Fig. 2.43.1. The time behavior of k(R ) for R ~ 10 GV for the SEP event 29 September, 1989. According to Dorman et al. (2005a,b).

From Fig. 2.43.1 can be seen that at the beginning of the event the obtained results are not stable, due to large relative statistical errors. After several minutes the amplitude of CR intensity increase becomes many times bigger than statistical error for one minute data a (about 1%), and we can see a systematical increase of the diffusion coefficient k(R ) with time. This result contradicts the conditions at which was solved the inverse problem in Section 2.42.2. Really the systematical increase of the diffusion coefficient with time reflects the increasing of k(r) with the diffusion propagation of solar CR from the Sun, i.e. reflects the increasing of

20 40 60 80 100 120

k(R ) with the distance from the Sun. It means that for the considered SEP event we need to apply the inverse problem described in Section 2.42.3, where it was assumed increasing of diffusion coefficient with the distance from the Sun according to Eq. 2.42.10.

2.43.2. The checking of the model when diffusion coefficient depends from the distance to the Sun

On the basis of the inverse problem solution described in Section 2.42.3, by using the first few minutes NM data of the SEP event we can determine the effective parameters 3 by Eq. 2.42.25, k (R) by Eq. 2.42.26, and No(r) by Eq. 2.42.27, corresponding to high rigidity, about 10 GV. In Fig. 2.43.2 the values of parameter k (R) are shown.

Fig. 2.43.2. Diffusion coefficient K (R) near the Earth's orbit (in units 10 cm sec )

in dependence of time (in minutes after 11.40 UT of September 29, 1989).

From Fig. 2.43.2 it can be seen that at the very beginning of event (the first point) the result is unstable: in this period the amplitude of increase is relatively small, so the relative accuracy is too low, and we obtain very big diffusion coefficient. Let us note, that at the very beginning of the event the diffusion model can be very hardly applied (more correct would be the application of kinetic model of SEP propagation). After the first point we have about stable result with accuracy ± 20 % (let us compare with Fig. 2.43.1, where the diffusion coefficient was found as effectively increasing with time). In Fig. 2.43.3 are shown values of parameter3 .

Fig. 2.43.3. Values of parameter 3 in dependence of time (in minutes after 11.40 UT of September 29, 1989).

It can be seen from Fig. 2.43.3 that again the first point is anomalously big, but after the first point the result become almost stable with average value 3 ~ 0.6 (with accuracy about ± 20%). Therefore, we can hope that the model of the inverse problem solution, described in Section 2.42.3 (the set of Eq. 2.42.10-2.42.27) reflects adequately SEP propagation in the interplanetary space.

2.43.3. The checking of the model by comparison of predicted SEP intensity time variation with NM observations

More accurate and exact checking of the solution of the inverse problem can be made by comparison of predicted SEP intensity time variation with NM observations. For this aim after determining of the effective parameters 3, k (R), and No (R) we may determine by Eq. 2.42.11 the forecasting curve of expected

SEP flux behavior for total neutron intensity. With each new minute of observations we can determine parameters 3, K1(R), and No(R) more and more exactly. It means that with each new minute of observations we can determine more and more exactly the forecasting curve of expected SEP flux behavior. We compare this forecasting curve with time variation of observed total neutron intensity (see Fig. 2.43.4 which contains 8 panels for time moments t = 10 min up to t = 120 min after 11.40 UT of 29 September, 1989).

Fig. 2.43.4. Calculation for each new minute of SEP intensity observations parameters ¡3, K (R), No (R) and forecasting of total neutron intensity (time t is in minutes after 11.40 UT of September 29, 1989; curves - forecasting, circles - observed total neutron intensity). From Dorman et al. (2005a,b).

Fig. 2.43.4. Calculation for each new minute of SEP intensity observations parameters ¡3, K (R), No (R) and forecasting of total neutron intensity (time t is in minutes after 11.40 UT of September 29, 1989; curves - forecasting, circles - observed total neutron intensity). From Dorman et al. (2005a,b).

From Fig. 2.43.4 it can be seen that it is not enough to use only the first few minutes of NM data (t = 10 min): the obtained curve forecasts too low intensity. For t = 15 min the forecast shows some bigger intensity, but also not enough. Only for t = 20 min (15 minutes of increase after beginning) and later (up to t = 40 min and more) we obtain about stable forecast with good agreement with observed CR intensity.

2.43.4. The checking of the model by comparison of predicted SEP intensity time variation with NM and satellite observations

The results described above, based only on NM on data, reflect the situation in SEP behavior in the high energy (more than 6 GeV) region. For extrapolation of these results to the low energy interval (dangerous for space-probes and satellites), we use satellite on-line data available through the Internet. The problem is how to extrapolate the SEP energy spectrum from high NM energies to very low energies detected by GOES satellite. The main idea of this extrapolation is the following: 1) the time of ejection for high and small energy ranges (detected by NM and by satellite) is the same, so it can be determined by using only NM data; 2) the source function relative to time is a 8-function, and relative to energy is a power function with an energy-dependent index Y = Yo + ln(Ek/Eko) with maximum at

Fig. 2.43.5 shows results based on the NM and satellite data of forecasting of expected SEP fluxes also in small energy intervals and comparison with observation satellite data.

Fig. 2.43.5 shows results based on the NM and satellite data of forecasting of expected SEP fluxes also in small energy intervals and comparison with observation satellite data.

Fig. 2.43.5. Predicted SEP integral fluxes for Ek > Emin = 0.1, 1.0, and 3.0 GeV. The forecasted integral flux for Ek > Emjn = 0.1 GeV is compared with the observed fluxes for Ek > 100 MeV on GOES satellite. The ordinate is log10 of the SEP integral flux (in cm-2sec-1sr-1), and the abscissa is time in minutes from 11.40 UT of September 29, 1989. From Dorman et al. (2005a,b).

Results of comparison presented in Fig. 2.43.4 and Fig. 2.43.5 show that by using on-line data from ground NM in the high energy range and from satellite in the low energy range during the first 30-40 minutes after the start of the SEP event, it is possible by using only CR data to solve the inverse problem by formulas in Sections 2.42.2 and 2.42.3: to determine the properties of SEP source on the Sun (time of ejection into solar wind, source SEP energy spectrum, and total flux of accelerated particles) and parameters of SEP propagation in the interplanetary space (diffusion coefficient and its dependence from particle energy and from the distance from the Sun).

2.43.5. The inverse problems for great SEP events and space weather

Let us note that the solving of inverse problems for great SEP events has important practical sense: to predict the expected SEP differential energy spectrum on the Earth's orbit and integral fluxes for different threshold energies up to many hours (and even up to few days) ahead. The total (event-integrated) fluency of the SEP event, and the expected radiation hazards can also be estimated on the basis of the first 30-40 minutes after the start of the SEP event and corresponding Alerts to experts operating different objects in space, in magnetosphere, and in atmosphere at different altitudes and at different cut-off rigidities can be sent automatically. These experts should decide what to do operationally (for example, for space-probes in space and satellites in the magnetosphere to switch-off the electric power for few hours to save the memory of computers and high level electronics; for jets to decrease their altitudes from 10 km to 4-5 km to protect crew and passengers from great radiation hazard, and so on). From this point of view especially important are the solving of inverse problems for great SEP by using on-line data of many NM and several satellites in the frame of models in which CR propagation described by the theory of anisotropic diffusion or by kinetic theory. The solving of these inverse problems will made possible on the basis of world-wide CR Observatories and satellite data (in real scale time, applicable from Internet) to made forecasting on radiation hazard for much shorter time after SEP event beginning. These important problems are formulated below, in Section "Conclusions and Problems" at the end of monograph.

2.44. The inverse problems for CR propagation in the Galaxy

The main parameters of CR propagation in the Galaxy can be determined by the solving the inverse problem for the some model of CR propagation (boxes model, diffusion model in disc or/and in halo, model with galactic wind, model of rotating Galaxy with galactic wind driven by pressure of CR, and so on). Partly these models we consider in Chapter 3 with account nonlinear effects (which are sufficient in case of CR propagation in the Galaxy) and experimental data on relative content of radioactive nuclei in CR 10Be and others (what determines the average time-life of CR in the Galaxy), contents of elements Li, Be, B in CR (determined the grammar of matter transferred by CR before they escape from the Galaxy), and data on gamma ray distribution (determined the distribution of CR sources). The paper of Bloemen et al. (1993) can be considered as a classical example of using these data for solving inverse problem of CR propagation in the disc and halo of Galaxy with account galactic wind (see in detail Chapter 3, Section 3.13). As result it was determined the diffusion coefficient and the velocity of galactic wind.

In Sections 3.14-3.16 we consider in detail other important inverse problems for CR propagation and distribution in the Galaxy with taking into account non-linear phenomena: CR pressure and kinetic stream instabilities; galactic wind driving by CR and generation of Alfvén turbulence by CR and its influence on CR

propagation; self-consistent problem for dynamic halo in rotating Galaxy for CR propagation and space-distribution, for formation of galactic wind and magnetic field; transport of random magnetic fields from the disc by galactic wind driven by CR and its influence on CR propagation; nonlinear Alfven waves generated by CR streaming instability and their influence on CR propagation in the Galaxy; the balance of Alfven wave generation by CR with damping mechanisms, and others.

2.45. The inverse problem for high energy galactic CR propagation and modulation in the Heliosphere on the basis of NM data

2.45.1. Hysteresis phenomenon and the inverse problem for galactic CR propagation and modulation in the Heliosphere

By the solving of the inverse problem for galactic CR propagation and modulation in the interplanetary space on the basis of observation data of CR-SA (solar activity) hysteresis phenomenon can be obtained important information on the main properties of the Heliosphere. The investigation of the hysteresis phenomenon in the connection between long-term variations in CR intensity observed at the Earth and SA, started about 50 years ago (Dorman, M1957; Forbush, 1958; Neher and Anderson, 1962; Simpson, 1963; Dorman, Ml 963a, Ml963b). In the middle of 60-th many scientists came to conclusion that the dimension of modulation region (or Heliosphere) is about 5 AU, and not more than 10-15 AU (Quenby, 1965; Kudo and Wada, 1968; Charakhchyan and Charakhchyan, 1968, 1971; Stozhkov and Charakhchyan, 1969; Pathak and Sarabhai, 1970). It was found that the radius ro of the CR modulation region is very small either by analysis of the intensity of coronal green line in some helio-latitude regions (as controlled solar activity factor; in this case was obtain ro ~ 5 AU), or by investigation the CR modulation as caused by sudden jumps in solar activity (ro ~ 10-15 AU). In Dorman and Dorman (1965, 1967a,b,c), Dorman

(M 1975b) the hysteresis phenomenon was investigated on the basis of neutron monitor (NM) data for about one solar cycle in the frame of convection-diffusion model of CR global modulation in the Heliosphere with taking into account time lag of processes in the interplanetary space relative to processes on the Sun. It was shown that the dimension of the modulation region should be about 100 AU (much bigger than accepted in those time in literature, 5-15 AU). These investigations were continued on the basis of CR and SA monthly average data for about four solar cycles in Dorman et al. (1997, 1999). Let us note that many authors worked on this problem, used sunspot numbers or other parameters of solar activity for investigations of CR long-term variations, but they did not take into account time lag of processes in the interplanetary space relative to processes on the Sun as integral action (see review in Belov, 2000). The method, described below, takes into account that CR intensity observed on the Earth at moment t is caused by solar processes summarized for the long period started many months before t. In recent paper Dorman (2001) was considered again by this method CR and SA data for solar cycles 19-22, but with taking into account drift effects according to Burger and Potgieter (1999). It was shown that including in the consideration drift effects (as depending from the sign of solar polar magnetic field (sign of parameter A) and determined by difference of total CR modulation at A > 0 and A < 0, and with amplitude proportional to the value of tilt angle between interplanetary neutral current sheet and equatorial plane) is very important: it became possible to explain the great difference in time-lags between CR and SA in hysteresis phenomenon for even and odd solar cycles.

2.45.2. Hysteresis phenomenon and the model of CR global modulation in the frame of convection-diffusion mechanism

It was shown in Dorman and Dorman (1965) that the time of propagation through the Heliosphere of particles with rigidity bigger than 10 GV (to which NM are sensitive) is not longer than one month. This time is at least about one order of magnitude smaller than the observed time-lag in the hysteresis phenomenon. It means that the hysteresis phenomenon on the basis of NM data can be considered as quasi-stationary problem with parameters of CR propagation changing in time. In this case according to Parker (1958, M1963), Dorman (1959c):

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