2. Energy losses of protons during acceleration in solar flares

Some researchers who study radiation and secondary particle fluxes consider an acceleration stage followed by a slowing down phase in the solar material once the action of the acceleration mechanism on particles has ceased (e.g. [86–89]); and they generally neglect the simultaneous occurrence of energy loss and acceleration.

However, particle acceleration is not performed in the vacuum but in the high density medium of flare regions; therefore, we shall study the local modulation of the acceleration spectrum as the protons are broken during the short-time scale of solar particle generation. The most important processes occurring in astrophysical plasmas capable of affecting the net energy change rate of particles in the range of kinetic energies of energetic solar protons (E�10<sup>6</sup> –10<sup>10</sup> eV) are:

#### 2.1. Collisional energy losses

electromagnetic flare emissions with those of energetic particles and coronal mass ejections (CME) is the method utilized to explore the physical conditions and processes taking place in the sources of particle generation. For example, results obtained from the SEPS server project and future HESPERIA HORIZON 2020 project. However, the study of the corpuscular radiation emitted in some flares can also provide us with very valuable information about the physical conditions and processes occurring in association with this solar phenomenon. It is known, for instance, that the processes involved in the generation of solar particles are probably of a non-thermal nature, because the intensity of particles usually decays more softly than an exponential of a the thermal type does, and so other properties may be deduced in order to investigate how and where multi-GeV solar protons originate, that means the source parameters and the parameters involved in the generation process of particle [69, 70]. In this chapter, we attempt to draw some inferences concerning solar sources by the analysis of 12 ground

It has been shown [40] that the best representation of the energy spectrum of solar protons through the whole energy domain explored experimentally at present is given by an inverse power law with an upper cutoff in its high energy portion. In fact, a good fit of the experimental data can be obtained with an exponential law in a limited energy band; however, a strong deflection is obtained with them as soon as a wider energy domain is involved. Besides, it has been established [11] that the measured differential intensity in solar proton events, as well as the source spectrum (inferred as an inverse power law in energy) are both velocity-dependent. Therefore, we infer that the acceleration rate of particles in the sun must provide the spectral shape and velocity dependence such as suggested by those results. This is the case with an

<sup>¼</sup> αβ<sup>W</sup> <sup>¼</sup> <sup>α</sup> <sup>W</sup><sup>2</sup> � Mc<sup>2</sup> <sup>2</sup> <sup>1</sup>=<sup>2</sup>

where β is the velocity of the particles in units of light velocity and W the total energy of particles. The parameter α denotes the efficiency of the acceleration mechanism, which in the case of solar sources may be considered as roughly constant when the acceleration process reaches the steady-state in a given event [79, 80]. It has been generally thought that the energy loss processes of solar particles acceleration stage are not important in practice, and have only been taken into account after the acceleration stage in order to explain some features of

In this chapter we shall consider, together with acceleration, energy loss processes occurring in the high density plasma of the solar source. It will be shown that energy losses in some proton flares can modulate the acceleration spectrum, thus implying that if such a small effect compared to the acceleration rate is able to modify the spectrum during the short lapse of the acceleration process, then the source spectrum is actually the result of a strong modulation due to local energy losses during acceleration and not only through interplanetary propagation; thus in Section 2, we discuss the basic equations of the more plausible energy loss processes in

(1)

level enhancements (GLE) of solar cycles 19 and 20.

dW dt 

acc

electromagnetic emissions in solar flares and heating of the chromosphere [87].

energy gain rate of the form

122 Cosmic Rays

These depend strongly on the density and temperature of the plasma; thus we assume that the main energy dissipation of particles must occur in the generation region, in the body of the flare itself. The rate of collisional losses in a medium of density n has been given in a simplified expression [37]

$$\left(\frac{dW}{dt}\right)\_{ion} = -\frac{7.62 \times 10^{-9} nL}{\beta} \text{ (eV/sec)}\tag{2}$$

where β= v/c is the particle velocity in terms of the light velocity, L is a unidimensional factor and logarithmically depending marginally on the particle energy. We shall assume a value of L � 27 for solar flare conditions, when the medium concentration is <sup>n</sup>�1012 – 1013 cm<sup>3</sup> . In Figure 1, the behavior of Eq. (2) with energy is shown. The complete description of collisional losses through the entire energy range including losses in the low energy portion (the so called nuclear stopping and electronic stopping) has been given by [10] for fully ionized hydrogen as:

Figure 1. Energy change rates of protons (acceleration for two different rates) and deceleration for collisional losses p–p nuclear collisions and adiabatic cooling in a medium of density n = 1012–1013 cm�<sup>3</sup> .

$$\frac{dE}{dt} = -\frac{1.57 \times 10^{-35} N}{\beta} \frac{Q^2}{A} H(\text{x}) \ln \Lambda \text{ (eV/ns)}\tag{2.1}$$

2.2. Energy degradation from proton-proton collisions

multiple pion yielding at high energies, p(p; aπ<sup>+</sup>

bπ<sup>0</sup>

agreement with [38]

At present, there are evidences of the occurrence of nuclear reactions between solar nuclei and solar material, producing high energy gamma rays although is not absolutely clear whether nuclear reactions of solar energetic particles and solar material take place, when protons are injected into the photosphere, or they pass through coronal condensations, or during their acceleration within the dense material of flare regions. We shall assume that nuclear interactions occur at least in the acceleration volume where very likely the motion of energetic particles is completely random with respect to the local solar material. The isotropic motion of the accelerated particles is suggested by an analysis of neutron fluxes [45]. For purposes of energy loss calculations, we do not take into account collisions protons with other nuclear species, because the maximum energy change in elastic scattering occurs when the colliding particles have similar mass. Although the energy dissipation from p: p collisions is believed to appear mainly from elastic scattering, however at high energies (>750 MeV), the inelastic crosssection becomes highly important [44] increasing up to a maximum at some GeV, where it remains practically constant. In fact, as pion production initiates at 285 MeV and a fraction ≥ 35% of the kinetic energy of the incident proton goes into pion energy, then, energy dissipation from inelastic p: p scattering is not negligible in a high density medium (n ≥ 10<sup>12</sup> cm<sup>3</sup>

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

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

Concerning inelastic p: p interactions, the gamma ray line at 2.2 MeV due to fast neutron production, seems to be strong evidence of the occurrence of p: p collisions in solar flares. All this depends strongly on the production model: The assumed geometry and the spectral shape considered [2]. In fact, the cross-section for the later interactions is 10: 100 times higher, that is, their threshold is ≤36 MeV/nucleon, while that for inelastic p: p scattering are 285 MeV. Nevertheless, it has been known for a long time from [12] that solar abundances of CNO and he are of the order of 1.5: 7% with respect to the local H, in such a way that this kind of equilibrium between local abundances and interaction cross-sections states a high probability for the occurrence of p: p collisions in the body itself of the solar flare material. The main problem related with these features is that some reactions, as for instance p(p; aπ<sup>0</sup>

) by π<sup>0</sup> decay produce high energy solar gamma rays (50 MeV) that have neither been detected to our knowledge nor their plausible absorption into the solar material satisfactorily explained. In fact, the predicted wide peak for these gamma rays ranging from 38.5: 118 MeV [6] could probably render their identification difficult due to the presence of high energy photons expected from bremsstrahlung of very high energy solar electrons. In addition, there is the fact that high energy p: p reactions must occur more frequently, since the inelastic crosssection rises progressively from 290 MeV up to a maximum of about 1 GeV where it remains practically constant. Refs. [14, 15] have reviewed the problems connected with secondary products of nuclear interactions in solar flares. Nevertheless we show later in this work that p: p collisions are only expected in some few GLE. Hence, although the measured flux of particles does not distinguish whether solar protons have suffered nuclear collisions or not, the modulation of the energy spectrum by their effects furnish available information about their occurrence. The importance of energy degradation from p: p collisions in cosmic rays physics has been pointed out for the first time by [129]. The energy loss rate by nuclear interactions is

)p or p(p; aπ, bπ<sup>0</sup>

).

125

)p and

,aπ,

, aπ<sup>+</sup>

)p or p(p;n, π<sup>+</sup>

where  $\mathbf{x} = 5.44 \times 10^4 \beta T^{-0.5}$ ,  $H(\mathbf{x}) = \xi\_1 H\_\epsilon(\mathbf{x}\_\epsilon) + \xi\_2 H\_p(\mathbf{x}\_p)$  with  $H\_\epsilon(\mathbf{x}\_\epsilon) = 0.88 \text{erf}(\mathbf{x}\_\epsilon) - \left(1 - 5.48 \times 10^{-4} / A\right) \mathbf{x}\_\epsilon e^{-\mathbf{x}\_\epsilon^2}$  for electrons,  $H\_p(\mathbf{x}\_p) = 0.88 \text{erf}(\mathbf{x}\_p) - \left(1 + \frac{1}{A}\right) \mathbf{x}\_p e^{-\mathbf{x}\_p^2}$  for protons, 
$$\xi\_1 = 1.097803296 \times 10^{27}, \xi\_2 = 5.979073244 \times 10^{23} and \ \Lambda = \left[4.47 \times 10^{16} A (T/N)^{0.5} \beta^2\right] / Q$$
.

For the task of simplicity and because we are dealing in this work with GLE (high energy protons), we will use preferentially Eq. (2).

#### 2.2. Energy degradation from proton-proton collisions

dE

βT�0:<sup>5</sup>

� � � <sup>1</sup> <sup>þ</sup> <sup>1</sup>

protons), we will use preferentially Eq. (2).

A � �xpe

Heð Þ¼ xe <sup>0</sup>:88erf xð Þ� <sup>e</sup> <sup>1</sup> � <sup>5</sup>:<sup>48</sup> � <sup>10</sup>�<sup>4</sup>

where <sup>x</sup> <sup>¼</sup> <sup>5</sup>:<sup>44</sup> � 104

� � <sup>¼</sup> <sup>0</sup>:88erf xp

Hp xp

124 Cosmic Rays

dt ¼ � <sup>1</sup>:<sup>57</sup> � <sup>10</sup>�<sup>35</sup><sup>N</sup> β

nuclear collisions and adiabatic cooling in a medium of density n = 1012–1013 cm�<sup>3</sup>

=A � �xee�x<sup>2</sup>

�x<sup>2</sup>

,H xð Þ¼ ξ1Heð Þþ xe ξ2Hp xp

<sup>p</sup> for protons,

For the task of simplicity and because we are dealing in this work with GLE (high energy

<sup>ξ</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>:<sup>097803296</sup> � 1027, <sup>ξ</sup><sup>2</sup> <sup>¼</sup> <sup>5</sup>:<sup>979073244</sup> � 1023and <sup>Λ</sup> <sup>¼</sup> <sup>4</sup>:<sup>47</sup> � 1016A Tð Þ <sup>=</sup><sup>N</sup> <sup>0</sup>:<sup>5</sup>

Q2

Figure 1. Energy change rates of protons (acceleration for two different rates) and deceleration for collisional losses p–p

� � with

<sup>e</sup> for electrons,

<sup>A</sup> H xð Þln<sup>Λ</sup> ð Þ eV=ns (2.1)

.

<sup>β</sup><sup>2</sup> h i

=Q

At present, there are evidences of the occurrence of nuclear reactions between solar nuclei and solar material, producing high energy gamma rays although is not absolutely clear whether nuclear reactions of solar energetic particles and solar material take place, when protons are injected into the photosphere, or they pass through coronal condensations, or during their acceleration within the dense material of flare regions. We shall assume that nuclear interactions occur at least in the acceleration volume where very likely the motion of energetic particles is completely random with respect to the local solar material. The isotropic motion of the accelerated particles is suggested by an analysis of neutron fluxes [45]. For purposes of energy loss calculations, we do not take into account collisions protons with other nuclear species, because the maximum energy change in elastic scattering occurs when the colliding particles have similar mass. Although the energy dissipation from p: p collisions is believed to appear mainly from elastic scattering, however at high energies (>750 MeV), the inelastic crosssection becomes highly important [44] increasing up to a maximum at some GeV, where it remains practically constant. In fact, as pion production initiates at 285 MeV and a fraction ≥ 35% of the kinetic energy of the incident proton goes into pion energy, then, energy dissipation from inelastic p: p scattering is not negligible in a high density medium (n ≥ 10<sup>12</sup> cm<sup>3</sup> ). Concerning inelastic p: p interactions, the gamma ray line at 2.2 MeV due to fast neutron production, seems to be strong evidence of the occurrence of p: p collisions in solar flares. All this depends strongly on the production model: The assumed geometry and the spectral shape considered [2]. In fact, the cross-section for the later interactions is 10: 100 times higher, that is, their threshold is ≤36 MeV/nucleon, while that for inelastic p: p scattering are 285 MeV. Nevertheless, it has been known for a long time from [12] that solar abundances of CNO and he are of the order of 1.5: 7% with respect to the local H, in such a way that this kind of equilibrium between local abundances and interaction cross-sections states a high probability for the occurrence of p: p collisions in the body itself of the solar flare material. The main problem related with these features is that some reactions, as for instance p(p; aπ<sup>0</sup> )p and multiple pion yielding at high energies, p(p; aπ<sup>+</sup> )p or p(p; aπ, bπ<sup>0</sup> )p or p(p;n, π<sup>+</sup> , aπ<sup>+</sup> ,aπ, bπ<sup>0</sup> ) by π<sup>0</sup> decay produce high energy solar gamma rays (50 MeV) that have neither been detected to our knowledge nor their plausible absorption into the solar material satisfactorily explained. In fact, the predicted wide peak for these gamma rays ranging from 38.5: 118 MeV [6] could probably render their identification difficult due to the presence of high energy photons expected from bremsstrahlung of very high energy solar electrons. In addition, there is the fact that high energy p: p reactions must occur more frequently, since the inelastic crosssection rises progressively from 290 MeV up to a maximum of about 1 GeV where it remains practically constant. Refs. [14, 15] have reviewed the problems connected with secondary products of nuclear interactions in solar flares. Nevertheless we show later in this work that p: p collisions are only expected in some few GLE. Hence, although the measured flux of particles does not distinguish whether solar protons have suffered nuclear collisions or not, the modulation of the energy spectrum by their effects furnish available information about their occurrence. The importance of energy degradation from p: p collisions in cosmic rays physics has been pointed out for the first time by [129]. The energy loss rate by nuclear interactions is agreement with [38]

$$\frac{dW}{dt} = -\sigma c n \beta W \text{ (eV/sec)}\tag{3}$$

Therefore, while the acceleration mechanism is in effect, and a fraction of particles are escaping from the flare region, the bulk of particles lose energy by adiabatic cooling due to the work that protons exert on the expanding material. Mechanisms for the expansion (or compression) of magnetic structures have been widely discussed (e.g. [96, 99]). It has been shown through energetic estimations that when particle kinetic density exceeds magnetic field pressure, the sunspot field lines are transported upward by the accelerated plasma; and thus, owing to the decrease of magnetic field density according to the altitude over the photosphere [1, 101], the magnetic bottles blow open at an altitude lower than 0.6: 1 Rs allowing particles] to escape into the interplanetary medium. Particles that have left the acceleration region before the magnetic bottle blows up may escape due to drift by following the field lines, or they remain stored therein losing energy losing energy until the magnetic structure is opened. We shall not consider this eventual deceleration during particle storage but only energy losses inside the acceleration volume. According to [46, 77], the energy change rate of particles by expansion (or compression) of magnetic fields producing adiabatic cooling or heating of the solar cosmic ray

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

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

gas, when the non-radial components of the plasma velocity are negligible is given as

where Vr and R are the velocity and distance of the plasma displacement, respectively, μ = 1+ ɣ

In order to estimate an approximate value for r = (2/3) (Vr/R) in flare conditions, we extend the following considerations: it is known that the hydromagnetic velocity of the coronal expansion

the direction of the expansion (e.g. [100, 101, 136]). Observations also show displacements with velocities of 650–2600 km s�<sup>1</sup> in association with type II burst [95] and expansion of flare knots in limb flares with velocities in the range 5.3–110 km s�<sup>1</sup> [54, 55, 83, 84]. Besides, it is also known that closed magnetic arches have a mean altitude of 0.6 Rs above the photosphere [122]. Therefore, assuming that the average velocity of 400 km s�<sup>1</sup> is a typical value of magnetic motions in the chromosphere and low corona and an average expanded distance of the source of 0.3 Rs

account the results usually associated with multi-Gev proton flares (GLE), then, magnetic loops expand � 30,000 km with a velocity of �45 km s�<sup>1</sup> at the time of the flare start, thus giving a

It is expected that if the physical conditions in the source of multi-GeV solar proton flares and processes acting on solar particles must be similar, the behavior of the theoretical source spectra of solar protons from event to event will be similar, and thus by comparing the rates (1)–(6) the influence of each process on the acceleration spectrum can be established. For

. Hence, in terms of total energy W the adiabatic deceleration rate in the expanding

<sup>R</sup> <sup>μ</sup><sup>E</sup> ð Þ eV=sec (5)

W ð Þ eV=sec (6)

–103 km s�<sup>1</sup> depending on

. On the other hand, if we take into

�<sup>1</sup> in Figure 1.

) and that in association with proton flares type IV sources

�1

�1

127

dE dt ad ¼ � 2 3 Vr

dW dt ¼ �rβ<sup>2</sup>

systematically appear expanding with velocities in the range of 102

value for r of the same order. We have illustrated Eq. (6) with r = 10�<sup>3</sup> s

while acceleration is operating, we obtain thus r ≈ 10�<sup>3</sup> s

and ɣ = W/Mc<sup>2</sup>

magnetic fields may be expressed as

is in average of the order 400 km s�<sup>1</sup>

where σ in p–p collisions is composed of σine <sup>p</sup>�<sup>p</sup> <sup>þ</sup> <sup>σ</sup>el <sup>p</sup>�<sup>p</sup>. As the inelastic cross-section is weakly energy dependent, it may be approximated to its mean value at high energies (σine <sup>p</sup>�<sup>p</sup> � 26 mb).� Concerning elastic collisions, a reasonable fit of the differential cross-section data by an analytical expression has been given by [91]. As the differential cross-section is highly isotropic, we can assume symmetry around 90�, such that their expression may be rewritten as σel <sup>p</sup>�<sup>p</sup> <sup>=</sup> hE�<sup>2</sup> + JE�<sup>1</sup> (if E ≤ 110 MeV) and σel <sup>p</sup>�<sup>p</sup> <sup>=</sup> hE�<sup>2</sup> + f (if E > 110 MeV), where h = 96.09 mb-MeV<sup>2</sup> , j = 5.497 � 103 mb MeV and f = 46.49 mb. We have then from Eq. (3):

$$
\left(\frac{dW}{dt}\right)\_{p-p} = -cn\left(h\overline{E}^2 + j\overline{E}\right)\beta W \text{ (if } E \le 110 \text{ MeV)}
$$

$$
\left(\frac{dW}{dt}\right)\_{p-p} = -cn\left(h\overline{E}^2 + f\right)\beta W \text{ (If } 110 < E < 290 \text{ MeV)}
$$

$$
\left(\frac{dW}{dt}\right)\_{p-p} = -\left[\eta + cn\left(h\overline{E} + f\right)\right]\beta W \text{ (If } E \ge 290 \text{ MeV), where } \eta = cn\sigma\_{\text{min}}
$$

So that the net energy change can be compacted as:

$$
\left(\frac{d\mathcal{W}}{dt}\right)\_{p-p} = -\left(hE^{-2} + jE^{-1} + f + \eta\right)\beta\mathcal{W} \text{ (eV/sec)}\tag{4}
$$

where h = 2.88 � <sup>10</sup>�<sup>15</sup> n Me <sup>2</sup> <sup>s</sup> �1 ,j= 1.65 � <sup>10</sup>�<sup>13</sup> n MeV s�<sup>1</sup> (if E <sup>≤</sup> 110 MeV), j = 0 and f = 1.39 � <sup>10</sup>�<sup>15</sup> n s�<sup>1</sup> (if E > 110 MeV), f = 0 (if E <sup>≤</sup> 110 MeV), <sup>η</sup> = cnσine <sup>p</sup>�<sup>p</sup> <sup>=</sup> 8.1 � <sup>10</sup>�<sup>16</sup> n s�<sup>1</sup> , (if E > 290 MeV) and η = 0 if (E < 290 MeV). We have plotted Eq. (4) in Figure 1 for two different values of the density n.

#### 2.3. Adiabatic deceleration at the source level

Adiabatic cooling of cosmic particles in the solar wind has been proved long ago (e.g. [34]). However, here we are dealing with adiabatic cooling at the sources of solar energetic protons in GLE and not in the interplanetary or interstellar media medium. It is well-known that great flares are associated with magnetic arches, such as loop prominences and flare nimbuses (e.g. [7, 97, 98]) which occur between regions of opposite-polarity in the photosphere. Observations show that magnetic flux tubes expand from flare regions [23, 66, 107, 109, 117]. These configurations identified as "magnetic bottles" are usually related to the development of flare phenomena (e.g. [14, 83, 84, 96, 104, 110, 123]), therefore, we shall investigate the relationship between these magnetic structures and the phenomenon of particle generation through the study of the energy spectra of solar protons in GLE: We assume the hypothesis that particles are enclosed within those "magnetic bottles", where they are accelerated up to high energies. Therefore, while the acceleration mechanism is in effect, and a fraction of particles are escaping from the flare region, the bulk of particles lose energy by adiabatic cooling due to the work that protons exert on the expanding material. Mechanisms for the expansion (or compression) of magnetic structures have been widely discussed (e.g. [96, 99]). It has been shown through energetic estimations that when particle kinetic density exceeds magnetic field pressure, the sunspot field lines are transported upward by the accelerated plasma; and thus, owing to the decrease of magnetic field density according to the altitude over the photosphere [1, 101], the magnetic bottles blow open at an altitude lower than 0.6: 1 Rs allowing particles] to escape into the interplanetary medium. Particles that have left the acceleration region before the magnetic bottle blows up may escape due to drift by following the field lines, or they remain stored therein losing energy losing energy until the magnetic structure is opened. We shall not consider this eventual deceleration during particle storage but only energy losses inside the acceleration volume. According to [46, 77], the energy change rate of particles by expansion (or compression) of magnetic fields producing adiabatic cooling or heating of the solar cosmic ray gas, when the non-radial components of the plasma velocity are negligible is given as

dW

j = 5.497 � 103 mb MeV and f = 46.49 mb. We have then from Eq. (3):

¼ �cn hE

¼ �cn hE

2 þ f 

2 þ jE 

p�p

dW dt 

p�p

So that the net energy change can be compacted as:

dW dt 

2.3. Adiabatic deceleration at the source level

p�p

�1

f = 1.39 � <sup>10</sup>�<sup>15</sup> n s�<sup>1</sup> (if E > 110 MeV), f = 0 (if E <sup>≤</sup> 110 MeV), <sup>η</sup> = cnσine

dW dt 

p�p

<sup>p</sup>�<sup>p</sup> <sup>þ</sup> <sup>σ</sup>el

Concerning elastic collisions, a reasonable fit of the differential cross-section data by an analytical expression has been given by [91]. As the differential cross-section is highly isotropic, we can assume symmetry around 90�, such that their expression may be rewritten as σel

energy dependent, it may be approximated to its mean value at high energies (σine

where σ in p–p collisions is composed of σine

hE�<sup>2</sup> + JE�<sup>1</sup> (if E ≤ 110 MeV) and σel

126 Cosmic Rays

dW dt 

where h = 2.88 � <sup>10</sup>�<sup>15</sup> n Me <sup>2</sup> <sup>s</sup>

values of the density n.

dt ¼ �σcnβ<sup>W</sup> ð Þ eV=sec (3)

<sup>p</sup>�<sup>p</sup> <sup>=</sup> hE�<sup>2</sup> + f (if E > 110 MeV), where h = 96.09 mb-MeV<sup>2</sup>

βW ð Þ if E ≤ 110 MeV

βW ð Þ If 110 < E < 290 MeV

¼ � hE�<sup>2</sup> <sup>þ</sup> jE�<sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>þ</sup> <sup>η</sup> β<sup>W</sup> ð Þ eV=sec (4)

,j= 1.65 � <sup>10</sup>�<sup>13</sup> n MeV s�<sup>1</sup> (if E <sup>≤</sup> 110 MeV), j = 0 and

<sup>p</sup>�<sup>p</sup> <sup>=</sup> 8.1 � <sup>10</sup>�<sup>16</sup> n s�<sup>1</sup>

¼ � <sup>η</sup> <sup>þ</sup> cn hE <sup>þ</sup> <sup>f</sup> <sup>β</sup><sup>W</sup> ð Þ If <sup>E</sup> <sup>≥</sup> 290 MeV , where <sup>η</sup> <sup>¼</sup> cnσin

E > 290 MeV) and η = 0 if (E < 290 MeV). We have plotted Eq. (4) in Figure 1 for two different

Adiabatic cooling of cosmic particles in the solar wind has been proved long ago (e.g. [34]). However, here we are dealing with adiabatic cooling at the sources of solar energetic protons in GLE and not in the interplanetary or interstellar media medium. It is well-known that great flares are associated with magnetic arches, such as loop prominences and flare nimbuses (e.g. [7, 97, 98]) which occur between regions of opposite-polarity in the photosphere. Observations show that magnetic flux tubes expand from flare regions [23, 66, 107, 109, 117]. These configurations identified as "magnetic bottles" are usually related to the development of flare phenomena (e.g. [14, 83, 84, 96, 104, 110, 123]), therefore, we shall investigate the relationship between these magnetic structures and the phenomenon of particle generation through the study of the energy spectra of solar protons in GLE: We assume the hypothesis that particles are enclosed within those "magnetic bottles", where they are accelerated up to high energies.

<sup>p</sup>�<sup>p</sup>. As the inelastic cross-section is weakly

<sup>p</sup>�<sup>p</sup> � 26 mb).�

<sup>p</sup>�<sup>p</sup> <sup>=</sup>

,

, (if

$$
\left(\frac{dE}{dt}\right)\_{\text{ad}} = \pm \frac{2}{3} \frac{V\_r}{R} \mu E \text{ (eV/sec)}\tag{5}
$$

where Vr and R are the velocity and distance of the plasma displacement, respectively, μ = 1+ ɣ �1 and ɣ = W/Mc<sup>2</sup> . Hence, in terms of total energy W the adiabatic deceleration rate in the expanding magnetic fields may be expressed as

$$
\left(\frac{dW}{dt}\right) = -\rho\beta^2 W \left(\text{eV/sec}\right) \tag{6}
$$

In order to estimate an approximate value for r = (2/3) (Vr/R) in flare conditions, we extend the following considerations: it is known that the hydromagnetic velocity of the coronal expansion is in average of the order 400 km s�<sup>1</sup> ) and that in association with proton flares type IV sources systematically appear expanding with velocities in the range of 102 –103 km s�<sup>1</sup> depending on the direction of the expansion (e.g. [100, 101, 136]). Observations also show displacements with velocities of 650–2600 km s�<sup>1</sup> in association with type II burst [95] and expansion of flare knots in limb flares with velocities in the range 5.3–110 km s�<sup>1</sup> [54, 55, 83, 84]. Besides, it is also known that closed magnetic arches have a mean altitude of 0.6 Rs above the photosphere [122]. Therefore, assuming that the average velocity of 400 km s�<sup>1</sup> is a typical value of magnetic motions in the chromosphere and low corona and an average expanded distance of the source of 0.3 Rs while acceleration is operating, we obtain thus r ≈ 10�<sup>3</sup> s �1 . On the other hand, if we take into account the results usually associated with multi-Gev proton flares (GLE), then, magnetic loops expand � 30,000 km with a velocity of �45 km s�<sup>1</sup> at the time of the flare start, thus giving a value for r of the same order. We have illustrated Eq. (6) with r = 10�<sup>3</sup> s �<sup>1</sup> in Figure 1.

It is expected that if the physical conditions in the source of multi-GeV solar proton flares and processes acting on solar particles must be similar, the behavior of the theoretical source spectra of solar protons from event to event will be similar, and thus by comparing the rates (1)–(6) the influence of each process on the acceleration spectrum can be established. For instance, it can be seen from Figure 1 that in the energy range 1–103 MeV and medium concentration n = 1013 cm<sup>3</sup> , the ratio r<sup>1</sup> = (dW/dt)p–p/(dW/dt)coll changes from r<sup>1</sup> = 1.7–16 and the ratio r<sup>2</sup> = (dW/dt)ad/(dW/dt)coll varies from r<sup>2</sup> = 4.6 10<sup>5</sup> –0.64; therefore if all processes would act simultaneously in solar flares, the acceleration spectrum is mainly affected by energy degradation from p–p collisions, whose effects are stronger in the high energy portion of the spectrum. Collisional losses are more important in the non-relativistic region, whereas adiabatic losses become important in the relativistic region of the spectrum. Using experimental data of several GLE of solar protons, we shall investigate if the same processes occur in all events, and thus similar physical conditions are prevalent at the sources, or if they vary from event to event, in which, case it is interesting to investigate why and how they vary.

for higher rigidities (> 0.44 GV) we have employed the 03:00 U.T. measurements on Balloon and N.M. data given by [39]. In the events of November18, 1968, February 25, 1969, March 30, 1969, November 2, 1969 and September 1, 1971, we have used the peak flux data in the (0.1–0.7) GV band, given by [47] from the IMP4 and IMP5 satellite measurements. For January 24, 1971 GLE, we have employed the 06:05 flux data and at 07:20 U.T. in the (0.28–0.7) GV band from [134] For August 4, 1972 event, we have considered the HEOS2 graphical fluxes in the (0.15–0.45) GV band at 16:00 U.T. by [61] which lie between the 09:57–22:17 U.T. data of [4] and is in good agreement with N.M. measurements; for the (0.6–1.02) GV band we have employed the balloon extrapolated data by [61]. For the high rigidity portion of the spectrum (> 1.02 Gy), we have made use of the measurements given by [41–43] from NM data, in the following form:

Jð Þ¼ > P K

the value was essentially the same of that of August 7 event.

therefore, can be directly related to the acceleration process

An excellent review of solar cosmic ray events has been given in [130].

law in kinetic energy Ð Em

ðPm P

where K is a constant, Pm the high rigidity cutoff and Φ the spectral slope of the differential fluxes. The values of Pm and Φ were taken through several hours around the peak flux of the event, as explained by the latter authors. The values of Φ were found to be systematically lower than other values furnished by GLE measurements due to the presence of the high rigidity cutoff parameter. For November 2, 1969 event we have taken the high rigidity power law spectrum as given by [61]; according to this data, we have considered a characteristic upper cutoff at 1.6 GV. In the case of August 4, 1972 event, we have taken the upper bound of Φ given for August 7 event by [43] considering that the particle spectrum became flatter with time during August 1972 events [4]. For the high rigidity cutoff, we have tested that within the error band,

The extrapolation of the high rigidity power laws to the integral fluxes of the lower rigidity branches, has allowed us to determine K from Eq. (7) and thus to construct the high rigidity branches of the proton fluxes. By smoothing fluxes of both branches we have obtained the experimental integral spectra, which we have represented in the kinetic energy scale with solid lines through Figures 2–4. We have verified the good agreement of the high energy power law shape deduced in this manner, with the corresponding integral slope of the differential power

it is systematically true that the best fit for the experimental points is given by such a power law, it is also true that there are some points that do not fit perfectly with that kind of curve; we have attempted to include these points in the experimental curves in the case of some GLE events. For January 28, 1967 event, we employed the integral spectrum deduced by [40] with the previously mentioned characteristics. It must be emphasized that the choice of these 12 multi-GeV proton events (GLE) follows from the fact that they furnish particle fluxes through a large range of energy bands and because of the information of the experimental value of Em in these cases, which unlike the other parameters of the spectrum is the only one that does not vary through the propagation of particles into the interplanetary space as shown by [40]) and

<sup>E</sup> <sup>E</sup>�<sup>Φ</sup>dE reported in several works by (e.g. [41–43]). However, although

P�<sup>Φ</sup>dP (7)

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

129

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