MODELING GRADUAL SEP
*Applied Physics Laboratory. The Johns Hopkins University, USA
Project sponsored by NASA
The largest long-lasting SEP events are associated with fast CMEs[Kahler, 2001]. Those SEP events last for several days or longer and are found to be associated with CMEs originating from virtually anywhere on the visible solar disk[Cliver et al., 1995;Reames, 1999]. The study of these events is important mainly for two reasons: their space weather implications [Kahler, 2001], and their dominant contribution to the fluence of energetic particles observed throughout a solar cycle[Lanzerotti, 1977;Shea and Smart, 1996]. The proposed scenario to account for these events involves the presence of a fast CME able to drive shocks. Modelers assume that the injection of energetic particles into the interplanetary medium starts when a perturbation, originated as a consequence of a solar eruption, generates a shock wave that propagates across the solar corona. If the conditions are appropriate, this shock is able to accelerate particles from the ambient plasma (or to accelerate particles from contiguous or previous solar events), and to inject them at the base of the IMF lines. These energetic particles stream out along these lines en route to earth and to spacecraft located in the interplanetary medium. The perturbation that generated the shock into the corona may also expand through the heliosphere driving a shock wave across the interplanetary medium. In order to explain observations of SEP events by multiple spacecraft magnetically connected to regions of the Sun distant from the parent solar active region, it is assumed that the shocks may extend up to 300 degrees in longitude near the corona [Cliver et al., 1995]. However, interplanetary shocks observed at 1 AU extend at most 180 degrees[Cane, 1988].
As the shock propagates away from the Sun, it crosses many IMF lines and may be responsible for accelerating particles out of the solar wind and/or from remnant particles of previous SEP events [Desai et al., 2001]. These energetic particles propagate along the IMF lines flowing outward from the shock. When these particles arrive at the spacecraft, particle intensity increases are detected which constitute the SEP events. The particle intensity profiles of the SEP events take different forms depending on (1) the heliolongitude of the source region with respect to the observer location, (2) the strength of the shock and its efficiency at accelerating particles, (3) the presence of a seed particle population to be further accelerated, (4) the evolution of the shock (its speed, size, shape and efficiency in particle acceleration), (5) the conditions for the propagation of shock-accelerated particles, and (6) the energy considered [Heras et al., 1988;Heras et al., 1995;Cane et al., 1988;Lario et al., 1998;Kahler et al., 1999].
The details of the proton flux profiles during these gradual SEP events are consistent with the presence of a traveling CME-driven shock which continuously injects energetic particles as it propagates away from the Sun. Figure 1 shows the proton flux profiles as a function of longitude for several recent events observed by the ACE and IMP-8 spacecraft. This figure is derived from Figure15 of Cane et al. . The concept of "cobpoint", defined by Heras et al.  as the point of the shock front which magnetically connects to the observer, is very useful to describe the different types of SEP flux profiles. Solar events from the western hemisphere have rapid rises to maxima because, initially, the cobpoint is close to the nose of the shock near the Sun. These rapid rises are followed by gradual decreasing intensities because the cobpoint is at the eastern flank of the shock just where and when the shock is weaker. The observation of the shock at 1 AU in these western events depends on the width and strength of the shock. Near central meridian the cobpoint is initially located on the western flank of the shock and progressively moves toward the nose of the shock. Low-energy proton fluxes usually peak at the arrival of the shock, being part of what are known as energetic storm particle (ESP) events. For events originating from eastern longitudes, connection with the shock is established just a few hours before the arrival of the shock and the cobpoint moves from the weak western flank to the central parts of the shock; connection with the shock nose is only established when the shock is beyond the spacecraft and, usually, it is at this time when the peak particle flux is observed. The evolution of the low-energy ion flow anisotropy profiles throughout the SEP events reflects also the cobpoint motion along the shock front [Domingo et al., 1989]. For additional examples see Cane et al. ; Heras et al. ; or Kahler .
The simulation of these particle events requires a knowledge of how particles and shocks propagate through the interplanetary medium, and how shocks accelerate and inject particles into interplanetary space. The modeling of particle fluxes and fluences associated with SEP events has to consider (1) the changes in shock characteristics as it travels through the interplanetary medium, (2) the different points of the shock where the observer is connected to, and (3) the conditions under which particles propagate. There have been several attempts to model these events. Each model presents its own simplifying assumptions in order to tackle the series of complex phenomena occurring during the development of SEP events. Two main approximations have been used to describe the particle transport: the cosmic ray diffusion equation [Jokipii, 1966] and the focusing-diffusion transport equation [Roelof, 1969;Ruffolo, 1995]. To describe the shock propagation, approximations range from considering a simple semicircle centered at the Sun propagating radially at constant velocity, to fully developed magnetohydrodynamic (MHD) models.
Lee and Ryan  adopted an analytical approach to solve the time-dependent cosmic ray diffusion equation for an evolving interplanetary shock which was modeled as a spherically-symmetric blast wave propagating into a stationary surrounding medium. Besides the inapplicability of the diffusion approximation outside the shock region, some strong assumptions were needed to retain a tractable model, in particular, very high blast wave velocities, the neglect of solar wind and a radial mean free path, independent of the particle energy, that increased with r2, where r denotes the radial heliocentric distance. None of the assumptions is especially well supported observationally in the inner heliosphere.
Heras et al.[1992; 1995] were the first to adopt the focused-diffusion transport equation, including a source term, Q, which represents the injection rate of particles accelerated at the traveling shock. The use of this transport equation is more adequate for these SEP events since it allows us to reproduce the large and long-lasting anisotropies usually observed at low-energies in gradual SEP events [Heras et al., 1994]. The injection of particles is considered to take place at the cobpoint. To track this point with time, the authors used an MHD model that describes the shock propagation from a given inner boundary close to the Sun up to the observer. The IMF is described upstream of the shock by the usual Parker spiral. This model has been refined by including solar wind convection and adiabatic deceleration effects into the particle transport equation and the corotation of the IMF lines [Lario et al., 1998]. It has been successfully applied to reproduce the low-energy (< 20 MeV) proton flux and anisotropy profiles of a number of SEP events simultaneously observed by several spacecraft [Heras et al., 1995;Lario et al., 1998].
Kallenrode and Wibberenz  and Kallenrode  adopted the same scheme as the previous works. However, these authors use a semicircle propagating radially from the Sun at constant speed to describe the shock. They also parameterize the injection rate, Q, in terms of a radial and azimuthal variation which represents the temporal and spatial dependences of the shock efficiency in accelerating particles. They allow also for particle propagation in the downstream region of the shock just by changing the magnitude of the focusing length, however they do not modify the actual IMF topology behind the shock which may lead to different results [Lario et al., 1999]. They also allow for a transmission of particles across the shock, but not a change of the particle energy when they are reflected and/or transmitted. On the other hand, Torsti et al.  and Antilla et al.  adopted a similar scheme as the above-mentioned works but assumed, in order to locate the cobpoint, that the distance of the cobpoint to the observer along the IMF line connecting with the observer decreases linearly with time. They also used a complex parametric function to describe, Q, including energetic, temporal and spatial dependences. Differences among the above models have been described in Sanahuja and Lario  and Kallenrode .
Ng et al. [1999a, 1999b, 2001] have used a similar approach to deal with SEP simulations. The authors have developed a numerical model where the particle transport includes proton-generated Alfvén waves. Whereas the above-described models assume that the scattering of particles may be parametrized by a given mean free path (which may depend on the particle energy and time), Ng et al. [1999a] consistently solve the focused-diffusion transport equation for the particles and the equation describing the evolution of differential wave intensity. Assuming that particles are accelerated out of a constant source plasma with a specific composition, Ng et al.  successfully describe the evolution of abundance ratios in some SEP events. No quantitative agreement of the predicted wave spectrum has yet been presented. Several simplifications were made in the model such as the assumption of radial IMF and the use of several phenomenological parameters in the equations. The shock was assumed to travel radially away from the Sun at a constant speed. The injection, Q, was also parametrized to account for temporal, radial and energy dependence. This model allows for a better description of self-generated scattering processes throughout the transport of particles of different species. Nevertheless, the use of radial IMF does not allow the reproduction of the longitudinal dependence shown in Figure 1. On the other hand, observation of shock speeds in different directions [Cane, 1988] and dynamic studies from MHD simulations [Smith and Dryer, 1990] indicate a decrease of the shock speed towards its flanks and a weakening of its front as it expands. Those models describing the shock as a semicircle propagating at constant speed oversimplify the shock geometry and evolution and therefore misplace the cobpoint and neglect the physical conditions at the point where particles are accelerated and injected.
None of the above models treats the fundamental nature of particle acceleration at the evolving interplanetary shocks. The details of how the MHD conditions at the shock front translate into an efficiency in particle acceleration and how it evolves as the shock expands are not completely understood. Lario et al.  proposed a parameterization to relate the evolution of the injection rate of shock-accelerated particles to the dynamic properties of the shock. That relation yields a quantification of the injection rate, its energy spectrum and its evolution; however, it does not address the physical mechanism of particle shock acceleration.
Recently, theoretical efforts have been addressed to incorporate the mechanisms of shock-acceleration of particles into traveling interplanetary shocks [Zank et al., 2000;Lee, 2001;Berezhko et al., 2001]. In particular, Zank et al.  have developed a dynamical time-dependent model of particle acceleration at the propagating shock. This model assumes a spherically symmetric solar wind into which a blast wave propagates. Both the wind and shock are modeled numerically using hydrodynamic equations and assuming a Parker spiral for the IMF. The local characteristics of the shock, such as the shock strength or the Mach number, are dynamically computed. Those parameters are used to determine the distribution of particles injected into the diffusive shock acceleration mechanism. Shock-accelerated particles propagate diffusively in the vicinity of the shock generating resonant Alfvén waves which are included in the model. However, when these particles are far enough upstream of the shock, they are allowed to freely escape without experiencing any scattering, deceleration or convection effects. In that way, Zank et al.  neglected the complications associated with a detailed transport model, such as those developed by Ruffolo , Lario et al. , and Ng et al. [1999a]. An interesting point of the Zank et al.  model is that, for extremely strong shocks, particle energies of the order of 1 GeV can be achieved when the shock is still close to the Sun. As the shock propagates outward, the maximum accelerated particle energy decreases sharply. Other shock acceleration models [Berezhko et al., 2001] also suggest the possibility that 1 GeV protons can be accelerated when extremely strong shocks are close to the Sun (< 3 solar radii). Comparisons of these models of particle shock-acceleration with specific observations have not yet been reported. For the moment, no theoretical and/or numerical model treats SEP acceleration and transport near its full complexity.
Antilla, A., L. G. Kocharov, J. Torsti, and R. Vainio, Long-duration high-energy proton events observed by GOES in October 1989, Ann. Geophysicae, 16, 921-930, 1998.
Berezhko, E. G., S. I. Petukhov, and S. N. Taneev, Shock acceleration of energetic particles in solar corona, Conf. Pap. Int. Cosmic Ray Conf. 27th, 8, 3215-3218, 2001.
Cane, H.V., The large-scale structure of flare-associated interplanetary shocks, J. Geophys. Res. 93, 1-6, 1988.
Cane, H. V., D. V. Reames, and T. T. von Rosenvinge, The role of interplanetary shocks in the longitude distribution of solar energetic particles, J. Geophys. Res. 93, 9555-9567, 1988.
Cliver, E. W. et al., Extreme "propagation" of solar energetic particles, Conf. Pap. Int. Cosmic Ray Conf. 24th, 4, 257-260, 1995.
Desai, M. I., et al., Acceleration of 3He Nuclei at Interplanetary Shocks, Astrophys. J., 553, L89-L92, 2001.
Domingo, V., B. Sanahuja, and A.M. Heras, Energetic particles, interplanetary shocks and solar activity, Adv. Space Res., 9, 191-195, 1989.
Heras, A. M., B. Sanahuja, V. Domingo, and J.A. Joselyn, Low-energy particle events generated by solar disappearing filaments, Astron. and Astrophys., 197, 297-305, 1988.
Heras, A. M., et al., The influence of the large-scale interplanetary shock structure on a low-energy particle event, Astrophys. J., 391, 359-369, 1992. [PDF]
Heras, A.M., et al., Observational signatures of the influence of the interplanetary shocks on the associated low-energy particle events, J. Geophys. Res. 99, 43-51, 1994.
Heras, A. M., et al., Three low-energy particle events: modeling the influence of the parent interplanetary shock, Astrophys. J., 445, 497-508, 1995. [PDF]
Jokipii, J. R., Cosmic-Ray Propagation. I. Charged Particles in a Random Magnetic Field, Astrophys. J., 146, 480-487, 1966.
Kahler, S.W., J.T. Burkepile, and D.V. Reames, Coronal/Interplanetary factors contributing to the intensities of E>20 MeV gradual SEP events, Conf. Pap. Int. Cosmic Ray Conf. 26th, 6, 248-251, 1999.
Kahler, S. W., Origin and properties of solar energetic particles in space, in Space Weather, edited by Song et al., AGU Monograph, 125 109-122, 2001.
Kallenrode, M.-B., Shock as a black box 2. Effects of adiabatic deceleration and convection included, J. Geophys. Res. 106, 24989-25003, 2001.
Kallenrode, M.-B., and G. Wibberenz, Propagation of particles injected from shocks: A black box model and its consequences for acceleration theory and data interpretation, J. Geophys. Res. 102, 22311-22334, 1997.
Lanzerotti, L.J., Measures of energetic particles from the Sun, in The Solar Output and Its Variation, O.R. White, ed., University of Colorado Press, Boulder, CO, 383-403, 1977.
Lario, D., B. Sanahuja, and A. M. Heras, Energetic particle events: efficiency of interplanetary shocks as 50 keV < E < 100 MeV proton accelerators, Astrophys. J., 509, 415-434, 1998. [PDF]
Lario, D., M. Vandas, and B. Sanahuja, Energetic Particle Propagation in the Downstream Region of Transient Interplanetary Shocks, in Solar Wind Nine, edited by S.R. Habbal, et al., AIP Conf. Proc. 471, 741-744, 1999.
Lee, M. A., Coupled hydromagnetic wave excitation and ion acceleration at traveling interplanetary shocks, J. Geophys. Res. 88, 6109-6119, 1983.
Lee, M. A., and J.M. Ryan, Time-dependent coronal shock acceleration of energetic solar flare particles, Astrophys. J., 303, 829-842, 1986.
Lee, M.A., The Theory of Particle Acceleration at Coronal/Interplanetary Shocks, EOS Trans. AGU, 82(47), Fall Meet. Suppl., Abstract SH12C-09, 2001.
Ng, C. K., D. V. Reames, and A. J. Tylka, Effects of proton-amplified waves on evolution of solar energetic particle composition in gradual events, Geophys. Res. Let. 26, 2145-2148, 1999a.
Ng, C. K., D. V. Reames, and A. J. Tylka, Model for the evolution of the elemental abundances of solar energetic particles, Conf. Pap. Int. Cosmic Ray Conf. 26th, 6, 151-154, 1999b.
Ng, C. K., D. V. Reames, and A. J. Tylka, Evolution of abundances and spectra in the large solar energetic particle events of 1998 Sep 30 and 2000 Apr 4, Conf. Pap. Int. Cosmic Ray Conf. 27th, 8, 3140-3143, 2001.
Reames, D. V., Particle acceleration at the Sun and in the heliosphere, Space Sci. Rev., 90, 413-491, 1999.
Roelof, E. C., Propagation of solar cosmic rays in the interplanetary magnetic field, in Lectures in High Energy Astrophysics, edited by H. Ögelman and J. R. Wayland, NASA Spec. Publ., SP-199, 111, 1969.
Ruffolo, D., Effect of adiabatic deceleration on the transport of solar cosmic rays Astrophys. J., 442, 861-874, 1995.
Sanahuja, B., and D. Lario, Low-energy cosmic rays: Modeling gradual events, in Conf. Proc. 16th European Cosmic Ray Symp., edited by J. Medina, Alcalá University Editions, 129-140, 1998.
Smith, Z., and M. Dryer, MHD study of temporal and spatial evolution of simulated shocks in the ecliptic plane within 1 AU, Solar Physics, 129,387-405, 1990.
Shea, M.A., and Smart, D.F., Solar proton fluxes as a function of the observation location with respect to the parent solar activity, Adv. Space Res., 17, 225-228, 1996.
Torsti, J., et al., The 1990 May 24 solar cosmic ray event, Solar Physics, 166, 135-158, 1996.
Zank, G. P., W. K. M. Rice, and C. C. Wu, Particle acceleration and coronal mass ejection driven shocks: a theoretical model, J. Geophys. Res. 105, 25079-25095, 2000.
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