33 RD I NTERNATIONAL C OSMIC R AY C ONFERENCE , R IO DE JANEIRO 2013 T HE A STROPARTICLE P HYSICS C ONFERENCE
Shock acceleration of solar energetic particles E.G. B EREZHKO , S.N. TANEEV, T.Y U . G RIGOR ’ EV Yu.G. Shafer Institute of Cosmophysical Research and Aeronomy, Lenin Ave. 31, Yakutsk, 677980 Russia.
[email protected] Abstract: The solar energetic particle (or solar cosmic ray (SCR)) acceleration by the shocks driven by coronal mass ejections is studied taking into account the generation of Alfv´en waves by accelerated particles. Detailed numerical calculations of the SCR spectra produced during the shock propagation through the solar corona have been performed within a quasilinear approach with a realistic set of coronal parameters. The resultant SCR energy spectrum is shown to include a power law part that ends with an exponential tail at maximal SCR energy which lies within the range εmax = 0.01 – 10 GeV, depending on the shock speed VS = 750 – 2500 km s−1 . The most efficient SCR acceleration takes place within the heliocentric distances r < 4R¯ . It is shown that calculated SCR energy spectra are consistent with observation. Keywords: plasma astrophysics, hydrodynamics and shock waves, solar corona, diffusive shock acceleration, solar cosmic rays, Alfv´en wave generation.
1 Introduction According to the existing views, the so-called ”gradual” solar energetic particle (below we will use the term ”solar cosmic ray” (SCR)) events are generated in the solar corona at the fronts of the shocks driven by coronal mass ejections (CMEs) (e.g. [1, 2]. The connection between gradual SCR events and CMEs directly follows from observations [3, 4, 5]. The process of diffusive shock acceleration originally established by Krymsky [6] is considered as the most appropriate SCR acceleration mechanism. The first attempts to apply the theory of diffusive shock acceleration to the solar coronal conditions [7, 8] showed the generation of SCR spectra with the required properties to be possible in principle. At the same time, these works, along with the succeeding ones [9, 10, 11]), contain a number of simplifications that limit their predictive capabilities. One of them, the plane-wave approximation, consists in neglecting the finite shock size and its time dependence. This approximation is applicable for the bulk of the accelerated particles that form a power law spectrum up to a maximal energy εmax . At the same time, it breaks down at high energies ε & εmax where the accelerated particle spectrum experiences an exponential cutoff. The finite shock size is the main physical factor determining the maximal SCR energy εmax and the spectrum shape at ε & εmax [12]. The increase of the CME-driven shock radius RS is followed by the decrease of the shock Alfv´enic Mach number MA and by an increase of the particle diffusion coefficient κ (ε , RS ) that leads to the decrease of particle acceleration efficiency [13, 14]. As a consequence the diffusive propagation of SCRs with highest energies begins to prevail over the shock expansion. Therefore, a remote observer records the SCR arrival considerably earlier than the shock arrival. This is consistent with the results of observations suggesting that SCRs with energies ε > 50 MeV are generated in the period of time when the shock radius is RS = (2 – 4)R¯ [15, 16, 17]. We studied the formation of the SCR spectrum due to diffusive shock acceleration in the solar corona in the linear approximation that disregarded the generation of Alfv´en waves by accelerated particles [13]. The Monte Carlo sim-
ulations of SCR acceleration [18, 19, 20] showed the importance of the selfconsistent generation of Alfv´en waves. However, since these calculations were performed for a very limited range of physical parameters, such as the range of heliocentric distances, it is premature to reach the final conclusion about the main peculiarities of the SCR spectrum formation by shocks in the solar corona. Ng and Reames [11] studied the selfconsistent SCR acceleration within the plane-wave approach for a set of physical parameters corresponding to the range of heliocentric distances RS = (3.5 – 5.8)R¯ . In contrast, according to our studies, the SCR acceleration is most efficient at r < 3.5R¯ [13]. In our previous paper [14] we studied the formation of SCR spectra within the quasilinear approach. It was shown that Alfv´en waves generated by accelerated particles influence the resultant SCR spectrum and that calculated selfconsistent SCR spectra are in a satisfactory way consistent with observation. Here we use the same quasilinear approach in order to explain the spectra of SCRs observed in large SCR event.
2 Model Quasilinear model which we use here was described in detail in our previous study [14]. It is based on the diffusive transport equation [21] for the particle (proton and α -particle) distribution function and the transport equation for Alfv´en turbulence in the upstream region (r > RS ) which are solved selfconsistently within the spherically symmetric approach. SCRs are initially originated at the shock front, where a fraction η ∼ 10−3 of gas particles are involved/injected into the acceleration. The background Alfv´en wave spectrum Ew0 (r0 , ν ) at the corona base r0 = 1.1R¯ can be determined from fact that according to the present view that the Alfv´en wave energy flux at the base of the corona Fw = W (3w R + 2cc ) is the main solar wind energy source. Here W = d ν Ew0 (ν ) is the magnetic energy density integrated over the wave spectrum, Ew0 (ν ) = ν −1 Ew0 (k)
Shock acceleration of SEPs 33 RD I NTERNATIONAL C OSMIC R AY C ONFERENCE , R IO DE JANEIRO 2013
is the spectral magnetic energy density of the Alfv´en waves, where the frequency ν and the wave number k are connected by the relation ν = k(w ± cA )/(2π ); the signs ± in this expression correspond to the waves Ew± (ν ) propagating away from the Sun (+) and toward the Sun (−); w is the medium (gas) speed; cc = cA (Ew+ − Ew− )/Ew is the speed √ of scattering centers relative to the gas; cA = B/ 4πρ is the Alfv´en velocity; B is magnetic field strength; ρ is the medium density; Ew (k) = d(δ B2 /8π )/d ln k is the differential magnetic energy density of Alfv´en waves. Energetic particles are scattered due to their interaction with the Alfv´en waves so that the increase of the wave intensity leads to the decrease of particle diffusion coefficient κ ∝ B2 /Ew (k = 1/ρB ) that in turn increase the particle acceleration rate. Here ρB is the particle giroradius, Ew = Ew+ + Ew− . Following [22], we assume that the wave spectrum at the base of the corona is Ew0 (r0 , ν ) ∝ ν −1 at 10−3 < ν < 5 × 10−2 Hz . At high frequencies ν > 5 × 10−2 Hz, the spectrum is expected to be steeper. We assume that in this inertial frequency range its form is the same as that in the solar wind [23]: Ew0 (r0 , ν ) ∝ ν −5/3 . Taking a typical values of wave energy flux Fw ≈ 106 erg cm−2 s−1 [22], plasma speed w = 0, and speed of scattering centers cc = 200 km s−1 at the base of the corona, we have W = 2.5 × 10−2 erg cm−3 and Ew0 (r = r0 , ν = 5 × 10−2 Hz) = 0.13 G2 /Hz . This energy is divided between the oppositely propagating − + waves according to the relations Ew0 = 0.7Ew0 and Ew0 = 0.3Ew0 . According to satellite measurements near the Earth’s orbit [23], Ew0 (r = 1 AU, ν = 5 × 10−2 Hz) = 10−12 G2 /Hz . Assuming a power law dependence of the wave energy density on heliocentric distance Ew0 (ν , r) ∝ r−δ , we have δ = 5. As a result, the spectral and spatial distribution of Alfv´en waves at frequencies ν > 5 × 10−2 Hz can be represented as Ew0 (r, k) = E0 (k/k0 )−β (r/R¯ )−δ , where β = 2/3, E0 = 6.5 × 10−3 erg cm−3 , and k0 = 2.4 × 105 cm−1 . Note that the role of scatterers for the shock accelerated ions is represented by the waves with frequencies ν > 5 × 10−2 Hz. We use the results of solar corona simulations [24] for a radial gas density profile ρ (r). The gas speed w is determined from the mass flux continuity condition w(r) = w0 [Ng (r)/Ng0 ](r/r0 )2 , where Ng = ρ /m is the proton number density, Ng0 = Ng (r0 ) = 108 cm−3 , w0 = w(r0 ) = 1 km s−1 . The magnetic field strength is taken in the form B = B0 (R¯ /r)2 , where B0 = 2.3 G. Besides the protons we also take into account the acceleration of α -particles by assuming the abundance of helium nuclei in the coronal plasma to be 10% of the hydrogen abundance. The above parameter values are slightly different compared with our previous consideration [14].
3 Acceleration efficiency The acceleration efficiency of high energy ions at the front of an evolving shock is determined by the values of three parameters [14]: power law index q = 3σef /(σef − 1), maximal momentum of particles accelerated at a given shock evolutionary epoch pmax (t) and parameter A = (VS − w)q−3 R3S Ng (RS ). Here σef = σ (1 − cc1 /u1 ) is the effective shock compression ratio, σ is the shock compression ration, u = VS − w is the speed of the medium relative to the shock front, quantity with subscript 1 corresponds to the point just ahead of the shock. Power law index q determines the shape of particle distribution function f ∝ p−q within the momentum range pinj ≤ p < pmax , where pinj = 4mu1 is the momentum of injected particles, m is their mass. The smaller the index q, the more high energy particles are produced by the shock at the current stage of its evolution, i.e., the higher the acceleration efficiency. Power law part of the spectrum ends with a quasi-exponential tail at p & pmax . The maximal momentum pmax changes during the shock evolution according to the relation [13]: ´ ³ δ −2β −3 1/(2−β ) pmax ∝ E0 u1 /RS . It decreases with the increase of the shock radius RS , that leads to the so-called escaping effect [25]. Its essence is that at each time of shock evolution t > 0, the particles with momenta p > pmax (t) that were produced at preceding stages when pmax was larger than the current value of pmax (t) are progressively accumulated in the upstream region. The propagation of these particles is weakly affected by the shock; the expansion rate of the volume occupied by them due to their diffusion exceeds the shock velocity. For this reason, they are called escaping particles. The third factor A determines the number of particles involved (injected) into acceleration at a given stage of shock evolution RS (t) and the contribution of stage RS (t) to the total accelerated particle spectrum N(ε ,t) =
4π p2 v
Z
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Here the integration is performing over the entire volume occupied by accelerated particles. The power law index q, the maximal accelerated particle energy εmax , and the parameter A, are plotted against the shock radius RS in Fig. 1 for a shock with the speed VS = 1500 km s−1 at RS < 10R¯ , while at RS > 10R¯ the speed VS decreases to 650 km s−1 at the Earth’s orbit. The power law index gradually increases from q = 4 at RS ≈ R¯ to q = 8 at RS ≈ 200R¯ due to increase of Alfv´en speed cA (RS ) at RS < 5R¯ and decrease of shock speed at RS > 10R¯ . The maximal accelerated particle energy εmax (RS ) gradually decreases from εmax ≈ 2 GeV at RS ≈ R¯ to εmax ≈ 1 MeV at RS ≈ 200R¯ . The parameter A (in arbitrary units) decreases from A = 10 at RS ≈ R¯ , A = 0.3 at RS = (5 − 80)R¯ to A ∼ 10−3 at RS ≈ 200R¯ .
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Fig. 1: Shock speed VS , power law index q, maximal energy εmax , and factor A versus shock size (radius) RS . Thus, it can be concluded that the behavior of the three important parameters q(RS ), pmax (RS ) and A(RS ) provide the most efficient acceleration of high energy particles at RS < 4R¯ . Therefore at late shock evolutionary phase RS À 4R¯ , the time profile of the SCR intensity is expected to contain two peaks. The first peak corresponds to the arrival of the bulk of the particles accelerated in the corona. The second peak coincides in time with the arrival of the shock front and contains the shock accelerated particles near the point of observation. The maximal energy of the particles in the second population is considerably lower than that in the first one. Therefore, our selfconsistent calculation is performed for the range RS < 4R¯ . It is assumed that at the shock evolutionary epoch RS = 4R¯ all the accelerated particles with energies εmax > 10 MeV escape from the acceleration region. We use the simple diffusion model to describe their further propagation in outer region r > 4R¯ (see [13] for details).
4 Results of calculations The results of our solution of the problem under consideration for an injection rate η = 10−3 are presented in Figs. 2,3. Figure 2 presents the total SCR energy spectra N(ε ) calculated for four different initial shock velocities, VS = 750, 1000, 1500 and 2500 km s−1 . These spectra were calculated at the corresponding shock radii RS /R¯ = 1.2, 2.2, 3.8 and 4, at which the efficient particle acceleration in the solar corona ends. It can be seen that the power law part of the accelerated proton spectrum N ∝ ε −γ with an index γ ≈ 1.8 at VS = 1500 km s−1 extends to εmax ≈ 2 GeV. At high energies ε > εmax , the power law part is followed by a quasi-exponential tail. Due to the decrease of selfconsistent particle diffusion coefficient, the maximum energy εmax ≈ 2 GeV is approximately an order of magnitude higher than that in the linear calculation. A qualitatively similar effect was established in the previous study [19]. As can be seen from Fig. 2, the power law index and the maximal energy depend significantly on the shock speed: the increase of the shock speed from VS = 750 km s−1 to VS = 2500 km s−1 leads to a decrease of power law index from γ = 3.5 to γ = 1.7 and to an increase of the SCR maximal energy from εmax ≈ 0.01 GeV to εmax ≈ 10 GeV. Note that calculations performed by Ng and Reames [11] at VS = 2500 km s−1 give much lower maximal energy of accelerated protons εmax ≈ 300 MeV compared with
Fig. 2: Total spectrum of the protons accelerated in the solar corona as a function of the kinetic energy for four shock velocities, VS = 750, 1000, 1500 and 2500 km s−1 . The spectra were calculated at the corresponding shock radii RS /R¯ = 1.2, 2.2, 3.8 and 4, at which the efficient particle acceleration in the solar corona ends. The thick and thin lines correspond to the linear and quasilinear approximations, respectively. our result. The reason of this discrepancy is that Ng and Reames considered SCR acceleration at the shock evolutionary stages RS = (3.5 − 5.8)R¯ whereas according to our consideration the highest energy particles are produced at the stage RS ≈ 2R¯ , which was not considered by Ng and Reames [11]. Alfv´en wave generation by accelerated particles provide the increase of maximal energy εmax compared with the linear case. This effect grows with the increase of shock speed (see Fig. 2) The second effect due to Alfv´en wave generation is the steepening of power law particle spectrum (see Fig. 2): since the accelerating particles generate outward propagating waves (Ew+ (k)) and absorb inward propagating waves (Ew− (k)) this leads to the increase of cc that in turn provides the increase of power law index q and makes the SCR spectrum steeper. We performed the calculation of SCR energy spectra expected at the Earth’s orbit in number of individual events. As an example we present in Fig. 3 the energy spectra of protons and α -particles observed at the Earth’s orbit by the IMP 8 CPME satellite [26] and the network of neutron monitors (NM) in the November 22, 1977 event [27] together with the results of our calculations performed for the injection rate η = 8 × 10−3 and the shock speed in the corona region VS = 560 km s−1 adopted from [17]. One can see that the calculated SCR spectra in satisfactory way agree with observations. To achieve agreement with the experiment the calculated SCR fluxes were reduced by a factor fre ≈ 10−3 . The necessity of such a renormalization may suggest that the actual injection rate is considerably lower than the adopted one. In this case, the generation of Alfv´en waves by accelerated particles is negligible (the linear case). The second possibility is that shock in the SCR generation region is predominantly quasiperpendicular where the injection of suprathermal particles is strongly suppressed. In this case, the renormalization factor fre is the fraction of the shock surface where it is quasiparallel and where relatively high injection rate η ∼ 10−3 takes place. The third important factor leading to a decrease of the SCR fluxes at large distance r À 10R¯ compared to our simplified calculation is the excitation of Alfv´en waves during their diffusive prop-
Shock acceleration of SEPs 33 RD I NTERNATIONAL C OSMIC R AY C ONFERENCE , R IO DE JANEIRO 2013
References
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( cm J , particles /
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10
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-particles November 22 , 1977
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Fig. 3: The intensity of protons (a) and α -particles (b) observed by the IMP 8 CPME satellite [26] and by network of neutron monitors (dotted region) in the November 22, 1977 event [27] are compared with the calculations.
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5 Conclusions Our analysis of the efficiency of charged particle acceleration by CME-driven shocks showed that the conditions for the most efficient acceleration of high energy particles (ε > 10 MeV) take place during the shock propagation through the solar corona at r < 4R¯ due to the highest level of Alfv´en turbulence and large Alfv´enic Mach number. The highest energy particles are produced at a stage of shock evolution RS ≈ 2R¯ . Our calculations performed for realistic parameters of the solar corona within quasilinear approach show that a power law spectrum of accelerated SCRs extends to a maximal energy εmax = 0.01 – 10 GeV at shock speeds VS = 750 – 2500 km s−1 respectively. Selfconsistent generation of Alfv´en waves provides the steepening of the accelerated particle spectrum and the increase of their maximal energy. The calculated SCR spectra for individual SCR events show them to be in satisfactory agreement with the experimental data. Thus, based on our calculations we can conclude that the main observed features of SCRs in the class of gradual events are consistent with their acceleration at CMEdriving shock in the corona region. Acknowledgment: This study was supported by the Ministry of Education and Science of the Russian Federation (contract no. 8404), the Russian Foundation for Basic Research (project no. 13-02-00941-a), Program no. 10 of the Presidium of the Russian Academy of Sciences, and a grant from the President of the Russian Federation (NSh-1741.2012.2).
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