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Astron. Astrophys. 346, 329-339 (1999)

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5. Discussion

Since light element production in the interstellar medium obviously requires a lot of energy in the form of supernuclear particles (i.e. with energies above the nuclear thresholds) as well as metals (especially C and O), it is quite natural to consider SNe as possible sources of the LiBeB observed in halo stars. We have analysed in detail the spallation nucleosynthesis induced by a SN explosion on the basis of known physics and theoretical results relating to particle shock acceleration. Two major processes can be identified, depending on whether the ISM or the ejecta are accelerated, respectively at the forward and reverse shocks. In the first case, the EPs consist mostly of protons and alpha particles and must therefore interact with C and O nuclei, which are much more numerous within the SNR than in the surrounding medium (especially at early stages of Galactic evolution). The process will thus last as long as the EPs stay confined in the SNR, i.e. approximately during the Sedov-like phase, but not more. In the second case, freshly synthesized CNO nuclei are accelerated, and Be production occurs through interaction with ambient H and He nuclei. The process is then divided into two, one stretching over the Sedov-like phase, with the particles suffering adiabatic losses, and the other one occuring outside the remnant, with only Coulombian losses playing a role.

We have calculated the total Be production in these three processes, taking the dynamics of the SNR evolution into account (dilution of the ejecta by metal-poor material and adiabatic losses). The results are shown in Fig. 6 for processes 1 and 2 (from Eqs. (9) and (18)). We find that with canonical values of [FORMULA], [FORMULA], [FORMULA] and a mean ambient density [FORMULA], the fraction of freshly synthesized CNO nuclei spalled into Be in these processes is [FORMULA] and [FORMULA], respectively, which is very much less than the value `required' by the observations, discussed in the introduction ([FORMULA]). Even allowing for unreasonably high values of the acceleration efficiency, [FORMULA], the total Be production by processes 1 and 2 would still be more than one order of magnitude below the observed value.

As suggested by Fig. 6 and our analytical study, higher densities improve the situation. However, even with [FORMULA] and acceleration efficiencies equal to 1, the Be yield is still unsufficient. Moreover, it should be noted that our calculations did not consider Coulombian energy losses (because they are negligeable as compared to adiabatic losses for usual densities), which become important as the density increases and therefore make the Be yield smaller. Finally, since we are trying to account for the mean abundance of Be in halo stars, as compared to Fe, we have to evaluate the Be production for an ambient density corresponding to the mean density encountered around explosion sites in the early Galaxy, which is very unlikely to be as high as [FORMULA]. It could even be argued that although the gas density might have been higher in the past than it is now (hence our `canonical value' [FORMULA]), the actual mean density about SN explosion sites could be lower than [FORMULA], because most SNe may explode within superbubble interiors, where the density is much less than in the mean ISM.

Thus, our conclusion is that processes 1 and 2 both fail in accounting for the Be observed in metal-poor stars in the halo of our Galaxy. Concerning the third process, adopting canonical values for the parameters again leads to unsufficient Be production, as noted in the previous section. While higher densities improve the situation by avoiding the adiabatic energy losses, one should nevertheless expect at least half of the EP energy to be lost in this way, for any reasonable density. This means that even if 10% of the explosion energy is imparted to EPs accelerated at the reverse shock, which is certainly a generous upper limit, the required number of [FORMULA] nuclei of Be per SN implies a spallation efficiency of [FORMULA] nucleus/erg. Now Fig. 4 shows that this requires an EP composition in which at least one particle out of ten is a CNO nucleus. In other words, the ejected mass of CNO must be of the order of that of H and He together. None of the SN explosion models published so far can reproduce such a requirement, and so there is clearly a problem with Be production in the early Galaxy.

The results presented here are in fact interesting in many regards. First, they show that it is definitely very difficult to account for the amount of Be found in halo stars. Consequently, we feel that the main problem to be addressed in this field of research is probably not the chemical evolution of Be (and Li and B) in the Galaxy, as given by the ensemble of the data points in the abundance vs metallicity diagrams (e.g. whether Be is proportional to Fe or to its square) but, to begin with, the position of any of these points. Are we able to describe in some detail one process which could explain the amount of Be (relative to Fe) present in any of the stars in which it is observed? The answer, we are afraid, seems to be no at this stage. It is however instructive to ask why the processes investigated here have failed. Concerning process 1 (acceleration of ISM, interaction with fresh CNO within the SNR), the main reason is that the CNO rich ejecta are `too much diluted' by the swept-up material as the SNR expands, so that the spallation efficiency is too low (or the available energy is too small). However, it seems rather hard to think of any region in the Galaxy where the concentration in CNO is higher than inside a SNR during the Sedov-like phase (especially in the first stages of chemical evolution)! So the conclusion that process 1 cannot work, even with a 100% acceleration efficiency, seems to rule out any other process based on the acceleration of the ISM, initially devoided of metals.

The other solution is then of course to accelerate CNO nuclei themselves, which provides the maximum possible spallation efficiency, independently of the ambient metallicity. Every energetic CNO will lead to the production of as much Be as possible given the spallation cross sections and the energy loss rates. The latter cannot physically be smaller than the Coulombian loss rate in a neutral medium, and this leads to the efficiency plotted in Fig. 4. Unfortunately, a significant amount of the CNO rich ejecta of an isolated SN can only be accelerated at the reverse shock at a time around the sweep-up time, [FORMULA]. This means that i) the total amount of energy available is smaller than the explosion energy (probably of order 10%, i.e. [FORMULA] erg), and ii) the accelerated nuclei will suffer adiabatic losses during the Sedov-like phase, reducing their energy by a factor of 2 or 3. As shown above, this makes process 2-3 incapable of producing enough Be, as long as the EPs have a composition reflecting that of the SN ejecta.

This suggest that a solution to the problem could be that the reverse shock accelerates preferentially CNO nuclei rather than H and He. For example, recent calculations have shown that such a selective acceleration arises naturally if the metals are mostly condensed in grains (Ellison et al. 1997). The proposition by Ramaty et al. (1997) that grains condense in the ejecta before being accelerated could then help to increase the abundance of CNO in the EPs. However, we have to keep in mind that any selective process called upon must be very efficient indeed, since as we indicated above, the data require that the EP composition be as rich as one CNO nuclei out of ten EPs, which means that CNO nuclei must be accelerated at least ten times more efficiently than H and He. This would have to be increased by another factor of ten if the energy initially imparted to the EPs by the acceleration process were only a factor 2 or 3 lower (i.e. [FORMULA] erg, which is more reasonable from the point of view of particle acceleration theory). Clearly, more work is needed in this field before one can safely invoke a solution in terms of selective acceleration.

As can be seen, playing with the composition to increase the spallation efficiency has its own limits, and in any case, Fig. 4 gives an unescapable upper limit, obtained with pure Carbon and Oxygen (at least for the canonical spectrum considered here - other spectra were also investigated, as in Ramaty et al. (1997), leaving the main conclusions unchanged). This would then suggest that another source of energy should be sought. However, the constraint that it should be more energetic than SNe is rather strong.

Another interesting line of investigations could be the study of the collective effects of SNe. Most of the massive stars and SN progenitors are believed to be born (and indeed observed, Melnik & Efremov 1995) in associations, and their joint explosions lead to the formation of superbubbles which may provide a very favourable environment for particle acceleration (Bykov & Fleishman 1992). Parizot et al. (1998) have proposed that these superbubbles could be the source of most of the CNO-rich EPs, and Parizot & Knoedlseder (1998) further investigated the gamma-ray lines induced by such an energetic component. The most interesting features of a scenario in which Be-producing EPs are accelerated in superbubbles is that i) when a new SN explodes, the CNO nuclei ejected by the previous SNe are accelerated at the forward shock , instead of the reverse shock in the case of an isolated SN, which implies a greater energy, and ii) no significant adiabatic losses occur, because of the dimensions and low expansion velocity of the superbubble. This makes the superbubble scenario very appealing, and it will be investigated in detail in a forthcoming paper.

However that may be, we should also keep in mind that when we say that a process does not produce enough Be, it always means that it does not produce enough Be as compared to Fe . Now it could also be that SN explosion models actually produce too much Fe. The point is that Be is compared to Fe in the observations, while it has no direct physical link with it. Indeed, Be is not made out of Fe, but of C and O. So to be really conclusive, the studies of spallative nucleosynthesis should compare theoretical Be/O yields to the corresponding abundance ratio in metal-poor stars. Unfortunately, the data are much more patchy for Be as a function of [O/H] than as a function of [Fe/H], especially in very low metallicity stars. The usually assumed proportionality between O and Fe could turn out to be only approximate, as recent observational works possibly indicate (Israelian et al. 1998; Boesgaard et al. 1998; these observations, however, still need to be confirmed by an independent method, all the more that they come into conflict with several theoretical and observational results; cf. Vangioni-Flam et al. 1998b). We shall address this question in greater detail in the attending paper (Parizot & Drury, 1999, Paper II). Note however that these observations may offer a possibility to rehabilitate the secondary Be production mechanism involving Galactic cosmic rays interacting in the ISM (Fields & Olive, 1999), although the energetics may also become problematic when new data are obtained at lower metallicity.

Finally, we wish to stress that the calculations presented in this paper rely on a careful account of the dynamics of the problem. More generally, time-dependent calculations are required to properly evaluate the spallation processes in environments where compositions and energy densities are evolving. In particular, as argued in Parizot (1999), no variation with density can be obtained with a stationary model, since an increase in the density induces an equivalent and cancelling increase in the spallation rates and the energy loss rates. By contrast, we have shown that all three of the processes considered here are more efficient at higher density - a result which could not have been found otherwise. Detailed, numerical time-dependent calculations will be presented in paper II, with conclusions similar to those demonstrated here.

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Online publication: May 6, 1999