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Astron. Astrophys. 346, 686-698 (1999)

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4. The results

4.1. LiBeB production by the EPs from the forward shock

The results we show in this section are obtained with the SN explosion models calculated by Woosley & Weaver (1995). We use their models Z, U and T, corresponding to stars with initial metallicity [FORMULA], [FORMULA], and [FORMULA], respectively, and keep the same labels as the authors to refer to specific models (e.g. model U15A corresponds to a star of 15 [FORMULA] with [FORMULA] initial metallicity and a standard explosion energy of [FORMULA] erg). We adopt the value [FORMULA] throughout, on the understanding that all the spallation rates are merely proportional to this parameter.

In Fig. 1, we show the typical evolution of the spallation rates for Be production as a function of time, for a SN exploding in a medium with mean density [FORMULA]. The main contribution is seen to come from reaction [FORMULA]O, which is due to the low [FORMULA] abundance ratio in the SN ejecta. For reactions involving alpha particles, this deficiency of carbon as compared to oxygen is compensated by a greater spallation efficiency. The general shape of the curves is easily understood if one refers to Eq. (6) and to the analysis of the preceding section. Indeed, the spallation rates are basically the product of the relevant cross section by the spectral density of energetic protons, [FORMULA], and the number density of Oxygen within the SNR. Now the latter is subject to dilution by the swept-up metal-free gas, and therefore decreases as [FORMULA], or [FORMULA], while [FORMULA] is merely the time integral of the injection function, Eq. (16) (at least as long as one can neglect the energy losses). We thus find [FORMULA], and the spallation rates:

[EQUATION]

which fits very well the curves in Fig. 1. Differentiating the above expression, we find the maximum production rates to occur at [FORMULA], which expresses the best compromise between Oxygen dilution in the SNR and a sufficient injection of EPs since the onset of the acceleration process.

[FIGURE] Fig. 1. Process 1 9Be production rate in numbers of nuclei per second through different spallation reactions as a function of time after the SN explosion. The SN model used is U15A (from Woosley & Weaver, 1995), and the ambient density is [FORMULA].

This behavior can be further observed on Fig. 2 where we plot the total production rates of Be as a function of time after explosion, for different values of the ambient density, ranging from 1 to [FORMULA]. The shortening of the Sedov-like phase already mentioned is clearly apparent on the figure, as is the behavior of [FORMULA] and [FORMULA]. The calculations also confirm that the position of the maximum is always at [FORMULA], although at the highest densities, this is very close indeed to the end of the adiabatic expansion phase, when the confinement of the EPs ceases and the whole process stops. The position of the maximum then varies as [FORMULA], while its height, obtained by replacing t by [FORMULA] in Eq. (24), is proportional to [FORMULA], and thus [FORMULA].

[FIGURE] Fig. 2. Process 1 total 9Be production rates as a function of time after the explosion of SN model U15A, for different ambient densities. Each curve starts shortly after the sweep-up time and ends at the adiabatic time, marking the end of the Sedov-like phase.

In Fig. 3 we show the evolution of the production rates for the five light element isotopes, either taking and not taking the adiabatic losses into account. The behavior of 6Li and 7Li is different from that of the other isotopes, because lithium is mainly produced through [FORMULA] reactions, as shown in Fig. 4, and these reactions are not sensitive to the dilution of the SN ejecta by the ambient material. The evolution of Li production rates therefore reflects directly the evolution of the EP fluxes. As just stated, this would be a pure logarithm if one could neglect the energy losses. It turns out that the adiabatic losses dominate the Coulombian losses for any reasonable ambient density. To see how they influence the EP fluxes, let us re-write Eq. (1) in the form:

[EQUATION]

where we dropped the destruction and second order terms. At energies of a few tens of MeV/n, Eq. (5) simplifies to give the expression for adiabatic losses:

[EQUATION]

Replacing in Eq. (25), we obtain:

[EQUATION]

where we recognized that a power-law for the injection function Q, with spectral index [FORMULA] ([FORMULA]), translates into a power-law for the EP spectral density N with the same index: [FORMULA]. This is a consequence of the proportionality between the energy loss rate and the energy itself. The equation for [FORMULA] is then straightforward:

[EQUATION]

from where we see that instead of the logaritmic increase [FORMULA] prevailing in the absence of energy losses, a steady-state value should be reached (if the Sedov-like phase last long enough) with:

[EQUATION]

So the adiabatic losses are important when both terms in the right hand side of Eq. (28) are of the same order, that is (evaluating the second term from its `no-loss value', and using [FORMULA] for the low-energy part of the spectrum):

[EQUATION]

or

[EQUATION]

This result is in very good agreement with the numerical results shown in Fig. 3. Likewise, the gap between the calculations with adiabatic losses turned on or off is increasing only logarithmically with time, so that the difference is rather small, even at the end of the Sedov-like phase. We find total Be production only a few tens of percent higher if we drop the adiabatic losses, and the difference even falls to zero when higher ambient densities are considered. This is of course because the Sedov phase is then considerably shortened.

[FIGURE] Fig. 3. Process 1 production rates of the five light element isotopes as a function of time after the explosion of SN model U15A, in a medium of density [FORMULA]. Results are shown for calculations taking adiabatic losses into account (full lines) as well as ignoring them (dashed lines).

[FIGURE] Fig. 4. Process 1 6Li production rate in numbers of nuclei per second through different spallation reactions as a function of time after the SN explosion. The SN model used is U15A and the ambient density is [FORMULA].

Although Figs. 1, 2 and 3 help us to clarify the dynamics of the process and understand the role of the different parameters, only the total, integrated light elements production is actually relevant to the Galactic chemical evolution. We show in Fig. 5 the results of the integration of the Be production rates over the whole Sedov-like phase, for different SN explosion models, as a function of the ambient density. Except for the case of the Z30A model, we find that for a given mass of the progenitor the total Be yield is independent of the initial metallicity of the star (zero, [FORMULA] or [FORMULA] times solar). The very small production of Be obtained with the Z30A model is in fact due to a very small amount of Oxygen expelled by the supernova. A model with a higher explosion energy (Z30B) gives results closer to those of T30A and U30A. Although yields significantly different are obtained for different masses of the progenitor, due to different compositions and masses of the ejecta, it is clear from Fig. 5 that the total amount of Be produced by process 1 (forward shock) is much too low to account for the Be observed in metal-poor star. Indeed, the results obtained for a [FORMULA] star with ambient density [FORMULA] are about three orders of magnitude too low, for our choice of [FORMULA]. This is in very good agreement with the analytical estimates presented in Paper I.

[FIGURE] Fig. 5. Integrated process 1 Be yields for different SN models as a function of ambient density. For the models with progenitor masses of 30 [FORMULA] and a density higher than a few [FORMULA], the sweep-up time [FORMULA] is greater than the adiabatic time [FORMULA], so that the Sedov-like phase does not exists.

Concerning the density dependence of the Be yields, the numerical results shown in Fig. 2 are also in good agreement with the analytical calculations. In particular, the yields increase with ambient density and reach a maximum at about a few [FORMULA], above which the Sedov-like phase becomes extremely short, and even vanishes for high mass progenitors (implying large ejected masses). Using Eq. (7) and (8) we can write this limiting density as:

[EQUATION]

4.2. LiBeB production by the EPs from the reverse shock

We now turn to the results obtained for the second mechanism, in which the SN ejecta are accelerated at the reverse shock at the onset of the Sedov-like phase. The 6Li and 9Be production rates are shown on Fig. 6 as a function of time, with and without adiabatic losses, for an ambient density of [FORMULA] and a progenitor corresponding to the U15A model of SN. As can be seen, the Be production rates are strongly dominated by inverse spallation reactions, i.e. reactions in which the projectile is the heavier nuclei. Moreover, since the abundance of C and O in the target suffers from dilution by ISM gas, the direct-to-inverse spallation efficiency ratio keeps decreasing during the Sedov-like phase. At the end of it, as already discussed, the direct reactions stop, while inverse ones are not affected. In Figs. 6b and 6d, the adiabatic losses have not been taken into account. The decrease of the direct spallation rates is thus due only to dilution, and we obtain the expected power law in [FORMULA], or [FORMULA]. In the meanwhile, the inverse spallation rates are almost constant, as the Sedov-like phase is much shorter than the time-scale for coulombian losses. This time-scale can literally be read from the figure. It is of order a few times [FORMULA] years for this set of parameters. Note however that the energy loss time-scale actually depends on the species and energy of the particle. Accordingly, what is observed on the spallation rates is in fact a mean coulombian time-scale, averaged over the EP energy spectrum, and more precisely the part of this spectrum which stands above the energy threshold of the cross-sections. This explains the slight variation observed for the different spallation channels.

[FIGURE] Fig. 6a-d. Process 2 (and 3) 6Li and 9Be production rates in numbers of nuclei per second through different spallation reactions as a function of time after the SN explosion. The SN model used is U15A and the ambient density is [FORMULA]. On the left, the calculations include adiabatic losses (U15A+); on the right, they do not (U15A-).

It is worth emphasizing that the time-scales that we obtain are much shorter than the confinement time-scales inferred from cosmic-ray propagation theories. This indicates that the leakage of the EPs out of the Galaxy has negligible influence on the spallation yields, and justifies our choice of neglecting it. Even for an ambient density of [FORMULA], the bulk of the light element production is contributed by nuclear reactions occuring within a few million years after the SN explosion, which is to be compared with Galactic confinement times of order a few [FORMULA] years.

Comparing Fig. 6a with Fig. 6b (or Fig. 6c with Fig. 6d), we can see the influence of the adiabatic losses on the nuclear rates. For inverse spallation reactions, we observe an almost perfect power law decrease, with logarithmic slope [FORMULA], in very good agreement with the value derived in Paper I. Indeed, the analytic treatment led us to expect spallation rates proportional to [FORMULA], or equivalently [FORMULA]. The slightly quicker decrease found in the numerical results is due to the contribution of the coulombian losses (whose effect is also visible on Figs. 6b and 6d), and to the shape of the spallation cross-sections close to their threshold. Likewise, the time evolution of direct spallation reactions is also very close to a power law, with logarithmic slope of [FORMULA], as expected.

As noted above, however instructive the examination of the spallation rates evolution may be, they cannot be directly compared to any observational data. We therefore calculated the (more relevant) integrated yields for different models corresponding to initial metallicities [FORMULA] (models Z), [FORMULA] (models U), and [FORMULA] (models T), and normalized them to the expected value, i.e. to the value required to explain the abundances observed in the metal-poor stars. Consequently, normalized yields respectively lower and higher than 1 are equivalent to under- and over-production of Be. A few words of explaination are however required as how the normalization is actually performed. The only assumption here is that the Galactic Be evolution is primary relative to both Fe and O. This means that the Be/Fe and Be/O ratios are approximately constant in metal-poor stars (as is consequently the O/Fe ratio). Then each supernova must lead, on average (over the IMF), to the same Be/Fe and Be/O ratios as those observed. These are thus the values we use to normalize our results. Now, as the Fe and O yields calculated by Woosley & Weaver (1995) are different for each of their SN models, we applied our normalization model by model and obtained the results shown in Fig. 7, as a function of the mass of the SN progenitor, for different initial metallicities.

[FIGURE] Fig. 7a-f. Normalized process 2 [Be/Fe] and [Be/O] yield ratios, as a function of the mass of the progenitor. Models Z, U and T correspond to the indicated initial metallicity of the stars. Models A, B and C correspond to different explosion energies (see text and Woosley & Weaver, 1995). The yield ratios are normalized to the value required by the observations as explained in the text.

As discussed earlier, the approximate constancy of the Be/Fe ratio is well established observationally, over two orders of magnitude in metallicity, from [FORMULA] to [FORMULA] times the solar value. On the other hand, we still lack similar measurements of the Be/O ratio in stars with [FORMULA] times the solar value, while the trend at higher metallicity seems to favour a slightly increasing Be/O, if one is to believe the recent observations by Israelian et al. (1998) and Boesgaard et al. (1998) (see also Fields & Olive, 1999). To this respect, it might seem that our normalization based on the primary behavior of Be is better justified for comparison to Fe than to O. In fact, it is just the opposite. Indeed, the models we are investigating (processes 1 and 2) predict a linear increase of Be as compared to O, whatever the Fe evolution may be. As already noted in Paper I, Be and Fe actually have no direct physical link, as the spallation reactions involve only C and O (and in fact mainly Oxygen, as we have shown; see Figs. 1 and 6). Both processes 1 and 2 could therefore account, in principle, for any value of the Be/Fe ratio, provided we can choose the Iron yield of the SNe (this is however not the case, and even if the SN explosion models entail possibly large uncertainties, the claim for and use of a constant Be/Fe ratio is in fact justified by the observations themselves). On the contrary, the Be/O ratio is entirely determined, at a fundamental level, by the processes we investigate here. A higher mass of Oxygen ejected by the supernova would indeed imply a larger Be yield as well, and conversely.

Except for a few `irregular models' which we shall discuss shortly, Fig. 7 shows that the Be yields obtained by process 2 are significantly smaller than the required values, by about two orders of magnitude when comparison is made with Fe, and roughly one order of magnitude when comparison is made with O. This is again in good quantitative agreement with the results of Paper I, so that we confirm that the processes considered here cannot be responsible for the majority of the Be production in our Galaxy. This conclusion has important implications which have been analysed in Paper I and will be summarized below. Let us now comment the figures in greater detail.

For each series of explosion models (Z, U and T), corresponding to different initial metallicities, Woosley & Weaver (1995) have calculated the yields of a number of elements for progenitors of different masses ranging from 12 to 40 [FORMULA]. For the more massive progenitors, they found that the yields of Fe, notably, greatly depended on the mass-cut, which in their models is directly linked to the explosion energy. For example, a 30 [FORMULA] model with a `standard' explosion energy of [FORMULA] erg ejects virtually no Iron at all. Explosion energies greater than the standard value have therefore been explored, leading to higher Fe yields for the most massive stars. We use the same notations as in Woosley & Weaver (1995), i.e. models A, B and C correspond to increasing explosion energies of order 1.2, 2 and [FORMULA] ergs, respectively. In fact, the explosion energy has been adjusted for higher mass progenitors in an ad hoc way in order to obtain approximately the `standard' Fe yield of [FORMULA]. Therefore, passing from model A to model B, and finally to model C as the progenitor's mass increases, amounts to ensure that the SN yields of both O and Fe do not vary in dramatic proportions. This is the reason why the curves for models A, B and C connect so smoothly on Figs. 7a-f. In particular, it is worth emphasizing that the results which we obtain for this `mixed model' (A, then B, then C), are remarkably similar whatever the initial metallicity and mass of the progenitor may be. We find in this way [FORMULA] and [FORMULA], where the brackets mean that the yield ratios have been normalized to the required value as described above.

It should be clear, however, that there is no special reason why we should increase the explosion energy for the most massive SN progenitors. In fact, the great sensitivity of the Fe yield to the explosion energy for these stars mostly means, to our opinion, that the SN explosion models are still unable to predict reliable yields (especially at the lowest metallicities; see the huge differences between the models in Fig. 7a). For instance, if we adopt the standard explosion energy (models A), then it is clear from Figs. 7a,c,e that the observed Be/Fe ratio is very easy to reproduce if one assumes that only the most massive stars formed in the early Galaxy. The reason for this success, however, is not that the massive stars (indirectly) produce a lot of Be, but rather that they produce extremely little Fe. In this case, then, a serious Fe underproduction problem will be encountered by the chemical evolution models, so that the high value of the [Be/Fe] should be regarded as somewhat artificial, and rather irrelevant to the question of Be production in the Galaxy. Moreover, such a behaviour is not expected to be found in the curves showing the Be production as compared to the Oxygen. Indeed, as already alluded to, if a particular SN model happens to not eject any substantial amount of O, then it will not lead to any significant Be production either, leaving the [Be/O] ratio virtually unchanged. This can be checked on Figs. 7b,d,f, where all the models are shown to give approximately the same results. The only exceptions arise at low metallicity for models A and can be easily understood. In these cases, indeed, the O yield becomes much lower than the C yield, so that the Be production is actually dominated by spallation reactions involving C. Consequently the Be yield is still quite substantial, while the O yield is very low, which brings about a situation very similar to that encountered with Fe.

However that may be, even if we trust the low (or even extremely low) Fe and O yields obtained from models A for high mass progenitors, the contribution of these high mass SNe still has to be weighted by their frequency among the type II SNe. In Figs. 8 and 9 we show the normalized [Be/Fe] and [Be/O] ratios, after averaging over a power-law IMF with logarithmic slope x ranging from 0.5 to 3. This allows us to explore the influence of varying the weight of the more efficient high mass stars relatively to the lower mass SN progenitors. A low IMF slope (towards 0.5) strongly favours high mass star formation, and is therefore expected to lead to a higher [Be/Fe] ratio than a high IMF slope (towards 3). This qualitative behavior is indeed observed on Figs. 8 and 9, but it can be seen that the effect is actually quite weak, even for such a large range of IMFs. Note that we used `IMFs by number' (of stars), and not `IMFs by mass', so that the Salpeter IMF corresponds to [FORMULA] in our notations. This means that a slope as low as [FORMULA] corresponds to an IMF in which more mass is locked in high mass than in low mass stars. Even for such an IMF, the Be/Fe ratio obtained is still less than a few percent of the observed value. Comparing Be to O, it is shown in Fig. 9 that the IMF slope has almost no influence on the normalized [Be/O] ratio, which is a consequence of the strong physical link between the ejected Oxygen and the Be production, as discussed above.

[FIGURE] Fig. 8. Normalized Be/Fe ratio calculated from SN models ZA, UA, and TA, averaged on the IMF, as a function of the IMF logarithmic slope. (Salpeter slope is 2.35). Models labeled with a `+' include adiabatic losses; those labeled with a `-' (dashed lines) do not.

[FIGURE] Fig. 9. Normalized Be/O ratio calculated from SN models ZA, UA, and TA, averaged on the IMF, as a function of the IMF logarithmic slope. (Salpeter slope is 2.35). Models labeled with a `+' include adiabatic losses; those labeled with a `-' (dashed lines) do not.

We have also shown, in Figs. 8 and 9, the results obtained without including adiabatic losses (dashed lines). Both [Be/Fe] and [Be/O] ratios are then found to be higher by a factor of about 3 to 4, which is in good quantitative agreement with the analytical calculations of Paper I (see Fig. 5 there). This result has two simple, but important implications. First, it points out the necessity of including the adiabatic losses in the calculations (unless explicitely shown that they do not apply), and therefore of using time-dependent models. Second, it indicates that a model in which the EPs do not suffer adiabatic losses has more chance to succeed in accounting for the observed amount of Be in the halo stars.

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
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