4. Spallation reactions after the Sedov-like phase
At the end of the Sedov-like phase, the EPs are no longer confined and leave the SNR to diffuse across the Galaxy. At the stage of chemical evolution we are considering here, there are no or few metals in the interstellar medium (ISM), so that energetic protons and particles accelerated at the forward shock will not produce any significant amount of Be after (although Li production will still be going on through reactions). In the case of the second process, however, the EPs contain CNO nuclei which just cannot avoid being spalled while interacting with the ambient H and He nuclei at rest in the Galaxy. This may be regarded as a third process for Be production, which lasts until either the EPs are slowed down by Coulombian interactions to subnuclear energies (i.e. below the spallation thresholds) or they simply diffuse out of the Galaxy. Since the confinement time of cosmic rays in the early Galaxy is virtually unknown, we shall assume here that the Galaxy acts as a thick target for the EPs leaving the SNR, an assumption which actually provides us with an upper limit on the spallation yields.
Unlike the first two processes evaluated above, this third process is essentially independent of dynamics. Thus, time-dependent calculations are no longer needed and, from this stage on, the calculations made by Ramaty et al. (1997) or any steady-state calculation is perfectly valid. In particular, the ambient density has no influence on light element production, since a greater number of reactions per second, as would result from a greater density, implies an equal increase of both the spallation rates and the energy loss rate. Once integrated over time, both effects cancel out exactly, and in fact, given the energy spectrum of the EPs, the efficiency of Be production (and Li, and B), expressed as the number of nuclei produced per erg injected in the form of EPs, depends only on their chemical composition.
Results are shown in Fig. 4 for different values of the source abondance ratios, and , allowing one to derive the spallation efficiency for any composition. Two-steps processes (such as followed by ) have been taken into account. Test runs show good agreement with the results of Ramaty et al. (1997).
As can be seen on Fig. 4, pure Carbon and Oxygen have a production efficiency of about , while this efficiency decreases by at least a factor of 10 for compositions with hundred times more H and He than metals (or about ten times more by mass). According to models of explosions for SN with low metallicity progenitors, the average ratio among the EPs should indeed be expected to be , unless selective acceleration occurs to enhance the abundance of the metals. As a consequence, efficiencies greater than should not be expected, so that a production of atoms of Be requires an energy of erg to be imparted to the EPs. This seems very unlikely considering that the total energy available in the reverse shock (the source of the EPs) should be of order one tenth of the SN explosion energy, not to mention the acceleration efficiency. Moreover, a significant fraction of the energy originally imparted to the EPs has been lost during the Sedov-like phase of the SNR evolution through adiabatic losses.
To evaluate the `surviving' fraction of energy, it sufficies to go back to Eq. (14), which indicates that when the radius of the shock is multiplied by a factor , the momentum p of all the particles is multiplied by a factor . It is worthwhile noting that, because of their specific momentum dependence, adiabatic losses do not modify the shape of the EP energy spectrum. In our case, , so that when all momenta p are divided by a factor , the distribution function is divided by the same factor . To see that, the easiest way is to work out the number of particules between momenta p and after the momentum scaling. This number writes , where is the new distribution function. Now must be equal to the number of particles that had momentum between and , which is, by definition, . Equating and yields the result .
Putting all pieces together, we find that when the shock radius R is multiplied by a factor , the distribution function and, thus, the total energy of the EPs are multiplied by . Now considering that R increases as during the Sedov-like phase, we find that the total energy of the EPs decreases as . Note that this is nothing but an other way to work out the decrease of the spallation rates for our second process during the Sedov-like phase (cf. Sect. 3.3). Finally, we find that a fraction of the initial energy imparted to the EPs is still available for spallation at the end of the Sedov-like phase. This factor is plotted on Fig. 5, as a function of the ambient density. It can be seen that for , the energy available to power our third process of light element production has been reduced by adiabatic losses to not more than one third of its initial value, and less than one half for densities up to . Clearly, high densities are favoured (energetically) because they tend to shorten the Sedov-like phase, and therefore merely avoid the adiabatic losses.
Although the total number of Be atoms produced is the relevant number to compare with the observations, one can also derive the number of Be produced per CNO nuclei ejected by the SN, , which makes easier the comparison between the third process and the two others. Let us do this in the case when the composition of the ejecta satisfies , which corresponds to about of CNO ejected for a total ejected mass of . The Be production is obtained from Fig. 4. With an initial energy of erg and an energy reduction factor , we get atoms of Be. Dividing by the number of CNO atoms freshly synthesized, we obtain . With and , we finally get . Estimating similarly in the case when , we obtain the same number again. The Be production efficiency is indeed higher in this case, but the number of CNO nuclei ejected is greater too. From this value of , we can claim that the third process is actually the most efficient in most cases (see Fig. 6), which is in itself an interesting conclusion.
© European Southern Observatory (ESO) 1999
Online publication: May 6, 1999