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

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3. Spallation reactions within SNRs

3.1. Qualitative overview

We now turn to the production of Li, Be and B (LiBeB) by spallation reactions within a SNR. As emphasized above, there are two obvious mechanisms. One is the irradiation of the CNO ejecta by accelerated protons and alphas. It is clear that the fresh CNO nuclei produced by the SN will, for the lifetime of the SNR, be exposed to a flux of energetic particles (EPs) very much higher than the average interstellar flux, and this must lead to some spallation production of light elements. This process starts at about [FORMULA] with a very intense radiation field and continues with an intensity decreasing roughly as [FORMULA] (where R is the radius of the SNR) until the remnant dies.

The second process is that some of the CNO nuclei from the ejecta are accelerated, either by the reverse shock in its brief powerful phase at [FORMULA] or by some of this material managing to get ahead of the forward shock. This later possibility is not impossible, but seems unlikely to be as important as acceleration by the reverse shock. Calculations of the Raleigh-Taylor instability of the contact discontinuity do suggest that some fast-moving blobs of ejecta can punch through the forward shock at about [FORMULA] (Jun & Norman 1996), and in addition Ramaty and coworkers have suggested that fast moving dust grains could condense in the ejecta at [FORMULA] and then penetrate through into the region ahead of the main shock. In all these pictures acceleration of CNO nuclei takes place only at about [FORMULA] and the energy deposited in these accelerated particles is certainly less than the explosion energy [FORMULA], although it might optimistically reach some significant fraction of that value (say [FORMULA]). Crucially the accelerated CNO nuclei are then confined to the interior of the SNR and will thus be adiabatically cooled on a rather rapid time-scale, initially of order [FORMULA].

3.2. Evaluation of the first process (forward shock)

From the above arguments, it is clear that SNe do induce some Be production. Now the question is: how much? Let us first consider the irradiation of the ejecta by particles (H and He nuclei) accelerated at the forward shock during the Sedov-like phase - process 1. We have already indicated that detailed studies of acceleration in SNRs show that the fraction of the explosion energy given to the EPs is roughly constant during the Sedov-like phase and of order 0.1 to 0.5 or so. Let [FORMULA] be that fraction. Since the EPs are distributed more or less uniformly throughout the interior of the remnant, the energy density can be estimated as

[EQUATION]

where R is the radius of the remnant and [FORMULA] is the explosion energy.

To derive a spallation rate from this we need to assume some form for the spectrum of the accelerated particles. Shock acceleration suggests that the distribution function should be close to the test-particle form [FORMULA] and extend from an injection momentum close to `thermal' values to a cut-off momentum at about [FORMULA]. The spallation rate per target CNO atom to produce a Be atom is then obtained by integrating the cross sections

[EQUATION]

with the normalisation [FORMULA]. Looking at graphs of the spallation cross-sections for Be (as given, e.g., in Ramaty et al. 1997), it is clear that these cross-sections can be well approximated as zero below a threshold at about 30-40 MeV/n and a constant value [FORMULA] above it. One then obtains roughly:

[EQUATION]

where [FORMULA] is the momentum corresponding to the spallation threshold and m refers to the proton mass. Fortunately, for this form of the spectrum the upper cut-off and the spallation threshold only enter logarithmically. A softer spectrum would lead to higher spallation yields and a stronger dependence on the spallation threshold.

Using Eq. (1) and the adiabatic expansion law for the forward shock radius, [FORMULA], we can now estimate the total fraction of the CNO nuclei which will be converted to Be during the Sedov-like phase as:

[EQUATION]

or

[EQUATION]

where as usual [FORMULA] denotes the density of the ambient medium into which the SNR is expanding and [FORMULA] is the total mass of the SNR ejecta. We now recall that the sweep-up time is given in terms of the SN parameters and the ambient number density, [FORMULA], as

[EQUATION]

where [FORMULA] is the velocity of the ejecta, or numerically:

[EQUATION]

Replacing in Eq. (8) and using canonical values of [FORMULA] and [FORMULA], we finally get:

[EQUATION]

Clearly this falls short of the value of order [FORMULA] required to explain the observations, even for values of [FORMULA] as high as 0.5. It might seem from Eq. (9) that very high ambient densities could help to make the spallation yields closer to the needed value. This is however not the case. First, the above estimate does not take energy losses into account, while both ionisation and adiabatic losses act to lower the genuine production rates. Second, and more significantly, the ratio [FORMULA] (and a fortiori its fifth root) becomes very close to 1 in dense environments, lowering [FORMULA] quite notably (see Fig. 6). In fact, it turns out that there is no Sedov-like phase at all in media with densities of order [FORMULA], the physical reason being that the radiative losses then act on a much shorter time-scale, eventually shorter than the sweep-up time.

3.3. Evaluation of the second process (reverse shock)

Let us now turn to the second process, namely the spallation of energetic CNO nuclei accelerated at the reverse shock from the SN ejecta and interacting within the SNR with swept-up ambient material. We have argued above that this reverse shock acceleration is only plausible at times around [FORMULA] and certainly the amount of energy transferred to CNO nuclei cannot be more than a fraction of [FORMULA]. Let [FORMULA] be the fraction of the explosion energy that goes into accelerating the ejecta at or around [FORMULA], and [FORMULA] the fraction of that energy that is indeed transferred to CNO nuclei. These particles are then confined to the interior of the remnant where they undergo spallation reactions as well as adiabatic losses. Let us again assume that the spectrum is of the form [FORMULA]. Then the production rate of Be atoms per unit volume is approximately

[EQUATION]

where [FORMULA] now refers to the accelerated CNO nuclei, the factor 14 comes from the mean number of nucleons per CNO nucleus and the factor 0.2, as before, from the [FORMULA] spectral shape (assuming the same upper cut-off position, but this only enters logarithmically). Integrating over the remnant volume, we obtain the spallation rate at [FORMULA]:

[EQUATION]

Now the adiabatic losses need to be evaluated rather carefully. It is generally argued that they act so that the momentum of the particles scales as the inverse of the linear dimensions of the volume occupied. Accordingly, in the expanding spherical SNR the EPs should lose momentum at a rate [FORMULA], reminiscent, incidentally, of the way photons behave in the expanding universe. In our case, however, the situation is complicated by the fact that the EPs do not push directly against the `walls' limiting the volume of confinement, which move at the expansion velocity, [FORMULA], but are reflected off the diffusion barrier consisting of magnetic waves and turbulence at rest with respect to the downstream flow, and thus expanding at velocity [FORMULA].

To see how this influences the actual adiabatic loss rate, it is safer to go back to basic physical laws. Adiabatic losses must arise because the EPs are more or less isotropised within the SNR and therefore participate to the pressure. Now this pressure, P, works positively while the remnant expands, implying an energy loss rate equal to the power contributed, given by:

[EQUATION]

where [FORMULA] is the total kinetic energy of the particles. Considering that [FORMULA] in the non-relativistic limit (NR) and [FORMULA] in the ultra-relativistic limit (UR), Eq. (12) can be re-writen as:

[EQUATION]

Finally, dividing both sides by the space density of the EPs and noting that [FORMULA] in the NR limit, and [FORMULA] in the UR limit, we obtain the momentum loss rate for individual particles, valid in any velocity range:

[EQUATION]

From this one deduces that at the time when the remnant has expanded to radius R, only those particles whose initial momenta at [FORMULA] were more than [FORMULA] are still above the spallation threshold. For a [FORMULA] distribution function the integral number spectrum decreases as [FORMULA] and thus the number of accelerated nuclei still capable of spallation reactions decreases as [FORMULA]. For a softer accelerated spectrum the effect would be even stronger because there are proportionally fewer particles at high initial momenta.

This being established, we can integrate Eq. (11) over time, to obtain the total production of Be atoms:

[EQUATION]

that is:

[EQUATION]

Dividing by the total number of CNO nuclei in the ejecta, [FORMULA], we get the final result:

[EQUATION]

Note that we assumed that the mass fraction of CNO in the ejecta is the same as the energy fraction of CNO in the EPs (which was the original meaning of [FORMULA]). Considering that all nuclear species have the same spectrum in MeV/n, and thus a total energy proportional to their mass number, this simply means that the acceleration process is not chemically selective, in the sense that the composition of the EPs is just the same as that of the material passing through the shock.

Numerically, again with [FORMULA] and [FORMULA], we finally obtain:

[EQUATION]

3.4. Relative contribution of the two processes

It is worth emphasizing the similarity between expressions (6) and (17) that we obtained for the spallation rates per CNO nuclei by the two processes considered here. This formal analogy allows us to write down their relative contributions straightforwardly:

[EQUATION]

As is often the case, this similarity is not fortuitous and has a physical meaning. The two processes may indeed be regarded as `dual' processes, the first consisting of the irradiation of the SN ejecta by the ambient medium, and the second of the ambient medium by the SN ejecta. The `symmetry' is only broken by the dynamical aspect of the processes. First, of course, the energy imparted to the EPs in both cases needs not be the same, for it depends on the acceleration efficiency as well as the total energy of the shock involved (forward or reverse). This is expressed by the expected ratio [FORMULA]. And secondly, in the first process one has to fight against the dilution of the ejecta - integration of [FORMULA], see Eq. (5) - while in the second process one fights against the adiabatic losses - integration of [FORMULA], see Eq. (15). This is expressed by the last factor in Eq. (19).

Clearly the latter decrease of the production rates is the least dramatic, and the reverse shock process must dominate the LiBeB production in supernova remnants. However, this conclusion still depends on the genuine efficiency of reverse shock acceleration, and once the relative acceleration efficiency [FORMULA] is given, the weight of the first process relative to the second still depends on the total duration of the Sedov-like phase, appearing numerically in Eq. (19) through the ratio [FORMULA]/[FORMULA], which in turn depends on the ambient density, [FORMULA]. The expression of [FORMULA] as a function of the parameters has been given in Eq. (8), so we are left with the evaluation of the time, [FORMULA], when the magnetic turbulence collapses and the EPs leave the SNR, putting an end to Be production. We argued above that [FORMULA] should correspond to the end 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.

In principle, the cooling rate can be derived from the so-called cooling function, [FORMULA]), which depends on the physical properties of the post-shock material, notably on its temperature, T, and metallicity, Z:

[EQUATION]

where n is the post-shock density, equal to [FORMULA] if the compression ratio is that of an ideal strong shock (nonlinear effects probably act to increase the compression ratio to values larger than 4). As for the dynamical time, we simply write

[EQUATION]

To obtain [FORMULA], we then need to solve the following equation in the variable t, obtained by equating [FORMULA] and [FORMULA] given above:

[EQUATION]

where it should be clear that the right hand side also depends on time through the temperature, T, and thus indirectly through the cooling function too. In the non-radiative SNR expansion phase, the function [FORMULA] is obtained directly from the hydrodynamical jump conditions at the shock discontinuity:

[EQUATION]

or numerically:

[EQUATION]

To solve Eq. (22), we still need to know the cooling function [FORMULA]. In the range of temperatures corresponding to the end of the Sedov-like phase, [FORMULA], it happens to depend significantly on metallicity, with differences up to two orders of magnitude for metallicities from [FORMULA] to [FORMULA] (Böhringer & Hensler 1989). Because we focus on Be production in the early Galaxy, we adopt the cooling function corresponding to zero metallicity, represented in Fig. 1 (adapted from Böhringer & Hensler 1989), which holds for values of Z up to [FORMULA].

[FIGURE] Fig. 1. Cooling function (bold line) as a function of the temperature for a medium with metallicity lower than [FORMULA]. The dashed lines illustrate the graphical determination of [FORMULA], for an ambient density [FORMULA] (see text).

For high enough ambient densities, the shock will become radiative early in the SNR evolution, when the temperature is still very high, say above [FORMULA] K. In this case, the cooling function is dominated by Bremsstrahlung emission and can be written analytically as:

[EQUATION]

Substituting from (24) and (25) in Eq. (22) and solving for t, we find:

[EQUATION]

To check the consistency of our assumption [FORMULA] (i.e. [FORMULA] K), let us now report Eq. (26) in (24) and write down the temperature [FORMULA] at the end of the Sedov-like phase:

[EQUATION]

which means that the above analytical treatment is valid only for ambient densities greater than about [FORMULA]. For lower densities, we must solve Eq. (22) graphically. First, we invert Eq. (24) to express t as a function of temperature, then we plot the function [FORMULA] on the same graph as [FORMULA] (see Fig. 1 for an example), find the value of T at intersection, and finally convert this value into the sought time [FORMULA] making use again of Eq. (24). The results, showing [FORMULA] as a function of the ambient density, are shown in Fig. 2.

[FIGURE] Fig. 2. Comparison of [FORMULA] and [FORMULA] as a function of the ambient density, [FORMULA], for two different values of the mass ejected by the supernova (10 and [FORMULA]). The dashed line shows the asymptotic analytic estimate of Eq. (26).

We now have all the ingredients to plot the efficiency ratio of the two processes calculated above. Fig. 3 shows the ratio [FORMULA] given in Eq. (19) as a function of the ambient density, assuming that [FORMULA]. Two different values of the ejected mass have been used, corresponding to different progenitor masses ([FORMULA]). It can be seen that low densities are more favourable to the reverse shock acceleration process. This is due to [FORMULA] being larger, implying a larger dilution of the ejecta (process 1 less efficient) and smaller adiabatic losses, which indeed decrease as [FORMULA] (process 2 more efficient). The part of the plot corresponding to [FORMULA] is not physical, because it requires [FORMULA], which simply means that the Sedov-like phase no longer exists and the whole calculation becomes groundless. Note however that in Fig. 3 the energy imparted to the EPs has been assumed equal for both processes, which is most certainly not the case. Actually, if [FORMULA] (e.g. [FORMULA] and [FORMULA]), then process 1 is found to dominate Be production during the Sedov-like phase, regardless of the ambient density.

[FIGURE] Fig. 3. Comparison of the Be (and B) production efficiency through the forward and reverse shock acceleration processes, for two values of the ejected mass (10 and [FORMULA]). The ratio [FORMULA] (see text) is plotted as a function of the ambient density, assuming that both processes impart the same total energy to the EPs ([FORMULA]).

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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999
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