power-law distribution for solar energetic proton events

POWER-LAW DISTRIBUTION FOR SOLAR ENERGETIC PROTON

EVENTS

S. B. GABRIEL Department of Aeronautics and Astronautics, University of Southampton, England

and

J. F E Y N M A N Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, U. S. A.

(Received 18 June, 1993; in final form 21 August, 1995)

Abstract. Analyses of the time-integrated fluxes of solar energetic particle events during the period 1965-1990 show that the differential distribution of events with flux F is given by a power law, with indices between 1.2 and 1.4 depending on energy. The power law represents a good fit over three to four orders of magnitude in fluence. Similar power-law distributions have been found for peak proton and electron fluxes, X-ray flares and radio and type III bursts. At fluences greater than 109 cm -2, the slope of the distribution steepens and beyond 101~ cm -2 the power-law index is estimated to be 3.5. At energies greater than l0 MeV, the slope of the distribution was found to be essentially independent of solar cycle, when the active years of solar cycles 20, 21, and 22 were analysed. The results presented are the first for a complete period of 27 years, covering nearly 3 complete solar cycles. Other new aspects of the results include the invariance of the exponent with solar cycle and also with integral energy.

1. Introduction

Interplanetary solar energetic particle events show a wide variation in their char- acteristics with flux values covering five to six orders of magnitude and durations of perhaps 1 day to 20 or 30 days. For the largest events (fluence > 109 cm -2 at E > 10 MeV), the durations of periods of high intensity proton flux show a large spread of between 5 and 33 days. During these events, electrons and protons are accelerated to very high energies with electrons reaching relativistic speeds and protons reaching energies in excess of 500 MeV. Although the events have been widely studied with several acceleration mechanisms proposed (see, for example, Ohki, 1989; and Cane, McGuire, and von Rosenvinge, 1986), a complete understanding of the acceleration mechanism or indeed the causal mechanism of the events is still lacking. While the events are usually associated with flares, it appears that a flare is not a prerequisite for the occurrence of a proton event (Kahler et al., 1986; Kahler, 1982). In this paper, we present results which show that the frequency of occurrence of solar proton events of a given size, as measured by the time-integrated flux over the event duration, is given by a simple power law of the form d N / d F = F -b with b-values between 1.2 and 1.4 depending on the integral energy. The results are consistent with power-law behaviour in other physical parameters associated with solar flares, peak hard and soft X-ray fluxes, radio

Solar Physics 165: 337-346, 1996. @ 1996 Kluwer Academic Publishers. Printed in Belgium.

338 S.B. GABRIEL AND J. FEYNMAN

and type III bursts, and peak proton and electron fluxes (Crosby, Aschwanden, and Dennis, 1993).

2. The Data

The data set used in our study consists of data collected from space probes. Since 1963 instruments have been observing proton fluxes in space. All of the feasible data from satellite observations have been collected and edited for valid particle responses and a tape has been prepared containing a nearly time-continuous record of daily average fluxes above the energy thresholds of 10, 30, and 60 MeV. Space- craft involved were IMP 1, 2, 3, OGO 1, and IMP 5, 6, 7, and 8. The details of the production of this data set and an analysis of some of the statistical properties of the proton fluxes are described in Armstrong, Brungardt, and Meyer (1983). For the present study, the daily average fluxes were integrated over the time of the event, which often covers several days. Fluences for the events between 1985 and 1989 at energies E > 10 MeV and for 1956-1989 at E > 30 MeV were obtained from daily average fluxes from the IMP 8 spacecraft (Gabriel, Feynman, and Spitale, 1991), as were the > 60 MeV fluences for the period 1965-1989. For the 1989 to 1990 period, hourly fluxes at/~ > 10, 30, and 60 MeV from the IMP 8 spacecraft were used to produce daily averages which, in turn, were used to calculate event fluences. Although the definition of an event was somewhat different for each of the periods (see Feynman et al., 1990a); Gabriel, Feynman, and Damon, 1991), the resulting events in terms of start and end times and fluence were compared and found to be in excellent agreement. The differences in the event lists are minor and will not influence the results of this study.

3. Results

Figure 1 shows the total number of events, in each fluence interval divided by that interval, dN/dF , for energies E > 10 MeV, during the period 1965-1990. Also shown in this figure is a straight-line fit, over the full fluence range of 106 to 10 l~ cln -2, using a least-squares fitting technique. The best fit is obtained with an exponent of 1.32 with a regression coefficient of 0.99. While the straight line fit over the full fluence range (4 order of magnitude) is very good as indicated by the high regression coefficient, it is evident from the graph that the last point deviates significantly from the fit. To avoid the problem with this highest fluence bin, it was removed and the best straight line fit to the remaining points is as shown in the figure. Table I summarises the fitting parameters and removing the highest point lowers the exponent but does not change the regression coefficient.

The errors shown were derived from a standard least-squares analysis using the software Mathematica. The 'linear regression' package within Mathematica uses

POWER-LAW DISTRIBUTION FOR SOLAR ENERGETIC PROTON EVENTS 339

E > 10 MeV Distribution

I E - 0 4

1E-05 -

1 E - 0 6 -

1E-07 -

Number o f events per unit fluence

1 E - 0 8 -

1E-09 -

1 E - 1 0 -

1 E - t l

+ ~-

+

........ I ........ I ........ I ........ I ........

1 E + 0 6 1 E + 0 7 1 E + 0 8 1 E + 0 9 1 E + 1 0 1 E + l l

Least-squares fit to full fluence range, 2.5 . 1 0 - - 6 - 2 . 0 �9 1010 cm - 2 , b = 1.32

Least-squares fit to fluence range, 2,5 �9 10 - 6 2 . 5 . 109 cm - 2 , b = 1.24

Fluence (cm -2)

Fig. 1. Fluence distribution for E > I0 MeV for period 1965 to 1990.

TABLE I

E > 10 MeV, 1965-1990; d N / d F ~ I s'-b

Fluence range (cm -2) b Regression coefficient

2.5 x 106-2.0 • 101~ 1,32 4- 0.05 0.99 2.5 x 106-2.5 x 109 1.24 4- 0.04 0.99

a conventional least-squares technique, estimating the regression coefficients that minimise the residual sum of squares and calculating the standard errors in both the slope and the intercept in the usual way (Montgomery and Peck, 1992). It should be noted that there are very few events at the highest fluence values and, in fact, in the bin 7 x 109 to 1 • 1010 c m 2 there were no events, only 2 in the bin 10 l~ to 4 x 101~ cm 2, and only 1 in the bin 4 x 109 to 7 to 109 cm 2. To overcome this, the bin size was increased to 4 x 109 to 4 x 10 l~ cm 2, but the problem of an insufficient number of events at the very highest fluences still remains. Reducing the bin size, to ensure that all bins were occupied by at least one event, had little effect on the value of the slope of the line, changing it to 1.30 over the full range of fluences.

Figure 2 shows the distribution of events for E > 30 MeV. Once again a straight line has been fitted to the data and yields an exponent of 1.27 which is less than that for the > 10 MeV value, but within the error estimates. As with the 10 MeV data, there are deviations from the straight line fit both at the lowest and highest fluences, with the effects this time being most pronounced at the lowest fluences.


E > 30 MeV Dist r ibut ion

1E-03

1E-04. -

I E - 0 5 -

1E-06 -

N u m b e r o f events per unit f luence

1E-07 -

1E-08 -

1E .09 -

1E-10

1 E + 0 5

" ' ,

+ + :.9,,,

~, .,.~

' ' ' '""1 . . . . . . . . . I . . . . . . . . I . . . . . . . . I . . . . . . .

1E+06 1E+07 1E+08 1E+09 1 E+ 10

Least-squares fit to full f luence

range, 2 , 5 . 1 0 - - 5 - 5 . 5 ' 109 c m - 2 ,

b = 1.27

Least-squares fit to f luence

range, 2.5. 10--6-5.5 . 109 crn -2 , b = 1.40

Fluence (cm - 2 )

Fig. 2. Fluence distributions for/~ > 30 MeV for period 1965 to 1990.

TABLE II

E > 30 MeV, 1965-1990; d N / d F ,,., F -b


2.5 x 105-5.5 x 109 1.27+0.06 0.98

2.5 • 106-5.5 x 109 1.40 • 0.06 0.99

To estimate the influence of these deviations, another fit was tried by eliminating the lowest 3 bins. The resulting straight line is shown in Figure 2 and the results for the > 30 MeV are summarized in Table II.

Over the fluence range of 2.5 x 10 6 to 5.5 • 10 9 c m -2 , the exponent is 1.40 and the regression coefficient is 0.99, so the fit has been improved and the exponent is identical, within the errors, to that for the > 10 MeV data over the full fluence range. For energies > 60 MeV, the distribution is similar to those at lower energies and is shown in Figure 3. Over the full range of fluences, 2.5 • 105- 8.5 x 108 cm-2, the best straight line fit has a slope of 1.32. The fitting parameters are shown in Table III. The behaviour at the highest and lowest fluences is very similar to that found for the > 30 MeV data and the exponents of 1.32 and 1.47 (Table III) are in good agreement with those for the > 10 and > 30 MeV analyses.


E > 03 MeV Distribution

i E-03

I E - 0 4 -

1E-05 -

Number of events per unit f luence

1E.06 -

1E-07 -

1E-08 -

1E-09

IE+05

i i r i r,,.i . . . . . . . . I . . . . . . . . I '

1 E + 0 6 1 E + 0 7 1E+08 I E+09

Least-squares fit to full fluence

range, 2 , 5 . 105 8 .5 . 108 cm - 2 ,

b : 1.32

Least-squares fit to fluence

range, 2 . 5 . I 0 - - 6 - 8 . 5 . 108 cm - 2 ,

b = 1.47

Fluence (cm -2)

Fig. 3. Fluence distribution for E > 60 MeV for period 1965 to 1990.

TABLE III

E > 60 MeV, 1965-1990; d N / d F ~ F - b


2.5 • 105-8.5 x 108 1.32 + 0.07 0.98 2.5 • 106-8.5 x 108 1 .47+0.07 0.98

4. Discussion

Most of the previous work (King, 1974; Feynman et al., 1990a, b; Goswami et al., 1988) on the distribution of solar proton events fluences has assumed a log-normal distribution, i.e., the log of the event fluence is distributed normally, although van Hollebeke, Ma Sung, and McDonald (1975) found that a power law matched the observed distributions for peak fluxes very closely. Nevertheless, the adoption of this type of distribution has been more for reasons of practicality, for the development of engineering models, rather than on the basis of physical correctness. Indeed, Feynman et al. (1990a, b) and Gabriel, Feynman, and Damon (1989, 1990) have pointed out that the distribution is clearly not log-normal since a log-normal distribution predicts a decrease in the numbers of events at fluences less than the mean value while the actual distribution shows the opposite effect. In addition, the question of the existence of a separate distribution of the very large events (i.e., fluences > 109 c m - 2 at E > 10 MeV) has been addressed by Feynman et al. (1990a, b) with the conclusion that since the events appeared to be part of the


TABLE IV

E > 10 MeV, active years, cycles 20, 21, and 22 (partial); d N / d F ~ F -b

Cycle b Regression coefficient

20 1.23 4- 0.07 0.98

21 1.28 4- 0,05 0.99 22 1.23 4- 0.05 0,96

same distribution, the acceleration mechanism is the same. Other work (Gabriel, Feynman, and Damon (1991)) has attempted to fit a Kappa distribution to the data, but with little, if any, improvements over the log-normal assumption. Kurt (1990) and Belovskiy et al. (1979) have found exponents of 1.35 and 1.45 using integral distributions for proton fluxes at energies greater than 25 MeV and 10 MeV, respectively. In contrast to the literature for the event integrated flux, Goswami et al. (1988) have found that a power-law distribution for all three solar cycles for the differential fluence variation, at > 10 MeV, in the range 107-1010 cm -2 has an exponent of 1.30 (-t- + 0.06) as opposed to Van Hollebeke, Ma Sung, and McDonald (1975), who find an exponent of 1.15. For cycles 20 and 21 Goswami et al. have calculated an exponent of 1.30 (:k0.12) and also find a break in the distribution at > 109 cm -2. This steepening of the distribution at around l09 cm -2 is consistent with results presented here and the exponent value of 1.30 -t- (0.12) is very close to the values shown in Table I for cycles 20, 21, and part of 22. It would be possible to use more complicated fits, for example two straight lines, one below 109 cm -2 and one above. However, this involves the use of more fitting parameters and therefore, is not guaranteed to produce a statistically significant improvement in the overall fit. Data from solar cycle 19 at E > 10 MeV were not included because measurements of the fluence at this energy for most of the events were not available. When Goswami et al. (1988) included this data, the break occurs at > 101~ cm -2, as was found by Lingenfelter and Hudson (1980) for events with fluence > 108 cm -2 during cycles 19 and 20.

A summary of the fitting parameters for the active parts of cycles 20, 21, and part of 22 is shown in Table IV.

The definition of the active years of a cycle has been taken from Feynman et al.

(1990b) as the period 2 years before to 4 years after the sunspot maximum (i.e., 1968.9, 1979.9, and 1990.9, respectively); for cycle 22 we have included data up to the end of 1990 and so this cycle is incomplete. Note that the exponents for the individual cycles 20, 21, and 22 are in good agreement with each other and close to, but slightly less than, the 1.3 value found by Goswami.

The break-point fluence of 109 cm -2 for E > 10 MeV is consistent with Goswami et al. 's result for cycles 20 and 21, but the results presented here include


part of cycle 22, which would suggest that this break-point is independent of solar cycle. But when Goswami et al. included cycle 19 data, the cut-off fluence increased to 10 m cm -2, as was found by Lingenfelter and Hudson for events with fluences > 108 cm -2 during cycles 19 and 20. It should be noted however, that there is a great deal of uncertainty associated with all of these calculations, because there are so few events with fluences > 101~ cm -2. Between 1963 and 1991 there were only two events with fluences greater than 101~ cm -2, those in August 1972 (1.13 x 10 m cm -2) and October 1989 (1.31 • 10 l~ cm -2) (Feynman etaL, 1993). If solar cycle 19 is included then there is a third event which occurred in November 1960 and had a fluence of 3.2 • 10 l~ cm -2 (Feynman et aL, 1990). The cut-off fluence proposed by Lingenfelter and Hudson, however, is only an estimate and in their paper they quote values between 3 x 109 to 2 x 101~ cm -2, based on the average flux over various periods. Using the straight line fit model shown in Table I, an expression for the frequency of events is given by

dN _ 2811.21F_1.32 (1) dF

where d N / d F is the number of particle events per solar cycle per unit fluence (p cm-2), for E > 10 MeV. Following the procedure given by Lingenfelter and Hudson we derive a relationship for the average flux and the break-point fluence, based on Equation (1) above:

= ~?0.68 �9 (> 10 MeV) (1.2 x 10 -5 p cm -2 s -1) ~ max , (2)

where ~5 is the average flux and Fmax is the break-point fluence. Using our data set for the 25 years between 1965 and 1990, the average flux, ~, is 87.5 p cm -2 s -1 giving a value for Fmax of 9 x 1 0 9 c m - 2 , which compares well with the value of about 10 l~ cm -a given by Lingenfelter and Hudson (1980).

To get an independent estimate of the gradient of the distribution above fluences of 10 l~ cm -2, we have used the probability curves derived by Feynman et al. (1993), which give the probability of a given fluence occurring in a given period of time. For a fluence level of 3 x 1011 cm -2 in 7 years (taken as one solar cycle), the probability is 0.005 and yields a d N / d F of 1.7 • 1 0 - 1 4 events per solar cycle per unit fluence (p cm-2). This is two orders of magnitude lower than that predicted by Equation (1). Assuming the break-point occurs a t 1010 c m - 2 gives an exponent of 3.5. This value is close to that of 4.0 given by Lingenfelter and Hudson, and within the uncertainties of the estimate, considering that we are extrapolating the proton model of Feynman et al. (1993) to fluences more than one of order of magnitude greater than the observational data.

Returning now to the differences between the exponent of 1.15 found by Van Hollebeke, Ma Sung, and McDonald (1975) and the values found herein which are between about 1.2 and 1.4. There are several potential reasons for these dis- crepancies but it should be noted that the values are in agreement to within the

344 S, B. GABRIEL AND J. FEYNMAN

TABLE V

E > 60 MeV, active years, cycles 21 (partial)


2.5 x 105-2.5 • 108 1.25 4-0.12 0.93

Corrected for heliolongitude

5.5 x 105-5.5 x 108 1.27 4-0.062 0.99

estimates of the uncertainties. The first is that the periods for which the data were analysed were different. Van Hollebeke, Ma Sung, and McDonald (1975) found a slope of 1.15 4- 0.05 for peak fluxes of 20-80 MeV protons for the period May 1967 to December 1972. Cliver et al. (1991) obtained a slope of 1.13 i 0.04 for 24- 43 MeV protons for the years 1977-1983. Both of these investigations limited the effects of propagation on the distribution by taking into acount the heliolongitude of an associated flare. The Van Hollebeke, Ma Sung, and McDonald data was for most of the active period of cycle 20 while the data analysed by Cliver et al. were taken during the majority of the active portion of cycle 21. Comparing these values with the values in Table IV for E > 10 MeV, we see that they are actually in good agreement, although consistently smaller, given the estimated uncertainties. This would suggest that the omission of any longitude effects in our analysis has only a very small effect on the slope of the distributions. In fact if one looks at the Van Hollebeke, Ma Sung, and McDonald analysis, there was actually no difference between the distribution slopes for events with longitudes in the range 20 ~ ~ W and those at 60 ~ to 20 ~ E, both having slopes of 1.10 i 0.05. The slope for all events was 1.15 -4- 0.05 which is very close to that for the longitudinally resolved distribution and indeed within the quoted uncertainties the two slopes could be concluded to be the same.

Nonetheless, we have attempted to correct the > 60 MeV data by identifying an associated flare with each of the proton events using the list of proton events and associated flare locations developed for the period June 1973 to August 1979 by Kahler (1982) and that from Shields, Armstrong, and Eckes (1985), for 1974 to 1981. The correction was derived, in a similar way to Kahler, by a least-squares fit to the log F versus A0 distribution, where A0 is the angular distance of the flare from 50 ~ W, the region of apparent optimum connection to the Earth (Van Hollebeke, Ma Sung, and McDonald, 1975). The derived correction to the fluence was exp(--0.78A0) with A0 in radians. The slope of the best fit line for the corrected data was 1.27 -t- 0.06 which was only slightly larger than that for the distribution of all the events (uncorrected for heliolongitude) during the active period of cycle 21 (see Table V for a summary of the fitting parameters).


But, as can be seen from Table V, there is an improvement in the goodness of the fit, with an increase in the regression coefficient from 0.93 to 0.99 and a decrease by a factor of 2 in the uncertainty of the slope.

Another reason for the differences in the exponents could be that the events investigated in this study are larger, in the sense that the threshold for event detection is approximately three orders of magnitude higher than that used by Van Hollebeke, Ma Sung, and McDonald (1975). So, while some of the events will be common to both studies, the data set used herein will not include the smaller events and this could cause the slope to be higher.

At E > 30 MeV and E > 60 MeV, the deviations from a straight line fit at the lowest fluences are probably due to an incomplete data set at the lower fluences. Events with fluences less than 105 cm -2 were not considered.

5. Conclusions

The frequency of solar proton events for the last 3 solar cycles has been found, in the main, to follow a power-law distribution as measured by the integral flux at energies greater than 10, 30, and 60 MeV over 3 to 4 orders of magnitude in fluence. This is a new result since it is the first time such an analysis has been performed on space-based observations covering the full three solar cycles, 20, 21, and 22. The scaling exponent was found to be between 1.21 and 1.39, allowing for uncertainties and variations with energy.

Deviations from power-law behaviour at the lowest fluences are thought to be caused by omission of the smallest events (< 105 cm-2). At fluences greater than 109 cm -2, but less than 101~ cm -2, the slope of the distribution appears to steepen, but quantitative modelling is not warranted due to the lack of data on events with fluences greater than 109 cm -2. Beyond 101~ cm -2 the slope is estimated to be 3.5, which is close to the previously published value of 4. The paucity of data at high fluences makes it difficult to determine the break-point fluence with much certainty. Probably the best that we can do is to say that it lies somewhere between 109 and 101~ cm -2. For the > 10 MeV fluences we have found for the first time that the slope of the power law is remarkably constant during the active parts of each of the 3 solar cycles.

We stress that our frequency distributions are based on measurements of the proton fluences at 1 AU and so will not be the same as those for the events on the Sun due to both coronal and interplanetary propagation effects. We also believe that power-law distributions for proton fluences are the more physically correct form for the distribution (rather than, for example, a single or multiple log-normal distributions). However, this new distribution will not significantly change the fluences calculated by the engineering proton models because these models were generated by a Monte-Carlo technique using a good fit to the largest proton events


which are the only ones that count (for example, for radiation design calculations, the largest events dominate the total fluence accumulated during the mission).

Acknowledgements

The solar particle data set relies on observations of the IMP satellite series. The majority of this information derives from the Johns Hopkins University/Applied

Physics Laboratory Charged Particle Measurement Experiment on IMP 8. We thank Professor T. E Armstrong for providing the data set.

Dr S. Gabriel acknowledges funding for part of this work from Dr Eamonn Daly at ESTEC and Dr J. Feynman acknowledges support of her contribution to

the research to the Jet Propulsion Laboratory, California Institute of Technology, under a contract with The National Aeronautics and Space Administration.

References

Armstrong, T. P., Brungarot, C., and Meyer, J. E.: 1983, in B. M. McCormack (ed.), Weather and Climate Response to Solar Variations, Colorado University Press, Boulder.

Belovskiy, M. N., Kurt, V. G., et al.: 1979, Soviet Cosmic Res. 17, 906. Cane, H. V., McGuire, R. E., and Von Rosenvinge, T. T.: 1986, Astrophys. J. 301, 448. Cliver, E., Reames, D., Kahler, S., and Cane, H.; 1991, Proc. 22rid Int. Cosmic Ray Conf., Dublin,

p. 25. Crosby, N. B., Aschwanden, M. J., and Dennis, B. R.: 1993, Solar Phys. 143, 275. Feynman, J., Armstrong, T. E, Dao-Gibner, L., and Silverman, S. M.: 1990a, J. Spacecraft Rockets

27, 403. Feynman, J., Armstrong, T. P., Dao-Gibner, L., and Silverman, S. M.: 1990b, Solar Phys. 126, 385. Feynman, J., Spitale, G., Wang, J., and Gabriel, S. B.: 1993, J. Geophys. Res. 98, 13 281. Gabriel, S. B., Feynman, J., and Spitale, G.: 1991, NATO ASI Conf. on Behaviour of Systems in the

Space Environment. Gabriel, S., Feynman, J., and Damon, E E.: 1989, Statistical Problems in Proton Event Predictionfor

Space Environments, 11th Conf. on Probability and Statistics in Atmospheric Sciences, p. 305. Gabriel, S. B., Feynman, J., Spitale, G., and Armstrong, T. P.: 1990, Proceedings oftheESA Workshop

on Space Environment Analysis, ESA Publication, WPP-23. Goswami, J. N., McGuire, R. E., Reedy, R. C., Lal, D., and Jha, R.: 1988, J. Geophys. Res. 93, 7195. Hudson, H. S.: 1978, Solar Phys. 57, 237. Kahler, S. W.: 1982, J. Geophys. Res. 87, 3439. Kahler, S. W., Cliver, E. W., Cane, H. V., McGuire, R. E., Stone, R. G., and Sheeley, N. R., Jr.: 1986,

Astrophys. J. 302, 504. King, J.: 1974, J. Spacecraft Rockets 11, 401. Kurt, V. G.: 1990, in E. R. Priest and V. Kirshnan (eds.), Basic Plasma Processes on the Sun, IAU,

p. 409. Lingenfelter, R. E. and Hudson, H. S.: 1980, in R. O. Pepin, J. A. Eddy, and R. B. Merrill (eds.),

Proceedings of Conference on the Ancient Sun, Pergamon Press, London. Montgomery, D. C. and Peck, E. A.: 1992, Introduction toLinearRegressionAnalysis, second edition,

John Wiley and Sons, New York. Ohki, K.: 1989, Space Sci. Rev. 51, 215. Shields, J. C., Armstrong, T. R, and Eckes, S. P.: 1985, J. Geophys. Res. 90, 9439. Van Hollebeke, M. A. I., Ma Sung, L. S., and McDonald, F. B.: 1975, SolarPhys. 41, 189.

power-law distribution for solar energetic proton events

Documents