Estimation of the solar flare neutron worst-case fluxes and fluences for missions traveling close to the Sun

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1 SPACE WEATHER, VOL. 10,, doi: /2011sw000732, 2012 Estimation of the solar flare neutron worst-case fluxes and fluences for missions traveling close to the Sun D. Lario 1 Received 7 September 2011; revised 30 December 2011; accepted 30 December 2011; published 6 March [1] A method to estimate the total fluence of solar flare neutrons at a spacecraft traveling in the innermost part of the heliosphere (at heliocentric radial distances of <1 AU) is presented. The results of the neutron production and emissivity codes of Hua and Lingenfelter (1987a, 1987b) scaled to one of the largest solar neutron events ever observed at the Earth are used to derive a conservative estimate of the energy spectrum of neutrons emitted from the Sun after a large solar flare. By taking into account the survival probability of a neutron to reach a certain heliocentric distance, we evaluate the observed time-integrated spectrum of solar neutrons as a function of the heliocentric distance of the observer. By considering (1) a working relationship between the soft X-ray class of a flare and the flare s production of solar neutrons, and (2) the number and size of soft X-ray flares that may occur during a mission traveling close to the Sun, we compute an upper limit for the total fluence of solar neutrons at energies >1 MeV, >10 MeV, >100 MeV and >1000 MeV to which such a mission may be exposed. We apply this method to the Solar Probe Plus mission. Although our method gives a conservative estimate of neutron fluxes, the predicted mission-integrated fluence of solar neutrons at Solar Probe Plus is orders of magnitude below that of solar energetic protons. Citation: Lario, D. (2012), Estimation of the solar flare neutron worst-case fluxes and fluences for missions traveling close to the Sun, Space Weather, 10,, doi: /2011sw Introduction [2] Radiation effects produced by secondary neutrons resulting from the interaction between primary charged particles and spacecraft materials have been studied for many years [e.g., Dyer et al., 1996; Shinn et al., 1999, and references therein]. These effects include, among others, enhancement in the total radiation dose due to primary particles and displacement damage energy depositions. Because of the finite lifetime of free neutrons (15 min mean lifetime), only the highest energy neutrons produced at the Sun have a significant probability of reaching 1 AU before decaying into protons (p), electrons (e ) and electron antineutrinos (n e )[Agueda et al., 2011, and references therein]. Therefore, primary solar neutrons have been usually neglected in the study of the interplanetary space radiation damage at 1 AU. However, a key question in this context is the contribution that primary solar neutrons may make to the total radiation dose for missions traveling close to the Sun, where the more abundant lower energy neutrons may easily reach the spacecraft. 1 Johns Hopkins University Applied Physics Laboratory, Laurel, Maryland, USA. [3] Solar flare neutrons are produced by the interaction of flare-accelerated ions with the solar atmosphere. In general, protons and heavy ions accelerated during solar flares propagate down into the denser lower solar atmosphere, shattering abundant heavy nuclei, producing numerous nuclear reactions and leaving nuclei in excited states that eventually decay into other nuclear species or emit g-rays [Ramaty et al., 1975]. Models of the neutron production in solar flares need to include flare-accelerated ion transport in the solar atmosphere, scattering processes and relevant nuclear reactions. Such a code was constructed by Hua and Lingenfelter [1987a, 1987b] and improved and updated by Hua et al. [2002]. The consequences of varying the ion acceleration and transport parameters of this code were described in detail by Murphy et al. [2007]. These codes predict angular distributions and energy spectra of the neutrons escaping from the solar atmosphere (see the above references for further details). [4] Techniques used to study the spectra of neutrons at the Sun include: (1) Direct detection of solar neutrons by spacecraft detectors and/or ground-level monitors on the Earth; (2) measurement of protons resulting from the beta decay of neutrons (n p + e + n e ); and (3) measurement of Copyright 2012 by the American Geophysical Union 1of11

2 Figure 1. Solar neutron survival probability P s to reach several heliocentric distances as a function of neutron energy. capture g-rays from neutrons absorbed by H nuclei [Chupp, 1988; Vilmer et al., 2011]. Ground-level neutron monitor observations provide information of the solar flare neutron spectrum at high energies ( 100 MeV). Satellite observations of solar neutrons [e.g., Ryan et al., 1993], of neutron-decay protons [e.g., Evenson et al., 1983; Chupp et al., 1987], and of solar flare g-rays [e.g., Ramaty et al., 1996; Watanabe et al., 2007] make it possible to extend the neutron spectra to lower energies. However, there are uncertainties about the low-energy end of the neutron production [e.g., Vilmer et al., 2011]. Observations of solar neutrons close to the Sun are essential to provide the information required to understand both the mechanisms of particle acceleration in flares and the production of low-energy solar neutrons [e.g., Vilmer et al., 2001; Share et al., 2011]. Unfortunately, the absence of neutron spectrometers on ESA s Solar Orbiter and NASA s Solar Probe Plus missions will prevent us from making neutron observations close to the Sun in the near future. [5] In this paper we present, from a radiation damage perspective, a method to estimate an upper limit of the mission-integrated fluence of solar neutrons for a probe traveling at heliocentric distances <1 AU. This method is based on the use of (1) an energy spectra of neutrons escaping from the Sun obtained from the parametric study of the model of Hua et al. [2002] performed by Murphy et al. [2007] but normalized to one of the largest 100 MeV neutron fluxes at the Sun as inferred by Watanabe et al. [2005], (2) a working relationship between the flux of neutrons at the Sun and the peak soft X-ray flux of the associated solar flare, (3) an estimation of the number and peak flux of soft X-ray flares that may occur during the lifetime of the mission, and (4) the probability that the solar neutrons emitted by each flare reach the heliocentric distance of the spacecraft on each day of the mission. [6] The sections of the paper are divided as follows. Section 2 describes the method adopted for computing the energy spectra of solar neutrons as a function of the heliocentric radial distance. In section 3 we adopt a working relationship between the soft X-ray class of the associated solar flare and the 100 MeV neutron flux at the solar surface based on neutron observations performed at Earth. This relationship is used only for predictive purposes to allow us to extend the slim database of solar neutron observations to a complete multiple-solar-cycle database. We do not attempt to establish a physical correlation between the soft X-ray flux and the neutron flux of solar flares but only a working relationship that allows us to relate a long-term data set, such as the soft X-ray flare classification, to the possible neutron production by flares. As described in section 3, this relationship is intended to be used only for purposes of spacecraft damage assessment, providing us an upper limit on neutron fluxes. In section 4 we estimate the number and peak soft X-ray flux of flares that may occur during the lifetime of a mission traveling close to the Sun. In section 5 we use the soft X-ray flare distributions estimated in section 4 and the working relationship found in section 3 to estimate the mission-integrated fluence of solar neutrons for the Solar Probe Plus mission. Finally, in section 6 we present the main conclusions of the present study. 2. Energy Spectrum of Solar Flare Neutrons as a Function of the Heliocentric Distance [7] The time-integrated spectrum of solar neutrons observed after a solar flare at a given distance R is usually expressed as f n ðe n ÞdE n ¼ R 2 Q ð En ; t s ÞP s ðe n ÞdE n ; ð1þ where Q (E n, t s ) is the differential neutron emissivity spectrum for neutrons emitted from the Sun at time t s and directed into a unit solid angle toward the observer [Chupp et al., 1987; Hua and Lingenfelter, 1987a], P s is the probability that the solar neutron will reach the distance R without decaying to a proton, electron and antineutrino, and E n is the kinetic energy of the neutron. The neutron survival function P s (E n ) is given by the relativistic expression [Lingenfelter et al., 1965] 2 3 P s ðe n Þ ¼ e t =t 6 R 7 ¼ exp4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5; ð2þ tc ½ðE n þ mc 2 Þ=mc 2 Š 2 1 where t is the time measured in a frame of reference moving with the neutron, t is the neutron mean life (886 s), and m is the neutron mass ( MeV/c 2 ). Figure 1 2of11

3 Figure 2. (a) Q (E, t) as inferred from the study by Murphy et al. [2007, Figure 15] and expressed by equation (3) (left ordinate axis) or by equation (4) (right ordinate axis). (b) Energy spectra of solar neutrons at different heliocentric distances. shows the neutron survival function P s as a function of the neutron kinetic energy for several heliocentric distances (note that we have subtracted 1 solar radii from the distance R under the assumption that most of the solar neutrons will escape from 1 solar radius; therefore, for an observer at heliocentric distance 9.5 solar radii, the distance that the neutrons have to travel is 8.5 solar radii). [8] The models of Hua et al. [2002] provide the quantity Q (E n, t s ) in units of neutrons MeV 1 sr 1 and as a function of the heliocentric angle q (see Vestrand et al. [1987] for a definition and a description of how this angle is computed). Our literature survey indicates that Murphy et al. [2007, Figure 15] show one of the highest published values of the differential emissivity function Q (E n, t s ) obtained from a combination of parameters in the model of Hua et al. [2002]. The units of Q (E n, t s ) provided by Murphy et al. [2007, Figure 15] are neutrons MeV 1 sr 1, with the normalization of one flare-accelerated proton per unit of area at energies >30 MeV. As a worst-case scenario, we decide to take an envelope over the highest values of Q (E n, t s ) shown by Murphy et al. [2007, Figure 15], regardless of the heliocentric angle q and the parameters used in the model of Hua et al. [2002]. We fit a 5-degree polynomial to the envelope that covers the highest values given by Murphy et al. [2007, Figure 15]. Figure 2a shows the resulting profile for the quantity Q (E n, t s ) given by the function log 10 ðq Þ¼ 4:09 þ 0:01 log 10 ðeþ 0:39 log 2 10 ðeþþ0:41 log3 10 ðeþ 0:30 log 4 10 ðeþþ0:05 log5 10 ðeþ; ð3þ with E in units of MeV and Q in units of neutrons MeV 1 sr 1 normalized to one flare-accelerated >30 MeV proton (N p (>30 MeV) = 1 in the left vertical axis in Figure 2a). Equation (3) applies to energies in the interval 1 < E < 1000 MeV. [9] Watanbe et al. [2005] summarized the characteristics of the solar neutron events observed in solar cycles 21, 22 and 23 by ground-based neutron monitors. Table 1 (adapted from Watanabe et al. [2005]) shows the properties of these events (see also Table 1 in Vilmer et al. [2001]). Column 1 gives the date when the solar event was observed. Columns 2 and 3 give the characteristics of the associated flare, in particular its GOES soft X-ray class (column 2) and its Ha site (column 3). The 100 MeV neutron flux at the Sun (column 4) and the power law index (assuming that the 100 MeV neutron flux follows a power Table 1. A List of Solar Neutron Events Adapted From Watanabe et al. [2005] a Date Flare Flare Location Flux at 100 Mev (10 28 /MeV/sr) Power Index 3 Jun 1982 X8.0 S09E May 1990 X9.3 N36W Mar 1991 X9.4 S26E Jun 1991 X12.0 N30E Nov 2000 X2.3 N22W Aug 2001 X5.3 S17E Oct 2003 X17.4 S16E Nov 2003 X8.3 S14W Nov 2003 X28 S19W Sep 2005 b X17 S06E a This list includes information on the flares and the energy spectra of the solar neutrons at the solar surface. b Information regarding the 2005 September 7 event is extracted from Watanabe et al. [2007]. 3of11

4 law E a ) are provided by Watanabe et al. [2005, 2007, and references therein]. According to the values of Table 1, the largest solar neutron flux at the Sun (in terms of the flux at 100 MeV) was the 24 May 1990 event (Vilmer et al. [2001] also classified this event as the largest one ever observed). Therefore, in a conservative case, we normalize the values of Q (E n, t s ) plotted in Figure 2a to the value neutrons MeV 1 sr 1 at 100 MeV, corresponding to the highest 100 MeV neutron flux at the Sun that is inferred from neutron monitor observations as shown in Table 1. The right vertical axis in Figure 2a gives the values of Q (E n, t s ) according to this normalization. Therefore, equation (3) may be rewritten as log 10 ðq Þ¼ 30:07 þ 0:01 log 10 ðeþ 0:39 log 2 10 ðeþþ0:41 log3 10 ðeþ 0:30 log 4 10 ðeþþ0:05 log5 10 ðeþ; ð4þ with E in units of MeV and Q in units of neutrons MeV 1 sr 1. Figure 2a shows that the energy spectrum above 100 MeV may be approximated by a power law E a (thick gray band) with spectral index a 3.02, which is close to the a = obtained by Watanabe et al. [2005] for the event on 24 May [10] Figure 2b shows the time-integrated spectra at several heliocentric distances, as expressed by equation (1), assuming that the neutron emission Q is isotropic and given by equation (4). In order to extend the spectra to energies higher than those shown in Figure 2a, we have assumed that Q (E n, t s ) extends according to the power law shown by the thick gray band in Figure 2a. In order to extend the spectra to energies lower than those shown in Figure 2a, we have assumed that at energies below 1 MeV, Q (E n, t s ) takes the same value as at 1 MeV, which is reasonable since the production spectrum of neutrons with energies of a few MeV is expected to roll over for some reactions or only increase slowly with decreasing energy, depending on the energy spectrum of accelerated ions [Hua and Lingenfelter, 1987b; Share et al., 2011]. [11] Figure 2b can be used to estimate the fluence of solar neutrons produced by a single large solar flare at different heliocentric distances. However, the occurrence of solar flares that produce solar neutrons can be sporadic throughout the lifetime of a mission. We do not know, a priori, when and where a solar flare that produces solar neutrons might occur. There have been attempts to set limits on the neutron production by small flares [e.g., Harris et al., 1992]; we can, however, estimate an upper limit on the neutron fluences by considering the possibility that small flares (i.e., soft X-ray flares of class M or below) also produce solar neutrons. The first claimed detection of 1 8 MeV solar neutrons inside 1 AU (at 0.48 AU by the MESSENGER spacecraft) was associated with a M2 flare (although as seen from the Earth, the flare, occurring beyond the east limb, at 102 E, was originally classified as a C8 flare, the X-ray observations from MESSENGER were used to reclassify the flare as M2 when propagated to Earth) [Feldman et al., 2010]. Therefore, it is possible that M-class flares may be able to produce solar flare neutrons. Note that Share et al. [2011] emphasized that the neutrons observed by MESSENGER were not solar neutrons, but rather resulted from secondary neutrons produced by solar energetic particle interactions with spacecraft materials (including 13 C). In particular, Share et al. [2011] argued that the flux of solar neutrons estimated by Feldman et al. [2010] would have implied a large number of flare accelerated protons. These protons would have also produced a large MeV g-ray line fluence. However, a M2 soft X-ray flare is not typically a prolific source of MeV line emission. In fact, the catalog of g-ray flares observed by the Solar Maximum Mission (SMM) collected by Vestrand et al. [1999] shows that the fluence of the MeV g-ray line for flares of class M5 or lower was very low. Therefore, the production of solar neutrons by soft X-ray flares of a low class (i.e., M2) may be considered small. [12] Our aim is to provide an upper limit for the solar neutron fluence to which a mission may be exposed during its lifetime. Therefore, we will assume that M class flares (even M1.0) may produce solar neutrons, although in lower numbers compared to more intense flares. To determine the neutron dosage incident on a spacecraft in the inner heliosphere throughout its lifetime, we shall use soft X-ray data to estimate the number of flares occurring throughout the mission, and adopt an empirically determined relationship between soft X-ray peak flux and neutron yield. This relationship is described in the next section. 3. A Working Relationship Between Solar Neutron Flux Upper Limits and Peak Soft X-Ray Fluxes [13] Watanabe et al. [2005] studied the solar neutron events associated with large solar flares in solar cycles 21, 22 and 23 and looked for a possible correlation between the soft X-ray peak flux of the associated solar flares and the integrated (between 50 and 1500 MeV) solar neutron fluxes at the solar surface. The correlation coefficient found by these authors was 0.46 [see Watanabe et al., 2005, Figure 2], which lead the authors to conclude that no convincing correlation exists between the soft X-ray flare classification and the solar neutron fluxes. Although all the flares included in the study by Watanabe et al. [2005] were X-class flares, these authors stated that there is no flareclass threshold for the occurrence of solar neutron events. [14] We have repeated the analysis performed by Watanabe et al. [2005], but considered only the flux of solar neutrons at 100 MeV at the solar surface (column 4 in Table 1). Figure 3 shows the correlation obtained between the peak flux of the soft X-ray flare and the flux of solar neutrons at 100 MeV on the solar surface for those events listed in Table 1. The open circle in Figure 3 identifies the neutron event used to scale the neutron fluxes in the conservative case shown in Figure 2 (24 May 1990). A least- 4of11

5 Figure 3. Correlation between the flux of 100 MeV solar neutrons at the solar surface and the soft X-ray flux of the associated solar flares. The open circle identifies the 24 May 1990 event. squares fit to the data points shown in Figure 3 gives the following equation: log 10 ðq 100 Þ ¼ 1:51 log 10 ðsxrþþ3:96; ð5þ where SXR is the soft X-ray peak flux in units of W m 2 and Q 100 is the flux of 100 MeV neutrons emitted from the solar surface in units of neutrons MeV 1 sr 1. The Pearson coefficient for the linear fit shown in Figure 3 is P r = For N = 10 data points and linear correlation coefficient P r = 0.54, the probability that the data come from an uncorrelated distribution is about 0.10 [Bevington and Robinson, 2003, Table C.3]. We regard this correlation as moderate, and proceed with our model using the fit indicated in Figure 3. This correlation shows the expected trend that the larger the flare (in terms of the soft X-ray peak flux), the more likely to produce a larger flux of neutrons. This is a consequence of what is known as the big flare syndrome, which states that large flares tend to be associated with phenomena that may not be directly related to each other [Kahler, 1982]. In other words, this correlation does not imply a physical association between the mechanisms that produce the soft X-ray emission and the solar flare neutrons. In fact, there are many X-class flares without evidence for nuclear emission [Shih et al., 2009]. Therefore, our use of equation (5) will overestimate the neutron fluxes. [15] Because observations of solar neutron events at 1 AU are rare [Vilmer et al., 2001, 2011; Watanabe et al., 2005], it is difficult to obtain the neutron fluence for an extended mission that spans over different levels of solar activity. The relationship shown in Figure 3, however, provides an opportunity to relate the production of solar neutrons to an extensive database of solar flares that spans more than three solar cycles. Therefore, in order to estimate the neutron fluxes produced by flares occurring during a given mission, we will use equation (5) that indicates that larger flares (in terms of SXR) may produce larger fluxes of neutrons. Another extreme option would be to adopt the energy spectrum shown in Figure 2 for all flares occurring during a mission. Both approaches will overestimate the neutron fluxes, and therefore they should be regarded as approximations for assessing spacecraft damage that solar neutrons may produce. [16] The value of the spectral index a = 3.02 for Q at energies >100 MeV (thick gray band in Figure 2a) represents a relatively hard spectrum with respect to the values inferred for the other neutron events listed in Table 1. Therefore, the use of this spectral index also leads to a conservative estimate of the high-energy neutron flux. [17] Figure 4 shows the results from using equation (5) to estimate the energy spectra of solar neutrons at the heliocentric radial distances of 9.5 solar radii (Figure 4b) and 1 AU (Figure 4c), for different classes of soft X-ray flares. Figure 4a shows the differential neutron emissivity spectrum Q as expressed by equation (4) (solid black line) and scaled according to equation (5) for flares of class C1.0 (red line), M1.0 (green line), X1.0 (dark blue line), X10.0 (light blue line) and a never-observed X100 flare (pink line). Figures 4b and 4c (measured using the right ordinate axis) show the energy spectra of neutrons at two heliocentric distances for different classes of soft X-ray flares. The normalization of Q shown in Figure 2 is based on data from the X9.4 flare on 24 May The inferred 100 MeV neutron flux at the solar surface for this flare departs from the correlation shown in Figure 3 (indicated by the open circle). Hence, the solid black lines in Figure 4 do not represent the fluxes that would have been obtained for an X9.4 flare in equation (5) but rather those that were used to normalize Q in Figure 2. [18] To extend the computed neutron fluxes over the lifetime of a mission, an estimation of the number and size (in terms of SXR) of the flares occurring during such a mission is required. Hereafter, we will use the mission profile (or ephemeris) of Solar Probe Plus (SPP) as representative of a mission traveling at distances R < 1 AU. 4. Estimates of the Number and Size of Soft X-Ray Flares Occurring During the SPP Mission [19] To estimate the number and size of soft X-ray flares occurring during a mission, we use the frequency distributions of soft X-ray flares observed over a number of solar cycles. Following Veronig et al. [2002], we collected tabulated data on soft X-ray flares observed by GOES in the nm wavelength band from 1 January 1976 to 31 December 2009 (ftp://ftp.ngdc.noaa.gov/stp/ SOLAR_DATA/SOLAR_FLARES/FLARES_XRAY/) and 5of11

6 Figure 4. (a) Q (E, t) inferred from Figure 2a and scaled according to the proportionality law inferred from Figure 3 for soft X-ray flares of different class. (b) Energy spectra of solar neutrons at 9.5 solar radii for the different soft X-ray solar flare classes. (c) Energy spectra of solar neutrons at 1 AU for the different soft X-ray solar flare classes. Figures 4b and 4c use the right ordinate axis. constructed the frequency distribution of the flares in terms of their peak flux (without background subtraction). This 34-year period spans more than three solar cycles and contains nearly flares, including 446 X-class, 5642 M-class, C-class and B-class flares. [20] Figure 5a shows the frequency distribution dn/dp of the peak fluxes (SXR) where dn denotes the number of events recorded during a given time period with the peak flux p in the interval [p, p + dp]. The distribution dn/dp can be approximated by a power law of the form Ap d dp. The straight line in Figure 5a is obtained by a least-squares fit to the solid circles of Figure 5a. The turnover of the distribution for less intense flares is due to the difficulty of detecting small flares, especially during solar maximum periods [e.g., Veronig et al., 2002]. The average number of flares of class above C2.5 (i.e., SXR Wm 2 ) per day over the three solar cycles (12419 days) is when using the result from the integration of the distribution dn = Ap d dp or when using the actual number of flares. [21] In order to compute the number of flares that may occur during the SPP mission, we need to estimate the probability of occurrence of a flare of a given size (i.e., of a given SXR) for every day of its orbit. Let us assume that the number of flares occurring per day follows a Poisson distribution with an average rate of and that the size of the flares follows the distribution dn = Ap d dp shown in Figure 5a. Under these assumptions and considering only flares of class above C2.5, we obtain the flare size distribution shown in Figure 5b for a period of 2688 days. The use of the Poisson distribution method, instead of using a simple factor correcting for the different time intervals, leads to slightly different functional forms in Figures 5a and 5b. The duration of 2688 days includes the nominal prime mission of SPP plus two additional full orbits of the spacecraft around the Sun. The end of the nominal SPP prime mission is defined by the time at which the spacecraft completes its third perihelion at 9.5 solar radii, which occurs on day 2510 of the mission [lws.larc.nasa.gov/solarprobe/]. The assumption that the number of flares per day follows a Poisson distribution assumes that the occurrence of a flare is independent of the time since the last flare occurred; therefore it does not consider recurrent solar flares or any solar cycle dependence of the rate at which solar flares occur. We note that assuming that the occurrence rate of flares obeys a Poisson distribution is commonly used in models that predict solar energetic particle fluences [e.g., Feynman et al., 2002]. [22] The mission of SPP extends over a period of 7 years which most likely will include both solar minimum and solar maximum periods. The occurrence frequency of solar flares during periods of solar maximum differs from that during periods of minimum solar activity. The solar-cycle dependence of the solar flare occurrence rate from 1 January 1976 to 31 December 2009 is shown in Figure 6. Figure 6a shows the soft X-ray peak intensity of the flares above C1.0 (note that before April 1978 the SXR 6of11

7 Figure 5. Frequency distribution of the number of flares as a function of the soft X-ray peak flux (a) during the period , and (b) during a time interval of 2688 days assuming that the occurrence of flares follows a Poisson distribution. The straight lines are least-squares fits to the solid circles. classification did not include decimals). Figure 6b shows the number of soft X-ray flares per day of all classes (black histogram) and of class equal to or greater than M1.0 (green histogram). Figure 6c shows the heliocentric radial distance of SPP, assuming it would have been launched on 1 January The SPP mission considered here consists of 2688 days, including two full orbits around the Sun after the third perihelion at 9.5 Rs (perihelion #24 in Figure 6c). Throughout this extended SPP mission we can compute the number and the class of flares that will occur. By moving the launch date of SPP forward in time (indicated by the arrow in Figure 6c), we can sample different phases of the solar cycle and thus compute the number and class of the soft X-ray flares occurring throughout the SPP mission under different conditions of the solar cycle. We name this approach the successive launches method. 5. Estimation of the Solar Neutron Fluence During the SPP Mission [23] Assuming that (1) the occurrence of flares above class C2.5 follows a Poisson distribution with an average rate of of flares per day and (2) the size distribution of flares follows the distributions shown in Figure 5, we compute the energy spectra of the emitted neutrons at the Sun for each flare using the proportionality law of equation (5). We then compute the energy spectra of neutrons at the distance of SPP for each day of the mission as shown in Figure 4. By adding the energy spectra of each single flare, we obtain the total spectra over the duration of the SPP mission as shown by the red histogram in Figure 7. This result assumes that equation (5) applies to all flares above class C2.5. By assuming that equation (5) applies only for those flares of class equal to or greater than M1.0, we obtain the red histogram shown in Figure 8. The shape of these histograms results from the integration of neutron fluxes observed at different distances throughout the mission. The high energy tail comes primarily from the larger events, while the flat top arises from flares occurring when the spacecraft is close to the Sun. [24] The legends in Figures 7 and 8 show the fluence of neutrons obtained by integrating the energy spectra above the indicated energy. The exclusion of those flares above class C2.5 but smaller than M1.0 leads to a reduction of the total of fluence between 15% for >1 MeV neutrons and 13% for >1000 MeV neutrons. When we assume that each single flare produces the same neutrons as the spectra shown by the black line in Figure 4a or Figure 2, we obtain the black integrated spectra shown in Figure 7 (for flares of class above C2.5) and Figure 8 (for flares of class above M1.0). We note that this can be the worst-case scenario since it considers that all flares produce the same number of neutrons as the largest solar neutron event ever observed. [25] Finally, by using the method of successive launches described in the last paragraph of section 4 to estimate the 7of11

8 Figure 6. (a) Soft X-ray peak intensity of the X-ray flares observed from 1 January 1976 to 31 December (b) Number of X-ray flares per day of all classes (black histogram) and of class equal to or greater than M1.0 (green histogram) observed from 1 January 1976 to 31 December (c) Heliocentric radial distance of SPP assuming a launch on 1 January number of flares occurring each day of the SPP mission, taking into account the solar variability, and using equation (5) to relate the SXR of each flare with the emitted solar neutron spectrum (applied as shown in Figure 4 and assuming that only flares of a class equal to or greater than M1.0 produce solar neutrons) we obtain the total fluence of neutrons above 1 MeV, 10 MeV, 100 MeV and 1000 MeV for each potential launch of SPP. Each one of these 8 of 11

9 Figure 7. Energy spectra of solar neutrons integrated throughout the SPP mission assuming that the occurrence of flares follows a Poisson distribution with average rate of flares per day above class C2.5 and distributed in class according to Figure 5b (red histogram) or assuming that all flares produce the same emissivity spectra as shown by the black line in Figure 4a (black histogram). The legend shows the fluence of solar neutrons obtained by integrating the energy spectra above the indicated energy. traveling at heliocentric radial distances <1 AU. The method is based on the estimation of the differential emissivity spectrum of solar neutrons for the largest solar neutron event ever observed from the Earth and a working relationship between the inferred 100 MeV neutron flux at the solar surface and the GOES soft X-ray flare classification. This relationship allows us to relate the large data set of soft X-ray flares with the slim data set of solar neutron events. The assumption that all soft X-ray flares of class above either C2.5 or M1.0 produce solar neutrons leads to an overestimation of neutron fluxes, since many soft X-ray flares (even of class X) have shown no evidence for neutron production [Shih et al., 2009]. [27] By taking into account the survival probability that a neutron will reach a certain heliocentric distance before decaying, we evaluate the time integrated spectrum of solar neutrons observed at a given heliocentric distance after a large flare. By using soft X-ray flare observations over the last three solar cycles as representative of the solar flare occurrence during the spacecraft mission, we estimate the number and size of soft X-ray flares occurring during the mission, and hence (via equation (5)) the number of solar neutrons emitted by these flares. The total fluence of solar flare neutrons of energies >1 MeV, >10 MeV, >100 MeV and >1000 MeV integrated over the duration of the flare is then computed. For the specific case of the Solar Probe Plus mission, this fluence is well below the due to solar energetic protons (SEPs), even for the conservative upper limit adopted in the present study. For example, the estimated mission-integrated fluence at SPP potential launches has been spaced 10 days over the whole data set of soft X-ray flares shown in Figure 6. A total of 1242 potential orbits have been considered. Figure 9 shows the distributions of the logarithms of the fluences of (a) >1 MeV, (b) >10 MeV, (c) >100 MeV and (d) >1000 MeV neutrons obtained using this method. The blue curves in Figure 9 show the probability of having a misison with total fluence exceeding a given value of the logarithm of the fluence (as indicated in the right ordinate axes of the respective panels in Figure 9). The red numbers indicate the value of the logarithm of the fluence at which the probability of exceeding this fluence is 0.05 (5%). In other words, they give the mission integrated fluence of >1 MeV (Figure 9a), >10 MeV (Figure 9b), >100 MeV (Figure 9c), and >1000 MeV (Figure 9d) neutrons at a confidence level of 95%, meaning that for 5 out of 100 times an SPP mission would have a fluence of neutrons above the indicated values. 6. Conclusions [26] We have developed a method to estimate an upper limit for the total fluence of solar neutrons for a spacecraft Figure 8. The same as Figure 7 but considering that only flares of class equal to or greater than M1.0 produce neutrons. 9of11

10 Figure 9. Neutron fluence distributions above (a) 1 MeV, (b) 10 MeV, (c) 100 MeV and (d) 1000 MeV obtained using the method of successive launches that considers the solar cycle variability in the occurrence of solar flares. of solar energetic >10 MeV protons using a range of radial dependences for SEP intensities, is always >10 11 protons cm 2 [Lario and Decker, 2011], whereas for solar neutrons of the same energy, we estimate a worst case of neutrons cm 2 (black lines in Figure 7). [28] We would like to clarify that the generation of secondary neutrons from interactions between spacecraft material and primary charged particles (including solar energetic protons and/or galactic cosmic rays) has not been considered in the present work. These secondary neutrons may produce a significant enhancement of the total radiation dose produced by SEPs (as estimated by Lario and Decker [2011]) and of the solar-flare neutron fluence estimated in the present work. 10 of 11

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(david. lario@jhuapl.edu) 11 of 11