Tilt Angle (degrees)
Tilt Angle (degrees)
Fig. 2.41.2. The ratio of electron differential intensities computed with the 2D and 3D drift models as a function of tilt angle a. Panel a shows the ratio for 1.94 GeV electrons for both the A > 0 and A < 0 polarity cycles at 1 AU and panel b shows the situation at 60 AU. Panels c and d display the situation for 0.30 GeV electrons at 1 AU and 60 AU respectively. From Ferreira et al. (1999a).
The modulation parameters in Fig. 2.41.2 are the same as for Fig. 2.41.1. Values are shown for 1.94 GeV electrons at 1 AU in panel a and at 60 AU in panel b. Panels c and d show the same situation but for 0.30 GeV electrons. From Fig. 2.41.2 follows that for 1.94 GeV electrons the ratio varies with < 1 % for both polarity cycles, at 1 AU and at 60 AU, for all tilt angles a. At 0.30 GeV, the A < 0 polarity cycle exhibits again a very small deviation from unity in the ratio, not more than ~5 %. For the A > 0 cycle, however, the ratio is > 1.0 at 1 AU, increasing with increasing a, with a maximum of 1.25. At 60 AU the ratio has a peculiar a dependence varying between 1.15 and 1.25 which is larger than at 1 AU, especially for a < 30°. The differences between the model solutions are obviously the largest for the intermediate to lower energies during the A > 0 cycle when the electrons drift in along the HCS. At energies below ~ 0.05 GeV the differences between the intensities dissipate quickly because electrons experience less and less drift effects with decreasing energy. The largest variation in the ratio between the two sets of solutions as a function of energy occurs at ~ 0.2 GeV and varies between 12% at 1 AU and 24% at 60 AU, with no difference at kinetic energies > 1 GeV. No qualitative differences were found between the solutions of the two models despite the difference in spatial dimensions and the different way the HCS was handled in the numerical schemes.
Hattingh (1998) indicated that the difference in the solutions of the 2D and 3D models using the same set of modulation parameters was somewhat dependent on the parameter values. The values used above correspond to solar minimum modulation conditions for which the steady-state models were develop. When more extreme variations were used the differences between the two models increased, indicating that some caution is required during periods of large modulation. Investigating this aspect further using electron modulation it was found that by increasing k_q, which has become a very important parameter in modulation models, a reduction in the differences between the 2D and 3D model solutions followed: see also Section 2.39 (Ferreira et al., 1999b) and Ferreira and Potgieter, 1999). This is expected because an increasing k_q causes less pronounced drift effects. It is worthwhile to mention that when no-drifts were used the two models produce identical solutions under all circumstances. Reducing the azimuthal, radial, polar and rigidity grid intervals, that is increasing the number of total grid points in the numerical scheme, resulted in only a slight reduction in the difference between the two models while the runtime in computing one solution increased considerably for the 3D model (Ferreira, 1999).
According to Ferreira et al. (1999a), comparing the solutions produced by the 2D and 3D numerical modulation models that were both developed by the Modulation Group in Potchefstroom, it was found that when examining electron spectra as a function of the HCS tilt angle a, no qualitative differences occurred between the two sets of solutions when using identical parameters. Quantitatively, in the inner Heliosphere the ratio between the two sets of solutions increased with increasing a, with 25% the largest difference at intermediate energies (~ 0.30 GeV) for a > 30° during the A > 0 cycle; at 60 AU the ratio varied with 15% to 25% with no clear trend in the a dependence of the intensities. At energies below ~ 0.05 GeV the differences between the intensities dissipate quickly because electrons experience diminishing drift effects with decreasing energy. For the A < 0 cycle, the solutions were essentially identical. Thus, with no qualitative differences and insignificant quantitative differences between the solutions of the 2D and 3D models, and taking into account the amount of computing time and resources needed for the 3D model, the use of the 2D drift model for modulation studies is still well justified.
In Sections 2.34-2.41 we considered in detail very important principal problems on galactic CR charged particles diffusion (especially enhanced perpendicular diffusion), convection, and drifts (gradient, curvature, and especially along Heliospheric current sheet) during their propagation and modulation in the Heliosphere as well as comparison with CR observation data (especially are important data near the Earth's orbit, on different distances from the Sun and Ulysses data on different helio-latitudes). In all these Sections it was not taken into account the time-lag of processes in the Heliosphere relative to corresponding causes processes on the Sun. The second what is also does not accounted in Sections 2.34-2.41 is the time lag caused by galactic CR particles penetrating into the inner Heliosphere. Below, in Sections 2.45 and 2.46 we will try to account these two points when we solve the inverse problems for CR propagation and modulation in the Heliosphere.
2.42. The inverse problem for solar CR propagation
2.42.1. Observation data and inverse problems for isotropic diffusion, for anisotropic diffusion, and for kinetic description of solar CR propagation
It is well known that Solar Energetic Particle (SEP) events in the beginning stage are very anisotropic, especially during great events as in February 1956, July 1959, August 1972, September-October 1989, July 2000, January 2005, and many others (Dorman, M1957, M1963a,b, M1978; Dorman and Miroshnichenko, M1968; Miroshnichenko, M2001). To determine on the basis of experimental data the properties of the SEP source and parameters of propagation, i.e. to solve the inverse problem, is very difficult, and it needs data from many CR stations. By the procedure developed in Dorman and Zukerman (2003), Dorman, Pustil'nik, Zukerman and Sternlieb, 2005; see review in Chapter 3 in Dorman, M2004), for each CR station the starting moment of SEP event can be automatically determined and then for different moments of time by the method of coupling functions to determine the energy spectrum of SEP out of the atmosphere above the individual
CR station. As result we may obtain the planetary distribution of SEP intensity out of the atmosphere and then by taking into account the influence of geomagnetic field on particles trajectories - the SEP angle distribution out of the Earth's magnetosphere. By this way by using of the planetary net of CR stations with online registration in real time scale can be organized the continue on-line monitoring of great ground observed SEP events (Dorman, Pustil'nik, Sternlieb et al., 2004; Mavromichalaki, Yanke, Dorman et al., 2004).
In paper Dorman (2005) we practically base on the two well established facts:
(i) the time of particle acceleration on the Sun and injection into solar wind is very short in comparison with time of propagation, so it can be considered as delta-function from time;
(ii) the very anisotropic distribution of SEP with developing of the event in time after few scattering of energetic particles became near isotropic (well known examples of February 1956, September 1989 and many others).
The paper of Dorman (2005), described below, is the first step for solution of inverse problem in the theory of solar CR propagation by using only one on-line detector on the ground for high energy particles and one on-line detector on satellite for small energies. Therefore we will base here on the simplest model of generation (delta function in time and in space) and on the simplest model of propagation (isotropic diffusion). The second step will be based on anisotropic diffusion, and the third - on kinetic description of SEP propagation in the interplanetary space.
The observed energy spectrum of SEP and its change with time are determined by the energy spectrum in the source, by the time of SEP ejection into the solar wind and by the parameters of SEP propagation in the interplanetary space in dependence of particle energy. Here we will try to solve the inverse problem on the basis of CR observations by the ground base detectors and detectors in the space to determine the energy spectrum of SEP in the source, the time of SEP ejection into the solar wind and the parameters of SEP propagation in the interplanetary space in dependence of particle energy. in general, this inverse problem is very complicated, and we suppose to solve it approximately step by step. In this Section we present the solution of the inverse problem in the frame of the simple model of isotropic diffusion of solar CR (the first step). We suppose that after start of SEP event, the energy spectrum of SEP at different moments in time is determined with good accuracy in a broad interval of energies by the method of coupling functions (see in detail in Chapter 3 in Dorman, M2004). We show then that after this the time of ejection, diffusion coefficient in the interplanetary space and energy spectrum in source of SEP can be determined. This information, obtained on line on the basis of real-time scale data, may be useful also for radiation hazard forecasting.
2.42.2. The inverse problem for the case when diffusion coefficient depends only from particle rigidity in this case the solution of isotropic diffusion for the pointing instantaneous source described by function
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