Estimation of solar energetic proton mission integrated fluences and peak intensities for missions traveling close to the Sun

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1 SPACE WEATHER, VOL. 9,, doi: /2011sw000708, 2011 Estimation of solar energetic proton mission integrated fluences and peak intensities for missions traveling close to the Sun D. Lario 1 and R. B. Decker 1 Received 7 July 2011; revised 6 September 2011; accepted 8 September 2011; published 10 November [1] A method to estimate both solar energetic particle mission integrated fluences and solar energetic particle peak intensities for missions traveling through the innermost part of the heliosphere (r < 1 AU) is presented. By using (1) an extensive data set of particle intensities measured at 1 AU over the last three solar cycles, (2) successive launch dates for the mission traveling close to the Sun over the time interval spanned by our data set, and (3) appropriate radial dependences to extrapolate fluences and peak intensities measured at 1 AU to the heliocentric radial distance of the mission at each specific time, we generate distributions of both mission integrated fluences and maximum peak intensities. From these distributions we extract the values of mission integrated fluence and maximum peak intensity at a required confidence level. Results of this method applied to the specific case of the nominal prime mission of Solar Probe Plus are shown. Citation: Lario, D., and R. B. Decker (2011), Estimation of solar energetic proton mission integrated fluences and peak intensities for missions traveling close to the Sun, Space Weather, 9,, doi: /2011sw Johns Hopkins University Applied Physics Laboratory, Laurel, Maryland, USA. 1. Introduction [2] Estimation of the radiation damage that solar energetic particles (SEPs) may produce to space vehicles traveling close to the Sun needs a careful estimation of both the occurrence probability of a SEP event during the spacecraft mission and the intensity and fluence of such events in the innermost part of the heliosphere. [3] Engineering models commonly used to estimate SEP fluences over spacecraft missions consider that the occurrence of SEP events is random in nature, with the occurrence of each event independent of the occurrence of previous events (see references cited by Jiggens and Gabriel [2009]). However, large SEP events tend to occur clustered in time when one or two powerful active regions appear on the solar disk and trigger a series of SEP events [e.g., Lario et al., 2005a]. Therefore, SEP events retain some memory of the past and are not completely random [Jiggens and Gabriel, 2009]. [4] Energetic particle data assembled during several solar cycles by near Earth spacecraft can be used as representative of the occurrence of SEP events in the future and for a given period of time [Jun et al., 2007]. Extrapolation of SEP intensities measured near Earth to a spacecraft traveling close to the Sun requires the use of radial dependences for peak intensities and fluences [Lario et al., 2007]. In situ observations of the energetic particle environment in the innermost part of the solar system are limited to data collected by the two Helios spacecraft between 0.28 and 0.98 AU during solar cycle 21. Studies of the radial gradients of SEP intensities and fluences are based on either statistical analyses combining Helios and 1 AU observations [e.g., Lario et al., 2006, and references therein] or models of SEP acceleration and transport [e.g., Shea et al., 1988; Ruzmaikin et al., 2005; Lario et al., 2007, and references therein]. [5] In this article we discuss a method developed to estimate mission integrated SEP fluences for spacecraft in orbit around the Sun and SEP peak intensities that spacecraft traveling in the innermost part of the heliosphere may observe. The method consists of the following: (1) accumulate SEP data measured at 1 AU over one or more past solar cycles; (2) assume that these particle intensities are representative of the particle intensities that will occur during the mission traveling close to the Sun; (3) assume that the launch date of the mission traveling close to the Sun moves over the time interval encompassed by our SEP data set; (4) adopt several radial dependences as described below to extrapolate peak intensities and fluences measured at 1 AU to the heliocentric distance of the mission at a Copyright 2011 by the American Geophysical Union 1of18

2 given time; (5) generate statistical distributions of either maximum peak intensities or mission integrated fluences from the set of hypothetical launch dates used in the third step; and (6) extract a value of either mission integrated fluence or maximum peak intensity with a given confidence level from the statistical distributions generated in the fifth step. [6] In section 2 we summarize previous studies on radial gradients of SEP intensities and fluences that can be used to extrapolate 1 AU intensities and fluences toward inner heliocentric distances. In section 3 we discuss the method developed to estimate total mission integrated SEP fluences, with special application to the case of the Solar Probe Plus (SPP) mission. Comparisons with standard methods used to estimate mission fluences (i.e., JPL 91 [Feynman et al., 1993]) are also discussed. In section 4 we discuss the method used to estimate the maximum peak intensities of SEP events observed at distances <1 AU. For completeness, worst case SEP fluxes obtained from the most intense SEP peak intensities observed at 1 AU during the last two solar cycles are also included. Finally, section 5 summarizes the main results of the present work. 2. Radial Dependences [7] The study of the radial dependence of SEP peak intensities and fluences has been addressed from both the observational and the modeling point of view. Lario et al. [2006] used energetic proton intensities measured by the IMP 8 spacecraft at 1 AU and the two Helios spacecraft traveling between 0.28 and 0.98 AU to analyze the peak intensities and fluences of SEP events observed simultaneously by at least two of these spacecraft at different radial distances. Lario et al. [2006] approximated the radial distributions of peak intensities and fluences of the ensemble of events by a functional form j ¼ j 0 r exp½ kðf F 0 Þ 2 Š; where r is the heliocentric radial distance of the spacecraft, F is the longitudinal angular distance between the nominal foot point of the field line connecting the observer to the Sun and the site of the active region that generated the event, F 0 is the centroid of the distributions, and j is either the peak intensity or the event fluence. Lario et al. [2006] found that the dominant parameter that determines the peak intensity and fluence of a SEP event is not the heliocentric radial distance of the observer but rather the longitudinal distance between the parent active region and the foot point of the magnetic field line connecting the observer to the Sun. Lario et al. [2006] found that, over the ensemble of events, a ranges from 2.7 to 1.9 for 4 13 MeV and MeV proton peak intensities, respectively, and from 2.1 to 1.0 for 4 13 MeV and MeV proton event fluences, respectively. Details of both the method used to determine these dependences and the values of the parameters defining the longitudinal variation of peak intensities and fluences are given by Lario et al. [2006]. ð1þ [8] From the modeling point of view, studies of the radial dependence of peak intensities and fluences must consider the mechanisms of energetic particle transport and acceleration through the development of the SEP events. If energetic particles are mainly injected from close to the Sun, the radial dependences of peak intensities and fluences are determined by the transport processes undergone by the particles along the interplanetary magnetic field (IMF) lines. Under the assumption that the particle transport is purely diffusive along the IMF with a spatial diffusion coefficient D = vl/3 (with v the particle speed and l the mean free path of the particles parallel to the magnetic field) and neglecting the effects of the solar wind expansion, the time integrated radial net flux scales like r 2, the maximum intensity for an impulsive injection scales like r 3, and the time integrated intensity or fluence scales like (rd rr ) 1, where D rr = D cos 2 y is the radial diffusion coefficient and y is the angle between the radial direction and the local magnetic field [Vainio et al., 2007]. The scaling laws obtained from the diffusion equation are not always valid since at least the following conditions have to be met: (a) the mean free path of the particles has to be much smaller than the heliocentric radial distance of the observer, i.e., l rr r, (b) the time when the maximum intensity is reached has to be much smaller than the adiabatic cooling time, i.e., r/l rr v/(2v sw ) (where V sw is the solar wind speed), (c) the duration of the injection at the Sun has to be shorter than the time of maximum intensity; and (d) the site of the particle injection r 0 has to be close to the Sun, i.e., r 0 r [Vainio et al., 2007]. For cases not meeting these conditions, it is necessary to use numerical transport models to investigate these radial dependences. [9] Hamilton [1988] used the spherically symmetric transport model of Parker [1965] to study the radial dependence of peak intensities and fluences in SEP events. This transport model includes the effects of spatial diffusion, convection and adiabatic energy loss under the assumption that particle distributions are isotropic and that the focusing effect of energetic particles along the magnetic field is negligible. Both assumptions are not well supported by SEP observations in the innermost part of the heliosphere where the focusing effect is the dominant factor in the SEP transport [Lario, 2007]. Large and long lasting anisotropies are usually observed at heliocentric radial distances smaller than or equal to 1 AU [Roelof, 1979; Heras et al., 1994]. These observations indicate that particle distributions are not isotropic and that SEP transport at these inner distances is far from being dominated by diffusion. [10] To provide guidelines for flux extrapolation with radial distances, Hamilton [1988] used his transport model with the assumptions that (1) energetic particles propagate in a flux tube whose cross section expands as r 2, and (2) the spatial radial diffusion coefficient is given by K r = K 0 r b where K 0 and b are parameters of the model (b <2).Whenthesolar wind speed is set to V sw = 0, this model reduces to a pure diffusion transport. In this case, the transport equation may be solved analytically to find that the peak intensity follows a r 3 dependence (independent of K 0 or b) andthefluence 2of18

3 Figure 1. Daily averages of (a) >5 MeV, (b) >10 MeV, (c) >30 MeV, and (d) >60 MeV proton fluences measured at 1 AU by IMP 8 and the series of GOES spacecraft from 1973/303 to 2008/182. (e) Monthly and monthly smoothed sunspot number (SSN) for the last three solar cycles. The dashed vertical lines indicate the start and end of each solar cycle. follows a r (b+1) dependence (if the cross section of the flux tube expands as r 2 ). If V sw 0, the variation of the peak intensity with radial distance is more accentuated because of the effects of energy loss; in this case, the variation with radial distance of peak intensities and fluences depends on both the values of b and K 0, as well as the energy spectrum of the particle intensities [Hamilton et al., 1990]. [11] Application of the Hamilton [1988] model to proton measurements in the energy range of 10 to 70 MeV from 1 to 5 AU was used to obtain laws that allow extrapolation of particle intensities and fluences measured at 1 AU to other radial distances [Shea et al., 1988]. Recommendations for radial extrapolation of peak fluxes and fluences, as documented by Feynman and Gabriel [1988] read as follows [see Smart and Shea, 2003]: (1) for flux extrapolations from 1 AU to >1 AU, use a functional form of r 3.3 and expect variations ranging from r 4 to r 3 ; (2) for flux extrapolations from 1 AU to <1 AU, use a functional form of r 3 and expect variations ranging from r 3 to r 2 ; and (3) for fluence extrapolations from 1 AU to other distances, use a functional form of r 2.5 and expect variations ranging from r 3 to r 2. [12] Because of the focusing effect due to the outwardly decreasing magnetic field magnitude in the inner heliosphere, the anisotropies become too large for the diffusion 3of18

4 Figure 2. Energy spectra of the solar proton fluences integrated over each one of the last three solar cycles. model to be applicable if the mean free path in the interplanetary medium is comparable to (or larger than) the observer s heliocentric radial distance. Lario et al. [2007] studied the radial dependence of peak intensities and fluences in the framework of the focused transport theory. They solved the focused diffusion transport equation that includes the effects of solar wind convection, adiabatic deceleration and pitch angle scattering [Ruffolo, 1995] assuming a Reid Axford time profile for the particle injection at the base of a flux tube described by an Archimedean spiral IMF whose cross section A(r) expands as r 2 cos(y). Both peak intensities and event fluences decrease with increasing radial distance. Lario et al. [2007] deduced functional forms to extrapolate peak fluxes and fluences with radial distance that depend on the energy of the particles, the pitch angle scattering conditions, and the duration of the particle injection. The main results found by Lario et al. [2007] are as follows: (1) The smaller the mean free path of the particles, the larger the decrease of both peak intensities and fluences with radial distance. (2) The smaller the energy of the particles, the larger the decrease of both peak intensities and fluences with radial distance. (3) Extended particle injections contribute to soften the decrease of the peak intensities with radial distance but have no influence on the event fluence. (4) The decreasing effect of focusing with radial distance or the increasing effect of pitch angle scattering with radial distance, or both, cause peak intensities and fluences to decrease more rapidly with radial distances >1.0 AU and less rapidly with those distances <1.0 AU (see details given by Lario et al. [2007]). [13] The power law dependences on helioradius derived from the focused diffusion transport equation with a fixed source close to the Sun [Lario et al., 2007] are, in general, gentler than those recommended for radial extrapolation of intensities and fluences deduced using diffusive SEP transport models [Feynman and Gabriel, 1988], especially within 1 AU of the Sun and for large mean free paths. In general, radial gradients for peak intensities and event fluences deduced by Lario et al. [2007] vary between r 1.3 and r 3 and between r 1.5 and r 0.7, respectively, with the exponent depending on the characteristics of the particles injection, the properties of the particle transport and the particle energy. [14] Whereas the results by Lario et al. [2007] apply to the case of a fixed source close to the Sun, Smart and Shea [2003] showed that SEP events where particle intensities are dominated by the effects of traveling interplanetary shocks have peak intensities that do not scale in radial distance as indicated by simple power law extrapolations. Models assuming the continuous contribution of particle acceleration by traveling shocks [e.g., Aran et al., 2005; Ruzmaikin et al., 2005; Vainio et al., 2007] obtain radial profiles of peak intensities and fluences that depend on both the energy of the particles and the parameters used in the model to characterize the shock formation, particle acceleration at the shock, particle injection from the shock into the interplanetary medium, and the transport of both SEPs and shocks. The heliolongitude of the parent solar event with respect to the observer s location seems to be the fundamental parameter that determines the radial variation of peak intensities and fluences (see details given by Aran et al. [2005]). [15] Ruzmaikin et al. [2005] used a model of diffusive shock acceleration by a traveling shock in which the upstream wave intensity driven by the accelerated particles is calculated from the steady state solution of the wave growth equation. This model calculates the maximum and minimum particle energies that the shock is able to accelerate as it propagates out from the inner boundary of the model at 0.1 AU. To study the radial dependence of particle intensities at a traveling shock, Ruzmaikin et al. [2005] considered the special case of a strong shock at 0.1 AU that decelerates as it moves outward to 1 AU. By approximating the radial dependence of the particle intensity at the shock front by a power law r a with an energy dependent exponent a(e), Ruzmaikin et al. [2005] deduced the energy dependence of a(e) for this specific case of a strong shock. The values of the exponent a deduced by Ruzmaikin et al. [2005] for the peak intensity at the shock are 2.46 for >5 MeV, 2.58 for >10 MeV, 2.75 for >30 MeV and 2.84 for >60 MeV protons (adapted from Ruzmaikin et al. [2005, Figure 4]). [16] Note that the domain of applicability of the model used by Ruzmaikin et al. [2005] is only from 0.1 to 1.5 AU, and that the results provided are given for a specific case. The fact that the index a increases with the particle energy indicates that the efficiency of the shock in accelerating, e.g., 60 MeV protons decreases faster with time and distance than the efficiency of the shock in accelerating 10 MeV protons. Whereas a shock is able to accelerate 60 MeV protons very efficiently when it is close to the Sun, it hardly can accelerate these protons when it arrives at 1 AU. In fact, very few transient shocks observed at 1 AU show >60 MeV proton intensity enhancements [Lario and Decker, 2001; Lario et al., 2005b], and the most intense SEP events are result of 4of18

5 Figure 3. (a) Daily averages of the >10 MeV solar proton fluences measured at 1 AU from 1973/303 to 2008/182. (b) Heliocentric radial distance of Solar Probe Plus (SPP) during its nominal prime mission. Launch date is successively stepped 1 day at a time over the whole energetic particle data set. (c) Monthly and monthly smoothed sunspot number (SSN). complex interplanetary configuration and not from a single CME driven shock propagating in the interplanetary medium [Kallenrode and Cliver, 2001; Lario and Decker, 2002]. Therefore, extrapolations of peak intensities measured at 1 AU to inner heliocentric distances must consider the fact that local shock acceleration may be not be present at 1 AU. Therefore, the extrapolation laws inferred from shockacceleration models applied to 1 AU measurements must be carefully used since the physics of shock acceleration may change from close to the Sun to the final observation at 1 AU. [17] As a note of caution, we would like to point out that the extrapolation laws for SEP peak intensities deduced by the works referred in this section are applicable only to peak intensities but not to other times within the SEP events. For example, during the decay phase of the events, it is common to observe similar intensities for spacecraft located at different radial distances [e.g., Lario et al., 2006, Figure 4]. Therefore, other extrapolation laws must be used for particle intensities measured in other time intervals within the SEP events, and in general, these 5of18

6 Figure 4. Statistical distribution of the >60 MeV proton mission integrated fluences obtained using the 12,663 hypothetical SPP missions using a radial dependence (a) r 1 and (b) r 2 to extrapolate daily fluences measured at 1 AU to the distance of SPP. The blue traces (right ordinate axis) give the probability P(F) of having a mission with a log fluence above a certain value F. The horizontal red lines and the values indicated in red mark the 95% confidence level F 95. radial dependences may vary throughout the same event and from event to event. 3. Estimates of Solar Energetic Proton Mission Integrated Fluences [18] The method developed to estimate mission integrated fluences consists first in accumulating an extensive data set of SEP fluences measured at 1 AU. In order to compute SEP fluences at 1 AU, we use daily averages of proton intensities measured by the Energetic Particle Sensor (EPS) on board the GOES 7, 8, 10 and 11 spacecraft [Sauer, 1993] in four integral energy ranges >5, >10, >30 and >60 MeV. The data we have considered cover the time interval from day 65 of 1987 (6 March 1987) to day 182 of 2008 (30 June 2008) and are available at spidr.ngdc.noaa. gov. In particular, we use data from GOES 7 for the time interval 1987/ /059; GOES 8 for 1995/ /101; GOES 10 for 2003/ /169; and GOES 11 for 2003/ /181. [19] In order to extend the data set back to day 303 of 1973 (30 October 1973), we use proton intensities for the integral energy ranges >4, >10, >30 and >60 MeV measured by the Charged Particle Measurement Experiment (CPME) on board the Interplanetary Monitor Platform (IMP 8) [Sarris et al., 1976] and available at sd IMP 8/CPME data gaps due to absence of tracking or due to removal of bad data (e.g., spikes) were filled by linear interpolation on the logarithm of the adjacent existing intensities. A MeV differential proton channel from the CPME (after removal of the few SEP events observed at these high energies) was also used to remove background due to penetrating cosmic rays. Similarly, the GOES data were de spiked, filled as necessary by using contemporary GOES or logarithm interpolation, and the >100 MeV channel (after removal of the few SEP events observed in this channel) was used to remove background. Background removal creates more homogeneous intensity time series constructed from multiple spacecraft, and also ensures that contributions from galactic cosmic rays, which have small radial gradients [Lario, 2007, and references therein], are not included in the helioradial scaling of SEP fluences. In this study we neglect the slight difference in energy between the >4 MeV IMP 8 channel and the >5 MeV GOES channel and consider the lowest energy channel as >5 MeV. [20] From daily integral proton intensities (in units of protons (cm 2 sr s) 1 ), we have computed daily fluences (in units of protons cm 2 ), by assuming that proton intensities are isotropic (i.e., multiplying the proton intensities by 4p and s). Figure 1 shows from top to bottom the background corrected proton daily fluences from 1973/303 through 2008/182 as measured at 1 AU in the integral energy ranges >5, >10, >30 and >60 MeV. A total of daily points compose our fluence data set. The lower panel shows the monthly averaged sunspot number for solar cycles The dashed vertical lines indicate the separation between the solar cycles: solar cycle 21 ( ), solar cycle 22 ( ) and solar cycle 23 ( ). Note that we have neglected the years 2009, 2010 and part of 2008 that formed part of the extended solar minimum between solar cycles 23 and 24. The inclusion of this long lasting period without SEP activity would 6of18

7 Figure 5. (a) Energy spectra of F 95 obtained using different radial dependences. Fitting of the energy spectra by a parabola F 95 =log 10 (f) = A + B log 10 (E) + C log 2 10 (E) with the coefficient C constrained to a value 0.3 give the following results: (r 1 ) A = 12.29, B = 0.74; (r 1 ;r 2 ) A = 12.56, B = 0.76; (r 1.5 )A=12.64,B= 0.74; (r 2 ;r 1 ) A = 12.77, B = 0.73; (r 2 )A=13.08,B= 0.75; (r 2.5 )A= 13.56, B = 0.74; (r 3 ) A = 14.13, B = Units are f in protons cm 2 and E in MeV. (b) F 95 values in Figure 5a plotted versus power law index a for each integral proton channel. Function F 95 (a) is well described by the linear function F 95 (a) =b + m a, with m 0.9. The four fits, which are shown in Figure 5b, allow one to estimate F 95 for any 0.5 a 3.5. have contributed to underestimate the fluence measured over a potential future mission that may have its lifetime during solar maximum conditions. [21] We have evaluated the SEP fluence for each energy range integrated over each solar cycle. Figure 2 shows the values of these fluences versus the energy threshold of each energy channel. Solar cycle 23 shows the largest fluence at all energy levels due to both the occurrence of large intense SEP events in this solar cycle [Lario et al., 2008] and the predominance of a few large SEP events in the computation of the total fluence integrated over a solar cycle [Shea and Smart, 1990]. [22] In order to evaluate the mission integrated SEP fluence for a spacecraft traveling through the innermost part of the heliosphere we will assume that the daily fluence measured by such spacecraft at a given time and helioradius is the daily 1 AU fluence multiplied by a function of only the helioradius of the spacecraft. Since the longitudinal separation between the spacecraft and possible active regions generating SEP events is unknown, we assume that fluences measured at 1 AU over the last three solar cycles contain a large number of events generated from a large variety of longitudes, and therefore the use of only a radial gradient includes also possible longitudinal dependences. The radial gradients used are discussed in section 2. [23] For the specific application of this method to an actual mission, we will use the Solar Probe Plus (SPP) mission whose orbit characteristics can be obtained from lws.larc.nasa.gov/solarprobe/. Figure 3 (top to bottom) shows the daily >10 MeV proton fluence measured at 1 AU, the heliocentric radial distance of SPP during its 2688 day nominal prime mission, and the monthly and smoothed monthly sunspot number during the last three solar cycles. The method consists in using consecutive launch days for the SPP mission (indicated by the arrow in Figure 3b as it moves over the data set) and compute the mission integrated fluence for each one of these hypothetical missions adopting an appropriate radial dependence to extrapolate the fluence at 1 AU to the radial position of SPP on every single day. A total of hypothetical SPP missions are used to generate statistical distributions of missionintegrated fluences. Note that for those missions launched 2688 days or less before the end of our data set we use the beginning of our data set to complete the SEP fluences that these missions would observe. Mission integrated fluences obtained by this method are dominated by the higher SEP intensities sampled near the Sun. The phasing between the occurrence of these large SEP events and the time when the spacecraft is near its perihelia depends on the exact launch date of the mission, implying that any statistical correlation between successive launches diminishes as the radial gradient used increases. The multiple perihelia passes of SPP imply that missions with large fluences may be separated by several days in their launch dates and statistical correlation between missions is then reduced. [24] Figure 4 shows the distribution of the >60 MeV proton mission integrated fluences using either a r 1 dependence (Figure 4a) or a r 2 dependence (Figure 4b) to 7of18

8 Table 1. Comparison of the Parameters m, s, and w Obtained by Feynman et al. [1993] (JPL 91), Rosenqvist et al. [2005], and Glover et al. [2008] for Each Energy Level a Parameter >4 MeV JPL 91 >10 MeV JPL 91 >10 MeV [Rosenqvist et al., 2005] >30 MeV JPL 91 >30 MeV [Glover et al., 2008] >60 MeV JPL 91 m s W a Adapted from Glover et al. [2008, Table 2] and Feynman et al. [1993, Table 2]. extrapolate 1 AU proton fluences to the SPP distances (i.e., a = 1 or 2 in equation (1)). For each one of the missionintegrated fluence distributions obtained for different energy channels and using different radial dependences, we compute the cumulative probability P(F) of having a given mission with log 10 (fluence) = F above a certain value. The blue traces in Figures 4a and 4b (measured using the right ordinate axis) show the probability P(F) of exceeding F. From these probability traces we can estimate a missionintegrated fluence with a given confidence level. For example, the red horizontal line and the value indicated in red in Figures 4a 4b indicates the value F at a confidence level of 95% (F 95 ) indicating that for 5 out of 100 times a SPP mission may have a mission integrated F above the value F 95. [25] Figure 5a shows the energy spectra obtained from the value of F 95 obtained at the energies >5, >10, >30 and >60 MeV using different radial dependences for the fluence extrapolations. As expected the steeper the radial gradient the larger the fluence. The hybrid radial dependences assumed for the cases b and d bound the case r 1.5. The physical motivation behind these broken power laws is as follows. The case with a steeper gradient outside of 0.25 AU (case d) might apply in the case that energetic particles undergo more scattering processes at large heliocentric distances whereas closer to the Sun particles propagate more directly along the IMF due to the focusing effect. Such radial dependence is supported by simulations of SEP propagation in the inner heliosphere [e.g., Lario et al., 2007]. By contrast, case b, with a steeper gradient inside the 0.25 AU, might apply if this region were, on average, a relatively more turbulent magnetized plasma, with transient disturbances such as CMEs and flares filling the vol- Figure 6. Energy spectra of the mission integrated fluence at a 95% confidence level for a mission lasting 2688 days permanently located at 1 AU obtained by applying the JPL 91 method following the distributions inferred by Feynman et al. [1993] (gray triangles), by Rosenqvist et al. [2005] (black star at 10 MeV), and by Glover et al. [2008] (black star at 30 MeV). Figure 7. Energy spectra of the mission integrated fluence at a 95% confidence level for a mission lasting 2688 days permanently located at 1 AU obtained by using the JPL 91 method following the Feynman et al. [1993] distributions (gray triangles) and by using our method as described in section 3 (black circles). 8of18

9 [29] A predictive engineering model widely used to estimate SEP fluences over long time intervals is the JPL 91 model developed by Feynman et al. [1993]. This model is based on a combined consideration of (1) the distribution of individual event fluences measured at 1 AU and (2) the probability of occurrence of an event over a given period. It is assumed that a normal distribution describes the distribution of the decimal logarithm of individual event fluences F = log 10 (f) $ % gðfþ ¼pffiffiffiffiffiffiffiffi 1 exp 1 ðf Þ ð2þ where m and s are the mean and standard deviation of the distribution of log 10 of the fluence values, respectively. It is also assumed that the probability of n events occurring in a time interval t follows a Poisson distribution w ðwþn pðn; wþ ¼e n! ð3þ Figure 8. Energy spectra of the SPP mission integrated fluence at a 95% confidence level obtained with the JPL 91 method following the Feynman et al. [1993] distributions and using a single average constant distance to extrapolate 1 AU fluences. ume with energetic particles and turbulence, and where intense MHD waves capable of scattering and impeding the progress of particles result in a steeper SEP radial gradient than outside 0.25 AU. Since the SPP orbit spends only 268 days inside 0.25 AU (out of the 2688 days of its nominal mission; see Figure 3b), the differences between cases b and d are minimal. [26] The solid lines in Figure 5a correspond to a fit using a quadratic form log 10 f = A + B log 10 E + C (log 10 E) 2 with the coefficient C constrained to a value 0.3 to prevent unphysical rollover of the spectra at energies lower than 5 MeV. The values of the coefficients A and B for the different cases are provided in the caption of the figure. [27] Figure 5b shows the log fluence F 95 values in Figure 5a plotted versus power law index a for each of the four integral proton channels. Function F 95 (a) is well described by the linear function F 95 (a)=b + m a, with m 0.9. The four fits, which are shown in Figure 5b, allow one to estimate F 95 of any 0.5 a Comparison With the JPL 91 Method [28] Deterministic estimates of the expected fluence on a mission cannot be easily made because over the time scale of a space mission, the fluence can be dominated by the contribution of a few rare and unpredictable highintensity events. Therefore, statistical approaches have been commonly used in order to estimate in advance fluences likely to be encountered by a space mission. where t is expressed in years and w is the long term average number of events per year. [30] By combining equation (2) and (3) it is possible to express the probability of exceeding a particular fluence level f =10 F during a mission lifetime t as Pð> F;Þ¼ X n¼1 pðn; wþqðf; nþ where Q(F,n) is the probability that the sum of all fluences due to n events (n = [1, ]) will exceed 10 F. [31] The parameters m, s and w have been estimated directly from data of event fluences collected from a number of different sources covering several years of the last two or three solar cycles [Feynman et al., 1993; Rosenqvist et al., 2005; Glover et al., 2008]. The selection criteria used to collect these data, select the SEP events, and compute SEP event fluences for each energy channel can be found in the references above. Table 1 lists the parameters m, s and w obtained from each one of these studies. [32] The data sets used by Feynman et al. [1993] for the integral energy channels >10 MeV and >30 MeV covered the period from 1963 to day 126 of 1991, whereas for the integral channels >4 MeV and >60 MeV the data sets covered the period between 1973 and Rosenqvist et al. [2005] used >10 MeV proton data for event fluences measured between January 1974 to May Glover et al. [2008] used the same time interval as Rosenqvist et al. [2005] to incorporate >30 MeV data. Glover et al. [2008] considered also >60 MeV data but given the reduced number of events reaching these high energies, these authors did not extend the study to energies higher than >30 MeV. The different data sets and SEP event selection by Feynman et al. [1993] and Rosenqvist et al. [2005] led to the different values of m, s and w shown in Table 1. ð4þ 9of18

10 Figure 9. Energy spectra of the SPP mission integrated fluence at a 95% confidence level obtained with the method described in section 3 (black circles) and with the JPL 91 method following the Feynman et al. [1993] distributions and using a single average constant distance to extrapolate 1 AU fluences (gray triangles) for different radial extrapolations laws. [33] In order to apply the JPL 91 method to a mission traveling in the innermost part of the heliopshere with a variable heliocentric distance, two different approaches can be considered. One approach consists in using the average value of r a over the whole mission as the factor to scale the fluence obtained for a mission at 1 AU of a given duration t. A different approach consists in dividing the trajectory of the mission in different radial bins and computing the time spent by the mission in each one of these bins. Hereafter we name the first method the constant distance approach and the second method the binned distance approach. [34] Application of the JPL 91 method to a mission lasting t = 2688 days located permanently at 1 AU, considering the parameters m, s and w listed in Table 1, at a 95% confidence level is shown in Figure 6. Note that the models of Rosenqvist et al. [2005] and Glover et al. [2008] (black stars in Figure 6) provide larger fluences than the JPL 91 model by Feynman et al. [1993] (gray triangles in Figure 6). The reason is the re calibration of data done in the update of the JPL of 18

11 Figure 10. Heliocentric radial distance of the nominal prime SPP mission with the number of days spent in each radial bin of a width of 0.1 AU. to compute event fluences and the division of events into multiple or single events (see discussion by Rosenqvist et al. [2005] and Glover et al. [2008]). Similarly, the fluences obtained by these methods for a mission at 1 AU lasting t = 2688 days are larger than the fluences obtained by integrating particle intensities over each one of the last three solar cycles (Figure 2). [35] The method described in section 3 can also be applied to a spacecraft located permanently at 1 AU for a time interval t = 2688 days (i.e., scaling the 1 AU fluences by a factor r 0 where r is the heliocentric distance of the spacecraft). Figure 7 compares the energy spectra obtained by the JPL 91 method (gray triangles) and our method (black solid triangles) at 95% of confidence level. JPL 91 yields larger fluences than our method. The difference between the fluences estimated by JPL 91 and our method increase with the particle energy. In order to select the SEP events to obtain the distribution of event fluences in the JPL 91 method, Feynman et al. [1993] used 7 years of data around solar maximum (two prior and four after the solar maximum). In order to compute the parameters m and s of equation (1), only the upper half of the distribution of the event fluences was used by Feynman et al. [1993]. With the aim of obtaining a conservative value of fluences, Feynman et al. [1993] did not consider a tail of events with low fluences in the lognormal fittings of the distribution of events. As shown in Feynman et al. [1993, Figure 3], this tail is more important at high energies than at low energies. Our method described in section 3 scans both solar maximum and solar minimum periods and uses all daily fluence values regardless of their value, and hence the lower fluences obtained in our method. [36] The constant distance approach applied to the SPP nominal mission requires one to compute the averages of the power laws of radial distances hr 1 i = 2.29; hr 1.5 i =4.33,hr 2 i = 10.05, etc. for the nominal mission of SPP. Figure 8 shows the energy spectra of the fluence obtained by applying the JPL 91 method as shown in Figure 6 but scaled using the average value hr a i with different values of a. [37] Figure 9 compares our method described in section 3 using several radial extrapolations laws and the JPL 91 method using the constant distance approach. Our method uses the actual radial distance of SPP with the possibility of large events occurring at small radial distances. Therefore, as the exponent a of the extrapolation law r a increases the difference between our method and JPL 91 decreases. For large values of a our method can predict larger values than the JPL 91 constant distance approach because of the finite probability of large events occurring at small heliocentric distances. [38] The binned distance approach consists in dividing the trajectory of the mission in different radial bins and computing the time spent by the mission in each bin. Figure 10 shows the heliocentric distance of SPP during its nominal mission and the number of days that SPP spends in each radial bin of a width of 0.1 AU. As can be seen, except for the intervals [0.6,0.7] AU and [0.7,0.8] AU, the time spent is less than 1 year. However, the use of JPL 91 requires long time intervals. Therefore, we add an additional step into the JPL 91 method that incorporates the possibility of having a spacecraft moving through different radial distances. First we consider a large number of missions (usually 10 6 missions) of a given duration t (t = 2688 days in the case of SPP). We use the Poisson distribution with the parameter w (long term average number of events per year) given by JPL 91 to obtain the number of events occurring in each one of these missions. Then we use the Gaussian distribution with the m and s parameters given by JPL 91 to determine the fluence of each one of these events. In order to determine where (or when) each event occurs, we randomly select a number between 0 and 1 that will give the radial bin where the event occurs according to the time spent by the mission in each bin [0, 20/t], [20/t,( )/t], [( )/t,( )/t], etc. The fluence of the event 11 of 18

12 Figure 11. Energy spectra of the SPP mission integrated fluence at a 95% confidence level obtained using JPL 91 under the constant distance approach (gray triangles) and the binned radial approach (open squares) for different radial extrapolations laws. is then scaled according to the assumed radial dependence r a where r is the averaged radial distance of each radial bin. [39] Figure 11 compares the results obtained using this binned distance approach (squares) and the constant distance approach (triangles). As expected, for a mission at 1 AU (i.e., r 0 ) both methods provide the same result, and as the exponent of the r a radial dependence increases, the fluences obtained with the binned radial distance approach are larger than the method using a single averaged radial distance. The differences between the mission integrated fluences obtained with the binned distance approach and the constant distance approach are larger at lower energies than at higher energies. The binned distance method considers the occurrence of events in radial bins that are closer to the Sun than the single averaged radial distance, and the probability of occurrence of events is larger at lower than higher energies. [40] Finally, Figure 12 compares the value of F 95 obtained by the three methods when using the same scale law of r 2. The two approaches used to apply the JPL 91 method exceed the total mission integrated fluence obtained by 12 of 18

13 Figure 12. Comparison of the energy spectra of F 95 obtained for the nominal prime SPP mission using our method (black circles), the JPL 91 method under the constant distance approach (gray triangles), and the JPL 91 method under the binned radial distance approach (open squares). using our method for exactly the same reasons given above when comparing each method separately. 4. Estimation of SEP Maximum Peak Intensities [41] The method applied to estimate SEP peak intensities for a mission traveling in the innermost part of the heliosphere is similar to the method used to estimate missionintegrated fluences described in section 3. However, instead of using daily averages of fluences, we use hourly averages of proton intensities measured by the series of GOES spacecraft from day 65 of 1987 to day 182 of 2008 as representative of the proton intensities that may occur during the mission traveling close to the Sun. [42] Figure 13 shows hourly averages of the corrected proton fluxes as downloaded from the Web site spdir.ngdc. noaa.gov. In particular, data comes from the following spacecraft: GOES 7 from 1987/065 to 1995/059; GOES 8 from 1995/060 to 2003/101; GOES 10 from 2003/102 to 2003/169; and GOES 11 from 2003/170 to 2008/182. Data gaps and spikes have been filled and replaced respectively, by using either data from contemporary GOES or logarithmic interpolation using existing adjacent data. Intensities of the GOES >100 MeV proton channels, with SEP events removed, were used to remove backgrounds using fit functions to scatterplots of the >5, >10, >30, and >60 MeV intensities versus the >100 MeV intensity. Background removal creates more homogeneous intensity time series constructed from multiple spacecraft, and also ensures that contributions from galactic cosmic rays, which have small radial gradients [Lario, 2007, and references therein], are not included in the helioradial scaling of SEP peak intensities. A total of hourly data points have been used. [43] The horizontal dashed lines in Figures 13a 13d indicate the maximum hourly averaged peak intensity observed at 1 AU during this time period. As described below in section 4.1, the use of intensities with a high time resolution is essential to properly determine peak intensities. The use of long term averages (e.g., daily) reduces the actual value registered as maximum peak intensity. For computational purposes we adopt the compromise of using hourly averages instead of a higher time resolution averages. In section 4.1 we provide the worst case maximum peak intensities observed in the largest SEP events using both hourly and 5 min averages. [44] As discussed in section 3, the method consists in launching hypothetical consecutive missions separated now hour by hour and registering the maximum mission intensity that each one of these hypothetical missions would detect. As in section 3 we apply this method to the specific case of SPP. In order to scale the measured GOES peak intensities with radial distance we use both the actual location of SPP on a given day of the mission and several radial dependences of peak fluxes as described in section 2. Note that the extrapolated laws deduced in the works referred in section 2 are applicable only to peak intensities of SEP events but not to other time intervals within the SEP events. Apart from registering the maximum peak intensity observed in each hypothetical mission, we can also identify the radial distance where this maximum peak intensity is observed. We then generate statistical distributions of both the maximum peak intensities registered over the set of SPP missions and the distances where these maximum peak intensities are detected. Figure 14 shows the distribution of >10 MeV proton maximum peak intensities found using a r 3 radial dependence (Figure 14a) and the distribution of radial distances where this maximum peak intensity is found (Figure 14b). The most probable distance where the maximum peak intensity may be observed depends on both the radial gradient used to extrapolate peak intensities and the time spent by the spacecraft in each radial bin. [45] For each one of the maximum peak intensity distributions, we compute the probability of having a mission with a maximum peak intensity above a certain intensity value (blue trace measured using the right ordinate axis in Figure 14a). We set a confidence level of 95% to show the results of the maximum peak intensities indicating that for 5 out of 100 times, a mission may observe a maximum peak intensity above the indicated intensity. The red horizontal line in Figure 14a and the value indicated in red shows the >10 MeV proton 95% confidence level (CL) maximum peak intensity in the case of a r 3 radial dependence. Table 2 shows the 95% CL maximum peak intensity obtained for different energy ranges and radial dependences. We have also applied this method to two proton differential energy 13 of 18

14 Figure 13. Hourly averages of proton intensities measured by the integral channels (a) >5 MeV, (b) >10 MeV, (c) >30 MeV, and (d) >60 MeV of the GOES series of spacecraft and (e) the monthly and the smooth monthly sunspot number (SSN) from 1987/065 to 2008/182. The horizontal dashed lines in Figures 13a 13d indicate the maximum intensity observed in this period at each energy channel. channels (9 15 MeV and MeV) available in the GOES data set. The radial dependences used range from r 1 to r 3.5. We have also incorporated a hybrid radial dependence (the column indicated by r 2 if <0.25 AU) and the value of peak intensities resulting from a model of particle acceleration at a strong shock extracted from Ruzmaikin et al. [2005, Figure 4] (the Ruzmaikin et al. [2005] column). [46] Figure 15 shows the energy spectra obtained for the 95% CL maximum peak intensities of the integral channels. The energy spectra shown in Figure 15 are fitted by a parabola log 10 (J) = a + b log 10 (E) + c log 2 10 (E) with the coefficients given in the figure caption. Note that the exponents a(e) used in the radial dependence r a deduced by Ruzmaikin et al. [2005] are energy dependent and hence that the shape of the energy spectrum obtained in this case differs from the other cases where the same scaling factor is used at all energies Worst Case SEP Peak Intensities [47] The horizontal dashed lines in Figures 13a 13d indicate the largest hourly averaged intensity measured by the series of GOES spacecraft at 1 AU from 1987/065 to 14 of 18

15 Figure 14. (a) Statistical distribution of the >10 MeV proton maximum peak intensity of the hypothetical SPP missions using a r 3 radial dependence. (b) Distribution of radial distances where these maximum peak intensities are observed. 2008/182. In this section we pay attention to the worst case SEP fluxes measured at 1 AU that can be used as extreme values of SEP intensities at different heliocentric radial distances. A comprehensive survey of SEP events measured by the series of GOES spacecraft during solar cycles 22 and 23 was performed by Lario et al. [2008] and we have used this study to select such events. Table 3 lists the maximum hourly integral intensities registered at the peak of the most intense SEP events of the last two solar cycles as obtained from our data set. [48] To the list of worst case maximum peak intensities observed at 1 AU shown in Table 3, we can add estimates of the peak intensities provided by Shea and Smart [1990] and Smart and Shea [2003] for the even more intense event on 4 August Shea and Smart [1990] and Smart and Shea [2003] provide the values 8.60e4 at >10 MeV and 1.90e4 at >30 MeV (units are protons cm 2 s 1 sr 1 ). [49] Table 4 provides the maximum peak intensities for the events listed in Table 2 but using 5 min averages and the same set of spacecraft instead of the hourly averages. The 5 min averages may differ substantially from the hourly averages especially in those cases where the peak flux occurs as a short duration spike. For the event with the highest fluxes (i.e., October 1989) the hourly and 5 min averages differ by less than a 10%. [50] Figures 16 and 17 show the energy spectra of the maximum peak differential and integral fluxes respectively, for the events on 20 October 1989, 24 March 1991, 14 July 2000, 29 October 2003 and 20 January 2005 using 5 min averages. Note that the peak fluxes of a given event in the different energy channels are not necessarily observed at the same time during the event. This is particularly significant for the event on 20 January 2005 where peak intensities at high energies ( 10 MeV) were measured during the Table 2. Maximum Peak Intensities at a 95% Confidence Level to Be Observed by the SPP Mission Using Different Radial Extrapolation Laws Proton Energy r 1 r 1.5 r 2 r 2.5 r 2 if < 0.25 AU, r 3 if >0.25 AU Ruzmaikin et al. [2005] a(2) r 3 r 3.5 >5 MeV b >10 MeV b >30 MeV b >60 MeV b MeV b MeV b a Power law indices used to extrapolate 1 AU peak intensities are extracted from Ruzamikin et al. [2005, Figure 4] as follows: 2.46 for >5 MeV, 2.58 for >10 MeV, 2.75 for >30 MeV, 2.84 for >60 MeV; 2.58 for 9 15 MeV; and 2.84 for MeV. b Units are log 10 (protons cm 2 sr 1 s 1 ) for integral channels and log 10 (protons cm 2 sr 1 s 1 MeV 1 ) for differential energy channels. 15 of 18