Donald D. Clayton
Cosmochronology may be defined as the scientific attempt to determine the age of the universe by determining the ages of some of its key components. In an expanding universe one may expect that galaxies formed some 10^{9} yr after the Big Bang. So the age of the universe can be inferred from the ages of galaxies. The age of our own Milky Way galaxy is in fact the objective of almost all methods, because the Galaxy is the only cosmologically significant object on which astronomers can obtain a large amount of detailed data. A correct model of the universal expansion requires that the age of the Galaxy fit sensibly to the inverse of the Hubble constant. So, for example, if q = 1/2, a sensible galactic age would be
where the last term, which is about 5-15% the first term, depending on the correct value of the Hubble constant H_{0}, 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 10^{9} 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 10^{9} 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 10^{9} years old.
The radioactive dating techniques suggest that the chemical elements in the solar system are 10-15 x 10^{9} 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.
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,
where t_{0} is the initial time when the number was N (t_{0}). 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 t_{1/2} = 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, ^{235}U and ^{238}U, having different half-lives t_{1/2} (235) = 0.704 Gyr and t_{1/2} (238) = 4.47 Gyr (where Gyr = 10^{9} yr), and having the lighter shorter-lived isotope less abundant In the ratio ^{235}U / ^{238}U = 0.00725, Rutherford suggested that they had initially been produced in equal abundance but that time had caused a much greater decline of the ^{235}U 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 ^{238}U, for example, to the abundance it would have had were it stable is called the remainder, r (^{238}U). 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:
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.
Nucleus | Half-life | Daughter | Production Ratio |
^{87}Rb | 48.0 Gyr | ^{87}Sr | r(87) = 0.92_{-0.07}^{+0.05} |
^{187}Re | 42.8 Gyr | ^{187}Os | r(187) = 0.90_{-0.03}^{+0.01 } |
^{232}Th | 14.1 Gyr | ^{208}Pb | p(232) / p(238) = 1.65 ± 0.15 r(232) unmeasurable |
^{238}U | 4.47 Gyr | ^{206}Pb | r(238) very uncertain |
^{235}U | 0.704 Gyr | ^{207}Pb | p(235) / p(238) = 1.34 ± 0.2 r(235) = 0.12 ± 0.05 |
^{244}Pu | 82 Myr | (^{232}Th) | ^{244}Pu / ^{238}U = 0.007 |
^{129}I | 16 Myr | ^{129}Xe | ^{129}I / ^{129}Xe = 1.5 x 10^{-4} |
^{107}Pd | 6.5 Myr | ^{107}Ag | ^{107}Pd / ^{107}Ag = 3 x 10^{-5} |
^{53}Mn | 3.7 Myr | ^{53}Cr | ^{53}Mn / ^{53}Cr = 3 x 10^{-4} |
^{26}Al | 0.75 Myr | ^{26}Mg | ^{26}Al / ^{26}Mg = 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
Second, these N (t) are not total interstellar abundances, which depend on the variable and unknown total mass of interstellar gas M_{G} (t), but are rather concentrations within solar matter (normally defined per 10^{6} Si atoms). Thus N_{A} (t) = X_{A} (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.
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 ^{238}U / ^{232}Th ratio in the solar system, and the radiogenic ^{187}Os. The status of the last two is summarized below.
^{238}U / ^{232}Th
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.
^{187}Re / ^{187}Os
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.
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. |
^{235}U / ^{238}U | 8-18 | Production ratio. History of disk growth. |
^{238}U / ^{232}Th | 9-16 | Production ratio. History of disk growth. Th/U abundance ratio. |
^{187}Os / ^{187}Re | 11-18 | ^{187}Re decay rate in stars. History of disk growth. ^{187}Os excited state. |
^{87}Sr / ^{87}Rb | 8-25 | Neutron cross section ratio. Branching in the s-process. |
^{207}Pb / ^{235}U | 7-20 | r-process production 207 < A < 235. Pb neutron cross sections. Pb abundance. |
^{207}Pb / ^{206}Pb | 9-20 | r-process production 207 < A < 235. Pb neutron cross sections, abundance. |