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

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2. Why we need to do time-dependent calculations

We intend to calculate the Li, Be and B (LiBeB) production induced by the interaction of energetic particles within a SNR. We shall first consider the fate of the particles accelerated out of the ambient, zero metallicity ISM entering the forward shock created by a SN explosion (process 1), and then turn to the acceleration of particles from the SN ejecta at the reverse shock, on a very short time scale around the so-called sweep-up time, [FORMULA] (process 2). It turns out that both of these processes are highly non-stationary, for a number of reasons which we now review.

2.1. EPs accelerated at the forward shock

Considering first process 1, we expect that the particle injection power be more or less proportional to the power of the shock, which is a decreasing function of time as the SNR evolves. Therefore the injection rate of the EPs is not constant, and no steady-state distribution function of the EPs within the SNR is ever reached. If everything else was constant in the problem, we could however calculate the total energy injected in the form of EPs during the whole process, and multiply it by the steady-state spallation efficiency (defined as the `number of nuclei synthesized per erg injected'), evaluated from standard steady-state calculations. This would provide us with the total spallation yields (i.e. the time integral of the spallation rates), which are the only observationally relevant quantities. This, however, cannot be done in the case we are considering, because the chemical composition of the target, namely the interior of the SNR, is also evolving during the expansion.

Indeed, as more and more metal-poor material is swept-up from the ISM by the shock, the metal-rich SN ejecta suffer stronger and stronger dilution, which makes the spallation of C and O less and less efficient. As a consequence, even though we can evaluate the total energy eventually imparted to EPs, we cannot deduce the spallation yields from it because we don't know what composition to choose for the target. Again, if this was the only non stationary feature in the process, we could still calculate the average target composition and compute the spallation yields from it. But since both the EP injection rate and the target composition are functions of time, steady-state models cannot be used in any consistent way, and a fully time-dependent calculation is required.

Qualitatively, it is easy to show that the yields which we obtain by integrating the time-dependent spallation rates must be appreciably higher than those derived from steady-state estimates using the total energy injected in the form of EPs and the (constant) mean target composition. Indeed, the latter amounts to assuming that the injection rate is also constant and equal to the average power of the EPs (i.e. the total energy divided by the duration of the process). However, in the time-dependent model, we take advantage of the fact that the EP power is higher at the beginning, when the target composition is richer in C and O. In other words, the spallation efficiency is higher when the EP fluxes are higher too, and conversely, less energy is imparted to the less efficient EPs accelerated towards the end of the process.

In addition to the sources of non-stationarity just mentioned (time-dependent injection and dilution of the ejecta), we also have to take into account the adiabatic losses suffered by the EPs as they wander inside the expanding volume of the SNR. Now these adiabatic losses are essentially function of time, becoming smaller and smaller as the SNR expands and the shock velocity gets lower. This again can only be taken into account in a time-dependent model.

2.2. EPs accelerated at the reverse shock

Coming now to the case of process 2, where particles from the enriched material ejected by the supernova are accelerated at the reverse shock, it is clear that the dilution effect mentioned above does not have any significant influence anymore. Indeed, the light element production is now dominated by the spallation of energetic C and O nuclei interacting with ambient H and He, instead of ambient C and O interacting with energetic H and He nuclei in the case of process 1 (see Figs. 1 and 4). The abundance of (non energetic) C and O in the target has therefore only a negligible influence, since these nuclei hardly contribute to the spallation yields. Nevertheless, the time dependence of the EP injection and the adiabatic losses still have to be taken into account, which is enough to make time-dependent calculations indispensable.

As has been argued in Paper I, the curve representing the power in the reverse shock, [FORMULA], as a function of time strongly peaks around the sweep-up time, [FORMULA], which is defined as usual as the time at which the swept-up mass is equal to the ejected mass. This also approximately marks the end of the free expansion phase and the beginning of the adiabatic (or Sedov-like) phase. In the absence of a motivated prescription for the reverse shock power function, [FORMULA], and on the understanding that its time scale is short as compared to the energy loss time scale, we shall consider below the injection of EPs as instantaneous in the case of process 2. We thus just could not be further away from a steady-state. However, as above, were this the only non-stationary feature of the process, we could still obtain the integrated spallation yields from steady-state calculations by merely multiplying the total energy injected in the form of EPs by the spallation efficiency, which in this case is almost independent of the target composition. Unfortunately, as already indicated, the adiabatic losses are also a function of time, and will therefore cause the aforementioned spallation efficiency to vary as the process goes on.

On the other hand, once the EPs leave the SNR at the end of the Sedov-like phase to interact with the surrounding ISM, they will only suffer the usual Coulombian losses, which are essentially independent of time. The above argument therefore does not apply anymore and this final part of process 2 (occuring outside the remnant) could be worked out with purely steady-state machinery. This is in fact what we did in Paper I (see its Sect. 4), in our study of what we then called process 3. Here, however, we shall not distinguish between the part of the process occuring inside the SNR, and the part occuring outside (former process 3), because our time-dependent numerical model allows us to treat both on the same footing. In particular, we obtain in this way not only the total LiBeB yields, but also their production rates as a function of time, whose time integral can be sucessfully checked to be equal to the steady-state yields.

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
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