The direct inversion method of Margolis is used to explore the dependence of Z less than or equal to 28 source abundance determinations on the choice of the pathlength distribution. The source abundances do not depend strongly on the form of the truncation used, although some truncation at the lower energies (compared to a leaky box) is necessary. The decrease of mean grammage with increasing energy is required by the observations. The effects of errors and he use of other secondary to primary ratios is discussed

Margolis, S. H.

English

NASA Technical Reports Server

38
OG 7.2-1
THE MODEL-INDEPENDENCE OF COSMIC RAY SOURCE DETERMINATIONS
STEVEN H. MARGOLIS
McDonnell Center for the Space Sciences
Washington University
St. Louis, MO 63130 USA
ABSTRACT
The direct inversion method of Margolis (1983) is used to explore the dependence of Z __28
source abundance determinations on the choice of the pathlength distribution. The source
abundances do not depend strongly on the form of the truncation used, although some
truncation at the lower energies (compared to a leaky box) is necessary. The decrease of
mean grammage with increasing energy is required by the observations. The effects of errors
and the use of other secondary to primary ratios is discussed.
1. Introduction. In the past, discussions of the source abundances of cosmic rays have always been in
the context of a particular propagation history (pathlength distribution) and a sequence of calculations
designed to correct an initial estimate of the source abundances to match observations. Margolis
_1983) showed that the propagation equations of the GCR through the interstellar medium permit
a direct inversion. This inversion method is used here to determine the source abundances of the
Galactic Cosmic Rays in a model-independent manner. The errors of observation set the criteria
for consistency between models. The implications of this method for determining the pathlength
distribution (PLD) and the abundances are discussed.
2. The Analytic Procedure. Since the details of the analysis are presented in detail elsewhere, only a
brief outline is provided for the reader's convenience. The steady-state diffusion equation governing the
propagation of the cosmic rays can be separated into 2 equations, one governing the spatial distribution
of the particles, and the other the nuclear physics of fragmentation and decay. This split is frequently
referred to as the "weighted slab method." Neglecting energy loss reduces the composition equations
to a set of linear equations with energy appearing as a parameter. By expressing the solution in terms
of matrix exponentials, Margolis (1983, 1985) showed that the observed abundances and the source
abundances could be related for an arbitrary PLD. A graphical analysis of these solutions allows not
only the determination of the "best" source abundances from a set of observations, but also the errors
associated with the determinations based on those measurements.
_. The Overall Source Abu_da_aces. Since cosmic ray observations sample different energies, a sep-
arate decomposition can be produced for each energy. Previous determinations have not yielded a
single source composition for all energies (Dwyer etal., 1981) nor a single choice of propagation pa-
rameters covering all energies (Garcia-Mufioz etal., 1981, 1984; Cesarsky, Koch, and Perron, 1981;
Ormes and Protheroe, 1981). Such variations are also seen here. For discussion purposes, consider
the truncated exponential distribution of Tsao, Shapiro, and Silberberg (1973), whose shape is that
of a linear rise to some grammage At matched to an exponential tail of scale As. Simultaneous use of
the ratios B/C and (Sc-Mn)/Fe allows the determination of a single decomposition and a single set
of source abundances for each energy. For observations, use the 9 energies abow 2.54 GeV/nucleon
from the HEAO C2 exp_,'iment (Engelmann etal., 1983). The resulting plots of As and At in Figure
I show a relatively uniform variation of the escape length and some fluctuations in the truncation
length. These results suggest that the truncation is significant but does not vary with energy. Figure
1 also shows an alternate set of tracks, determined by looking at the ratio palr.B/C and V/Fe. The
escape lengths match reasonably well, but this second set suggests a truncation which decreases with
energy. How, then, should these results be compared, and what confidence can be placed on these
results?
An advantage of this formulation is the direct calculation of estimated errors from the errors of
observation, if reasonable values for these are known. For the data used here, the formal errors at the
source are generally smaller than the scatter of the computed values. The true errors are probably
somewhat larger. The Silicon measurements, whose average was used to normalize the observations
(Engelmann et al., 1983) show a scatter 4 times the size of the formal error of the average. A
comparison of the two sets of source abundances described above shows that, at worst, the difference
https://ntrs.nasa.gov/search.jsp?R=19850026560 2020-03-20T16:59:39+00:00Z
39
OG 7.2-1
6 B/C : 0 Figure 1. The parameters A, and
A_ are plotted as functions of en-(Sc-Mn)/Fe =05 _--x_-_ _ ergy per nucleon for two different
_, ---- V/Fe =0 model selection criteria. Note that
W(.94 A e "_.. the decrease of Ae is relatively uni-
<_ I _'_"_ __ form as a function of energy, but
'_ that the variation of the trunca-
3[ , /"'"- tion At is quite noisy. Both sets
0::<_ .... -- of parameters are determined us-2
L_. ,^_ ing the null B/C source ratio. The'At,__ solid lines are the parameters de-[ ----------'_ ",_ _-- termined by the null contour of the
0 _ "'1"" .... It Ix_ sub-Fe/Fe source ratio, and thedashed lines were based on the null
0.4 0.6 0.8 I 1.2 1.4 V/Fe source ratio.
LOG(ENERGY)
between the separate determinations is only twice the formal error. Given that, it seems reasonable
to produce the GCRS abundances shown in Table 1. These were formed from the simple averages
of the two sets of 9 decompositions, and the formal errors shown are derived under the assumption
of independence. The table also shows the errors derived from the scatter of the individual source
abundances, which shall be denoted as the informal errors. These two estimates serve as confidence
indicators.
Comparison of the abundances with both the formal and informal errors shows quite clearly
the distinction between well- and poorly-determined elements. The formal errors, when compared
with the abundances, indicate the significance to which the source abundances are determined within
the propagation model. The informal errors, when compared with the formal errors, indicate the
variation of the abundances across the separate determinations. The informal error is the better
indication of the significance of &source abundance, but as noted above, the intrinsic scatter of the
original normalizgtion is about 4 times the formal, statistical error. C (Z = 6) and Fe (Z = 26),
for example, have abundances which stand out quite clearly from the errors. The abundances of F
(Z = 9) and K (Z = 19), though negative, are comparable to the errors. Significant values cannot be
determined for those elements whose values are clearly consistent with zero abundance. The difficulty
comes in assessing a level of significance relative to the errors. The scatter of the normalization
suggests that an abundance is significant only when its value is more than 4 times its associated error.
The abundances which are significant under this criterion axe marked with an asterisk (*).
4. Discussio#. The contrast between the variations in the parameters selected and the abundances
calculated can be understood from the complementary variations displayed in Figure 1. The mirroring
of changes in the general trends of the escape and truncations lengths imply a well-defined average
grammage and a non-pathological PLD underlying the abundances. The observations can be matched
approximately by several models. Despite some uncertainty in the PLD, the source abundances vary
slowly. This uncertainty suggests two questions. How much error does the choice of a particular PLD
introduce? To what extent can the shape of the PLD be investigated using these techniques?
The choice of the form of the PLD need not be a significant source of error. The "no short
pathiengths" model employed by Garcia-Mufioz et al., (1984) yields an alternate sequence, with a
truncation parameter about half that of the corresponding parameter for the linear rise model; this
seems almost intuitive when the linear rise model is viewed as including a triangular distribution
with half the area of the rectangle ezcinded by the no short paths model. The more extreme form of
the truncation does not significantly change the source abundances estimated here. The abundance
values of Table I also compare quite well with others derived from the same measurements using other
distributions and other methods (Koch-Miramond et al., 1983). (Note again that the errors presented
40
OG 7.2-1
The Cosmic Ray Source Abundances
Atomic Abundance Formal Informal
Number Si = l0 s Error Error
4 10.2 15.2 51.7
5 1.7 26.2 12.0
6 * 4270.0 43.0 212.0
7 * 374_.0 24.5 24.6
8 * 505C.0 44.6 143.0
0 -5.6 3.6 7.1
10 * 660.0 9.7 26.9 Table 1. The Cosmic Ray Source Abun-
U * 60.8 4.9 0.1 dances determined using the truncated
12 * 1050.0 11.7 32.7 exponential distribution. The asterisks13 * 117.0 5.3 8.0
mark those elements with clearly signifi-14 * 1000.0 11.1 0.0
cant abundances. The Formal Errors are
15 7.8 2.5 2.1 those calculated from the errors of the16 * 143.0 5.5 8.3
18 separate energy/model combinations.17 2.8 2.7 2.7
The Informal Errors represent the scatter18 * 15.5 3.6 2.5
of those separate points.19 -1.1 3.5 5.4
20 * 63.7 4.9 5.0
21 -3.2 2.5 3.7
22 -11.8 4.3 8.5
23 0.9 3.1 2.4
24 11.1 4.7 7.8
25 * 20.5 3.8 3.9
26 * 058.0 12.7 34.5
27 3.1 1.0 0.6
28 * 55.1 3.3 4.5
do not include the cross section errors, but are only the measurement errors as modified by
effects of propagation.) There is a well-defined source composition which can be related to the
observations by a PLD whose average grammage decreases with increasing energy. Can anything more
specific be determined about the form of the PLD?
This method might allow some discrimination between distributions. The internal scatter, the
informal error, of the determined abundances differs among distributions. The consistency of a run
abundances with energy is an important measure of the quality of a particular decomposition. The
informal errors, examined over all charges, could offer a quantitative reason to select one distribution
shape over another. General features of the PLD might be distinguished on a comparative basis;
while not truly model-independent, such a procedure does not introduce bias beyond that involved in
formulating the propagation problem. The observations used here do not provide sumcient discrimi-
nation to distinguish between different forms of truncation, but only to confirm the necessity for some
the lower energies.
Determinations of a variable truncation in a strictly phenomenological way do not explain the
propagation of the cosmic ray nuclei. With some decrease in the associated errors, the methods
described here should be able to provide a test for quantitative theories such as that of Margolis
(1981) and Margolis and Bussard (1983) for the origin of the energy-dependent PLD. They point out
scattering from Alfv_n waves generated by the flux of cosmic rays leads naturally to a truncation
which decreases with energy and an average grammage which would decrease and then level off
increasing energy. Taking the results of Garcia-Mufioz et al. (1984) as evidence for a variable
truncation, one must look to the overall variation of grammage with energy as the next key test. The
prediction of Margolis and Bussard that the average grammage should level off as energy increases is
suggested, but not proved, by the curves of Figure 1. In the future, more sensitive decompositions at
41
OG 7.2-1
higher energy will resolve this question.
Much of the sensitivity of any decomposition depends on the range of elements through the
variation of the total inelastic cross section. Although the various types of distributions lead to
different values of precision, the increases in accuracy are comparable for models in the range allowed
by the observations. The potential accuracy for the linear rise, truncated exponential models used
here is increased by only about 20% if an S/P ratio based on the Te-Ba peak elements is substituted for
the sub-Fe/Fe ratio. A ratio based on Pt-Pb group elements might permit a 25% increase compared
to the sub-Fe/Fe ratios, all other factors being equal.
Unfortunately, all other factors are not equal. Currently available observations at the higher
charges are so much less precise that the overall selectivity is much poorer. The general increase
of partial cross sections with atomic number helps a little, but the errors associated with poorly
determined cross sections and multiple primary contributions to secondary elements outweigh all such
gains. This also limits the utility of the obvious generalization of the two-parameter search to more
complex distributions. At present, the search for the GCRS would be aided most by a more reliable
normalization of the observed abundances across the dynamic ranges of charge and energy.
5. Summary. The source abundances of the Galactic Cosmic Rays can be calculated by a reasonably
model-independent procedure. The decomposition depends on the relations between the production of
secondary elements and the relative abundances of the primaries. The analytic nature of the procedure
allows the straightforward assignment of errors to the source abundances calculated from the errors
associated with the observations. Although a model for the PLD must be used in the procedure,
there is no effective restriction on the form of the model, and the dispersion of the source abundances
calculated at several energies can be used as a measure of quality to select between distributions.
The technique has been demonstrated by calculating the source abundances of the nuclides Z < 28.
The source abundances are consistent with those determined by others using different methods. The
average grammage traversed by the cosmic rays decreases with increasing energy. The PLD required
by the observations is deficient in short pathlengths at lower energies. There are indications that the
trancation decreases with energy, but the errors associated with the observations are a little too large
to make a definitive assessment at this time.
6. Acknowledgements. The work reported here was supported by NASA Grant NAGS-448 and by
McDonnell Center research funds.
7. References.
Cesarsky, C. J., Koch, L., and Perron, C. 1981. Proc. 17th Int. Cosmic Ray Conf. (Paris), 2, 22.
Dwyer, R. D., Garcia-Mufioz, M., Guzik, T. G., Meyer, P., Simpson, J. A., and Wefel, J. P. 1981.
Proc. 17th Int. Cosmic Ray Conf. (Paris), 9, 222.
Engelmann, J., Goret, P., Juliusson, E., Koch-Miramond, L., Masse, P., Soutoul, A., Byrnak, B.,
Lund, N., Peters, B., Rasmussen, I. L., Rotenberg, M., and Westergaard, N. J. 1983. Proc. 18th
Int. Cosmic Ray Conf. (Bangalore), 2, 1T.
Garcia-Mu6oz, M., Guzik, T. G., Margolis, S. H., Simpson, J. A., and Wefel, J. P. 1981. Proc. 17th
Int. Cosmic Ray Conf. (Paris), 9, 195.
Garcla-Mufioz, M., Guzik, T. G., Simpson, J. A., and Wefel, J. P. 1984. Ap. J. (Letters), 280, L13.
Koch-Miramond, L., Engelmann, J., Goret, P., Julisson, E., Masse, A P., Soutoul, A., Perron, C.,
Lund, N., and Rasmussen, I. L. 1983. (Proc.) 18th Int. Cosmic Ray Conf. (Bangalore), 9, 275.
Margolis, S. H. 1981. Pro¢. 17th Int. Cosmic Ray Conf. (Paris), 9, 215.
Margolis, S. H., 1983. Pro¢. 18th Int. Cosmic Ray Conf. (Bangalore), 9, 267.
Margolis, S. H. 1985. Ap. J., (in press).
Margolis, S. H., and Bussard, R. W. 1983. Pro¢. 18th Int. Cosmic Ray Conf. (Bangalore), 9, 271.
Ormes, J. F., and Protheroe, R. J. 1981. Pro¢. 17th Int. Cosmic Ray Conf. (Paris), 3, 31.
Tsao, C. H., Shapiro, M. M., and Silberberg, R. 1973. Pro¢. 18th Int. Cosmic Ray Conf. (Denver),
1, 10T.


The model-independence of cosmic ray source determinations

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