Monte Carlo simulations of interplanetary transport are employed to study adiabatic ...... technique among those applicable to a strongly non-uniform system.
ADIABATIC DECELERATION OF SOLAR ENERGETIC PARTICLES AS DEDUCED FROM MONTE CARLO SIMULATIONS OF INTERPLANETARY TRANSPORT L. KOCHAROV1 , R. VAINIO1 , G. A. KOVALTSOV2 and J. TORSTI1 1 Space Research Laboratory, Turku University FIN-20014, Finland 2 Ioffe Physical-Technical Institute, St. Petersburg 194021, Russia
(Received 4 December, 1997; accepted 26 March 1998)
Abstract. Monte Carlo simulations of interplanetary transport are employed to study adiabatic energy losses of solar protons during propagation in the interplanetary medium. We consider four models. The first model is based on the diffusion-convection equation. Three other models employ the focused transport approach. In the focused transport models, we simulate elastic scattering in the local solar wind frame and magnetic focusing. We adopt three methods to treat scattering. In two models, we simulate a pitch-angle diffusion as successive isotropic or anisotropic small-angle scatterings. The third model treats large-angle scatterings as numerous small-chance isotropizations. The deduced intensity–time profiles are compared with each other, with Monte Carlo solutions to the diffusion-convection equation, and with results of the finite-difference scheme by Ruffolo (1995). A numerical agreement of our Monte Carlo simulations with results of the finite-difference scheme is good. For the period shortly after the maximum intensity time, including deceleration can increase the decay rate of the near-Earth intensity essentially more than would be expected based on advection from higher momenta. We, however, find that the excess in the exponential-decay rate is time dependent. Being averaged over a reasonably long period, the decay rate of the near-Earth intensity turns out to be close to that expected based on diffusion, convection, and advection from higher momenta. We highlight a variance of the near-Earth energy which is not small in comparison with the energy lost. It leads to blurring of any fine details in the accelerated particle spectra. We study the impact of realistic spatial dependencies of the mean free path on adiabatic deceleration and on the near-Earth intensity magnitude. We find that this impact is essential whenever adiabatic deceleration itself is important. It is also found that the initial angular distribution of particles near the Sun can markedly affect MeV-proton energy losses and intensities observed at 1 AU. Computations invoked during the study are described in detail.
1. Introduction It is recognized that interplanetary transport effects must be taken into account at any analysis of solar particle data. The diffusion-convection equation provides a classical groundwork for the analysis (Parker, 1965). The diffusion-convection equation considers, among other processes, the effect of adiabatic deceleration on the cosmic-ray distribution. This treatment assumes a nearly-isotropic particle distributions. For anisotropic distributions, a framework of focused transport can be used (Earl, 1976, 1984). A general description of adiabatic deceleration of cosmicray particles in the focused transport theory was recently given by Ruffolo (1995), Solar Physics 182: 195–215, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.
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who also provided a brief but comprehensive review of previous publications. Ruffolo employed a finite-difference method to deduce solutions for a single set of parameters and for a very limited time interval. Different numerical methods may be adopted to solve the diffusion-convection equation or to simulate the focused transport. In many studies of the particle transport and acceleration, the Monte Carlo method was employed (e.g., Palmer and Jokipii, 1981; Ellison, Jones, and Reynolds, 1990; Earl et al., 1995). However, to our knowledge, no Monte Carlo simulations of solar wind effect on the focused transport in the Archimedean magnetic field have been reported before. Ruffolo (1995) found that the proton intensity decay due to ‘switching on’ deceleration can be 75% faster than one would expect on the basis of momentum advection alone, because of an interplay between deceleration and diffusion. This finding raises questions about how important may the precise calculation of adiabatic deceleration be for analysis of real experimental data, and what numerical models should be used. We address these issues in the present paper. In this work we present four alternative numerical codes employing Monte Carlo simulations to trace particle propagation in the interplanetary medium. We consider both the traditional diffusion-convection model and three models based on the focused transport. We compare results of our Monte Carlo simulations and results of the finite-difference scheme of Ruffolo (1995). Finally, the sensitivity to model parameters is studied. Being experimentally oriented, our calculations incorporate a realistic spatial dependency of the particle mean free path and a longer time interval than that hitherto calculated by means of a finite-difference method. We also study the effect of initial angular distribution of particles near the Sun on the intensity–time profiles observed near the Earth.
2. Monte Carlo Solutions to the Diffusion-Convection Equation A Fokker–Planck or diffusion-convection equation of nonrelativistic particles (Parker, 1965; Dolginov and Toptyghin, 1966) is of the form 4u ∂ 1 ∂ 1 ∂ Q(E) ∂F ∂F 2 δ(r − r0 )δ(t) , = 2 r Dr − 2 (r 2 uF ) + (EF ) + ∂t r ∂r ∂r r ∂r 3r ∂E 4π r 2 (1) where F is the number of particles per unit of volume and unit of energy, Dr = D cos2 9 is the radial diffusion coefficient (a cross-field diffusion is neglected), 9 is the angle between the radius vector r and the magnetic field B, D = λv/3 is the parallel diffusion coefficient, E, v, and λ are the particle kinetic energy, velocity, and mean free path, respectively, u is the solar wind velocity, and Q(E) is the particle spectrum at an instant source positioned at the radial distance r = r0 . This equation describes transport of particles along an interplanetary magnetic field line (Figure 1), when the variable r is employed instead of the spiral distance along the
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u ψ
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Figure 1. Illustration of the interplanetary magnetic field line, the solar wind velocity and the variables used.
magnetic field line, z (e.g., Kunow et al., 1991). Scattering must be strong enough to cause the particles’ pitch-angle distribution to be nearly isotropic. Along with scattering, magnetic focusing is hidden in the 3D-diffusion-type operator, because, in the case of Parker’s model of magnetic field, ∂ ∂N ∂ D A ∂ 2 ∂F r Dr = D − N, 2 r ∂r ∂r ∂z ∂z ∂z L
(2)
where A is the cross-sectional area of a tube of magnetic flux, N is the number of particles per unit of magnetic line length and unit of energy, and L = −B/(dB/dz) is the focusing length. From the microscopic point of view, the last term in (2) describes streaming due to the effect of focusing. The conservation of the first adiabatic invariant results in an increase of the velocity component parallel to the magnetic field: dvk /dt = v⊥2 /(2L). Over one length of the mean free path, particles get the average parallel velocity, U = (hv⊥2 i/v) (λ/L)/2, due to the focusing. Hence, the term, D/L = λv/(3L), is recognized to be a bulk velocity of particles caused by focusing. The effect of focusing of solar cosmic rays is always essential when compared to scattering, no matter how small the value of the mean free path may be (Kocharov et al., 1996). Note that it is equally possible to consider diffusion with respect to the spatial variable z, so that the focusing term may be directly seen in the transport equation (Equation 10 of Earl, 1984). As long as the transport equation (1) takes the form of a Fokker–Planck equation, one can apply standard
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numerical methods to solve it; in particular, the very flexible Monte Carlo method may be used (Gardiner, 1985). The finite-difference solutions of Ruffolo (1995) demonstrate that the nearEarth intensity decay can be faster than a simple expectation based on the adiabatic energy losses at r = 1 AU. The effect was qualitatively explained in terms of a diffusive decay rate proportional to ∂ 2 N/∂z2 . If so, the effect should be also observed in the diffusion-convection equation solutions. It is also interesting to study the effect at a longer time scale, because, during the period s = V t < 4 AU considered by Ruffolo (1995) for λ = 0.3 AU, only 2-fold intensity decay occurs, while, in experimental practice a longer time series can be also analyzed. Following Ruffolo, we use the distance traveled, s = V t, as a temporal variable, where V is the proton velocity, V = V (E0 ), and throughout the paper, excluding Equation (4), E0 = 2 MeV. To test the effect and to study directly its physical nature, two sets of solutions have been studied. In the first set of simulations, we injected mono-energetic particles at the Sun and registered their energy near the Earth’s orbit. The second set of calculations was performed for a power-law solar injection: Q(E) ∼ E −3 . In all calculations of this section, a constant radial mean free path is adopted, λr ≡ λ cos2 9 = 0.1 AU. For simplicity, the mean free path is assumed to be also constant in rigidity. Kallenrode (1993) considered a number of proposed spatial dependencies of the mean free path and decided to use λr = constant. It seems to be more reasonable than the adoption of a constant parallel mean free path λ = constant. However, in later sections, we also provide a comparative study of the possibilities λr = constant and λ = constant. More details about solving Equation (1) may be found in Appendix A. We consider a mono-energetic solar injection of protons with an initial energy mean free E0 = 2 MeV, and a constant radial R R path of 0.1 AU. The average energy of near-Earth protons, hEi = EN dE/ N dE, as a function of distance traveled, s, is shown in the upper panel of Figure 2. We also show the ±1σ statistical R corridor for the energy and the energy-integrated proton intensity, Iint = I dE. The exponential-decay rate for the average energy, d(lnhEi) 1 , =− V TE ds
(3)
is shown in the lower panel of the figure. The horizontal dashed line illustrates the local estimation, when all particles are constantly kept at r = 1 AU: 1/V TE = ( 43 u/V )/r = 0.027 AU−1 , u = 400 km s−1 . It is seen that the observed energydecay rate for solar particles is not constant in time. It coincides with the locallyexpected decay rate at s ≈ 20 AU. At that time, the particle intensity falls to four times lower than the peak level. Being averaged over the period s = 5–33 AU, i.e., up to the intensity 10-fold-decay, the energy decay rate is 0.0306 AU−1 , which
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Figure 2. Average energy of protons arriving at r = 1 AU, hEi, and the energy-integrated intensity, Iint , vs time for monoenergetic proton injection with initial kinetic energy E0 = 2 MeV. Time is measured in units of distance traveled, s = tV (E0 ). The lower panel shows the temporal change in the inverse e-folding time of the average proton energy. Results are obtained by solving the diffusion-convection equation at a constant radial mean free path, λr = 0.1 AU.
exceeds the locally estimated value by only 13%. The temporal dependence of TE may be approximated as s −0.38 1 ≈ 1.18 × 10−5 s−1 . (4) TE 1 AU The accuracy of this approximation is better than 8% at s ≤ 25 AU. At the same λr = 0.1 AU, the approximation (4) was found to be equally valid for the different initial energies, E0 = 4, 8, 16, 32, and 64 MeV, if the corresponding initial velocity is used to calculate the distance traveled: s = V (E0 )t. Hence, if the mean free path
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Figure 3. Intensity of 2 MeV protons vs time for the power-law injection spectrum at the Sun. The profiles are solutions of the diffusion-convection equation that included no solar wind effects (plus signs), convection only (open circles) and both convection and deceleration (filled circles) at a constant radial mean free path, λr = 0.1 AU. Inverse exponential-decay times, 1/(T V ), are shown as heavy (upper) and light (lower) dashed curves for intensity-time profiles that include all solar wind effects and only convection, respectively. The difference between the exponential decay rates is shown as a solid curve. The horizontal dashed line illustrates the difference estimated using local energy losses at r = 1 AU.
is fixed, the exponential-decay rate, 1/TE , depends mainly on the distance traveled by particles. In other words, the locally observed adiabatic deceleration depends on a global distribution of particles in the interplanetary medium. For the case of the power-law solar injection, Q(E) ∼ E −3 , the near-Earth 2 MeV proton intensities are shown in the upper part of Figure 3. These, and
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all the other curves for the power-law injection seen throughout the paper, are normalized to 1032 protons per solar hemisphere in the entire 1.5–3.5 MeV energy range. In Figure 3, proton intensity–time profiles are shown for the simulations that included either no solar wind effects or only convection or both convection and deceleration. Also shown are the exponential-decay rates for the intensities and a difference between the rates obtained with and without deceleration. The horizontal dashed line illustrates the locally estimated difference in the decay rates, (γ − 1) ( 43 u/V )/r = 0.054 AU−1 . Similar to the energy decay rate of fixed particles (Figure 2), the locally estimated difference is observed at s ≈ 20 AU. Because the deceleration rate is spatially nonuniform and the particles occupy a volume that depends on time, it turns out that the apparent deceleration rate and, correspondingly, the intensity-decay rate difference depend on time. Shortly after the intensity maximum one can obtain, say, a 60% enhancement in the decay-rate difference, but later in time the difference may be close to its locally estimated value. Being averaged over the 10-folding time, s = 5–19 AU, the difference is 0.167 − 0.097 = 0.07 AU−1 , which is 30% larger than the locally estimated difference of 0.054 AU−1 . This implies only a 10%-error in the total decay rate, when the local estimation for the deceleration is used.
3. Comparison of Four Monte Carlo Treatments of Particle Transport We simulate the interplanetary transport with four methods: (i) we solve the diffusion-convection equation (DCE-model described in Section 2 and Appendix A); (ii) we keep track of particles in the focused transport framework when particles have a chance to be isotropized after each small time-step (the small time-step isotropization model or SSI-model); (iii) we trace particles in the focused transport framework proposing that particles experience numerous successive small-angle scatterings with a scattering frequency, ν, which is independent of particle pitch angle (the isotropic small-angle scattering model or IAS-model); (iv) we keep track of particles in the focused transport framework when particles experience numerous successive small-angle scatterings with ν depending on the particle pitch angle, θ (the anisotropic small-angle scattering model or AAS-model). In the last three models, we simulate particle streaming, focusing and pitch-angle scatterings. The scatterings are performed in the local solar wind frame. Properties of scattering in the SSI, IAS and AAS models are different: we simulate a large-angle scattering in the SSI model and a pitch-angle diffusion in the IAS and AAS models (for more details see Appendix B). Two spatial dependencies of the mean free path are considered: (i) a constant parallel mean free path, λ = 0.1 AU and λ = 0.3 AU; (ii) a constant radial mean free path, λr = 0.1425 AU. The latter was selected to provide the near-Earth parallel mean free path λ(r = 1 AU) = 0.3 AU. Solar injection of protons is proposed to be either a power law in energy with spectral index γ = 3 or mono-energetic with
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s=t*V, AU Figure 4. Proton intensity – time profiles at 2 MeV obtained for a power-law injection energy spectrum and a constant parallel mean free path, λ = 0.1 AU, by solving the diffusion-convection equation (plus signs) and by Monte Carlo simulations of focused transport that employed small-chance isotropizations (solid curve).
E0 = 2 MeV. In this section, we propose that particles are injected isotropically in the solar wind frame. As an alternative, a strongly anisotropic injection along the magnetic field will be considered in the next section and in Appendix B. In Figure 4 we show the near-Earth 2 MeV-proton intensity-time profiles calculated for the power-law injection spectrum and for the constant parallel mean free path λ = 0.1 AU. The intensity shown here and throughout the paper is the directional average (excluding Figure 7). The value of λ is small and, correspondingly, the time period under the study is long. This implies a long calculation time. For this reason, in the frame of focused transport, we select the SSI model which is less time consuming. The profile deduced in the SSI model is compared with the corresponding solution of the diffusion-convection equation (DCE). It is seen that agreement is good almost everywhere. The only exception is the very early onset of the proton event, where a diffusion-type equation never works well. In Figure 5, we increase the value of λ up to 0.3 AU. The disagreement between the DCE and SSI models extends now through the entire rise phase, i.e., the DCE approach can not be used to calculate the intensity–time profile here. In the decay phase of the event, however agreement is good. At λ = 0.3 AU, we also performed calculations in IAS and AAS models of pitch-angle diffusion. Comparison of the curves demonstrates
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s=t*V, AU Figure 5. Intensity – time profiles of 2 MeV protons obtained by solving the diffusion-convection equation (plus signs) and by Monte Carlo simulations of focused transport that employed either small-chance isotropizations (heavy solid curve) or isotropic small-angle scatterings (dotted curve) or anisotropic small-angle scattering (light solid curve) for a power-law injection energy spectrum and a constant parallel mean free path, λ = 0.3 AU.
that near the time of maximum the profile is sensitive to the model adopted. At the same time, the decay phase looks very similar in all models. In the next set of simulations, we adopt the mono-energetic solar injection, E0 = 2 MeV, and calculate the energy of protons observed near the Earth. Results are presented in Figure 6 for the constant parallel mean free path λ = 0.3 AU and for the constant radial mean free path λr = 0.1425 AU. Both cases have been calculated for the AAS model. For comparison, the corresponding solutions of the diffusion-convection equation are also shown. It is seen that DCE essentially overestimates adiabatic losses in the beginning of the event when the proton angular distribution is very anisotropic, while later, in the decay phase of the event, very similar profiles are obtained. At s > 3 AU, the DCE and ASS profiles demonstrate equal decay rates of the near-Earth proton energy. These exponential-decay rates decrease with time, as has been already discussed in Section 2. At last, in Figure 7, we compare angular distributions obtained in different models of the pitch-angle scattering, the SSI, IAS, and AAS models. For all profiles, the constant parallel mean free path λ = 0.3 AU was assumed. Profiles are time and energy integrated. Note the strong gradient of distribution around µ = 0 in
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s=t*V, AU Figure 6. Average energy of protons arriving at r = 1 AU, hEi, vs time for a mono-energetic proton injection with initial kinetic energy, E0 = 2 MeV. Profiles are obtained by solving the diffusion-convection equation (dashed and dotted curves for λr = 0.1425 AU and λ = 0.3 AU, respectively) and by Monte-Carlo simulations of focused transport that employs anisotropic small-angle scattering (the heavy and light solid lines for λr = 0.1425 AU and λ = 0.3 AU, respectively).
the case of anisotropic scattering, and a sharp peak at µ = 0.9–1 in the case of the isotropization. During the period s = 0–4 AU, a difference in the profiles is clearly seen. The value of the first-order anisotropy, hµi, is however very similar in every case. We find no difference between the SSI and IAS model distributions integrated over the intensity decay phase, s = 4–12 AU. 4. Results of Monte Carlo Simulations of the Focused Transport For experimental applications, it is not only important to calculate with sufficient accuracy particle profiles in a fixed transport model, but it is also important to study the sensitivity of the results to a model parameters. For this reason, we compare intensity–time profiles obtained under different propositions on spatial dependence of the mean free path and on the initial angular distribution of protons near the Sun. Both parameters are matters of rather large uncertainties. In Figure 8 we show the 2 MeV intensity–time profile deduced in the IAS model for the power-law injection spectrum, γ = 3. The upper curve in Figure 8 does not
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µ =cos(θ θ) Figure 7. Time- and energy-integrated distributions of protons at r = 1 AU vs the pitch-angle cosine as deduced from Monte Carlo simulations of focused transport that employed either small-chance isotropizations (light solid curve) or isotropic small-angle scatterings (dotted curve) or anisotropic small-angle scatterings (heavy solid curve) for a power-law injection energy spectrum in the range E0 = 1.5 – 3.5 MeV and a constant parallel mean free path, λ = 0.3 AU. The integration time interval is 0 – 30 000 s.
include any solar wind effects. The next two curves are calculated with solar wind effects included for isotropic and anisotropic solar injection at a constant parallel mean free path λ = 0.3 AU. In the case of anisotropic injection, initial pitch angle cosines are uniformly distributed from 0.9 to 1. The last curve is calculated with solar wind effects included for isotropic injection and constant radial mean free path λr = 0.2 AU. This value has been adjusted to have approximately the same onset phase and maximum-intensity time. We also consider a case of equal parallel mean free paths near the Earth (Figure 9). It is seen from Figures 8 and 9 that the parameter sensitivity is rather high. The difference between a ‘solar-wind-on’ curve and a ‘solar-wind-off’ curve at s < 4 AU may be of the same order of magnitude as that between two ‘solar-wind-on’ curves corresponding to different parameters suggested. To calculate a tail of the profile properly, one has to adopt an adequate spatial dependence of the mean free path and to take into account adiabatic deceleration during the transport. Recall that a far tail of the event can be properly calculated even with most simple models, with model of the small-chance isotropizations or with the diffusion-convection equation. Finally, in Figure 10, we show the near-Earth proton energy spectrum. The solar injection spectrum is a power law with γ = 3, terminated by a cutoff energy,
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(s − 1), AU Figure 8. Intensity – time profiles of 2 MeV protons obtained by Monte Carlo simulations of focused transport that employed isotropic small-angle scattering and a power-law injection energy spectrum. Profiles are shown for a constant radial mean free path, λr = 0.2 AU, and isotropic injection (heavy solid curve), for a constant parallel mean free path, λ = 0.3 AU, and isotropic injection (light solid curve), for a constant parallel mean free path, λ = 0.3 AU, and beam-type injection (dashed curve). For the case of isotropic injection and λ = 0.3 AU, we also show results that do not include any solar wind effects (dotted curve).
Ec = 3.5 MeV, so that no particles are injected with energy E > Ec . We keep track of particles under constant radial mean free path λr = 0.2 AU during 3 × 104 s after the injection. The near-Earth, time-integrated energy spectrum reveals a fast but smoothed fall above 2.8 MeV. This notable smoothing is caused by variance of energy lost during the propagation (Figure 2). The variance of energy comes from the variance of particle trajectories, so that different particles experience different energy losses. In other words, the smoothing is caused by an interplay of stochastic spatial diffusion and spatially non-constant regular deceleration. 5. Discussion Our results show once again that the diffusion-convection ‘equation is valid if there is enough scattering by the magnetic irregularities to keep the energetic-particle distribution function nearly isotropic in the frame moving with the scattering centers’ (quotation after Jokipii and Morfill, 1993). Discrepancy between the results of the focused transport and diffusion-convection frameworks occurs only when particle angular distribution is strongly anisotropic. This happens in the beginning of a solar particle event (Figures 4 and 5). Note that it is impossible for particles
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s=t*V, AU Figure 9. Intensity – time profiles of 2 MeV protons obtained by Monte Carlo simulations of focused transport that employed isotropic small-angle scattering for a constant parallel mean free path (light solid curve) and for a constant radial mean free path (heavy solid curve). In both cases, parallel mean free paths at r = 1 AU are equal to 0.3 AU; the solar injection is of the power-law form in energy and isotropic.
to reach the Earth before s = 1.15 AU, but the DCE results show a significant intensity for s < 1 AU (Figure 5). This is a well-known artifact of the diffusion approximation. During a late phase of the event, when distributions are more or less isotropic even at λ/L ∼ 13 , both frameworks may lead to a very similar time profiles (Figures 5 and 6). It was previously found in the framework of focused transport, that including adiabatic deceleration can increase the decay rate of the near-Earth intensity by 75% more than would be expected based on an estimation of advection from higher momenta (Ruffolo, 1995). This means that the exponential-decay rate of the 2 MeV proton intensity as function of s is proposed to be a sum of three terms: 1 1 1 1 + + , ≈ VT V Tssf V Tc V Td
(5)
where Tssf is the exponential-decay time constant for streaming, scattering, and focusing, i.e., in the absence of solar wind effects; Tc is that due to including solar wind convection; and Td is that due to including deceleration. At the constant parallel mean free path λ = 0.3 AU, and with the power-law energy spectrum at
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E, MeV Figure 10. Proton energy spectrum at r = 1 AU obtained by Monte Carlo simulations of focused transport that employed isotropic small-angle scattering for a power-law injection spectrum with a cutoff energy Ec = 3.5 MeV, a constant radial mean free path, λr = 0.2 AU, and an isotropic injection. The integration period is 0– 30 000 s.
the Sun, these four terms were previously found by Ruffolo (1995) to be 0.2480, 0.1327, 0.0345, and 0.0871 AU−1 , respectively, so the above approximation is in error by 0.0063 AU−1 , because of interaction between the effects of deceleration and convection. The last number, 1/(V Td ) = 0.0871, is 75% larger than a simple estimate resulting from adiabatic deceleration at r = 1 AU. We have studied this effect using the diffusion-convection equation. We deduced 1/(V Td ) by a comparison of the exponential decay rates for the solutions that include either only convection or both convection and deceleration (Figure 3). We find the following: (i) shortly after the event maximum, the difference, 1/(V Td ), really exceeds the simple estimation for r = 1 AU; (ii) as time elapses and the intensity decays, the value of the difference decreases approaching at s ≈ 20 AU the locally expected value; (iii) being averaged over the intensity 10-folding time, the deviation from the simple estimation is 30%. The physical reason for the deviation at the beginning of the event is that we are observing particles that have experienced losses mainly at r < 1 AU. Later in the event, the particles travel farther from the Sun, and the extra decay disappears. No temporally constant difference in the intensity decay rates, 1/(V Td ), exists. In the framework of focused transport, we have performed Monte Carlo simulations for the case of anisotropic small-angle scatterings, AAS model, at the constant parallel mean free path λ = 0.3 AU (Figure 11). A similar case was earlier
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s=t*V, AU Figure 11. Intensity – time profiles of 2 MeV protons obtained by Monte Carlo simulations of focused transport that employed anisotropic (solid curves) or isotropic (dotted curves) small-angle scattering for a constant parallel mean free path, λ = 0.3 AU. The two lower curves are for simulations with all solar wind effects included. The two upper curves do not include any solar wind effects. All curves were calculated for the anisotropic injection along the magnetic field with a power-law spectrum in energy.
calculated with a finite-difference scheme by Ruffolo (1995). We have checked our results against those of Ruffolo (see Appendix B). The good agreement achieved indicates that both numerical schemes work well. Then, we restricted our consideration to isotropic small-angle scatterings, IAS model. In this model, we vary injection and transport parameters, and also consider a three-fold longer time interval (Figures 8 and 9). Based on these new calculations, we conclude the following: (i) the initial angular distribution of protons at the Sun can affect the magnitude of the near-Earth intensity even at λ = 0.3 AU; (ii) adopted spatial dependence of the mean free path affects both the magnitude and the decay rate of the nearEarth intensity; (iii) when a constant radial mean free path, λr = constant, is adopted instead of λ = constant, the maximum intensity decreases, and the decay rate increases. The peak intensity is also sensitive to the angular scattering model (Figures 5 and 11). The magnitude of proton intensity at 1 AU depends on adiabatic deceleration near the Sun. The deceleration depends on proton angular distributions. For these reasons, the initial angular distribution of solar protons and angular/spatial depen-
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dencies of scattering can affect the near-Earth intensity. Scattering models differ in proton angular distributions observed around the maximum intensity time (Figure 7). Such distributions can be studied with the ERNE/HED detectors on board SOHO (Torsti et al., 1997). In particular, it is possible to check the SSI model, the model of large-angle scatterings, which may be important in wave-particle interactions (Quest, 1988). Precise measurements of the proton angular distributions could ascertain angular properties of interplanetary scattering and limit uncertainties in calculations of adiabatic losses during particle propagation from the Sun to the near-Earth observer. The last point we would like to highlight is a variance of energy during the deceleration (Figure 2). The variance is not small in comparison with the energy lost. It leads to blurring of possible fine details in the accelerated particle spectra. For instance, the near-Earth energy spectrum shown in Figure 10 corresponds to a power-law source spectrum with cutoff energy Ec = 3.5 MeV. A marked smoothing of the spectrum at ≈ 3 MeV is clearly seen. It is caused by the variance of deceleration experienced by particles during the interplanetary propagation. In some solar energetic particle events, small dips in helium spectra near 1 MeV nucl−1 were observed (Möbius et al., 1982). Evidently it would be risky to fit theoretical spectra to such experimental data with no interplanetary transport considered. Acknowledgements We acknowledge benefits from conversations with I. N. Toptygin and D. Ruffolo at the 25th ICRC in Durban. The Academy of Finland is thanked for the financial support for the Space Research Laboratory of the University of Turku. Appendix A. Monte Carlo Method for Solving the Diffusion-Convection Equation Here we describe the Monte Carlo procedure adopted to solve Equation (1). We consider an impulsive injection of particles at radial distance r = r0 = 0.01 AU from the center of the Sun. A particle gets either a fixed or a uniformly distributed initial energy, E0 , and an initial weight, W0 . The initial weight is proportional to the injection spectrum, Q(E0 ). We calculate, per one Monte Carlo time step, 1t, the change of the particle energy, 1E, and the change of the particle coordinate, 1r, as 1E = −
4 uE 1t, 3 r
1r = 23 λv cosn 9
1t + u1t + RG r
q
2 λv 3
cos2 91t ,
(A1)
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where n = 4 for λ = constant and n = 2 for λr = constant, RG is the Gaussian random number with unit variance. Throughout this paper, we use the Archimedean field model of Parker (1958), so that 1 , cos 9 = p 1 + (/u)2 r 2
(A2)
where is the angular velocity of solar rotation. The solar wind speed, u, is 400 km s−1 . We employ a time step linear in r, 1t = αr, where the parameter α was picked by trial. Particles are registered when they are found inside some thin layer, r⊕ − 1⊕ /2 < r < r⊕ + 1⊕ /2, where r⊕ ≡ 1 AU is the observation point, 1⊕ is the width of the layer. Widths of the registration time and energy bins are 1T and 1E , respectively. At registration, we ascribe to a particle the final weight W = W0 v1t/(1T 1⊕ 1E ), so that a registration matrix is proportional to the near-Earth particle intensity, I (E, t, r = r⊕ ). Several simulations were performed to test the accuracy of the computer code. First of all, convection and energy losses were switched off (u = 0), and numerical solutions were compared with well-known analytical solutions of the radial diffusion equation. Then, for u 6 = 0, we tested the case Dr = D0 r and W0 = KE −γ , where an analytical solution is also available (Fisk and Axford, 1968). In all cases, agreement between the results of our Monte Carlo simulations and the analytical solutions was found to be good, limited only by statistical fluctuations.
Appendix B. Monte Carlo Simulations of Focused Transport with Solar Wind Effects Included Previously we described Monte Carlo simulations of the focused transport of highenergy particles, where no solar wind effects were considered (Torsti et al., 1996; Kocharov et al., 1997). Now we present a description of a numerical code applied by L. Kocharov to include solar wind effects into a focused transport model. With no solar wind effects included, the calculations of the particle propagation are based on the model of focused transport illustrated by the equation (Roelof, 1969; Earl, 1976) ∂ ∂ v ∂ ∂f ∂f + vµf + (1 − µ2 )f − (1 − µ2 )ν = Q(z, µ, t), ∂t ∂z ∂µ 2L ∂µ ∂µ
(A3)
where f = f (z, µ, t) is the number of particles per unit of magnetic line length and per unit of solid angle in the velocity space; z is the coordinate of the observer (along the magnetic field line, Figure 1); µ is the cosine of the particle pitch angle; v is the particle velocity; L(z) is the focusing length, ν is the scattering frequency which determines the parallel mean free path, λ; Q(z, µ, t) is the source function.
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For the isotropic scattering (IAS model), ν does not depend on µ. In the case of anisotropic scattering (AAS model), we adopt ν = ν(µ) as ν(µ) =
3v (µ2 + (1 − µ20 )µ2 )(q−1)/2, 2(2 − q)(4 − q)λ0 0
(A4)
where q is the spectral index of MHD turbulence; the parameter λ0 coincides with the mean free path, λ, if µ0 = 0. In this paper, we adopt the values q = 1.5 and µ0 = 0.04. It is convenient to describe the pitch-angle scattering in terms of the parallel mean free path, λ, (e.g., Toptygin, 1985) given by 3v λ= 8
Z1 −1
1 − µ2 dµ . ν(µ)
(A5)
We use this mean free path for the parameterization of the scattering frequency. At each Monte Carlo time step, 1t, we simulate a scattering through a random small angle with respect to the direction of the particle movement at the moment. A detailed description of our Monte Carlo code, with no solar wind effects included, was given earlier by Torsti et al. (1996) and by Kocharov et al. (1997) for isotropic (q = 1) and anisotropic (q > 1) scattering, respectively. As an additional method to simulate the focused transport, we use the small time-step isotropization approach (SSI model). The rate of the scatterings is determined by the characteristic isotropization time, τ = λ/v. We use a simulation time step, 1t, that is much smaller than τ . After each time step, the particle has a small probability to be scattered, PI S = 1t/τ . Whether or not the scattering occurs is decided via a random number generator. After scattering, pitch-angle cosine, µ, is uniformly distributed from −1 to 1. The SSI approach is the most time-saving technique among those applicable to a strongly non-uniform system. It may not describe all details of a real scattering, but allows one to generate a correct spatial diffusion coefficient, D = λv/3. Note that isotropization of each particle at the long time step, 1t = λ/v, would generate a wrong spatial diffusion coefficient, D = λv/6. In our previous and present calculations, a volume registration of particles is employed. This means that we originally calculate the proton distribution function and all other parameters of arriving protons as averaged over a small volume in the vicinity of the Earth. If we injected N0 particles per 1 sr of solar surface, the near-Earth intensity is in the form I (µ, t, r, r⊕ ) =
N0 v 1t Aˆ , 1⊕ r⊕2 cos 9 1T 2π 1µ
(A6)
where Aˆ is the µ−t matrix of particle registration where particles are counted with weight W0 depending on the injection energy spectrum proposed; 1µ and 1T are
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the registration bin width for the pitch-angle cosine and for the time, respectively, 1µ = 0.1. One must use a large enough registration volume in order to count at least once each particle passing the Earth’ orbit, 1⊕ > v1t, and to collect good statistics, 1⊕ v1t. We adopt, 1⊕ = 0.04 AU at λ = 0.3 AU and 1⊕ = 0.02 AU at λ = 0.1 AU. We also use 1T 1t, because the time resolution can not be better than the Monte Carlo time step, and statistical fluctuations should be minimized. We typically have 100 equally wide time bins for the entire simulated period. Note that another method of registration might be applied, where particles are recorded each time they cross the point of the observer. In that case, the recorded particle matrix is proportional to particle flux instead of intensity, and poor statistics at small µ results. If one does not neglect the solar wind effects, two reference frames should be considered: the fixed (corotating) frame, where the Sun is always at the origin, and the local solar wind frame is co-moving with the solar wind velocity at a given point. Because the large-scale structure of the magnetic field is taken to be static in the fixed frame, the process of focusing conserves the absolute value of the particle velocity there, i.e., it is a natural frame to simulate the focusing. The smallscale irregularities in the field can be considered to be frozen in the solar wind. For this reason, the process of scattering should be simulated in the solar wind frame. Thus, we use two frames simultaneously. A similar approach was earlier employed, e.g., by Vasiliev, Toptygin, and Fridgan (Toptygin, 1985, Section 13.3), but they considered a radial magnetic field and a large-angle scattering after each long time step, 1t = λ/v. We consider an Archimedean magnetic field, and either small-angle scatterings or small-chance isotropizations at each short time step, 1t λ/v. Between scatterings, we keep track of particles in the fixed frame (similar to Torsti et al., 1996). We use a fixed frame that is corotating with the Sun (Parker, 1958), so that the magnetic field does not depend on time and the solar wind velocity, uC , is parallel to the magnetic field at each point (Figure 1); uC = u/ cos 9, where u is the solar wind velocity in the nonrotating frame. Consequently, we have no systematic betatron energy loss to be included into the numerical scheme. Injection and scattering however are performed in the local solar wind frame. Before and after a scattering, particle velocity and pitch angle cosine must be transformed to the solar wind frame and to the fixed frame, respectively. Expressions for the transformation are well known and must be used with solar wind velocity uC = u/ cos 9. The rate of the scatterings is determined in the solar wind frame. In the solar wind frame, the scatterings are performed identically to the procedure described by Kocharov et al. (1997). Any model of pitch angle scattering may be employed, SSI-model, IAS-model, or AAS-model. However, the combination of the Galileo/Lorenz transformations and anisotropic scatterings turns out to be the most computer time consuming. For this reason, special care should be taken about the optimal choice of the Monte Carlo time step in the AAS-model. If we adopt a scattering frequency in the form of (A4), the angular scale of the scattering
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frequency change, θν = |ν/(dν/dθ)|, strongly depends on µ = cos θ. We require an average scattering angle to be much less than θν . For this reason, the Monte Carlo time step is taken as a function of the current pitch angle, θ. The adopted angular dependence of the current time step is in the form q µ2b + µ3 , (A7) 1t = (1t)0 1 − µ2 + µm where (1t)0 = 0.0005/ν(µ = 1), µb = 0.03, and µm = 0.2. The particle registration weight in the matrix Aˆ is changing linearly qwith the time step to keep
normalization constant for all values of µ, W = W0 µ2b + µ3 /(1 − µ2 + µm ), where W0 is an initial weight depending on the injection energy spectrum. The matrix Aˆ is three-dimensional now because we also register particle energy in the 1E bins. For these reasons, we have to replace in Equation (A6) 1t → (1t)0 and 1µ → 1µ 1E . We adopt 1E = 0.1 MeV. Several simulations were performed to test the accuracy of the code. In particular, we calculated the 2 MeV-proton intensity-time profile for the power law solar injection, Q(E) ∼ E −3 , constant parallel mean free path λ = 0.3 AU and q = 1.5. A similar case was calculate by Ruffolo (1995) with a finite-difference method. Calculations are performed with and without solar wind effects included. Solar injection is proposed to be strongly anisotropic, so that all particles are uniformly distributed between µ = 0.9 and µ = 1. The directional average intensity of 2 MeV protons is calculated as a mean value for the 1.9–2.1 MeV energy interval. We kept track of 2 × 106 particles per the entire 1.8–2.8 MeV range. Our profiles are shown in Figure 11 (for q = 1.5 see solid curves). Agreement between the two numerical schemes is found to be satisfactory. For instance, the exponential decay times reported by Ruffolo (1995) are 1/(V Tssf ) = 0.1327 and 1/(V T ) = 0.2480 for the solar wind effects excluded and included, respectively. Our results are 1/(V Tssf ) = 0.132 ± 0.002 and 1/(V T ) = 0.248 ± 0.002 for the period s = 2.5–4 AU. It is seen that the results agree within the statistical 1σ limits. We also show profiles for the case of isotropic scattering (q = 1). Note that the 1–2% disagreement between the amplitude and the onset time of our profiles and those by Ruffolo (1995) comes from a slightly different description of the angular dependence of the scattering frequency ν(µ). Our dependence is slightly less anisotropic. At a fixed value of λ, this implies fewer scatterings at µ=1, which leads to a slightly earlier arrival of the first particles. A qualitatively similar but much stronger effect is seen from comparison of profiles calculated for q = 1.5 and profiles for q = 1 (Figure 11).
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References Dolginov, A. Z. and Toptyghin, I. N.: 1966, Zh. Eksper. Teor. Fiz. (Soviet Phys. JETP) 51, 1771. Earl, J. A.: 1976, Astrophys. J. 205, 900. Earl, J. A.: 1984, Astrophys. J. 278, 825. Earl, J. A., Ruffolo, D., Pauls, H. L., and Bieber, J. W.: 1995, Proc. 24th Int. Cosmic Ray Conf., Rome 4, 341. Ellison, D. C., Jones, F. C., and Reynolds, S. P.: 1990, Astrophys. J. 360, 702. Fisk, L. A. and Axford, W. I.: 1968, J. Geophys. Res. 73, 4396. Gardiner, C. W.: 1985, Handbook of Stochastic Methods, Springer-Verlag, Berlin. Jokipii, J. R. and Morfill, G. M.: 1993, Proc. 23rd Int. Cosmic Ray Conf., Paris 3, 179. Kallenrode, M.-B.: 1993, J. Geophys. Res. 98, 19 037. Kocharov, L. G., Torsti, J., Vainio, R. and Kovaltsov, G. A.: 1996, Solar Phys. 165, 205. Kocharov, L. G., Torsti, J., Laitinen, T. and Vainio, R.: 1997, Solar Phys. 175, 785. Kunow, H., Wibberenz, G., Green, G., Müller-Mellin, R., and Kallenrode, M.-B.: 1991, in R. Schwenn and E. Marsch (eds.), Physics of Inner Heliosphere, Vol. 2, Springer-Verlag, Berlin, p. 262. Möbius, E., Scholer, M., Hovestadt, D., Klecker, B., and Gloeckler, G.: 1982, Astrophys. J. 259, 397. Palmer, I. D. and Jokipii, J. R.: 1981, Proc. 17th Int. Cosmic Ray Conf., Paris 3, 381. Parker, E. N.: 1958, Astrophys. J. 128, 664. Parker, E. N.: 1965, Planetary Space Sci. 13, 9. Quest, K. B.: 1988, J. Geophys. Res. 93, 9649. Roelof, R. C.: 1969, in H. Ögelman and J. R. Wayland (eds.), Lectures in High-Energy Astrophysics, NASA SP-199, p 111. Ruffolo, D.: 1995, Astrophys. J. 442, 861. Toptygin, I. N.: 1985 Cosmic Rays in Interplanetary Magnetic Fields, D. Reidel, Publ. Co., Dordrecht, Holland. Torsti, J., Kocharov, L. G., Vainio, R., Anttila, A., and Kovaltsov, G. A.: 1996, Solar Phys. 166, 135. Torsti, J., Laitinen, T., Vainio, R., Kocharov, L., Anttila, A., and Valtonen, E.: 1997, Solar Phys., 175, 771.