3. Experimental integral spectra of multi-GeV solar proton events

The description of the spectral distribution of solar particle fluxes of a given event is concerned, the result is a strong spread of spectral shape representations, according to the different detection methods employed, the energy bands and time intervals studied. The most plausible spectral shapes are described either by inverse power laws in kinetic energy or magnetic rigidity and exponential laws in magnetic rigidity (e.g. [53]). One of the most popular methods was developed by Forman et al, published in Ref. [59].

For example, in the case of the GLE of January 28, 1967, for which experimental measurements of fluxes through a wide energy range are available, several different spectral shapes have been analyzed: from the study of the relativistic portion of the spectrum, [60–62] proposes an exponential rigidity law { exp.(P/0.6(GV)} and alternatively a differential power law spectrum in rigidity ( <sup>P</sup><sup>5</sup> ); [8] proposed a differential spectrum of the form (<sup>P</sup>4.8) for relativistic protons of the event. Taking into consideration data from balloon, polar satellite and neutron monitors (N.M.), [3] gives an integral spectrum of the form ( <sup>P</sup><sup>4</sup> ); similarly, [40] deduced an integral spectrum as a power law in kinetic energy (<sup>E</sup><sup>2</sup> ) with an upper cutoff at Em = 4.3 GeV or in magnetic rigidity <sup>P</sup> as (<sup>P</sup>3.1) with an upper cutoff at Pm <sup>=</sup> 5.3 GV. These authors have shown that as far as the whole energy spectrum through the different energy bands is concerned, any spectral shape that does not take into an upper cutoff is strongly deflected from the experimental data.

It would seem, therefore, that the description of energy spectra of solar particles is one of the most particular topics connected with solar cosmic ray physics: that is, owing to the lack of global measurements of the whole spectrum at a given time and to the lack of simultaneity in the measurements of differential fluxes, the integral spectra must be constructed with the inhomogeneous data available for each event. Therefore, in order to do so for 12 GLE during solar cycles 19 and 20, we have used low rigidity data (high latitude observations) for the following events: for September 3, 1960 event we have employed the 14:10 U.T. data from Rocket Observations [18] in the (0.1–0. 7) GV band. For November 12 and 15, 1960 GLE's, we have used the 18:40 U.T. and 05:00 U.T. data, respectively, from rocket observations in the (6.16–1.02) GV band [73]. For July 7, 1966 GLE, we have used the 19:06 U.T. data given by [57, 58] in the (0.13–0.19) GV band, and the spectrum given by [118] in the (0.19–0.44) GV band;

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:

instance, it can be seen from Figure 1 that in the energy range 1–103 MeV and medium

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.

3. Experimental integral spectra of multi-GeV solar proton events

methods was developed by Forman et al, published in Ref. [59].

(N.M.), [3] gives an integral spectrum of the form ( <sup>P</sup><sup>4</sup>

spectrum as a power law in kinetic energy (<sup>E</sup><sup>2</sup>

The description of the spectral distribution of solar particle fluxes of a given event is concerned, the result is a strong spread of spectral shape representations, according to the different detection methods employed, the energy bands and time intervals studied. The most plausible spectral shapes are described either by inverse power laws in kinetic energy or magnetic rigidity and exponential laws in magnetic rigidity (e.g. [53]). One of the most popular

For example, in the case of the GLE of January 28, 1967, for which experimental measurements of fluxes through a wide energy range are available, several different spectral shapes have been analyzed: from the study of the relativistic portion of the spectrum, [60–62] proposes an exponential rigidity law { exp.(P/0.6(GV)} and alternatively a differential power law spectrum in

of the event. Taking into consideration data from balloon, polar satellite and neutron monitors

magnetic rigidity <sup>P</sup> as (<sup>P</sup>3.1) with an upper cutoff at Pm <sup>=</sup> 5.3 GV. These authors have shown that as far as the whole energy spectrum through the different energy bands is concerned, any spectral shape that does not take into an upper cutoff is strongly deflected from the experimental data.

It would seem, therefore, that the description of energy spectra of solar particles is one of the most particular topics connected with solar cosmic ray physics: that is, owing to the lack of global measurements of the whole spectrum at a given time and to the lack of simultaneity in the measurements of differential fluxes, the integral spectra must be constructed with the inhomogeneous data available for each event. Therefore, in order to do so for 12 GLE during solar cycles 19 and 20, we have used low rigidity data (high latitude observations) for the following events: for September 3, 1960 event we have employed the 14:10 U.T. data from Rocket Observations [18] in the (0.1–0. 7) GV band. For November 12 and 15, 1960 GLE's, we have used the 18:40 U.T. and 05:00 U.T. data, respectively, from rocket observations in the (6.16–1.02) GV band [73]. For July 7, 1966 GLE, we have used the 19:06 U.T. data given by [57, 58] in the (0.13–0.19) GV band, and the spectrum given by [118] in the (0.19–0.44) GV band;

); [8] proposed a differential spectrum of the form (<sup>P</sup>4.8) for relativistic protons

, the ratio r<sup>1</sup> = (dW/dt)p–p/(dW/dt)coll changes from r<sup>1</sup> = 1.7–16 and

–0.64; therefore if all processes would

); similarly, [40] deduced an integral

) with an upper cutoff at Em = 4.3 GeV or in

concentration n = 1013 cm<sup>3</sup>

128 Cosmic Rays

rigidity ( <sup>P</sup><sup>5</sup>

the ratio r<sup>2</sup> = (dW/dt)ad/(dW/dt)coll varies from r<sup>2</sup> = 4.6 10<sup>5</sup>

$$J(>P) = K \int\_{P}^{P\_m} P^{-\Phi}dP\tag{7}$$

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

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 law in kinetic energy Ð Em <sup>E</sup> <sup>E</sup>�<sup>Φ</sup>dE reported in several works by (e.g. [41–43]). However, although 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 therefore, can be directly related to the acceleration process

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

Figure 2. Theoretical and observational integral energy spectrum of hot events.

4. Theoretical spectra of solar protons in the source

Figure 4. Theoretical and observational integral energy spectrum of warm events.

approach.

In order to deduce the velocity and time dependent theoretical spectrum of the accelerated protons, one must take into account the various processes which affect particles during the remaining time within the acceleration volume. The main processes acting on particles during acceleration in a high density plasma are related either to catastrophic changes of particle density from the accelerated flux or to energy losses. Whereas the first kind of processes affect mainly the number density of the spectrum, energy losses entail a shift of the particle distribution toward lower energies, and a certain degradation of the number density due to thermalization of the less energetic particles. The number density changes on the accelerated proton flux may occur from catastrophic particle diffusion out of the flare source or by nuclear disintegration or creation of solar protons by nuclear reactions. Given the lack of knowledge about the exact magnetic field configuration and thus of the confinement efficiency of these fields, we do not consider here the effects of plausible escape mechanisms [26, 27, 104] on the theoretical spectrum. Therefore, to make a clear distinction between the energy loss effects (Section 2) on the spectrum of acceleration, we shall also neglect nuclear transformation during acceleration, local modulation post-acceleration and interplanetary modulation [67, 68] in this

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

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

131

Figure 3. Theoretical and observational integral energy spectrum of cold events.

Exploration of Solar Cosmic Ray Sources by Means of Particle Energy Spectra http://dx.doi.org/10.5772/intechopen.77052 131

Figure 4. Theoretical and observational integral energy spectrum of warm events.

Figure 2. Theoretical and observational integral energy spectrum of hot events.

130 Cosmic Rays

Figure 3. Theoretical and observational integral energy spectrum of cold events.

## 4. Theoretical spectra of solar protons in the source

In order to deduce the velocity and time dependent theoretical spectrum of the accelerated protons, one must take into account the various processes which affect particles during the remaining time within the acceleration volume. The main processes acting on particles during acceleration in a high density plasma are related either to catastrophic changes of particle density from the accelerated flux or to energy losses. Whereas the first kind of processes affect mainly the number density of the spectrum, energy losses entail a shift of the particle distribution toward lower energies, and a certain degradation of the number density due to thermalization of the less energetic particles. The number density changes on the accelerated proton flux may occur from catastrophic particle diffusion out of the flare source or by nuclear disintegration or creation of solar protons by nuclear reactions. Given the lack of knowledge about the exact magnetic field configuration and thus of the confinement efficiency of these fields, we do not consider here the effects of plausible escape mechanisms [26, 27, 104] on the theoretical spectrum. Therefore, to make a clear distinction between the energy loss effects (Section 2) on the spectrum of acceleration, we shall also neglect nuclear transformation during acceleration, local modulation post-acceleration and interplanetary modulation [67, 68] in this approach.

In addition, we shall not take into account spatial spread in the energy change rates within the acceleration process such that energy fluctuations [81, 82] which are considered minor for the purpose of this work.

It must be emphasized that since we are dealing with solar energetic particles, the well-known phenomena of Forbush decreases are rather related with galactic cosmic rays but not necessarily with solar energetic protons (e.g. [20]).

To establish the particle spectrum, we shall follow the assumptions that under the present simplified conditions lead to similar results that are obtained by solving a Fokker-Planck type transport equation on similar conditions [36, 81], that is, when the steady-state is reached in the source: we assume that a suprathermal flux with similar energy or a Maxwellian particle distribution is present in the region where the acceleration process is operating and a fraction N<sup>0</sup> of them can be accelerated during the time interval in which the stochastic acceleration mechanism is acting [93]. The selection of particles follows to the fact that their energy must be ≥ than a critical energy, Ec, determined by the competition of acceleration and by local energy losses. By analogy with radioactive decay the energy distribution of cosmic ray particles is assumed as an exponential distribution in age of the form

$$N(E)dE = N(t)dt = \frac{N\_0}{\tau} \exp\left(-t/\tau\right)dt\tag{8}$$

which in terms of the Lorentz factor is expressed as

$$N(\gamma)d\gamma = \left(1/Mc^2\right)N(t)\text{dt}\tag{8.1}$$

Jð Þ¼ > E N0e

4.1. The spectrum of acceleration

t Eð Þ<sup>c</sup> <sup>=</sup><sup>τ</sup> e

For the case in�which energy losses are completely unimportant within the acceleration time scale, the net energy change rate is determined by the acceleration rate, Eq.(1), which for

the condition (dγ/dt) = (dγ/dt)acc–(dγ/dt)loss = 0 gives γ<sup>c</sup> = 1 (and hence Ec = 0), such that by

simplicity's sake, we shall represent hereafter in terms of the Lorentz factor γ as

dγ

integration of (12) we obtain the acceleration time up to the energy E = Mc(γ�1) as

Now, by substitution of (13) in Eq. (8.1), we obtain the following differential spectrum

<sup>W</sup>�ð Þ <sup>1</sup>þ1=ατ <sup>¼</sup> <sup>N</sup><sup>0</sup>

N<sup>0</sup>

ατMc<sup>2</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � ��1=<sup>2</sup>

ατ

When the parameter β is considered outside of the integrating equations a somewhat different

The corresponding integral spectrum of the accelerated particles appears from Eqs. (11)–(13) as

αβτ Mc<sup>2</sup> � �<sup>1</sup>=αβτW�ð Þ <sup>1</sup>þ1=αβτ

� <sup>γ</sup><sup>m</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>m</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> h i�1=ατ � � (15)

� Em <sup>þ</sup> Mc<sup>2</sup> <sup>þ</sup>

� � <sup>q</sup> �I=ατ ( ) (15.1)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>m</sup> þ 2Mc<sup>2</sup>Em

E2

<sup>t</sup> <sup>¼</sup> <sup>1</sup> α

Nð Þ¼ γ

N Wð Þ¼ <sup>N</sup><sup>0</sup>

<sup>J</sup>ð Þ¼ <sup>&</sup>gt; <sup>E</sup> <sup>N</sup><sup>0</sup> <sup>γ</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> h i�1=ατ

the integral spectrum expressed in terms of kinetic energy becomes,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>E</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>Mc<sup>2</sup><sup>E</sup> h i p �I=ατ

which in terms of total energy W is expressed as

Mc<sup>2</sup> � �<sup>1</sup>=ατ <sup>1</sup> <sup>þ</sup> <sup>β</sup> � ��1=ατ

β

N Wð Þ¼ <sup>N</sup><sup>0</sup> ατ

expression is obtained:

(where) <sup>γ</sup><sup>m</sup> <sup>¼</sup> Em <sup>þ</sup> Mc<sup>2</sup> � �=Mc<sup>2</sup>

<sup>J</sup>ð Þ¼ <sup>&</sup>gt; <sup>E</sup> <sup>N</sup><sup>0</sup> Mc<sup>2</sup> � �<sup>1</sup>=ατ <sup>E</sup> <sup>þ</sup> Mc<sup>2</sup> <sup>þ</sup>

�t Eð Þ=<sup>τ</sup> � <sup>e</sup>

�t Eð Þ <sup>m</sup> <sup>=</sup><sup>τ</sup> h i (11)

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

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

dt � � <sup>¼</sup> α γ<sup>2</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> (12)

ln <sup>γ</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> h i (13)

<sup>γ</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> h i�1=ατ

<sup>W</sup><sup>2</sup> � Mc<sup>2</sup> ð Þ<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> (14.1)

Mc<sup>2</sup> � �<sup>1</sup>=ατ <sup>W</sup> <sup>þ</sup> <sup>W</sup><sup>2</sup> � Mc<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> n o � � <sup>1</sup>=ατ

(14)

133

where tis the necessary time to accelerate particles up to the energy E and τ is considered as a mean confinement time of particles in the acceleration process. Eq. (8) represents hence the differential spectrum of the accelerated particles; to obtain the integral spectrum we take the integration of (8) up to the maximum energy of the accelerated protons, Em (corresponding to the upper cutoff in the particle spectrum) the existence of which has been shown by [43] as discussed before.

$$J(>E) = \int\_{E}^{E\_m} N(E)dE = \int\_{t}^{t\_m} N(t)dt = N\_0 \int\_{t}^{t\_m} \frac{e^{-t/\tau}}{\tau}dt = N\_0 \left[e^{-t/\tau} - e^{-t\_m/\tau}\right] \tag{9}$$

where tm is the acceleration time up to the high energy cutoff. Because the acceleration process is competing with energy loss processes, the net energy gain rate is effectively fixed on particles, only beginning at a certain threshold value, Ec defined by (dE/dt) = 0, such that only particles with E>Ec are able to participate in the acceleration process (the flux N0). Thus the acceleration time t is defined as

$$t = \int\_{E\_c}^{E} \left(\frac{dE}{dt}\right) dt = t(E) - t(E\_c) \tag{10}$$

Similarly the constant value tm, representing the acceleration time up to the high energy cutoff, Em defined as tm = t(Em) - t(Ec), where t(Ec) denotes the time of the acceleration onset. Therefore, Eq. (9) can be rewritten as

Exploration of Solar Cosmic Ray Sources by Means of Particle Energy Spectra http://dx.doi.org/10.5772/intechopen.77052 133

$$J(>E) = N\_0 e^{t(E\_c)/\tau} \left[ e^{-t(E)/\tau} - e^{-t(E\_m)/\tau} \right] \tag{11}$$

#### 4.1. The spectrum of acceleration

In addition, we shall not take into account spatial spread in the energy change rates within the acceleration process such that energy fluctuations [81, 82] which are considered minor for the

It must be emphasized that since we are dealing with solar energetic particles, the well-known phenomena of Forbush decreases are rather related with galactic cosmic rays but not necessar-

To establish the particle spectrum, we shall follow the assumptions that under the present simplified conditions lead to similar results that are obtained by solving a Fokker-Planck type transport equation on similar conditions [36, 81], that is, when the steady-state is reached in the source: we assume that a suprathermal flux with similar energy or a Maxwellian particle distribution is present in the region where the acceleration process is operating and a fraction N<sup>0</sup> of them can be accelerated during the time interval in which the stochastic acceleration mechanism is acting [93]. The selection of particles follows to the fact that their energy must be ≥ than a critical energy, Ec, determined by the competition of acceleration and by local energy losses. By analogy with radioactive decay the energy distribution of cosmic ray particles is

purpose of this work.

132 Cosmic Rays

ily with solar energetic protons (e.g. [20]).

assumed as an exponential distribution in age of the form

which in terms of the Lorentz factor is expressed as

Jð Þ¼ > E

acceleration time t is defined as

fore, Eq. (9) can be rewritten as

ðEm E

N Eð ÞdE ¼

N Eð ÞdE <sup>¼</sup> N tð Þdt <sup>¼</sup> <sup>N</sup><sup>0</sup>

particle spectrum) the existence of which has been shown by [43] as discussed before.

N tð Þdt ¼ N<sup>0</sup>

dE dt � �

where tm is the acceleration time up to the high energy cutoff. Because the acceleration process is competing with energy loss processes, the net energy gain rate is effectively fixed on particles, only beginning at a certain threshold value, Ec defined by (dE/dt) = 0, such that only particles with E>Ec are able to participate in the acceleration process (the flux N0). Thus the

Similarly the constant value tm, representing the acceleration time up to the high energy cutoff, Em defined as tm = t(Em) - t(Ec), where t(Ec) denotes the time of the acceleration onset. There-

ðtm t

e�t=<sup>τ</sup> τ

ðtm t

t ¼ ðE Ec

where tis the necessary time to accelerate particles up to the energy E and τ is considered as a mean confinement time of particles in the acceleration process. Eq. (8) represents hence the differential spectrum of the accelerated particles; to obtain the integral spectrum we take the integration of (8) up to the maximum energy of the accelerated protons, Em (corresponding to the upper cutoff in the

<sup>τ</sup> exp ð Þ �t=<sup>τ</sup> dt (8)

�t=<sup>τ</sup> � <sup>e</sup> �tm=<sup>τ</sup> h i

dt ¼ t Eð Þ� t Eð Þ<sup>c</sup> (10)

(9)

<sup>N</sup>ð Þ <sup>γ</sup> <sup>d</sup>γ<sup>¼</sup> <sup>1</sup>=Mc<sup>2</sup> � �N tð Þdt (8.1)

dt ¼ N<sup>0</sup> e

For the case in�which energy losses are completely unimportant within the acceleration time scale, the net energy change rate is determined by the acceleration rate, Eq.(1), which for simplicity's sake, we shall represent hereafter in terms of the Lorentz factor γ as

$$
\left(\frac{d\gamma}{dt}\right) = \alpha \left(\gamma^2 - 1\right)^{1/2} \tag{12}
$$

the condition (dγ/dt) = (dγ/dt)acc–(dγ/dt)loss = 0 gives γ<sup>c</sup> = 1 (and hence Ec = 0), such that by integration of (12) we obtain the acceleration time up to the energy E = Mc(γ�1) as

$$t = \frac{1}{\alpha} \ln \left[ \gamma + \left( \gamma^2 - 1 \right)^{1/2} \right] \tag{13}$$

Now, by substitution of (13) in Eq. (8.1), we obtain the following differential spectrum

$$N(\boldsymbol{\gamma}) = \frac{N\_0}{\alpha \pi M c^2} \left(\boldsymbol{\gamma}^2 - \mathbf{1}\right)^{-1/2} \left[\boldsymbol{\gamma} + \left(\boldsymbol{\gamma}^2 + \mathbf{1}\right)^{1/2}\right]^{-1/\alpha \pi} \tag{14}$$

which in terms of total energy W is expressed as

$$N(\mathcal{W}) = \frac{\mathcal{N}\_0}{a\pi} \left(\mathcal{M}c^2\right)^{1/a\pi} \frac{\left(1+\beta\right)^{-1/a\pi}}{\beta} \mathcal{W}^{-\left(1+\left(a\pi\right)\right)} = \frac{\mathcal{N}\_0}{a\pi} \left(\mathcal{M}c^2\right)^{1/a\pi} \frac{\left\{\mathcal{W} + \left(\mathcal{W}^2 - \left(\mathcal{M}c^2\right)^{1/2}\right)\right\}^{1/a\pi}}{\left(\mathcal{W}^2 - \left(\mathcal{M}c^2\right)^2\right)^{1/2}} \tag{14.1}$$

When the parameter β is considered outside of the integrating equations a somewhat different expression is obtained:

$$N(W) = \frac{N\_0}{\alpha \beta \tau} \left(Mc^2\right)^{1/\alpha \beta \tau} W^{-\left(1 + 1/\alpha \beta \tau\right)}$$

The corresponding integral spectrum of the accelerated particles appears from Eqs. (11)–(13) as

$$J(>E) = N\_0 \left[ \left[ \gamma + \left( \chi^2 - 1 \right)^{1/2} \right]^{-1/\alpha \tau} - \left[ \gamma\_m + \left( \chi\_m^2 - 1 \right)^{1/2} \right]^{-1/\alpha \tau} \right] \tag{15}$$

(where) <sup>γ</sup><sup>m</sup> <sup>¼</sup> Em <sup>þ</sup> Mc<sup>2</sup> � �=Mc<sup>2</sup>

the integral spectrum expressed in terms of kinetic energy becomes,

$$J(>E) = N\_0 \left(Mc^2\right)^{1/a\pi} \left\{ \left[ E + Mc^2 + \sqrt{E^2 + 2Mc^2}E \right]^{-l/a\pi} - \left[ E\_m + Mc^2 + \sqrt{E\_m^2 + 2Mc^2}E\_m \right]^{-l/a\pi} \right\} \tag{15.1}$$

#### 4.2. The modulated spectrum in the acceleration region

In order to study local modulation of spectrum (14) or (15) during acceleration, we shall proceed to consider energy loss processes together with the energy gain rate (12), according to the processes discussed in Section 2.

#### 4.2.1. Modulation by collisional losses

When collisional losses are not negligible during acceleration, the net energy change rate is determined by (2) and (12) as

$$\frac{d\gamma}{dt} = a\left(\gamma^2 - 1\right)^{1/2} - \left(b/Mc^2\right)\gamma\left(\gamma^2 - 1\right)^{-1/2} \tag{16}$$

Jð Þ¼ > E N<sup>0</sup> exp

with <sup>p</sup> <sup>¼</sup> <sup>E</sup><sup>1</sup> <sup>E</sup>1þ2Mc<sup>2</sup> ½ � ð Þ <sup>1</sup>=<sup>2</sup>

dγ

C = �A + bMc<sup>2</sup>

C = �A + bMc<sup>2</sup>

<sup>t</sup> <sup>¼</sup> <sup>1</sup>

a2 2 8 ><

>:

(a3�a1)+a<sup>3</sup>

t Eð Þ<sup>i</sup> τ

ατð Þ E<sup>1</sup> � E<sup>2</sup>

� exp � <sup>E</sup><sup>2</sup> �E<sup>2</sup> � <sup>2</sup>Mc<sup>2</sup> � � � � <sup>1</sup>

� � <sup>ε</sup>þEþMc<sup>2</sup>

ατð Þ <sup>E</sup>1�E<sup>2</sup> , <sup>¼</sup> <sup>E</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>Mc<sup>2</sup><sup>E</sup> � �<sup>1</sup>=<sup>2</sup>

that spectrum (19.1) reduces a spectrum (15.1) when b = 0.

The corresponding particle energy spectrum to Eq. (2<sup>0</sup>

4.3. Modulation by proton-proton nuclear collisions

dt <sup>¼</sup> α γ<sup>2</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> � <sup>b</sup>=Mc<sup>2</sup> � �γ γ<sup>2</sup> � <sup>1</sup> � ��1=<sup>2</sup> � h Mc<sup>2</sup>

�h, D = A + jMc<sup>2</sup>

last one but with A=(α�f�η)(Mc<sup>2</sup>

<sup>λ</sup> ln 2 <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup>

� � �

� <sup>A</sup><sup>2</sup> <sup>1</sup> � <sup>a</sup><sup>2</sup>

2

2 � �<sup>1</sup>=<sup>2</sup>

6 4

Mc<sup>2</sup> � � �

2

� � �

eration and <sup>E</sup>1, E<sup>2</sup> correspond respectively to <sup>b</sup> � <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>4</sup>α<sup>2</sup> Mc<sup>2</sup> � �<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> � �=2<sup>α</sup>

adding Eq. (4), the net energy rate (16) turns into the following expression

equation of the form Aγ<sup>3</sup> + Bγ<sup>2</sup> + Cγ +D= 0 with A = α(Mc<sup>2</sup>

) 2

this critical value up to the energy E is obtained from Eq. (20) as

� � �

sin �<sup>1</sup> <sup>a</sup>2<sup>γ</sup> � <sup>1</sup> j j γ � a<sup>2</sup> � �

þ 2γ

� � � � �

(" �<sup>p</sup>

tan �<sup>1</sup> E E<sup>2</sup> <sup>þ</sup> Mc<sup>2</sup> � � <sup>þ</sup> Mc<sup>2</sup>E<sup>2</sup> <sup>ε</sup>E<sup>2</sup> �E<sup>2</sup> � <sup>2</sup>Mc<sup>2</sup> ð Þ � �!# � exp �t Eð Þ <sup>m</sup>

In the event that proton-proton collisions are important during the acceleration process. By

The critical value γc for acceleration resulting when (dγ/dt) = 0 is obtained by solving a cubic

η, such than when a medium concentration n is fixed, the basic dependence remains on α. Given that for the bulk of the involved parameters the conditions a<sup>1</sup> > 1, a<sup>2</sup> ≤ �1 and 0 < a<sup>3</sup> ≤ 1 are systematically satisfied through all the energy ranges the relation Ec = Mc<sup>2</sup> (γ<sup>c</sup> – 1) states a<sup>1</sup> as the critical value for effective acceleration. The acceleration time of particles beginning with

> <sup>A</sup>1a1þA2a2þA3a<sup>3</sup> γ � a<sup>1</sup> 2 a<sup>2</sup> <sup>1</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup>

> > <sup>þ</sup> <sup>A</sup><sup>3</sup> <sup>1</sup> � <sup>a</sup><sup>2</sup>

where the constants. A<sup>1</sup> = (a1�1)(a2�a3)/ξ, A<sup>2</sup> = (a2�1)(a3�a1)/ξ and A<sup>3</sup> = (a3�1) (a1�a2)/ξ

<sup>A</sup><sup>1</sup> <sup>a</sup><sup>2</sup> ð Þ <sup>1</sup>�<sup>1</sup> <sup>1</sup>=<sup>2</sup> <sup>2</sup>

3 � �<sup>1</sup>=<sup>2</sup>

(a1�a2), and take on different values according to the energy range concerned;

� � � � �

� � � �

emerge from the integration by partial fractions of Eq. (20), with ξ = a<sup>1</sup>

ð Þ <sup>γ</sup> � <sup>1</sup> �<sup>2</sup> <sup>þ</sup> j Mc<sup>2</sup>

�<sup>h</sup> if <sup>E</sup> <sup>≤</sup> 110 MeV, or, A=(α�f)(Mc<sup>2</sup>

�h, D=A�h if 110 < E ≤ 290 MeV and for the range E > 290 MeV similar to the

�1=ατ ð Þ� <sup>E</sup>�E<sup>1</sup> <sup>ε</sup><sup>þ</sup> <sup>E</sup><sup>1</sup> <sup>E</sup>1þ2Mc<sup>2</sup> ½ � ð Þ <sup>1</sup>

ð Þ� <sup>E</sup>�E<sup>1</sup> <sup>ε</sup>� <sup>E</sup><sup>1</sup> <sup>E</sup>1þ2Mc<sup>2</sup> ½ � ð Þ <sup>1</sup>

2

� � � � �

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

, Ei = b/2 α is the threshold value for effective accel-

� �. It can be seen

) is developed in the Appendix.

ð Þ <sup>γ</sup> � <sup>1</sup> � ��<sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>þ</sup> <sup>η</sup> h i h i ð Þ <sup>γ</sup> � <sup>1</sup> <sup>1</sup>=<sup>2</sup>

> ) 2

. Therefore, the roots a1, a<sup>2</sup> and a<sup>3</sup> depend on α, b, h, j, f and

<sup>γ</sup>ð Þ <sup>2</sup> � <sup>1</sup> <sup>1</sup>=<sup>2</sup> <sup>þ</sup> <sup>2</sup>a1<sup>γ</sup> � <sup>2</sup>

sin �<sup>1</sup> <sup>a</sup>3<sup>γ</sup> � <sup>1</sup> j j γ � a<sup>3</sup> ) 2

> � � � � �

� t γ<sup>c</sup> � �)

τ � �)

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

ð Þ 19:1

135

(20)

,

,

, B = �A�(b + j)Mc<sup>2</sup>

, B = �A�bMc<sup>2</sup>

3 7 5

2

(a2�a3) +

(21)

2

where b = 7.62 � <sup>10</sup>�<sup>9</sup> nL, then, the solution of (16) is easily performed by employing a change of variable of the form x ¼ ½ � ð Þ γ � 1 =ð Þ γ þ 1 [90], such that the acceleration time from the critical energy E<sup>c</sup> up to the energy E, in terms of the Lorentz factor is

$$t = \ln \left| \frac{1+\mathbf{x}}{1-\mathbf{x}} \right|^{1/a} \left| \frac{\phi^{1/2}\mathbf{x} - (-Y\_2)^{1/2}}{\phi^{1/2}\mathbf{x} - (-Y\_2)^{1/2}} \right|^p + \xi \tan^{-1} \left[ \mathbf{x} \left( \phi/Y\_1 \right)^{1/2} \right] \Big|\_{\mathbf{x}\_\mathcal{E}}^\mathbf{x} = t(\mathbf{x}) - t(\mathbf{x}\_\varepsilon) \tag{17}$$

with φ = b/Mc<sup>2</sup> , Y<sup>1</sup> = 2α + (4α<sup>2</sup> + φ<sup>2</sup> ) 1/2, <sup>Y</sup><sup>2</sup> <sup>=</sup> <sup>2</sup>α�(4α<sup>2</sup> <sup>+</sup> <sup>φ</sup><sup>2</sup> ) 1/2, p=Y3/[2(�Y2) 1/2φ1/2], Y<sup>3</sup> = (2φ/α) [(φ-Y2)/(Y1-Y2)], Y<sup>4</sup> = (2φ/α)[(Y1-φ)/(Y1-Y2)], ζ = Y4/(φY1) 1/2 and xc= [(ɣc-1)/(ɣ<sup>c</sup> + 1)]1/2, where γ<sup>c</sup> = (b/2αMc<sup>2</sup> ) + 1 is the critical value for acceleration determined by (dγ/dt) = 0, and the constant value t xð Þ<sup>c</sup> corresponds to the value of t(Ec) appearing in Eq. (10). The differential spectrum of particles is obtained by substituting of (Eq. 17) in Eq. (8<sup>0</sup> ) as follows

$$N(\mathbf{y}) = \frac{N\_0}{\pi M c^2} e^{\phi(\mathbf{x}\_1)/\pi} \frac{\left(\mathbf{y}^2 - \mathbf{1}\right)^{1/2}}{\left[a(\mathbf{y}^2 - \mathbf{1}) - \phi\mathbf{y}\right]} \left(\frac{\mathbf{1} + \mathbf{x}}{\mathbf{1} - \mathbf{x}}\right)^{-1/a\pi} \left[\frac{\phi^{1/2}\mathbf{x} - (-\mathbf{Y}\_2)^{1/2}}{\phi^{1/2}\mathbf{x} + (-\mathbf{Y}\_2)^{1/2}}\right]^{-\mathcal{Q}/2} \exp\left[(-\eta/\pi)\tan^{-1}\left[\mathbf{x}\left(\phi/\mathbf{Y}\_1\right)^{1/2}\right]\right] \tag{18}$$

The integral spectrum is then from Eq. (11) and Eq. (17)

$$\begin{split} I(>E) &= N\_0 \exp\left(t(\mathbf{x}\_c)/\tau\right) \left\{ \begin{pmatrix} \frac{1+\chi}{1-\chi} \end{pmatrix}^{-1/d\tau} \begin{pmatrix} \phi^{1/2} \mathbf{x} - (Y\_2)^{1/2} \\ \phi^{1/2} \mathbf{x} + (Y\_2)^{1/2} \end{pmatrix}^{-p/2} \exp\left[ \left(-\frac{q}{\tau}\right) \tan^{-1} \left[ \mathbf{x} (\phi(Y\_1)^{1/2}) \right] \right] \\\\ -\exp\left(-t(\mathbf{x}\_m)/\tau\right) \end{pmatrix} \end{split} \tag{19}$$

where t(xm) corresponding to t(Em) in Eq. (11), appearing from the evaluation of Eq. (17) in the constant value xm <sup>¼</sup> <sup>γ</sup><sup>m</sup> � <sup>1</sup> � �<sup>=</sup> <sup>γ</sup><sup>m</sup> <sup>þ</sup> <sup>1</sup> � � � � <sup>1</sup>=<sup>2</sup> : It can be seen that spectra (18) or (19)reduces to (14) or (15) when b = 0. The integral spectrum in terms of kinetic energy is expressed as

#### Exploration of Solar Cosmic Ray Sources by Means of Particle Energy Spectra http://dx.doi.org/10.5772/intechopen.77052 135

$$I(>E) = N\_0 \exp\left(\frac{t(E\_i)}{\tau}\right) \left\{ \left[\frac{\left|\varepsilon + E + Mc^2\right|}{Mc^2}\right]^{-1/\alpha} \frac{\left|(E - E\_i) - \varepsilon + \left[E\_1(E\_1 + 2Mc^2)\right]^2\right|}{\left(E - E\_i\right) - \varepsilon - \left[E\_1(E\_1 + 2Mc^2)\right]^2} \right]^{-p}$$

$$\times \exp\left(\frac{-\left[E\_2\left(-E\_2 - 2Mc^2\right)\right]^{\frac{1}{2}}}{\alpha \tau (E\_1 - E\_2)} \tan^{-1}\left(\frac{E\left(E\_2 + Mc^2\right) + Mc^2E\_2}{\varepsilon E\_2 \left(-E\_2 - 2Mc^2\right)}\right)\right) \right] - \exp\left(\frac{-t(E\_m)}{\tau}\right) \right\} \qquad (19.1)$$

with <sup>p</sup> <sup>¼</sup> <sup>E</sup><sup>1</sup> <sup>E</sup>1þ2Mc<sup>2</sup> ½ � ð Þ <sup>1</sup>=<sup>2</sup> ατð Þ <sup>E</sup>1�E<sup>2</sup> , <sup>¼</sup> <sup>E</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>Mc<sup>2</sup><sup>E</sup> � �<sup>1</sup>=<sup>2</sup> , Ei = b/2 α is the threshold value for effective acceleration and <sup>E</sup>1, E<sup>2</sup> correspond respectively to <sup>b</sup> � <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>4</sup>α<sup>2</sup> Mc<sup>2</sup> � �<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> � �=2<sup>α</sup> � �. It can be seen that spectrum (19.1) reduces a spectrum (15.1) when b = 0.

The corresponding particle energy spectrum to Eq. (2<sup>0</sup> ) is developed in the Appendix.

#### 4.3. Modulation by proton-proton nuclear collisions

4.2. The modulated spectrum in the acceleration region

dγ

critical energy E<sup>c</sup> up to the energy E, in terms of the Lorentz factor is

x � �ð Þ Y<sup>2</sup>

x � �ð Þ Y<sup>2</sup>

)

spectrum of particles is obtained by substituting of (Eq. 17) in Eq. (8<sup>0</sup>

1 þ x 1 � x

� ��1=ατ ϕ<sup>1</sup>=<sup>2</sup>

� ��1=ατ <sup>ϕ</sup>1=2x�ð Þ <sup>Y</sup><sup>2</sup> <sup>1</sup>=<sup>2</sup>

1=2

� � � � �

p

1/2, <sup>Y</sup><sup>2</sup> <sup>=</sup> <sup>2</sup>α�(4α<sup>2</sup> <sup>+</sup> <sup>φ</sup><sup>2</sup>

constant value t xð Þ<sup>c</sup> corresponds to the value of t(Ec) appearing in Eq. (10). The differential

ϕ<sup>1</sup>=<sup>2</sup>

<sup>ϕ</sup>1=2xþð Þ <sup>Y</sup><sup>2</sup> <sup>1</sup>=<sup>2</sup> � ��p=<sup>2</sup>

where t(xm) corresponding to t(Em) in Eq. (11), appearing from the evaluation of Eq. (17) in the

(14) or (15) when b = 0. The integral spectrum in terms of kinetic energy is expressed as

1=2

to the processes discussed in Section 2.

4.2.1. Modulation by collisional losses

determined by (2) and (12) as

134 Cosmic Rays

<sup>t</sup> <sup>¼</sup> ln <sup>1</sup> <sup>þ</sup> <sup>x</sup> 1 � x

with φ = b/Mc<sup>2</sup>

γ<sup>c</sup> = (b/2αMc<sup>2</sup>

N<sup>0</sup> <sup>τ</sup>Mc<sup>2</sup> <sup>e</sup>

Nð Þ¼ γ

� � � � � � � �

t xð Þ<sup>c</sup> <sup>=</sup><sup>τ</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> α γð Þ� <sup>2</sup> � <sup>1</sup> ϕγ � �

<sup>J</sup>ð Þ¼ <sup>&</sup>gt; <sup>E</sup> <sup>N</sup><sup>0</sup> exp ð Þ t xð Þ<sup>c</sup> <sup>=</sup><sup>τ</sup> <sup>1</sup>þ<sup>x</sup>

� exp ð Þ �t xð Þ <sup>m</sup> =τ

constant value xm <sup>¼</sup> <sup>γ</sup><sup>m</sup> � <sup>1</sup> � �<sup>=</sup> <sup>γ</sup><sup>m</sup> <sup>þ</sup> <sup>1</sup> � � � � <sup>1</sup>=<sup>2</sup>

<sup>1</sup>=<sup>α</sup> ϕ<sup>1</sup>=<sup>2</sup>

� � � � �

, Y<sup>1</sup> = 2α + (4α<sup>2</sup> + φ<sup>2</sup>

ϕ<sup>1</sup>=<sup>2</sup>

[(φ-Y2)/(Y1-Y2)], Y<sup>4</sup> = (2φ/α)[(Y1-φ)/(Y1-Y2)], ζ = Y4/(φY1)

The integral spectrum is then from Eq. (11) and Eq. (17)

1�x

In order to study local modulation of spectrum (14) or (15) during acceleration, we shall proceed to consider energy loss processes together with the energy gain rate (12), according

When collisional losses are not negligible during acceleration, the net energy change rate is

where b = 7.62 � <sup>10</sup>�<sup>9</sup> nL, then, the solution of (16) is easily performed by employing a change of variable of the form x ¼ ½ � ð Þ γ � 1 =ð Þ γ þ 1 [90], such that the acceleration time from the

<sup>þ</sup> <sup>ξ</sup> tan �<sup>1</sup> <sup>x</sup> <sup>ϕ</sup>=Y<sup>1</sup>

) + 1 is the critical value for acceleration determined by (dγ/dt) = 0, and the

<sup>x</sup> � �ð Þ <sup>Y</sup><sup>2</sup> <sup>1</sup>=<sup>2</sup>

<sup>x</sup> þ �ð Þ <sup>Y</sup><sup>2</sup> <sup>1</sup>=<sup>2</sup> " #�∅=<sup>2</sup>

)

dt <sup>¼</sup> α γ<sup>2</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> � <sup>b</sup>=Mc<sup>2</sup> � �γ γ<sup>2</sup> � <sup>1</sup> � ��1=<sup>2</sup> (16)

� �<sup>1</sup>=<sup>2</sup> h i�

exp � <sup>q</sup> τ � �

) (19)

: It can be seen that spectra (18) or (19)reduces to

<sup>1</sup>=<sup>2</sup> h i h i (

� � x xc

1/2, p=Y3/[2(�Y2)

¼ t xð Þ� t xð Þ<sup>c</sup> (17)

1/2 and xc= [(ɣc-1)/(ɣ<sup>c</sup> + 1)]1/2, where

) as follows

exp ð Þ �q=<sup>τ</sup> tan �<sup>1</sup> <sup>x</sup> <sup>ϕ</sup>=Y<sup>1</sup> � �<sup>1</sup>=<sup>2</sup> h i h i

tan �<sup>1</sup> <sup>x</sup>ðϕð Þ <sup>Y</sup><sup>1</sup>

1/2φ1/2], Y<sup>3</sup> = (2φ/α)

(18)

In the event that proton-proton collisions are important during the acceleration process. By adding Eq. (4), the net energy rate (16) turns into the following expression

$$\frac{d\boldsymbol{\gamma}}{dt} = \boldsymbol{a} \left(\boldsymbol{\gamma}^2 - 1\right)^{1/2} - \left\{\boldsymbol{b} / \mathcal{M}c^2\right\} \boldsymbol{\gamma} \left(\boldsymbol{\gamma}^2 - 1\right)^{-1/2} - \left[\boldsymbol{h} \left[\mathcal{M}c^2 \left(\boldsymbol{\gamma} - 1\right)^{-2} + j \left[\mathcal{M}c^2 \left(\boldsymbol{\gamma} - 1\right)\right]^{-1} + \boldsymbol{f} + \boldsymbol{\eta}\right]\right] \left(\boldsymbol{\gamma} - 1\right)^{1/2} \tag{20}$$

The critical value γc for acceleration resulting when (dγ/dt) = 0 is obtained by solving a cubic equation of the form Aγ<sup>3</sup> + Bγ<sup>2</sup> + Cγ +D= 0 with A = α(Mc<sup>2</sup> ) 2 , B = �A�(b + j)Mc<sup>2</sup> , C = �A + bMc<sup>2</sup> �h, D = A + jMc<sup>2</sup> �<sup>h</sup> if <sup>E</sup> <sup>≤</sup> 110 MeV, or, A=(α�f)(Mc<sup>2</sup> ) 2 , B = �A�bMc<sup>2</sup> , C = �A + bMc<sup>2</sup> �h, D=A�h if 110 < E ≤ 290 MeV and for the range E > 290 MeV similar to the last one but with A=(α�f�η)(Mc<sup>2</sup> ) 2 . Therefore, the roots a1, a<sup>2</sup> and a<sup>3</sup> depend on α, b, h, j, f and η, such than when a medium concentration n is fixed, the basic dependence remains on α. Given that for the bulk of the involved parameters the conditions a<sup>1</sup> > 1, a<sup>2</sup> ≤ �1 and 0 < a<sup>3</sup> ≤ 1 are systematically satisfied through all the energy ranges the relation Ec = Mc<sup>2</sup> (γ<sup>c</sup> – 1) states a<sup>1</sup> as the critical value for effective acceleration. The acceleration time of particles beginning with this critical value up to the energy E is obtained from Eq. (20) as

$$t = \frac{1}{\lambda} \left\{ \ln \left[ \left| 2 \left( \mathbf{y}^2 - \mathbf{1} \right)^{1/2} + 2 \mathbf{y} \right|^{A\_1 a\_1 + A\_2 a\_2 + A\_3 a\_3} \left| \frac{\mathbf{y} - \mathbf{a}\_1}{2 \left( a\_1^2 - 1 \right)^{1/2} \left( \mathbf{y}^2 - \mathbf{1} \right)^{1/2} + 2 a\_1 \mathbf{y} - 2} \right|^{A\_1 \left( a\_1^2 - 1 \right)^{1/2}} \right] \right\} \tag{21}$$
 
$$- \left[ A\_2 \left( 1 - a\_2^2 \right)^{1/2} \sin^{-1} \left( \frac{a\_2 \mathbf{y} - \mathbf{1}}{\left| \mathbf{y} - a\_2 \right|} \right) + A\_3 \left( 1 - a\_3^2 \right)^{1/2} \sin^{-1} \left( \frac{a\_3 \mathbf{y} - \mathbf{1}}{\left| \mathbf{y} - a\_3 \right|} \right) \right] - t \left\langle \boldsymbol{\chi}\_c \right\rangle \right]$$

where the constants. A<sup>1</sup> = (a1�1)(a2�a3)/ξ, A<sup>2</sup> = (a2�1)(a3�a1)/ξ and A<sup>3</sup> = (a3�1) (a1�a2)/ξ emerge from the integration by partial fractions of Eq. (20), with ξ = a<sup>1</sup> 2 (a2�a3) + a2 2 (a3�a1)+a<sup>3</sup> 2 (a1�a2), and take on different values according to the energy range concerned; λ = α (if E ≤ 110 MeV), λ = α � f (if 110 < E ≤ 290 MeV) and λ = α – f � η (if E > 290 MeV). The differential spectrum in this case follows from Eqs. (8.1) and (20) as

very weakly on r, and where E = α(Mc<sup>2</sup>

)

following acceleration time

<sup>k</sup> ln z<sup>2</sup> <sup>þ</sup> <sup>R</sup>1<sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup> �

where <sup>K</sup><sup>1</sup> <sup>¼</sup> ð Þ <sup>2</sup>C<sup>2</sup> � <sup>R</sup>1C<sup>1</sup> <sup>=</sup>2Δ<sup>1</sup>=<sup>2</sup>

differential spectrum of the form

K<sup>4</sup> ¼ ð Þ 2C<sup>8</sup> � R7C<sup>7</sup> =ð Þ �Δ<sup>4</sup>

0 @

2 4 8 < :

� �

�

ð Þ �Δ<sup>2</sup> <sup>1</sup>=<sup>2</sup> !

1=2

<sup>þ</sup>k<sup>2</sup> tan �<sup>1</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>3</sup>

<sup>c</sup>1=<sup>2</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> �

� �

and I = � <sup>r</sup>(Mc<sup>2</sup>

<sup>t</sup> <sup>¼</sup> <sup>1</sup>

Nð Þ¼ γ

N<sup>0</sup> Mc<sup>2</sup>kτ e

n�

� � � � �

� <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> �

� <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>1</sup> � ð Þ <sup>Δ</sup><sup>1</sup> <sup>1</sup>=<sup>2</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>Δ</sup><sup>1</sup> <sup>1</sup>=<sup>2</sup>

� �

<sup>110</sup> < E <sup>≤</sup> 290 MeV, E = (α�f)(Mc2

spectrum according Eq. (11) is,

� �θ<sup>1</sup>

> � � � � �

�θ<sup>5</sup> z�θ<sup>6</sup> 1

�

) 2

particle spectrum, we have simplified Eq. (24) by changing variable Z = <sup>γ</sup>�(γ<sup>2</sup>

�

E > 290 MeV). As in the preceding cases, the substitution of Eq. (25) in (8<sup>0</sup>

� �θ<sup>2</sup>

z<sup>8</sup> þ Jz<sup>7</sup> þ Mz<sup>6</sup> þ Nz<sup>5</sup> þ Pz<sup>4</sup> þ Qz<sup>3</sup> þ Rz<sup>2</sup> þ Sz þ V � �

�

<sup>A</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>

<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>

� �

ð Þ �Δ<sup>2</sup> <sup>1</sup>=<sup>2</sup> !

Θ<sup>1</sup> = c1/2κτ, Θ<sup>2</sup> = c3/2κτ, Θ<sup>3</sup> = c5/2κτ, Θ<sup>4</sup> = c7/2κτ, Θ<sup>5</sup> = K1/2κτ, Θ<sup>6</sup> = c9/2κτ, Θ<sup>7</sup> = (�K2)/κτ, Θ<sup>8</sup> = (�K3)/κτ and Θ<sup>9</sup> = (�K4)/κτ,J= 2(F + I)/V, M = (4E + 4G + 2I)/V, N = (6F + 8H�GI)/V, P = (GE + 8G)/V, Q = (GP + 8H + GI)/V, R = (4E + 4c�2I)/I, S = 2 (F�I)/V, V = (E + I)/V and V=E�I. The values of E, F, G, H, I in the range E < 110 MeV are the values given above; in the range

,F=E�bMc<sup>2</sup>

the range E > 290 MeV the only difference with the precedent range is E=(α�f�η)(Mc<sup>2</sup>

� �θ<sup>3</sup>

" !

�

,G= �E + bMc<sup>2</sup>

t zð Þ<sup>c</sup> <sup>=</sup><sup>τ</sup> �z<sup>8</sup> <sup>þ</sup> <sup>2</sup>z<sup>7</sup> � <sup>2</sup>z<sup>5</sup> <sup>þ</sup> <sup>2</sup>z<sup>4</sup> � <sup>2</sup>z<sup>3</sup> <sup>þ</sup> <sup>2</sup><sup>z</sup> � <sup>1</sup>

) 2

constant t(Zc) is the evaluation of (25) in the threshold value Zc = γc�(γ<sup>c</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>

� �

<sup>þ</sup> <sup>k</sup><sup>3</sup> tan �<sup>1</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>5</sup>

,F= �E�(b + j)Mc<sup>2</sup>

defined in the low energy range, the wide interval 1.0 ≤ γ ≤ 1.1 states a unique value of γ<sup>c</sup> for any acceleration parameter α when the values of n and r are fixed. In order to deduce the

obtaining in this way a rational function which integration by partial fractions gives the

<sup>c</sup>3=<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> �

> ð Þ �Δ<sup>3</sup> <sup>1</sup>=<sup>2</sup> !

Δ<sup>3</sup> and Δ<sup>4</sup> their discriminants, corresponding to two real and six complex roots of the nine roots of the rational function denominator, and C1, C2,. C<sup>9</sup> are the coefficients of the linear factors. For a given value of the acceleration efficiency α all the quantities involved in (25) become constants and take on different values according to the three energy intervals studied. The factor κ is give as κ = α + r (if E ≤ 110 MeV), κ = α�f�η (if 110 < E ≤ 290 MeV) and κ = α�f�η +r (if

<sup>1</sup> , K<sup>2</sup> ¼ ð Þ 2C<sup>4</sup> � R3C<sup>3</sup> =ð Þ �Δ<sup>2</sup>

� �

<sup>2</sup> in the range E ≤ 110 MeV. Therefore, since critical energy for acceleration is

�

<sup>c</sup>5=<sup>2</sup> <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>R</sup>7<sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>8</sup> �

<sup>þ</sup> <sup>k</sup><sup>4</sup> tan �<sup>1</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>7</sup>

1=2

; R1, R2, … R8 are the coefficients of the quadratic factors Δ1, Δ2,

<sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>R</sup>7<sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>8</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>

� �

� �θ<sup>4</sup>

ð Þ �Δ<sup>3</sup> <sup>1</sup>=<sup>2</sup>

� �

ð Þ �Δ<sup>4</sup> <sup>1</sup>=<sup>2</sup> !#

� c7=<sup>2</sup> � � � � �

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

� t zð Þ<sup>c</sup> )

, K<sup>3</sup> ¼ ð Þ 2C<sup>6</sup> � R5C<sup>5</sup> =ð Þ �Δ<sup>3</sup>

,G= �E�h + bMc<sup>2</sup>

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

,H=E�h + jMc<sup>2</sup>

�1) 1/2 , thus,

> � � � � �

k1

1 A 137

(25)

1=2 K1

) furnishes us with a

<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>

,H=E�h and I = <sup>r</sup>(Mc<sup>2</sup>

2 �1) 1/2

ð Þ �Δ<sup>4</sup> <sup>1</sup>=<sup>2</sup> !#)

(26)

) 2 . In

) 2 . The

. The integral

zc<sup>9</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>1</sup> � ð Þ <sup>Δ</sup><sup>1</sup> <sup>1</sup>=<sup>2</sup> <sup>2</sup><sup>z</sup> <sup>þ</sup> <sup>R</sup><sup>1</sup> <sup>þ</sup> ð Þ <sup>Δ</sup><sup>1</sup> <sup>1</sup>=<sup>2</sup>

$$N(\boldsymbol{\gamma}) = \frac{N\_0 Mc^2}{\pi} e^{i(\boldsymbol{\gamma}\_c)/\tau} \left| 2(\boldsymbol{\gamma}^2 - 1)^{1/2} + 2\boldsymbol{\gamma} \right|^{-\delta} \left| \frac{\boldsymbol{\gamma} - \boldsymbol{a}\_1}{2(\boldsymbol{a}\_1 \boldsymbol{\gamma} - \boldsymbol{1}) + 2\left(\boldsymbol{a}\_1^2 - 1\right)^{1/2} \left(\boldsymbol{\gamma}^2 - 1\right)^{1/2}} \right|^{-b\_1} \tag{22}$$
 
$$\exp\left[ -\delta\_2 \sin^{-1} \left( \frac{a\_2 \boldsymbol{\gamma} - 1}{|\boldsymbol{\gamma} - \boldsymbol{a}\_2|} \right) - \delta\_3 \sin^{-1} \left( \frac{a\_3 \boldsymbol{\gamma} - 1}{|\boldsymbol{\gamma} - \boldsymbol{a}\_3|} \right) \right] \frac{\left(\boldsymbol{\gamma} - 1\right) \left(\boldsymbol{\gamma}^2 - 1\right)^{1/2}}{A \boldsymbol{\gamma}^3 + B \boldsymbol{\gamma}^2 + C \boldsymbol{\gamma} + D}$$

where δ = (A1a<sup>1</sup> + A2a2+ A3a3)/λτ, δ<sup>1</sup> = A1(a<sup>1</sup> 2 �1) 1/2 λτ, δ<sup>2</sup> = A2(1 � a<sup>2</sup> 2 ) <sup>1</sup>/22/λτ and <sup>δ</sup> = A3(1�a<sup>3</sup> 2 ) 1/2 / λτ; therefore, the integral spectrum is given from (Eq. 11) and Eq. (21) as

$$\begin{split} f(\boldsymbol{>E}) &= N\_0 \{ \boldsymbol{M} \boldsymbol{\varepsilon}^2 \} ^2 \left\{ \boldsymbol{\varepsilon}^{\ell(\boldsymbol{\gamma}\_c)/\tau} \Big| \mathcal{Q} \{ \boldsymbol{\gamma}^2 - 1 \}^{\frac{1}{2}} + 2\boldsymbol{\gamma} \Big| \begin{matrix} \boldsymbol{\gamma} - \boldsymbol{a}\_1 \\ \frac{2(a\_1 \boldsymbol{\gamma} - 1) + 2\left(a\_1^2 - 1\right)^{\frac{1}{2}} (\boldsymbol{\gamma}^2 - 1)^{\frac{1}{2}}} \end{matrix} \Big| \begin{matrix} \boldsymbol{\gamma} \\ \boldsymbol{\gamma} \end{matrix} \\ &\times \exp\left[ -\delta\_2 \sin^{-1} \left( \frac{a\_2 \boldsymbol{\gamma} - 1}{|\boldsymbol{\gamma} - \boldsymbol{a}\_2|} \right) - \delta\_3 \sin^{-1} \left( \frac{a\_3 \boldsymbol{\gamma} - 1}{|\boldsymbol{\gamma} - \boldsymbol{a}\_3|} \right) \right] \exp\left[ -t \left( \boldsymbol{\gamma}\_m \right) / \tau \right] \end{matrix} \right\} \tag{23}$$

which in terms of kinetic energy becomes,

$$I(>E) = N\_0 \exp\left(\frac{t(E)}{\pi}\right) \left\{ \left[ \left| \frac{2}{M^2} \left( \left( E^2 + 2Mc^2 E \right)^{1/2} + E + Mc^2 \right) \right|^{-\delta\_3} \frac{\left| 2(\mathfrak{c}\_1^2 - 1) \left( E^2 + 2Mc^2 E \right)^{1/2} + 2\mathfrak{c}\_1 E + 2Mc^2 (\mathfrak{c}\_1 - 1) \right|^{\delta\_3}}{E + Mc^2 (1 - a\_1)} \right|^{\delta\_3} \right. \\ \left. + \left. \left( \exp\left[ A\_2 \left( 1 - a\_2^2 \right)^{1/2} \sin^{-1} \left( \frac{a\_2 E + (a\_2 - 1) Mc^2}{|E + (1 - a\_2) Mc^2|} \right) \right] + A\_3 \left( 1 - a\_3^2 \right)^{1/2} \sin^{-1} \left( \frac{a\_2 E + (a\_2 - 1) Mc^2}{|E + (1 - a\_2) Mc^2|} \right) \right] \right. \\ \left. + \left. \left( \left[ \left( \mathfrak{c}\_1 - \mathfrak{c}\_2 \right) \left( E + Mc^2 \right)^{1/2} \right] + A\_2 \left( 1 - a\_3^2 \right)^{1/2} \sin^{-1} \left( \frac{a\_2 E + (a\_2 - 1) Mc^2}{|E + (1 - a\_3) Mc^2|} \right) \right] \right] \right. \\ \left. + \left. \left( \left[ \mathfrak{c}\_2 - \mathfrak{c}\_1 \right] \left( E + Mc^2 \right)^{1/2} \right) \right] \right\} \left( 23.1 \right) \end{cases}$$

where

<sup>δ</sup><sup>1</sup> <sup>¼</sup> Mc<sup>2</sup> � �<sup>2</sup> =Qτ h ið Þ <sup>a</sup>1A<sup>1</sup> <sup>þ</sup> <sup>a</sup>2A<sup>2</sup> <sup>þ</sup> <sup>a</sup>3A3<sup>þ</sup> , <sup>δ</sup><sup>2</sup> <sup>¼</sup> Mc<sup>2</sup> � �<sup>2</sup> =Qτ h iA1ð Þ <sup>a</sup><sup>2</sup> � <sup>1</sup> <sup>1</sup>=<sup>2</sup> , <sup>δ</sup><sup>3</sup> <sup>¼</sup> Mc<sup>2</sup> � �<sup>2</sup> =Qτ and Q, A1, A2, A3, a1, a2, a3, are constants that depend on α, b, η, h, j and f which emerge from the integration by partial fractions and take different values throughout the three different range considered.

#### 4.4. Modulation by adiabatic processes

Under the consideration of adiabatic deceleration of protons while the acceleration mechanism is acting, the net energy change rate Eq. (20), is transformed by addition of Eq. (6) in

$$\begin{split} \frac{d\gamma}{dt} &= a\left(\boldsymbol{\gamma}^{2} - 1\right)^{1/2} - \left(M\boldsymbol{c}^{2}\right)\boldsymbol{\gamma}\left(\boldsymbol{\gamma}^{2} - 1\right)^{-1/2} - \left\{h\left[M\boldsymbol{c}^{2}\left(\boldsymbol{\gamma} - 1\right)\right]^{-2} + j\left[M\boldsymbol{c}^{2}\left(\boldsymbol{\gamma} - 1\right)^{-1} + f + \eta\right]\right\} \\ &\times \left(\boldsymbol{\gamma}^{2} - 1\right)^{1/2} - \rho\left(\boldsymbol{\gamma}^{2} - 1\right)\boldsymbol{\gamma}^{-1} \end{split} \tag{24}$$

The condition (dγ/dt) = 0 for determining γ<sup>c</sup> in this case, leads to a transcendental equation of the form <sup>E</sup>γ<sup>4</sup> + Fγ<sup>3</sup> + Gγ<sup>2</sup> + H<sup>γ</sup> + I(γ�1)(γ<sup>2</sup> �1) <sup>3</sup>/<sup>2</sup> = 0, whose solution depends only on α, n and very weakly on r, and where E = α(Mc<sup>2</sup> ) 2 ,F= �E�(b + j)Mc<sup>2</sup> ,G= �E�h + bMc<sup>2</sup> ,H=E�h + jMc<sup>2</sup> and I = � <sup>r</sup>(Mc<sup>2</sup> ) <sup>2</sup> in the range E ≤ 110 MeV. Therefore, since critical energy for acceleration is defined in the low energy range, the wide interval 1.0 ≤ γ ≤ 1.1 states a unique value of γ<sup>c</sup> for any acceleration parameter α when the values of n and r are fixed. In order to deduce the particle spectrum, we have simplified Eq. (24) by changing variable Z = <sup>γ</sup>�(γ<sup>2</sup> �1) 1/2 , thus, obtaining in this way a rational function which integration by partial fractions gives the following acceleration time

λ = α (if E ≤ 110 MeV), λ = α � f (if 110 < E ≤ 290 MeV) and λ = α – f � η (if E > 290 MeV). The

� � � � �

j j γ � a<sup>3</sup>

�<sup>δ</sup> γ � a<sup>1</sup> <sup>2</sup>ð Þþ <sup>a</sup>1<sup>γ</sup> � <sup>1</sup> <sup>2</sup> <sup>a</sup><sup>2</sup>

λτ, δ<sup>2</sup> = A2(1 � a<sup>2</sup>

� � � � �

� � � �δ

j j γ � a<sup>3</sup>

� � �

�δ<sup>1</sup> <sup>2</sup> <sup>a</sup><sup>2</sup> ð Þ <sup>1</sup>�<sup>1</sup> <sup>E</sup><sup>2</sup>

� � � �

sin �<sup>1</sup> <sup>a</sup>3Eþð Þ <sup>a</sup>3�<sup>1</sup> Mc<sup>2</sup> Eþð Þ 1�a<sup>3</sup> Mc<sup>2</sup> j j � �i�<sup>δ</sup><sup>3</sup>

=Qτ h i

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

<sup>2</sup> þ 2γ

� <sup>δ</sup><sup>3</sup> sin �<sup>1</sup> <sup>a</sup>3<sup>γ</sup> � <sup>1</sup>

<sup>þ</sup> <sup>E</sup> <sup>þ</sup> Mc<sup>2</sup>

and Q, A1, A2, A3, a1, a2, a3, are constants that depend on α, b, η, h, j and f which emerge from the integration by partial fractions and take different values throughout the three different

Under the consideration of adiabatic deceleration of protons while the acceleration mechanism

The condition (dγ/dt) = 0 for determining γ<sup>c</sup> in this case, leads to a transcendental equation of

�1)

is acting, the net energy change rate Eq. (20), is transformed by addition of Eq. (6) in

<sup>1</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup>

2 )

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

� �=<sup>τ</sup> � �)

EþMc2ð Þ 1�a<sup>1</sup>

exp �t γ<sup>m</sup>

<sup>þ</sup>2Mc<sup>2</sup> ð Þ<sup>E</sup> <sup>1</sup>=<sup>2</sup>

<sup>A</sup>1ð Þ <sup>a</sup><sup>2</sup> � <sup>1</sup> <sup>1</sup>=<sup>2</sup>

<sup>þ</sup> j Mc<sup>2</sup>

n o h i

<sup>3</sup>/<sup>2</sup> = 0, whose solution depends only on α, n and

Aγ<sup>3</sup> þ Bγ<sup>2</sup> þ Cγ þ D

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

<sup>1</sup>/22/λτ and <sup>δ</sup> = A3(1�a<sup>3</sup>

� � � � �

<sup>þ</sup>2a1Eþ2Mc2ð Þ <sup>a</sup>1�<sup>1</sup>

� exp �t Eð Þ <sup>m</sup> τ � �)

, <sup>δ</sup><sup>3</sup> <sup>¼</sup> Mc<sup>2</sup> � �<sup>2</sup>

ð Þ <sup>γ</sup> � <sup>1</sup> �<sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>þ</sup> <sup>η</sup>

� � � � δ2

–δ<sup>1</sup>

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

�δ<sup>1</sup>

(22)

2 ) 1/2 /

(23)

ð Þ 23:1

=Qτ

(24)

þ 2γ

� <sup>δ</sup><sup>3</sup> sin �<sup>1</sup> <sup>a</sup>3<sup>γ</sup> � <sup>1</sup>

λτ; therefore, the integral spectrum is given from (Eq. 11) and Eq. (21) as

e

j j γ � a<sup>2</sup> � �

Mc<sup>2</sup> <sup>E</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>Mc<sup>2</sup><sup>E</sup> � �<sup>1</sup>=<sup>2</sup>

� �

ð Þ <sup>a</sup>1A<sup>1</sup> <sup>þ</sup> <sup>a</sup>2A<sup>2</sup> <sup>þ</sup> <sup>a</sup>3A3<sup>þ</sup> , <sup>δ</sup><sup>2</sup> <sup>¼</sup> Mc<sup>2</sup> � �<sup>2</sup>

8 < : 2 �1) 1/2

<sup>t</sup> <sup>γ</sup>ð Þ<sup>c</sup> <sup>=</sup><sup>τ</sup> <sup>2</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �<sup>1</sup>

� � �

� � � �

� h i �

<sup>þ</sup>A<sup>3</sup> <sup>1</sup> � <sup>a</sup><sup>2</sup> 3 � �<sup>1</sup>=<sup>2</sup>

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

� � �

differential spectrum in this case follows from Eqs. (8.1) and (20) as

<sup>t</sup> <sup>γ</sup>ð Þ<sup>c</sup> <sup>=</sup><sup>τ</sup> <sup>2</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup>

� � �

j j γ � a<sup>2</sup> � �

where δ = (A1a<sup>1</sup> + A2a2+ A3a3)/λτ, δ<sup>1</sup> = A1(a<sup>1</sup>

<sup>J</sup>ð Þ¼ <sup>&</sup>gt; <sup>E</sup> <sup>N</sup><sup>0</sup> Mc<sup>2</sup> � �<sup>2</sup>

which in terms of kinetic energy becomes,

h � �

4.4. Modulation by adiabatic processes

� <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> � <sup>r</sup> <sup>γ</sup><sup>2</sup> � <sup>1</sup> � �γ�<sup>1</sup>

the form <sup>E</sup>γ<sup>4</sup> + Fγ<sup>3</sup> + Gγ<sup>2</sup> + H<sup>γ</sup> + I(γ�1)(γ<sup>2</sup>

t Eð Þ<sup>i</sup> τ � �

2 � �<sup>1</sup>=<sup>2</sup>

=Qτ h i

� exp �δ<sup>2</sup> sin �<sup>1</sup> <sup>a</sup>2<sup>γ</sup> � <sup>1</sup>

2

sin �<sup>1</sup> <sup>a</sup>2Eþð Þ <sup>a</sup>2�<sup>1</sup> Mc<sup>2</sup> Eþð Þ 1�a<sup>2</sup> Mc<sup>2</sup> j j

dt <sup>¼</sup> α γ<sup>2</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> � Mc<sup>2</sup> � �γ γ<sup>2</sup> � <sup>1</sup> � ��1=<sup>2</sup> � h Mc<sup>2</sup>

�

("

Nð Þ¼ γ

136 Cosmic Rays

Jð Þ¼ > E N<sup>0</sup> exp

<sup>∙</sup> exp <sup>A</sup><sup>2</sup> <sup>1</sup> � <sup>a</sup><sup>2</sup>

<sup>δ</sup><sup>1</sup> <sup>¼</sup> Mc<sup>2</sup> � �<sup>2</sup>

range considered.

where

dγ

N0Mc<sup>2</sup> τ e

exp �δ<sup>2</sup> sin �<sup>1</sup> <sup>a</sup>2<sup>γ</sup> � <sup>1</sup>

$$t = \frac{1}{k} \left\{ \left[ \ln \left( \left| z^2 + R\_1 z + R\_2 \right|^{5/2} \left| z^2 + R\_3 z + R\_4 \right|^{5/2} \left| z^2 + R\_5 z + R\_6 \right|^{5/2} \left| z^2 + R\_7 z + R\_8 \right|^{5/2} \frac{\left| 2z + R\_1 - (\Lambda\_1)^{1/2} \right|^{\Lambda\_1}}{\left| 2z + R\_1 + (\Lambda\_1)^{1/2} \right|} \right) \right] \right\} \tag{25}$$

$$+ k\_2 \tan^{-1} \left( \frac{2z + R\_3}{(-\Delta\_2)^{1/2}} \right) + k\_3 \tan^{-1} \left( \frac{2z + R\_3}{(-\Delta\_3)^{1/2}} \right) + k\_4 \tan^{-1} \left( \frac{2z + R\_7}{(-\Delta\_4)^{1/2}} \right) \right] - t(z\_c) \right\} \tag{25}$$

$$\text{where } \mathbf{K}\_1 = (2\mathbf{C}\_2 - \mathbf{R}\_1 \mathbf{C}\_1) / 2\boldsymbol{\Delta}\_1^{1/2}, \mathbf{K}\_2 = (2\mathbf{C}\_4 - \mathbf{R}\_3 \mathbf{C}\_3) / (-\boldsymbol{\Delta}\_2)^{1/2}, \mathbf{K}\_3 = (2\mathbf{C}\_6 - \mathbf{R}\_5 \mathbf{C}\_5) / (-\boldsymbol{\Delta}\_3)^{1/2} \mathbf{K}\_1$$

K<sup>4</sup> ¼ ð Þ 2C<sup>8</sup> � R7C<sup>7</sup> =ð Þ �Δ<sup>4</sup> 1=2 ; R1, R2, … R8 are the coefficients of the quadratic factors Δ1, Δ2, Δ<sup>3</sup> and Δ<sup>4</sup> their discriminants, corresponding to two real and six complex roots of the nine roots of the rational function denominator, and C1, C2,. C<sup>9</sup> are the coefficients of the linear factors. For a given value of the acceleration efficiency α all the quantities involved in (25) become constants and take on different values according to the three energy intervals studied. The factor κ is give as κ = α + r (if E ≤ 110 MeV), κ = α�f�η (if 110 < E ≤ 290 MeV) and κ = α�f�η +r (if E > 290 MeV). As in the preceding cases, the substitution of Eq. (25) in (8<sup>0</sup> ) furnishes us with a differential spectrum of the form

$$N(\mathbf{y}) = \frac{N\_0}{Mc^2 k \tau} e^{(z\_2)/\tau} \left( \frac{-z^8 + 2z^7 - z^3 + 2z^4 - 2z^3 + 2z - 1}{z^8 + \left| z^7 + Mz^6 + Nz^5 + Pz^4 + Qz^3 + Rz^2 + Sz + V} \right| \right)$$

$$\times \left\{ \left( \left| z^2 + R\_1 z + R\_2 \right|^{-\theta\_1} \left| z^2 + R\_3 z + R\_4 \right|^{-\theta\_2} \left| z^2 + R\_5 z + R\_6 \right|^{-\theta\_3} \left| z^2 + R\_7 z + R\_8 \right|^{-\theta\_4} \right. \tag{2.7}$$

$$\times \left| \frac{2z + R\_1 - (\Delta\_1)^{1/2}}{2z + R\_1 + (\Delta\_1)^{1/2}} \right|^{-\theta\_5} z^{-\theta\_6} \right) \exp\left[ \theta\_7 \tan^{-1} \left( \frac{2z + R\_3}{(-\Delta\_2)^{1/2}} \right) + \theta\_8 \tan^{-1} \left( \frac{2z + R\_5}{(-\Delta\_3)^{1/2}} \right) + \theta\_9 \tan^{-1} \left( \frac{2z + R\_7}{(-\Delta\_4)^{1/2}} \right) \right] \right\} \tag{2.8}$$

Θ<sup>1</sup> = c1/2κτ, Θ<sup>2</sup> = c3/2κτ, Θ<sup>3</sup> = c5/2κτ, Θ<sup>4</sup> = c7/2κτ, Θ<sup>5</sup> = K1/2κτ, Θ<sup>6</sup> = c9/2κτ, Θ<sup>7</sup> = (�K2)/κτ, Θ<sup>8</sup> = (�K3)/κτ and Θ<sup>9</sup> = (�K4)/κτ,J= 2(F + I)/V, M = (4E + 4G + 2I)/V, N = (6F + 8H�GI)/V, P = (GE + 8G)/V, Q = (GP + 8H + GI)/V, R = (4E + 4c�2I)/I, S = 2 (F�I)/V, V = (E + I)/V and V=E�I. The values of E, F, G, H, I in the range E < 110 MeV are the values given above; in the range <sup>110</sup> < E <sup>≤</sup> 290 MeV, E = (α�f)(Mc2 ) 2 ,F=E�bMc<sup>2</sup> ,G= �E + bMc<sup>2</sup> ,H=E�h and I = <sup>r</sup>(Mc<sup>2</sup> ) 2 . In the range E > 290 MeV the only difference with the precedent range is E=(α�f�η)(Mc<sup>2</sup> ) 2 . The constant t(Zc) is the evaluation of (25) in the threshold value Zc = γc�(γ<sup>c</sup> 2 �1) 1/2 . The integral spectrum according Eq. (11) is,

$$\begin{split} f(\boldsymbol{>E}) &= N\_{0}e^{\boldsymbol{\theta}(\boldsymbol{z}\_{i})/\tau} \Big{{}\Big{(}\left|\boldsymbol{z}^{2} + \boldsymbol{R}\_{1}\boldsymbol{z} + \boldsymbol{R}\_{2}\right|^{-\theta\_{1}} \left|\boldsymbol{z}^{2} + \boldsymbol{R}\_{3}\boldsymbol{z} + \boldsymbol{R}\_{4}\right|^{-\theta\_{2}} \left|\boldsymbol{z}^{2} + \boldsymbol{R}\_{5}\boldsymbol{z} + \boldsymbol{R}\_{6}\right|^{-\theta\_{3}} \left|\boldsymbol{z}^{2} + \boldsymbol{R}\_{7}\boldsymbol{z} \\ &+ \boldsymbol{R}\_{8} \Big{|}^{-\theta\_{4}} \left|\frac{2z + \boldsymbol{R}\_{1} - (\boldsymbol{\Delta}\_{1})^{1/2}}{2z + \boldsymbol{R}\_{1} + (\boldsymbol{\Delta}\_{1})^{1/2}}\right|^{-\theta\_{5}} \left|\boldsymbol{z}^{-\theta\_{6}}\right| \Bigg{)} \exp\left[\theta\_{7}\tan^{-1}\left(\frac{2z + \boldsymbol{R}\_{3}}{\left(-\boldsymbol{\Delta}\_{2}\right)^{1/2}}\right) + \theta\_{8}\tan^{-1}\left(\frac{2z + \boldsymbol{R}\_{5}}{\left(-\boldsymbol{\Delta}\_{3}\right)^{1/2}}\right)\right] \\ &+ \theta\_{9}\tan^{-1}\left(\frac{2z + \boldsymbol{R}\_{7}}{\left(-\boldsymbol{\Delta}\_{4}\right)^{1/2}}\right) \Bigg{]} - \exp\left(-t(z\_{m})/\tau\right)\end{split} \tag{27}$$

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

energies.

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

5. Procedure and results

� � � � � �

<sup>E</sup> <sup>þ</sup> Mc<sup>2</sup> � �=Mc<sup>2</sup> � ��ω=<sup>τ</sup> � <sup>e</sup>

B@

<sup>E</sup> <sup>þ</sup> Mc<sup>2</sup> <sup>α</sup> <sup>E</sup> <sup>þ</sup> Mc<sup>2</sup> ð Þþ <sup>r</sup> <sup>E</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>EMc<sup>2</sup> � �<sup>1</sup>=<sup>2</sup>

� � � � � �

�t EmþMc<sup>2</sup> ð Þ<sup>=</sup>Mc<sup>2</sup> ½ �<sup>=</sup><sup>τ</sup>

It is worth mentioning that although it is expected that the critical energy for acceleration Ec increases while adding energy loss process to the net energy charge rate, nevertheless, the value of Ec resulting from Eq. (24) is essentially the same as that obtained from Eq. (20). This can be understood from Figure 1, because adiabatic cooling is practically negligible at low

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

This assumption locates the acceleration region in chromospheric densities in agreement with

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 implica-

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 normal-

Enor <sup>¼</sup> q Jð Þ <sup>&</sup>gt; <sup>E</sup> earth � �

Enor (32)

the proton population as ne ≈ nH = 1013 cm�<sup>3</sup> (e.g. [19, 35, 56, 113, 114, 116, 118]).

some analysis of the charge spectrum of solar cosmic rays [64, 92].

<sup>J</sup>ð Þ <sup>&</sup>gt; <sup>E</sup> acc � �

tions of this assumption in the next section.

ization energy, Enor

�χ=τ

�ψ=<sup>τ</sup> 0

�

� � � � � �

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

<sup>E</sup> <sup>þ</sup> Mc<sup>2</sup> � � <sup>þ</sup> <sup>E</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>EMc<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> <sup>E</sup> <sup>þ</sup> Mc<sup>2</sup> ð Þ� <sup>E</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>EMc<sup>2</sup> � �<sup>1</sup>=<sup>2</sup>

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

� � � � � � 139

(31.1)

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> <sup>m</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> corresponding to the high energy cutoff value in the acceleration process.

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> <sup>m</sup><sup>þ</sup> � <sup>2</sup>EmMc<sup>2</sup><sup>Þ</sup> 1=2 .

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 acceleration and compression of the local material

$$
\sigma(d\gamma/dt) = \alpha(\gamma^2 - 1)^{-1/2} + \rho(\gamma^2 - 1)\gamma^{-1} \tag{28}
$$

As in the case of Eq. (12) the threshold for acceleration is meaningless, and thus the acceleration time up to the energy E is given as

$$t = \ln\left(\left|\frac{\gamma}{a\gamma + \rho(\gamma^2 - 1)^{\frac{1}{2}}}\right|^{\chi} \left|\frac{\gamma + (\gamma^2 - 1)^{\frac{1}{2}}}{\gamma - (\gamma^2 - 1)^{\frac{1}{2}}}\right|^{\psi} \gamma^{\mu} |a|^{\chi}\right) \tag{29}$$

where <sup>¼</sup> <sup>r</sup> <sup>α</sup>2�r<sup>2</sup> ð Þ , <sup>ψ</sup> <sup>¼</sup> <sup>α</sup> <sup>2</sup> <sup>α</sup>2�r<sup>2</sup> ð Þ and <sup>ω</sup> <sup>¼</sup> <sup>r</sup> <sup>r</sup>2�<sup>α</sup>2, consequently, the differential spectrum of particles is

$$N(\boldsymbol{\gamma})d\boldsymbol{\gamma} = \frac{N\_0}{mc^2 \pi |\boldsymbol{a}|^{d/2}} \left| \frac{\boldsymbol{\gamma}}{a\boldsymbol{\gamma} + \rho(\boldsymbol{\gamma}^2 - 1)^{\frac{1}{2}}} \right|^{-\chi/\pi} \left| \frac{\boldsymbol{\gamma} + (\boldsymbol{\gamma}^2 - 1)^{\frac{1}{2}}}{\boldsymbol{\gamma} - (\boldsymbol{\gamma}^2 - 1)^{\frac{1}{2}}} \right|^{-\psi/\pi} \frac{\boldsymbol{\gamma}^{(1-\boldsymbol{a}/\tau)} d\boldsymbol{\gamma}}{(\boldsymbol{\gamma}^2 - 1)^{\frac{1}{2}} [a\boldsymbol{\gamma} + \rho(\boldsymbol{\gamma}^2 - 1)^{\frac{1}{2}}]} \tag{30}$$

and then the integral spectrum is simply given as

$$f(>E) = \frac{N\_0}{|\alpha|^{\chi/2}} \left( \left| \frac{\gamma}{a\gamma + \rho(\gamma^2 - 1)^{\frac{1}{2}}} \right|^{-\chi/\tau} \left| \frac{\gamma + (\gamma^2 - 1)^{\frac{1}{2}}}{\gamma - (\gamma^2 - 1)^{\frac{1}{2}}} \right|^{-\psi/\tau} \gamma^{-\omega/\tau} - e^{-t\left(\gamma\_w\right)/\tau} \right) \tag{31}$$

which in terms of kinetic energy becomes,

$$f(>E) = \frac{N\_0}{|a|^{\chi/2}} \left( \left| \frac{E + Mc^2}{\left[\alpha(E + Mc^2) + \rho\left(E^2 + 2EMc^2\right)^{1/2}\right]} \right|^{-\chi/\tau} \left| \frac{\left(E + Mc^2\right) + \left(E^2 + 2EMc^2\right)^{1/2}}{\left(E + Mc^2\right) - \left(E^2 + 2EMc^2\right)^{1/2}} \right|^{-\psi/\tau} \right)$$
 
$$\left[ \left(E + Mc^2\right)/Mc^2 \right]^{-\omega/\tau} - e^{-l\left[\left(E\_n + Mc^2\right)/Mc^2\right]/\tau}$$

It is worth mentioning that although it is expected that the critical energy for acceleration Ec increases while adding energy loss process to the net energy charge rate, nevertheless, the value of Ec resulting from Eq. (24) is essentially the same as that obtained from Eq. (20). This can be understood from Figure 1, because adiabatic cooling is practically negligible at low energies.
