size distrtibutions of the >10 mev solar proton events

SIZE DISTRIBUTIONS OF THE >10 MeV SOLAR PROTON EVENTS

L. I. MIROSHNICHENKO∗, B. MENDOZA and R. PÉREZ ENRÍQUEZ∗∗Instituto de Geofísica UNAM, 04510 México, D.F., México

(Received 29 September 1999; accepted 19 March 2001)

Abstract. As has been recognized recently, data on size (frequency) distributions for different sets ofsolar flare parameters are very helpful in modeling flare and acceleration processes. Relying upon anew arising paradigm of particle acceleration at different sources at/near the Sun (flares, shock waves,etc.), in this paper, we analyze long-term data (1955–1996) from several Catalogues of Solar ProtonEvents (SPEs). Above 1 p.f.u. (proton cm−2 s−1 sr−1) of the >10 MeV protons, we have separatedin all 320 events associated with identified sources (flares). Then, within this database of flare-relatedevents, a second group (a subgroup) has been formed of the 159 events, additionally having a certainor probable sudden storm commencement (SSC) association (SSC-related events). The basic resultis that the power-law slope of size distribution for the 320 flare-related events at integral energyintensities is about 1.37 ± 0.05 over the entire range of the proton intensities, from 1.0 to 105 p.f.u.This slope is in general agreement with earlier analyses of integral energy distributions, but steeperthan that for differential energy distributions. A second result is that the SSC-associated events havea double power-law distribution with two different exponents, near 1.00 ± 0.04 and 1.53 ± 0.03below and above 103 p.f.u., respectively. The longitude distributions of the proper sources for thesetwo groups display different behaviour suggesting different origins of the two particle populations.A certain difference was also found to exist in the slopes of integral size distributions at >10 MeVand >500 MeV. This may point to a dependence of slope on the proton energy under consideration.

1. Introduction

Data on size (frequency) distributions for different sets of solar flare parameters(peak fluxes and/or energy fluences in X-ray and radio wave bursts, in proton andelectron emissions, etc.) have been extensively reported in recent decades (e.g.,Kurt, 1989, 1990; Crosby, Aschwanden, and Dennis, 1993, and references therein).These data were recognized to be very helpful for the resolution of some problemsrelated to flare modeling (e.g., Rosner and Vaiana, 1978; Lu and Hamilton, 1991;Litvinenko, 1996, 1998; Wheatland and Sturrock, 1996; Wheatland and Glukhov,1998) and particle acceleration (e.g., Hudson, 1978; Miroshnichenko, 1995; Litvi-nenko, 1996; Aschwanden, Dennis, and Benz, 1998). In particular, it has beenfound (Crosby, Aschwanden, and Dennis, 1993) that the frequency distributions ofvarious solar flare phenomena show a power-law shape consistent with the stochas-tic model of Rosner and Vaiana (1978), suggesting that the flare energy build-upis governed by exponential growth. The measured distributions of flares are also

∗On leave from IZMIRAN, Troitsk, Moscow Region, 142190, RUSSIA∗∗Campus UNAM-Juriquilla, 73200, Juriquilla, Queretaro, MEXICO

Solar Physics 202: 151–171, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

152 L. I. MIROSHNICHENKO, B. MENDOZA, AND R. PEREZ ENRIQUEZ

consistent with those predicted by computer simulations of avalanche models (Luand Hamilton, 1991) that are governed by the principle of self-organized criticality.

On the other hand, in the development of the avalanche model of solar flares,Wheatland and Sturrock (1996) suggested to take into account the finite size ofactive regions and then compared their model to the distribution of hard X-raybursts observed by the ICE space probe. Recently, this work has been modified byWheatland and Glukhov (1998) to include a growth rate of free energy in activeregions. The energy release through magnetic reconnection in multiple currentsheets is used by Litvinenko (1996) as an alternative suggestion to the avalanchemodel for flares (Lu and Hamilton, 1991). Notably, a power-law flare distributionwith a slope of 1.5 can be deduced from scaling-law arguments only as followsfrom dimensional analysis by Litvinenko (1998).

Unlike flare electromagnetic emissions, data on interplanetary particle eventsare still rather poor and discrepant, their distribution functions being discerniblydifferent from those for flare electromagnetic emissions. For instance, the sizedistributions of electron events (the peak electron flux) reveal the following slopes:1.50 ± 0.20 at Ee > 17 keV (Potter, Lin, and Anderson, 1980) and Ee > 45 keV(Belovsky, Kurt, and Ochelkov, 1979); 1.35 ± 0.15 (Ee > 70 keV; Belyakov et al.,1984; Daibog et al., 1989); 1.46 ± 0.15 (Ee = 0.5–1.1 MeV; Belovsky, Kurt, andOchelkov, 1979); 1.30 ± 0.07 (Ee = 3.6–18.5 MeV; Cliver, Reames, and Kahler,1991). These values, however, may be reconciled with those for energy fluencesof flare electromagnetic emissions (e.g., Kurt, 1989, 1990; Crosby, Aschwanden,and Dennis, 1993). This is true, at least, for electrons with energy >70 keV, whichin the non-thermal interpretation are considered to be responsible for hard X-raygeneration. In summarizing the results on size distributions of electromagneticfluences and fast electron fluxes, Kurt (1990) has concluded that both types ofemissions can be described, in general, by a differential power-law function witha slope of 1.45 ± 0.15. More accurate and extended analysis of all available datashows (Crosby, Aschwanden, and Dennis, 1993) that solar flares, indeed, exhibitvery similar distributions at different wavelengths, such as in radio, soft X-rays orhard X-rays. The slope of the distribution functions, however, is dependent on theflare parameter under study. Typically, the slopes are of 1.7–1.8 for the peak countrate (or peak flux), 1.4–1.6 for flare energies, and about 2.0 for flare duration.

As to the proton peak flux distributions at the Earth’s orbit, they turn out tobe significantly flatter than those obtained for other parameters of solar flaresmore representative of the total flare energy. Setting the differential distributionin a power-law form, the following slopes have been obtained: 1.15 ± 0.15 inthe energy range of 20–80 MeV (van Hollebeke, Ma Sung, and McDonald, 1975);1.40 ± 0.15 at >10 MeV (Belovsky and Ochelkov, 1979); 1.35 ± 0.15 at >25 MeV(Kurt, 1989); 1.13 ± 0.04 in the range of 24–43 MeV (Cliver, Reames, and Kahler,1991); 1.47 and 2.42 at >10 MeV (Smart and Shea, 1997) at the peak flux intensitybelow and above 103 p.f.u., respectively; 1.27 ± 0.02 and 1.38 ± 0.03 (Mendoza,Melendez-Venancio and Miroshnichenko, 1997) at the peak fluxes of the >10 MeV

SIZE DISTRIBUTIONS OF PROTON EVENTS 153

protons above 1 p.f.u. and 10 p.f.u., respectively (1 p.f.u. = 1 proton flux unit =1 proton cm−2 s−1 sr−1). It is appropriate to mention here an apparent distinctionbetween the slopes of distributions for differential energy intensities and those forintegral energies.

The clear differences between the slopes of size distributions for proton, elec-tron and electromagnetic flare emissions were shown to be very important (e.g.,Miroshnichenko, 1995) when interpreting an initial stage of acceleration of so-lar cosmic rays (SCR). More recently, in the light of a new arising paradigm ofparticle acceleration (e.g., Reames, 1995, 1996) at different sources at/near theSun (flares, shock waves, etc.), we started an extended statistical study of solarproton events (SPEs) (Mendoza, Melendez-Venancio and Miroshnichenko, 1997;Melendez-Venancio, Mendoza, and Miroshnichenko, 1998). The main purposes ofthis paper are: (1) to compile a homogeneous sample of the >10 MeV proton dataover a long period of time; (2) to obtain the size (frequency) distributions for protonevents relying upon more abundant SPE statistics than in previous works, and (3)to outline some new possibilities to treating a slope problem.

Using peak fluxes for the >10 MeV protons measured near the Earth duringthe period 1955–1996, we separate, first of all, a group of 320 events associatedwith flares (flare-related events). Then, within this sample, a second group (a sub-group) is formed of 159 events which have, additionally, a certain or probablesudden storm commencement (SSC) association (SSC-related, or shock-associatedevents). Also, we draw into consideration some other data for the >10 MeV pro-tons (Kahler et al., 1991; Smart and Shea, 1997), as well as peak fluxes of the>500 MeV protons (Miroshnichenko et al., 1995) derived from the observations ofthe Ground Level Enhancements (GLEs) of solar cosmic rays (Section 2). Our data-base of 320 events is analyzed in Section 3 where we obtain two size distributionsfor both groups (320 and 159 events), depending on the peak proton intensity, overthe entire mentioned period. Furthermore, we construct an integral distribution fortotal statistics of 320 events and, for a comparison, two other integral distributionsare obtained, for more limited data samples of Kahler et al. (1991) and Mirosh-nichenko et al. (1995). Our preliminary results (Mendoza, Melendez-Venancioand Miroshnichenko, 1997; Melendez-Venancio, Mendoza, and Miroshnichenko,1998) are partly revised and specified here. In Section 4 we compare our resultswith previous ones and outline some ways of possible interpretation. Section 5contains a brief summary and the main conclusions of this study.

2. Data Selection

Several problems usually arise when performing this kind of study, the principalthree of them being the galactic cosmic ray (GCR) background near the Earth’sorbit, SPE selection from measurements, and identification of SPE sources at theSun. In principle, the galactic background may fix the limit of detectability of event


intensity. For example, to separate the SPEs, van Hollebeke, Ma Sung, and Mc-Donald (1975) assumed a limit value of 10−4 p.f.u. MeV−1 to be the backgroundproton flux above 20 MeV, their study being limited to 163 small and moderatesized events (up to 10 p.f.u. MeV−1) measured on IMP 4 and 5, and only 107 eventshave been identified with proper sources at the Sun. Notice, however, that in fact,to minimize the effect of the spectral dependence upon longitude, van Hollebeke,Ma Sung, and McDonald (1975) used the peak intensity measured at � 40 MeVas a characteristic of the event size. Cliver, Reames, and Kahler (1991) considered92 events with a peak fluxes from >10−3 to 102 p.f.u. MeV−1, though the quietbackground on IMP 8 in the range of 24–43 MeV was <2 × 10−4 p.f.u. MeV−1.According to recent model estimates (Nymmik, Panasyuk, and Suslov, 1995), GCRbackground in the range of 20–80 MeV changes from 1.5 × 10−6 to 6.0 × 10−5

p.f.u. MeV−1 depending on the level of solar activity. In general, the situation issuch that the larger detectors can achieve better signal to noise ratio and lowerSPE thresholds. Besides, the recorded background can be effectively eliminated byspecial methods for background reduction (e.g., Valtonen, Kecskemety, and Kiraly,1999).

To select SPEs appropriate for the present study, we used a threshold (digit)integral intensity >1 p.f.u. at the peak time of the event and threshold energy of>10 MeV. Such a standard criterion was accepted in several SPE catalogues pub-lished in 1975–1998 (Dodson, Hedeman, and Kreplin, 1975; Akinyan, Bazilevs-kaya, and Ishkov, 1983; Bazilevskaya, Vashenyuk, and Ishkov, 1986, 1990; Slad-kova, Bazilevskaya, and Ishkov, 1998). The first of them compiled by Dodson,Hedeman and Kreplin (1975) contains 352 events measured in 1955–1969, withthe >10 MeV proton flux in excess of 0.1 p.f.u. and 0.01 p.f.u. before and since De-cember 1965, respectively, including about 200 events in excess 1 p.f.u. Four othercatalogues (Akinyan, Bazilevskaya, and Ishkov, 1983; Bazilevskaya, Vashenyuk,and Ishkov, 1986, 1990; Sladkova, Bazilevskaya, and Ishkov, 1998) comprise onlythe >1 p.f.u. events, having a common numeration, with a total number of 334SPEs for the period 1970–1996.

When preparing those catalogues, it was a very serious problem to identify acertain event with a proper source at/near the Sun. In fact, each catalogue is aresult of tedious research work carried out by the same standard rules and pro-cedures. The authors of the catalogues undertook a systematic survey of the SPEdata accumulated from extended measurements by numerous spacecraft during thelast four cycles of solar activity, taking care in selecting data from instruments thathad a large dynamic range and were not likely saturated. Those instruments weredesigned to either directly measure the >10 MeV proton flux or provide data fromwhich equivalent integral flux values could be derived. For some of the early events(see, e.g., Dodson, Hedeman, and Kreplin, 1975; Smart and Shea, 1997) the authorsused spectral fitting procedures to construct the >10 MeV flux from available dataat other energies. Thus, a series of procedures of this kind, together with some othermethodical precautions (for details see, e.g., the descriptions of the catalogues by


Bazilevskaya, Vashenyuk, and Ishkov, 1990), provide the homogeneity of the dataset for the proposed statistical analysis, in spite of a great variety of different datasources, instruments, methods of measurements, etc.

For the present study we selected only the >1 p.f.u. events with either a certainor probable association with a flare, according to the standard catalogues’ criterion,the overwhelming majority of originating flares having occurred within the range ofheliolongitude of 90◦ E–90◦ W. As a result, our basic statistics comprise 320 SPEs,including 118 events of 1955–1969 (Dodson, Hedeman, and Kreplin, 1975), 63events of 1970–1979 (Akinyan, Bazilevskaya, and Ishkov, 1983; Bazilevskaya,Vashenyuk, and Ishkov, 1986), 71 events of 1980–1986 (Bazilevskaya, Vashenyuk,and Ishkov, 1990), and 68 events of 1987–1996 (Sladkova, Bazilevskaya, andIskov, 1998). Notice that our database of 320 SPEs contains also the data on 40Ground Level Enhancements (GLEs) of solar cosmic rays (SCR) from the sourceson the visible solar disk, as well as on the 11 GLEs from identified (derived)behind-the-limb flares. Then, within the sample of 320 events, a second group (asubgroup) was formed of 159 events which have, additionally, a certain or probablesudden storm commencement (SSC) association (SSC-related, or shock-associatedevents).

To select shock-associated events we used the results of their identificationgiven in the SPE Catalogues where the prompt and shock components have beenassigned to a single complex SPE. As a typical case, it may be mentioned theevent of 4 August 1972 (Akinyan, Bazilevskaya, and Ishkov, 1983; Bazilevskaya,Vashenyuk, and Ishkov, 1986). In all those cases we took into account the only one(highest) value of intensity as a measure of the event size. It was also implied thatcomplex proton events are probably caused by combined effect (superposition)of the proper flare and CME-driven shock (e.g., Cliver, 1996), the latter beingmanifested at the Earth as a SSC. At the same time, we kept in mind the certainevidence (e.g., Reames, 1996) that the largest and most energetic particle eventsat the Earth’s orbit are associated with the CME-driven shocks. In the subsequentanalysis, the two groups of events at the peak intensity of >1 p.f.u. will be consid-ered separately, all 320 SPEs being treated as flare-related and 159 of them, besidesthat, as shock-associated events.

Due to space limitations for this paper, we do not reproduce here all our sta-tistics in the form of separate tables. Nevertheless, to be sure of the validity ofthe proposed separation and to demonstrate selected data samples, we constructedthe longitude distributions of identified sources for both groups of SPEs, includingthose associated with the behind-the-limb flares. Figure 1 shows the distributionsfor all 320 events and, separately, for 76 and 159 shock-associated events observedin 1970–1986 and 1955–1996, respectively. One can see a distinct difference in thesource locations on the solar disk between the SPEs of the first and second groups:a number of events of the first group grow steadily with longitude to the westlimb of the Sun, the events of the second group being distributed more uniformly,with a maximum at about 30◦ W. Although the histograms in Figure 1 have been


Figure 1. Histograms of heliolongitude distributions for identified sources of the 320 flare-relatedevents (light, 1955–1996), 76 shock-associated events observed in 1970–1986 (filled), and 159shock-associated events detected at the Earth’s orbit in 1955–1996 (hatched).

obtained by more extended sets of data than in the earlier works, they are not yetinformative enough for more detailed analysis because we restricted ourselves toonly one energy threshold of solar protons (>10 MeV) at the intensity level of>1 p.f.u.. Also, on account of rather poor statistics of the events within differentlongitude intervals, we did not use those data for doing power-law fits on differentlongitude bins (anyway, this task is out of the scope of the present study).

For a comparative analysis, we also used 66 events by Kahler, Shea, and Smart(1991) for the >10 MeV protons measured in 1976–1990, with a peak intensity of>10−2 p.f.u. Furthermore, we considered 196 events of Smart and Shea (1997) forthe >10 MeV protons detected from January 1967 to February 1994, with a peakintensity of >10 p.f.u. corresponding to the conventional criterion of large SPE(e.g., Smart and Shea, 1989). In this context, it is necessary to mention a problemof using the proton data measured by the GOES spacecraft and compiled by theNOAA Space Weather Operations (SWO). As was noted by Kahler (1993), somelarge proton events may have a combined temporal structure due to the fact that theGOES low-energy particle detectors can be penetrated by high-energy particlesearly in the rise phase of the event. The NOAA gives a special caution at theend of the SPE list (NOAA, 1998). Proton fluxes in the list are integral 5-min


averages, given in p.f.u., for energies >10 MeV measured by the GOES spacecraftin geosynchronous orbit. The SWO defines the start of a proton event to be the firstof three consecutive data points with fluxes greater than or equal to 10 p.f.u. Theend of an event is the last time the flux was greater than or equal 1 to 10 p.f.u. Thisdefinition, motivated by SWO customer needs, allows multiple proton flares and/orinterplanetary shock proton increases to occur within one SWO proton event. Itmeans that the total number of SPEs selected by the criterion >10 p.f.u. may beundercounted. This deficiency was a reason that the NOAA data have been kept outof the scope of our basic analysis. Nevertheless, as argued in Section 3, those datahave a certain methodical interest. The authors of the above-mentioned cataloguesdrew into consideration many additional (indirect) solar and interplanetary data tomore completely resolve any individual proton event.

In the light of apparent distinction between the slopes of distributions for differ-ential energy intensities and those of integral energy intensities (see Introduction),it is important to look at the distribution of proton events at higher energies, such asE > 500 MeV measured in the GLEs (e.g., Miroshnichenko, Rogriguez-Frias, anddel Peral, 1995). Our paper of 1995, in fact, contains only differential spectrumparameters for the 31 GLEs observed in 1949–1991 by the ionization chambers(IC) and neutron monitors (NM). Since then, however, the list of GLEs has beenextended to 35 events, the spectrum data having been revised and considerablyspecified. In Table I we give all those data, including the normalization factor, D0,and exponent, γ , in the power-law rigidity spectrum, D(R) = D0R

−γ , togetherwith estimated integral intensities, Im(>1 GV), for relativistic solar protons, RSP(the energy E > 500 MeV, or rigidity R > 1 GV) measured in 1949–1992, withthe peak intensity of >10−1 p.f.u. Most estimates for the GLEs observed after1955 have been obtained with the data of several SPE Catalogues (see Section 2).Also some other sources of the GLE-RSP data have been taken into account (e.g.,Duggal, 1979; direct Meteor satellite data, etc.). Preliminary statistical analysis ofthe GLE size distribution was carried out by Miroshnichenko, Rogriguez-Frias,and del Peral, 1995). Of course, a number of important aspects concerning Table Iare out of the scope of the present paper.

It may be pointed out here that all previous works used far fewer samples ofSPE data than the 320 events of the current study. Although this number is notvery impressive in comparison with the data on flare electromagnetic emissions(e.g., Crosby, Ashwanden, and Dennis, 1993; Aschwanden, Dennis, and Schwartz,1998), nevertheless, as shown below, it allows us to obtain new interesting infor-mation about the size distributions of solar proton events at a long time scale.

3. Method and Results

The proton peak intensity distributions were constructed separately for differentsets of data in the differential power-law form (e.g., Crosby, Aschwanden, and


Dennis, 1993)

dN = Ax−α dx, (1)

where dN is a number of events recorded with the parameter x of interest betweenx and x + dx; A and α are constants determined by a least-squares fit to the data.Integral frequency distributions of the form

N(> x) =∞∫

x

dN =∞∫

x

Ax−αdx = Ax−α+1/(α − 1), for α > 1 (2)

were also used as the best fit for studies with poor statistics. The differential distri-butions are preferable as all bins of the histograms are independent of each other,thus allowing the power-law fit and the uncertainties to be calculated using stan-dard least-squares procedures (e.g., Reiff, 1990). The results of current statisticalanalysis are shown in Figures 2 and 3 for differential and integral distributions,respectively. In addition, Table II contains a summary of size distributions obtainedby several research groups since 1975 for different samples of proton event data.

Owing to recent publication of a new catalogue of SPEs by Sladkova, Bazilevs-kaya and Ishkov (1998), an opportunity appeared to revise the basic statistics of305 events above 1 p.f.u. (1955–1993) that we analyzed in two preliminary studies(Mendoza, Melendez-Venancio and Miroshnichenko, 1997; Meléndez-Venancio,Mendoza, and Miroshnichenko, 1998). The main changes here were reduced to theextension of total statistics to 320 events and specification of peak fluxes for theevents of 1987–1996. At the same time, the statistics on SSC-related events hasbeen enlarged. The point is that in our papers of 1997–1998 we used 76 eventsonly (see Figure 1) that, in fact, covered the period from 1970 through 1986. Ifwe expand our list for the periods of 1955–1969 and 1987–1996, the numberof events with a certain and probable SSC association reaches 159. If one takesinto consideration also the events with a possible but somewhat doubtful SSC-flareassociation, the total statistics may be enlarged to 181. When constructing workingplots for all three groups (76, 159 and 181 events), the best fit was obtained in thesecond case (159 events).

A differential plot for all 320 events in Figure 2 (filled diamonds) is consistentwith a single slope of 1.37 ± 0.05 (the best fit with a weighted alpha value) overthe entire range of the proton intensities, from >1 p.f.u. up to >103 p.f.u. (line 9in Table II). As to the SSC-related events, it is remarkably that in all three casesmentioned above, the event distributions are identical (within the error limits),though in the case of 181 events a scatter of observational points was found to belarger than in two other ones. This last shortcoming seems to be due to somewhatuncertain identifications of the sources and sizes of the events by the authors ofthe first SPE Catalogue (Dodson, Hedeman, and Kreplin, 1975), especially in theperiod 1955–1960 when direct particle measurements were absent in about 78%


TABLE I

Integral fluxes of solar protons at rigidity above 1 GV

Date Time Rigidy D0 γ Fm (>1 GV) Im (>1 GV)

of SPE (UT) R (GV) (cm−2 s−1 GV−1) (cm−2 s−1) (cm−2 s−1)

1. 28 Feb. 1942 13:00 >1 8.33 × 102 4–5 3.8 × 101 1.21 × 101

2. 7 Mar. 1942 06:00 >1 1.04 × 103 4–5 4.8 × 101 1.53 × 101

3. 25 July 1946 18:53 >1 Integral – 7.1 × 101 2.26 × 101

IC data

4. 19 Nov. 1949 12:00 >1 2.78 × 103 4–5 1.3 × 102 4.14 × 101

5. 23 Feb. 1956 05:00 1.5–5.0 1.25 × 104 6.8 8.0 × 102 2.55 × 102

6. 4 May 1960 10:50 2–5 6.30 × 101 3.4 2.6 × 101 8.27 × 100

7. 12 Nov. 1960 20:00 1.0–3.5 1.70 × 102 5.2 3.2 × 101 1.02 × 101

8. 15 Nov. 1960 04:00 1.5–4.0 1.55 × 102 5.0 3.7 × 101 1.18 × 101

9. 28 Jan. 1967 12:00 0.5–1 1.25 × 101 4.5 1.4 × 100 4.45 × 10−1

10. 18 Nov. 1968 11:00 1.6–5. 1.57 × 101 5.0 4.0 × 100 1.27 × 100

11. 25 Feb. 1969 10:00 1.0–4.4 9.50 × 100 4.1 3.1 × 100 9.86 × 10−1

12. 30 Mar. 1969 14:00 1–3 2.45 × 100 4.0 8.2 × 10−1 2.60 × 10−1

13. 24 Jan. 1971 24:00 1.0–5. 1.66 × 101 5.0 4.2 × 100 1.34 × 100

14. 1 Sept.1971 22:00 1–5 1.57 × 101 5.5 3.6 × 100 1.14 × 100

15. 4 Aug. 1972 16:00 1.0–1.6 2.04 × 101 8.0 2.9 × 100 9.23 × 10−1

16. 7 Aug. 1972 17:00 1–3 7.00 × 100 4.0 3.2 × 100 1.02 × 100

17. 29 Apr. 1973 22:15 >1 Integral – 4.8 × 10−1 1.52 × 10−1

NM data

18. 30 Apr. 1976 21:40 1.0–1.7 1.40 × 100 3.7 1.6 × 100 5.09 × 10−1

19. 19 Sept. 1977 14:00 >1 2.40 × 10−1 4.0 6.0 × 10−1 1.90 × 10−1

20. 24 Sept. 1977 10:12 1.0–6.3 4.00 × 100 3.4 1.7 × 100 5.41 × 10−1

21. 22 Nov. 1977 12:00 2.3–4.0 5.00 × 102 5.5 3.3 × 100 1.05 × 100

22. 7 May 1978 03:45 2.1–6.2 4.08 × 102 4.1 3.1 × 100 9.86 × 10−1

23. 23 Sept. 1978 11:15 >1 1.88 × 101 4.8 5.0 × 100 1.59 × 100

24. 21 Aug. 1979 07:00 >1 5.73 × 100 4.6 1.60 × 100 5.09 × 101

25. 10 Apr. 1981 17:30 >1 1.72 × 100 4.5 4.9 × 10−1 1.55 × 10−1

26. 10 May 1981 10:00 >1 2.00 × 100 4.3 6.0 × 10−1 1.90 × 10−1

27. 12 Oct. 1981 10:00 >1 1.37 × 101 4.4 4.1 × 100 1.30 × 100

28. 26 Nov. 1982 04:55 >1 5.67 × 100 4.1 1.8 × 100 5.72 × 10−1

29. 8 Dec. 1982 00:45 >1 8.62 × 101 5.5 1.9 × 101 6.05 × 100

30. 16 Feb. 1984 09:15 >1 7.25 × 100 4.3 3.2 × 101 1.02 × 101

31. 29 Sept. 1989 12:17 1–4 9.33 × 100 2.9 9.5 × 101 3.02 × 101

32. 24 Mar. 1991 04:39 >1 Integral – 3.5 × 10−1 1.10 × 10−1

Meteor data

33. 11 June 1991 01:56 1–4 1.55 × 101 5.5 3.5 × 100 1.11 × 100

34. 15 June 1991 08:10 1–4 6.19 × 101 6.0 1.3 × 101 4.14 × 100

35. 25 June 1992 00:32 >1 Integral – 3.8 × 10−1 1.20 × 10−1

Meteor data


Figure 2. Differential size distributions of 320 flare-related SPEs (open diamonds) and of 159SSG-related proton events (open triangles) from 1955 through 1996. For comparison, a size dis-tribution of 45 events of 1965 –1996 (open circles) from the list of Smart and Shea (1997) is alsoshown.

of the events. Therefore, we plot in Figure 2 a size distribution for the 159 shock-associated events only (open triangles), and their longitudinal distribution is shownin Figure 1, too. Unlike the plot for 320 basic events, the 159 shock-associatedevents display a two power-law behaviour, with slopes of 1.00 ± 0.04 (line 11)and 1.53 × 0.05 (line 13) below and above 103 p.f.u., respectively, the differencebetween the slopes being evidently out of the limits of approximation errors.

Recently, a similar break in slope was noted by Smart and Shea (1997) in theintegral numbers of the >10 MeV proton events per solar cycle. These workershave analyzed large proton events (>10 p.f.u.) observed in cycles 20, 21, and 22.Their primary sources of data were mainly direct measurements of the >10 MeVprotons on the IMP spacecraft. Unfortunately, in the original paper a total number


of the events under study was not given. However, judging by the plot of SPE sizedistribution, in all about 170 events observed in 1965–1996 at >10 p.f.u. wereused, and only 26 of them had a peak flux of >103 p.f.u. (7, 6, and 13 events persolar cycles 20, 21, and 22, respectively). Also, one misleading point of their workshould be mentioned: they show an integral size distribution with the slopes 0.47and 1.42 below and above 103 p.f.u., respectively, without making that fact obviouseven in their comparison with differential distributions. In fact, their differentialdistribution exponents are 1.47 and 2.42, respectively, and those values are moreconsistent with other results in Table II.

To clearify the nature of the distribution break at the peak flux of 103 p.f.u. re-ported by Smart and Shea (1997), their data summary with 45 events at >300 p.f.u.observed from January 1967 to February 1994 was re-examined here, and at thebottom of Figure 2 we present one additional plot (open circles) obtained in thecurrent study. According to our estimate, the best fit of those data is consistent witha slope of 2.12 ± 0.03 (line 12 in Table II) above 103 p.f.u., considerably differentfrom the original value of about 2.42 (line 6) by Smart and Shea (1997). To avoidoverlapping of three plots in Figure 2 and to make them well-organized, the distri-butions of 320 flare-related events and 45 events by Smart and Shea (1997) wereshifted upwards and downwards, respectively, for about one order of magnituderelative to the plot of 159 shock-associated events.

In view of an evident distinction between the alpha values in Table II for the sizedistributions of proton events detected in differential and integral energy ranges, itis of great interest to compare the distribution slopes at different proton energies.At this point it would be important, first of all, to separate SPEs according to accel-eration mechanisms based on some peculiarities of the proton energy spectra fromdifferent sources (Miroshnichenko, Mendoza, and Pérez-Enríquez, 1999). Anotherimportant problem is that the proton energy spectra vary with heliolongitude (e.g.,van Hollebeke, Ma Sung, and McDonald, 1975). This suggests a possible energydependence of the exponents in power-law fits if one uses the event data overbroad longitude ranges, such as in the current study. As mentioned above, thisproblem was avoided by van Hollebeke, Ma Sung, and McDonald (1975) andCliver, Reames, and Kahler (1991) by taking a differential energy band, but isinherent in the integral >10 MeV data, like that taken from the SPE Catalogues of1955–1996.

In searching for an appropriate way to make this problem more clear, we paidattention to the two independent sets of SPE data, by Kahler, Shea, and Smart(1991) and Miroshnichenko, Rogriguez-Frias, and del Peral (1995). In the firstpaper, the researchers have summarized the peak fluxes for 66 so-called ‘mixed’,i.e, impulsive and gradual, events for the >10 MeV protons measured at the thresh-old intensity of 10−2 p.f.u. in 1976–1990. Note that their database also includes19 GLE. On the other hand, Miroshnichenko, Rogriguez-Frias and del Peral (1995)estimated the peak fluxes for the >500 MeV protons in 31 GLEs observed in 1949–1991, at the threshold intensity of 10−1 p.f.u.. So, there was a hope to compile


Figure 3. Integral size distributions of proton events obtained from the large database of 320 protonevents (filled diamonds), using the data of Kahler, Shea and Smart (1991) for the >10 MeV protons(filled triangles, 43 events) and the data from Table II for the >500 MeV protons (filled circles, 20events).

two comparable sets of proton intensity data for different energy thresholds (>10MeV and >500 MeV). With this purpose, we revised those data sets once more. Inparticular, the data on relativistic solar protons (RSP) were significantly specifiedand their statistics was somewhat enlarged (see Table I). Finally, it was foundreasonable to construct three integral distributions in the same Figure 3, usingthe large database of 320 events (filled diamonds), 43 events from the paper byKahler, Shea, and Smart (1991) (filled triangles), and 20 GLEs from Table I (filledcircles). Notice that all three distributions are given and compared in Figure 3 usingthe same threshold intensity of 1 p.f.u.. From their empirical fittings (by nonlinearapproximations), one can see that a difference between three distributions surpassesby far the limits of existing uncertainties. Manifestly, the middle plot (43 mixedevents) is similar to the upper one (320 events), and both of them display a rathersmooth fall over the entire range of comparable intensities between 1 p.f.u. and 103

p.f.u. At the same time, the lower curve (20 GLEs) steeply slopes down between1 p.f.u. and 102 p.f.u. This may point to a certain dependence of slope on theproton energy range under consideration.

For the estimates of uncertainties in the alpha values listed in Table II, we useda standard linear correlation analysis to fit data points to a power law through thecorrelation matrix (e.g., Reiff, 1990) with weighting data on the number of eventsper unit of intensity. In some cases, however, also the uncertainties in the peak fluxwere important. For example, an integral distribution of the >500 MeV protonevents (Figure 3) has been obtained with an initial uncertainty in the intensityvalues of about ± 50% (Miroshnichenko, Rodriguez-Frias, and del Peral, 1995).


As it turned out in the course of this study, the event distributions, at thresholdintensity >10 p.f.u., in addition to our basic results, are of interest. For this reason,we keep in Table II corresponding alpha estimates (lines 5, 8, 10 and 14), in spite ofthe preliminary nature of our previous results (Mendoza, Melendez-Venancio andMiroshnichenko, 1997), absence of approximation errors in the paper by Smartand Shea (1997), and disputable applicability of the NOAA data for the scientificcomparison. Indeed, those estimates seem to be consistent with a single slope ofabout 1.43 within the error limits. Also, the alpha value of 1.27 (line 7) at thresholdintensity >1 p.f.u. (Mendoza, Melendez-Venancio, and Miroshnichenko, 1997) isimportant when discussing a tendency to a deviation from power-law behaviour ofthe event distributions at low intensities (see below). Anyway, those data may behelpful for future studies of this kind.

4. Discussion

To begin with a discussion, we return once more to the plots, represented in Fig-ure 1. The heliolongitude distributions of the parent solar flares for various sets ofSPEs have been examined in many papers (e.g., Cane, Reames, and von Rosen-vinge, 1988; Reames, 1995; Smart and Shea, 1996; Bazilevskaya and Sladkova,1997).

In particular, the flat distribution obtained by Reames (1995) for so-called grad-ual SPEs is considered as strong evidence for the acceleration of particles by aCME-driven shock. As was noted in Section 2, our statistics display an apparentdifference in the source locations on the solar disk between the SPEs of differentgroups in Figure 1: the number of flare-related events grows steadily with longitudeto the west limb of the Sun, meanwhile the shock-associated events are distributedmore uniformly, with a maximum at about 30◦W.

At first sight, such a result seems to be similar to what one could expect fromsome earlier works (e.g., Cane, Reames, and von Rosenvinge, 1988), especially forthe flare-associated events. Notice, however, that Cane, Reames, and von Rosen-vinge (1988) relied in their study upon different observational and methodical base.First of all, they constructed the heliolongitude distribution for 235 events with athreshold intensity of >10−3 p.f.u. MeV−1 at energies >20 MeV; further, their datameasured on IMP 4, 5, 7, and 8 and ISEE 3 covered only a 19.7-year period com-mencing mid-May 1967; finally, the events associated with interplanetary shocks(evidenced at the Earth by the SSCs) display their distribution maximum between0◦ –30◦E, in contrast to the findings of the present study. Anyway, the histogramsin Figure 1 seem to be useful for further analysis of heliolongitude distributions ofdifferent sets of the SPEs.

As follows from the above results, it is not realistic to expect that size dis-tributions of proton events (Table II) should match those of flare electromagneticemissions (e.g., Crosby, Aschwanden, and Dennis, 1993). In fact, SPE distributions


TAB

LE

II

Sum

mar

yof

size

dist

ribu

tion

sof

sola

rpr

oton

even

ts

Sou

rce

Obs

erva

tion

Ene

rgy

Thr

esho

ldN

umbe

rP

ower

-law

Ref

eren

ce

ofda

tape

riod

MeV

inte

nsit

yof

even

tsin

dex

(−α

)

(p.f

.u.)

a

1.IM

P4,

519

67–

1972

20–

80b

>10

−416

31.

15±

0.05

van

Hol

lebe

ke,M

aS

ung

and

McD

onal

d(1

975)

2.E

xplo

rer

34,4

119

67–

1972

>10

>10

−187

1.40

±0.

15B

elov

sky

and

Och

elko

v(1

979)

3.V

ener

a13

,14

1981

–19

82>

25>

7×

10−4

361.

45±

0.05

Kur

t(19

89)

4.IM

P8

1977

–19

8324

–34

b>

10−3

921.

13±

0.04

Cliv

er,R

eam

esan

dK

ahle

r(1

991)

5.IM

PS

pace

craf

t19

65–

1996

>10

>10

170

1.47

±0.

??S

mar

tand

She

a(1

997)

6.IM

PS

pace

craf

t19

65–

1996

>10

>10

326

2.42

±0.

??S

mar

tand

She

a(1

997)

7.S

PE

Cat

alog

ues

1955

–19

93>

10>

130

51.

27±

0.02

Men

doza

,Mel

ende

z-V

enan

cio

and

Mir

oshn

iche

nko

(199

7)

8.S

PE

Cat

alog

ues

1955

–19

93>

10>

1022

81.

38±

0.03

Men

doza

,Mel

ende

z-V

enan

cio

and

Mir

oshn

iche

nko

(199

7)

9.S

PE

Cat

alog

ues

1955

–19

93>

10>

132

01.

37±

0.05

Thi

sw

ork

10.

SP

EC

atal

ogue

s19

55–

1996

>10

>10

233

1.43

±0.

03T

his

wor

k

11.

SP

EC

atal

ogue

s19

55–

1996

>10

>1

134c

1.00

±0.

03T

his

wor

k

12.

IMP

Spa

cecr

aft

1955

–19

96>

10>

750

322.

12±

0.03

Thi

sw

ork

13.

SP

EC

atal

ogue

s19

55–

1996

>10

>10

325

c1.

53±

0.04

Thi

sw

ork

14.

NO

AA

list

1976

–19

97>

10>

1013

41.

47±

0.06

Thi

sw

ork

a 1p.

f.u.

=1

prot

onfl

uxun

it=

1pr

oton

cm−2

s−1

sr−1

.b T

hres

hold

inte

nsit

yin

the

line

s1

and

4is

give

nin

p.f.

u.M

eV−1

(dif

fere

ntia

lflux

),in

othe

rli

nes

inp.

f.u.

(int

egra

lflux

).c S

SC

-rel

ated

even

ts.


may reflect both the features of the source(s) of accelerated particles (flares andshocks) and different conditions of the particle release and propagation. Therefore,unlike most previous studies, we search for a difference rather than for a likelihoodbetween the distributions of SPEs and that of other parameters of the sources. Asto the distribution functions themselves, of paramount importance is a differencebetween their slopes for total statistics of 320 events and 159 shock-associatedSPEs of 1955–1996 at the peak intensity of >1 p.f.u. (Figure 2). Being larger thanthe limits of approximation errors, this difference may suggest different sources (ortheir combinations) for both groups of events; in particular, the events of the secondgroup may occur due to particle acceleration by a CME-driven shock. Moreover,the shock-associated SPEs display a distinct double power-law behaviour, withslopes of 1.00 ± 0.04 (line 11) and 1.53 ± 0.05 (line 13) below and above 103 p.f.u.,respectively.

Further, we draw attention to the fact of similarity of the slopes for size distrib-utions at >10 p.f.u. (lines 5, 8, 10, and 14 in Table II). Though being obtained withdifferent data sets, the alpha values seem to be the same (about 1.43 ± 0.03), withinthe limits of approximation errors. Of course, one cannot exclude that this casualcoincidence has a methodical origin, due to different criteria of data selection incorresponding studies. Indeed, the estimates in lines 8 and 10 rely upon the samedatabase of 320 events; Smart and Shea (1997) analyzed the numbers of eventsper solar cycle; our estimate in line 14 is based on the NOAA list that includesonly events with sizes >10 p.f.u., without any corrections for possible multipleproton increases within one SWO proton event (see Section 2). Notice, however,that the NOAA data are compiled for space weather applications, not for scientificpurposes. So, it is unrealistic to expect that the NOAA list, even if corrections aremade, can be the basis of scientific comparison. Nevertheless, the similarity of theslopes at >10 p.f.u. seems to be of interest for future studies, in particular, if oneattempts to subdivide the events based on the size threshold (>1, >10, >100 p.f.u.,etc.).

Finally, as was found earlier (e.g., Mendoza, Melendez-Venancio, and Mirosh-nichenko, 1997), size distributions of the >10 MeV proton events show a dis-cernible tendency to a deviation from power-law behaviour at low intensities. Sucha tendency may be seen, for example, at the peak flux below 10 p.f.u. in Figure 2(diamonds), in spite of repeated revision of the 320 event database. This deviationseems to be a consequence of the procedure of data selection. The possibility ofsystematic effects (errors and artifacts) in routine identification of the SPEs at lowintensity was first noticed by Smart and Shea (1989). It is especially true for the‘pre-spacecraft era’ (1955–1965), when there was only indirect ionospheric (ri-ometer) information about the >10 MeV proton intensity. In addition, uncertaintiesin the event numbers before 1966 might arise because of the coarse method oftheir classification by proton intensity (e.g., Dodson, Hedeman, and Kreplin, 1975).Thus, we do not exclude that our statistics of 320 events contains systematic errors


before 1966. It may provide a useful caution against possible misunderstandings inuse of early data on the SPEs.

In contrast, van Hollebeke, Ma Sung, and McDonald (1975) and Cliver, Reames,and Cane (1991) seemed to avoid this shortcoming, though their results are con-cerned with very low intensity thresholds, 10−4 and 10−3 p.f.u. MeV−1, respec-tively. Also, Belovsky, Kurt, and Ochelkov (1979) did not find any turnover in thedistribution of the electron events in the interval of energies of 0.5–1.1 MeV overthe entire range of their maximum intensities, from 1.0 to 102 p.f.u. MeV−1. Thisdistinction, probably, is mostly due to more homogeneous sets of data for differen-tial energies, such as those the authors used. Meanwhile, an effect of turnover in thedistributions was clearly found for electrons of >30 keV and >45 keV (Belovsky,Kurt, and Ochelkov, 1979), >70 keV (Belyakov, Daibog, and Dyachkov, 1984;Daibog, Kurt, and Logachev, 1989), hard X-rays and other flare emissions (Kurt,1990; Crosby, Aschwanden, and Dennis, 1993). The turnover is suggested to becaused by certain methodical reasons, for example, due to detector sensitivities atlow intensities. Crosby, Aschwanden, and Dennis (1993) cannot exclude, however,some physical constraints, for example, the existence of lower limits in charac-teristic parameters of an elementary instability of the coronal plasma (Lu andHamilton, 1991). In a more recent study by Aschwanden, Dennis, and Schwartz(1998), spatial sizes of elementary acceleration cells and elementary time scalesconsistent with observed power-law frequency distributions have been estimatedin terms of the logistic avalanche model.

A real question of a change of slope is occurring, however, at the oppositeend of the intensity scale, where Smart and Shea (1997) discovered a break in theSPEs distribution at 103 p.f.u., for a number of the >10 MeV proton events withpeak flux >10 p.f.u. (see lines 5 and 6 in Table II). Meanwhile, the distributionof 320 events at >1 p.f.u. (diamonds in Figure 2) does not seem to support thebreak at 103 p.f.u. Such a conclusion evidently contradicts the result by Smart andShea (1997). On the other hand, independently, our estimate from their originaldata at >750 p.f.u. (circles in Figure 2) confirms such a feature, though with adifferent slope (2.12 ± 0.03, line 12 in Table II). Moreover, large shock-associatedevents (triangles in Figure 2) clearly demonstrate a sharp break in the slope above103 p.f.u.

These results are qualitatively consistent with those obtained by Reedy (1996)for the fluence distribution, N(>F), of Earth-sensed SPEs from 1954 to 1991. Theintegral distribution of the number of events, N, per year was shown to have aform of ∼ F−0.4 in the range of low fluences (up to ∼1010 cm−2) and of ∼F−0.9

at high fluences (≥ 1011 cm−2) of the >10 MeV protons. A similar tendency wasfound by Nymmik (1999) for the >30 MeV protons: their fluence distribution inthe solar cycles 20–22 can be described by a power-law function with exponentialsteepening for large fluences.

In the light of these results, several new aspects seem to arise. First of all, onemay assume that the turnover effect below 10 p.f.u. is an artifact due to uncertain-


ties at low proton intensity, i.e., it is methodical in nature and characterizes theearlier observed proton events only. On the other hand, if this effect really existsat low intensity, it can significantly influence the estimates of the alpha values athigher intensities as well, as one can see from Figures 2 and 3. In addition, asfollows from Figure 2, there is evidence of a sharp break in slope at the highestintensities. Anyhow, the great variety of slopes listed in Table II for different datasets, implies that the underlying physics of the proposed turnover and break effectsmay be more instructive than was thought earlier.

Independently of subtle details about SPE distributions, the results obtained bydifferent research groups (Table II) over the entire period of the observations maybe evidence that the energy released in the form of accelerated particles is notbound to obey a linear dependence upon total flare energy, as stated, for exam-ple, by Kuznetsov and Kurt (1991). Their conclusion that protons with energiesEp < 20 MeV account for a proportionate fraction of the total flare energy budgetwas not confirmed, in particular, by recent findings of Cliver, Reames, and Kahler(1991) for >20 MeV protons. At any rate, the flatter size distribution found byCliver, Reames and Kahler (1991) contradicts the arguments by Kuznetsov andKurt (1991) that similar size distributions exist for flare electromagnetic and protonemissions. Such a similarity would imply a single class of flares, contrary to exist-ing modern two-class classification as impulsive and gradual ones (e.g., Reames,1995, 1996).

In order to adequately treat a change of slope occurring at high intensity, oneshould invoke proposed or existing mechanisms of restrictions imposed upon thenumbers and maximum intensities of observed proton events. In turn, the restric-tions are dependent on conditions of particle acceleration, release and propaga-tion in interplanetary space (e.g., Miroshnichenko, Mendoza, and Pérez-Enríquez,1999). It is widely believed that the most severe constraints on particle accelerationmodels are imposed during impulsive flare events (e.g., Miller, Cargill, and Emslie,1997). Those events tend to be compact and occur low in the corona, while gradualevents occur at greater heights and correlate with CMEs. An important role ofCME-driven shocks in gradual proton events is widely recognized (e.g., Reames,1996; Kahler, 1996). At the same time, it is also true that some gradual eventspossess a confined ‘core’ of flare-accelerated particles surrounded by a ‘halo’ ofCME/shock particles (e.g., Cliver, 1996), or, in other words, those events are ‘mix-tures’ of impulsive and gradual ones (e.g., Mandzhavidze and Ramaty, 1993) thatcomplicates significantly treating the event size distributions.

The observational data and present knowledge of plasma processes at the Sunare still not extensive enough to derive the proper restrictions on size distributionsof the proton events in terms of acceleration mechanisms. In other words, thereis no possibility to distinctly separate the proton events according to their sources(flares, shocks, etc.), though the difference in the distributions between the flare-associated and SSC-associated events in Figure 2 may be evidence of their differentorigins. Meanwhile, there is an obvious interplanetary effect to explain, at least,


the change in slope at the 103 p.f.u. value for shock-associated events. That isso-called streaming-limited saturation of SEP events (Reames and Ng, 1998, andreferences therein). In a proposed nonlinear scenario, protons streaming outwardfrom an intense source near the Sun reach a maximum-intensity plateau (Ng andReames, 1994) due to particle scattering by self-generated Alfvén waves. Accord-ing to Reames and Ng (1998), at energies of a few MeV the limiting intensity isattained for a dozen or more events per solar cycle. From the GOES data in the 110–500 MeV interval for the four large proton events in September–October 1989,however, it was demonstrated (Reames and Ng, 1998) that this effect may be tracedto the energies of >100 MeV. Such a streaming limit may be closely linked withthe particle acceleration at interplanetary shock waves that gives rise to the largegradual SEP events. This is especially important in view of the distinct manifesta-tion of the slope break in the distribution of the shock-associated >10 MeV events(Figure 2).

5. Conclusions

To the best of our knowledge, this is the first attempt to accomplish such an ex-tended statistical analysis of solar proton events observed at 1 AU from 1955 to1996 at >10 MeV, including 320 flare-related SPEs, 159 events related also to asudden storm commencement (SSC), as well as 35 GLEs at >500 MeV. Summingup the main results of this study we note that, together with appropriate resultspublished since 1975 (Table II), our findings provide new important diagnosticinformation about some features of the Sun’s proton productivity and its relationto existing problems of particle acceleration at/near the Sun.

(1) As was expected, the longitude distributions show a difference in the sourcelocation on the solar disk between the events above 1 p.f.u. of >10 MeV protonsnot related and related to SSCs, the events of the second group being distributedmore uniformly, with a maximum at about 30◦ W. Notice that at energies >2 MeVand intensities of >10−3 p.f.u. MeV−1, the events associated with interplanetaryshocks (evidenced at the Earth by the SSCs) display their distribution maximumbetween 0◦ –30◦ E (Cane, Reames, and von Rosenvinge, 1988).

(2) The best power-law fit for the basic sample of 320 proton events is attainedat the slope of 1.37 ± 0.05 over the entire range of the proton intensities, 100 –105 p.f.u., which contradicts the main conclusion of Smart and Shea (1997) abouta sharp break in the slope at about 103 p.f.u.. Meanwhile, the corresponding distrib-ution for the 159 SSC-related events demonstrates double power-law features, withthe slopes of 1.00 ± 0.04 and 1.53 ± 0.05 below and above 103 p.f.u., respectively.This is qualitatively consistent with the principal result of Smart and Shea (1997).Independently, such a break is supported by our examination of their data above300 p.f.u.


(3) Integral size distributions at >10 MeV and >500 MeV, by their empiri-cal non-linear approximations, demonstrate that a difference between them sur-passes by far the limits of existing uncertainties. Manifestly, the distributions ofthe >10 MeV events display a rather smooth fall over the entire range of proton in-tensities between 1.0–105 p.f.u. At the same time, the corresponding curve for the>500 MeV events (GLEs) steeply slopes down between 1.0–102 p.f.u. This maypoint to a certain dependence of slope on the proton energy range under consider-ation. Such a tendency, however, needs to be confirmed by the event distributionsat intermediate energies (for instance, at >100 MeV).

(4) Irremovable uncertainties in the event numbers at low intensity for earlierobserved events may influence a slope of power-law distribution in the entire rangeof proton intensities, 100 –105 p.f.u. At the same time, it cannot be excluded thatthe low-intensity turnover effect is not only methodical in nature, but also bears thetraces of constraints imposed upon the particle acceleration and release processesat/near the Sun, i.e., upon the intensity and occurrence rate of proton events.

(5) It is evidenced that the high-intensity break effect may be real and physicallymeaningful for the shock-associated proton events. If so, an effect of streaming-limited saturation of SEP events seems to be very important, especially for the large‘gradual’ proton events. At the same time, it becomes obvious that more detailedstudies of these effects are worth pursuing with more refined criteria for the SEPevents.

Acknowledgements

This work was supported partially by the CONACyT (Mexico), State Scientific andTechnical Program ‘Astronomy’ (Russia, Grant No. 4–129) and Russian Founda-tion of Basic Research (Grant No.96-02-16827a). We also thank R. Melendez-Venancio (UNAM, México) for his contribution to this work at its preliminarystage, and Dr A. Lara Sanchez (UNAM, México) for his interest and discussionof some methodical and computational aspects of this study. Our special thanksare to Dr A. I. Sladkova (Moscow State University) for providing us with the datafrom the SPE catalogue 1987–1996 before its publication. The authors appreciatevery much the insightful and detailed comments of two anonymous referees whichhelped to improve the paper.

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size distrtibutions of the >10 mev solar proton events

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