4. The spectra of secondaries
Secondaries are produced when heavy nuclei break up in collisions with thermal nuclei, mostly hydrogen in the interstellar medium. If the production is throughout the time of transport then the secondaries are expected to show a steeper spectrum; if on the other hand the spallation is mostly already in the source region, then the secondary spectrum ought to be the same as that of the primaries.
The theory for the transport and the production of secondaries is well developed (e.g. Garcia-Munoz et al. 1987), and uses the steady state leaky box model to describe the spectrum as well as the secondary production; a recent review and summary of the problems inherent in this endeavor has been given by Shibata (1995). In a steady state leaky box model the ratio of the spectra of the secondary elements to the primary elements such as the boron/carbon or the sub-Fe/Fe ratio directly give the energy dependence of the diffusion time scale out of the Galaxy. Such a calculation gives an energy dependence for the time scale of (e.g. Engelmann et al. 1990); one difficulty with this result is that it would lead to anisotropies of the cosmic rays and would not allow a direct extrapolation to energies beyond the knee. There is no other evidence for the relatively small scales in the interstellar medium of any turbulence other than consistent with a Kolmogorov spectrum, and also there is no evidence for a particular length scale corresponding to the Larmor radius at the knee. Therefore, we have proposed (Biermann 1996, 1997b), that the production of secondaries has three independent stages:
a) First of all, the giant molecular clouds themselves have a time evolution which cannot be neglected; they form out of smaller cloudlets faster than the Alfvénic time scale, and so trap the cosmic ray particles. These particles then leak out, with an energy dependence given by the Alfvénic turbulence in the cloud. We note that this corresponds to length scales far below what can be directly inferred from high angular resolution observations in clouds. Assuming that this turbulence also corresponds to a Kolmogorov spectrum (see, e.g., Goldstein et al. 1995), we then have a production rate of secondaries, which itself is a function of energy, and decreases with energy as . These particles are then released into the interstellar medium, where they are subject to diffusion out of the Galaxy, and so acquire another energy dependence of again , so as to give a total energy dependence of the secondary to primary ratio of , very close to that observed. This is of course limited to that energy range where diffusion is a useful concept in clouds, and when the time scale of diffusion is actually significantly shorter than the lifetime of the giant molecular cloud. Thus there is a critical energy per particle, which we estimate to near 20 Z GeV, above which this process becomes irrelevant. The detailed analytical derivation of this argument is given in full in Biermann (1996), and will be expanded upon in further communications.
b) Above this critical energy of an estimated 20 Z GeV, the Galaxy and its molecular clouds behave as stationary targets for cosmic ray interaction, and we come back to the canonical model, such as explained in Garcia-Munoz et al. (1987). Therefore, for these particles the secondary to primary ratio just acquires the energy dependence of interstellar turbulence, and so the ratio is expected to be . This dependence is a smooth continuation of the steeper dependence at lower energies.
c) There is an additional contribution, which arises from the source, and which should be energy independent: As shown in Nath & Biermann (1994b) supernovae that explode into winds, hit a surrounding molecular shell, and then produce secondaries with an approximate grammage of order 1 - 3 gm/cm2 (= column density traversed in the zig-zag path of charged particles in an inhomogeneous magnetic field). This then leads to a ratio of secondaries to primary particles which is energy independent, an aspect that has also been remarked by others (Drury et al. 1993).
We note that the ratio of these processes depends on the spallation cross section of the element considered; the relative strength of process (c) versus process (b) clearly depends on the element. For primary elements whose spallation cross section corresponds approximately to the effective grammage of process (c), the spallation of such an element is strongly affected by this process, while the spallation in the much higher grammage of processes (a) and (b) would then proceed to also influence the first generation secondaries, so as to finally produce secondaries of many generations down the element sequence.
Translating this result into the language common in the literature, this means that escape length as measured in gm/cm2 and escape time can no longer used synonymously. The escape time is proportional to in the relativistic range of particle energies. The escape length as a means to describe interaction has three different regimes, and the one relevant in the GeV/nucleon range is about , and here, in our simplistic model, .
All three contributions (a), (b), and (c) can be looked for in the data; the data are normally shown as escape length, which basically is the energy dependence of the secondary to primary ratio in the steady state leaky box model (see, e.g., Fig. 20 in Shibata 1995). With our model we can plot the same data as Shibata (1995) and can check whether we can fit either process (a) above combined with process (b), or process (c). We do this in Fig. 2.
In the first model combination (process (a) and (b)) we can satisfactorily fit the data with a grammage of about 19.8 gm/cm2 for process (a) and a grammage of about 6.6 gm/cm2 for process (b); the transition rigidity would be at 27 5 GV.
In the second model combination (process (a) and (c)) we can also satisfactorily fit the data with a grammage of about 29 gm/cm2 for process (a), rather close to canonical values, and a grammage of for process (c).
A judgement which model combination is a better to the data overall will be made below. The result is that model combination (a) and (c) appears to match the high energy data better for secondary elements; this means that spallation in time-dependent molecular clouds and in the molecular shells around massive star winds are the dominant contributors to the spallation observed.
We note again, as already emphasized in paper CR I that we use a turbulence spectrum in the interstellar medium, which has a single powerlaw over the entire range of length scales relevant for cosmic ray scattering, corresponding to energies up to a few EeV, and have argued that such a powerlaw is best approximated by a Kolmogorov law (Wiebel-Sooth et al. (1995); Wiebel-Sooth et al. (paper CR VI), in prep.; Biermann (1997b)).
4.1. The elements Li, Be and B
The nuclei lithium, beryllium, and boron in energetic cosmic rays are produced mostly from the breakup of carbon and oxygen nuclei. The combined spectrum is shown in Fig. 3.
We note that the spectrum has a fairly large error range, but is quite consistent with a source-related component. However, the individual spectrum of Boron suggests a steepening by , and so may be consistent with process (b), the spallation in clouds at energies which are so high as to render the time evolution of clouds irrelevant. On the other hand, the error bars for all three elements are so large, that a certain conclusion cannot be drawn, other than that processes (b) and (c) are both compatible with these high energy data, and process (a) is hard to reconcile with the data.
4.2. The sub-Fe group
The elements scandium through manganese, Sc, Ti, V, Cr and Mn are mostly produced in the breakup of the Fe-group elements Fe, Co and Ni. Their combined spectrum is shown in Fig. 4. Here we also show the spectrum, which would result from spallation in the interstellar medium at large, using the notion that for these high energy particles the time dependence of molecular clouds is no longer relevant (process (b)), but still dominant over source-related spallation (process (c)).
It seems that process (c), the source related spallation gives a much better fit to the data. But taken into account the statistical errors and systematic uncertainties involved in the data measurements, a final decision between the different models cannot be made at the moment. Within the errors, both model combinations are compatible with the data.
The spectral index for the sub-Fe elements is the same as that for the Fe-group elements, consistent with the notion that at these particle energies the spallation is dominated by source interaction.
The explanation put forward by Nath & Biermann (1994b) for the COMPTEL-observation of -ray emission lines from the Orion star forming region suggests that the interaction of the supernova shock running at first through the wind, and then hitting a shell of dense material provides a certain amount of near-source spallation of a grammage of order . Here we wish to estimate this grammage from the data to check for consistency. Since the charge Z for sub-Fe and Fe-group elements are very close, it is not important whether we use in this estimate energy/charge or energy/particle; we will continue to use the latter framework.
The spallation cross section of Fe to sub-Fe is about 300 mbarn, which corresponds to a grammage of , and so the ratio of for sub-Fe to Fe-group elements gives us in numbers for an inferred grammage of order 1 gm/cm2, quite consistent with the earlier estimate. When deriving the spallation during the transport this source-related grammage has to be subtracted.
© European Southern Observatory (ESO) 1998
Online publication: January 8, 1998