A theoretical model of the inner proton radiation belt

Transcription

1 SPACE WEATHER, VOL. 5,, doi: /2006sw000275, 2007 A theoretical model of the inner proton radiation belt R. S. Selesnick, 1 M. D. Looper, 1 and R. A. Mewaldt 2 Received 17 August 2006; revised 31 October 2006; accepted 2 November 2006; published 6 April [1] A theoretical calculation provides inner radiation belt proton intensities as a function of time and of the three adiabatic invariants, M, K, andl, in the kinetic energy range from 10 MeV to 4 GeV and the L range from 1.1 to 2.4. Long residence times for trapped protons of up to several thousand years require similarly long input time series for the geomagnetic field, solar activity, and solar proton fluences. Additional inputs include galactic cosmic ray spectra, nuclear scattering cross sections, and the neutral and plasma densities in the atmosphere, ionosphere, and plasmasphere. Trapped proton sources are cosmic ray albedo neutron decay (CRAND), calculated from a Monte Carlo particle transport simulation, and solar proton injection using a derived empirical injection efficiency that is 10 4 at 10 MeV. Radial diffusion provides inward transport of injected solar protons. Calculated intensities at energies ]100 MeV and for L ^ 1.3 are dominated by solar protons, CRAND being the dominant source otherwise. Losses are by ionization of the neutral atmosphere, energy transfer to plasma electrons, and inelastic nuclear scattering. Numerical trajectory tracing determines trapping limits and drift shell averages of the albedo neutron intensity and of neutral and plasma densities for loss rate calculations. Geomagnetic secular variations cause adiabatic energy and drift shell changes. Intensities are greater than they would be in a constant geomagnetic field by factors up to 10, a result of long proton residence times and the presently decreasing geomagnetic dipole moment. Citation: Selesnick, R. S., M. D. Looper, and R. A. Mewaldt (2007), A theoretical model of the inner proton radiation belt, Space Weather, 5,, doi: /2006sw The Aerospace Corporation, Los Angeles, California, USA. 2 Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California, USA. 1. Introduction [2] An accurate numerical model of the inner radiation belt would be of practical value for specifying the radiation environment in low- and medium-altitude satellite orbits where energetic protons are a known hazard [Stassinopoulos and Raymond, 1988; Dyer, 2002]. Empirical models, such as NASA s widely used AP-8 model [Sawyer and Vette, 1976], have provided useful input to satellite design but are known to be uncertain, particularly with regard to intensities of highly energetic (>100 MeV) protons and variability of the intensity at all energies [Lauenstein and Barth, 2005]. Significant improvement to empirical models will require new observations of the inner radiation belt using advanced instrumentation. In the short-term absence of such new data sets, progress can be made in developing theoretical, or physics-based, inner radiation belt models. [3] The theoretical models calculate the radiation belt intensity by numerical solution of particle transport equations using known and measured inputs [e.g., Freden and White, 1960; Dragt, 1971; Farley and Walt, 1971; Jentsch and Wibberenz, 1980; Jentsch, 1981; Beutier et al., 1995; Albert et al., 1998; Vacaresse et al., 1999; Miyoshi et al., 2000]. In addition to their practical value, they are of scientific interest for establishing the mechanisms that populate the radiation belt. The most accurate results in the long term are likely to be obtained from a semiempirical approach combining theoretical models with new and existing data sets using data assimilation techniques [Naehr and Toffoletto, 2005]. [4] The goals of this work are to outline the physical processes that should be included in an accurate theoretical model, to construct a prototype model of the inner radiation belt protons using available inputs, and to establish directions for future improvement. Simultaneous evaluation of various particle transport processes in the model provides new insight into their relative significance, and we investigate in particular the role of geomagnetic secular variation, the relative contributions of inward radial diffusion versus the local neutron decay source, and the efficiency of solar proton injection. Detailed comparisons of the model results with radiation belt observa- Copyright 2007 by the American Geophysical Union 1of19

2 Figure 1. Geomagnetic dipole moment versus time from the CALS7K.2 [Korte and Constable, 2005] and IGRF-10 [Maus et al., 2005] models, with an uncertainty range for the CALS7K.2 dipole moment (dotted curves). The epochs for which the trapped proton model inputs were calculated are also indicated. tions and possible data assimilation are not addressed at this stage. 2. Transport Equation [5] The trapped proton intensity j is normally measured by a particle detector at a fixed location on a satellite orbit, at a given time t, and as a function of the kinetic energy E (or equivalently speed v) and local pitch angle a. A transport equation for j, that can be derived from the Boltzmann equation [Dragt, j de dt j v ¼ S j vt where collision terms have been combined into a source rate S and lifetime t, and a continuous energy change is represented by the rate de/dt. The derivation includes averaging over the phases of the trapped proton gyration, bounce, and drift motions, so that j is described in three dimensions, such as E, a, and L shell, plus time t. [6] The solution to equation (1) is obtained by integrating along E-t paths [Selesnick, 2001]: ð1þ Z t R je; ð tþ ¼ v SE ð 0 ; t 0 Þe ð t 00dE dt þ1 tþdt 00 dt 0 ð2þ t 0 where de/dt and t are each evaluated at E 00 and t 00, and where E 0 or E 00 is the energy at time t 0 or t 00, respectively. The initial condition is j =0att = t 0, which is the time when E is equal to the maximum trapped particle energy or trapping limit (see section 3.3). The solution (2) is conveniently evaluated by numerical integration if de/dt, S, and t are known. Therefore the principal modeling task is to evaluate these three quantities over the extent of the E-t paths (which may be thousands of years). After some preliminary considerations we describe them each in turn. 3. Geomagnetic Field [7] The coordinate system for the trapped proton model is based on the geomagnetic field, which varies continuously. Model inputs are evaluated for a set of epochs based on corresponding magnetic field models, and inputs for intermediate times are obtained by linear interpolation. The IGRF-10 model [Maus et al., 2005] is used for six recent epochs from A.D to A.D and the CALS7K.2 model [Korte and Constable, 2005] is used for ten earlier epochs from 1700 B.C. to A.D In all cases the spherical harmonic expansion is continued up to and including degree 7 (except in the geomagnetic cutoff calculations, see section 4.1.3, where it is continued up to and including degree 10). [8] The dipole moments from both the CALS7K.2 and IGRF-10 models are shown versus time in Figure 1, including the full time resolution available. The discrete epochs chosen for this work are also indicated, as is the uncertainty range for the CALS7K.2 model Adiabatic Invariants [9] The three adiabatic invariants of trapped particle motion are not conserved by radiation belt protons because of gradual energy loss, nuclear scattering, and radial diffusion. They are nevertheless useful as the model coordinate system because they are conserved by the slow geomagnetic secular variation. One disadvantage of this choice is that a time series of model intensities at constant values of E or a, or at a fixed location, must be obtained by interpolation. [10] The versions of the adiabatic invariants used are [Roederer, 1970] p2 M ¼ 2mB m ð3þ 2of19

3 Figure 2. (top) Minimum drift shell altitude and (bottom) trapped proton kinetic energy versus time for the indicated values of the three adiabatic invariants, M, K, and L, color coded by K. Z s 0 m K ¼ ½B m Bs ðþš 1=2 ds ð4þ s m L ¼ 2pm2000 E R E F where p is momentum, m is the proton rest mass, B is the local magnetic field, B m is the mirror point magnetic field, s is distance along a magnetic field line, the integration is along the magnetic field between mirror point locations s m and s m 0, F is the magnetic flux inside the drift shell, m 2000 E is the Earth s magnetic dipole moment in A.D. 2000, and R E is the Earth s radius. A constant dipole moment is necessary in equation (5) for L to be invariant as the geomagnetic field varies. The year A.D is an arbitrary choice and the L values do not correspond exactly with those computed using other dipole moments, ð5þ but L is a more convenient choice for the third invariant than F. The first two invariants, M and K, replace E and a in the evaluation of j. [11] The radiation belt model is calculated on a grid in M, K, and L that remains fixed as the geomagnetic field varies. The grid contains 15 L values from 1.1 to 2.4 and 50 K values from 0 to 1.70 G 1/2 R E, each with closer spacing at the lower end of the range, and 20 M values logarithmically spaced from 50 to 10 5 MeV/G. The kinetic energy range varies with K and L but is 10 MeV to 4 GeV. Each pair of K and L values corresponds to a guiding center drift shell. In many cases part of that drift shell is below the Earth s surface so that no calculations are necessary for those grid points. For example, maximum K values for which the entire drift shell is above the Earth in A.D are G 1/2 R E for L = 1.1 and 0.5 G 1/2 R E for L = 1.5. In addition, for each drift shell there is a maximum M value, 3of19

4 Figure 3. Drift averaged number densities of atmospheric neutral oxygen atoms (O + 2 O 2 ) versus K for A.D. 2005, L = 1.2, and F 10.7 = 140, color coded by M value. corresponding to the trapping limit, above which calculations are unnecessary. [12] Constant values of the adiabatic invariants correspond to varying drift shell locations and trapped particle energies because of the varying geomagnetic field. This is illustrated in Figure 2, which shows the minimum drift shell (or mirror point) altitude and proton kinetic energy E versus time for constant M and L and various constant K values. As shown, the current decrease in the Earth s dipole moment (Figure 1) leads to decreasing mirror point altitudes and increasing kinetic energies Drift Averaging [13] Several of the model inputs, such as atmospheric densities, are drift averaged prior to their use in calculating source or loss rates. This eliminates spatial dependencies for a given drift shell. The drift average of any quantity u is hui ¼ 1 Z s2 uds ð6þ s 2 s 1 s 1 where s is now the path length along a trapped proton trajectory and the total path length covered is s 2 s 1. The trajectories are calculated numerically by integrating the equation of motion for a given M starting from a mirror point that corresponds to a given K and L. The integral must cover at least one full drift around the Earth, but for high-energy protons the gyroradii may be sufficiently large that several drifts are required for a reliable average. The drift averages are repeated for each of the model M, K, and L values in each geomagnetic field model. [14] For drift shells with low mirror points (a few hundred km or less) the M dependence of the drift averages for a given K and L may be strong because of gyroradius effects. Therefore it is essential that the averages be taken over the true trajectories rather than just the guiding center trajectories. This is illustrated in Figure 3 which shows drift averaged neutral oxygen densities from the model atmosphere (see section 7.1) at L = 1.2 in the A.D magnetic field versus K for each of the model M values. At any given K the densities increase substantially with increasing M, caused by the protons with larger gyroradii reaching deeper into the atmosphere Trapping Limits [15] The maximum trapped particle rigidity R, or momentum per charge, for a given guiding center drift shell is also calculated by numerical integration of particle trajectories. For protons, the maximum R corresponds to a maximum M for a given K and L. A trajectory is followed until one of three conditions is satisfied: (1) it completes 12 full drifts around the Earth, (2) it reaches below the surface of the Earth, or (3) it reaches an altitude at least two gyroradii below the minimum altitude of the guiding center drift shell. In the first case it is considered to be trapped; in the latter two cases it is considered not to be. If a trapped trajectory is found then the initial gyrophase of the trajectory is changed by 180 (from radially outward to inward) and the calculation repeated with the same R value, the results of this second calculation being conclusive. A binary search determines the maximum R. Results of this procedure are shown in Figure 4 for 4of19

5 Figure 4. Calculated trapping limits, or maximum trapped proton kinetic energies, versus L shell for A.D. 2005, color coded by K value. Also shown are data points representing the energy at which the proton intensity observed at SAMPEX is 10% of its peak value, a lower bound on the trapping limit [Looper et al., 1996]. The data points are labeled with their approximate K values for each L. the A.D geomagnetic field model, where the maximum rigidity is converted to proton kinetic energy. The calculation predicts that the highestenergy trapped protons are in the L range of 1.2 to 1.4 with E 3 to 4 GeV for K = 0. For protons with the highest K values, or lowest mirror point altitudes, the trapping limit is a factor 10 lower at a given L. Similar results are obtained for each of the field model epochs. [16] Also shown in Figure 4 are the outer limits of the proton radiation belt at which the intensity is 10% of the peak intensity as determined from data taken on the SAMPEX satellite [Looper et al., 1996]. These points are expected to be close to, but somewhat below, the trapping limit. The SAMPEX altitude of 600 km means that protons are observed only with relatively high K values as shown in Figure 4. As expected, the calculated trapping limits are somewhat higher than the SAMPEX data points for comparable K values. The difference is more significant for the higher L values (L ^ 2) and it is likely that the calculated values there are too high, meaning that the trapped proton model would extend too high in energy for that region. [17] Loss of trapping with increasing R may be a result of either the gyroradius becoming so large that the trajectory intersects the solid Earth (condition 2), or the trajectory becomes nonadiabatic (condition 3). The latter case is characterized by a varying mirror point altitude so that the trajectory will also eventually intersect the Earth. The transition to nonadiabatic motion is not well resolved by the trapping criteria defined above, being somewhat sensitive to a trajectory s initial gyrophase angle (hence the 180 reversal described above). This accounts for the fluctuations of the curves in Figure 4. A more accurate description of this transition would be a decreasing lifetime, t in equation (1), with increasing R. For simplicity a sharp transition is adopted and the calculated maximum R as a function of K and L provides the upper energy, or M, limit for the radiation belt model. 4. Trapped Proton Sources [18] The total source rate is the net effect of cosmic ray albedo neutron decay (CRAND), solar proton injection, and radial diffusion: S ¼ S n þ S p þ S D 4.1. CRAND [19] Cosmic ray albedo neutron decay (CRAND) is the principal trapped proton source at high energies (E ^ 100 MeV) [Singer, 1958; Farley and Walt, 1971; Jentsch, 1981]. The source rate is S n ¼ hj ni vgt n where j n is the neutron intensity which is drift averaged according to equation (6), t n = 887 s is the neutron decay lifetime, and g is the relativistic time dilation factor for a given neutron energy. The neutron intensity j n is evaluated ð7þ ð8þ 5of19

6 Figure 5. (left) Sample albedo neutron omnidirectional energy spectra calculated for incident isotropic and monoenergetic H and He with the indicated kinetic energies. (right) Albedo neutron zenith angle distributions for the same incident H energy (He not shown), color coded by the neutron kinetic energy. at the geographic location where the negative tangent to the trapped proton trajectory intersects the top of the atmosphere (100 km altitude). If there is no intersection for a given point on the trajectory then j n = 0. This assumes that neutron decay does not appreciably reduce the neutron flux in the radiation belt region and that the resulting proton attains the full kinetic energy and moves in the same direction as the decaying neutron, both good approximations. Calculating the drift average of j n in this fashion avoids the necessity for including an approximate geometric injection coefficient [Dragt et al., 1966] Neutron Albedo [20] The albedo neutron flux as a function of neutron kinetic energy E n and zenith angle z is j n ðe n ; z; R cv ; WÞ ¼ X i Z N i ðe 0 ; E n ; zþt i ðe 0 ; R cv Þj GCR i ðe 0 ; WÞdE 0 where R cv is the geomagnetic vertical cutoff rigidity, W is sunspot number, N i is the neutron energy and zenith angle distribution for a given incident cosmic ray energy E 0, T i is the cosmic ray geomagnetic transmission as a function of E 0 for a given R cv, and j i GCR is the galactic cosmic ray energy spectrum for element i parameterized by W. [21] The albedo function N i was obtained for incident GCR H and He from Monte Carlo simulations using the particle transport code Geant4 (version 7.1.p01) [Agostinelli et al., 2003]. The GCRs were sampled from an ð9þ isotropic distribution at 23 selected kinetic energies from 20 MeV to 100 GeV for H and at 16 selected kinetic energies from 20 MeV/nucleon to 7 GeV/nucleon for He (the He energy range was artificially extended to 100 GeV/ nucleon by using the proton results at equivalent energy per nucleon). The atmosphere on which the particles were incident was represented in this simulation by flat layers 1 km thick up to 400 km altitude. The composition of each layer was taken from the model atmosphere (see section 7.1) at 0 latitude and F 10.7 = 140 because the atmosphere does not vary significantly with location or solar input at the lower altitudes (100 km) where most of the interactions take place. The transport and interaction models used in the Geant4 simulation were those of the SLAC Space Electronics Physics List (D. Wright, physics_lists/micro/physlistdoc.html). The distributions N i of upward albedo neutrons were compiled and examples are shown in Figure 5. [22] Sample albedo neutron spectra obtained from equation (9), combining the results of the Monte Carlo simulation with the GCR spectra (see section 4.1.2) at a given geomagnetic cutoff, are shown in Figure 6. Vertical neutron flux, R j n cos z sinz dz, is shown for comparison with measurements taken at a fixed geomagnetic latitude [Kanbach et al., 1974; Preszler et al., 1976]. The CRAND source for the inner radiation belt comes mostly from low geomagnetic latitude, or high cutoff rigidity, where neutron data are not available. The lowlatitude neutron flux is relatively insensitive to solar 6of19

7 Figure 6. Calculated albedo neutron vertical flux versus kinetic energy for indicated values of the vertical geomagnetic cutoff rigidity, color coded by F Also shown are data for comparison with the calculations at the 4.83 GV cutoff. cycle modulation, parameterized by solar F 10.7 (see section 7.3), because of the high cutoff rigidity Galactic Cosmic Rays [23] Only GCR H and He are included in the albedo calculation. Their energy spectra are a simplification of the ISO model [International Organization for Standardization, 2002]: j GCR i ¼ A ic i b ai 1 Z i 1GV gi R R 5:5 ð10þ RþR 0 where A i and Z i are the nuclear mass and charge numbers respectively, b = v/c, c is the speed of light, R is rigidity, R 0 = W 1.45 GV, and, for {H, He}, C i = { , } (m 2 sr s GeV/nuc) 1, g i = {2.74,2.77}, and a i = {2.85, 3.12}. Spectra for selected sunspot numbers W are shown in Figure Geomagnetic Cutoff [24] The cutoff rigidity may be approximated using Størmer s dipole formula [Størmer, 1955; Smart and Shea, 2005] m R c ¼ E c cos 4 l h i 2 ð11þ r 2 1 þ ð1 sin z sin x cos 3 lþ 1=2 where m E is Earth s dipole moment, r is the distance from the dipole center, l is the magnetic latitude, z is the zenith angle, and x is the azimuthal angle measured clockwise from magnetic north. The vertical cutoff is obtained by setting z =0: R cv ¼ m Ec 4R 2 E L2 m ð12þ where, to improve accuracy, the McIlwain L shell [McIlwain, 1966; Roederer, 1970], L m, replaces the dipole L shell, L d = r/(r E cos 2 l). The L m is calculated using the full multipole geomagnetic field model of degree 10 and the dipole moment m E appropriate to each epoch for geographic locations at the 100 km altitude of the albedo simulations. Sample maps of R cv for A.D and A.D are shown in Figure 8. The A.D map is in reasonable agreement with more accurate results obtained by trajectory tracing techniques [Smart and Shea, 2005]. Cutoff maps are calculated similarly for each of the field model epochs. [25] The transmission function is approximated from the solid angle of allowed trajectories T i ¼ 1 ð 2 1 þ sin x SCÞ ð13þ where x SC is the Størmer cone angle obtained from equation (11) by setting sin z = 1 and solving for sin x. This neglects the penumbra formed by trajectories intersecting the Earth that would otherwise be allowed [e.g., Smart and Shea, 2005] Solar Protons [26] Observations show that solar protons and heavy ions are injected into the radiation belt to L ^ 2 during 7of19

8 Figure 7. Model galactic cosmic ray (GCR) energy spectra for H (solid curves) and He (dashed curves), color coded by sunspot number W. Figure 8. Contour maps of the vertical geomagnetic cutoff rigidity, labeled in GV, for (top) A.D and (bottom) A.D of19

9 Figure 9. Cumulative interplanetary solar proton fluence for kinetic energy E > 30 MeV versus time. The fluence is reset to zero when approaching the upper limit. Data are from McCracken et al. [2001] for times prior to A.D and from Shea and Smart [1990], Mewaldt et al. [2005], and Reedy [2006] for later times. large solar proton events and magnetic storms [Lorentzen et al., 2002; Hudson et al., 2004; Mazur et al., 2005]. Trajectory tracing simulations show how solar ion trapping and transport to low L occurs [Hudson et al., 1997, 2004; Kress et al., 2005] but statistical studies of the injection efficiency as a function of L, E, and equatorial pitch angle are not yet available. The solar proton source rate is approximated as S p ðe; t 8 1 X < F i ðeþdðt t i Þ for L 2 Þ ¼ vt p : i 0 for L < 2 ð14þ where the summation extends over all solar proton events, F i is the fluence energy spectrum for event i at time t i, and d is the Dirac delta function. The trapping timescale t p is adjusted to provide reasonable agreement (see section 8) between the final model and empirical trapped proton intensities: t p ¼ E 2:5 60 s ð15þ 1 MeV [27] The energy dependence of t p leads to an injected solar proton spectrum that is considerably softer than the interplanetary source spectrum, as is observed qualitatively for solar ion injections [Mazur et al., 2005]. The softening may be a result, at least in part, of the arrival time of the solar protons at Earth relative to the interplanetary shock, with most of the high-energy protons arriving prior to the shock and subsequent trapping. [28] A trapping efficiency may be defined from the estimated trapping timescale as = t B /t p. It represents the approximate fraction of solar protons reaching low L that become trapped. For E = 10 and 100 MeV the trapped proton bounce periods at L =2aret B 1.6 and 0.5 s, corresponding to 10 4 and 10 7, respectively. [29] The measured solar proton cumulative integral fluence for E > 30 MeV since A.D is shown in Figure 9. Data for times prior to A.D are derived from nitrate-rich layers in polar ice [McCracken et al., 2001] and for later times are observed indirectly from Earth or directly from spacecraft [Shea and Smart, 1990; Mewaldt et al., 2005; Reedy, 2006]. Steps at the times t i of each solar proton event show that most of the fluence is accumulated from large events which are also those most likely to reach the low L shells (L 2) and be injected efficiently into the inner radiation belt. This provides some justification for the simplified source function in equation (14). [30] For most of the large solar proton events since A.D (20 events) the fluence spectra F i (E) are obtained by fits to individual event data from the ACE, GOES, and SAMPEX spacecraft using functional forms and methods described by Mewaldt et al. [2005]. For the rest of the known large events since A.D. 1560, the fluences of Figure 9 are used to normalize an average spectrum. It was obtained by fitting a double power law form to the observed total fluence spectrum in the energy range of 0.1 to 200 MeV using data from ACE and GOES 8, 10, and 11 in the period from September, A.D through A.D. 2005, which includes all major events from solar cycle 23. Prior to A.D the same average spectral shape is 9of19

10 combined with a constant flux density of 25.6 cm 2 s 1 for E > 30 MeV obtained by averaging the fluence rate over the entire interval of Figure 9. The resulting average solar proton intensity in (cm 2 s sr MeV) 1 is ( j p ¼ 41:0 E 1:458e E=35:79 MeV 1MeV for E 111:02 MeV 4: E 4:56 1MeV for E > 111:02 MeV ð16þ and equation (14) is replaced by S p (E,t) =j p /(vt p ) for L 2 at those early times. [31] Solar proton albedo neutron decay (SPAND) is an additional source of inner zone trapped protons. By replacing j GCR i in equation (9) with the average solar proton intensity j p (and similarly for the solar He intensity) it is easy to show that the average SPAND source is insignificant relative to CRAND because of the relatively soft solar proton spectrum, particularly so at the higher geomagnetic cutoff rigidities [see also Dragt et al., 1966]. Therefore it is not included in the model Radial Diffusion [32] Radial diffusion redistributes trapped protons in L, but in terms of the transport equation solution (equation (2)) it is treated as a source that may be either positive or negative depending on the radial dependence of the phase space density f = j/p 2 [e.g., Schulz and Lanzerotti, 1974]: S D ¼ p2 D LL ð17þ where the partial derivatives are at constant M and K. The diffusion coefficient is from on the empirical estimate of Jentsch [1981]: D LL ¼ L 9 sin 2:7 1 MeV a 0 3: s 1 ð18þ E where sin 2 a 0 = B 0 /B m, a 0 is the equatorial pitch angle, and B 0 is the equatorial magnetic field which is the same as B m for K =0. [33] The diffusion source term S D is evaluated at each time step using the j values from the partial evaluation of the t 0 integral in equation (2) up to that time step (with a correction factor to appropriately change the upper limit of the t 00 integral). This requires that the solutions for all M and L values be compiled simultaneously with interpolation in energy to evaluate the partial derivatives at constant M. The solution is stable for sufficiently small time steps. The boundary conditions, used in the numerical evaluation of the radial derivatives, are f = 0 beyond the inner and outer model L values (in some cases f =0at interior points because of the trapping limits). 5. Gradual Energy Change [34] The total rate of gradual energy change is the sum of ionization and free electron losses, and of adiabatic heating or cooling: de dt ¼ de dt n þ de dt þ de e dt B ð19þ 5.1. Ionization Energy Loss [35] The energy loss rate due to ionization of the neutral atmosphere is de v dt n¼ X de hn i im i dx ð20þ i i where n i is the atom number density of atmospheric component i with atomic mass M i, and x i is path length times mass density. Tables of de/dx i as a function of incident proton energy on a target gas of each atmospheric constituent are obtained from the SRIM-2003 software package [Ziegler, 2004] Free Electron Energy Loss [36] The energy loss rate to free electrons from the ionosphere and plasmasphere is calculated in the highvelocity quantum limit [deferrariis and Arista, 1984, equations (1) and (8a)]: de ¼ w2 p Z2 e 2 ln 2m ev 2 dt e v hw p ð21þ where w p =(4phn e ie 2 /m e ) 1/2 is the drift averaged plasma frequency, n e is the plasma (or free electron) number density, e is the electron charge, m e is the electron rest mass, h is the reduced Planck s constant, and Z =1is the incident proton charge number Adiabatic Energy Change [37] The rate of energy gain or loss by, respectively, adiabatic compression or expansion of a drift shell in the varying geomagnetic field is obtained from conservation of the first adiabatic invariant: de ¼ dt B EEþ2mc2 2ðE þ mc 2 Þ d ln B m dt ð22þ [38] The mirror magnetic field B m is known for each drift shell in each magnetic field model and a linear rate of change d ln B m /dt is assumed between the field models before and after any given time t. Earth s currently decreasing dipole moment (Figure 1) causes adiabatic heating [Heckman and Lindstrom, 1972; Schulz and Paulikas, 1972]. 10 of 19

11 Figure 10. Total inelastic cross sections for protonnucleus reactions color coded by element (from Meyer [1972] for H and from Letaw et al. [1983] for heavier elements). 6. Nuclear Scattering [39] Radiation belt protons are lost by nuclear reactions with nuclei in the neutral atmosphere, ionosphere, or plasmasphere. The lifetime t is given by 1 t ¼ v X i hn i is i ð23þ where the sum is over all of the elements in the atmosphere and plasma density models (see section 7), n i is the combined number density for nuclei of element i, and s i is the total inelastic scattering cross section for protons of a given energy incident on those nuclei. The cross sections are shown versus proton energy in Figure 10 [Meyer, 1972; Letaw et al., 1983]. [40] Other scattering processes may be considered: the same inelastic nuclear reactions are a source of lowerenergy protons, and elastic nuclear scattering is both a source and loss (that is, trapped protons are redistributed in energy and pitch angle). While losses by inelastic scattering are significant, these other processes are assumed to be negligible [Dragt, 1971] and are not included in the model, but they warrant further study for more detailed calculations. 7. Neutral and Plasma Densities [41] Neutral densities are used in the calculation of neutron albedo, ionization energy loss, and nuclear scattering; plasma densities are used in the calculation of free electron energy loss and nuclear scattering. Prior to the energy loss and scattering calculations, densities are drift averaged for each M, K, and L in each geomagnetic field model Neutral Atmosphere [42] Neutral densities are from the NRLMSISE-00 model atmosphere [Picone et al., 2002]. This provides number densities for atmospheric H, He, N, N 2,O,O 2, and Ar. To simplify its evaluation, annual, local time, and daily variations are turned off, and Ap is fixed at an average value of 4. The parameterization by solar 10.7 cm radio flux, F 10.7, is retained to represent the solar cycle dependence of the atmospheric densities using a discrete set of 5 values for both the daily and 80-day average F 10.7 inputs. The densities are evaluated as a function of geographic altitude and latitude, averaging over longitude. Sample neutral densities versus altitude are shown in Figure 11 for each of the selected F 10.7 values. After drift averaging, densities are obtained for any given time and corresponding F 10.7 value by interpolating between geomagnetic field model epochs and within the set of discrete F 10.7 values using the logarithm of the densities Ionosphere and Plasmasphere [43] Plasma densities are obtained from the GCPM model of the ionosphere and plasmasphere [Gallagher et al., 2000]. It provides number densities of H +,N +, and O +, with the sum of these being the plasma electron density. They are evaluated as a function of altitude and latitude, averaging over magnetic local time, with the same F 10.7 parameterization as for the neutral model. Sample densities versus altitude are shown in Figure Solar Activity [44] The F 10.7 solar radio flux parameterizes the neutral atmosphere and plasma densities. Direct observations of Figure 11. Number densities of atmospheric neutral N 2 (solid curves) and H (dashed curves) versus altitude, color coded by F 10.7, from the NRLMSISE-00 model [Picone et al., 2002]. Other neutral species included in the calculations are listed. 11 of 19

12 averages are derived from reconstructed sunspot numbers obtained from tree ring 14 C data [Solanki et al., 2004]. A compilation of the F 10.7 values from all of these sources is shown versus time in Figure 13. Figure 12. Number densities of H + (solid curves), O + (dashed curves), and He + (dash-dotted curves) ions versus altitude in the ionosphere and plasmasphere color coded by F 10.7, from the GCPM model [Gallagher et al., 2000]. The model plasma electron density is the sum of these ion densities. F 10.7 are available since A.D and monthly averages are used. For the period from A.D to A.D. 1947, yearly averages are derived from the directly observed sunspot number W using the statistical relationship (NorthWest Research Associates, com/spawx/comp.html). F 10:7 ¼ 63:74 þ 0:727W þ 0:000895W 2 ð24þ This is also used to convert the parameterization of the GCR spectrum (equation (10)) from W to F (Both F 10.7 and W are provided by NGDC gov/stp/solar).) Prior to A.D. 1600, F 10.7 decadal 8. Results [45] Trapped proton energy versus time (E-t) paths are obtained by integrating equation (19) over time. Samples are shown in Figure 14 for selected L and K values. Generally, the timescale for energy loss is faster, due to higher drift averaged densities, at lower L, higher K, and higher F It is also faster at lower M (or E) because of the nature of the ionization and free electron loss. Trapped proton residence times neglecting scattering effects, that is, the length of the E-t paths, vary from less than a year to 4000 years. Fluctuations in the energy can be seen on decadal timescales, caused by solar cycle variations, and on century or longer timescales, caused by geomagnetic field variations. The E-t paths begin at the trapping limit for the current t and end at energies corresponding to the M values chosen for evaluation of the model intensity. The paths are stored for later integration of the source and loss functions to evaluate the proton intensity according to equation (2). The paths begin at times not much earlier than 1700 B.C. because the particularly low, and subsequently increasing, geomagnetic dipole moment at that epoch (Figure 1) causes rapid energy loss and it is therefore a convenient starting point for the required model inputs. [46] To evaluate the model intensity at a given time, E-t paths are calculated for all L, K and M values in the model and the integrations carried out over those paths. This is repeated at each time for which the model is to be evaluated. Sample results in Figure 15 show intensity versus time for selected L, K, and M values during the period from A.D Figure 13. Solar F 10.7 versus time including monthly averages of the directly observed data since A.D. 1947, yearly averages of the sunspot number reconstructed data since A.D. 1600, and decadal averages of the 14 C reconstructed data for the previous 10,000 years [Solanki et al., 2004]. 12 of 19

13 Figure 14. Calculated trapped proton kinetic energies versus time for the indicated constant L and K values. Colors simply distinguish the curves. The energy rate of change de/dt includes atmospheric ionization, energy transfer to free electrons, and adiabatic heating or cooling caused by the geomagnetic secular variation. to A.D covering approximately 3 solar cycles. The F 10.7 is also shown for comparison. [47] The intensity at L = 1.2 is seen to vary with the solar cycle, being highest near solar minimum (low F 10.7 ) and lowest near solar maximum (high F 10.7 ). These variations are most significant at the higher K values for which the proton energy loss rates are less than, or comparable to, a solar cycle timescale (Figure 14). The smoother curves at lower K values and higher intensity values represent integrations of the source and loss rates over timescales longer than a single solar cycle. As this timescale increases the intensity maximum is delayed relative to the time of solar minimum because of the longer proton residence times [Dragt, 1971], an effect that has also been observed in low-altitude satellite data [Huston and Pfitzer, 1998]. In addition to the solar cycle and shorter-timescale variations caused by changes in F 10.7, there is a gradual trend of decreasing intensity caused by the lowering mirror point altitudes and increasing energies (Figure 2). This trend would not be apparent if the location and energy were constant rather than M, K and L. [48] The intensity at L = 2 does not display any solar cycle variations except at the highest K values where the energy loss timescales are short enough. Instead, the intensity varies with the sudden arrival of solar proton injections and their subsequent decay due to radial diffusion and the other loss mechanisms. The injections are more apparent at lower K values for a fixed M because of the correspondingly lower E. In reality the efficiency of solar proton injections is variable, so that unusually intense injections such as that observed in March, 1991 [Mullen et al., 1991; Hudson et al., 1997] are not accurately modeled. [49] At L = 1.5 the intensity shows solar cycle variations at high K values and is relatively steady at the lower K values where the energy loss timescales are long (Figure 14). There are some slow variations at low K caused by the delayed inward diffusion of the solar proton injections that were seen much earlier at L =2. [50] Sample energy spectra for selected L and K values at a fixed time, A.D. 2005, are shown in Figure 16. Energy spectra for K = 0, that is, for equatorially mirroring protons, and for selected L values are shown in Figure 17. Also shown there for comparison are spectra calculated without the solar proton source (dashed curves) and equatorial spectra for the same L values obtained from the AP-8 MIN empirical model. [51] The CRAND source generally produces hard spectra at low L relative to the softer solar proton injected spectra at higher L. In Figure 16 the solar proton spectra 13 of 19

14 Figure 15. Model trapped proton intensities versus time for a constant M = MeV/G for three indicated L values, color coded by K value. The monthly averaged F 10.7 is shown at the bottom. are seen at L = 1.7 for E ] 100 MeV having diffused inward from higher L, with the harder CRAND spectra at higher E. The spectra at L = 1.2 are solely from CRAND. This variation with L is seen more clearly in Figure 17 where the CRAND spectra may be compared directly with the those including both CRAND and inward diffusing solar protons. [52] The AP-8 spectra at L = 1.3 and 1.4 have greater relative intensity at low energies than the corresponding model spectra, suggesting that the solar proton source is more significant for those L values than the model predicts. This may be indicative of a model radial diffusion coefficient (equation (18)) that is too small at low L. [53] To investigate the significance of the varying geomagnetic field, ratios of trapped proton intensities calculated with and without the geomagnetic secular variation are shown versus E for A.D. 2005, L = 1.3, and selected K values in Figure 18. For the case without secular variation the A.D geomagnetic field was used throughout the calculation. The intensities are generally higher with the varying geomagnetic field because the decreasing dipole moment during the last century (Figure 1) caused energization (equation (22)) and contraction of the drift shells [Farley et al., 1972]. Larger drift shells at earlier times had lower drift averaged densities and correspondingly smaller energy and scattering loss rates. The effect is greater at higher E (or M) and higher K values because the intensity is more sensitive to the contracting drift shells, even though the residence times are shorter. A small difference in M or K at those high values can significantly change the residence time (see the top plot of Figure 14) which explains the sometimes erratic changes in the intensity ratio of Figure 18. It is apparent 14 of 19

15 Figure 16. Model trapped proton energy spectra for A.D at two indicated L values, color coded by K value. that the inclusion of the geomagnetic secular variation is essential for obtaining accurate trapped proton intensities. [54] Intensity versus L for selected M values at a fixed time, A.D. 1990, and for K = 0 are shown in Figure 19. Also shown is the fraction of the total intensity that is due to CRAND, where the CRAND intensity is calculated without the solar proton source. The separation at the higher L values between the injected solar protons with the soft spectrum at low M and the CRAND protons with the hard spectrum at high M is apparent. The CRAND source is dominant over the solar proton source for all L at high M, while for low M the solar protons are seen to diffuse into L 1.3 [Jentsch, 1981]. The transition at high L from solar proton dominated to CRAND dominated is at E 100 MeV (Figure 17). The CRAND population is generally steady while the solar proton component varies considerably at the higher L values as a result of the solar proton injection history (Figure 15), reflected for the particular time of Figure 19 in the shapes of the low M curves. As the solar protons diffuse inward their intensity represents an average over longer periods so that the details of particular injection events are lost. [55] Radial diffusion is generally thought of as a slow process for the inner radiation belt because of the small value of D LL at low L (equation (18)), with timescales 100 to 1000 years (Figure 14) [Schulz and Lanzerotti, 1974]. Comparable timescales for the other source and loss processes allow solar protons injected with low K to reach low L. For high K the timescales of the other processes are shorter so that radial diffusion is less significant. However, near the inner L boundary for any given K and M, the radial phase space density L is large and the 15 of 19

16 Figure 17. Equatorial (K = 0) trapped proton energy spectra color coded by L value from 1.2 to 1.8. (left) The model spectra for A.D are calculated with (solid curves) and without (dashed curves) the solar proton source. (right) The AP-8 MIN empirical model spectra are shown for comparison. rate of radial diffusion along this gradient is significant despite the small value of D LL. Such steep gradients, seen at low L for K = 0 (Figure 19), exist at higher L for higher K values. They result in higher intensity and less solar cycle variability at low mirror point altitudes than would exist without radial diffusion. This explains the smooth curves at the highest K values for L = 1.2 in Figure 15, which resemble those at much lower K. Figure 18. Ratios of model trapped proton energy spectra calculated with and without geomagnetic secular variations, for A.D and L = 1.3, color coded by K value. 16 of 19

17 Figure 19. (top) Model equatorial (K = 0) trapped proton intensities versus L for A.D. 1990, color coded by M value. (bottom) Fraction of the intensity that is due to CRAND rather than solar proton injection. 9. Discussion [56] The long residence times of inner zone trapped protons (Figure 14) require integration of the source and loss terms over 1000 year timescales. Such a calculation may be fraught with uncertainty [Schulz and Lanzerotti, 1974, p. 112], but recent advances in our knowledge of the historical geomagnetic field (Figure 1), of solar activity records (Figure 13), and of solar proton events (Figure 9), as well as improved atmosphere and plasma density models (Figures 11 and 12), and nuclear transport codes (Figure 5), have at least made it feasible. Such long integration times are not required in all cases. For mirror point altitudes at or below a few hundred km the residence times may be comparable to or shorter than a solar cycle, the model inputs are relatively well known, and the trapped proton intensities vary inversely with the solar activity level (Figure 15). In such instances a solution of a steady state transport equation using a suitable average solar activity level input (F 10.7 ) would suffice. For the equatorial or high-altitude proton intensities at the heart of the inner radiation belt (L 1.2 to 1.8) and at high kinetic energies (E ^ 100 MeV) the time-dependent solutions with long integration times are required. The calculations described above represent a first attempt to construct an accurate theoretical model of these trapped protons. [57] The sensitivity of the model output to uncertainties in the various inputs is a complex issue because of the varying timescales and spatial regimes involved. A detailed study has not been attempted but the issue is illustrated in the case of the geomagnetic field input by a comparison with a calculation using an unvarying field model (Figure 18). The two calculations can differ by factors up to 10 depending in a complex fashion on the M, K, and L values. An accurate geomagnetic history is required in many cases. [58] Statistics of solar proton injections into the radiation belt are not well known and a simple model was adopted. The requirement for reasonable long-term low-energy equatorial trapped proton intensities compared to empirical estimates (Figure 17) resulted in an estimated energydependent average trapping efficiency that can be compared with observations from specific solar proton events. In reality the efficiency varies with L, proton energy, solar wind and geomagnetic conditions [Hudson et al., 2004; Mazur et al., 2005], so that specific events may lead to trapping with higher or lower intensity than predicted. [59] The calculated and empirical (AP-8) equatorial trapped proton intensities (Figure 17) are in reasonably agreement at low E or at low L, but the calculated values are somewhat higher at high E and high L. The 17 of 19

18 discrepancy is most evident at energies of several hundred MeV and greater where the model spectra are harder than the empirical spectra. The AP-8 model may be inaccurate at these energies for which the intensities are extrapolated above the maximum observation energies (AP-8 is thought to underestimate proton intensity for E > 10 MeV [Lauenstein and Barth, 2005]). It is also possible that the calculated CRAND source is too strong, or there is a missing loss mechanism. Losses due to geomagnetic storms are not included in the model calculation and are likely to be a significant factor at the higher L values (L ^ 2) [Vacaresse et al., 1999; Looper et al., 2005; Mazur et al., 2005]. It is also likely that a more accurate radial diffusion coefficient would lead to better agreement at lower L values [Albert et al., 1998]. [60] The present model may be improved in the future by the addition of physical processes such as elastic nuclear scattering. Empirical inputs such as the radial diffusion coefficient and the solar proton injection efficiency are examples of parameters that could be improved by data assimilation techniques, leading to more accurate modeled intensities. Such techniques may also help to overcome shortcomings due to unmodeled processes such as losses during geomagnetic storms, or new physical insight into such processes may be obtained. Such improvements could lead to a practical environmental model of the inner radiation belt. [61] Acknowledgments. For help and advice we thank M. Hudson, D. Gallagher, G. Ginet, M. Korte, B. Kress, J. Mazur, P. O Brien, M. Schulz, M. Shea, and D. Smart. The work at Aerospace was supported by NSF grant ATM under the National Space Weather Program. The work at Caltech was supported by NASA under grants NNG04GB55G and NAG References Agostinelli, S., et al. (2003), Geant4---A simulation toolkit, Nucl. Instrum. Methods Phys. Res., Sect. A, 506, Albert, J. M., G. P. Ginet, and M. S. 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