Suprathermal and energetic particle flux variability can be statistically characterized in a number of ways. As time histories of flux integrals (cumulative fluences) are relevant both for practical reasons (radiation effects) and for a better understanding of the production and propagation processes, simple quantitative methods are of some interest. Previous studies (e.g. Mewaldt et al., 2001)[1] showed that the character of cumulative fluence plots changed substantially with energy. While at several MeV/nuc a few solar particle events predominate, at much lower energies similar contributions from many separate events of various origins were found. We shall now use a simple parametric method for comparisons, and mention some other possibilities. As in the Kolmogorov hypothesis test, the maximum vertical distance of normalized cumulative plots from the straight line valid for a constant flux will be shown to be useful. Small values of that parameter (called K here for Kolmogorov) indicate variation in many small steps, while large K-values correspond to the dominance of a small number of large events. Below a few MeV/nuc K-parameters will be shown to decrease with decreasing energies. Extrapolations to small energies will be mainly discussed

Kiraly, P.

Mason, G. M.

Mewaldt, R. A.

Posner, A.

Rodriquez-Posner, J.

Caltech Authors - Main

29th International Cosmic Ray Conference Pune (2005)  1, 71-74 
 
Energy dependence of cumulative suprathermal and energetic 
par ticle fluence plots  
 
P. Királya, G.M. Masonb, R.A. Mewaldtc , A. Posnerd and J. Rodríguez-Pachecoe 
(a) KFKI Research Institute for Particle and Nuclear Physics, H-1525 Budapest P.O.B. 49, Hungary,  
(b) Dept. of Physics, University of Maryland, College Park, MD 20742, USA;  
now at Applied Physics Laboratory, Johns Hopkins University, Laurel, MD 20723,USA 
(c) Space Radiation Lab, California Inst. of Technology, Pasadena, CA 91125, USA 
(d) Southwest Research Institute, San Antonio, TX 78238-5166, USA 
(e) Space Research Group, Dept. de Física, Univ. Alcalá, 28871 Alcalá de Henares, Spain 
Presenter: P. Király (pkiraly@sunserv.kfki.hu), hun-kiraly-P-abs2-sh12-oral 
 
Suprathermal and energetic particle flux variability can be statistically characterized in a number of ways. As 
time histories of  flux integrals (cumulative fluences) are relevant both for practical reasons (radiation 
effects) and for a better understanding of the production and propagation processes, simple quantitative 
methods are of some interest.  Previous studies (e.g. Mewaldt et al., 2001)[1] showed that the character of 
cumulative fluence plots changed substantially with energy. While at several MeV/nuc a few solar particle 
events predominate, at  much lower energies similar contributions from many separate events of various 
origins were found. We shall now use  a simple parametric method for comparisons, and mention some other 
possibilities. As in the Kolmogorov hypothesis test, the maximum vertical distance of normalized 
cumulative plots from the straight line valid for a constant flux will be shown to be useful. Small values of 
that parameter (called K here for Kolmogorov) indicate variation in many small steps, while large K-values 
correspond to the dominance of a small number of large events. Below a few MeV/nuc K-parameters will be 
shown to decrease with decreasing energies. Extrapolations to small energies will be mainly discussed.  
   
 
1. Introduction 
 
Cumulative ion fluence plots for energies around 1 MeV/nuc are dominated by distinct steps, corresponding 
to coronal mass ejections (CMEs), to some particularly large flares, and sometimes to corotating interaction 
regions (CIRs). Due to the clustering of flares and CME's to active periods (corresponding to active regions 
on the Sun), and also to propagation effects, steps are somewhat smoothed out, and also display some 
substructure. Because of the huge (5 to 7 orders of magnitude) differences between extreme hourly fluxes, 
quiet time fluxes and instrumental background provide no appreciable contribution to cumulative fluences. 
 
Above several 100 MeV/nuc cumulative fluences are mostly due to galactic CRs, and their rate is much 
more uniform than for the SH origin. Although Forbush decreases, global merged interaction region effects, 
and also some relatively fast changes in heliospheric structure show up on the yearly cumulative plots, they 
are characterized by a steady, almost straight-line increase due to a slowly changing flux. 
 
We shall discuss how cumulative ion fluence plots change with energy around and particularly below 1 
MeV, and how those changes vary with the phase of the solar cycle. Apart from some outlook at the end, 
mainly ISEE-3 DFH-EPAS, and IMP-8 CPME data will be used.  For DFH-EPAS hourly mean ion fluxes a 
relatively short period (1978 to 82) will be used, when ISEE-3 was at L1, and not much affected by the 
magnetosphere. For IMP-8 data the elimination of magnetospheric effects is harder, particularly at the 
lowest energy (0.29 to 0.5 MeV). In a preliminary analysis done last year we found that more than 120 
hourly flux values (out of about 170 thousand) of the then available data had to be eliminated for fairly 
obvious magnetospheric interference or for occasional bit errors. Recently cleaned flux data sets put on the 
web by the CPME team in CVS format appear to have eliminated those problems. We are now using daily  
72          P. Király et al. 
  
mean fluxes from 1973 to 2000, and no data cleaning "by hand" was necessary. For the DFH-EPAS data set 
(for a more detailed discussion see Király and Rodríguez-Pacheco, 1999)[2], merely 7 hours of data had to 
be omitted for errors or inconsistencies, out of a grand total of more than 30 thousand. 
 
Although a huge amount of ion flux data obtained by more sophisticated instruments exists now, in this 
paper we restrict discussion to the above two. On the one hand, IMP-8 CPME data cover an extremely long 
time period, useful for looking at solar cycle effects (actually we have not yet used data for the last few 
years, because of yet incomplete cleaning). On the other hand, DFH-EPAS data at L1 appear to have an 
exceptionally good quality, as proved by the very regular flux distributions shown in our 1999 paper. We 
plan to return to a more comprehensive discussion of existing data sets at a later stage. 
 
 
2. Cumulative ion fluences and the K parameter  
 
In order to make cumulative fluence plots for different energies more readily comparable, first we normalize 
the plots by dividing fluences integrated up to a certain time by the total fluence for the chosen period (e.g. 
for a year or any pre-defined period). Then we compare a plot with the straight line that would be obtained if 
the same total fluence was produced in equal steps (corresponding to a constant flux). The plots thus 
obtained are illustrated on the example of 1978 DoY 228 to 365 ISEE-3 ion data for 8 energy bins (Fig. 1).  
Time
Fl
ue
nc
e
Fr
ac
ti
on
238 to 384 keV 384 to 620 keV 620 to 1000 keV 1000 to 1600 keV
36 to 56 keV 56 to 91 keV 91 to 147 keV 147 to 238 keV
Figure 1. Illustrative example  of  normalized fluence plots for suprathermal and energetic ions measured by the DFH-
EPAS instrument in late 1978, far upstream of the terrestrial  magnetosphere. It is clear that for that period maximum 
vertical distances between the two lines do decrease with decreasing energy, while the number of steps hardly changes. 
 
One measure of the deviation of the two lines is their maximum vertical distance. In fact, Kolmogorov (and 
Smirnov) used that measure in their famous hypothesis test for integral probability distributions, thus we call 
that measure "K parameter" (introduced in the present context by Király, 2004)[3]. Of course other measures 
could also be applied, e.g. the area between the two lines, the standard deviation of differences, the sum of 
maximum vertical distances on the positive and negative sides (for a discussion on the analogous problem in 
hypothesis tests, see Press et al., 1992[4]). At present we show the usefulness of K, leaving it open whether 
other methods might prove even more efficient. 
 
3. Energy dependence of K 
 
As qualitatively seen in Fig. 1, during that particular period K appears to decrease somewhat with decreasing 
energy. It is important to check how general that tendency is. The far upstream (mostly near-L1) data of 
DFH-EPAS contain 9 similar sets of mean hourly fluxes, for roughly half-year periods between late 1978 
and late1882. For each of the 9 periods and 8 energies K has been calculated. Although K values differ 
considerably for different periods, almost all of the 63 differences of K values for the same period and for  
Cumulative particle fluence plots                                         73 
 
subsequent energies are positive, i.e. higher energies were found to correspond to larger K. Averaging the 9 
K values for each energy, the plot given in Fig. 2 has been obtained. It is remarkable how linearly K appears 
to depend on log energy in the energy range covered 
by the DFH-EPAS detectors. That linearity can 
certainly not extend to a very wide energy range, 
because the value of K is restricted to the (0,1) 
interval. It appears still interesting that the linear fit, 
extrapolated downward in energy, has a zero 
intersect close to SW energies. Preliminary results 
for ACE data suggest a similarly monotonous and 
smooth, but less linear dependence of K on log E. 
Both the energy dependence and the absolute values 
of K for given energy, however, may also depend on 
the phase of the solar cycle, thus further analysis is 
justified.  
0 0.5 1 1.5 2 2.5 3
Log Energy in keV
0.1
0.2
0.3
0.4
Me
an
K
Pa
ra
me
te
r
Figure 2. Mean K parameter values for all DFH ion energies 
 
Daily mean ion flux data for the three lowest energies (0.29 to 0.5, 0.5 to 1, and 1 to 2 MeV) of the CPME 
instrument aboard IMP-8 have also been analyzed for the period 1974 to 2000. With very few exceptions, K 
parameters were again found to depend monotonously on energy. In addition to the mean K parameter 
(averaged over 28 years for each energy), also the mean SD parameters have been calculated, representing 
standard deviations of differences between normalized cumulative fluence plots and the straight lines for 
steady fluxes (see Fig. 1). Both sets of data points and their linear extrapolations are given in Fig. 3. It is 
interesting that downward extrapolations point again towards SW energies. 
0 0.5 1 1.5 2 2.5 3
Log Energy in keV
0.025
0.05
0.075
0.1
0.125
0.15
0.175
Me
an
SD
Pa
ra
me
te
r
0 0.5 1 1.5 2 2.5 3
Log Energy in keV
0
0.1
0.2
0.3
0.4
0.5
Me
an
K
Pa
ra
me
te
r
Figure 3. Dots represent mean K parameter values (left) and mean SD parameters (right)  for the three lowest IMP-8 
CPME ion energies, averaged over 28 years.  Linear extrapolations point to vanishing values at about 1 keV. 
 
 
4. Var iation over  the solar  cycle 
 
Because of the long time period covered, IMP-8 CPME data also provide useful information about the solar 
cycle dependence of both annual fluences and K parameters. Annual fluences were calculated from the 
normalization constants (the sums of available daily mean fluences times number of seconds in a day), and 
corrected for the fractional number of available days. Possible instrument saturation effects were not taken 
into account. K parameters were calculated as described above, without averaging over time. Fig. 4 displays 
the results. It is remarkable that K parameters appear to peak more at solar minima than at maxima. This can 
be ascribed to the presence of relatively large numbers of comparable events at solar maxima, while at solar  
 
74     P. Király et al. 
 
minima contributions from a few rare events may predominate. Some indications for Gnevyshev-type effects 
for solar maximum periods can also be seen on both plots. 
1975 1980 1985 1990 1995 2000
7
7.5
8
8.5
9
9.5
10
10.5
Lo
g
Ye
ar
ly
Fl
ue
nc
e
1975 1980 1985 1990 1995 2000
0.2
0.3
0.4
0.5
0.6
0.7
0.8
K1
,
K2
,
K3
Pa
ra
me
te
rs
Figure 4. Ion fluences (left) and K parameters (right) for 3 CPME energies (solid: lowest, dash-dotted: highest energy) 
 
5. Fur ther  outlook 
 
As stated earlier, the inclusion of further and more sophisticated data sets (particularly of those covering 
both the suprathermal component and various nuclei for long time periods) is justified. Also, the relative 
merits of various alternatives to the K parameter should be checked out. The possible relevance of the 
intriguing downward extrapolations discussed above to 
suprathermal tail formation should be examined in more detail. As 
to extrapolations to a wider energy range, linear functions of log E 
are certainly not applicable. A tentative Gaussian fit to K values 
derived from 28 years of CPME data for all 10 energies is 
displayed in Fig. 5. At energies much higher than a few MeV, 
contributions from slowly changing CR fluxes become gradually 
predominant. In fact, preliminary checks on neutron monitor data 
yield very small K values, decreasing with energy, in spite of 
clearly visible modulation effects. 
-3 -2 -1 0 1 2 3 4
Log Energy in MeV
0
0.1
0.2
0.3
0.4
0.5
Me
an
K
Pa
ra
me
te
r
  
Figure 5. Tentative Gaussian fit to mean K values for 10 CPME energies 
 
On the low-energy side, K values for SW fluences at 1 AU are also well below 0.1 (for Voyager SW fluence 
data, however, an increase of K values is found with increasing heliocentric distance, due to merged 
interaction regions, and to the gradual change of the correlation between SW speed and density from 
negative to positive values). In the eV region of typical solar photon energies, variations of the solar 
"constant" yield yearly K values as small as 10-5 to 10-4.   
 
 
6. Acknowledgements 
 
Financial and organizational support of the International Space Science Institute (Bern) is gratefully 
acknowledged. ISEE-3 DFH-EPAS and IMP-8 CPME teams are acknowledged for making data available.  
 
References: 
[1] Mewaldt, R.A. et al., AIP Conference Proc., Vol. 598,  p. 165 (2001) 
[2] Király, P. and J. Rodríguez-Pacheco, Proc. 28th ICRC, Vol. 6, p. 171 (1999) 
[3] Király, P., Int. J. for Modern Phys. A, accepted for publication (2005)  
[4] Press, W.H. et al., Numerical Recipes in Fortran, Second Edition (1992) 


Energy dependence of cumulative suprathermal and energetic particle fluence plots

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Energy dependence of cumulative suprathermal and energetic particle fluence plots

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