrum of only those nuclei that will ultimately appear in the
solar system where the measurements will be made. Furthermore, it is
the birthdate spectrum of the co-produced stable nuclei, rather than
of the surviving radioactive ones. Fourth, the conceptual requirement of
a birthdate spectrum of the interstellar nuclei when (and where) the
Sun formed requires that the total galactic production rate in the
solar neighborhood be adjusted for nuclei locked up within stars and
for the interplay between the rate of growth of total mass in the solar
neighborhood to the total rate of star formation there. If the
accretion of new disk mass at the Sun's galactocentric distance
continues well after the birth of globular clusters, the mean age of
solar system nuclei may be substantially less than half the age of
the globular clusters. In other words, a rather complete description
of the growth and chemical evolution of the galactic disk is a
prerequisite for relating the mean age of solar nuclei to the actual age
of the Galaxy. It is also necessary to decide whether the solar
composition is truly typical of a well-mixed interstellar medium.
Some indications (e.g., oxygen richness) exist that suggest that the
Sun may in fact be atypical. If one assumes the interstellar gas in a
galactic annulus containing the Sun to be well mixed, that composition
is described by a set of coupled differential equations relating the
star formation rate to the mass and composition of the gaseous medium.
This is an elaborate computer program. However, if the star formation
rate is taken to be proportional to the mass of gas in the same given
galactic annulus, instructive analytic solutions of the entire problem
exist (see Additional Reading). These analytic solutions reveal the
interplay among the conventional astronomical observables of chemical
evolution, the rate of mass growth of the solar annulus, and the
nuclear cosmochronologies for solar material.
A unique method for nonsolar matter finds its natural context also
within the chemical evolution of the solar neighborhood. It stems from
the observed concentration of Th in solar-type stars (G dwarfs), which
can themselves have any age up to 15 Gyr. Astronomical challenges are
the measurement and interpretation of the Th line strength and that of
a stable nucleus (e.g., Nd) in these faint dwarfs and a calculation of
the ages of those stars from their luminosities and colors. The latter
is quite uncertain. Given those quantities, however, the method works
as follows. The Th/Nd ratio declines with time in the evolving
interstellar medium, so that later-forming G dwarfs form with smaller
initial ratios. After the stars form, however, the Th/Nd ratios in
their atmospheres decline even faster than that in the interstellar
gas, because the latter ratio is held up by fresh nucleosynthesis
whereas the Th decays exponentially in the stellar atmospheres. As a
consequence, the trend of Th/Nd in their atmospheres today versus the
ages of the stars gives the age of the galactic disk in the solar
neighborhood. This method currently suggests that age to be 9-14 Gyr;
however, major uncertainties clouding this method are the actual ages
of the dwarf stars, the line strength interpretation, and the relative
rates of nucleosynthesis of Th and the comparison element. The latter
is an important unsolved problem in the chemical evolution of the
Galaxy, which is itself a general topic that has often been underrated
in cosmochronological calculations.
RECENT RESULTS
Each method of cosmochronology continuously evolves. New observations
and new developments in the theory of nucleosynthesis continuously
improve knowledge at the same time that they illuminate the
uncertainties. At the present time this writer judges the best three
nuclear cosmochronological techniques to be the Th/Nd ratio in G dwarf
stars, the 238U / 232Th ratio in the solar system,
and the radiogenic
187Os. The status of the last two is summarized below.
238U / 232Th
The production ratio in r-process nucleosynthesis is
dominated by the relative numbers of nonfissioning radioactive parents
of these nuclei. 238U has 3.1 parents (A = 238, 242,
246, and 10% of 250), whereas 232Th has 5.9 parents (A
= 232, 236, 240, 244, 91% of
248, and 97% of 252). If the fission branches in the r-process are
identical to those in the laboratory, and if each parent had equal
intrinsic abundance, then the production ratio is p(232) /
p(238)=
1.89. Nucleosynthesis calculations, though uncertain, suggest a
slightly smaller production ratio. The relative abundances of U and Th
must be obtained from meteorites, where differing ratios are
found in differing meteoritic types. If the abundance ratio
from Type-I carbonaceous chondrites is adopted, the ratio of remainders
for these two long-lived nuclei at the time the solar system
formed was r (238) / r (232) = 0.73 ± 0.07. That is,
the fraction of 238U
nuclei surviving until the formation of the solar system was 73%
of the fraction of longer-lived 232Th that survived the same
history.
To see what this remainder ratio requires for an age for the solar
neighborhood, one must take an astrophysical model. A popular simple
model is to assume gradual conversion of initial gas to stars at a
rate proportional to the remaining gas. In that case the disk age is
10 ± 1.5 Gyr. However, if infall increased the disk mass by a factor of
2 or more over an infall epoch greater than 3 Gyr, the same remainder
ratio implies a disk age between 10 and 15 Gyr. To each model of
galactic evolution corresponds a ``best range of ages'' for the
remainder ratio. To obtain a more specific answer requires a stricter
specification of the nature of the history of the galactic disk. That
in turn is a problem for conventional astronomy, which thereby
strongly impacts what might at first glance seem to be a nuclear
technique rather than an astronomical one.
187Re / 187Os
With careful neutron cross-section measurements and the
application of s-process theory it has been possible to show that the
cosmoradiogenic portion of 187Os, defined as that part owing its
existence to 187Re decay during presolar history, is 47
± 5% of total
187Os. That argument, first advanced by this writer in 1964, places
quantitative bedrock under one of the powerful methods of determining
the time of the beginning of disk nucleosynthesis. The major
impediment to its application lies in uncertainty over the appropriate
187Re half-life. That uncertainty arises from the expectation
that the
isotope decays faster within stars than in the laboratory. The latest
study of this suggests that the half-life is reduced from 62 to 44 Gyr
by this effect (which is in itself novel). With these assumptions,
however, the simple constant-mass model of the solar neighborhood
gives a galactic age 12-2+4 Gyr, whereas models
allowing for
several-gigayear growth of the disk mass yield 17 ± 3 Gyr. Again, the
detailed age depends upon the detailed model.
Summary of Methods
Table 2 lists the cosmochronological techniques described in this
entry Both the most likely result and a range estimate are given. A
third column gives the major uncertainties that plague a precise
result from that method. Those with the smallest age spread are
clearly more reliable. That estimate of uncertainty is somewhat
subjective, representing this writer's evaluation of the scientific
literature. It will be clear that no definite answer is yet
available; but the most comfortable compromise may be a Galaxy that
began about 15 Gyr ago and a solar neighborhood disk that had matured
to the point of having most of its mass in place about 12 Gyr ago.
The future holds plenty of room for improvements and/or conflicts.
Table 2. Cosmochronological Estimates of Galactic
Age
|
Method
| Best Value (Gyr)
| Major Uncertainity
|
|
Globular clusters
| 15-17
| Evolution without mass loss?
|
Th in G dwarfs
| 9-14
| Ages of the G stars. Evolution of interstellar Th/Nd ratio.
|
Faintest white dwarfs
| 9-12
| Luminosity function of white dwarfs. Cooling rate at low
luminosity.
|
235U / 238U
| 8-18
| Production ratio. History of disk growth.
|
238U / 232Th
| 9-16
| Production ratio. History of disk growth. Th/U abundance ratio.
|
187Os / 187Re
| 11-18
| 187Re decay rate in stars. History of disk
growth. 187Os excited state.
|
87Sr / 87Rb
| 8-25
| Neutron cross section ratio. Branching in the s-process.
|
207Pb / 235U
| 7-20
| r-process production 207 < A < 235. Pb
neutron cross sections. Pb abundance.
|
207Pb / 206Pb
| 9-20
| r-process production 207 < A < 235. Pb
neutron cross sections, abundance.
|
|
Additional Reading
Butcher, H.R. (1987). Thorium in G-dwarf stars as a chronometer
for the galaxy. Nature 328 127.
Clayton, D.D. (1988). Nuclear cosmochronology within analytic
models of the chemical evolution of the solar neighborhood.
Mon. Not. R. Astron. Soc. 234 1.
Fowler, W.A. and Hoyle, F. (1960). Nuclear cosmochronology.
Ann. Phys. 10 280.
Sandage, A. (1986). The population concept, globular clusters,
subdwarfs, ages, and the collapse of the Galaxy. Ann. Rev.
Astron. Ap. 24 421.
Winget, D.E., Hansen, C.J., Liebert,J., Van Horn, H.M., Fontaine,
G., Nather, R.E., Kepler, S.O., and Lamb, D.Q. (1987). An
independent method for determining the age of the universe. Ap.
J. Lett. 315 L77.
See also Cosmology, Observational Tests; Galactic Structure,
Globular Clusters; Galaxy, Chemical Evolution; Star Clusters,
Stellar Evolution; Stars, White Dwarf, Structure and Evolution.