characterizing solar proton energy spectra for radiation effects applications

2218 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 47, NO. 6, DECEMBER 2000

Characterizing Solar Proton Energy Spectra forRadiation Effects Applications

M. A. Xapsos, Senior Member, IEEE, J. L. Barth, Senior Member, IEEE, E. G. Stassinopoulos, Senior Member, IEEE,S. R. Messenger, R. J. Walters, G. P. Summers, and E. A. Burke, Member, IEEE

Abstract—The Weibull distribution for smallest values is shownto be a useful description for solar proton event energy spectra.One advantage is its compact analytic expression, which allowseasy conversion between differential and integral spectra. Anotheris its versatility, which is necessary for describing the highlyvariable spectra of concern. Furthermore, the Weibull distribu-tion appears to be appropriate for use over broad energy rangesextending out to GeV. Examples are shown and comparisonsto previously used distributions are made. An especially usefulconsequence of this approach for radiation effects applications isthat it allows both predictive model spectra and observed spectrato be described by the same distribution. This allows spectra tobe systematically ranked by severity of radiation damage causedin microelectronics. It further allows observed spectra to berelated to predictive model parameters such as confidence levels.These points are demonstrated by evaluating the ionization dosedeposited by various spectra in silicon behind aluminum shieldingappropriate for spacecraft.

I. INTRODUCTION

SOLAR proton events are a possible source of significantdamage for spacecraft in geostationary or polar orbits as

well as those on interplanetary missions. In order to quantita-tively assess the radiation damage that can occur, an estimate ofthe probability of encountering events is needed, along with theevent magnitudes and energy spectra over the mission time ofinterest.

The energy spectrum is one of the most difficult character-istics of a solar event to describe accurately. Individual eventsare notoriously variable with respect to this characteristic. Manydistributions have been suggested for spectra based upon empir-ical or theoretical grounds. These include an exponential in en-ergy [1], [2] exponential in magnetic rigidity [1], [2] a lognormaldistribution in characteristic magnetic rigidity [3], a power func-tion in energy based on a shock acceleration model [4] and aBessel function expression derived from stochastic accelerationarguments [5]. Variations of the original acceleration model ex-pressions have also been reported [6].

Manuscript received July 24, 2000. This work was supported by the NASAMarshall Space Environments and Effects Program.

M. A. Xapsos, R. J. Walters and G. P. Summers are with the NavalResearch Laboratory, Code 6825, Washington, DC 20375 USA (e-mail:[email protected]).

J. L. Barth and E. G. Stassinopoulos are with NASA GoddardSpace Flight Center, Code 562, Greenbelt, MD 20771 USA (e-mail: [email protected]).

S. R. Messenger and E. A. Burke are with SFA Inc., Largo, MD 20774 USA.Publisher Item Identifier S 0018-9499(00)11249-3.

In this paper the Weibull probability distribution is shownto be applicable to solar proton energy spectra over wide en-ergy ranges. The following section summarizes the sources ofsolar proton event data used. Then the Weibull distribution isdescribed as applied to solar proton energy spectra. It is then il-lustrated by application to some observed events, including bothdifferential and integral distributions. A discussion follows onextending the Weibull description to very high proton energies.The applicability of the distribution to predictive model spectrafor various confidence levels and mission lengths is then noted.The fact that both observed spectra and model spectra are welldescribed by a common formulation allows the two to be tiedtogether. This is demonstrated for both worst case and cumula-tive event spectra. In both cases, transport calculations throughspacecraft shielding are done to compare the ionization dose de-posited by various spectra.

II. SOLAR PROTON DATA SOURCES

Most of the data used in this study are the same as that usedfor theEmission ofSolar Protons (ESP) model [7]–[9]. This isa new solar proton event model that increases predictive capa-bilities for spacecraft design in several ways beyond those ofprevious models. The model energy spectra used in this paperwere obtained by fitting the ESP fluence vs. energy calcula-tions to the Weibull distribution. The ESP model is based onsatellite measurements for the last 3 complete solar cycles, pro-cessed at NASA Goddard Space Flight Center. The data coverthe time period 1963 through 1996. Cycle 20 data are from theIMP-3 through IMP-8 series of satellites. Cycle 21 data are fromIMP-8. Cycle 22 data are from the GOES-5 through GOES-7satellites. A number of other analyzes of solar proton eventdata were also reviewed for this model including those by King[1], Goswami [3], Armstrong [10], Feynman [11] and Shea andSmart [12]. Further details about the data, spacecraft and detec-tors are discussed elsewhere [7]. Unless otherwise indicated thedata for observed events shown in this paper are also from thesesources.

III. SOLAR PROTON ENERGY SPECTRA AND THEWEIBULL

DISTRIBUTION

The representation of solar proton energy spectra that we rec-ommend is given in integral form by

(1)

where can be either the proton fluence or proton flux havingenergy that exceeds a threshold energy. We take the units

0018–9499/00$10.00 © 2000 IEEE

XAPSOSet al.: CHARACTERIZING SOLAR PROTON ENERGY SPECTRA FOR RADIATION EFFECTS APPLICATIONS 2219

Fig. 1. Comparison of data for the differential peak flux energy spectrumobserved on July 20, 1981 (points) to the best fit Bessel function distribution(dashed line) and the best fit Weibull distribution (solid line). Data are from [5].

of to be cm when it represents integral fluence andwhen it represents integral flux. The units of

are MeV. The constants , and are determined by anonlinear regression fit to the energy spectrum of interest. (1) isequivalent to the Weibull distribution for smallest values whenthe location parameter is set equal to zero [13]. The constantis related to the event magnitude whileand are related to thespectrum hardness. The latter constant is also referred to as theshape parameter in connection with the Weibull distribution.

The differential energy spectrum is easily found by differen-tiating with respect to . This results in

(2)

Differential fluences are in units of and differ-ential fluxes are in . We have found thatthe 3 fitting parameters can easily be extracted from either ofthe above 2 equations using a standard Marquardt–Levenbergnonlinear regression routine [14], [15].

The Weibull distribution can be interpreted generally in termsof survival [16]. This interpretation is qualitatively applicableto solar proton energy spectra. For example, as protons are ac-celerated in a field caused by a shock wave, there is progres-sively more resistance against attaining higher energies. Thus,the proton flux decreases with increasing energy, and the en-ergy spectrum can be interpreted as a survival distribution. Suchan interpretation may be related to the proposed scattering ofthe protons by self-generated waves [17], [18]. However, fur-ther investigation is needed to determine if this type of physicalprocess is related to the Weibull distribution.

A. Application to Observed Events

It has been noted previously that solar proton event spectraare highly variable from one event to the next. However, theWeibull distribution as described above has been found to bebroadly applicable. Some examples are illustrated in this sec-tion. Fig. 1 shows the differential peak flux distribution of asolar proton event on July 20,1981. The data are taken from

Fig. 2. Comparison of data for the integral fluence energy spectrum for thesolar proton event of August 4, 1972 (points) to the best fit exponential functionin magnetic rigidityR (dashed line) and the best fit Weibull distribution (solidline).

TABLE IWEIBULL PARAMETERS FORSOME NOTABLE SOLAR PROTONEVENTS DURING

SOLAR CYCLES 20–22. SEE EQUATIONS (1) AND (2)

McGuire and Rosenvinge [5]. The authors of [5] compared fitsof 7 such spectra using Bessel functions, power functions andexponential functions of magnetic rigidity. They concluded thatthe Bessel functions provided the best description. We havecompared these results to fitted Weibull distributions. It wasfound that the Weibull description was at least as good a fit tothe data, and in most cases better than the Bessel function fit.This is based on both visual inspection and on examination ofthe correlation coefficients. A typical example is shown in Fig. 1where the Weibull distribution is shown by the solid line and theBessel function is shown by the dashed line. It is clear that theWeibull description is applicable over a greater energy range.Examination of the 10 differential spectra published by Mazuret al. [19] have led us to similar conclusions.

It is well known that the often cited solar proton event ofAugust 4, 1972 had an unusual energy spectrum with an inte-gral form that is approximated by an exponential in energy [1],[2]. Inspection of (1) shows that the Weibull distribution canapproximate this when the parameteris close to 1. Resultsobtained for fitting this energy spectrum are shown in Fig. 2.The fit to the Weibull distribution is shown by the solid lineand the corresponding parameters are given in Table I. Theseresults are very close to an exponential in energy, illustratingthe versatility of the distribution. Also shown in Fig. 2 by thedashed line is a fit based on the commonly used exponentialin magnetic rigidity. It can also be shown that the Weibull dis-tribution approximates an exponential in magnetic rigidity for


Fig. 3. Comparison of data for the integral fluence energy spectrum forthe solar proton event of November 2, 1992 (points) to the best fit Weibulldistribution (solid line). Data are from [20].

proton energies less than about 100 MeV when the parameteris near 0.5. This is roughly the energy range that the exponentialin magnetic rigidity has been found to be most useful for.

Table I lists results obtained for some of the largest eventsof the last 3 completed solar cycles (20–22). The last 4 eventslisted are from the very active cycle 22 and were fitted over theenergy range of 1 to 300 MeV. The Table shows the onset dateof the event, the 3 fitted Weibull parameters, and the correla-tion coefficient, . The correlation coefficients reported are theoutput of a nonlinear regression routine [14]. The high valuesobtained are indicative of the high quality of the fits.

B. Extension to Very High Proton Energies

As discussed in the previous section, other distributions thathave been used to describe solar proton energy spectra appearto be more limited in the energy range over which they are ac-curate. Applications invariably arise that require knowledge ofincreasingly higher energy ranges, so it is useful to examine theapplicability of the Weibull distribution to very high energies.Unfortunately, the amount of data available at energies in thehundreds of MeV and GeV range is rather limited. However, 2representative examples are shown here.

Tylka et al. have analyzed 3 solar proton events out to ener-gies of about 500 MeV [20]. In this work, proton integral flu-ences in the 1 to 100 MeV range were determined from theMEPAD instrument on the GOES-7 satellite. Integral fluencesfor 3 higher energies were determined from the HEPAD instru-ment on GOES-6. We find that the Weibull distribution gives agood fit to each of the 3 reported integral spectra. An exampleof this is shown in Fig. 3 for the event of November 2, 1992.

Measurements of spectra can be extended into the GeVenergy range by using neutron monitors to study groundlevel events [21]. An example of an energy spectrum with astrikingly broad energy range is the September 29, 1989 datarecently reported by Lovellet al. [22]. The differential protonflux spectrum is reproduced here in Fig. 4. Results out toabout 250 MeV were deduced from IMP-8 and GOES-7. Theremaining data out to 10 GeV, shown by the shaded region,were obtained using neutron monitors. The size of the region

Fig. 4. Comparison of data for the differential flux energy spectrum observedon September 29, 1989 to the best fit Weibull distribution (solid line). Thedata were obtained from IMP (squares), GOES (circles) and neutron monitors(shaded region). Data are from [22].

TABLE IIWEIBULL PARAMETERS FORESP MODEL WORSTCASE EVENTS DURING ONE

SOLAR ACTIVE YEAR. SEE EQUATIONS (1) AND (2)

indicates the uncertainty in the flux measurement. A Weibullfit, shown by the line, describes the entire data set remarkablywell over its full range of about 4 orders of magnitude in energyand 9–11 orders of magnitude in differential flux. Thus, theWeibull distribution appears to be suitable for describing eventswith fluxes that extend to very high energies.

IV. RELATING MODEL SPECTRA TOMEASUREDSPECTRA FOR

SPACECRAFTAPPLICATIONS

In previous work, we have shown that (1) is suitable for de-scribing integral fluences calculated with the ESP model. Forexamples of this, the reader is referred to [8] for worst case eventmodel spectra and [9] for cumulative model spectra. Such exam-ples will not be repeated here. However, it is important to tie themodel results and measured results together to help make real-istic assessments of the radiation effects. This also allows eventsto be ranked in order of severity of radiation damage. This isdone in the following 2 subsections for worst case event spectraand for cumulative event spectra. For both cases the ionizationdose deposited in silicon surrounded by various thicknesses ofaluminum shielding is evaluated.

A. Worst Case Event Spectra

Results similar to those in Table I have been obtained for ESPmodel worst case event energy spectra, and are shown in Table II


Fig. 5. Dose deposited in silicon by solar proton event energy spectra as afunction of aluminum shield thickness. The solid lines and points are for worstcase spectra obtained from the ESP model [7] at the confidence levels shown.The dashed lines and unfilled circles are for the October 19, 1989 and March23, 1991 events.

as a function of confidence level,. The time period for thesecalculations was taken as one solar active year. Thus, the tabu-lated parameters combined with (1) define the worst case energyspectrum that is expected to be encountered during a one solaractive year period at the given level of confidence. The 100%confidence level is not a physically based limit but a practicallimit for design considerations [8].

Comparison of the parameter values listed in Tables I and IIindicate that the and values cover a wider range for the ob-served events than those obtained from the ESP model spectra.This reflects the fact that the shape of the spectrum for any givenevent is highly variable. However, when the complete sampleof solar proton data is processed in the manner employed bythe model a more consistent and uniform behavior emerges asshown.

Now that both worst case model events and several of themost severe observed events are described in this convenientformalism, the corresponding radiation damage can be assessedmore systematically. This will not be done here for specific de-vice structures. Instead, a general assessment will be given forthe ionization dose that would be observed for spacecraft ap-plications. In these calculations the following assumptions aremade. First, the energy spectrum of the solar proton event isgiven by the Weibull distribution having parameters given inTables I and II. Second, the energy spectrum is isotropically in-cident upon a spherical shell of aluminum shielding having aspecified thickness. Third, the ionization dose of interest is thatfor a small volume of silicon at the center of the shielding.

Transport calculations through the aluminum shielding weredone using a method described in detail previously [23]. Thisis a modified approach of that described by Haffner [24] andBurrell [25]. Results of several calculations were checkedagainst those obtained from the code Space Radiation [26], andclose agreement was observed. Fig. 5 shows results for dose de-posited by solar proton event spectra in krad(Si) vs. aluminumshield thickness. Calculations were done for all spectra shownin Tables I and II incident on 25, 50, 75, 100, 125 and 150 mil

TABLE IIIPOWER FUNCTION PARAMETERS FORSOME NOTABLE SOLAR PROTONEVENT

SPECTRADURING SOLAR CYCLES 20–22. SEE EQUATION (3)

TABLE IVPOWER FUNCTION PARAMETERS FORESP MODEL WORSTCASE EVENT

SPECTRADURING ONE SOLAR ACTIVE YEAR. SEE EQUATION (3)

thick aluminum. The Figure compares the dose deposited by theworst case model events at various confidence levels rangingfrom 50% to 100% and the dose deposited by the events ofOctober 19, 1989 and March 23, 1991. It is seen that the eventof October 19, 1989 falls just above the 90% confidence leveland the event of March 23, 1991 falls just above the 80%confidence level over the range of shield thickness used. Whilethese two severe events closely parallel the model dose-depthcurves in the Figure, it should not be concluded that all eventswould. These results are clearly dependent on the event energyspectrum, which is highly variable. The shape of the dose-depthcurves in Fig. 5 is strongly influenced by that portion of thespectrum where the proton ranges are comparable to the shieldthickness. In this situation the proton energies out to ’s ofMeV are the most important.

Several interesting observations can be made about the resultsshown in Fig. 5. First, it is clear from the straight line behaviorthat the deposited dose is a power function of the shield thick-ness. In other words, the deposited dose,, is given by

(3)

where is the shield thickness and and are fitted parame-ters. Results of the transport calculations were fitted to equation(3) for all spectra listed in Tables I and II. The fitted results areshown in Table III for the observed spectra and Table IV forthe model spectra. The parameters used correspond to units ofmils of aluminum for and krad(Si) for . Thus, for example(3) can be used to calculate a dose of about 5.9 krad(Si) forthe October 19, 1989 event for a 60 mil thick shield. The factthat a simple power function results from these calculations is apleasant surprise because the deposited dose is proportional tothe product of 2 rather complex functions over a broad energyrange—the transmitted proton fluence and the proton stoppingpower. From a practical point of view, this means that (3) canbe conveniently used to extrapolate and interpolate to different


Fig. 6. Dose deposited in silicon by solar proton event energy spectra asa function of aluminum shield thickness. The solid lines and points are forcumulative spectra for a time period of 7 solar active years obtained from theESP model [7] at the confidence levels shown. The dashed line and unfilledcircles are for the cumulative spectrum of the 7 solar active years for solarcycle 22.

shield thicknesses. Further examination of this power functiondependence indicated it is valid for a range of shield thicknessfrom a few mils out to about 200 mils. Another consequenceof the power function result is that it provides a straightfor-ward method of characterizing an event’s spectral hardness bymeans of the power index,, over the range of shield thick-ness in which the power function is valid. The larger the powerindex, the more rapidly the transmitted spectrum drops off withincreased shielding, and the softer the spectrum. As discussedabove, this applies out to energies ’s of MeV. This is clearlya relevant characterization of spectral hardness for spacecraftapplications.

B. Cumulative Event Spectra

Transport calculations were also done for cumulative solarproton event energy spectra obtained from the ESP model fora 7 solar active year period and confidence levels ranging from50% to 99%. Results are shown in Fig. 6. These are comparedto results obtained using the cumulative spectrum for the 7 solaractive years of solar cycle 22 measured by the GOES satellites.It is seen that the cycle 22 spectrum equates approximately tothe 70% confidence level. This seems reasonable given that themodel incorporates data from the last 3 complete solar cyclesand cycle 22 was the most active of the 3. This Figure also illus-trates the amount of safety margin built into choosing high levelsof confidence. For example, consider a 100 mil thick shield. Ifthe commonly used 90% confidence level had been used for amission during solar cycle 22, the safety margin would havebeen about a factor of 2.2 in cumulative dose. If the 99% confi-dence level had been used, the safety margin would have beenabout a factor of 6.6 in cumulative dose.

Finally, it should be noted that these results for cumulativeevent spectra presented in Fig. 6 follow a power function be-havior similar to that discussed for worst case events. These

TABLE VPOWER FUNCTION PARAMETERS FORCUMULATIVE SOLAR PROTON EVENT

SPECTRA FOR7 SOLAR ACTIVE YEARS. SEE EQUATION (3)

results were also fitted to (3), and the parameters are listed inTable V.

V. CONCLUSION

Knowing the worst case and cumulative energy spectra ofsolar proton events is a necessity for radiation effects assess-ments. It has been shown that both differential and integral en-ergy spectra for solar proton events can be conveniently de-scribed by the Weibull distribution for smallest values. This ap-plies to observed spectra and to model spectra. With regard tothe ESP model [7] this adds a new feature, which other solarproton event models do not incorporate.

The Weibull distribution can approximate many of the fea-tures of other functions that have been suggested for this pur-pose but which are of more limited utility. The broad applica-bility of this distribution enhances the ability to calculate radia-tion effects due to solar proton events. Furthermore, it enhancesthe ability to make comparisons between model spectra and ob-served spectra. An example of the utility of this approach wasshown by calculating the dose deposited by various spectra as afunction of shield thickness appropriate for spacecraft applica-tions. The results obtained were expressed as a power functionof the shield thickness. This should be a convenient result forspacecraft designers.

ACKNOWLEDGMENT

The authors thank Dr. A. Tylka of NRL for making availablethe data shown in Fig. 3.

REFERENCES

[1] J. H. King, “Solar proton fluences for 1977–1983 space missions,”J.Spacecraft, vol. 11, pp. 401–408, 1974.

[2] L. J. Lanzerotti, D. W. Maurer, H. H. Sauer, and R. D. Zwickl, “Largesolar proton events and geosynchronous communication spacecraft solararrays,”J. Spacecraft, vol. 28, pp. 614–616, 1991.

[3] J. N. Goswami, R. E. McGuire, R. C. Reedy, D. Lal, and R. Jha, “Solarflare protons and alpha particles during the last three solar cycles,”J.Geophys. Res., vol. 93, no. A7, pp. 7195–7205, 1988.

[4] D. C. Ellison and R. Ramaty, “Shock acceleration of electrons and ionsin solar flares,”Ap. J., vol. 298, pp. 400–408, 1985.

[5] R. E. McGuire and T. T. von Rosenvinge, “The energy spectra of solarenergetic particles,”Adv. Sp. Res., vol. 4, pp. 117–236, 1984.

[6] P. P. Majewski, E. Normand, and D. L. Oberg, “A new solar flare heavyion model and its implementation through MACREE, an improved mod-eling tool to calculate single event effect rates in space,”IEEE Trans.Nucl. Sci., vol. 42, pp. 2043–2050, Dec. 1995.


[7] M. A. Xapsos, J. L. Barth, E. G. Stassinopoulos, E. A. Burke, and G.B. Gee, “Model for emission of solar protons (ESP)—Cumulative andworst case event fluences,” NASA-Marshall Space Flight Center SEEProgram, http://see.msfc.nasa.gov.

[8] M. A. Xapsos, G. P. Summers, J. L. Barth, E. G. Stassinopoulos, andE. A. Burke, “Probability model for worst case solar proton event flu-ences,”IEEE Trans. Nucl. Sci., vol. 46, pp. 1481–1485, Dec. 1999.

[9] M. A. Xapsos, G. P. Summers, J. L. Barth, E. G. Stassinopoulos, andE. A. Burke, “Probability model for cumulative solar proton event flu-ences,”IEEE Trans. Nucl. Sci., vol. 47, no. 3, pp. 483–490, June 2000.

[10] T. P. Armstrong, C. Brungardt, and J. E. Meyer, “Satellite observationsof interplanetary and polar cap solar particle fluxes from 1963 topresent,” inWeather and Climate Responses to Solar Variations, B. M.McCormac, Ed. Niwot, Colorado: Colorado Associated UniversityPress, 1983.

[11] J. P. Feynman, T. P. Armstrong, L. Dao-Gibner, and S. M. Silverman,“New interplanetary proton fluence model,”J. Spacecraft, vol. 27, pp.403–410, 1990.

[12] M. A. Shea and D. F. Smart, “A summary of major solar proton events,”Solar Phys., vol. 127, pp. 297–320, 1990.

[13] E. Castillo,Extreme Value Theory in Engineering. New York, NY:Academic Press, Inc., 1988.

[14] SigmaStat version 2.03. Chicago, IL: SPSS Inc., 1995.[15] W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery,Numer-

ical Recipes in Fortran—The Art of Scientific Computing, 2nd ed. NewYork, NY: Cambridge University Press, 1992.

[16] R. C. Elandt-Johnson and N. L. Johnson,Survival Models and DataAnalysis. New York, NY: John Wiley & Sons, 1980.

[17] M. A. Lee, “Coupled hydromagnetic wave excitation and ion accelera-tion upstream of the earth’s bow shock,”J. Geophys. Res., vol. 87, p.5063, 1982.

[18] D. V. Reames and C. K. Ng, “Streaming-limited intensities of solar en-ergetic particles,”Astrophys. J., vol. 504, pp. 1002–1005, Sept. 1998.

[19] J. E. Mazur, G. M. Mason, B. Klecker, and R. E. McGuire, “The energyspectra of solar flare hydrogen, helium, oxygen and iron: Evidence forstochastic acceleration,”Ap. J., vol. 401, pp. 398–410, 1992.

[20] A. J. Tylka, W. F. Dietrich, P. R. Boberg, E. C. Smith, and J. H. AdamsJr., “Single event upsets caused by solar energetic heavy ions,”IEEETrans. Nucl. Sci., vol. 43, pp. 2758–2766, Dec. 1996.

[21] D. F. Smart and M. A. Shea, “Comment on the use of GOES solar protondata and spectra in solar proton dose calculations,”Radiat. Meas., vol.30, pp. 327–335, 1999.

[22] J. L. Lovell, M. L. Duldig, and J. E. Humble, “An extended analysis ofthe September 1989 cosmic ray ground level enhancement,”J. Geophys.Res., vol. 103, no. A10, pp. 23 733–23 742, Oct. 1998.

[23] S. R. Messenger, M. A. Xapsos, E. A. Burke, R. J. Walters, and G. P.Summers, “Proton displacement damage and ionizing dose for shieldeddevices in space,”IEEE Trans. Nucl. Sci., vol. 44, pp. 2169–2173, Dec.1997.

[24] J. W. Haffner,Radiation and Shielding in Space. New York, NY: Aca-demic Press, 1967.

[25] M. O. Burrell, “The calculation of proton penetration and dose rates,” inProceedings of the Second Symposium on Protection Against Radiationsin Space, 1964, NASA Publication SP-71.

[26] Space Radiation version 4.0: Space Radiation Associates.

characterizing solar proton energy spectra for radiation effects applications

Documents