Estimation of the Particle Radiation Environment at the L1 Point and in Near-Earth Space

The interplanetary and near-Earth environments are characterized by several main ionizing radiation components with different origins, e.g., galactic cosmic rays (GCRs), solar energetic particles (SEPs), and trapped particles. GCRs and SEPs are driven by the interplanetary magnetic field, evolving on timescales ranging from a few hours up to years, mainly controlled by both short- and long-term processes related to solar activity, e.g., solar rotational periodicities, quasi-biennal oscillations, and Schwabe cycle (Laurenza et al. 2009; Vainio et al. 2009; Arrmano et al. 2018). In particular, GCRs provide a continuous high-energy proton flux, while SEP events represent short-lived hazards, and their fluxes differ in their energy spectra and in their opposite relation to the solar activity cycle. On the other hand, trapped particles, also known as Van Allen radiation belts, are confined in the near-Earth space by the Earth's magnetic field and are locally accelerated by both magnetospheric and solar wind processes (Singer 1958). Investigating radiation components and the high-energy particle background is of primary interest for any space mission profile, both in near-Earth and interplanetary space, as well as for space weather purposes (see, e.g., Reid 1986; Letaw et al. 1987; Townsend 2005; Vainio et al. 2009; Alam et al. 2017; Gastaldello et al. 2017; Panasuyk et al. 2017; Malandraki & Crosby 2018). Indeed, spacecraft instrument damage, failures, and malfunctions are related to variations in radiation components, which can increase up to five orders of magnitude, according to the different possible conditions (see, for instance, Baker & Kanekal 2008), e.g., during SEP events, for which several forecasting models have been developed (e.g., Laurenza et al. 2009; Alberti et al. 2017b; Laurenza et al. 2018, and references therein). Moreover, ionized particles also affect the communications (on a wide range of frequencies), controls, and operations of spacecraft as well as ionization and displacement damages, due to energetic particles reaching the mirrors and being concentrated toward the focal plane instruments. These phenomena occur, producing different effects and variations, for low Earth orbit or geosynchronous orbit satellites as well as for spacecraft located near the Lagrangian point L1.

Several space/solar missions, e.g., WIND, Solar and Heliospheric Observatory, Advanced Composition Explorer (ACE), and Earth observatories, e.g., Deep Space Climate Observatory, are currently located near the Lagrange point L1, at a distance of ∼1.5 million kilometers from Earth. This point is a very good position for, e.g., monitoring the solar wind, which reaches it about one hour before reaching Earth. For this reason, it has been used to provide in situ measurements and monitor solar wind plasma (speed, temperature, direction, electron and ion distributions, protons, and alpha-particles) and fields, to study energetic particles, accelerated in situ and/or near the Sun.

One of the main requirements for space missions is the estimation of the energetic particle background near the regions where spacecraft will be located. For instance, protons with energies in the range from decades to hundreds of keVs (so-called soft protons) are of particular interest for the Advanced Telescope for High Energy Astrophysics (ATHENA), an ESA large-class mission that will be launched in 2030 (Nandra et al. 2018), and consisting of a large X-ray telescope (Barcons et al. 2015) and two focal plane instruments with complementary capabilities: the Wide Field Imager (Meidinger et al. 2017), and the X-ray Integral Field Unit (X-IFU) (Pajot et al. 2018). The X-IFU is a cryogenic imaging spectrometer with unprecedented spectral capabilities (ΔE < 2.5 eV at 6 keV), based on a large array of about 4000 TES microcalorimeters. Soft protons can interact with the X-ray optics and are funneled toward the focal plane where they impact the detectors. Hence they represent a severe threat to the possibility of exploiting scientific data, reducing the available exposure times and introducing a poorly reproducible background component. Indeed, previous and current X-ray space telescopes encountered several operational problems due to increased soft proton background fluxes (see, e.g., Katayama et al. 2004; Carter & Read 2007; Budjas et al. 2017, and references therein). Given the large collecting area of the ATHENA mirrors, the soft proton contribution to the background is expected to be even higher than that in previous missions. However, a systematic study of the observed energetic particle background in this energy range has yet to occur.

It is well known that high-energy particles are hazardous conditions in space. For instance, ≥100 MeV protons are capable of penetrating spacecraft shielding materials and directly affecting the operation of electronic components within the vehicle (Vainio et al. 2009). They may induce the production of secondary species that are capable of reducing the operational lifetimes of spacecraft electronics as well as causing immediate data loss or damage. Particularly, ≥100 MeV protons are as relevant for the ATHENA background as soft protons. Moreover, solar and galactic particles with energies greater than 100 MeV/n can charge the test masses (Grimani et al. 2004; Araujo et al. 2005) of interferometers on board several future missions (e.g., LISA and ASTROD) for the detection of low-frequency gravitational waves. As the test mass charging constitutes one of the most important sources of noise for the experiments in the low-frequency range, several studies (Grimani et al. 2012, 2014; Arrmano et al. 2018) have been devoted to the estimation of the ≥100 MeV particle flux for the LISA-Pathfinder technological test mission for LISA.

Several models have been proposed in the last ∼30 yr to predict the occurrence of such high-energy SEP events at satellite working time, mainly focused on event fluence (see, e.g., Storini et al. 2008; Vainio et al. 2009; Jiggens et al. 2018, and references therein) to characterize the radiation environment during SEP events. One of the first of these models was proposed by Modisette et al. (1965) and is based on the probability distribution functions (PDFs) of SEP events with energies >30 and >100 MeV identified during the maximum phase of solar cycle (SC) No. 19. Then, the SOLPRO model (King 1974; Stassinopoulos 1975) was developed to separately consider "ordinary" and "anomalously large" SEP events that behave in two different particle populations. Moreover, the so-called JPL-85 model (Feynman et al. 1990) first used a log-normal distribution to fit the solar proton event size distribution, with the upgraded version JPL-91 allowing the selection of a lower threshold value to start proton fluence computation in several integrated energy channels (Feynman et al. 1993). Several models also used the log-normal approach (Xapsos et al. 1999, 2000; Escudier et al. 2002; Xapsos et al. 2007) to derive SEP fluences and predict integral omnidirectional solar proton fluences at 1 au. Both models, called ONERA and ESP, as well as the extended version PSYCHIC, are capable of predicting cumulative solar proton fluences and worst-case scenarios. Nevertheless, it is extremely important to also correctly characterize the interplanetary energetic environment during quiet periods (e.g., when the GCR intensity is maximum) and investigate its variation with solar activity.

In this paper we performed a long-term analysis of the proton flux background near the Lagrangian point L1 and in near-Earth space to characterize the radiation environment at soft protons and high energy (greater than 100 MeV) ranges, as they both pose serious hazards for space missions, including ATHENA. Nevertheless, the proton flux variability at energies between ∼5 and 100 MeV has already been assessed by Laurenza et al. (2019) using IMP-8 observations to complement this study.

We used statistical tools (i.e., cumulative distribution functions, CDFs) applied to the proton flux measured by different instruments on different spacecraft over extended periods of time. The EPAM/LEMS120 instrument on board the ACE spacecraft is used to characterized the proton flux variability over SC No. 23 and SC No. 24, while data recorded by IMP-8 have been used to provide insights for previous SCs (i.e., SC No. 21, SC No. 22, and part of SC No. 23). The data used and CDF computation are presented in Sections 2 and 3, respectively. A comparison between the different instrument data in the energy range ∼50–300 KeV (at energies >300 keV) is presented in Section 4 (Section 5). Finally, both the ACE and Geotail proton spectra at 50% and 90% cumulative probabilities (CPs) have been obtained, fitted ,and compared to investigate the behavior of the proton flux as a function of the differential energy. Worst-case scenarios are derived in Section 6 and conclusions are drawn in Section 7.

2.1. ACE EPAM/LEMS120 Data

We used hourly particle flux data (www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA_EPAM.html), covering the time period from 1997 August 14 to 2014 December 31 (i.e., SC No. 23 and part of SC No. 24), from the EPAM/LEMS120 telescope (Gold et al. 1998) on board the ACE spacecraft. Note that the lowest energy channels of the other detector (LEMS30) cannot be used because of a strong increase in the background rate from 2002. The LEMS120 sensor is oriented at 120° from the spin axis, which is always between 4° and 20° of the Sun, with a half opening angle of 255. Thus, LEMS120 provides measures in directions that are roughly consistent with ATHENA's pointing direction that will be within ±36° of 90° from the Sun. The LEMS120 sensor records ion flux data in 8 differential energy channels (see Table 1) in the range 0.047–4.8 MeV, mostly constituted by protons. Hence, their flux can be considered, in a first approximation, as representative of the proton flux at L1 where the ACE spacecraft is orbiting (at the L1 point, the orbital period of the spacecraft becomes exactly equal to Earth's orbital period).

Table 1.  Energy Channels for Each Instrument, with Corresponding Energy Ranges

ACE EPAM/LEMS120		IMP-8/CPM CMPE		Geotail EPIC-ICS
Label	Energy Range (MeV)	Label	Energy Range (MeV)	Label	Energy Range (MeV)
P1'	0.047–0.068	P1	0.29–0.50	G1	0.058–0.077
P2'	0.068–0.115	P2	0.50–0.96	G2	0.077–0.107
P3'	0.115–0.195	P3	0.96–2.00	G3	0.107–0.154
P4'	0.195–0.321	P4	2.00–4.60	G4	0.154–0.227
P5'	0.321–0.580	P5	4.60–15.0	G5	0.227–0.341
P6'	0.580–1.060	P7	15.0–25.0	G6	0.341–0.522
P7'	1.060–1.900	P8	25.0–48.0	G7	0.522–0.813
P8'	1.900–4.800	P9	48.0–96.0	G8	0.813–1.560
		P10	96.0–145.0	G9	1.560–3.005
		P11	145.0–440.0

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The EPAM/LEMS120 Level 2 data at hourly resolution are shown in Figure 1 for each energy channel. A clear quasi-periodic modulation is evident in P1' measurements. It is most likely due to instrument temperature fluctuations (Haggerty et al. 2006), i.e., of unphysical origin, not present for higher-energy channels, as well as a clear increasing monotonic trend. On the other hand, a SC modulation, characterized by a reduction in the proton flux during the period 2008–2010, corresponding to the minimum phase of SC No. 23, is mostly visible for high-energy channels (i.e., from P5' to P8'), less evident in low-energy ones (only a small signature in P2'–P4'), and almost completely invisible for P1' due to the unphysical modulation. Finally, as expected, low-energy channels measure greater proton fluxes than high-energy ones, with the P1'–P4' measurements showing from one to three orders of magnitude proton flux increases with respect to P5'–P8' ones.

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As the P1' measurements are characterized by a quasi-periodic modulation and by an increasing monotonic trend, they must be investigated in more detail because they could affect the background estimation. For this reason, we used the Hilbert–Huang Transform (HHT) to filter out both the unphysical modulation and the monotonic trend from P1' measurements (Huang et al. 1998). Details of the analyses performed to obtain a filtered P1' signal are reported in the Appendix.

2.2. IMP-8 CPME Data

We used the IMP-8 data covering the period 1974 January 1–2001 October 26 to characterize the near-Earth particle environment in SC No. 21 and SC No. 22, before the launch of the ACE spacecraft. The IMP-8 satellite is the last in a family of 10 Interplanetary Monitoring Platforms (IMPs) orbiting in a nearly circular orbit around the Earth at a radius of about 35 Earth radii with an orbital period of 12 days. During its orbit, it spent 60% or more of its time in the solar wind, while the remaining time was passed in the magnetosheath and magnetosphere. IMP-8 data, when it was located in the solar wind and outside the foreshock, can be freely downloaded from http://sd-www.jhuapl.edu/IMP/imp_cpme_data.html. The Charged Particle Measurement Experiment (CPME) data measured the proton flux in 10 energy differential channels, as reported in Table 1. Here, because the CMPE data mainly span outside the energy range of both ACE EPAM/LEMS120 and Geotail EPIC-ICS, for comparison we used the P1 (0.29–0.50 MeV) which is closest to the soft proton energies. Moreover, we also used data from the P11 (145–440 MeV) energy channel (i.e., the highest one), which can be used to obtain information about extreme SEP events, as well as the GCR background.

Figure 2 shows the proton flux data recorded in the P1 and P11 energy channels when IMP-8 was located in the interplanetary space, as they can be used as representatives of proton fluxes measured near the L1 point. From Figure 2 a clear SC modulation is apparent: the P1 flux (second panel in Figure 2) is in phase with the solar activity, represented by the sunspot number proxy (top panel of Figure 2), as the highest fluxes are mainly observed during and around the maximum phases. Thus, they can reasonably be associated with energetic particles of solar or interplanetary origin. In particular, in channel P1, the flux fluctuates by several orders of magnitude, and "quiet periods" alternate with "active periods," although the former are relatively shorter. Similarly, in channel P11 the solar particles become dominant only in the "active periods," when the flux increases by three orders of magnitude with respect to "quiet periods." The above features are consistent with the fact that protons with energies lower than 1 MeV are frequently emitted at the Sun or accelerated in interplanetary space (e.g., by propagating shocks), but more rarely (in connection with flares or shocks driven by coronal mass ejections) those with energies of hundreds of MeV. In channel P11, the flux during "quiet periods," of the order of 10⁻³ cm⁻² s⁻¹ sr⁻¹ MeV⁻¹, shows an anti-correlation with the sunspot number. The quiet level, observed in channel P11, should be largely identified with the GCRs (Laurenza et al. 2009). Indeed, the P11 flux profile resembles the GCR intensity one (bottom panel in Figure 2), as recorded by the Rome neutron monitor called the Studio Variazioni Intensità di Raggi COsmici Observatory (INAF/IAPS—UNIRoma3 collaboration). Conversely, the GCR contribution is nearly negligible at energies lower than 1 MeV, so fluxes observed in channel P1 are, at any time, primarily of solar and interplanetary origin. Thus, the long-term variations of the proton fluxes in the two energy ranges depend on the different behaviors of the different particle populations.

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2.3. Geotail EPIC-ICS Data

We used the Geotail data measured from 1995 January to 2001 December during SC No. 23. It is a satellite moving on an elliptic orbit, with an apogee of 30 Earth radii, flying for a large part (∼7 months) of each year inside the solar wind, upstream and on the flanks of the bow shock.

The Geotail measurements can be seen as representatives of the L1 environment if two conditions are satisfied: the spacecraft has to be (i) inside the solar wind but (ii) outside the foreshock, i.e., located upstream of the last field line connected to the shock. By making use of a data set specifically corresponding to the time intervals during which Geotail was inside the solar wind, produced by the NASA at the GFSC (ftp://spdf.gsfc.nasa.gov/pub/data/geotail/merged/sw_min_merged_ascii), and by assuming a shock-simplified model (from Cairns et al. 1995; Farris et al. 1999) we considered only the time intervals in which the spacecraft was located inside the solar wind but outside the foreshock. These conditions are verified by examining magnetic field data from both Geotail and IMP-8 and by comparing these measurements with corresponding observations made by both ACE and WIND, which are located inside the solar wind. Table 2 reports the time periods when Geotail was in the solar wind.

Table 2.  Time Periods When Geotail Was in the Solar Wind

Time Intervals
1995 Mar 15–1995 Oct 8
1996 Apr 3–1996 Oct 21
1997 Apr 16–1997 Nov 4
1998 Apr 25–1998 Nov 11
1999 May 9–1999 Dec 2
2000 May 23–2000 Dec 16
2001 Jun 8–2001 Dec 26

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In particular, we used the Geotail EPIC-ICS experiment data related to the proton flux measured in 9 different energy channels, from 58 keV to 3 MeV, characterized by 16 azimutal sectors (Williams et al. 1994). By averaging the fluxes over all azimutal sectors we computed the omnidirectional fluxes at hourly resolution. Table 1 shows the details of the EPIC-ICS energy ranges.

Figure 3 shows the proton flux for the different energy channels. As expected, low-energy channels measure greater proton fluxes than high-energy ones: typically, G1–G4 show one (up to two) order of magnitude proton flux increases with respect to G5–G9. Finally, a reduction in the proton flux during the period 1995–1997, corresponding to the minimum phase of SC No. 22, is observed at high energies.

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To investigate the variability of the background proton fluxes we used a statistical approach based on the evaluation of the CDF for each energy channel. For each considered data set from ACE, IMP-8, and Geotail, the PDFs, from which CDFs can be obtained, have been evaluated using a non-parametric approach based on the kernel density estimator technique (e.g., Silverman 1998; Alberti et al. 2014, 2017b, for more details) with the definition of a non-negative function, called a kernel function, with zero mean and integrating to one, and by choosing an appropriate smoothing parameter, called the bandwidth. From the class of kernel functions (Gaussian, Epanechnikov, Triangular, and so on) a good choice is the Epanechnikov kernel, which is optimal in a minimum variance sense. Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. Thus, when PDFs have been evaluated, CDFs are obtained as the probability that F' will take a value less than or equal to a value F (with being F the proton flux). In the case of a continuous distribution, it gives the area under the probability density function from minus infinity to F:

$$\begin{eqnarray}&&\mathrm{CDF}(F^{\prime} )=\mathrm{Prob}(F^{\prime} \lt F)={\int }_{-\infty }^{F}\mathrm{PDF}(F^{\prime} ){dF}^{\prime} .\end{eqnarray} \tag{ 1 }$$

In what follows we describe the CDF for each data set.

3.1. ACE EPAM/LEMS120 Data Analysis

Figure 4 shows the CDFs obtained for the eight energy channels of the EPAM/LEMS120 instrument.

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We note that the CDFs obtained from P2'–P8' measurements present similar shapes, while a different shape is obtained for the P1' data (black line in Figure 4). This is related to the quasi-periodic modulation observed in the P1' time series, which affects both shape and particle flux values. Using filtered data (dashed–dotted line in Figure 4) a clear difference is found between P1' and the filtered P1' CDFs, the latter being similar in shape to the other energy channels.

By using CDFs we are able to evaluate proton flux levels that are not exceeded for a given percentage of the total time coverage, as reported in Table 3 for each energy channel. From a statistical point of view, the characterization of the energetic environment in the L1 point can be obtained by estimating the 90% CP of observing a flux lower than a maximum flux F, i.e., the proton flux maximum value that is not exceeded for 90% of the total time coverage. This means that, during the complete period of measurement (i.e., 1997–2014), each energy channel measures proton fluxes below a value F, which, as expected, decreases as energy increases (for each selected CP). Moving from low (∼50 keV) to high (∼5 MeV) energies, the maximum value decreases by about three orders of magnitude at the 90% CP.

Table 3.  Proton Flux Maximum Values (in Units of cm⁻² s⁻¹ sr⁻¹ MeV⁻¹) Obtained from Data in the Energy Channels of EPAM/LEMS120 on board ACE for Different Cumulative Probabilities Computed over the Period 1997 August 14–2014 December 31

Cumulative Probability (%)	P1'	${\rm{P}}{1}_{\mathrm{EMD}}^{{\prime} }$	P2'	P3'	P4'	P5'	P6'	P7'	P8'
10	354	280	25	14	5	1	0.5	0.1	0.03
20	680	390	30	18	7	2	0.6	0.2	0.04
50	1300	645	60	34	13	3	1	0.3	0.05
80	2510	1454	567	280	109	40	13	4	0.7
90	6688	3900	2681	1143	553	219	66	21	6

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The third column of Table 3 shows the correct values from using the CDF of the P1' filtered data below which the proton flux is observed for a given percentage of time: the unphysical modulation tends to increase the maximum values of a factor between 1.2 and 2, increasing as CP increases. This means that the proton flux for P1' could be significantly overestimated due to the presence of the instrumental background.

In addition, fits have been performed using the Ellison–Ramaty functional form (Ellison & Ramaty 1985) for proton flux maximum spectra at two chosen CPs as reported in Table 5, where E is the energy (expressed in keV). Note that the obtained fit function coefficients are slightly different from those of Lotti et al. (2018), because the former have been computed using the P1' filtered data, whereas the latter are from using the original ones (compare column two and three in Table 3). Using the above fit functions it is possible to estimate the proton flux maximum values observed for 90% and 50% of the total time coverage at any energy in the considered range. We have also computed CDFs for data of the 8 energy channels of EPAM/LEMS120, the proton spectra at the 50% and 90% CPs and their fits over the period 1997–2009, i.e., almost all SC No. 23, as they are more suitable to show the solar activity variability. Results are reported in Table 4 and column three of Table 5.

Table 4.  Proton Flux Maximum Values (in Units of cm⁻² s⁻¹ sr⁻¹ MeV⁻¹) Obtained from the Energy Channels of EPAM/LEMS120 on board ACE for Different Cumulative Probabilities Computed over the Period 1997 August 14–2009 December 31 (SC No. 23)

Cumulative Probability (%)	${\rm{P}}{1}_{\mathrm{EMD}}^{{\prime} }$	P2'	P3'	P4'	P5'	P6'	P7'	P8'
10	387	31	18	7	2	0.7	0.15	0.04
20	501	37	23	9	3	0.9	0.23	0.05
50	872	74	41	18	4	1.3	0.35	0.06
80	2132	716	336	116	44	15	5	1
90	5124	3415	1770	722	291	89	37	11

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Table 5.  Fit Functions for the Proton Spectrum Obtained by EPAM/LEMS120 on board ACE for Two Cumulative Probabilities Computed Both over the Period 1997 August 14–2014 December 31 and for the Time Interval 1997 August 14–2009 December 31 (Total Time—SC No. 23)

Cumulative Probability (%)	Total Time	SC No. 23
50%	8444 × E−2.44 ${e}^{(2.4\times {10}^{-4}E)}$	4312 × E−2.29 ${e}^{(1.08\times {10}^{-4}E)}$
90%	2096 × E−1.51 ${e}^{(-2.6\times {10}^{-4}E)}$	2854 × E−1.52 ${e}^{(-1.17\times {10}^{-4}E)}$

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3.2. IMP-8 CPME Data Analysis

Figure 5 shows the CDFs of the proton flux data recorded in the P1 and P11 energy channels of the CPME instrument on board IMP-8. We note that the P1 energy channel CDF has a similar shape to those obtained from ACE data (see Figure 4), while a lower particle flux variation range is observed for P11: the proton flux maximum value only increases from 1.1 × 10⁻³ at the 50% CP to 1.8 × 10⁻³ at the 90% CP.

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Moreover, we also found (not shown) that data recorded during 1997–2001 in SC No. 23, have lower CDFs values with respect to comparable phases in SC No. 21 and SC No. 22 in both P1 and P11, as SC No. 23 has been less intense than SC No. 21 and SC No. 22 (e.g., Alberti et al. 2017b). In Table 6 the computed values for the separated complete SCs are reported.

Table 6.  Proton Flux Maximum Values (in Units of cm⁻² s⁻¹ sr⁻¹ MeV⁻¹) Obtained from IMP-8 P1 and P11 Energy Channels for Two Cumulative Probabilities over Different Time Periods: 1976–1986 (SC No. 21), 1987–1996 (SC No. 22) and 1997–2001 (Increasing Phase of SC No. 23), and 1974 January–2001 October (Total Time)

	P1				P11 (×10−3)
Cumulative	SC	SC	SC	Total	SC	SC	SC	Total
Probability [%]	No. 21	No. 22	No. 23	Time	No. 21	No. 22	No. 23	Time
50	10	9	7	7.4	1.3	1.1	0.9	1.0
90	1196	812	630	663	1.9	1.7	1.4	1.6

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3.3. Geotail EPIC-ICS Data Analysis

Figure 6 shows the CDFs obtained for each energy channel of EPIC-ICS on board Geotail.

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As expected, the proton flux decreases as energy increases, from ∼10² cm⁻² s⁻¹ sr⁻¹ MeV⁻¹ for G1 in the range 58.1–77.3 keV to ∼10⁻¹ cm⁻² s⁻¹ sr⁻¹ MeV⁻¹ for G9 in the range 1560.8–3005.5 keV at the 50% CP. Generally, the Geotail EPIC-ICS measurements span up to 3–4 orders of magnitude with decreasing energy for each selected CP above 50%.

For ATHENA operational purposes it is important to investigate and provide proton flux estimates for a specific energy range, between few tens and few hundreds of keV, i.e., so-called soft protons. Geotail observations can be compared at low energies, i.e., <300 keV, with ACE measurements using the G1–G5 and P1'–P4' channels, over a common acquisition period (from 1997 August 14 to 2001 December 26) when Geotail was in the solar wind (see Table 2). Figure 7 shows a comparison between the CDFs obtained for energies <300 keV. In this case, a quiet good superposition is found between the Geotail and ACE CDFs for comparable energy channels (e.g., G2 and P2'), when CPs ≳50%–60% are considered. For the 90% CP the proton flux maximum value is between ∼10³ and 10⁴ cm⁻² s⁻¹ sr⁻¹ MeV⁻¹ for energies from ∼200 to ∼50 keV. However a discrepancy is found at lower CPs (i.e., ≲50%) to be less than one order of magnitude on similar energy ranges, possibly due to the different instrument sensitivity.

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A quantitative comparison, which takes into account the difference in the energy channels ranges between the two spacecraft, can be done by combining measurements from Geotail G1–G5 energy channels and the ACE P1'–P4' ones to obtain two equivalent energy channels in the range ∼50–300 keV, using the following procedure:

• 1.  
  the proton flux values in each energy channel k were multiplied by the corresponding energy range ΔE_k and geometric factor g such that we define ${p}_{k}^{{\prime} }\,={p}_{k}\ast {\rm{\Delta }}{E}_{k}\ast g$, where p_k is the proton flux in the kth energy channel;
• 2.  
  ${p}_{k}^{{\prime} }$ were summed over the k channels (i.e., G1–G5 for Geotail, P1'-P4' for ACE);
• 3.  
  the sum is divided by the value of the energy range ΔE (ΔE = 283 keV for Geotail, ΔE = 274 keV for ACE) and by the geometric factor g (g = 0.2 cm² sr for Geotail, g = 0.428 cm² sr for ACE).

Obviously, the 50–300 KeV proton fluxes are mainly dominated by the low energies <100–150 KeV. However, this range of energies is of great interest for the ATHENA-XIFU background evaluation because protons with these energies can cause detector failures and optical damage. In this way, we can evaluate the CDFs from measurements collected inside the equivalent energy channels with bounds around 50 keV and 300 keV for both sets of measurements, as reported in Figure 8. A significant agreement is found between the Geotail and ACE CDFs when >50% CPs are considered, their proton flux ratio being 0.64 (65/101) at the 50% CP and 1.08 (3845/3557) at the 90% CP. Thus, both spacecraft measure almost the same proton flux in the above equivalent energy channel for time percentages higher than 50%, with agreement increasing with the CP.

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Finally, the flux variation with energy can be estimated by computing the proton spectra at the 50% and 90% CPs, allowing us to obtain proton flux maximum values at different energies, as shown in Figure 9. Assuming a power-law behavior (F ∼ E^−β) for the spectral shape, a small difference is found in the spectral slope exponent, from a value of β = 0.9 (β = 1.9) for the 50% CP to a value of β = 1.1 (β = 1.6) when the 90% one is considered for the Geotail (ACE) data. Obviously, larger particle flux values have been found for the 90% CP, with a difference of about two orders of magnitude with respect to the 50% one. Nevertheless, there seems to be at least a 50% ratio between the ACE and GEOTAIL flux values at 90% CP, and it increases with energy. Note that this ratio reaches more than one order of magnitude above 300 keV, possibly due to the contamination of magnetospheric and/or foreshock particles in the Geotail data. Hence, the Geotail measurements can be correctly used as representative of the L1 proton environment within a factor 2 only at energies <300 keV, e.g., when ACE was not yet in orbit, and at >50% CP. This can be attributed to the different locations of the two spacecraft. Indeed, when Geotail was in the solar wind, its maximum distance from the Earth was about 30 Earth radii (close to the bow shock), while ACE is located near the Lagrangian point L1, at about 220 Earth radii.

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A comparison between different spacecraft can be also done for higher energies, i.e., >300 keV, by comparing the Geotail measurements with those of IMP-8 on several energy channels over a common period of acquisition (i.e., from 1995 March 15 to 2001 October 26, when both spacecraft were in the solar wind, see Table 2). In particular, we compared the Geotail measurements from the G6 to G9 energy channels and the IMP-8 energy channels from P1 to P4. From Figure 10 we note that good agreement is obtained inside the energy range 200–900 keV (i.e., G5–G7 and P1–P2), while poorer agreement is found for higher-energy channels. This is due to the fact that the latter span different wider energy ranges with different bounds.

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We compared the CDFs obtained from the equivalent energy channels of IMP-8 P1, Geotail G6, and ACE P5' over the common period during 1997–2001 and only during time intervals when they were simultaneously in the solar wind (see Figure 11).

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They are found to have similar flux maximum values for a percentage of time greater than 50%, with small discrepancies at lower ones. This discrepancy could be a reflection of the larger geometric factor of the EPIC-ICS instrument on board Geotail that is able to measure lower fluxes than both the ACE and IMP-8 instruments. However, the ACE and IMP-8 measurements in this energy range are almost equivalent (within a 4%) at the 90% CP, so they can be used in an interchangeable way to explore different conditions. Tables 7 and 8 report the maximum values below which the IMP-8/P1 and ACE/P5' proton flux, respectively (which can be considered to be equivalent, see Figure 11), are observed for 50% and 90% of the total observing time period. The flux relative variation between the 1981–1982 period and the ACE total time coverage is inferred to be about a factor of 10 (13) for the 50% (90%) time percentage. Finally, comparing the ACE P5' data with the IMP-8 P1 ones during comparable maximum phases of SC No. 21 and SC No. 23 (1981–1982 and 2000–2001, respectively), a proton flux higher of about a factor of 3 (2) is found in the former for the 90% (50%) CP. This is a further indication that SC No. 21 was characterized by more intense solar activity.

The evaluation of the proton flux obtained in Sections 3.1–3.2 was performed over the total observation time and separated SCs for ACE and IMP-8, respectively. Hence, they represent averages, possibly underestimating the radiation environment during periods characterized by strong solar activity. The worst-case scenarios for soft protons and high energies (145–440 MeV) over a wide period, comparable to a life mission, has been investigated by computing the CDFs only for the maximum phase of each SC. The highest flux levels using ACE data are found in the years 2000–2001 during the maximum phase of SC No. 23. Table 9 shows the parameters obtained by fitting the ACE data with an Ellison–Ramaty function (second column) and a power law (third column), both for the 1997–2014 and 2000–2001 (worst-case) periods. Figure 11 depicts the obtained spectra and fits. In the 90% CP spectrum, an increase of about a factor 5 for filtered P1' data during 2000–2001 is apparent, while it varies between 3 and 5 for the other channels.

Table 9.  Fit Functions for the Proton Spectrum Obtained from the EPAM/LEMS120 on board ACE at the 90% Cumulative Probability for the 1997–2014 and 2000–2001 Time Periods

Cumulative Probability (%)	Ellison–Ramaty Function	Power Law
90%	2096*E−1.51 ${e}^{(-2.6\times {10}^{-4}E)}$	4235*E−1.67
90% worst-case	6734*E−1.45 ${e}^{(-2.67\times {10}^{-4}E)}$	15032*E−1.62

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The worst case for the "quiet periods" at energies 145–440 MeV is obtained over two years (1975–1976) during the minimum phase of SC No. 21, using the IMP-8 P11 cumulative distribution: at the 50% and 90% CPs, proton flux values of 2.1 × 10⁻³ pr cm⁻² s⁻¹ sr⁻¹ MeV⁻¹ and 3.8 × 10⁻³ pr cm⁻² s⁻¹ sr⁻¹ MeV⁻¹ are found, respectively. A comparison with Table 6 indicates small variations within a factor of 2 with respect to other time periods. Obviously, these limits do not include the contribution of high-energy SEP events that reach much higher flux values, as they are observed for no more than 10% of the observational time. Thus, the above estimates represent the contribution of the low-energy GCRs to the radiation environment.

A long-term analysis of the ACE data covering the 1997 August 14–2014 December 31 interval (SC No. 23 and part of SC No. 24) has been performed. CDFs have been produced for the eight differential energy channels of the EPAM/LEMS120 experiment, covering the energy range 0.047–4.8 MeV. Since the energy channel P1' is affected by an unphysical quasi-periodic modulation, it has been corrected using the HHT approach through which we produced a corrected (filtered) P1' signal to evaluate correct proton flux estimates. It has been shown that the unphysical modulation tends to increase flux maximum values of a factor 1.2–2, according to the different CPs considered. Particle spectra have been derived related to the percentages of time spent below different flux values by computing the CDFs of each energy channel as reported in Tables 3 and 4 for the whole time coverage and SC No. 23, respectively.

The same analysis has been performed on the proton flux data recorded on board Geotail (when it was in the solar wind and outside the foreshock during the 1995–2001 period) in a similar energy range (0.058–3 MeV), and compared with results obtained from ACE data. In addition, we computed the cumulative distributions of the IMP-8 proton flux data in the 0.29–0.50 MeV (P1) and 145–440 MeV (P11) differential energy channels, covering the period 1974 January 1–2001 October 26 (SC No. 21, SC No. 22, part of SC No. 23). The former can provide valuable information about the proton flux variation during different SCs when compared with ACE and Geotail results, whereas the IMP-8 P11 channel measures the high-energy particles, which can compromise the success of any space mission.

Quite good agreement has been found between Geotail and both IMP-8 and ACE measurements at comparable energies. In particular, a comparison between Geotail and ACE in the soft proton range 50–300 keV, of great interest for the ATHENA mission, shows that Geotail measurements in near-Earth space at about 20 Earth radii are close to those in the L1 environment within a factor of 2 (5), only at energies <300 keV and at 50% (90%) CP (Figure 9). This could be a consequence of the Geotail location, close to the Earth environment, from which an additional particle contribution might increase the proton flux at higher energies.

The worst-case scenario over a wide period has been investigated by computing the CDFs only for the maximum phase of each considered SC. The highest flux thresholds at energies ∼50–4800 keV are obtained for two years (from 2000 to 2001) during the maximum phase of SC No. 23 of the ACE data collection. The proton spectra observed during 2000–2001 can be considered an upper limit for the ACE measurements as a result of the solar activity variability, which tends to increase the proton flux by about a factor 5 with respect to the ACE total time coverage (1997–2014). For instance, an upper limit for the omnidirectional proton flux of ∼4 × 10³ pr cm⁻² s⁻¹ sr⁻¹ MeV⁻¹ is found, over the whole data acquisition time, at the 90% CP when the 47–68 KeV energy range is considered. This value increases up to ∼2 × 10⁴ pr cm⁻² s⁻¹ sr⁻¹ MeV⁻¹ during solar maximum conditions (see Figure 12). More generally, the ∼0.05–5 MeV proton spectrum at the 90% CP, computed from ACE data, increased during the maximum phase of SC No. 23 by about a factor from 3 to 5, depending on the energy, with respect to that computed over the total time coverage.

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Data recorded in the IMP-8 P1 energy channel have been compared with those in its equivalent energy channel ACE P5' over a common acquisition period, from 1997 August 14 to 2001 October 26. The results show very good agreement, both having almost equal proton flux maximum values for CPs greater than 50%. Thus, IMP-8 P1 data for SC No. 21 and SC No. 22 are also used to explore the proton flux variability related to solar activity in different SCs. It has been found that for CPs ≥50%, when IMP-8 and ACE can be considered equivalent, the flux values are higher for SC No. 21 with respect to those in SC No. 23 (comparing Tables 7 and 8) in the energy range 0.29–0.50 MeV (by about a factor 3 and 2 at the 90% and 50% CPs, respectively). The observed differences could be due to the lower large-scale solar magnetic activity (i.e., active regions and sunspots) and/or the weak interplanetary magnetic field (decreasing the efficiency of particle acceleration mechanisms) in SC No. 23 with respect to previous SCs, manifested in a smaller production of energetic particles. As a matter of fact, in SC No. 23 many activity parameters were reduced; for instance, a lower sunspot number (top panel of Figure 2), sunspot area, and interplanetary magnetic field intensity (e.g., Laurenza et al. 2012, Figure 6), as well as the line-of-sight component of the solar magnetic field (Krainev 2012). This aspect is even more striking in SC No. 24, as it is characterized by unusually low solar and geomagnetic activity both during its minimum (Diego et al. 2010; Otkidychev 2014) and maximum (Richardson 2013; Singh & Tonk 2014; Wang & Colaninno 2014) with respect to the preceding 100 yr, possibly in connection with the about 80–120 yr Gleissberg cycle of solar activity (e.g., Vecchio et al. 2017). Indeed, during the SC No. 24 minimum we found very low fluxes of energetic particles in almost all ACE EPAM/LEMS120 energy channels, as illustrated in Figure 1. This also shows that generally, during most of SC No. 24, particle fluxes were lower than those in SC No. 23, which is consistent with the paucity observed in SC No. 24 of sunspots and high-energy SEP events (Gopalswamy et al. 2015), number, and intensity of soft X-ray and radio bursts and >10 MeV SEP events (Prasanna Subramanian & Shanmugaraju 2016; Alberti et al. 2017b).

Finally, we found that variations with solar activity of the high-energy (145–440 MeV) proton flux, at both the 90% and 50% CPs, are less than a factor 2. In particular, the worst-case scenario for the "quiet periods," representative of the GCRs variations, is found in the years 1975–1976, during which the proton flux increase is of about a factor of 2 (3.8 × 10⁻³/1.8 × 10⁻³) at the 90% CP with respect to the total investigated period.

The research leading to these results has received funding from: ASI (Italian Space Agency) through the Contract Nos. 2015-046-R.0 and 2018-11-HH.0, the European Union's Horizon 2020 Programme under the AHEAD (Integrated Activities in the High Energy Astrophysics Domain) project (grant agreement No. 654215), and the ESA AREMBES project, contract No 4000116655/16/NL/BW. ACE EPAM/LEMS120 data have been retrieved from http://www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA_EPAM.html. IMP-8 CPME data have been retrieved from http://sd-www.jhuapl.edu/IMP/imp_cpme_data.html. The GEOTAIL EPIC database was built at the CDPP (www.cdpp.eu, the French Data Center for Plasma Physics hosted at IRAP) in close collaboration with the Applied Physics Laboratory (APL) PI of the GEOTAIL/EPIC experiment. The SVIRCO Observatory is supported by the INAF/IAPS-UNIRoma3 collaboration.

The first step of the HHT is the empirical mode decomposition (EMD), an algorithmic procedure through which oscillations that present a common local timescale are iteratively extracted from the data (Huang et al. 1998). This adaptive and a posteriori decomposition method was developed to process nonlinear and non-stationary data and it is based on the so-called sifting process through which a finite number n of intrinsic oscillatory functions c_i(t), referred to as intrinsic mode functions (IMFs), is extracted. So, a time series x(t) can be written as

$$\begin{eqnarray}&&x(t)=\displaystyle \sum _{i=1}^{n}{c}_{i}(t)+r(t),\end{eqnarray} \tag{ 2 }$$

where r(t) is the final residue of the decomposition from which no more IMFs can be extracted (see, e.g., Huang et al. 1998; Alberti et al. 2014, for more details).

The second and last step of the HHT is the Hilbert spectral analysis applied to the previously obtained IMFs through which an analytical signal is built using the IMFs and their Hilbert transforms (Huang et al. 1998). This allows us to write each IMF as an oscillating function modulated both in amplitude and in phase (or frequency), ${c}_{i}(t)={a}_{i}(t)\cos [{{\rm{\Phi }}}_{i}(t)]$, with Φ_i(t) being the instantaneous phase of the ith mode (the instantaneous frequency can be derived as ${f}_{i}(t)=d{{\rm{\Phi }}}_{i}(t)/{dt}$) (Huang et al. 1998). In this way, several limitations of other decomposition techniques that do not consider a time-dependent frequency are avoided and a suitable description of non-stationarity properties of time series is obtained (Huang et al. 1998). Then, a typical average timescale τ_i can be estimated for all the IMFs as ${\tau }_{i}=2\pi /\langle {f}_{i}(t){\rangle }_{t}$ (with being $\langle ...{\rangle }_{t}$ the time average).

This method has been successfully applied in both solar physics (e.g., Laurenza et al. 2014; Vecchio et al. 2017, and references therein) and space physics (e.g., Piersanti et al. 2017, and references therein), with different purposes, e.g., a filtering approach (Alberti et al. 2016), turbulence studies (Consolini et al. 2017; Carbone et al. 2018), solar wind-magnetosphere-ionosphere coupling (e.g., Alberti et al. 2017a, 2018; Consolini et al. 2018), and near-Earth space current systems characterization (e.g., Alberti 2018).

A.1. Filtering P1' Measurements: Corrected Background Values

A rough look at P1' time series seems to suggest that the unphysical modulation is confined to a timescale range 1.0 ± 0.5 yr (as also pointed out in Haggerty et al. 2006), and due to its clear non-stationary behavior it has been spread over different modes (as shown in Figure 13) where they (or reconstructions of them) are superposed on the raw P1' data, together with the empirical mode associated with the 11 yr SC contribution (magenta line).

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It can be reasonably reproduced using empirical modes with j = 22–24. Hence, the P1' signal has been reconstructed by excluding such modes, i.e., the unphysical modulation has been filtered out, and compared with both P2' and P4' (see Figure 14).

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We note that filtered data are in agreement with other energy channels (as P2' and P4'), although, as expected, ${\rm{P}}{1}_{\mathrm{EMD}}^{{\prime} }$ admits higher values than other channels.