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

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3. Description of the theoretical and numerical model

It has been shown in the previous section that the spallative production of light elements associated with the explosion of a SN in the ISM is essentially a dynamical process, and therefore requires non-stationary calculations. A general time-dependent model for the interaction of EPs in the ISM has been developed and presented in Parizot (1999), so we shall use it here extensively, recalling only the results relevant to our specific problem and calculating the required inputs for processes 1 and 2.

3.1. The mathematical formalism and the physical ingredients

In each case, we separate the acceleration of the energetic particles (EPs) from their propagation and interaction within the SNR. This is legitimate because the time scale for acceleration up to the energies we are concerned with is very much smaller than any other time scale in the problem, whether dynamical (SNR evolution) or physical (energy loss rate, spallation rates). Consequently, our calculations apply to the EPs once they have been `injected' inside the SNR (from the region close to the shock). Let us assume for the moment that we have determined the so-called injection function , [FORMULA], which we define as the number of particles of species i introduced at energy E and time t, per unit energy and time (in [FORMULA]). The EP distribution function, [FORMULA] then satisfies the usual propagation equation (see Parizot 1999):

[EQUATION]

where [FORMULA] is the energy loss rate for the nuclei of species i at energy E (in [FORMULA]), [FORMULA] is the production rate of nuclei i as secondary particles, and [FORMULA] is the time scale for catastrophic losses, such as nuclear destruction or escape from the region under study.

Since we are concerned with spallation reactions involving the nuclei of the CNO group, we can neglect the two-step processes such as [FORMULA] [FORMULA] followed by [FORMULA] [FORMULA]. We indeed found, using a steady-state model, that the omission of the two-step processes leads too an error of at most [FORMULA], in good agreement with Ramaty et al. (1997) calculations. Since this is smaller than the other observational and theoretical uncertainties, and their implementation in a time-dependent model greatly complicates the situation, we shall neglect them here (Note that in any case, even if there were no other uncertainty in the problem, it is much more accurate to do time-dependent calculations without two-step processes than steady-state calculations including two-step processes, as our simulations have shown). To state this in a more physical way, we can claim that the spallative production of carbon amounts to at most a few percents of the initial CNO supply from the supernova explosion. To the level of precision of the SN models, to mention that only, this correction is of no significance, so we shall simply drop [FORMULA] in Eq. (1).

Concerning the catastrophic loss time, [FORMULA], it is obtained for stable nuclei as:

[EQUATION]

where [FORMULA] is the escape time, and [FORMULA] is the destruction time. The latter is derived from semi-empirical formulas giving the total inelastic cross sections [FORMULA] for a projectile i in a target of species j (Silberberg & Tsao 1990), according to:

[EQUATION]

where [FORMULA] is the velocity of the energetic particle and [FORMULA] is the number density of target species j at time t.

Following the above qualitative analysis (see Paper I for more details), we assume that the time of escape out of the SNR is infinite during the Sedov-like phase of the SNR expansion, and `zero' afterwards. This merely translates the fact that the EPs are confined within the SNR during the adiabatic phase (at least those of lowest energy, which produce most of the spallative LiBeB), and then leak out on a very short time scale. Once the EPs have escaped from the SNR, we need to distinguish between our two processes. In the first case (acceleration at the forward shock), the EPs are deprived of CNO and will not give rise to enough spallation reactions out of the SNR to raise the LiBeB production in any significant way. This is due to the very low ambient metallicity. In the second case, however, the EPs are made of the supernova ejecta themselves and are thus rich in CNO. As a consequence, as far as LiBeB production is concerned, there is no difference whether they interact within or outside the SNR, as interactions with H and He nuclei dominate anyway. We must therefore follow these accelerated nuclei after the end of the adiabatic phase, and compute the corresponding contribution to the total production of light elements.

Concerning the energy loss rate, [FORMULA], we need to take into account both ionisation (Coulombian) and adiabatic losses. The former are very common and just cannot be avoided as soon as energetic particles are to be interacting in the ISM. The latter, however, must be included here because the EPs are confined within the SNR where their velocities are randomized. As a consequence, they do participate to the internal pressure which drives the remnant during the Sedov-like phase, and suffer the adiabatic losses like any other particle working outward when reflected at the expanding shell. Quantitatively, these adiabatic losses have been calculated in Paper I. They are given by Eq. (14) there, namely:

[EQUATION]

where p is the momentum of the particle and [FORMULA] is the radius of the shock. Assuming the Sedov-like expansion law ([FORMULA]) and writing the loss rate in terms of energy, we obtain:

[EQUATION]

This energy loss rate does not depend on the EP species, but is clearly a function of time. On the other hand, the ionisation losses, [FORMULA], do depend on the nuclear species, as well as on time, indirectly, through the density and composition of the ambient medium. Indeed, it has to be realised that the medium in which the EPs are `propagating', namely the interior of the SNR, is initially very rich in freshly synthesized CNO nuclei, and then gets poorer and poorer in metals as the ejecta are being diluted in the ambient, metal-poor, swept up material.

This dilution effect is most important for the calculation of the total LiBeB production through our first mechanism (acceleration of the ISM at the forward shock). Indeed, the instantaneous production rates are directly proportional to the density of CNO within the remnant at time t, which goes like [FORMULA], i.e. [FORMULA]. Quantitatively, the LiBeB production rates are obtained by integrating the spallation cross sections over the EP distribution functions:

[EQUATION]

where [FORMULA] is the cross section for the reaction [FORMULA], and [FORMULA] is the number density of nuclei j in the target (here, the interior of the SNR).

The total LiBeB production is then obtained for the first mechanism by integrating these production rates from [FORMULA] to [FORMULA], which marks the end of the Sedov-like phase as well as the end of the confinement of the EPs inside the SNR. For the second mechanism, we need to integrate from [FORMULA] to the confinement time of the cosmic rays within the Galaxy. As we shall see below, integrating up to infinity only leads to a small overestimate of the total LiBeB production, since the low energy cosmic rays responsible for most of that production have anyway a short lifetime above the spallation thresholds.

The sweep-up time, [FORMULA], is obtained straightforwardly from its definition as a function of the SN parameters and the ambient number density, [FORMULA]:

[EQUATION]

The determination of [FORMULA] is more difficult and somewhat arbitrary, even in the approximation of a perfectly homogeneous circumstellar medium. We argued above and in Paper 1 that [FORMULA] should more or less coincide with the end of the Sedov-like phase, when the shock induced by the SN explosion becomes radiative, that is when the cooling time of the post-shock gas becomes of the same order as the dynamical time. This depends on the cooling function which in turn depends on the density and metallicity of the post-shock gas. Such details and their influence on [FORMULA] have been considered in Paper I. Here, we only give the asymptotic result, valid in the limit of large ambient densities, [FORMULA]:

[EQUATION]

Comparing the dependence of [FORMULA] and [FORMULA] on density, we find that the Sedov-like phase gets shorter when [FORMULA] is increased, and thus the duration of process 1 decreases.

3.2. The injection function at the forward shock

We now turn to the determination of the injection function, [FORMULA], in the case of our first mechanism. As suggested by shock acceleration calculations, we assume that the distribution function of the accelerated particles is [FORMULA], so that the number of protons injected inside the SNR per unit time between momenta p and [FORMULA], irrespective of their direction, is:

[EQUATION]

from thermal values up to [FORMULA] eV/c. This leaves us only with the calculation of the normalisation, [FORMULA], as a function of time.

Following again the most widely accepted theoretical ideas, we assume that the total energy injected per unit time in the form of energetic particles at time t is equal to a constant fraction, [FORMULA], of the power, [FORMULA], flowing through the shock at that time (recalling that the acceleration time scale is small as compared to the dynamical one). Mathematically, this normalisation condition reads:

[EQUATION]

where [FORMULA] is the energy of a proton of impulsion p. Integrating the left hand side (LHS) of Eq. (10), one finds:

[EQUATION]

where [FORMULA] and [FORMULA] is the number

[EQUATION]

Typical values for [FORMULA] and [FORMULA] are [FORMULA], and [FORMULA]. Then, to first order:

[EQUATION]

depending on [FORMULA] only logarithmically.

Combining Eqs. (10) and (11), we obtain the injection function at the forward shock as:

[EQUATION]

or in terms of energy:

[EQUATION]

The asymptotical behavior is thus: [FORMULA] for [FORMULA], and [FORMULA] for [FORMULA].

It should be clear that the above injection function is indeed a function of time, through the incoming power [FORMULA]. To evaluate it, one can make use of the well known formulas giving the time evolution of the shock radius, [FORMULA], and velocity, [FORMULA], during the Sedov-like phase, and calculate [FORMULA]. However, since the Sedov phase is a similarity solution, we know that the result will be nothing else but [FORMULA], where [FORMULA] is the explosion energy. The time-dependent injection function is then finally:

[EQUATION]

As can be seen, the power injected in the form of energetic particles decreases as [FORMULA] as the SNR expands. This is not a futile result, since it happens that the earliest times are also the most favourable to the spallative production of light elements in a SNR. Indeed, as was discussed in Sect. 2, the CNO nuclei suffer a rapid dilution as the remnant expands, lowering the spallation rates. Ignoring the enhancement of the EPs when the SNR is still rather small would thus leads one to significantly underestimate the LiBeB production.

In the above derivation, we did not worry about the chemical composition of the EPs. Clearly the injection function still has to be weighted by the relative abundance of each nuclear species present in the ISM swept up by the SNR. As already mentioned, we are interested only in the LiBeB production at low ambient metallicity, since this is when the observed proportionality between Be and Fe abundances is the most striking and unexpected. According to the assumption that we are testing here, each supernova leads to the same amount of 9Be production, whatever the ambient metallicity. Therefore, all our calculations are made with a zero ambient metallicity. The EPs accelerated out of the ISM are thus made of H and He only, with their primordial relative abundances.

3.3. The injection function at the reverse shock

In the case of the acceleration of the supernova ejecta through the reverse shock, the injection function can be written straightforwardly as:

[EQUATION]

where it is assumed that the acceleration takes place instantaneously at [FORMULA]. This may be justified by noting that the genuine acceleration and reverse shock evolution time scales are certainly smaller than EP evolution time scales (nuclear interactions and energy losses). The relative abundance of the different nuclei in the accelerated particles just reflects that of the supernova ejecta, [FORMULA], and the shape of [FORMULA] is the same as above. This time, however, the injection function has to be normalised to:

[EQUATION]

where [FORMULA] is the fraction of the explosion energy which goes into the EPs accelerated at the reverse shock. This can be phenomenologically expressed as the product of two coefficient: [FORMULA], where [FORMULA] is the fraction of the shock energy imparted to the EPs (i.e. [FORMULA], defined above), and [FORMULA] is the fraction of the explosion energy which goes into the reverse shock. In our calculation, we adopt the `canonical values' of [FORMULA] and [FORMULA], and thus [FORMULA]. It should be clear, however, that these values are only indicative, and that the results simply scale proportionally to [FORMULA] and [FORMULA].

The time integration in Eq. (18) is straightforward, and with [FORMULA], we get:

[EQUATION]

where [FORMULA] has been given in Eq. (12) and (13). Note that the mass m appearing in the above expressions is always the proton mass, and that correlatively the energies are expressed in MeV/n for all the nuclear species.

3.4. The formal solution for the EP distribution function

The formal solution of the time-dependent propagation Eq. (1) is (Parizot 1999):

[EQUATION]

where

[EQUATION]

This solution, however, only considers the time-dependence of the injection function, [FORMULA], and not that of the conditions of propagation, namely the energy losses and the destruction time. Now it is clear that the adiabatic losses do depend on time as well as the ionisation losses and the nuclear destruction time, through the chemical composition within the SNR. One then needs to divide the whole process into sufficiently short phases so that these parameters stay approximately constant during each phase, and put together the solutions (20) for each phase in a proper way (for details, see Parizot, 1999). For the present calculations, it proved sufficient to divide the Sedov-like phase into 15 successive phases.

In the case of our second injection function, Eq. (17), corresponding to the reverse shock acceleration, the time delta function allows us to integrate Eq. (20) to obtain:

[EQUATION]

where [FORMULA] is the solution of:

[EQUATION]

In other words, [FORMULA] is the energy at which a particle of species i must have been accelerated at time [FORMULA] in order to have slowed down to energy E at time t. Similarly, the exponential factor in Eqs. (20) and (22) is nothing but the survival probability of a particle i from its injection at energy [FORMULA] (or [FORMULA]) to the current energy, E.

The above solutions allow us to calculate the EP distribution function for both of our injection functions, Eqs. (16) and (17)-(19). Eq. (6) can then be used to compute the LiBeB production rates at any time after the beginning of acceleration, at [FORMULA]. The results are presented in the following section.

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
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