- Email: [email protected]
Charge state behaviour of projectiles under an acceleration mechanism in a hot plasma
Astroparticle Physics 17 (2002) 415–420 www.elsevier.com/locate/astropart
Charge state behaviour of projectiles under an acceleration mechanism in a hot plasma L. del Peral *, M.D. Rodrıguez-Frıas, R. G omez-Herrero, J. Rodrıguez-Pacheco, J. Gutierrez Departamento de Fısica, Universidad de Alcal a, Apartado 20, E-28871 Alcal a de Henares (Madrid), Spain Received 28 January 2001
Abstract We have studied the ionization states of most ions in solar flares when a stochastic acceleration mechanism is present. Calculations using a computer code (ESCAPE) designed to find the charge states of heavy and light ions under stochastic acceleration have shown an energy dependence on the ionization states, stronger for heavy ions. Moreover the charge states depend on source parameters as density or temperature, but if an acceleration mechanism is taken into account in impulsive solar energetic particles events, the same source temperature may not be inferred for all the ions. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Charge state of solar energetic particles; Cosmic rays; Acceleration in solar flares
1. Introduction At laboratory plasma experiments, the projectile beam is injected into the plasma target, and the charge-exchange processes take account while projectiles lose energy [1]. In hot plasmas (T 107 K) found in astrophysical sources as for example solar flares, the charge state of ions evolves while they are accelerated at their sources. The problem is that particle acceleration cannot be seen as only the opposite process to power stopping, because power stopping depends on the medium parameters such as density and temperature, while acceleration depends on magnetohydrodynamic parameters, such
*
Corresponding author. E-mail address: [email protected] (L. del Peral).
as the turbulence spectrum and the electromagnetic fields involved. Therefore to really have information on the ionic charge states in impulsive solar energetic particles (ISEP) events, the acceleration mechanism involved has to be carefully taken into account. Evidence for particle acceleration in hot plasmas as solar flares is provided by direct solar energetic particle (SEP) measurements by detectors on board spacecrafts and by the detection of neutral radiation, produced by accelerated particles interacting with the solar atmosphere. To account for the charge states behaviour of SEP under an acceleration mechanism, a huge amount of acceleration scenarios may be postulated, either with continuous acceleration in 1-phase or episodical acceleration in two- and three-acceleration phases. One-phase acceleration is frequently
0927-6505/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 6 5 0 5 ( 0 1 ) 0 0 1 7 0 - 0
416
L. del Peral et al. / Astroparticle Physics 17 (2002) 415–420
associated to direct electric field acceleration, while two- and three-acceleration phases are rather associated with stochastic and shock wave acceleration. For an extensive review see Ref. [2]. On the previous basis, ISEP events are better described in terms of acceleration by stochastic turbulence low in the solar corona, whereas gradual solar energetic particles (GSEP) are usually explained by shock wave acceleration [3]. From the ionization states sampled at Earth, information about source parameters as the temperature or density and the conditions at the acceleration site may be inferred. Experimentally, it has been found that heavier ions at the same energy per nucleon penetrate to lower latitudes than lighter ones. From this situation, it can be deduced that heavier ions have larger mass-to-charge ratios that the lighter ones, i.e., they are not as full stripped of their electrons. Moreover new direct measurements of SEP ionic charge states, with high sensitivity of the new instrumentation have been obtained recently, in particular from ACE [4,5] and SAMPEX [6]. These new experiments have provided charge states information in a wider energy range, even up to 60 MeV/n, and for single SEPs, instead of the event averages provided by earlier measurements. Up to now they have mainly reported on ionic charge state distributions of GSEP, while those from ISEP have been scarce, mainly due to the low ion statistic in this kind of events. The first numerical work dealing with statistical acceleration in turbulent plasmas was that of Moussas et al. [7] and Newman et al. [8] deal with the numerical study of shock acceleration in turbulent plasmas. Concerning the computational approaches to the study of these topics, only a few models have been found in the literature. The model of Barghouty and Mewaldt [9] and that of Stovpyuk and Ostryakov [10] had an energy diffusion term in the Fokker–Planck equation. They include shock-induced acceleration and have been proposed for typical large solar events (GSEP) while the model developed by Kartavykh and Ostryakov [13] and that of Rodrıguez-Frıas et al. [11,12] include stochastic acceleration to account for ISEP events. These two last ISEP models differ in the turbulence chosen. As it is very well known
many types of turbulence are ineffective in accelerating particles because only energy diffusion takes place. A detailed description of the model developed by Rodrıguez-Frıas et al. is presented in Ref. [12] where they have chosen as turbulence a second-order Fermi type magnetohydrodynamic turbulence instead of an Alfven wave turbulence as Kartavykh and Ostryakov [13] have done.
2. ESCAPE code and the stochastic acceleration mechanism Energised ions travelling inside a plasma at velocity v may undergo two charge exchange processes. They can capture or lose electrons while they interact with the ambient plasma. Therefore the following processes have to be considered: electron ionization, autoionization after electron excitation, radiative recombination and dielectronic recombination. Moreover, these energised ions lose energy due to Coulomb collisions with the electrons of the medium, where the Bethe– Bloch equation gives the energy loss rate due to ionization. Therefore the charge state distribution of the projectiles have been obtained by the interaction of the ion projectile with the free plasma electrons, while ion–ion interactions have been neglected. For a detailed description of the ESCAPE code see Ref. [11]. Here, our analysis is focused on projectile ions accelerated from the background thermal plasma, where their initial velocities and charge states correspond to that of the thermal plasma. For the thermal charge states, qth , we merely rely in calculations based on astrophysical plasma ionization fractions given by Arnaud and Rothenflug [14] and updated for Fe ions by Arnaud and Raymond [15], as tables of equilibrium ionization of plasma ions for coronal conditions. As acceleration mechanism, the Fermi type acceleration has been implemented, that accounts mainly for ISEP events, better described in terms of acceleration by stochastic turbulence. Deceleration effects have been explicitly taken into account [11]. The stochastic acceleration may be described by either a diffusion equation in momentum space or a Fokker–Planck equation in energy space. The
L. del Peral et al. / Astroparticle Physics 17 (2002) 415–420
a parameter is obtained from the diffusion coefficient DðpÞ of the momentum diffusion equation, that for stochastic acceleration is [16] p2 ð1Þ 3b where p is the ion momentum, a is the efficiency of the acceleration mechanism involved and b is the ion velocity in terms of the light speed. The convection and diffusion coefficients A and D, respectively, of the Fokker–Planck equation are DðpÞ ¼ a
A¼
dE dt
ð2Þ
D¼
d2 E dt2
ð3Þ
where E is the projectile kinetic energy. Both coefficients may be related to DðpÞ in the momentum space, obtaining 1 oðvp2 DðpÞÞ A¼ 2 p op
ð4Þ
D ¼ 2v2 DðpÞ
ð5Þ
Therefore from Eqs. (1), (2) and (4) we can obtain the acceleration rate as
417
dE 1 oðvp2 DðpÞÞ 4 4 ¼ 2 ¼ apc ¼ aðE2 þ 2mc2 EÞ1=2 dt p op 3 3 ð6Þ As can be seen a has dimension T1 and is the acceleration efficiency. The a parameter depends on the specific MHD turbulence, the wave number, the total turbulent energy density and the magnetic energy density, and can roughly be taken as a timeindependent and energy-independent parameter. There are two relevant time scales at a given energy E: the time scale sE E=ðdE=dtÞ to account for acceleration to that energy E, that as can be seen is inversely proportional to a and the time scale sesc for escaping from the acceleration region. 3. Results and discussion We have focused our study on 12 astrophysically abundant elements (C, N, O, Ne, Na, Mg, Al, Si, S, Ca, Fe, Ni) and we have covered a very wide energy range, from the thermal equilibrium up to 1 GeV/n. As source plasma, we are interested in the solar flares, regions of low b (b 103 ), hot (T ¼ 107 K) plasma with high density (n ¼ 109 cm3 ), short magnetic confinement times (s ¼ 0:01–10 s),
Fig. 1. (a) Fractional mean charge state, q=Z, of 56 Fe ions versus kinetic energy E (eV/n) for T ¼ 8 106 K, n ¼ 5 108 cm3 and three different acceleration efficiencies: a ¼ 0:1, 0.05 and 0.01 s1 . (b) Temporal profile of the fractional charge state of 56 Fe ions for T ¼ 8 106 K, n ¼ 5 108 cm3 and three different acceleration efficiencies: a ¼ 0:1, 0.05 and 0.01 s1 .
418
L. del Peral et al. / Astroparticle Physics 17 (2002) 415–420
high magnetic field (B ¼ 100 G) and high Alfven speed (vA 104 km s1 ). To model the projectile behaviour under acceleration, the source plasma has been modelled as a plasma of protons and free electrons, where the heavy ions are just test particles. 3.1. Implications on charge dependence of the acceleration mechanism To show how the efficiency of the acceleration mechanism affects the charge state behaviour, we have plotted in Fig. 1 the evolution of the fractional charge states of Fe and Si ions while they are accelerated under an acceleration mechanism. Information on particle acceleration may be inferred from high energy charge states, since higher energies generally require longer trapped times in the acceleration site. Fig. 1(a) shows how for low acceleration efficiencies (i.e. a ¼ 0:01 s1 ) the acceleration takes place slowly and the projectile has time to become completely stripped. From Fig. 1(b) it can be seen how under such low efficiencies, a ¼ 0:01 s1 , the ion charge state remains invariant during the first second, due to the equilibrium between electron capture and ionization rate. Later, ionization dominates the electron capture and finally the ion becomes completely ionized. In Fig. 2 it can be seen how the obtained mean charge states may either be enhanced or depressed
depending on the acceleration efficiency of the acceleration mechanism involved. These acceleration efficiencies, lower than 0.1 s1 , have no physical meaning and have been plotted only to show that this stochastic acceleration model allows the projectiles to pick-up electrons, instead of lose them, therefore depending on the parameters one can reproduce with the model, situations where the charge state diminishes, and therefore electron capture instead of ionization is dominant. Of course we have to move in the range of a parameters consistent with solar source conditions, to reproduce the particle energy spectrum experimentally observed.
Fig. 2. Evolution of the energy dependence of 28 Si charge states under different acceleration efficiencies, a.
Fig. 3. Temporal evolution of the charge state of Fe and Si ions and mean charge states versus kinetic energy for T ¼ 8 106 K and two density numbers, n.
L. del Peral et al. / Astroparticle Physics 17 (2002) 415–420
419
Experimentally, it has been found that the acceleration mechanism is selective with the charge states. The abundances vary from one solar flare to another one, therefore it has been pointed out that the acceleration efficiency depends on the ionic charge states. Our concluding remark is that following our results, the sources of ISEP events seem to accelerate almost fully stripped ions up to Si. We suggest that the Fermi type acceleration mechanism has enough acceleration efficiency to strip ions during their acceleration in a high temperature (T > 5 106 K) environment. 3.2. Implications on source parameters To analyse the dependence of the ionization states on source parameters as the density or the temperature, we have plotted in Fig. 3 the temporal evolution of the charge states of Fe and Si ions for two different source densities. Also the energy dependence of the charge states under different densities are shown. As can be seen higher densities mean higher ionization states, and for Si ions they become fully stripped at lower energies and earlier in time than when they are accelerated in a lower density source. It is usual to assume a single equilibrium temperature for the source of the ISEP events, from the experimental charge states values measured, following the calculations of Arnaud and Rothenflug [14] updated for Fe ions for Arnaud and Raymond [15]. Fig. 4(a) shows the equilibrium temperature consistent with the mean charge state at 1 MeV/n obtained for each ion. We have found that when an acceleration mechanism is taken into account in ISEP events, the same source temperature may not be inferred for all the ions of the ISEP event. Therefore the ionization state values obtained here, under an acceleration mechanism, are not consistent with any single equilibrium temperature in the solar corona. 3.3. Implications on energy dependence of the charge states We have found energy variations in the ionization states as can be seen in Table 1 for E ¼ 1, 10 and 50 MeV/n. This energy dependence is more
Fig. 4. (a) Equilibrium temperature inferred from Ref. [14] and for Fe ions from Ref. [15], for the ionization states obtained. Arrows mean that the equilibrium temperatures obtained are only lower levels due to the fact that these ions are full stripped. (b) Open circles are the charge states obtained from Ref. [14] and for Fe ions [15] consistent with a source temperature of T ¼ 8 106 K. Black squares are the ionization states obtained for E ¼ 1 MeV/n and black triangles for E ¼ 50 MeV/n. A light shift in the Z parameter has been applied to avoid the overlap that appears mainly for light ions.
pronounced for heavier ions than for lighter ones as can be seen also in Fig. 4(b). 3.4. Comparison with previous works Meanwhile for GSEP events many papers have reported on ionization states for major elements, for ISEP events those are scarce and only for Si and Fe ions. The ionic charge states for 3 He–Ferich ISEP events are significantly higher than those observed at 1 MeV/n in typical GSEP events [17]. As can be seen in Fig. 4(b) the mean ionic charge states obtained for 1 and 50 MeV/n are higher than
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L. del Peral et al. / Astroparticle Physics 17 (2002) 415–420
Table 1 Ionic mean charge states for different kinetic energies and acceleration times, in a source of T ¼ 8 106 K and n ¼ 5 108 cm3 , under an acceleration efficiency a ¼ 0:1 s1 Ion
E (Mev/n) 1
Ni Fe Ca Ar S Si Al Mg Na Ne O N C
10
50
q
ta (s)
q
ta (s)
q
ta (s)
21.4 19.4 18.7 17.1 15.1 13.5 12.8 12.0 11.0 10.0 8.0 7.0 6.0
0.6 0.6 0.5 0.6 0.4 0.5 0.5 0.4 0.3 0.2 0.0 0.0 0.0
22.0 20.0 19.4 17.7 15.8 14.0 13.0 12.0 11.0 10.0 8.0 7.0 6.0
2.0 1.9 1.6 1.9 1.6 1.1 0.6 0.4 0.3 0.2 0.0 0.0 0.0
22.1 20.1 19.6 17.8 16.0 14.0 13.0 12.0 11.0 10.0 8.0 7.0 6.0
2.4 2.0 2.6 2.4 3.0 2.1 0.6 0.4 0.3 0.2 0.0 0.0 0.0
ta is the acceleration time that spends the ion to reach a charge state q.
those consistent with an equilibrium temperature of T ¼ 6 108 K, confirming that the ionization states obtained for 3 He–Fe-rich events point to a higher source region temperature for these ISEP events than for large GSEP events. Moreover, ionization states ranging between 19.4 and 20.0 for Fe ions and between 13.5 and 14 for Si ions, in the (1–50) MeV/n energy range, are consistent with a qSi ’ 14 and qFe ’ 20 reported for impulsive events [18,19].
[3]
[4] [5] [6] [7] [8] [9]
Acknowledgements
[10]
This work has been supported by the Spanish Comisi on Interministerial de Ciencia y Tecnologıa (CICYT), project Ref. BXX2000-0784 and by Universidad de Alcal a under project Ref. E021/ 2001.
[11]
References [1] R.C. Isler, Plasma Phys. Control. Fusion 36 (1994) 171– 208. [2] J. Perez-Peraza, Rayos C osmicos 98, in: J. Medina, L. del Peral, M.D. Rodrıguez-Frıas, J. Rodrıguez-Pacheco, A. de Henares (Eds.), Proceedings of the 16th ECRS, Servicio de
[12] [13]
[14] [15] [16] [17] [18] [19]
Publicaciones de la Universidad de Alcala, 1998, pp. 97– 112. B. Klecker, in: Dingus et al. (Eds.), 26th International Cosmic Ray Conference, Salt Lake City, AIP Conf. Proc. 516, 1999. E.C. Stone et al., Space. Sci. Rev. 86 (1998) 1. E. M€ obius et al., in: Proceedings of the 26th International Cosmic Ray Conference, vol. 6, 1999, pp. 87–90. J.E. Mazur et al., Geophys. Res. Lett. 26 (1999) 173–176. X. Moussas et al., Solar Phys. 112 (1987) 365–382. P.L. Newman et al., Astron. Astrophys. 255 (1992) 443. A.F. Barghouty, R.A. Mewaldt, in: Proceedings of the 26th International Cosmic Ray Conference, vol. 6, 1999, pp. 183–186. M.F. Stovpyuk, V.M. Ostryakov, in: Proceedings of the 26th International Cosmic Ray Conference, vol. 6, 1999, pp. 66–69. M.D. Rodrıguez-Frıas, L. del Peral, J. Perez-Peraza, J. Phys. G: Nucl. Part. Phys. 26 (2000) 259–264. M.D. Rodrıguez-Frıas, L. del Peral, J. Perez-Peraza, J. Geophys. Res. 108 (2001) 15657–15664 (No. A8). Y.Y. Kartavykh, V.M. Ostryakov, in: Proceedings of the 26th International Cosmic Ray Conference, vol. 6, 1999, pp. 272–275. M. Arnaud, R. Rothenflug, Astron. Astrophys. Suppl. Ser. 60 (1985) 425–457. M. Arnaud, J. Raymond, Ap. J. 398 (1992) 394–406. R. Ramaty, in: Particle Acceleration Mechanisms in Astrophysics, AIP, New York, 1979, p. 135. G.M. Mason et al., Ap. J. 452 (1995) 901–911. B. Klecker et al., Ap. J. 281 (1984) 458–462. A. Luhn et al., Ap. J. 317 (1987) 951–955.
Charge state behaviour of projectiles under an acceleration mechanism in a hot plasma L. del Peral *, M.D. Rodrıguez-Frıas, R. G omez-Herrero, J. Rodrıguez-Pacheco, J. Gutierrez Departamento de Fısica, Universidad de Alcal a, Apartado 20, E-28871 Alcal a de Henares (Madrid), Spain Received 28 January 2001
Abstract We have studied the ionization states of most ions in solar flares when a stochastic acceleration mechanism is present. Calculations using a computer code (ESCAPE) designed to find the charge states of heavy and light ions under stochastic acceleration have shown an energy dependence on the ionization states, stronger for heavy ions. Moreover the charge states depend on source parameters as density or temperature, but if an acceleration mechanism is taken into account in impulsive solar energetic particles events, the same source temperature may not be inferred for all the ions. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Charge state of solar energetic particles; Cosmic rays; Acceleration in solar flares
1. Introduction At laboratory plasma experiments, the projectile beam is injected into the plasma target, and the charge-exchange processes take account while projectiles lose energy [1]. In hot plasmas (T 107 K) found in astrophysical sources as for example solar flares, the charge state of ions evolves while they are accelerated at their sources. The problem is that particle acceleration cannot be seen as only the opposite process to power stopping, because power stopping depends on the medium parameters such as density and temperature, while acceleration depends on magnetohydrodynamic parameters, such
*
Corresponding author. E-mail address: [email protected] (L. del Peral).
as the turbulence spectrum and the electromagnetic fields involved. Therefore to really have information on the ionic charge states in impulsive solar energetic particles (ISEP) events, the acceleration mechanism involved has to be carefully taken into account. Evidence for particle acceleration in hot plasmas as solar flares is provided by direct solar energetic particle (SEP) measurements by detectors on board spacecrafts and by the detection of neutral radiation, produced by accelerated particles interacting with the solar atmosphere. To account for the charge states behaviour of SEP under an acceleration mechanism, a huge amount of acceleration scenarios may be postulated, either with continuous acceleration in 1-phase or episodical acceleration in two- and three-acceleration phases. One-phase acceleration is frequently
0927-6505/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 6 5 0 5 ( 0 1 ) 0 0 1 7 0 - 0
416
L. del Peral et al. / Astroparticle Physics 17 (2002) 415–420
associated to direct electric field acceleration, while two- and three-acceleration phases are rather associated with stochastic and shock wave acceleration. For an extensive review see Ref. [2]. On the previous basis, ISEP events are better described in terms of acceleration by stochastic turbulence low in the solar corona, whereas gradual solar energetic particles (GSEP) are usually explained by shock wave acceleration [3]. From the ionization states sampled at Earth, information about source parameters as the temperature or density and the conditions at the acceleration site may be inferred. Experimentally, it has been found that heavier ions at the same energy per nucleon penetrate to lower latitudes than lighter ones. From this situation, it can be deduced that heavier ions have larger mass-to-charge ratios that the lighter ones, i.e., they are not as full stripped of their electrons. Moreover new direct measurements of SEP ionic charge states, with high sensitivity of the new instrumentation have been obtained recently, in particular from ACE [4,5] and SAMPEX [6]. These new experiments have provided charge states information in a wider energy range, even up to 60 MeV/n, and for single SEPs, instead of the event averages provided by earlier measurements. Up to now they have mainly reported on ionic charge state distributions of GSEP, while those from ISEP have been scarce, mainly due to the low ion statistic in this kind of events. The first numerical work dealing with statistical acceleration in turbulent plasmas was that of Moussas et al. [7] and Newman et al. [8] deal with the numerical study of shock acceleration in turbulent plasmas. Concerning the computational approaches to the study of these topics, only a few models have been found in the literature. The model of Barghouty and Mewaldt [9] and that of Stovpyuk and Ostryakov [10] had an energy diffusion term in the Fokker–Planck equation. They include shock-induced acceleration and have been proposed for typical large solar events (GSEP) while the model developed by Kartavykh and Ostryakov [13] and that of Rodrıguez-Frıas et al. [11,12] include stochastic acceleration to account for ISEP events. These two last ISEP models differ in the turbulence chosen. As it is very well known
many types of turbulence are ineffective in accelerating particles because only energy diffusion takes place. A detailed description of the model developed by Rodrıguez-Frıas et al. is presented in Ref. [12] where they have chosen as turbulence a second-order Fermi type magnetohydrodynamic turbulence instead of an Alfven wave turbulence as Kartavykh and Ostryakov [13] have done.
2. ESCAPE code and the stochastic acceleration mechanism Energised ions travelling inside a plasma at velocity v may undergo two charge exchange processes. They can capture or lose electrons while they interact with the ambient plasma. Therefore the following processes have to be considered: electron ionization, autoionization after electron excitation, radiative recombination and dielectronic recombination. Moreover, these energised ions lose energy due to Coulomb collisions with the electrons of the medium, where the Bethe– Bloch equation gives the energy loss rate due to ionization. Therefore the charge state distribution of the projectiles have been obtained by the interaction of the ion projectile with the free plasma electrons, while ion–ion interactions have been neglected. For a detailed description of the ESCAPE code see Ref. [11]. Here, our analysis is focused on projectile ions accelerated from the background thermal plasma, where their initial velocities and charge states correspond to that of the thermal plasma. For the thermal charge states, qth , we merely rely in calculations based on astrophysical plasma ionization fractions given by Arnaud and Rothenflug [14] and updated for Fe ions by Arnaud and Raymond [15], as tables of equilibrium ionization of plasma ions for coronal conditions. As acceleration mechanism, the Fermi type acceleration has been implemented, that accounts mainly for ISEP events, better described in terms of acceleration by stochastic turbulence. Deceleration effects have been explicitly taken into account [11]. The stochastic acceleration may be described by either a diffusion equation in momentum space or a Fokker–Planck equation in energy space. The
L. del Peral et al. / Astroparticle Physics 17 (2002) 415–420
a parameter is obtained from the diffusion coefficient DðpÞ of the momentum diffusion equation, that for stochastic acceleration is [16] p2 ð1Þ 3b where p is the ion momentum, a is the efficiency of the acceleration mechanism involved and b is the ion velocity in terms of the light speed. The convection and diffusion coefficients A and D, respectively, of the Fokker–Planck equation are DðpÞ ¼ a
A¼
dE dt
ð2Þ
D¼
d2 E dt2
ð3Þ
where E is the projectile kinetic energy. Both coefficients may be related to DðpÞ in the momentum space, obtaining 1 oðvp2 DðpÞÞ A¼ 2 p op
ð4Þ
D ¼ 2v2 DðpÞ
ð5Þ
Therefore from Eqs. (1), (2) and (4) we can obtain the acceleration rate as
417
dE 1 oðvp2 DðpÞÞ 4 4 ¼ 2 ¼ apc ¼ aðE2 þ 2mc2 EÞ1=2 dt p op 3 3 ð6Þ As can be seen a has dimension T1 and is the acceleration efficiency. The a parameter depends on the specific MHD turbulence, the wave number, the total turbulent energy density and the magnetic energy density, and can roughly be taken as a timeindependent and energy-independent parameter. There are two relevant time scales at a given energy E: the time scale sE E=ðdE=dtÞ to account for acceleration to that energy E, that as can be seen is inversely proportional to a and the time scale sesc for escaping from the acceleration region. 3. Results and discussion We have focused our study on 12 astrophysically abundant elements (C, N, O, Ne, Na, Mg, Al, Si, S, Ca, Fe, Ni) and we have covered a very wide energy range, from the thermal equilibrium up to 1 GeV/n. As source plasma, we are interested in the solar flares, regions of low b (b 103 ), hot (T ¼ 107 K) plasma with high density (n ¼ 109 cm3 ), short magnetic confinement times (s ¼ 0:01–10 s),
Fig. 1. (a) Fractional mean charge state, q=Z, of 56 Fe ions versus kinetic energy E (eV/n) for T ¼ 8 106 K, n ¼ 5 108 cm3 and three different acceleration efficiencies: a ¼ 0:1, 0.05 and 0.01 s1 . (b) Temporal profile of the fractional charge state of 56 Fe ions for T ¼ 8 106 K, n ¼ 5 108 cm3 and three different acceleration efficiencies: a ¼ 0:1, 0.05 and 0.01 s1 .
418
L. del Peral et al. / Astroparticle Physics 17 (2002) 415–420
high magnetic field (B ¼ 100 G) and high Alfven speed (vA 104 km s1 ). To model the projectile behaviour under acceleration, the source plasma has been modelled as a plasma of protons and free electrons, where the heavy ions are just test particles. 3.1. Implications on charge dependence of the acceleration mechanism To show how the efficiency of the acceleration mechanism affects the charge state behaviour, we have plotted in Fig. 1 the evolution of the fractional charge states of Fe and Si ions while they are accelerated under an acceleration mechanism. Information on particle acceleration may be inferred from high energy charge states, since higher energies generally require longer trapped times in the acceleration site. Fig. 1(a) shows how for low acceleration efficiencies (i.e. a ¼ 0:01 s1 ) the acceleration takes place slowly and the projectile has time to become completely stripped. From Fig. 1(b) it can be seen how under such low efficiencies, a ¼ 0:01 s1 , the ion charge state remains invariant during the first second, due to the equilibrium between electron capture and ionization rate. Later, ionization dominates the electron capture and finally the ion becomes completely ionized. In Fig. 2 it can be seen how the obtained mean charge states may either be enhanced or depressed
depending on the acceleration efficiency of the acceleration mechanism involved. These acceleration efficiencies, lower than 0.1 s1 , have no physical meaning and have been plotted only to show that this stochastic acceleration model allows the projectiles to pick-up electrons, instead of lose them, therefore depending on the parameters one can reproduce with the model, situations where the charge state diminishes, and therefore electron capture instead of ionization is dominant. Of course we have to move in the range of a parameters consistent with solar source conditions, to reproduce the particle energy spectrum experimentally observed.
Fig. 2. Evolution of the energy dependence of 28 Si charge states under different acceleration efficiencies, a.
Fig. 3. Temporal evolution of the charge state of Fe and Si ions and mean charge states versus kinetic energy for T ¼ 8 106 K and two density numbers, n.
L. del Peral et al. / Astroparticle Physics 17 (2002) 415–420
419
Experimentally, it has been found that the acceleration mechanism is selective with the charge states. The abundances vary from one solar flare to another one, therefore it has been pointed out that the acceleration efficiency depends on the ionic charge states. Our concluding remark is that following our results, the sources of ISEP events seem to accelerate almost fully stripped ions up to Si. We suggest that the Fermi type acceleration mechanism has enough acceleration efficiency to strip ions during their acceleration in a high temperature (T > 5 106 K) environment. 3.2. Implications on source parameters To analyse the dependence of the ionization states on source parameters as the density or the temperature, we have plotted in Fig. 3 the temporal evolution of the charge states of Fe and Si ions for two different source densities. Also the energy dependence of the charge states under different densities are shown. As can be seen higher densities mean higher ionization states, and for Si ions they become fully stripped at lower energies and earlier in time than when they are accelerated in a lower density source. It is usual to assume a single equilibrium temperature for the source of the ISEP events, from the experimental charge states values measured, following the calculations of Arnaud and Rothenflug [14] updated for Fe ions for Arnaud and Raymond [15]. Fig. 4(a) shows the equilibrium temperature consistent with the mean charge state at 1 MeV/n obtained for each ion. We have found that when an acceleration mechanism is taken into account in ISEP events, the same source temperature may not be inferred for all the ions of the ISEP event. Therefore the ionization state values obtained here, under an acceleration mechanism, are not consistent with any single equilibrium temperature in the solar corona. 3.3. Implications on energy dependence of the charge states We have found energy variations in the ionization states as can be seen in Table 1 for E ¼ 1, 10 and 50 MeV/n. This energy dependence is more
Fig. 4. (a) Equilibrium temperature inferred from Ref. [14] and for Fe ions from Ref. [15], for the ionization states obtained. Arrows mean that the equilibrium temperatures obtained are only lower levels due to the fact that these ions are full stripped. (b) Open circles are the charge states obtained from Ref. [14] and for Fe ions [15] consistent with a source temperature of T ¼ 8 106 K. Black squares are the ionization states obtained for E ¼ 1 MeV/n and black triangles for E ¼ 50 MeV/n. A light shift in the Z parameter has been applied to avoid the overlap that appears mainly for light ions.
pronounced for heavier ions than for lighter ones as can be seen also in Fig. 4(b). 3.4. Comparison with previous works Meanwhile for GSEP events many papers have reported on ionization states for major elements, for ISEP events those are scarce and only for Si and Fe ions. The ionic charge states for 3 He–Ferich ISEP events are significantly higher than those observed at 1 MeV/n in typical GSEP events [17]. As can be seen in Fig. 4(b) the mean ionic charge states obtained for 1 and 50 MeV/n are higher than
420
L. del Peral et al. / Astroparticle Physics 17 (2002) 415–420
Table 1 Ionic mean charge states for different kinetic energies and acceleration times, in a source of T ¼ 8 106 K and n ¼ 5 108 cm3 , under an acceleration efficiency a ¼ 0:1 s1 Ion
E (Mev/n) 1
Ni Fe Ca Ar S Si Al Mg Na Ne O N C
10
50
q
ta (s)
q
ta (s)
q
ta (s)
21.4 19.4 18.7 17.1 15.1 13.5 12.8 12.0 11.0 10.0 8.0 7.0 6.0
0.6 0.6 0.5 0.6 0.4 0.5 0.5 0.4 0.3 0.2 0.0 0.0 0.0
22.0 20.0 19.4 17.7 15.8 14.0 13.0 12.0 11.0 10.0 8.0 7.0 6.0
2.0 1.9 1.6 1.9 1.6 1.1 0.6 0.4 0.3 0.2 0.0 0.0 0.0
22.1 20.1 19.6 17.8 16.0 14.0 13.0 12.0 11.0 10.0 8.0 7.0 6.0
2.4 2.0 2.6 2.4 3.0 2.1 0.6 0.4 0.3 0.2 0.0 0.0 0.0
ta is the acceleration time that spends the ion to reach a charge state q.
those consistent with an equilibrium temperature of T ¼ 6 108 K, confirming that the ionization states obtained for 3 He–Fe-rich events point to a higher source region temperature for these ISEP events than for large GSEP events. Moreover, ionization states ranging between 19.4 and 20.0 for Fe ions and between 13.5 and 14 for Si ions, in the (1–50) MeV/n energy range, are consistent with a qSi ’ 14 and qFe ’ 20 reported for impulsive events [18,19].
[3]
[4] [5] [6] [7] [8] [9]
Acknowledgements
[10]
This work has been supported by the Spanish Comisi on Interministerial de Ciencia y Tecnologıa (CICYT), project Ref. BXX2000-0784 and by Universidad de Alcal a under project Ref. E021/ 2001.
[11]
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