THE STANDARD ENERGY SPECTRA FOR SOLAR ENERGETIC
PARTICLE MODELS
D.Mottl and R.Nymmik
Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia
ABSTRACT
The work will demonstrate that the >30 MeV solar energetic particle energy spectra are power-law functions of
particle momentum, in conformity with the theoretical concept of particle acceleration by a shock in a turbulent
medium. At <30 MeV, the particle propagation processes en route to the Earth orbit make the spectra gradually
harder. Some characteristics of the set of energy spectra, which are important in terms of constructing the solar
energetic particle models, are discussed.
INTRODUCTION
All the present-day energetic proton flux models are directly or indirectly relevant to the concept of the
particle energy spectra. This follows from the fact that all of the models are constructed using the integral particle
flux databases calculated from the measured differential fluxes on some assumptions concerning the form of the
energy spectra. Any particular spectral form has an impact on the confidence level of all the models to be
constructed. This is irrespective of how an integral particle flux dataset is used to construct a particular model for
solar energetic particles, whether it is the database of different-energy particle fluence or peak flux distributions,
whether it is the database of energy spectra distributions.
The solar energetic particle (SEP) energy spectra vary from event to event, with their form and parameters
depending on the measurement time and location. The particle flux simulation is usually restricted to the energy
spectra recorded in the Earth orbit beyond the magnetosphere, basing on two types of the spectra, namely the
energy spectra of the peak fluxes and fluences (the latter are the total particle number in a given event).
Different analyses of the experimental SEP flux data to find the functional form of the particle energy spectra
have yielded incompatible results. For example, having analyzed the energy spectra of protons and alpha particles
in separate SEP events, Freier and Webber (1963) concluded that the particle rigidity exponent was the best to
describe the energy spectra:
 R
dF
 C exp   
dR
 Ro 
(1)
This energy spectrum form is adopted in USA (NASA) and in Russia when describing the SEP fluxes influence.
At the same time, having analyzed an extensive dataset of the 4-80 MeV SEP event spectra, Van Hollebeke et
al. (1975) concluded that the dataset can also be described properly by the power-law energy function:
dF
 CE 
dE
(2)
The authors of the SEP peak flux catalogues (Bazilevskaya et al., 1986, 1990; Sladkova et al., 1998) are of the
same opinion, but they have to assume one or a few spectral knees in the power law spectra.
Having analyzed the particle fluxes in a set of SEP events, Nymmik (1993) demonstrated that the power-law
proton rigidity (R [MV]) function
dF
 R 
 C

dE
 239 

dE

(3)
was the best to describe the 30 MeV solar proton energy spectra. Later, Nymmik (1998,1999) demonstrated that
the power-law function of particle momentum (per nucleon)
 p 
dF

 C 
dE
 po 

dE ,

(4)
was the energy spectrum form capable of describing both solar protons and heavy ions. Here p  E ( E  2mc 2 ) is
momentum per nucleon,   p / p 2  m 2 is relative velocity; p0 = 239 MV/nucleon corresponds to E=30
MeV/nucleon.
Functions (3) and (4) were shown to be also valid in the <30 MeV range (p <239 MV/nucleon) on assumption
that

E ,

 30 
   o
(5)
where  o is the spectral index at E30 MeV;  is spectral droop index. This form of the energy spectrum may be
assumed to be due to such an acceleration mechanism as particle acceleration by a shock in a turbulent plasma
(Blandford and Ostriker, 1978). At lower energies, the spectrum gets harder due primarily to propagation of
particles from their generation point on the Sun to the Earth, because the hardening degree of the spectrum depends
of the positional relationship of the two points (Reames et al., 1997).
It is obvious, meanwhile, that formula (1) cannot be used to describe the >100 MeV energy spectra (see
below, Figure 1) ant that, in an attempt to describe the energy spectra by formula (2), any sufficiently broad energy
range has to be divided into intervals of different spectral indices.
It should be noted that the conclusions drawn from the results of studying the nonrelativistic and relativistic
energy spectrum ranges prove to be sometimes different. For example, the analysis of the energy spectra of the <80
MeV protons and the <80 MeV/nucl He, <40 MeV/nucl O, and <20 MeV/nucl Fe ions (Mazur et al., 1992) has lead
the authors to conclude that the spectra can best be described by the modified Bessel functions and are described
worse by a power-law energy function with exponential energy turnoff. In the latter case, the characteristic turnoff
level derived from experimental data proves to be below, or about, 10 MeV/nucl. An earlier analysis of proton
fluxes in five SEP events at <120 MeV (Ellison and Ramaty, 1985) found a 25 MeV turnoff effect in but a single of
the events. It should be noted that the latter conclusion is explicitly at variance with the results of the satellite (IMP,
GOES, METEOR) measurements of the proton energy spectra in the intermediate (30-500 MeV) range, where any
exponential turnoff has never been observed in the SEP event spectra.
The exponential cutoffs are also sought for in the relativistic energy spectra. For example, Lovell et al. (1998)
found the exponential turnoff with a characteristic energy Eo=77090 MeV for the peak flux spectra of the 29
September 1989 SEP event. At the same time, the analysis of experimental high-energy data (Lockwood et al.,
1990) did not indicate any turnoff at energies of up to 10 GeV.
In our opinion, the reliability of determining the energy spectrum form depends essentially on the energy
range selected in an analysis to distinguish between the spectral forms. The reliability seems also to be dependent
on the analysis methods, which must permit eliminating the impact of the statistical and methodological errors and
must be made within a sufficiently broad energy range. One of the possible techniques for reducing the ambiguity
in analyzing the SEP energy spectra is to use all the available experimental data simultaneously.
In such a way, we can extend the energy range of determining the spectral form, while the methodological
errors in separate measurement runs (due to differences in instruments and experimental techniques) can be found
by comparing among readings of different instruments.
The present analysis of the solar proton energy spectra differ from the earlier analyses in that not only a
broader energy range was analyzed, but also a set of SEP fluxes were subjected to a simultaneous statistical
analysis of lot of the events measured in different experiments.
THE PROTON ENERGY SPECTRA OF SEP GROUND LEVEL EVENTS (GLE)
Analysis of separate events
The functional form of the SEP energy spectra is analyzed basing on 13 ground level events (GLE) of Cycle
22 (see Table 1). The events have been supported by all the available experimental datasets of the integral peak
proton fluxes measured on IMP-8 (Internet, IMP-8), GOES (Internet, GOES) and METEOR satellites, on
stratospheric balloons, and with ground-based neutron monitors (all these data can be found in Sladkova et
al.(1998). The energy range involved extends to about four orders (from 1 MeV to 10 GeV). It should be noted that
the spectra of our selected SEP events are characterized by the hardest of the observed energy spectra because the
neutron monitors can detect the SEP fluxes if only the GeV solar particles are sufficiently abundant.
The experimental data were analyzed in terms of two functional dependences, namely, the power-law energy
spectrum and the power-law momentum spectrum. The exponential spectral form (Freier and Webber, 1963) is
disregarded here because it is evidently inapplicable to describing the experimental data (see Figure 1).
First of all, the energetic spectra of each event were analyzed. The spectral parameters (C, o, and α in Eqs. (4)
and (5)) were Monte-Carlo calculated by minimizing the standard deviation of the experimental data from an
approximating function. As an example, Figure 1 presents the experimental data of the 21 May 1990 SEP event and
shows the approximations of the data by the power-law functions of energy and momentum and by the senseless
rigidity exponent function.
Fig. 1. The integral proton peak fluxes of the 21 May 1990 SEP event. The markers are for METEOR (the
circles), GOES (the squares), IMP8 (the black circles), balloons (the triangles), and neutron monitors (the black
squares). The energy spectra are presented as power-law functions of momentum (rigidity) (Curve 1) and
energy (Curve 2), rigidity exponent (Curve 3).
From Figure 1 it follows that the approximation of experimental data by power-law energy function FE differs
little from the approximation by power-law momentum function Fp. In 12 of 13 events, nevertheless, the least
standard deviation is smaller in the case of approximation by power law function of momentum (F p) than energy
(FE).
Simultaneous analysis of a GLE set
The difference between the power law spectra of energy and momentum becomes evident in a simultaneous
comparison among all events. The analysis of the spectra with different spectral indices is possible in case we
compare among the deviations of the experimental data from their approximations. The approximations were
realized for the >30 MeV range and interpolated then to the smaller energies with a constant spectral index γo.
Figure 2 is a plot of the logarithms of the ratio of experimental data Fexp to the approximating functions Fp and
FE versus energy for the total dataset of 13 events.
Fig. 2. The energy dependences of log(Fexp/Fp) (the
left-hand ordinate scale and the lower part of the data
field) and log(Fexp/FE) (the right-hand ordinate and the
upper part of the data field). The dots are the
experimental data for all events and all energies.
Fig. 3. The energy dependences of log(Fexp/Fp)
(the left-hand ordinate scale and the lower part of the
data field) and log(Fexp/FE) (the right-hand ordinate
and the upper part of the data field. The circles are
the experimental dots averaged over energy ranges,
the asterisk is the 46-100 MeV range point.
.
On averaging the seemingly chaotic data sets over energy and over all events (Figure 3), we can conclude that,
in the case of Fp, the deviations of the 30 MeV experimental data are chaotically distributed on either sides of the
zero line, thereby indicating that the chosen approximation Fp is a good fit to the experimental data.
In the case of FE, the deviations of experimental data from FE are characterized by the systematic trend, which
conforms to the energy dependence of log(Fp/FE) (the solid curve in Figure 3).
The data displayed in Fig. 3 permit some important conclusions.
First, The energy spectrum above 30 MeV is a power-law momentum (rigidity in the case of protons) function
for the total set of SEP events analyzed.
Second, at E<30 MeV there occurs a spectral droop, compared with the power-law function at higher energies.
So, 30 MeV is the crucial energy for describing the energy spectra by power-law functions with constant spectral
indices in must cases on the Earth’s orbit. Whenever you wish to describe the experimental data by power-law
functions in an energy range that includes energies of 30 MeV or below, you take the risk of coming at numerous
unfounded conclusions (see below), if we additionally keep in main the insufficient experimental data exactness
and narrow energy range of measured experimental data.
Third, compared with Fp, a spectral turnoff is not observed within errors at relativistic energies, contrary to
what must be observed due to the restrictions imposed on the process of acceleration by a solar shock wave (Ellison
and Ramaty, 1985). Our estimated data of Figure 3 show that the index of the exponential restriction (if any) is
Ee>50 GeV (see the minor deviation of the lower dashed line from zero at ~10 GeV).
The fourth conclusion follows from the significant deviation of experimental data from the approximation in
the 46-100 MeV range (the dot within the asterisk), which comprises the satellite-measured proton fluxes. In this
case, the meters show definite methodological errors, which consist essentially in an increased additional counting
of foreign particles as energy rises.
The given methodological effect leads to a seemingly rising hardness of the energy spectrum of particles.
THE ENERGY SPECTRA MEASURED IN DIFFERENT EXPERIMENTS
We have systematically studied the above effect and calculated the indices for the total set of 13 events using
the results of measuring the differential peak fluxes in the events by CPME (IMP-8, energy range 0.46440 MeV)
and HEPAD (GOES-6 and 7, 4.2500 MeV, uncorrected and corrected), the integral peak fluxes on Meteor
satellite, and the peak fluxes measured by neutron monitors (430 MeV).
Fig. 4. The differential peak fluxes in the 25 July and 29 September 1989 events measured on IMP-8 (CPME). The
plot shows the approximations of the experimental data by power-law momentum functions (the solid lines)
together with the differential energy spectra of the two events for the total experimental dataset. The triangles
indicate the low-energy side of the neutron monitor energy range.
For the each event we calculated the energy spectra to be power law function of particle momentum and
evaluated the spectral indexes. Figure 4 shows the differential peak fluxes measured on IMP-8 (the CPME data,
0.46440 MeV) in two SEP events. Figures 5 and 6 show the same fluxes measured on GOES-7 (the HEPAD data,
uncorrected and corrected by authors).
The data displayed in the above figures permit some important conclusions.
First, the functions (4) and (5) approximate the experimental data perfectly, without leaving even the least
possibility for any exponential turnoff to be discerned in the spectra.
Second, the spectral indices inferred from the CPME and HEPAD (corrected) data (see Figures 4 and 5) of the
25 July (γIMP =2.56, γGOES*=2.10) and 29 September (γIMP =2.53, γGOES*=3.07) 1989 events are much smaller
compared with what was inferred from the total experimental dataset (4.09 and 4.02, respectively). This is quite
obvious, as demonstrated also by comparing the count rates of the utmost high-energy channels of the electronics
with the data of the low-energy-sensitive neutron monitors, in which case the difference reaches an order. At the
same time, the spectral indices inferred from the uncorrected HEPAD are close (γGOES=3.38, 25 July 1989) to, or
coincide (γGOES=4.05, 29 September 1989) with, the spectral indices determined from the total dataset (see Figure
6).
Fig. 5. The differential peak fluxes in the 25 July and 29 September 1989 events measured on GOES-7 (HEPAD,
corrected data). The plot shows the approximations of the experimental data by power-law momentum functions
(the solid lines) together with the differential energy spectra of the two events for the total experimental dataset.
The squares indicate the low-energy side of the neutron monitor energy range.
Fig. 6. The differential peak fluxes in the 25 July and 29 September 1989 events measured on GOES-7 (HEPAD,
uncorrected data). The plot shows the approximations of the experimental data by power-law momentum functions
(the solid lines) together with the differential energy spectra of the two events for the total experimental dataset.
The squares indicate the low-energy side of the neutron monitor energy range.
The peculiarity noted above implies that the given difference is due to the contribution from the foreign
particles (i.e., the particles incoming from beyond solid angle, or the secondaries produced in the matter of
telescope), which hit the semiconductor telescope detectors that record the highest-energy particles. Considering
the above, and having introduced the correction C for these effects to the input experimental data, we must have a
corrected high-energy channel count rate Nc below the uncorrected count rate Nu:
Nc=Nu-C<Nu ,
quite in agreement with the correction procedure as described on Internet site (INTERNET, GOES).
However, the comparison of the uncorrected GOES-7 data with the corrected data of the last three detection
channels shows that Nc>Nn. over the total GOES-6,7 dataset. This explains the great difference of such abnormally
corrected GOES data from the analysis results over the total experimental dataset with a significant fraction of the
neutron monitor data.
Table 1 presents the results of calculating the spectral indices inferred from the data of all the instruments used
in our analysis.
Table.1. The data on the parameters of 13 SEP events of Cycle 22. The numerals in the first columns are the year,
month, and day of the event commencement; r is the shock compression ratio; the symbol * indicates that the
measurement data have been corrected by the respective authors
Event
890725
890616
890929
891019
891022
891024
891115
900521
900524
900526
900528
910611
910615
Mean

total
4.09
4.96
4.02
5.20
5.56
4.58
4.03
4.55
4.31
4.28
4.14
4.77
4.92
4.570.14
r
3.75
2.53
3.94
2.36
2.17
2.90
3.91
2.93
3.29
3.34
3.63
2.69
2.56

NM
4.25
5.00
4.40
5.38
6.12
4.94
3.95
4.41
4.33
4.73
3.44
4.76
5.29
4.690.19

IMP-8
2.56
2.79
2.53
2.22
3.14
2.39
2.73
2.17
2.26
1.39
1.92
2.70
2.400.14

GOES-7
3.38
4.83
4.05
6.47
4.30
4.38
3.71
4.33
3.24
3.17
3.37
6.12
4.22
4.260.28

GOES-7*
2.10
3.91
3.07
5.53
3.47
3.49
2.79
3.42
2.43
1.98
2.48
5.63
3.30
3.350.73

METEOR
4.20
4.43
4.61
4.45
4.65
4.20
4.11
4.52
4.08
4.12
4.02
4.69
3.22
4.250.36
Analyzing the IMP-8 and corrected GOES-7 data shows that the spectral indices calculated from the IMP8measured differential peak fluxes and from the corrected GOES-7 data are at variance with the METEOR,
uncorrected GOES-7, and neutron monitor results, as well as with the total dataset. Table 1 presents the results of
calculating the spectral indices of the differential peak energy spectra (p), showing that the neutron monitors, the
METEOR, and uncorrected GOES-7 data (the GOES-6 data are the same) lead to the spectral indices that are close
to the indices obtained for the total experimental dataset. At the same time, the IMP-8 data and the corrected
GOES-6,7 data lead to the spectral indices that are almost two times as small on average.
The key point is that the total dataset lead to the energy spectra that are quite in agreement with the hypothetic
generation of particles by shocks, while the IMP-8 and corrected GOES-7 datasets of the same events exclude that
mechanism outright.
Indeed, in case the SEPs are accelerated by a shock in a turbulent plasma (Blandford and Ostriker, 1978) the
relation
r

p
p
 3
(6)
holds between the index p of the power-law momentum spectrum and the shock compression ratio r.
Considering the observed velocities of shocks near the Sun (Sheeley et al.,1985), which rarely exceed 2000
km/sec and given this particular shock acceleration mechanism, the spectral index p of accelerated particle fluxes
can but rarely be below 3.5, which is the fact if the total experimental dataset is analyzed. The METEOR,
uncorrected GOES-7 (and GOES-6), and neutron monitor data lead to the spectral indices that are close to the
indices obtained for the total experimental dataset. Analysis of the IMP-8 and corrected GOES-7 data shows that
the spectral indices calculated from the peak flux data of the satellites are at variance with each other and have to
be seriously analyzed and, maybe, corrected.
Irrespective of the measurement data confidence level, however, all the experimental data analyzed permit a
mathematically-correct approximation by power-law momentum functions, thus making that particular spectral
form promising to analyze the SEP event particle fluxes.
DOES ANY SPECTRAL TURNOFF EXIST AT LOW ENERGIES?
As shown above, with increasing energy, the relativistic-energy solar proton energy spectra measured in the
Earth orbit do not exhibit any marked softening of the power-law spectrum of proton momenta predicted in paper
Ellison and Ramaty (1985). Here we shall demonstrate that the spectrum does not soften with increasing energy at
low energies either. Fig. 7 shows the measured proton fluxes of the 7 and 21 June 1980 events, which were
presented in Ellison and Ramaty (1985) to confirm the exponential turnoff of the spectra at >10 MeV. We
approximated the fluxes by the functions (4) and (5) and by the power-law momentum functions (4) divided by the
exponent, which is believed (Ellison and Ramaty, 1985) to describe the spectral turnoff at as low energies as >10
MeV.
By Monte-Carlo varying the spectral parameters and least-squares determining their combination, we have
found the spectra presented in Figure 7 for both approximations.
In case formulas (4) and (5) are used, the standard deviations are smaller (0.037 and 0.020) compared with the
their values inferred from the formulas of Ellison and Ramaty (1985) (0.046 and 0.027). In the 7 and 21 June 1980
events, therefore, we see the typical SEP event power-law proton momentum spectra that get harder at <30 MeV,
as takes place for all other SEP spectra. Therefore, the power-law spectra with turnoff as energy increases in the
range of tens of MeV seem to be a groundless assumption that has never be realized for SEP spectra. It should also
be emphasized that the indices of the power-law momentum spectra of the 7 and 21 June 1980 events (p=6.99 and
p=4.72) are quite common in the power-law spectra of particle momentum.
Fig. 7. The measured peak proton fluxes in the 21 (the upper dots) and 7 (the lower dots) June 1980 events
from Ellison and Ramaty (1985). Approximations of the measurements by the functions (1) and (2) (the solid
lines) and by the power-law momentum functions (the dashed lines) with the exponential spectral droop as
energy increases, according to Ellison and Ramaty (1985). The A’-A and B’-B straight lines are the power-law
momentum spectra without spectral softening droop at <30 MeV.
Figure 8 shows the 7 and 21 June 1980 SEP events energy spectra together with the 29 September 1989 event
spectra. They are quite alike, namely, a power-law momentum function at >30 MeV with gradual hardening at <30
MeV. A comparison among the absolute intensities of the spectra can well be avoided because the 7 and 21 June
1980 event spectra were measured at 0.5 a.u., and the 29 September 1989 spectra at 1 a.u. The only principal
difference between the spectra is in the energy range, namely, the 7 and 21 June 1980 flux measurements were
restricted to high energies, thus making it possible to explain the experimental data at will. Figure 8 illustrates also
the physical cause of the droop effect at low energies as an effect of SEP propagation. The droop in the 29
September 1989 event is evidently larger compared with the same effect in the SEP events of 1980, which were
measured at a smaller distance from the Sun and whose low energy fluxes are less distorted (drooped).
Fig. 7. The peak proton fluxes in the 21 (the upper dots) and 7 (the lower dots) June 1980 events (Ellisson and
Ramaty, 1985) together with the 29 September 1989 event spectrum. The solid lines are approximations by Eqs.
(4) and (5). The dashed lines are the constant power law momentum index spectra extrapolated to low energies.
It remains only to note that the fairly numerous recent works postulate an exponential turnoff of SEP energy
spectra and derive the various conclusions on solar physics from the low-energy turnoff. At the same time, even a
passing acquaintance with the above-discussed experimental data shows that all the cases of alleged turnoffs have
never been supported by any actual turnoff occurrence and that at >30 MeV we have the ordinary power-law
momentum spectrum of SEP event protons or ions. Obviously, this fact makes the conclusions of the above works
contestable.
CONCLUSION
Any reliable analysis of experimental data has to be based on such a description of the SEP event particles
(protons and ions) that would be adequate to the nature of experimental data. This approach permits any erroneous
conclusion to be avoided when the measurement results suffer not only statistical, but also methodological errors,
while the particle fluxes proper are often recorded within a narrow energy range. In the circumstances, the standard
and repeatedly tested functional form of describing the spectra, namely, the power-law particle momentum function
with a constant spectral index above 30 MeV/(nucleon), is becoming more and more important to use. The
description is free of any distorting simplifications at relativistic and nonrelativistic energies and permits the
present-day experimental fluence and peak flux data to be described within a very broad energy range at a very
high confidence level. When used to analyze experimental data, the given spectral form makes it possible to avoid
such erroneous concepts as the exponential turnoff in the particle energy spectra at nonrelativistic and moderately
relativistic energies (<10 GeV).
ACKNOWLEDGMENT
This work has been supported partly by the grant INTAS-00-629
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E-mail address of R.A.Nymmik nymmik@sinp.msu.ru
Manuscript received: 19 October 2002, revised
; accepted