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On the Galactic Evolution of $D$ and $^3He$



UMN-TH-1206/93 July 1993

arXiv:astro-ph/9310021v1 12 Oct 1993

On the Galactic Evolution of D and 3 He

Elisabeth Vangioni-Flam1 , Keith A. Olive2 , and Nikos Prantzos1
1 2

Institut d’Astrophysique de Paris, 98bis Boulevard Arago, 75014 Paris, France

School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA

Abstract The determined abundances of primordial 4 He and 7 Li provide a basis with which to test the standard model of big bang nucleosynthesis in conjunction with the other two light element isotopes D and 3 He, also produced in the big bang. Overall, consistency in the standard big bang nucleosynthesis model is best achieved for a baryon-to-photon ratio of typically 3 × 10?10 for which the primordial value of D is ?ve times greater than the present observed abundance and about three times greater than the pre-solar value. We consider various models for the chemical evolution of the Galaxy to test the feasibility for the destruction of D without the overproduction of 3 He and overall metallicity. Models which are capable of achieving this goal include ones with a star formation rate proportional to the gas mass fraction or an exponentially decreasing star formation rate. We discuss the e?ect of parameters that govern the initial mass function and of surviving fractions of 3 He in stars between one and three solar masses.

1

Introduction

As part of the foundation of our understanding of the early Universe, the need to test and scrutinize the standard model of big bang nucleosynthesis (SBBN) is essential. Fortunately, SBBN is a testable theory. It predicts the primordial abundances of the light elements D , 3 He, 4 He and 7 Li (see e.g. Walker et al. 1991). Unfortunately, one can not directly measure the primordial abundances of these elements. With a certain degree of con?dence however, the abundances of 4 He and 7 Li may be extracted from observations. In the case of 4 He, systematic observations of low metallicity, extragalactic HII regions (see e.g. Pagel et al. 1992) have led to a fairly well determined value for primordial 4 He (Olive, Steigman and Walker 1991; Pagel et al. 1992). In the case of 7 Li, the observation of a well de?ned abundance in old halo dwarf stars (Spite & Spite 1982a,b 1986; Hobbs and Duncan 1987; Rebolo, Molaro and Beckman 1988), seemingly independent of temperature (for > T ? 5500 K) and metallicity (for [F e/H] < ?1.3) is generally regarded as a good indicator for the primordial abundance. Though there remains the problem of lithium depletion in stars, standard stellar models (see e.g. Deliyannis, Demarque, & Kawaler 1990) support the connection between the observed abundances in halo dwarfs and the primordial value. The case for D and 3 He is somewhat more complicated. Deuterium is destroyed in stars. Despite the fact that all observed deuterium is primordial, the primordial abundance cannot be determined from observations. Nevertheless a useful upper limit to the baryon-to-photon ratio, η, is established from the lower limit to the deuterium abundance (Gott et al. 1974). A comparison between the predictions of the SBBN model and observed solar and interstellar values of deuterium must be made in conjunction with models of galactic chemical evolution (Audouze and Tinsley 1974). The problem concerning 3 He is even more di?cult. Not only is primordial 3 He destroyed in stars but it is very likely that low mass stars are net producers of 3 He. Thus the comparison between theory and observations is complicated not only by our lack of understanding regarding chemical evolution but also by the uncertainties in production of 3 He in stars. Once again a useful lower limit to η is obtained by assuming that at least some of the 3 He in stars survives (Yang et al. 1984). SBBN has also been indirectly confronted by recent observations of Be and B (Rebolo et al. 1988; Ryan et al. 1990; Gilmore, Edvardsson & Nissen 1992; Ryan et al. 1992; Gilmore et al. 1992; 1

Duncan, Lambert & Lemke 1992; Rebolo et al. 1993). In the big bang, these elements are produced at abundance levels which are orders of magnitude below that of the observations (Thomas et al. 1993; Delbourgo-Salvador and Vangioni-Flam 1993). Though it is highly likely that that both Be and B are produced by cosmic-ray spallation (Reeves et al. 1970; Meneguzzi et al. 1971) some 7 Li is also partly produced by galactic cosmic-ray nucleosynthesis providing a potential constraint on big bang nucleosynthesis (Steigman & Walker 1992; Olive & Schramm 1992). The combination of a double source for 7 Li and the low value for the abundance in the 7 Li plateau all lead to a small value for the primordial 7 Li abundance. A higher primordial 7 Li abundance can be tolerated in conjunction with stellar models which deplete 7 Li (see e.g. Pinsonneault, Deliyannis, & Demarque 1992), however, these models are constrained (Steigman et al. 1993) by recent observations of 6 Li in a halo dwarf (Smith, Lambert & Nissen 1993). The results of SBBN with regard to constraints placed on what is perhaps the single remaining parameter of SBBN, the baryon-to-photon ratio, η, can be summarized as follows (Walker et al. 1991): Extrapolations to zero metallicity of the helium abundance in extragalactic HII regions leads to a best value for the primordial helium mass fraction (Olive et al. 1991; Pagel et al. 1992), Yp ? 0.23 ± 0.01
< η10 ? 4

(1)

where η10 = η × 1010 . Note that only the upper limit from 4 He is used and one must be aware that relaxing the upper bound to 0.245 relaxes the bound on η10 to ? 6. The best estimate for the pre-solar deuterium abundance (Geiss 1993) gives the upper bound on η (D/H)⊙ ? (2.6 ± 1.0) × 10?5
< η10 ? 7

(2)

In this case, it is only the lower bound on D/H which is useful. By assuming that at least 25% of the primordial 3 He survives stellar processing the upper limit on (D + 3 He)/H gives a lower bound to η, (D +3 He)/H < 10?4 η10 > 2.8 (3) Finally with regard to 7 Li, if we assume neither 7 Li depletion, nor a contribution from cosmic-ray nucleosynthesis (these two e?ects work in opposite directions), the mean 7 Li abundance in the plateau of halo dwarfs is
7

Li/H = (1.2 ± 0.1) × 10?10 2

2.1 < η10 < 3.4

(4)

When all of the light elements produced in the big bang are considered, consistency is achieved when η is in the range, 2.8 < η10 < 3.4 (5) In our forthcoming discussion concerning the destruction of D and 3 He, we will for the most part keep η10 ?xed at the value η10 = 3. At this value of η, the primordial values of D/H and 3 He/H are approximately, 7.5 × 10?5 and 1.5 × 10?5 respectively. < Larger values of η10 , such as η10 ? 4, are possible given the uncertainty in some of the nuclear reaction rates involving 7 Li (Walker et al. 1991). With a larger value of η10 , smaller initial values of D and 3 He are required. For example, at η10 = 4, (D/H)p = 4.7×10?5 and (3 He/H)p = 1.3×10?5 . With this value of D/H, less deuterium destruction is required, thus relaxing this constraint on chemical evolution models. As has been shown by Steigman and Tosi (1992) there are satisfactory models with higher values of η. Given the SBBN as a presumption, we plan to study the evolution of D and 3 He. With the above value of D/H, D evolution is a necessity. The abundances of D and 3 He have been reviewed recently in Geiss (1993). The pre-solar D abundance, recall, is not measured directly. Instead, a comparison is made between the 3 He abundance in carbonaceous chondrites (in the noble gas component of meteorites which are una?ected by solar deuterium burning) whose values are low and the higher 3 He abundances measured in gas-rich meteorites, the lunar soil and solar wind. The former is representative of the true pre-solar 3 He abundance, while the latter represents the sum of pre-solar (D + 3 He). Amazingly, the pre-solar abundances of these isotopes has remained quite stable. Measurements of 3 He/4 He in the solar wind with the ISEE-3 satellite (Coplan et al. 1984) show real ?uctuations in the 3 He/4 He ratio. Though the average value is the same as the older measurements, there is most certainly a greater uncertainty associated with the pre-solar (D + 3 He) value. Geiss has also increased the uncertainty in the pre-solar 3 He abundance (in carbonaceous chondrites) due to e?ects of fractionization. In what follows, we adopt Geiss’ pre-solar values for D and 3 He (uncertainties are given at the 2 σ level): ((D +3 He)/H)⊙ = (4.1 ± 1.0) × 10?5 (3 He/H)⊙ = (1.5 ± 0.3) × 10?5 (D/H)⊙ = (2.6 ± 1.0) × 10?5 3 (6) (7) (8)

Deuterium is also measured in the ISM using Lyman absorption spectra in nearby stars, thus we have a handle on the present day D abundance as well. Overall these measurements give a present D/H ratio of 0.5 to 2 × 10?5 (see e.g. Vidal-Madjar 1991, Ferlet 1992). A recent high precision measurement of D/H was made in the direction of Capella using the HST Goddard High Resolution Spectrograph (Linsky et al. 1992). The measured value (and the one we will adopt for the present day D/H ratio) is (D/H)o = 1.5+.07 × 10?5 ?.18 (9) Recent determinations (Bania, Rood & Wilson 1993) of 3 He in the ISM yield values which range from 1.1 to 4.5 × 10?5 , a domain which is still too broad to fully constrain models of chemical evolution. It seems timely to update, re?ne and generalize the analysis of the destruction of deuterium in the course of its galactic evolution since the observed abundances of D (as well as that of 4 He) have increased with respect to those considered by Vidal-Madjar and Gry (1984) and DelbourgoSalvador et al. (1985). Indeed, in the 1980’s, it seemed necessary to invoke a small surviving fraction of D to obtain a primordial abundance in agreement with the big bang prediction. For that reason, Vangioni-Flam and Audouze (1988) developed speci?c models aimed at destroying D by a factor ten or even more. This problem, however, now seems less severe and a milder destruction factor (4 to 5) is su?cient. Moreover, the recent measurements of the D abundance in the local ISM (Eq. 9) and the protosolar ratio (Eq. 8) indicate that D has decreased since the birth of the sun. We also have better limits on the present gas mass fraction, σ, which is a key parameter in galactic evolutionary models. A typical estimate of the surface density of total matter at the solar circle from dynamical arguments is 54 ± 8 M⊙ /pc2 (Kuijken and Gilmore 1991). This value combined with the gas surface density of 6 to 10 M⊙ /pc2 , (Solomon 1993) leads to an estimate of σ between 0.1 and 0.2. Finally with regard to 3 He, the situation remains vexing: observational problems persist due to the dispersion of the measured interstellar abundances; and above all, there are considerable theoretical uncertainties on the production and destruction of 3 He in low mass stars. Our goal in this paper therefore, is to explore models of chemical evolution which have the possibility of accounting for the destruction of deuterium from a primordial value of ? 7.5 × 10?5 to the pre-solar and present day values above, which in addition avoid overproducing 3 He. The same 4

question was probed recently by Steigman and Tosi (1992). Starting with a speci?c set of chemical evolution models (Tosi 1988), they constrained the degree of deuterium destruction and hence the primordial deuterium abundance and η. Here, we take a di?erent approach. Given the level of uncertainty in models of chemical evolution, which result from our lack of knowledge regarding the initial mass function (IMF) and perhaps more importantly the star formation rate (SFR), we investigate to what extent plausible senarios of the chemical history of the Galaxy can be reconciled with the factor of ? 5 of total destruction of D .

2

The Destruction of Deuterium

Our goal in this section is to determine the conditions for which deuterium may be destroyed by a factor of ? 2 ? 4 from its primordial value to its pre-solar value and by a factor of ? 5 to its present value. The destruction of deuterium in connection with models of galactic chemical evolution has been discussed somewhat extensively in the literature (Audouze & Tinsley 1974; Ostriker & Tinsley 1975; Audouze et al. 1976; Gry et al. 1984; Clayton 1985; Delbourgo-Salvador et al. 1985; Vangioni-Flam and Audouze 1988; Steigman & Tosi 1992). There is no question that the degree of deuterium destruction is model dependent. Indeed the ratio D/Dp (where Dp is the primordial abundance) depends on most aspects of a chemical evolution model, which include the IMF, the SFR (Vangioni-Flam and Audouze 1988), return fraction R (Ostriker and Tinsley 1975; Clayton 1985), the infall rate (Audouze et al. 1976; Clayton 1985), the composition of the infalling gas (Gry et al. 1984; Delbourgo-Salvador et al. 1985), and even computational approximations such as the often used instantaneous recycling approximation (see below). To calculate the abundance of deuterium as a function of time, even in a simpli?ed closed-box model with no infall, one must still specify the IMF, SFR, and return fraction. One can write down a simple analytic expression for D/Dp which involves only the gas mass fraction, σ, and R (Ostriker and Tinsley 1975) in the instantaneous recycling approximation (IRA). The gas mass evolves as dσ = ?(1 ? R)ψ(t) dt (10)

5

where ψ(t) is the SFR, and the return fraction is de?ned in terms of the IMF, φ, as R=
Msup M1

(M ? Mrem )φ(M)dM

(11)

In (11), M1 ≈ 0.85 is the present main-sequence turn-o? mass and Msup is the upper mass limit for φ. Mrem is the remnant mass: Mrem = M for M < 0.5M⊙ , = 0.45+0.11 M/M⊙ for 0.5 < M/M⊙ < 9.0 (Iben and Tutukov 1984) and = 1.5M⊙ otherwise. Correspondingly, the evolution of the deuterium mass fraction is d(XD σ) = ?XD ψ(t) (12) dt which can be combined with (10) and easily solved D = σ R/(1?R) Dp (13)

If we take σo ≈ 0.1 as a representative present-day value, then simple (power-law) IMF’s taken with Mrem from above give R ≈ 0.2 which in turn yields (D/Dp )o ? 1/2. A total deuterium destruction factor of 2-3 is common in many models (Audouze and Tinsley 1975; Steigman and Tosi 1992). However from the simple expression above (13), for models with a rapidly decreasing SFR, the resulting IMF (as determined from the present-day mass function) in general yields a larger value for R. When R ? 0.5, deuterium will be reduced by a factor of 10. Infall introduces another parameter which a?ects the deuterium abundance. The degree of deuterium destruction depends on the composition of the infalling gas. For a primordial composition, the total amount of destruction is limited and in many cases one ?nds a rise in the deuterium abundance from the pre-solar value to its present value (Gry et al. 1984; Steigman and Tosi 1992), which appears to be in contradiction with the data. In the models considered by Steigman and Tosi (1992) the net destruction (primordial to present) of deuterium was typically no larger than a factor of 2. Thus better determinations of both the pre-solar and ISM values of D/H can be a valuable tool in limiting the amount of infalling gas. If instead, the infalling gas composition is that of processed material, the D/Dp ratio can be much smaller, D/Dp ? (1/10) ? (1/40) (Gry et al. 1984; Delbourgo-Salvador et al. 1985). Because of the apparent decrease in D/H with time, and for the purpose of simplicity as well as the lack of observational evidence, we will not include infall in the subsequent discussion. We simply note that the assumption of substantial (i.e. non 6

negligible) infall with primordial composition in the disk during the last ?5 Gyr may lead to an increase in the D abundance and to large (D + 3 He)/H values and as such would be inconsistent with the observations. Depending on the speci?c model, the use of the IRA may also a?ect the degree to which D is destroyed. For example, in Fig. 1, we show the evolution of D as a function of time with and without the use of the IRA. In Fig. 1a, we have chosen a single slope IMF (see below for a more complete description of these models) and a SFR, ψ(t), which is proportional to the gas mass fraction, σ. As one can see, the e?ect of the IRA on the destruction of D is reasonably small. In Fig. 1b, we have chosen the Scalo (1986) IMF and an exponentially decreasing SFR with a time constant of 3 Gyr. This is a rather extreme case, leading to a current SFR much lower than the past average SFR (see below) and much lower than observations of the current SFR in the solar neighborhood (indicating that the SFR is ?3-5 M⊙ pc?2 Gyr?1 , for a surface gas density of ?10 M⊙ pc?2 ). We adopt it as an extreme example of a large D depletion. It can be seen from Fig. 1b that in this case the e?ect of the IRA is signi?cantly more important. It is interesting to analyze the e?ect of removing the IRA. It turns out that D destruction is correlated to the evolution of the gas fraction σ(t). In Fig. 2a we see that the evolution of σ is approximately the same with and without the IRA in the case where ψ(t) = νσ. This can be explained in the following way: since the gas evolves relatively slowly, the amount of (D poor) ejecta is small compared to σ(t) at any time. D destruction depends then little on the assumption of an instantaneous recycling. On the other hand, when ψ(t) = exp(?tGyr /3), σ declines very sharply early on and in the IRA, the amount of matter ejected becomes signi?cant with respect to σ. Indeed, in Fig. 2b one ?nds signi?cantly more gas at intermediate times (t ?a few Gyr) when the IRA is made, which is largely composed of the, instantly returned, D poor ejecta; consequently, the IRA leads to a larger D depletion at that period. At late times, however, the IRA and non-IRA give similar amounts of gas (Fig 2b); indeed, for such late times all of the stars that can return an important fraction of their mass, have enough time to do it. But the IRA stars, being created in small numbers at late times (small SFR) cannot considerably dilute the D abundance of the gas with their ejecta; on the contrary, the large number of long lived stars that were created early on in the non-IRA case, return a large (with respect to the late gas) D free amount of matter, considerably diluting the D abundance at late times. Thus, the IRA approximation in the case of 7

a rapidly decreasing SFR overestimates the D depletion at early times and underestimates it at late times. The di?erence in the behaviour of D (and 3 He) in the case of the IRA and non-IRA calculations is much more apparent when the results are plotted as a function of the gas fraction; this is done in Fig. 3, which nicely illustrates the previous discussion. The impact of the SFR on the degree of deuterium astration was studied extensively by VangioniFlam and Audouze (1988, VFA). We summarize that work here as it will serve as a basis to our present work and its general philosophy remains pertinent. In VFA, two kinds of solutions had been proposed to astrate D e?ciently: a) - A high SFR in the early galaxy, with a normal IMF( model II in VFA). In this case, the D poor gas is ejected, on average, after a long delay, due to the large number of low mass stars. The overproduction of 3 He is avoided under the reasonable assumption that about 30% of the original D + 3 He survives in the form of 3 He in stars between 1 and 3 M⊙ . This model had been subsequently discarded after it was put to the test using the G-dwarf metallicity distribution (Francois, Vangioni-Flam and Audouze 1989); b) - A modi?ed IMF favoring massive stars (model IV in VFA). In this case, the D -free gas is released almost instantaneously by massive stars, and the IMF must be adjusted to avoid excess production of 16 O and metals. It is worth mentioning that model I (in VFA) with a SFR proportional to the gas fraction destroys D by a factor of 3.3, which is not too far o? from the new required value. Perhaps the time has come to reconsider this kind of simple model. Exponentially decreasing SFRs have also been explored by Olive, Thielemann & Truran (1987, OTT) in the framework of the IRA. A constant IMF derived from the present-day mass function (PDMF) was used together with its associated SFR (Scalo 1986). Apparently, this is a good candidate because this special combination of SFR and IMF o?ers an e?cient way to lower the D abundance and a possible solution to the G-dwarf problem (Olive 1986) at the same time. There are some similarities between these models and model IV of VFA. The overproduction of O and metals is avoided at the expense of imposing a cuto? at the high mass end of the IMF. We will also consider a substitution of Scalo’s IMF by a power law one in order to try to avoid as much as possible an excess of metals produced by massive stars.

8

3

3

He Production and Destruction

Even more complicated than the history of deuterium, is that of 3 He. Not only does the abundance of 3 He, as a function of time, depend on standard galactic evolution parameters such as the IMF, SFR, etc., but also on the production of 3 He inside a star and its return to the ISM. While there is little debate that in more massive stars (M > 5 ? 8M⊙ ) 3 He is e?ciently destroyed, in low mass stars (1M⊙ < M < 2M⊙ ) 3 He is perhaps produced, some of which will be returned to the ISM. It is precisely because of the likelihood that not all of the primordial 3 He is destroyed in stars, that the measurement of pre-solar D + 3 He can be used to set a lower limit on η. It was noted in Rood, Steigman and Tinsley (1976), that by requiring that 3 He not be overproduced, a lower limit to η could be set. The limit disappears if D and 3 He destruction is complete. The argument yielding a lower limit to η based on pre-solar D + 3 He was ?rst given in Yang et al. (1984). The argument runs as follows: First, during pre-main-sequence collapse, essentially all of the primordial D is converted into 3 He. The pre-main-sequence produced and primordial 3 He will < survive in those zones of stars in which the temperature is low, T ? 7 × 106 K. In these zones 3 He may even be produced by p ? p burning. At higher temperatures, (up to 108 K), 3 He is burned to
4

He. If g3 is the fraction of 3 He that survives stellar processing, then the 3 He abundance at a time t is at least 3 D + 3 He D He ≥ g3 ? g3 (14) H t H H t p

The inequality comes about by neglecting any net production of 3 He. Of course, Eq. (14) can be rewritten as a upper limit on (D + 3 He)/H in terms of the observed pre-solar abundances (t = ⊙) and g3 . The models of Iben (1967) and Rood (1972) indicate that low mass stars, M < 2M⊙ are net producers of 3 He. For stars with mass M < 8M⊙ , Iben and Truran (1978) have estimated the ?nal surface abundance of 3 He, ( He/H)f = 1.8 × 10
3 ?4

M⊙ M

2

+ 0.7 (D + 3 He)/H

i

(15)

indicating that g3 > 0.7, notwithstanding the uncertainties involved in determining (15). For more massive stars, Dearborn, Schramm & Steigman (1986) have estimated g3 for a variety of metallicities 9

and 4 He abundances. Overall, they ?nd g3 in the range 0.1 to 0.5 for stars with M > 8M⊙ . For the purposes of obtaining a lower limit to η based on (D + 3 He)/H, it is necessary to estimate a lower limit to g3 . Without this lower limit, heavy destruction of deuterium and 3 He would allow for very low values of η (Olive et al., 1981). A troubling aspect of this argument has always been the lack of observational support. Recently however, Ostriker and Schramm (1993) have argued that on the basis of observations by Hartoog (1979) of 3 He in horizontal branch stars, a lower limit of g3 > 0.3 could be inferred. Also, the high value of 3 He/H observed in a planetary nebula by Rood, Bania & Wilson (1992) seem to support the idea that 3 He is in fact not completely destroyed and may be produced. In our subsequent calculations, we will use the values of g3 as given in Dearborn et al. (1986). However, for the mass range 1 to 3 M⊙ , which we ?nd crucial for determining the pre-solar deuterium plus 3 He abundance, we will consider several possibilities. We will refer to g3 as a set of three values corresponding to estimates of g3 at (1, 2, 3)M⊙ respectively. For comparison, Dearborn et al. (1986) used g3 = 1 for M < 3M⊙ , and g3 = 0.7 for 3 < M/M⊙ < 8. Delbourgo-Salvador et al. (1985) used g3 = 0.7 for M < 2M⊙ , g3 = 0.25 for 2 < M/M⊙ < 5, and g3 = 0 for M > 5M⊙ . For a more complete discussion on estimates of g3 we refer the reader to Yang et al. (1984) and Dearborn et al. (1986). It is important to note that the 3 He survival factors considered previously and here are all signi?cantly lower than the Iben and Truran value at 1M⊙ , g3 = (1.8 × 10?4 )/[(D + 3 He)/H]i + 0.7, which even for the relatively high value [(D + 3 He)/H]i = 9 × 10?5 , gives g3 = 2.7. As we will see, such a large survival fraction will prove to be irreconcilable with the pre-solar abundance determination. Unfortunately, we can o?er no solution as to why g3 should be lower other than the constraints imposed by the pre-solar D + 3 He data. In addition to its sensitivity to g3 , the 3 He abundance as a function of time is also quite dependent on the parameters of the galactic chemical evolution model (as for D ). In the IRA, an analogous expression to (13) was derived in Olive et al. (1990) (D + 3 He) = (D + 3 He)p D Dp
g3 ?1

(16)

so that the model dependence of D/Dp feeds into (D + 3 He). 10

From Fig. 1, we see how (D + 3 He) is a?ected by the IRA. In both cases (1a and 1b) the sum of (D + 3 He) is correlated, although not in a straightforward way, to the fate of D . In Fig. 1a there is more D with the IRA in the end than without the IRA (as explained in the previous section). Since D is not severely depleted in that case (it constitutes ?half of the (D + 3 He) amount), the ?nal (D + 3 He) abundance is also larger in the IRA case. On the other hand, in Fig. 1b, D su?ers a severe depletion early on, and its weight in the ?nal (D + 3 He) sum is small. What matters then is the ?nal 3 He amount. In the non-IRA case a lot of 3 He at late times comes from the early created long-lived stars that were enriched in D ; 3 He is then quite abundant, as is the sum (D + 3 He). In the IRA case most of 3 He comes from more recently created stars, that are D poor; its ?nal abundance is then smaller than in the IRA case, and the same is true for the sum (D + 3 He). We see then that the IRA has an opposite e?ect on the (D + 3 He) sum in those two cases. In any case, the abundance of (D + 3 He) compared to the pre-solar value (6) will turn out to be among the toughest challenges to overcome. In short, it is not the destruction of deuterium that is problematic, but rather the overproduction of 3 He.

4
4.1

Galactic Evolution Models
Constraints

The point now is clear: as shown over the last ten years, the destruction of deuterium is highly model dependent. Our objective is to reduce the SBBN D abundance by a factor ?5 over the galactic lifetime, reproducing at the same time the solar oxygen abundance and global metallicity Z, for a galactic age of ?14 Gyr and a current gas fraction 0.1 < σ < 0.2. [O/F e] vs [F e/H] and [F e/H] vs time are also consistent with observations in all of the models proposed in this study and before (e.g Francois et al. 1989, Vangioni-Flam, Prantzos & Chauveau 1993). Previous work has also shown that the adopted formalism complemented by speci?c treatment of cosmic ray spallation accounts for the evolution of Be and B as well ( Prantzos et al. 1993). In this context, the evolution of 9 Be/H vs. [F e/H] and 11 B/H vs. [F e/H] are largely independent of the IMF and SFR (Prantzos et al. 1993; Olive et al. 1993). Consequently, we ?nd similar results as before.

11

Finally as we are considering models with a time-varying SFR, it is worthwhile to note the constraints on the history of the SFR. For this purpose it is useful to de?ne the relative birthrate, b(t) = ψ(t)/ ψ where the average SFR is de?ned by ψ = 1 To
To 0

ψ(t)dt

(17)

and To is the age of the Galaxy. For a constant SFR b = 1, while for an exponentially decreasing SFR, b(To ) = (To /τ )(eTo /τ ? 1)?1 . For To = 14 Gyr and τ = 3, 5, and 10 Gyr, b(To ) = 0.044, 0.18, and 0.46 respectively. When ψ = νσ, with ν = 0.25, b(To ) = 0.33. Limits on b(To ) have been reviewed by Scalo (1986): among the strongest limits is b(To ) > 0.4 derived from stellar age distributions. Though there is a certain degree of uncertainty in this bound (see eg. Tinsley 1977, Twarog 1980), it does give us an indication that τ = 3 Gyr should be viewed as an extreme value. However, in bimodal models of star formation (Larson, 1986; VFA), where only one component (the massive end) has a rapidly decreasing SFR, the value τ = 3 may still be plausible and satisfy the age constraints.

4.2

Models and yields

The yields of Woosley (1993) are adopted in this work. They are not very di?erent from other recent works (Arnett 1991; Thieleman et al. 1993; Weaver and Woosley 1993) at least as far as oxygen is concerned (see Prantzos 1993 for a comparison and implications). The oxygen yield is determined within a factor of ? 2 due to uncertainties in the 12 C(α, γ)16 O reaction rate and in the treatment of convection in massive stars. (For a discussion see OTT, and Weaver and Woosley 1993). We performed a limited check of the impact of those yields by running a standard model of chemical evolution (closed box, power-law IMF between 0.4 and 100 M⊙ , SFR ∝ σ), the results of which appear in Table 1. The iron yield from core collapse supernovae (originating from massive stars) is unfortunately still uncertain except for the 20M⊙ case (?0.07 M⊙ of Fe is produced, after the interpretation of the light-curve of SN1987a). The adopted yields of Woosley have been adjusted to get reasonable values of [O/F e] (?0.5) in low metallicity stars, as was done in our previous work. Also uncertain is the past rate of supernovae of type Ia. We have chosen a constant SNIa rate of 0.2 per century, 12

with each ejecting 0.6 M⊙ of F e. The present ratio SNIa/SNII ? 0.1 is required to be reproduced and this simple procedure is generally su?cient to ?t the steepening of the [O/F e] vs. [F e/H] curve beyond [F e/H] ? ?1. Finally, the destruction of 3 He in low mass stars (1 ≤ M/M⊙ ≤ 3) which is poorly known (see section 3) has been treated as a free parameter within reasonable limits. Two sets of models have been selected, di?ering by their SFR. In model I, the SFR ψ(t), is assumed to be proportional to the gas mass fraction. The constant of proportionality, or astration rate is ν = 0.25 Gyr?1 . The IMF is parametrized as φ(M) ∝ M ?(1+x) with the normalization
Msup Minf

Mφ(M)dM = 1

(18)

where the slope x, and the mass limits, Minf and Msup are taken to be variable. In model Ia , a single slope IMF is considered between the limits 0.4 < M/M⊙ < 100. We have also tested these models with various choices of the 3 He survival fraction, g3 . In model Ib , the Tinsley (1980) IMF is used between 0.1 < M/M⊙ < 100. This traditional model is worth rehabilitating in the present context for D destruction. Model II, inspired by Scalo(1986), Larson (1986) and OTT, features an exponentially decreasing SFR, ψ(t) ∝ e?t/τ . In Larson’s model (a model of bimodal star formation) the steeply decreasing SFR for high mass stars leads to a large density of low-luminosity white dwarfs. The price to pay in such a model however, is a somewhat unusually low value for Msup to avoid the overproduction of metals, primarily oxygen. In Larson’s model, Msup = 16M⊙ . Using an exponentially decreasing (non-bimodal) SFR, as we have considered here, OTT derived limits on Msup as a function of τ . Arguing further that if the yields of massive stars were well understood, the abundance patterns such as [C/F e] ≈ 0, [O/F e] ≈ 0.5, and [(Ne + Mg + Si)/F e] ≈ 0.5 could place a strong limit on τ and b(t). For example the yields of Arnett (1978) typically require the presence of massive > > (M ? 40M⊙ ) stars, leading to a limit b(t) ? 0.75. However the yields of Woosley and Weaver (1986)
> require only the presence of ? 15M⊙ stars and the limit on b(t) drops to b(t) ? 0.03. In model IIa , we considered several values of τ using the Scalo (1986) IMF. In model IIb the single slope IMF was

again employed. A summary of the di?erent parameters used in the two sets of models is shown in Table 2. 13

Figure 4 shows the cumulative metallicity distribution of the models in the galactic disk phase > (for [F e/H] ? ? 0.7), compared to observations; the agreement is satisfactory, (though in model Ia it is less so) i.e. the G-dwarf problem is solved, since the disk starts with an initial metallicity enrichment from the previous (halo) phase.

5

Results

In Fig. 5a we show the evolution of D/H as a function of time for the four basic models considered. As one can see from Fig. 5a, there is a great variability in the degree of destruction of D . With the exception of model IIa,a , (chosen to be extreme), all of the models give a perfectly adequate picture for the time evolution of D . (The present value inferred from model Ia is perhaps slightly high compared to the HST measured ISM abundance.) Thus one of our primary goals is achieved: Starting with a primordial abundance D/H = 7.5 × 10?5 , we are in fact able to obtain destruction factors of 3-5 to agree with the pre-solar and present-day measurements. Models without an extremely large return fraction (R ? 0.40 in model Ia and R ? 0.54 in model Ib ) and a “reasonably smooth” SFR (∝ σ) reproduce the observations quite satisfactorily. This is also true for model IIb (also with R ? 0.40) which has a rapidly decreasing SFR (∝ exp(?tGyr /3)), and depletes D somewhat more than the other two. On the other hand, models with a steeply decreasing SFR (models IIa,a and IIa,b with R ? 0.65 and 0.52 respectively), can be “lethal” to D . This is the case of models IIa,a (SFR ∝ exp(?tGyr /3)) and IIa,b (SFR ∝ exp(?tGyr /5)),which destroy D by a factor of ?100, as can be seen in Fig. 5b. Model IIa,c (with R ? 0.42) has a slowly decreasing SFR (∝ exp(?tGyr /10)) and gives a quite acceptable ?t to the D observations (Fig. 5b again). Finally, Fig. 5c illustrates the role of the IMF in the depletion of D , for the case of a power-law IMF with x = 1.7 and a smooth SFR (= 0.25 σ), i.e. model Ia,e . When Minf = 0.1M⊙ (R ? 0.17) the depletion of D is small (less than a factor of two); it becomes compatible with the observations when Minf = 0.4M⊙ (and R ? 0.40). As noted earlier, it is more di?cult to satisfy (with any model) the constraint imposed by the pre-solar (D + 3 He)/H value. Figs. 6a, 6b and 6c show the evolution of (D + 3 He) corresponding to the models presented in Figs. 5a, 5b and 5c, respectively. From all of the models in Fig. 5a (that reproduce the D evolution well), only one (Ib ) can also 14

satisfy the (D + 3 He) constraint when the 3 He survival fraction is g3 = (1.0, 0.7, 0.7). In the other two cases (Ia,e and IIb ) it was necessary to reduce g3 in order to bring about agreement with the data. This di?culty in obtaining an acceptable pre-solar (D + 3 He) is not shared by models with a steeply decreasing SFR, as can be seen in Fig. 6b. Models IIa,a and IIa,b reproduce nicely the observed (D + 3 He) value, even with the larger value of g3 value for the survival fraction of 3 He. The reason is, of course, that they destroy so early (and e?ciently) their D , that even if a considerable fraction of it survives in the form of 3 He, the sum is still reasonably low. But, as seen in Fig. 5b, those two models destroy too much D , and as such should be excluded. On the contrary, model IIa,c (reasonably reproducing the D evolution) needs again a lower g3 value (0.5,0.3,0.3) in order to reproduce marginally the pre-solar (D + 3 He)/H value. Finally, Fig. 6c illustrates the e?ect of the IMF on the (D + 3 He) evolution. The low value of Minf = 0.1M⊙ leads to a pre-solar (D + 3 He) much larger than observed. A larger Minf = 0.4M⊙ brings about a better agreement with the observations. It now becomes straightforward to understand the excess (if g3 is not lowered) D + 3 He in models Ia and IIb . It all has to do with the distribution of stellar masses. Too much mass in low mass stars will release an excess of 3 He. As we saw, this e?ect is displayed by lowering Minf from 0.4M⊙ to 0.1M⊙ in model Ia,e . In this case because of the single slope IMF, when Minf = 0.1M⊙ there is signi?cantly more mass in very low mass stars which do not evolve. The result is clear. Much less D is destroyed, and there is an excess of D + 3 He as was shown in Figs. 5c and 6c. Model Ia does worse than Ib despite the fact that model Ib goes down to lower masses because the Tinsley (1980) IMF begins to turn over (though less sharply than does the Scalo (1986) IMF) and has a much ?atter slope (x = 0.25) compared to the x = 1.7 slope in Ia . The shape of the IMF at low mass is of critical importance in determining the late-time behavior of D and 3 He. Similarly, IIb does worse than IIa with respect to D + 3 He. Fig. 7 presents the evolution of 3 He for several of the models discussed in this section, and helps understanding the results on (D + 3 He) presented in Figs. 6a, 6b and 6c. The survival fraction of
3

He used in models IIa,a and Ib leads naturally to a large increase in the 3 He abundance. On the other hand, the small g3 value adopted in models IIb and Ia,e (as to lead to an acceptable pre-solar

D + 3 He), leads to a very slight increase in the abundance of 3 He during galactic evolution. 15

The importance of the 3 He survival factor was already recognized by Truran and Cameron (1971). They showed that the assumption of 3 He survival in low mass stars stars had a profound e?ect on the comparison to data at the time of the formation of the solar system. The potential for excess 3 He was also discussed in Rood et al. (1976). In Fig. 8, we perform a more systematic investigation of this e?ect, lowering g3 from the more conservative value used in model Ia,a to almost half of it in model Ia,e . As expected, the e?ect is rather large, but only for the lowest value is a (marginal) agreement to observations obtained. A renewed e?ort in better understanding this parameter is certainly needed. An overall summary of our results is displayed in Table 3.

6

Conclusions

We have shown that the degree of D astration during galactic evolution depends crucially on the adopted stellar IMF and the star formation rate. We found that it is not very di?cult to destroy D in the context of relatively standard models of galactic chemical evolution without infall. Without knowing anything about the primordial infall rate, it seems that this might be a good approximation, especially in light of the recent D observations which indicate that the D abundance has decreased since the formation of the solar system. Future observations will certainly help our understanding of this issue. However, in spite of the ease in destroying D , one should remain cautious regarding the evolution of the other elements, such as 3 He and 16 O. Among the most di?cult of the constraints to satisfy < is the limit (D + 3 He)/H ? 5 × 10?5 on the D and 3 He pre-solar abundance. In particular 3 He production in stars with mass ? 1M⊙ should be minimal. It would be very useful to study in detail the production and destruction of this element in the evolution of low mass stars including the Asymptotic Giant Branch phase, since it is ultimately more constraining than D . Considering the new requirements for the destruction of D , our model I, with the SFR proportional to the gas mass fraction, is a good candidate for D(t) as well as Z(t) if 3 He is not overproduced in low mass stars. Model II, on the other hand, can signi?cantly destroy D without a?ecting 3 He, but the overproduction of O requires cutting o? the high mass end of the IMF. Thus our main conclusion is that a destruction of D by factor of 2 ? 3 at the time of the formation of 16

the solar system and a factor ? 5 today, can be achieved with relative ease in a variety of models without infall. However, our understanding of the net production of 3 He in stars of about one solar mass is crucial to determining the overall viability of these models. Somehow, g3 ≤ 1 for M ? 1M⊙ , i.e. there should not be signi?cant net 3 He production in such stars, and thus must be below the estimate of Iben and Truran (1978). A potential cop-out to this predicament may yet be that the solar abundance of 3 He is not representative of the average ISM abundance at that time (Rood et al. 1976). Indeed, at t = To , none of our models produce 3 He in excess of the current limits on the present-day ISM abundance as measured by Bania et al. (1993). Perhaps the solar system is anomalously low in 3 He.

17

Acknowledgements We would like to thank Jean Audouze, Michel Cass?, David Schramm and Jim Truran for very e helpful discussions. KAO would like to thank the Institut d’Astrophysique de Paris where this work was started for their hospitality. The work of KAO was supported in part by DOE grant DE-AC02-83ER-40105. and by a Presidential Young Investigator Award. The work of NP and EV-F was supported in part by PICS no 114, “Origin and Evolution of the Light Elements”, CNRS. We are grateful to Yvette Oberto for her help in the calculations.

Table 1. Abundance comparison at the birth of the solar system using the yields of Arnett (1991) and Woosley (1993) in the case of model Ia Yields from Woosley 1.4 2 0.85 Yields from Arnett 1.4 1.4 0.75

Z/Z⊙ O/O⊙ F e/F e⊙

18

Table 2. The list of models explored in our study. model Ia,a Ia,b Ia,c Ia,d Ia,e Ib IIa,a IIa,b IIa,c IIb SFR 0.25σ ” ” ” ” ” ?t/3 e e?t/5 e?t/10 e?t/3 IMF M ?2.7 ” ” ” ” Tinsley Scalo ” ” M ?2.7 Minf 0.4 ” ” ” ” 0.1 0.1 ” ” 0.4 Msup 100 ” ” ” ” ” 20 ” 100 100 g3 (1.0,0.7,0.7) (1.0,0.5,0.5) (0.7,0.5,0.5) (0.7,0.3,0.3) (0.5,0.3,0.3) (1.0,0.7,0.7) ” ” (0.5,0.3,0.3) (0.5,0.3,0.3)

19

Table 3. Results of four basic models compared with observations in the solar system as well as in the ISM for D and σ. σ (D/H)⊙ (Dp /D)⊙ (Dp /D)o
(D+ 3 He) H ⊙

O/O⊙ F e/F e⊙ Z/Z⊙

Observations 0.1 to 0.2 (2.6 ± 1.0) × 10?5 ?3 ?5 (4.1 ± 1.0) × 10?5 1 1 1

Model Ia,e 0.13 3.4 × 10?5 2.2 3.8 5.1 × 10?5 2.0 0.9 1.4

Model Ib 0.20 2.6 × 10?5 3 5 5 × 10?5 2.3 1.0 1.7

Model IIa,a 0.20 0.0

4 × 10?5 1.1 0.8 1.3

Model IIb 0.15 1.9 × 10?5 4 5 3.8 × 10?5 2.6 1.1 1.8

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23

Figure Captions Figure 1: a: Impact of the IRA on the evolution of D/H (1) and (D +
3

He)/H (2) as a

function of time for the model Ia,e . The data points correspond to Eqs. (6), (8) and (9). Dashed curves correspond to calculation made with the IRA, and solid curves are calculated without the IRA. b: As in Fig. 1a, for the model IIa,a . Figure 2: a: The evolution of the gas mass fraction as a function of time, with (dashed curve) and without (solid curve) the IRA for model Ia . b: As in Fig. 2a, for model IIa,a . Figure 3: The abundances of D (1), and 3 He (2) as a function of the gas mass fraction σ with (dashed curve) and without (solid curve) the IRA for model IIa,a . Figure 4: Metallicity distribution in the Galactic disk (dashed lines) for three models: Ia , IIa,a , and IIb . (Model Ib is quite similar to Ia .) The data shown as a histogram (solid line) are taken from Norris & Ryan (1991). Figure 5: a: The evolution of the D/H ratio as a function of time for the di?erent models: Model Ia,e with Ψ(t) = 0.25σ and φ(M) ∝ M ?2.7 , 0.4 ≤ (M/M⊙ ) ≤ 100; Model Ib with Ψ(t) = 0.25σ and the IMF from Tinsley (1980); Model IIa,a with Ψ(t) = e?t/τ , τ = 3Gyr, and the IMF from Scalo (1986), 0.1 ≤ (M/M⊙ ) ≤ 20; Model IIb with Ψ(t) = e?t/τ , τ = 3Gyr, and φ(M) ∝ M ?2.7 , 0.4 ≤ (M/M⊙ ) ≤ 100. Figure 5: b: as in 5a for models IIa : IIa,a with τ = 3 Gyr; IIa,b with τ = 5 Gyr; IIa,c with τ = 10 Gyr. Figure 5: c: The impact of the lower limit of the IMF Minf on the evolution of (D/H) using model Ia,e . The curve labeled 1 (2) uses Minf = 0.4M⊙ (Minf = 0.1M⊙ ). Figure 6: a: As in Figure 5a, for the evolution of (D + 3 He)/H. Figure 6: b: As in Figure 5b, for the evolution of (D + 3 He)/H. Figure 6: c: As in Figure 5c, for the evolution of (D + 3 He)/H. Figure 7: Evolution of 3 He/H for di?erent galactic models 24

Figure 8: The evolution of (D + 3 He)/H as a function of time for Model Ia with di?erent sets of values of the 3 He survival fraction, g3 (sets of values refer to stellar masses (1, 2, 3) M⊙ , respectively: Model Ia,a with g3 = (1, 0.7, 0.7); Model Ia,b with g3 = (1, 0.5, 0.5); Model Ia,c with g3 = (0.7, 0.5, 0.5); Model Ia,d with g3 = (0.7, 0.3, 0.3); Model Ia,e with g3 = (0.5, 0.3, 0.3).

25

This figure "fig1-1.png" is available in "png" format from: http://arXiv.org/ps/astro-ph/9310021v1

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