5. Procedure and results

<sup>J</sup>ð Þ¼ <sup>&</sup>gt; <sup>E</sup> <sup>N</sup>0et zð Þ<sup>c</sup> <sup>=</sup><sup>τ</sup> <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>R</sup>1<sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup> �

> � � � �θ<sup>5</sup> z�θ<sup>6</sup> � � � � �

energy cutoff value in the acceleration process.

ation and compression of the local material

tion time up to the energy E is given as

<sup>α</sup>2�r<sup>2</sup> ð Þ , <sup>ψ</sup> <sup>¼</sup> <sup>α</sup>

mc<sup>2</sup>τ αj j<sup>χ</sup>=<sup>2</sup>

<sup>J</sup>ð Þ¼ <sup>&</sup>gt; <sup>E</sup> <sup>N</sup><sup>0</sup>

� � � � �

and then the integral spectrum is simply given as

� � � � �

0 @

j j <sup>α</sup> <sup>χ</sup>=<sup>2</sup>

which in terms of kinetic energy becomes,

�θ<sup>4</sup> <sup>2</sup>zþR1�ð Þ <sup>Δ</sup><sup>1</sup> <sup>1</sup>=<sup>2</sup> <sup>2</sup>zþR1þð Þ <sup>Δ</sup><sup>1</sup> <sup>1</sup>=<sup>2</sup>

> ð Þ �Δ<sup>4</sup> 1=2 !#

� � �

<sup>þ</sup>θ<sup>9</sup> tan �<sup>1</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>7</sup>

þR8j

138 Cosmic Rays

<sup>2</sup>EmMc<sup>2</sup><sup>Þ</sup>

where <sup>¼</sup> <sup>r</sup>

<sup>N</sup>ð Þ <sup>γ</sup> <sup>d</sup><sup>γ</sup> <sup>¼</sup> <sup>N</sup><sup>0</sup>

1=2 . � �

where t(Zm) is the evaluation of Eq. (25) in <sup>Z</sup> <sup>¼</sup> <sup>γ</sup><sup>m</sup> � <sup>γ</sup><sup>2</sup>

� exp ð Þ �t zð Þ <sup>m</sup> =τ

��θ<sup>1</sup> <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>R</sup>3<sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>4</sup> �

exp <sup>θ</sup><sup>7</sup> tan �<sup>1</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>3</sup>

)

In order to express the previous equation as a function of the kinetic energy E, the variable Z should be written as Z Eð Þ¼ <sup>E</sup> <sup>þ</sup> Mc<sup>2</sup> � � � <sup>E</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>EMc<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> and Z Eð Þ<sup>m</sup> <sup>¼</sup> Em <sup>þ</sup> Mc<sup>2</sup> � � � <sup>E</sup><sup>2</sup>

It is also interesting to analyze the opposite case, when instead of an expansion of the source materials, there is a compression of the source medium (e.g. [101–103]) with a consequent adiabatic acceleration of the flare particles, which entail a change of sign in the last term of the net energy change rate (24). Let us develop the situation for which energy losses are completely negligible in relation to the acceleration rate during the stochastic particle acceler-

As in the case of Eq. (12) the threshold for acceleration is meaningless, and thus the accelera-

2

� � � � �

χ

� � � � �

<sup>γ</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �<sup>1</sup>

<sup>γ</sup> � <sup>γ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>1</sup>

<sup>γ</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �<sup>1</sup>

<sup>γ</sup> � <sup>γ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>1</sup>

2

� � � � �

�ψ=τ

2

� � � � �

�ψ=τ

2

2

<sup>γ</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �<sup>1</sup>

<sup>γ</sup> � <sup>γ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>1</sup>

2

� � � � � ψ

<sup>γ</sup><sup>ω</sup>j j <sup>α</sup> <sup>χ</sup>

<sup>r</sup>2�<sup>α</sup>2, consequently, the differential spectrum of particles is

<sup>γ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>1</sup>

<sup>γ</sup>�ω=<sup>τ</sup> � <sup>e</sup>

2

ð Þ¼ <sup>d</sup>γ=dt α γ<sup>2</sup> � <sup>1</sup> � ��1=<sup>2</sup>

αγ <sup>þ</sup> <sup>r</sup> <sup>γ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>1</sup>

2

γ αγ <sup>þ</sup> <sup>r</sup> <sup>γ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>1</sup>

� � � � �

�χ=τ

2

� � � � �

�χ=τ

� � � � �

� � � � �

<sup>t</sup> <sup>¼</sup> ln <sup>γ</sup>

<sup>2</sup> <sup>α</sup>2�r<sup>2</sup> ð Þ and <sup>ω</sup> <sup>¼</sup> <sup>r</sup>

γ αγ <sup>þ</sup> <sup>r</sup> <sup>γ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>1</sup>

� � � � �

0 @ � �

� n�

ð Þ �Δ<sup>2</sup> 1=2

!

��θ<sup>2</sup> <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>R</sup>5<sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>6</sup> �

" !

� �

<sup>þ</sup> <sup>θ</sup><sup>8</sup> tan �<sup>1</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>5</sup>

ð Þ �Δ<sup>3</sup> 1=2

<sup>m</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> corresponding to the high

<sup>þ</sup> <sup>r</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �γ�<sup>1</sup> (28)

1

γð Þ <sup>1</sup>�ω=<sup>τ</sup> dγ

<sup>2</sup> αγ <sup>þ</sup> <sup>r</sup> <sup>γ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>1</sup>

�<sup>t</sup> <sup>γ</sup>ð Þ<sup>m</sup> <sup>=</sup><sup>τ</sup>

1

2 h i (30)

A (31)

A (29)

��θ<sup>3</sup> <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>R</sup>7<sup>z</sup> �

(27)

<sup>m</sup><sup>þ</sup> �

As seen in the preceding section, the calculation of our theoretical spectra, Eqs. (15),(19), (23), (27) and (31) requires three fundamental parameters, one of them directly related to the physical state of flare regions, that is, the medium concentration n, and the others concerning the acceleration mechanism itself, that is, the acceleration efficiency α and the mean confinement time τ. These last two depend of course on some of the physical parameters of the source, which we attempt to estimate from the appropriate values of α and τ. In the case of the solar source, we have considered the mean value of the electron density and a conservative value for the proton population as ne ≈ nH = 1013 cm�<sup>3</sup> (e.g. [19, 35, 56, 113, 114, 116, 118]).

This assumption locates the acceleration region in chromospheric densities in agreement with some analysis of the charge spectrum of solar cosmic rays [64, 92].

Besides, since our expressions contain the acceleration parameter as the product ατ and since we are dealing with particles of the same species, for the sake of simplicity we have adopted the assumption τ = 1�s which allows us to separate the behavior of the acceleration efficient α in order to analyze it through several events and several source conditions. In any event, this value falls within the generally accepted range (e.g. [130, 131]); we shall discuss the implications of this assumption in the next section.

The determination of α has been carried out through the following procedure: in order to represent the theoretical spectrum within the same scale as that of the experimental curve, we have normalized both fluxes at the minimum energy for which available experimental data are effectively trustworthy, in such a way as to state the maximum flux of particles at the normalization energy, Enor

$$\left[\left[f(>E)\_{acc}\right]\_{E\_{mr}}=\eta\left[\left[f(>E)\_{earth}\right]\_{E\_{mr}}\right]\_{E\_{mr}}\tag{32}$$

where q is the normalization factor. Since our expressions do not directly furnish the source integral spectrum but rather J(>E)/N0, we have deduced in this way a normalization flux K0, keeping the same proportion with the differential flux N<sup>0</sup> appearing in our expressions

$$N\_0 = qk\_0 = \text{protons} / 4\pi R\_{\text{SE}}^2 \text{s} \quad \text{(protons/cm}^2 \text{str s)}\tag{33}$$

experimental curve under the same conditions, we could proceed to fix the value of the acceleration parameters in advance, which would entails making a priori inferences about the physical parameters of the source involved in the acceleration process of a given solar event; furthermore, this would result in a bias for the interpretation of the phenomenology involved in each event depending on the selected value of the efficiency α; that is, high values would give systematically the best fit with spectrum (27), whereas low values would show a systematically better fit with spectrum (15). Therefore, we proceeded conversely by determining the appropriate parameters of the source from the value of α in the theoretical spectra that best represents the experimental curve. The optimum values of α, obtained for each of the theoretical curves allows us to determine the critical energy E<sup>c</sup> and the normalization flux Ko appropriate to each case. We have tabulated the values of α, Ec and Ko obtained for every event through calculations of the spectra (15) (19), (23), (27) and (31) in Tables 1–3. We have illustrated the optimum theoretical curves on Figures 2–4. From an examination of these results, it can be observed that no general conclusion can be drawn about the behavior of our theoretical spectra by the simple comparison of energy change rates (1), (2), (4) or (6) at different energy values 7 as if the medium density n were the only important parameter in determining the processes occurring at the source. Other factors must intervene, as can be seen from the fact that spectra behavior changes from event to event. Nevertheless, according to the behavior of particle spectra, we can group the solar events in three groups of similar characteristics: those illustrated in Figure 2, which we shall denominate hot events, where it can be seen that theoretical spectra progressively approach the experimental curves while adding energy loss processes to the acceleration rate. Therefore, the physical processes taking place at the source in those events are described by spectrum (27) indicating that adiabatic cooling of protons together with energy degradation from p–p collisions and collisional losses may have taken place. In this case spectrum (31) (illustrated only in the January 28, 1967 event) is systematically the more deflected curve, showing the absence of adiabatic compression, at least during the acceleration period. Figure 3 shows the second group which we will call cold events, and

Table 3. Characteristic parameters of the acceleration process in solar protons warm events: acceleration efficiency α, high energy cutoff Em, normalization energy En, flux of accelerated particles in the source N<sup>0</sup> and heliographic coordinates of the

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flare according to different reports.

where RSE <sup>¼</sup> <sup>1</sup>:<sup>5</sup> � 1013cm = sun-earth distance. We have listed Enor for every event on columns 8 of Tables 1–3.

The value of N<sup>0</sup> for every event is tabulated on columns 10 of Tables 1–3.

Assuming that the theoretical curve among Eqs. (15), (19), (23), (27) and (31) is near the experimental curve in a given event, describes the kind of phenomena occurring at the source better, we have proceeded to perform this intercomparison according to the following criterion: first, the condition stated by Eq. (32) at the normalization energy and, second, that J(>E) ≈ 0 at the high energy cutoff Em. In order to compare each one of the theoretical spectra with an


Table 1. Characteristic parameters of the acceleration process in solar protons hot events: acceleration efficiency α, high energy cutoff Em, normalization energy En, flux of accelerated particles in the source N<sup>0</sup> and heliographic coordinates of the flare according to different reports.


Table 2. Characteristic parameters of the acceleration process in solar protons cold events: acceleration efficiency α, high energy cutoff Em, normalization energy En, flux of accelerated particles in the source N<sup>0</sup> and heliographic coordinates of the flare according to different reports.


where q is the normalization factor. Since our expressions do not directly furnish the source integral spectrum but rather J(>E)/N0, we have deduced in this way a normalization flux K0, keeping the same proportion with the differential flux N<sup>0</sup> appearing in our expressions

where RSE <sup>¼</sup> <sup>1</sup>:<sup>5</sup> � 1013cm = sun-earth distance. We have listed Enor for every event on columns

Assuming that the theoretical curve among Eqs. (15), (19), (23), (27) and (31) is near the experimental curve in a given event, describes the kind of phenomena occurring at the source better, we have proceeded to perform this intercomparison according to the following criterion: first, the condition stated by Eq. (32) at the normalization energy and, second, that J(>E) ≈ 0 at the high energy cutoff Em. In order to compare each one of the theoretical spectra with an

Table 1. Characteristic parameters of the acceleration process in solar protons hot events: acceleration efficiency α, high energy cutoff Em, normalization energy En, flux of accelerated particles in the source N<sup>0</sup> and heliographic coordinates of

Table 2. Characteristic parameters of the acceleration process in solar protons cold events: acceleration efficiency α, high energy cutoff Em, normalization energy En, flux of accelerated particles in the source N<sup>0</sup> and heliographic coordinates of

SE<sup>s</sup> protons=cm<sup>2</sup> str <sup>s</sup> (33)

<sup>N</sup><sup>0</sup> <sup>¼</sup> qk<sup>0</sup> <sup>¼</sup> protons=4πR<sup>2</sup>

The value of N<sup>0</sup> for every event is tabulated on columns 10 of Tables 1–3.

8 of Tables 1–3.

140 Cosmic Rays

the flare according to different reports.

the flare according to different reports.

Table 3. Characteristic parameters of the acceleration process in solar protons warm events: acceleration efficiency α, high energy cutoff Em, normalization energy En, flux of accelerated particles in the source N<sup>0</sup> and heliographic coordinates of the flare according to different reports.

experimental curve under the same conditions, we could proceed to fix the value of the acceleration parameters in advance, which would entails making a priori inferences about the physical parameters of the source involved in the acceleration process of a given solar event; furthermore, this would result in a bias for the interpretation of the phenomenology involved in each event depending on the selected value of the efficiency α; that is, high values would give systematically the best fit with spectrum (27), whereas low values would show a systematically better fit with spectrum (15). Therefore, we proceeded conversely by determining the appropriate parameters of the source from the value of α in the theoretical spectra that best represents the experimental curve. The optimum values of α, obtained for each of the theoretical curves allows us to determine the critical energy E<sup>c</sup> and the normalization flux Ko appropriate to each case. We have tabulated the values of α, Ec and Ko obtained for every event through calculations of the spectra (15) (19), (23), (27) and (31) in Tables 1–3. We have illustrated the optimum theoretical curves on Figures 2–4. From an examination of these results, it can be observed that no general conclusion can be drawn about the behavior of our theoretical spectra by the simple comparison of energy change rates (1), (2), (4) or (6) at different energy values 7 as if the medium density n were the only important parameter in determining the processes occurring at the source. Other factors must intervene, as can be seen from the fact that spectra behavior changes from event to event. Nevertheless, according to the behavior of particle spectra, we can group the solar events in three groups of similar characteristics: those illustrated in Figure 2, which we shall denominate hot events, where it can be seen that theoretical spectra progressively approach the experimental curves while adding energy loss processes to the acceleration rate. Therefore, the physical processes taking place at the source in those events are described by spectrum (27) indicating that adiabatic cooling of protons together with energy degradation from p–p collisions and collisional losses may have taken place. In this case spectrum (31) (illustrated only in the January 28, 1967 event) is systematically the more deflected curve, showing the absence of adiabatic compression, at least during the acceleration period. Figure 3 shows the second group which we will call cold events, and where it can be seen that energy losses are not important within the time scale of the acceleration process because theoretical curves get progressively separate from the experimental one while adding energy loss processes. Actually the best systematic approach in these cases is obtained with spectrum (31) (illustrated only for November 12, 1960 event) indicating that acceleration of protons by adiabatic compression could have taken took place. The third group that we shall distinguish as warm events is represented in Figure 4, where we can observe that there is no systematic tendency as compared to the previous groups. Nevertheless, it can be seen that at least at low energies the best approach to the experimental curve is described by spectrum (23), whereas at high energies the best fit is obtained with spectrum (15), thus indicating that to greater or lesser degree energy losses by collisional losses and protonproton collisions may be important on low energy protons but they become negligible in relation to the acceleration rate in high energy particles. The point where this change may occur varies from very low energies in some events (July 7, 1966) to very high energies in others (January 24, 1971). The larger deflection from the experimental curve in these cases is obtained with spectrum (27), indicating that adiabatic expansion do not take place; furthermore, the fact that spectrum (31) (illustrated only for the November 18, 1968 event) is systematically deflected in relation to the acceleration spectrum (15) indicates that there is no adiabatic compression either. The values of the parameters describing the most adequate theoretical spectrum of events of Figures 2–4 are tabulated on columns 7, 3 and 6 of Tables 1–3, respectively.

In order to estimate the amount of local plasma particles that must be picked up by the acceleration process to produce the observed spectrum, are must know the value of No in (8) when t = 0. Therefore, roughly assuming that at least for events of (Figure 3, Table 2), the picked up protons originate in a thermal plasma where the velocities distribution is of a Maxwellian-type, or that they appear from a preliminary heating related to turbulent thermal motions, then, it can be inferred that the primary differential flux is given as, related with the flux defined in Eq. (33).

$$N\_0 = \left[\theta / (2\pi)^{3/2}\right] (k/M)^{1/2} e^{3/2} nT^{1/2} \tag{34}$$

ii. The January 28, 1967 event follows the same tendency as the preceding event up to 800 MeV, with an exception at very low energies (≤ 30 MeV) where it can be seen that spectrum (23) is slightly better than (27). Beyond 800 MeV spectrum (23) becomes the more deflected curve. The low particle energy flux tail is noticeably similar to the

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The best fit of the experimental curve is systematically given by spectrum (31) and (15) (e.g. the November 12, 1960 event), whereas spectrum (27) is systematically the most deflected one.

a. The theoretical curve which best approximates the experimental one at low energies is

b. At given energy (from 500 to 3000 MeV) the previous tendency is abandoned, such

d. Spectrum (31) is systematically deflected in relation to spectrum (15) (e.g. November

e. The July 7, 1966 event, however, by following the feature (a) at E ≤ 25 MeV, beyond this energy spectrum (15) comes nearer to the experimental curve than spectrum (23), whereas spectrum (19) through a progressive, separation becomes the most deflected

a. For a given event the obtained value of acceleration efficiency α is the same with spectrum (31) and (15) (columns 3 and 4 of Tables 1 and 3) contrary to the events of

b. Examination of a given spectrum (same column 5, or, 6 or 7) shows that α and Ec

c. For a given event, the values of α in the events of Tables 2 and 3 (columns 4, 5, 6, and 7) increase monotonically while adding energy loss processes to the acceleration rate, with the exception of the events of Table 1, in which case the obtained values of α with

d. For a given event of Table 1, the value of Ec increases monotonically with the addition of an energy loss process to the net energy change rate, whereas in the events of Tables 2 and 3 the value of Ec obtained from (27) (column 7) decreases in relation to

e. The obtained value of K0, (column 10) is related only to the magnitude of the event (i.e.

f. There is no correlation between Em and the other parameters of the tables α, Ec, K0, or heliographic coordinate; neither is there any correlation between the maximum flux at

minimum theoretical energy for effective acceleration (Ec 12 MeV).

that spectrum (15) interchanges sequential order with spectrum (23). c. Spectrum (27) is systematically the most deflected curve at all energies.

Table 2, in which case α is lower with spectrum (31) than with (15).

spectrum (27) decrease in relation to the value of α from spectrum (23).

2. The events of Figure 3 show that:

18, 1968 event).

curve beyond 2000 MeV.

3. The events of Figure 4 show the following characteristics

spectrum (23) followed by spectrum (19).

4. Examination of Tables 1–3 shows the following features:

behave in an inversely proportional manner.

the values obtained from spectrum (23).

the value of J(>E) at En).

En and α or Ec.

where M is the mass of protons and k the of Boltzman's constant. Then, by assuming that Ko is related to the flux of protons involved in the acceleration process and the flux No related to the original concentration of the medium, we have estimated from Eq. (33) the fraction of the local plasma particles that were accelerated in each event and tabulated them on columns 10 of Tables 1–3. In evaluating (34), we have assumed a different value of temperature T for each one of the 3 groups of events, before discussing them in the next section.

Now let us summarize the results which emerge from Figures 2–4 and Tables 1–3, before extending their interpretation in next section:

	- i. In September 1, 1971 event, the best fit of the experimental spectrum is obtained with (27) whereas the worst fit is given by (15) and (31).

where it can be seen that energy losses are not important within the time scale of the acceleration process because theoretical curves get progressively separate from the experimental one while adding energy loss processes. Actually the best systematic approach in these cases is obtained with spectrum (31) (illustrated only for November 12, 1960 event) indicating that acceleration of protons by adiabatic compression could have taken took place. The third group that we shall distinguish as warm events is represented in Figure 4, where we can observe that there is no systematic tendency as compared to the previous groups. Nevertheless, it can be seen that at least at low energies the best approach to the experimental curve is described by spectrum (23), whereas at high energies the best fit is obtained with spectrum (15), thus indicating that to greater or lesser degree energy losses by collisional losses and protonproton collisions may be important on low energy protons but they become negligible in relation to the acceleration rate in high energy particles. The point where this change may occur varies from very low energies in some events (July 7, 1966) to very high energies in others (January 24, 1971). The larger deflection from the experimental curve in these cases is obtained with spectrum (27), indicating that adiabatic expansion do not take place; furthermore, the fact that spectrum (31) (illustrated only for the November 18, 1968 event) is systematically deflected in relation to the acceleration spectrum (15) indicates that there is no adiabatic compression either. The values of the parameters describing the most adequate theoretical spectrum of events of Figures 2–4 are tabulated on columns 7, 3 and 6 of Tables 1–3,

In order to estimate the amount of local plasma particles that must be picked up by the acceleration process to produce the observed spectrum, are must know the value of No in (8) when t = 0. Therefore, roughly assuming that at least for events of (Figure 3, Table 2), the picked up protons originate in a thermal plasma where the velocities distribution is of a Maxwellian-type, or that they appear from a preliminary heating related to turbulent thermal motions, then, it can be inferred that the primary differential flux is given as, related with the

> ð Þ <sup>k</sup>=<sup>M</sup> <sup>1</sup>=<sup>2</sup> e 3=2

where M is the mass of protons and k the of Boltzman's constant. Then, by assuming that Ko is related to the flux of protons involved in the acceleration process and the flux No related to the original concentration of the medium, we have estimated from Eq. (33) the fraction of the local plasma particles that were accelerated in each event and tabulated them on columns 10 of Tables 1–3. In evaluating (34), we have assumed a different value of temperature T for each

Now let us summarize the results which emerge from Figures 2–4 and Tables 1–3, before

i. In September 1, 1971 event, the best fit of the experimental spectrum is obtained with

nT<sup>1</sup>=<sup>2</sup> (34)

N<sup>0</sup> ¼ 9=ð Þ 2π

one of the 3 groups of events, before discussing them in the next section.

1. The events illustrated in Figure 2, show the following features:

(27) whereas the worst fit is given by (15) and (31).

extending their interpretation in next section:

<sup>3</sup>=<sup>2</sup> h i

respectively.

142 Cosmic Rays

flux defined in Eq. (33).

The best fit of the experimental curve is systematically given by spectrum (31) and (15) (e.g. the November 12, 1960 event), whereas spectrum (27) is systematically the most deflected one.

	- a. The theoretical curve which best approximates the experimental one at low energies is spectrum (23) followed by spectrum (19).
	- b. At given energy (from 500 to 3000 MeV) the previous tendency is abandoned, such that spectrum (15) interchanges sequential order with spectrum (23).
	- c. Spectrum (27) is systematically the most deflected curve at all energies.
	- d. Spectrum (31) is systematically deflected in relation to spectrum (15) (e.g. November 18, 1968 event).
	- e. The July 7, 1966 event, however, by following the feature (a) at E ≤ 25 MeV, beyond this energy spectrum (15) comes nearer to the experimental curve than spectrum (23), whereas spectrum (19) through a progressive, separation becomes the most deflected curve beyond 2000 MeV.
	- a. For a given event the obtained value of acceleration efficiency α is the same with spectrum (31) and (15) (columns 3 and 4 of Tables 1 and 3) contrary to the events of Table 2, in which case α is lower with spectrum (31) than with (15).
	- b. Examination of a given spectrum (same column 5, or, 6 or 7) shows that α and Ec behave in an inversely proportional manner.
	- c. For a given event, the values of α in the events of Tables 2 and 3 (columns 4, 5, 6, and 7) increase monotonically while adding energy loss processes to the acceleration rate, with the exception of the events of Table 1, in which case the obtained values of α with spectrum (27) decrease in relation to the value of α from spectrum (23).
	- d. For a given event of Table 1, the value of Ec increases monotonically with the addition of an energy loss process to the net energy change rate, whereas in the events of Tables 2 and 3 the value of Ec obtained from (27) (column 7) decreases in relation to the values obtained from spectrum (23).
	- e. The obtained value of K0, (column 10) is related only to the magnitude of the event (i.e. the value of J(>E) at En).
	- f. There is no correlation between Em and the other parameters of the tables α, Ec, K0, or heliographic coordinate; neither is there any correlation between the maximum flux at En and α or Ec.

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 Table 3.

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

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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

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)

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

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

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

high energies was not very important during the time scale of the acceleration.

gives the best fit to the experimental curves (e.g. November 12, 1960 event).

deflects in relation to (19) (e.g. the February 25, 1969 and January 24, 1971 events).

degradation by p–p collisions during the acceleration time.

produced.

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 events will be discussed in Section 6.
