.347H .212. The Astrophysical Journal, 212:347-359, 1977 March 1 . © 1977. The American Astronomical Society. All rights reserved. Printed in U.S.A. 1977ApJ. COMMENTS ON GALACTIC EVOLUTION AND NUCLEOCOSMOCHRONOLOGY Kem L. Hainebach and David N. Schramm Enrico Fermi Institute, University of Chicago Received 1976 June 9; revised 1976 August 23 ABSTRACT Long-lived nucleochronologies are calculated for several recently proposed models of the chemical evolution of the Galaxy. Special attention is paid to the 187Re/187Os chronometer, for which important data have recently become available. It is found that although the rate of star formation does vary with time in the different models, the quantity 0 exp (v), which is related to the effective net rate of nucleosynthesis, is constant for most of the physically plausible recent evolution models examined. This constant net rate implies that for these models the age of the Galaxy at the time the solar system formed is twice the mean age of the stable elements. This mean age can be estimated by the parameter Amax, in which case the age of the Galaxy is 2Amax + 9 max i©, where tQ is the age of the solar system (4.6 x 10 yr). The present uncertainties in A yield an age for the Galaxy of 7 to 20 x 109 years. However, this range could be significantly reduced by an accurate measurement of the half-life of 187Re and more knowledge on the effect of stellar temperatures on the 186Os/187Os neutron-capture cross section ratio. In fact, experi- ments which could be carried out in the next few years can reduce these uncertainties tremendously and enable an age determination to be made which might severely restrict cosmological models. Subject headings: nucleosynthesis — stars : abundances — stars : evolution I. INTRODUCTION: DATING THE GALAXY The dating of meteorites, lunar rocks, and terrestrial rocks by the use of radioactive clocks has become relatively straightforward, with the primary difficulties involving the accurate measurement of the abundances of the radio- nuclides. For these objects nucleosynthesis has ceased, so there are no sources (except perhaps from the decay of other radionuclides), and one need only keep track of the decay. An additional complexity is added when one tries, by radioactive dating, to date the Galaxy, or to determine the history of Galactic nucleosynthesis, since the relative abundances will be affected not only by decay but also by the time-dependent rates of production. In fact, these production terms are just the history one is trying to determine. One can try to construct a self-consistent model for the physical and chemical evolution of the Galaxy which, among other constraints, produces the observed abundances of the elements, including the radionuclides. Several such attempts by various workers are discussed in this paper. These detailed Galactic evolution models follow the abundances of the elements as they leave the interstellar gas to be incorporated into stars, and as stars produce and either retain them or return them to the gas in either a gentle or violent manner. A great difficulty in making such models is that there exists no good physical theory of star formation. It would therefore be desirable to be able to elicit from the nucleochron- ologies some information about the history of star formation and nucleosynthesis in our Galaxy which is true for a large class of reasonable models. Schramm and Wasserburg (1970, hereafter called SW) proposed that for a large class of Galactic chemical evolution models, the mean age of the elements, measured backward from the time of solar system formation, can be well approximated by a quantity Amax, which is a function only of a few measurable or otherwise known quantities belonging to a long-lived radionuclide chronometer pair, a pair consisting of two radionuclides or a radionuclide and its decay product. The quantity Amax is not a function of the model. In the models they explored they found that in addition to the mean age, Amax gives the age of the Galaxy at the time of solar system formation to a factor of 2. Tinsley (1975) has shown that certain observational data suggest that some of the assumptions which defined SW’s class of evolution models may be violated in our Galaxy, and has reexamined the usé of the mean age as determined by Amax for estimating the total age of the Galaxy. The purpose of this paper is to calculate the age of the Galaxy using several nucleochronometer pairs, for Tinsley’s (1975) “standard model” as well as other recent proposed models of Galactic chemical evolution, and to compare the model ages so obtained with the model-independent Amax. For the limit of long-lived chronometer pairs it is found that the age of the Galaxy, T, at the time the solar system formed is 2Amax. This is found to be true of nearly all of the observationally allowed models studied, not just of those which include assumptions which require it to be true. Since these models have all been created to describe our Galaxy, this suggests that the relationship T ~ 2Amax may be a feature of our Galaxy, and thus of any model which correctly describes it. Our analyses take note of the recently measured neutron-capture cross sections of 186Os and 187Os (Browne 347 © American Astronomical Society • Provided by the NASA Astrophysics Data System .347H .212. 348 HAINEBACH AND SCHRAMM Vol. 212 . 1975; Browne and Berman 1975, 1976) and the recently recalculated neutron-capture cross section of the 187Os first excited state, likely to be populated in a stellar ^-process environment (Woosley et al 1976), which allow one to use 187Re and 187Os as a cosmochronometer (Clayton 1964). The Re/Os chronometer now tends to lean toward 1977ApJ. greater ages for the Galaxy than previously thought, for all models examined. This has been discussed in Haine- bach and Schramm (1976). The present paper will fully describe the Galactic evolution and nucleosynthesis models and will concentrate on how the two topics interact. II. GALACTIC CHEMICAL EVOLUTION: FORMALISM Schramm (1974) has reviewed nucleocosmochronology and looked at radioactive species which might be of use as chronometers. At the present time the only radionuclides which have been fully developed for use as cosmochronometers are those synthesized by the r-process {rapid neutron-capture process). The r-process chronom- eters work particularly well since they are presumed to be synthesized along with the bulk of the heavy elements, and since their abundance ratios at the time of production are not totally unknown. With the r-process nature of the chronometers in mind, let us look at the Galactic chemical evolution models. Tinsley’s (1975) equations of Galactic evolution will be used with small modifications. In particular, following Talbot (1973), the possibility that the average metallicity of newly formed stars may be greater than that of the gas and dust in the disk is allowed: ZS = FSZ, ZS = FSZ, (1) where Zs is the mass fraction of metals in stars, and Z is that in the gas; Xs is the mass fraction of a radionuclide of interest in stars, X is that in the gas; and Fs is the enhancement factor in stars, assumed to be the same for all heavy nuclides. Another modification allows matter accreted by the Galaxy to be metal-rich: Zf = FfZ, Xf = FfX, (2) where / refers to matter falling into the disk. The equations of Galactic evolution, assuming instantaneous recycling (i.e., stars which eject matter do so almost immediately after they form), are : dmtot _ ~ dt J 9 (3) where is the total mass in the disk, or in some unit of area of the disk, and/is the rate mass is falling into the disk; (4) where mg is the mass of gas in the disk, 0 is the rate mass is being converted from gas into stars, and R is the fraction of their mass which stars return to the gas (by supernovae, etc.); and ^ = —ZS0(1 - R) + yzM -R) + Zrf, (5) where mz = Zmg is the mass of metals in the gas, and yz is the mass of metals produced and ejected by stars per unit of mass retained by stars. Using equation (4), one can rewrite equation (5) in terms of the metallicity Z: f , _ iOjiA) 0(1 - *) f (v z ZJJ + Vz + z z (5a) dt mg Mg < '- >;t One also has for radionuclides: = -Am* - XS0(1 — R) + ^0(1 - R) + Xff, (6) where mx = Xm is the mass, in the gas, of a radionuclide of interest, and A is its decay rate. Writing mx = AmaN, where A = atomic weight, mH = mass of a hydrogen atom, and N = number of atoms of the radionuclide in the gas, equation (6) becomes: dN ¿(1 - -R) 0(1 — R) Xs f XfX = -\N + yx (7) dt AmH mg X ntg X\ ' © American Astronomical Society • Provided by the NASA Astrophysics Data System .347H .212. No. 2, 1977 NUCLEOCOSMOCHRONOLOGY 349 . Defining the production rate P = yx(} — R)¡AmK, and a parameter o>, which describes the movement of metals in and out of the gas for reasons other than nucleosynthesis: 1977ApJ. ^ f z f (8) mg Z nig Z and using the assumptions of equations (1) and (2) that XS¡X — Zs/Z, etc., equation (7) becomes dN = — XN + Pip - wN, dt (9) where N, t/i, and a> are, in general, functions of time. Following Tinsley, we define (10) and write the solution of equation (9): [A + v(i>1 M + Hn N(t) = í>- i* e ' Pip(t') . ai) Jo SW define an age parameter for any long-lived chronometer pair of radionuclides ij: 1 ■ In R(Ü), (12) A, where Fi/Py (13) w [^Ir- T is the time when the solar nebula was isolated from Galactic nucleosynthesis, and A is the period between T and the time solid particles began to condense.