3. The data
Here we describe the basic data set as available in the literature for individual elements, give several plots, and tables. The sources for the data are given in Table 4 of the Appendix. The spectra have been fitted above an energy of GeV, where Z is the charge of the nucleus, in order to minimize the effect of solar modulation; we repeated the exercise with 20 and GeV as the lower boundary without finding a significant change (see, e.g. Garcia-Munoz et al. 1986; Evenson & Meyer 1984; Seo et al. 1991). The numbers given in the tables are from the first boundary above. For the fit we have used the CERN-library (MINUIT - Function Minimization and Error Analysis, CERN Program Library D506). The spectra fitted are of the form
where E is the energy per particle.
We note that the spectral fit takes the experimental errors as given by the original authors into full account. The fairly small final error is just due to the assembly of all existing data and their critical combination. We have divided the results into three tables. Table 1 gives the results from experiments where the elements can be separated. Table 2 gives the results where groups of elements have been measured together. Table 3 gives the results where we have added the data for groups of elements together, both based on individual measurements as well as from grouped measurements. In addition the spectral indices for the various elements as a function of nuclear charge number Z are shown in Fig. 1.
The data for He illustrate systematic errors in the normalization of the flux: the lower energy data by Seo et al. (1991) give a spectral index of , here we obtain , which is quite consistent. The graphs in Biermann et al. (1995a) clearly show that the slopes of the spectra of He for individual measurement campaigns agree, while their overall normalization is different. When combining data from different measurement techniques and campaigns this can lead to erroneous spectral indices, since different experiments cover different energy ranges. For example, when we apply the fit by omitting the data of Ryan et al. (1972), we obtain a spectral index for He of .
Table 2 shows the results of the fits in the case when experiments measured groups of elements directly. The comparison of Table 1 and 2 illustrates systematic errors still inherent in the data; for instance the individual elements Cl to Ca have all been measured by Engelmann et al. (1985, 1990), with Ca also measured by Ichimura et al. (1993a, b, c) and Kawamura et al. (1990). The group of elements Cl to Ca has been measured by Simon et al. (1980), Asakimori et al. (1991), Burnett et al. (1990a, b), JACEE (1993), and Ichimura et al. (1993a, b, c). Thus the two data sets are independent and allow to estimate errors. The difference illustrates that the systematic errors can be larger than the statistical errors, and that any agreement between spectral indices of prediction and data, as well as data for different elements has to be taken with some caution. In view of the diversity of the various experiments that contribute to the individual element spectra and the difficulties to give correct estimates of the systematic uncertainties, we consider the values for obtained for the simple powerlaw fits as quite satisfactory. With this in view we combine elements to determine the power law parameters for groups of elements as shown in Table 3.
Table 3 shows the fits to all data available for the various groupings. We note that the particle spectra are close to the prediction: for hydrogen the prediction was , and the data give ; for the heavier elements He through Ni the prediction was , and the data give .
We conclude that despite the many simplifications in the theoretical approach the data are quite consistent with the predictions.
We note that for the Fe-group elements we have ignored here the spallation correction from interaction in the interstellar medium, which produces a slight flattening; although not significant, the data are consistent with such a moderate flattening of these nuclei.
© European Southern Observatory (ESO) 1998
Online publication: January 8, 1998