6. Discussion

It has been said that we cannot give a general interpretation of our theoretical source spectra behavior on the sole basis of the relationships between the energy change rates (1)–(6) since their behavior in the events of Figure 2 is different from that in Figure 3 and both differ from that in Figure 4, implying that the kind of processes, their sequence of occurrence and their importance is not the same from event to event To interpret this behavior we cannot remit ourselves to the amount of traversed material, positing that particles originated in the invisible side of the sun or in the eastern hemisphere have lost more energy, because in that case events as such as the March 30, 1969 or February 2, 1969 ones would behave like the events of Table 1. Moreover, our hypothesis does not consider deceleration of particles after acceleration, while they traverse the solar atmosphere. Therefore, we believe that the explanation is on the basis of the parameter temperature: that is, we argue that solar proton flares develop under three main different temperature regimes, a low one that we shall denominate cold events (T≈103 –105K) (Table 3), an intermediate regime that we shall call warm events (≈10<sup>5</sup> –l0<sup>7</sup>K) (Table 4), and a high temperature regime that we shall call hereafter hot events (T > 107K) (Table 3). On the basis of this conjecture, let us discuss the main results of the preceding section:

Concerning points 1(a), 1(b) and 1(c), we can comment that as the medium was very hot, collisional losses were very high, making spectrum (18) better than spectrum (15); due to the high temperature and high density in the source nuclear reactions took place and thus spectrum (23) is even closer than (18) to the experimental curve.

Furthermore, the fact that the best fit is given by (27) seems to indicate that beyond a certain temperature, the source material is able to expand and consequently particles which have not escaped the source are adiabatically cooled. In addition, since spectrum (15) is better than (31) it is assumed that compression of the medium did not take place in high temperature regions, and so neither did adiabatic heating of protons. The irregular behavior of spectrum (23) at E ≤ 30 MeV and E ≥ 800 MeV in the January 28, 1967 event in relation to the tendency outlined in the last section, may be interpreted as indicating that the low energy protons observed in this event did not originate in. the same process, which explains why the observations show a high flux of protons at energy lower than the threshold acceleration value for in a medium of density n≈1013cm<sup>3</sup> • Therefore, these particles may form part of the high energy tail of a preliminary heating process which were not transported by the expanding material. This would mean that only deceleration by collisional losses and p–p collisions took place during the acceleratory process. At high energies, although energy losses from p–p collisions are stronger than collisional losses (Figure 1), it can be speculated that the low flux of high energy protons escape very fast from the acceleration region, so that the contribution of this process at high energies was not very important during the time scale of the acceleration.

g. If we ignore the fact that the assumed heliographic position of the flare associated to the January 28, 1967 event is relatively uncertain, it can be noted that there is a south asymmetry in the what we designate as hot events (Table 1), a north asymmetry in cold and warm events (Table 2) and a certain west and north asymmetry among the events of

h. The critical energy Ec from cold and warm events is correlated with the temperature of the source in the sense that their values increase from cold to warm and from warm to hot events. The significance of the association of the parameter temperature to solar proton

It has been said that we cannot give a general interpretation of our theoretical source spectra behavior on the sole basis of the relationships between the energy change rates (1)–(6) since their behavior in the events of Figure 2 is different from that in Figure 3 and both differ from that in Figure 4, implying that the kind of processes, their sequence of occurrence and their importance is not the same from event to event To interpret this behavior we cannot remit ourselves to the amount of traversed material, positing that particles originated in the invisible side of the sun or in the eastern hemisphere have lost more energy, because in that case events as such as the March 30, 1969 or February 2, 1969 ones would behave like the events of Table 1. Moreover, our hypothesis does not consider deceleration of particles after acceleration, while they traverse the solar atmosphere. Therefore, we believe that the explanation is on the basis of the parameter temperature: that is, we argue that solar proton flares develop under three main different temperature regimes, a low one that we shall denominate cold events (T≈103

high temperature regime that we shall call hereafter hot events (T > 107K) (Table 3). On the

Concerning points 1(a), 1(b) and 1(c), we can comment that as the medium was very hot, collisional losses were very high, making spectrum (18) better than spectrum (15); due to the high temperature and high density in the source nuclear reactions took place and thus spec-

Furthermore, the fact that the best fit is given by (27) seems to indicate that beyond a certain temperature, the source material is able to expand and consequently particles which have not escaped the source are adiabatically cooled. In addition, since spectrum (15) is better than (31) it is assumed that compression of the medium did not take place in high temperature regions, and so neither did adiabatic heating of protons. The irregular behavior of spectrum (23) at E ≤ 30 MeV and E ≥ 800 MeV in the January 28, 1967 event in relation to the tendency outlined in the last section, may be interpreted as indicating that the low energy protons observed in this event did not originate in. the same process, which explains why the observations show a high flux of protons at energy lower than the threshold acceleration value for in a medium of

preliminary heating process which were not transported by the expanding material. This

• Therefore, these particles may form part of the high energy tail of a

(Table 3), an intermediate regime that we shall call warm events (≈10<sup>5</sup>

trum (23) is even closer than (18) to the experimental curve.

basis of this conjecture, let us discuss the main results of the preceding section:

–105K)

–l0<sup>7</sup>K) (Table 4), and a

Table 3.

144 Cosmic Rays

6. Discussion

density n≈1013cm<sup>3</sup>

events will be discussed in Section 6.

Concerning point 2 of the last section, we assume that the acceleration process in the events of Figure 3 was carried out in a low temperature regime so that collisional losses were completely unimportant in relation to the acceleration rate, and nuclear reactions did not take place, at least within the acceleration phase. Furthermore, a compression of the local material is associated with low temperature regimes as indicated by the fact that spectrum (31) systematically gives the best fit to the experimental curves (e.g. November 12, 1960 event).

Points 3(a)–3(d) are interpreted as follows: the temperature and density associated with the acceleration region was high enough to favor nuclear reactions, but not the expansion of source material; consequently, collisional losses of low energy protons were important in the events of Figure 4, providing spectrum (23) with a better description of the experimental curve. Also, because the higher temperature does not allow for a compression of the material, spectrum (31) is systematically deflected in relation to spectrum (15). Furthermore, the sudden change in the order of the sequence of curves (15) (19) and (23) is the combined effect of the temperature associated to each event and the importance of the accelerated flux of high energy protons as discussed above with respect to the January 28, 1967 event; the lower the temperature the faster spectrum (19) deflects in relation to (15) (e.g. the November 15, 1960 and November 18, 1968 events); and the higher the flux of the accelerated high energy protons, the later spectrum (23) deflects in relation to (19) (e.g. the February 25, 1969 and January 24, 1971 events).

Related to point 3(e) of last section, it would appeal that the temperature associated with this event was not very high, so that collisional losses were significant only on the low energy protons. Because of the low flux of the accelerated protons in this event, the effect of p–p collisions diminishes as energy increases. This event behaves almost like the cold events of Figure 3, since energy losses are negligible in relation to the acceleration rate of high energy protons. The reason why beyond 2 GeV spectrum (19) is more deflected than (27) is that the latter includes the p–p contribution to this event and collisional losses are unimportant on high energy particles (Figure 1). Interpretation of 3(b) and 3(e) must also consider the fact that high energy particles escape faster from the acceleration volume, and so, they are subject to energy degradation by p–p collisions during the acceleration time.

The interpretation of 4(a) follows from the fact that in cold events the contribution of the adiabatic heating is translated into a lower effort of the acceleration mechanism; however, in the hot and warm events (Tables 1 and 3) adiabatic heating did not occur, and so no effect was produced.

In relation to the interpretation of 4(b) to 4(d) it must be pointed out that the inverse proportionality between α and Ec follows from the fact that for a given situation the requirement for effective acceleration is lowered while the acceleration efficiency becomes progressively higher. On the other hand, the addition of energy losses to a given situation (same row in the

Tables) generally entails an increase in the requirement of energy Ec, and thus an increase of α in order to exceed the new barrier. However, the irregularities synthetized in points 4(c) and 4 (d) of last section, which can be seen on Tables 1–3, that may be explained in the following manner: the critical energy, Ec, is defined at low energies where the effect of adiabatic deceleration is negligible in relation to the other processes involved (Figure 1), and thus for a same value of α the values of Ec from (19) and (23) are remarkably similar. Nevertheless, the decrease of the values of α in column 7 of Table 1 may be explained by the fact that although the requirement for acceleration is the same, as in column (6), a supplementary process is acting on the particles, and efficiency of the process is being lowered. Since Ec and α behave inversely, the value of Ec appears to increase; but in fact the real value of Ec in this event was 11.6 MeV. Besides, we see from columns 6 and 7 of Tables 2 and 3 that under the hypothetical situation of the presence of adiabatic cooling in these events, the efficiency α appears higher in relation to that of column 6, given that there is an additional barrier to overtake. The value of Ec should behave similarly, but since the value of Ec in (13) is the same as that in (19), then, this hypothetical increase of α shown in column 7 in relation to that of column (6) implies a decrease of the value of Ec in column 7; this in fact does not occur because adiabatic cooling did not take place and thus the real values of α and Ec in events of Tables 1 and 3 were those of columns 3 and 6 respectively. The interpretation of 4 (e) follows from the definitions of Eqs. (31) and (32), whereas points 4(f) and 4 (g) cannot have a coherent interpretation, what can be attributed to the complexity and variability of conditions from flare to flare (e.g. the medium density, temperature, conductivity, magnetic field strength, magnetic topologies, etc.). In relation to point 4(h) it must be mentioned that deduce the same result by discussing three main different temperature regimes in the acceleration region of solar particles [105]; they estimate threshold values for proton acceleration of 1, 2.7 and 5.5 MeV for a cold region, an intermediate one and a hot region. These values are slightly lower than ours, since they do not take into account all the energy loss processes wedid. In any event, as we discussed previously, the threshold value Ec increases with the temperature because energy loss processes are increased with this parameter.

temperature of the medium, and to suggest that the acceleration regions must be associated alternately with the hot and cold aspects present during a flare or even in a pre-flare state, but

Exploration of Solar Cosmic Ray Sources by Means of Particle Energy Spectra

Related with the expansion and compression of the source medium, there are some observa-

author in [102, 103] has studied hydromagnetic criteria for expansion and compression of the sunspot magnetic lines, which he distinguishes as two different phases of the flare development; although he shows that sometimes the expansion phase may not present itself according to our findings such as we found in warm and cold events. However, in Sakurai's model acceleration occurs during the compression phase, whereas our results indicate that expansion of the source material may also occur during the acceleration process; moreover, our analysis does not show indications of expansion and compression during the same event during the phase of particle acceleration. Nevertheless, we see that, with exception of the November 12, 1960 event, the acceleration efficiency is very high where there is a compression (cold event), presumably due to the strong spatial variations of the of the longitudinal and transversal field

It must be emphasized that we have taken into account that expansion of closed structures occurs only within a height lower than 0.6 to 1 solar radius, and thus expansions beyond this distance may be associated with propagation of shock waves generated in relation to type II burst or CME; therefore, our assumptions concern only adiabatic cooling through the local

In the specific case of the November 18, 1968 event, for which our results do not indicate any expansion of the source, observations reported a loop expansion; however s it is usually supported the fact that there is no mass motion but only a traveling excitation front. It must also be mentioned that it is generally accepted that low energy protons are much more likely to be subject to adiabatic cooling since high energy protons are rather dominated by drifts and scattering in field inhomogeneities [27, 33]; Moreover, according to [131, 132, 133] adiabatic deceleration disappears as the density of the accelerated particles decreases, so that when particle velocity is much higher than both the velocity of the medium and the Alfven velocity, the adiabatic cooling is null. This would imply that in the case of our hot events (Figure 2) protons of energy much higher than 670 MeV should not be adiabatically cooled in a medium of T > 10<sup>8</sup>K, however, our results show that even higher energy protons were adiabatically decelerated. Therefore, we claim that at least in these two events, our results support the hypothesis that particles were accelerated in closed magnetic field lines with high

Now turning to the problem of p–p nuclear collisions in some solar flares: we had mentioned that the value of NH1013 cm<sup>3</sup> was an average value in flare regions, since in fact concentrations as high as 1016 cm<sup>3</sup> have been reported (e.g. [118]) which implies that Eq. (23) and Eq. (27) will remain near the observational curves. This feature leads us to speculate that some flares have a high proton concentration medium (e.g. January 24, 1971), whereas in others the concentration is much lower (e.g. July 7, 1966), and that a great spread in high energy gamma rays and neutron fluxes is expected from flare to flare. The difference between observational and theoretical fluxes of gamma ray and neutrons is not a matter of discussion here, we only

K for expansion. The

147

http://dx.doi.org/10.5772/intechopen.77052

certainly under very different temperature regimes from flare to flare.

expansion of the source and not in higher the solar envelope.

lines, as suggested by [101, 102].

confinement efficiency.

tional indications [84] which propose a minimum value of <sup>3</sup> <sup>10</sup><sup>7</sup>

In addition to the suggestion of three temperature regions in acceleration regions extended by [105], several other suggestions have been presented in this direction: the author in [78] has discussed temperatures of 104K suggested by the central peak of hydrogen emission lines, up to more than 108K suggested by thermal emissions of X-rays. Furthermore, the flare phenomenon has usually been interpreted on basis of a dual character): the optical flare of <sup>T</sup>104K and high electron density, and on the other hand, the high energy flare plasma of <sup>T</sup>107 –109K and relatively low electron density. The existence of several temperature regimes during a given flare has also been evoked by suggesting that the emitting regions have a filamentary and intermingling structure with hot filaments about 1 km. of diameter imbedded in cooler material [113, 115], or by suggesting a cooling of a hot region during the flare development [17, 135]. Some other models for explaining the flare energy output suggest several phases of the phenomenon, each associated with a different temperature; for example, a of relatively low temperature thermal phase followed by an explosive high temperature phase [13, 50–52, 111] posit similar models. We have not attempted to place our results into the framework of what of any of these interpretations of the flare phenomenon, but rather only to demonstrate that the generation of solar particles is accompanied by, several processes whose occurrence is narrowly related to, the temperature of the medium, and to suggest that the acceleration regions must be associated alternately with the hot and cold aspects present during a flare or even in a pre-flare state, but certainly under very different temperature regimes from flare to flare.

Tables) generally entails an increase in the requirement of energy Ec, and thus an increase of α in order to exceed the new barrier. However, the irregularities synthetized in points 4(c) and 4 (d) of last section, which can be seen on Tables 1–3, that may be explained in the following manner: the critical energy, Ec, is defined at low energies where the effect of adiabatic deceleration is negligible in relation to the other processes involved (Figure 1), and thus for a same value of α the values of Ec from (19) and (23) are remarkably similar. Nevertheless, the decrease of the values of α in column 7 of Table 1 may be explained by the fact that although the requirement for acceleration is the same, as in column (6), a supplementary process is acting on the particles, and efficiency of the process is being lowered. Since Ec and α behave inversely, the value of Ec appears to increase; but in fact the real value of Ec in this event was 11.6 MeV. Besides, we see from columns 6 and 7 of Tables 2 and 3 that under the hypothetical situation of the presence of adiabatic cooling in these events, the efficiency α appears higher in relation to that of column 6, given that there is an additional barrier to overtake. The value of Ec should behave similarly, but since the value of Ec in (13) is the same as that in (19), then, this hypothetical increase of α shown in column 7 in relation to that of column (6) implies a decrease of the value of Ec in column 7; this in fact does not occur because adiabatic cooling did not take place and thus the real values of α and Ec in events of Tables 1 and 3 were those of columns 3 and 6 respectively. The interpretation of 4 (e) follows from the definitions of Eqs. (31) and (32), whereas points 4(f) and 4 (g) cannot have a coherent interpretation, what can be attributed to the complexity and variability of conditions from flare to flare (e.g. the medium density, temperature, conductivity, magnetic field strength, magnetic topologies, etc.). In relation to point 4(h) it must be mentioned that deduce the same result by discussing three main different temperature regimes in the acceleration region of solar particles [105]; they estimate threshold values for proton acceleration of 1, 2.7 and 5.5 MeV for a cold region, an intermediate one and a hot region. These values are slightly lower than ours, since they do not take into account all the energy loss processes wedid. In any event, as we discussed previously, the threshold value Ec increases with the temperature because energy loss processes are

In addition to the suggestion of three temperature regions in acceleration regions extended by [105], several other suggestions have been presented in this direction: the author in [78] has discussed temperatures of 104K suggested by the central peak of hydrogen emission lines, up to more than 108K suggested by thermal emissions of X-rays. Furthermore, the flare phenomenon has usually been interpreted on basis of a dual character): the optical flare of <sup>T</sup>104K and

relatively low electron density. The existence of several temperature regimes during a given flare has also been evoked by suggesting that the emitting regions have a filamentary and intermingling structure with hot filaments about 1 km. of diameter imbedded in cooler material [113, 115], or by suggesting a cooling of a hot region during the flare development [17, 135]. Some other models for explaining the flare energy output suggest several phases of the phenomenon, each associated with a different temperature; for example, a of relatively low temperature thermal phase followed by an explosive high temperature phase [13, 50–52, 111] posit similar models. We have not attempted to place our results into the framework of what of any of these interpretations of the flare phenomenon, but rather only to demonstrate that the generation of solar particles is accompanied by, several processes whose occurrence is narrowly related to, the

–109K and

high electron density, and on the other hand, the high energy flare plasma of <sup>T</sup>107

increased with this parameter.

146 Cosmic Rays

Related with the expansion and compression of the source medium, there are some observational indications [84] which propose a minimum value of <sup>3</sup> <sup>10</sup><sup>7</sup> K for expansion. The author in [102, 103] has studied hydromagnetic criteria for expansion and compression of the sunspot magnetic lines, which he distinguishes as two different phases of the flare development; although he shows that sometimes the expansion phase may not present itself according to our findings such as we found in warm and cold events. However, in Sakurai's model acceleration occurs during the compression phase, whereas our results indicate that expansion of the source material may also occur during the acceleration process; moreover, our analysis does not show indications of expansion and compression during the same event during the phase of particle acceleration. Nevertheless, we see that, with exception of the November 12, 1960 event, the acceleration efficiency is very high where there is a compression (cold event), presumably due to the strong spatial variations of the of the longitudinal and transversal field lines, as suggested by [101, 102].

It must be emphasized that we have taken into account that expansion of closed structures occurs only within a height lower than 0.6 to 1 solar radius, and thus expansions beyond this distance may be associated with propagation of shock waves generated in relation to type II burst or CME; therefore, our assumptions concern only adiabatic cooling through the local expansion of the source and not in higher the solar envelope.

In the specific case of the November 18, 1968 event, for which our results do not indicate any expansion of the source, observations reported a loop expansion; however s it is usually supported the fact that there is no mass motion but only a traveling excitation front. It must also be mentioned that it is generally accepted that low energy protons are much more likely to be subject to adiabatic cooling since high energy protons are rather dominated by drifts and scattering in field inhomogeneities [27, 33]; Moreover, according to [131, 132, 133] adiabatic deceleration disappears as the density of the accelerated particles decreases, so that when particle velocity is much higher than both the velocity of the medium and the Alfven velocity, the adiabatic cooling is null. This would imply that in the case of our hot events (Figure 2) protons of energy much higher than 670 MeV should not be adiabatically cooled in a medium of T > 10<sup>8</sup>K, however, our results show that even higher energy protons were adiabatically decelerated. Therefore, we claim that at least in these two events, our results support the hypothesis that particles were accelerated in closed magnetic field lines with high confinement efficiency.

Now turning to the problem of p–p nuclear collisions in some solar flares: we had mentioned that the value of NH1013 cm<sup>3</sup> was an average value in flare regions, since in fact concentrations as high as 1016 cm<sup>3</sup> have been reported (e.g. [118]) which implies that Eq. (23) and Eq. (27) will remain near the observational curves. This feature leads us to speculate that some flares have a high proton concentration medium (e.g. January 24, 1971), whereas in others the concentration is much lower (e.g. July 7, 1966), and that a great spread in high energy gamma rays and neutron fluxes is expected from flare to flare. The difference between observational and theoretical fluxes of gamma ray and neutrons is not a matter of discussion here, we only want to note that these fluxes are mainly generated from the most energetic protons which are in fact the first to escape and do not frequently interact with the medium, as discussed previously in relation with some events of Figures 2 and 4. This implies that depending on the magnetic confinement efficiency in each flare, the expected flux of the secondary radiation will be of greater or lesser importance. According to Figures 2 and 4 a high gamma ray flux must be generated in the February 25, 1969, January 24, 1971 and September 1, 1971 events, whereas a lower flux should be expected from the July 7, 1966 event and no gamma-ray fluxes from nuclear collision in the acceleration volume must be expected in the events of Figure 3. The variability of the expected high energy gamma-ray fluxes has been previously discussed in [25]. Concerning neutron fluxes we argue that they are strongly absorbed by a neutron capture reaction (n+ H<sup>3</sup> <sup>e</sup> ! <sup>H</sup><sup>3</sup> + p).

an average value for different energies of protons, we shall estimate the average value of ι for a 50 MeV proton and assume that the value of ι is typical of the acceleration region configura-

Exploration of Solar Cosmic Ray Sources by Means of Particle Energy Spectra

obtained are α = 0.1 and 1.54 s�<sup>1</sup> leading to the following values: ι =10 Km and 0.84 Km respectively, which are of the same order as the values found by Perez-Peraza (1975) for multi-GeV solar protons. To estimate τ in a magnetic field (H) where the field gradient is ≈H/ ι, we use the fact that τ = L<sup>2</sup> /vι, where L is the linear size of the acceleration region; an approximate' value of L may be deduced by the fact that the volume of flare regions varies from 10<sup>25</sup> to 1029cm3 from flare to flare [19, 54, 55], and hence a linear dimension of �109 cm may be considered as a typical value [30, 31] Assuming that the acceleration volume cannot be greater, than the flare volume, we shall consider L = 108 cm as a typical linear dimension for acceleration regions [116]. In such conditions we obtain τ = 1 and 12.6 s. for solar events where α= 0.13 and 1.54 s�<sup>1</sup> respectively. We should say that if a shorter length scale L than the assumed one were taken values of τ <1 could be obtained, and hence our theoretical fluxes J (>E) would come closer to experimental curve as discussed above. In fact, it can be observed in Figure 3, that the theoretical curve corresponding to α=0.13 and thus to a low value of τ (the November 12, 1960 event) is nearer the experimental curve than to the theoretical curve corresponding to higher values of α, where it is supposed that τ must be higher. It must be noted that a higher value of α in one event with respect to another event does not imply a shorter escape time for particles in the former with respect to the latter, because the source conditions are not the same from one event to the other, as can be seen from the fact that magnetic inhomogeneities are much closer between them in events of high acceleration efficiency. We have considered a second-order Fermi-type mechanism to illustrate that even in the extreme case of such low efficiency the acceleration process may be performed within the flare time scale and to show that the assumption of τ = 1 s is well justified. If instead of a secondorder Fermi mechanism we consider a first-order Fermi-type process in a shock wave, such as is usually attributed to the acceleration of solar particles (e.g. [32, 110]) the resulting value of τ is then lower than 1 s. From the study of heavy nuclei overabundances in solar cosmic rays it can be predicted that the value of τ is comprised between 0.1 and 0.4 s; these values when included in our calculations result in a much better fit of the theoretical spectra to the observa-

The acceleration time scale of protons in solar flares, can be estimated from the following

in low temperature regimens

f Eð Þ¼ <sup>d</sup> � hE�<sup>2</sup> � jE�<sup>1</sup> � �β<sup>W</sup> � <sup>b</sup>=<sup>β</sup> in intermediate temperature regions

f Eð Þ¼ <sup>d</sup> � hE�<sup>2</sup> � jE�<sup>1</sup> � �<sup>β</sup> � <sup>r</sup><sup>2</sup> � �<sup>W</sup> � <sup>b</sup>=<sup>β</sup> in high temperature regimens

f Eð Þ : In the energy range 106 <sup>≲</sup> <sup>E</sup>≲1010 eV we have according our results

, the extreme values of α

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tion; hence for a field strength of 500 G and density n = 10<sup>13</sup> cm�<sup>3</sup>

tional curves that the one illustrated with τ = 1 s.

expression: <sup>t</sup> <sup>¼</sup> <sup>Ð</sup> <sup>E</sup>

f Eð Þ¼ αβ<sup>W</sup>

(where) d ¼ α � f � η

�

Ec dE

discussed in last section that,

αβ <sup>þ</sup> <sup>r</sup>β<sup>2</sup> � �<sup>W</sup>

It must be pointed out that the need of protons for a minimum energy in order to overtake energy losses and to be accelerated upwards, measured energies may not be a strong requirement since the temporal and spatial sequence of phenomena in a flare seem to indicate the occurrence of a two-step acceleration of solar particles (e.g. [19, 16, 123]). A great variety of preliminary acceleration processes capable of accelerating particles up to some MeV has been suggested (e .g. [104, 112], etc.). It �can be assumed that a certain portion of the low energy tail of the particle spectrum may belong to the first acceleration step. By smoothing the experimental data we have obtained a peculiar shape for this low energy tail of some spectra, although a similar shape is predicted from the theoretical point of view [5]. Moreover, authors in [94] discuss a noticeable deviation of the power spectrum below ≈ 4 MeV in low energy proton events, which they attribute to collisional losses during storage in the ionized medium of the low corona. We are aware of the difficulty of estimating the exact shape of the low energy spectrum, due to the strong modulation of these particles either within or outside of the source. Therefore, we argue that in addition to energy losses, this particular slope change in the low energy tail of some spectra may be due to an upper cutoff in the preliminary acceleration process.

Now let us discuss the assumption made in Section 5 in taking τ as a constant value: although it is expected that the mean confinement time varies according to particle rigidity, it is not clear if the escape mechanism from the source occurs through leakage, by thin or thick scattering, by curvature drifts, by gradient drifts or even by a sudden catastrophic disruption of a closed magnetic structure at the source; therefore, we opted for a mean value τ = 1 sec. Whose implications can be seen as follows: we note from Eq. (11) that if the value of τ increases, then J(>E) increases, whereas if τ decreases, then J(>E) decreases and so the theoretical spectra will approximate the experimental curves. At any rate, what can be deduced is that if τ is either lower or higher than the assumed value, the sequence of theoretical spectra does not change or consequently our conclusions are not altered. In order to evidence that the value of τ is in general of the order assumed, we shall develop the following considerations: if we make the extreme assumption that acceleration of solar protons is performed by a low efficiency process, such as a second-order Fermi-type mechanism then we know that in these cases the acceleration efficiency is given as α= V<sup>2</sup> <sup>a</sup> /vι, where v is the velocity of protons, ι the acceleration step within the acceleration volume, and Va the hydromagnetic velocity of the magnetic field irregularities. Taking into account that our values of α in a given event can be considered as an average value for different energies of protons, we shall estimate the average value of ι for a 50 MeV proton and assume that the value of ι is typical of the acceleration region configuration; hence for a field strength of 500 G and density n = 10<sup>13</sup> cm�<sup>3</sup> , the extreme values of α obtained are α = 0.1 and 1.54 s�<sup>1</sup> leading to the following values: ι =10 Km and 0.84 Km respectively, which are of the same order as the values found by Perez-Peraza (1975) for multi-GeV solar protons. To estimate τ in a magnetic field (H) where the field gradient is ≈H/ ι, we use the fact that τ = L<sup>2</sup> /vι, where L is the linear size of the acceleration region; an approximate' value of L may be deduced by the fact that the volume of flare regions varies from 10<sup>25</sup> to 1029cm3 from flare to flare [19, 54, 55], and hence a linear dimension of �109 cm may be considered as a typical value [30, 31] Assuming that the acceleration volume cannot be greater, than the flare volume, we shall consider L = 108 cm as a typical linear dimension for acceleration regions [116]. In such conditions we obtain τ = 1 and 12.6 s. for solar events where α= 0.13 and 1.54 s�<sup>1</sup> respectively. We should say that if a shorter length scale L than the assumed one were taken values of τ <1 could be obtained, and hence our theoretical fluxes J (>E) would come closer to experimental curve as discussed above. In fact, it can be observed in Figure 3, that the theoretical curve corresponding to α=0.13 and thus to a low value of τ (the November 12, 1960 event) is nearer the experimental curve than to the theoretical curve corresponding to higher values of α, where it is supposed that τ must be higher. It must be noted that a higher value of α in one event with respect to another event does not imply a shorter escape time for particles in the former with respect to the latter, because the source conditions are not the same from one event to the other, as can be seen from the fact that magnetic inhomogeneities are much closer between them in events of high acceleration efficiency. We have considered a second-order Fermi-type mechanism to illustrate that even in the extreme case of such low efficiency the acceleration process may be performed within the flare time scale and to show that the assumption of τ = 1 s is well justified. If instead of a secondorder Fermi mechanism we consider a first-order Fermi-type process in a shock wave, such as is usually attributed to the acceleration of solar particles (e.g. [32, 110]) the resulting value of τ is then lower than 1 s. From the study of heavy nuclei overabundances in solar cosmic rays it can be predicted that the value of τ is comprised between 0.1 and 0.4 s; these values when included in our calculations result in a much better fit of the theoretical spectra to the observational curves that the one illustrated with τ = 1 s.

want to note that these fluxes are mainly generated from the most energetic protons which are in fact the first to escape and do not frequently interact with the medium, as discussed previously in relation with some events of Figures 2 and 4. This implies that depending on the magnetic confinement efficiency in each flare, the expected flux of the secondary radiation will be of greater or lesser importance. According to Figures 2 and 4 a high gamma ray flux must be generated in the February 25, 1969, January 24, 1971 and September 1, 1971 events, whereas a lower flux should be expected from the July 7, 1966 event and no gamma-ray fluxes from nuclear collision in the acceleration volume must be expected in the events of Figure 3. The variability of the expected high energy gamma-ray fluxes has been previously discussed in [25]. Concerning neutron fluxes we argue that they are strongly absorbed by a neutron capture

It must be pointed out that the need of protons for a minimum energy in order to overtake energy losses and to be accelerated upwards, measured energies may not be a strong requirement since the temporal and spatial sequence of phenomena in a flare seem to indicate the occurrence of a two-step acceleration of solar particles (e.g. [19, 16, 123]). A great variety of preliminary acceleration processes capable of accelerating particles up to some MeV has been suggested (e .g. [104, 112], etc.). It �can be assumed that a certain portion of the low energy tail of the particle spectrum may belong to the first acceleration step. By smoothing the experimental data we have obtained a peculiar shape for this low energy tail of some spectra, although a similar shape is predicted from the theoretical point of view [5]. Moreover, authors in [94] discuss a noticeable deviation of the power spectrum below ≈ 4 MeV in low energy proton events, which they attribute to collisional losses during storage in the ionized medium of the low corona. We are aware of the difficulty of estimating the exact shape of the low energy spectrum, due to the strong modulation of these particles either within or outside of the source. Therefore, we argue that in addition to energy losses, this particular slope change in the low energy tail of some

Now let us discuss the assumption made in Section 5 in taking τ as a constant value: although it is expected that the mean confinement time varies according to particle rigidity, it is not clear if the escape mechanism from the source occurs through leakage, by thin or thick scattering, by curvature drifts, by gradient drifts or even by a sudden catastrophic disruption of a closed magnetic structure at the source; therefore, we opted for a mean value τ = 1 sec. Whose implications can be seen as follows: we note from Eq. (11) that if the value of τ increases, then J(>E) increases, whereas if τ decreases, then J(>E) decreases and so the theoretical spectra will approximate the experimental curves. At any rate, what can be deduced is that if τ is either lower or higher than the assumed value, the sequence of theoretical spectra does not change or consequently our conclusions are not altered. In order to evidence that the value of τ is in general of the order assumed, we shall develop the following considerations: if we make the extreme assumption that acceleration of solar protons is performed by a low efficiency process, such as a second-order Fermi-type mechanism then we know that in these cases the accelera-

within the acceleration volume, and Va the hydromagnetic velocity of the magnetic field irregularities. Taking into account that our values of α in a given event can be considered as

<sup>a</sup> /vι, where v is the velocity of protons, ι the acceleration step

spectra may be due to an upper cutoff in the preliminary acceleration process.

reaction (n+ H<sup>3</sup>

148 Cosmic Rays

<sup>e</sup> ! <sup>H</sup><sup>3</sup> + p).

tion efficiency is given as α= V<sup>2</sup>

The acceleration time scale of protons in solar flares, can be estimated from the following expression: <sup>t</sup> <sup>¼</sup> <sup>Ð</sup> <sup>E</sup> Ec dE f Eð Þ : In the energy range 106 <sup>≲</sup> <sup>E</sup>≲1010 eV we have according our results discussed in last section that,

$$\begin{aligned} f(E) &= \begin{cases} a\beta W \\ (a\beta + \rho\beta^2)W \end{cases} \text{ in low temperature regimes} \\\\ f(E) &= \left(d - hE^{-2} - jE^{-1}\right)\beta W - b/\beta \text{ in intermediate temperature regions} \\\\ f(E) &= \left[\left(d - hE^{-2} - jE^{-1}\right)\beta - \rho^2\right]W - b/\beta \text{ in high temperature regimes} \\\\ f(\text{where}) & d = \alpha - f - \eta \end{aligned}$$

Therefore, a consideration of the parameters obtained α and Ec for a medium density n = 1013 cm�<sup>3</sup> give acceleration times much lower than the time scale of the explosive phase of the flare phenomenon. For instance, for a low efficiency event (α = 0.14) in a high temperature regime, the time necessary to accelerate a proton from 10 MeV to 5000 MeV, is only of the order of 8 sec.

It is interesting to comment on the estimated parameter ι on the basis of our results of the parameter α: as pointed out by [102] the time scale of the explosive phase in solar flares, is �10<sup>3</sup> s, and it is believed to be that of the stored magnetic energy dissipation, which is given as

$$
\pi\_d = 4\pi\sigma l^2 / c^2 \tag{35}
$$

 a half of their energy before escaping into interstellar space. Moreover, the acceleration of particles in interplanetary space [21, 22, 85] may strongly disturb the spectrum. Given the strong modulation of solar particles at different levels, one cannot expect a good fit between the predicted source spectrum and the experimental one. Nevertheless, we believe that the kind of intercomparison performed here permits the clarification of ideas about the processes

Exploration of Solar Cosmic Ray Sources by Means of Particle Energy Spectra

http://dx.doi.org/10.5772/intechopen.77052

151

In order to provide some answers to the numerous questions associated with the generation of solar particles (e.g. [24, 26, 71, 102, 119]) we have attempted to study the physical processes and physical conditions prevailing in solar cosmic ray sources by separating source level effects from interplanetary and solar atmospheric effects. On this basis, we have drawn some inferences from the intercomparison of the predicted theoretical energy spectra of protons in the acceleration region with the experimental spectra of multi-GeV proton events. Concerning this kind of events a number of modern techniques have been recently developed (e.g. [72]) and the, the PGI group in Apatity, Mursmansk, Russia [124–128]. In some of GLE it has been frequent to discern two particles populations: a prompt component and a delayed one. A new kind of classification has been proposed, GLE's and SubGLE's depending the number of station

We have chosen to study this particular kind of solar events (GLE) because they allow the study of the behavior of local modulation on protons, through the widest range of solar particle energies. Although one should expect that local modulation by particle energy losses at the source should follow the behavior illustrated in Figure 1, our results on source energy spectra indicate that is not the general case, but local modulation varies from event to event, depending on the particular phenomena that take place at the source according to the particular physical parameters prevailing in each event, such as density, temperature, magnetic field strength as well as the acceleration efficiency and particle remaining time before they escape

In drawing conclusions about the physical processes at the source, we have assumed a fixed value of the parameter n, taking into account that although spectroscopic measurements show a variation in the value of n from flare to flare, these fluctuations are nonetheless very near the value n = 1013 cm<sup>3</sup> [115], and thus our conclusions about energy loss processes in the acceleration region are not significantly altered by small fluctuation on this parameter. Moreover, an analysis of the electromagnetic emission associated with flares indicate a spread of

chosen to fix the parameter n in order to concentrate our analysis on the parameter temperature. On the other hand, in drawing conclusions about the physical parameter of the acceleration process we have selected a mechanism with an energy gain rate proportional to particle energy as is the case of stochastic acceleration by MHD turbulence [36]; nevertheless, we believe that our results can in general be considered as valid, in the sense that whatever the

–108K), hence we have

that register the earth level enhancement, location and latitude of NM stations.

several decades on the medium temperature in flare regions (104

related to the generation of solar flare particles.

7. Concluding remarks

from the source.

where l is the characteristic length of the system and σ the electrical conductivity in flare material is of the order of 2.1 � 1012–2.4 � 1014 <sup>s</sup> �1 . A single calculation with (35) shows us that <sup>l</sup> = 1.7 � 104 –1.8 � l0<sup>5</sup> cm which agrees well with the values estimated in this work and previously deduced by [79].

It worth comment on the discrepancy between the predicted theoretical energy spectra at the source and the experimental spectra measured in the earth environment: first we note that the physical processes that can occur in a medium as dense as the sun's atmosphere are undoubtedly very diverse, and so, we do not claim to have included in our treatment all loss processes for charged particles, but only those of greatest interest that can affect protons within the energy range we are concerned with and during the short time scale of the acceleration durability. In fact, although Cerenkov losses are included in Eq. (2) we have ignored other losses from collective effects, however, some of them, such as energy 10 s by plasma perturbations see to be negligible for protons o f E > 23 MeV; also we have not considered energy losses caused by viscosity and Joule dissipation as suggested by [120]. On the other hand, we have not included nuclear transformation within the acceleration volume, as for instance proton production by neutron capture, nor loss of particles from the accelerated flux as leakage from the acceleration volume. Therefore, it is expected that the consideration of these neglected processes, together with a lower value of τ as discussed above and a higher proton concentration of the medium would depress our theoretical fluxes in greater congruency with the experimental curves. Again, local modulation of particles at the source level after acceleration are not examined here, either by an energy degradation step in a closed magnetic structure, or while traversing the dense medium of the solar atmosphere as studied by [121].

In fact, observations of low energy particles indicate the existence of a strong modulation within a small envelope of � 0.2–0.3 A.U. (e.g. [34]). Furthermore, studies of relativistic solar flare particles during the May 4,.1960 and November 18, 1968 events have shown that particles diffuse in the solar envelope (< 30 Rs) [9, 21, 22, 63] which entails a modulation of the solar fluxes. Evidences of partic1e storage in the sun, where particles can be strongly decelerated, have been widely mentioned in the literature (e.g. [1, 65, 106]). Modulation in interplanetary space is a complicated process (e.g. [28, 29]) which provokes both the depression in the number density of particles and their strong deceleration: estimations of [74] indicate that particles lose � 10–64% of their energy through propagation, while [75, 76] sustains a loss of  a half of their energy before escaping into interstellar space. Moreover, the acceleration of particles in interplanetary space [21, 22, 85] may strongly disturb the spectrum. Given the strong modulation of solar particles at different levels, one cannot expect a good fit between the predicted source spectrum and the experimental one. Nevertheless, we believe that the kind of intercomparison performed here permits the clarification of ideas about the processes related to the generation of solar flare particles.
