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Monte-Carlo Simulations of Shock Acceleration in Solar Turbulence

Explore shock acceleration of solar energetic particles using Monte Carlo simulations in self-generated turbulence. Study large solar particle events and wave growth phenomena for efficient particle acceleration.

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Monte-Carlo Simulations of Shock Acceleration in Solar Turbulence

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  1. Rami Vainio Dept of Physical Sciences, University of Helsinki, Finland Timo LaitinenDept of Physics, University of Turku, Finland Monte-Carlo simulations of shock acceleration of solar energetic particles in self-generated turbulence COST Action 724 is thanked for financial support

  2. Large Solar Particle Events Reames & Ng 1998

  3. GOES Proton flux 1986-1997 Hourly fluence (protons/cm2 sr) 104 105 106 107 108 104 105 106 107 104 105 106 10 1 N ~ F -0.41 0.1 Fraction of time (%) 0.01 0.001 Most of the IP proton fluence comes from large events Reames (2003)

  4. v = velocity in solar-wind frame v dv/dt < 0 → wave growth dv/dt > 0 → wave damping v' = const. v v|| VA Streaming instability and proton transport Outward propagating AWs amplified by outward streaming SEPs → stronger scattering

  5. W2 Vsh v = particle velocity in the ambient AW frame downstream →upstream Vsh v upstream → downstream W1 = u1+vA1 dv/dt > 0 → particle acceleration v' = const. v2 > v1 v1 v|| ΔW = W2 - W1 Particle acceleration at shocks Particles crossing the shockmany times (because of strongscattering) get accelerated

  6. Self-generated Alfvén waves • Alfvén-wave growth rateΓ= ½π ωcp · pr Sp(r,pr,t)/nvA pr= m ωcp/|k| Sp= 4π p2 ∫dμ vμ f(r,p, μ,t) = proton streaming per unit momentum • Efficient wave growth (at fixed r,k) during the SEP event requires1 << ∫dt Γ(t) = ½π (ωcp/nvA) pr∫dt Sp(r,pr,t) = ½π (ωcp/nAvA) pr dN/dpr→ p dN/dp >> (2/π) nAvA/ωcp = 1033 sr-1 (vA/vA) (n/2·108cm-3)½where A = cross-sectional area of the flux tube dN/dp = momentum distr. of protons injected to the flux tube sr Vainio (2003)

  7. p dN/dp [sr-1] Vainio (2003) solar-wind model with a maximum of vA in outer corona 1034 most efficient wave growth 1033 1 10 100 r [Rsun] Self-generated waves (cont'd) • Threshold spectrum for wave-growth p dN/dp|thr =1033 sr-1 (n/2·108cm-3)½ (vA/vA(r))lowestin corona • Apply a simple IP transport model: radial diffusion → @ 1 AU, dJ/dE|max =15·(MeV/E)½/cm2·sr·s·MeVfor p dN/dp =1033 sr-1. • Thus, wave-growth unimportant • for small SEP events • at relativistic energies • Only threshold spectrum released “impulsively”, waves trap the rest → streaming limited intensities

  8. Coupled evolution of particles and waves Protons Alfvén waves t = t1 p dNp/dr weak scattering (Λ > LB) weak scattering log P(r) Γ(r) p Sp(r) r r t = t2 > t1 p dNp/dr weak scattering weak scattering log P(r) impulsive release of escaping protons turbulent trapping with gradual leakage p Sp(r) Γ(r) r r

  9. Numerical modeling of coronal DSA • Large events exceeding the threshold for wave-growth require self-consistent modeling • particles affect their own scattering conditions • Monte Carlo simulations with wave growth • SW: radial field, W = u + vA = 400 km/s • parallel shock with constant speed Vs and sc-compression ratio rsc • WKB Alfvén waves modified by wave growth • Suprathermal (~ 10 keV) particles injected to the considered flux tube at the shock at a constant rate • waves P(r,f,t) and particles f(r,p,μ,t) traced simultaneouslyΓ= π2 fcp · pr Sp(r,pr,t)/nvA <(Δθ)2>/Δt = π2 fcp · fr P(r,fr,t)/B2pr = fcp mpV/f fr = fcp mpV/p Vs B u

  10. Examples of simulation results • Shock launched at R = 1.5 Rsun at speed Vs = 1500 km/s in all examples. • Varied parameters: • Ambient scattering mean free path @ r = 1.5 Rsun and E =100 keVΛ0 = 1, 5, 30 Rsun • Injection rate q = Ninj/tmax << qswwhere qsw = ∫ n(r)A(r) dr /tmax = 2.2·1037 s-1 • Scattering center compression ratio of the shock, rsc = 2, 4

  11. rsc = 2, q ~ 4.7·1032 s-1, Λ0 = 1 Rsun - Proton acceleration up to 1 MeV in 10 min - Hard escaping proton spectrum (~ p–1 ) - Very soft (~ p–4) spectrum at the shock - Wave power spectrum increased by 2 orders of magnitude at the shock at resonant frequencies

  12. rsc = 4, q ~ 4.7·1032 s-1, Λ0 = 1 Rsun - Proton acceleration up to ~20 MeV in 10 min - Hard escaping proton spectrum (~ p–1) - Softer (~ p–2) spectrum at the shock - Wave power spectrum increased by > 3 orders of magnitude at the shock at resonant frequencies

  13. rsc = 4, q ~ 1.9·1033 s-1, Λ0 = 5 Rsun - Proton acceleration up to ~20 MeV in < 3 min - Hard escaping proton spectrum (~ p–1) - Softer (~ p–2) spectrum at the shock - Wave power spectrum increased by ~ 4 orders of magnitude at the shock at resonant frequencies

  14. rsc = 4, q ~ 3.9·1032 s-1, Λ0 = 30 Rsun - Proton acceleration up to ~100 MeV - Hard escaping proton spectrum (~ p–1) - Softer (~ p–2) spectrum at the shock - Wave power spectrum increased by > 5 orders of magnitude at the shock at resonant frequencies

  15. Comparison with the theory of Bell (1978) Qualitative agreement at the shock below cut-off Good agreement upstream behind escaping particles

  16. Escaping particles (Λ0 = 1 Rsun) threshold for wave-growth NOTE: Observational streaming- limited spectrum somewhat softer than the simulated one (~ E-1/2).

  17. Cut-off energy log f @shock • Simulations consistent with analytical modeling: • proton spectrum at the shock a power law consistent with Bell (1978) • escaping particle spectrum a hard power law consistent with Vainio (2003):pdN/dp|esc ~ 4·1033 sr–1 • Power-laws cut off at an energy, which depends strongly on the injection rate q = Ninj/tmax Ec ~ qa with a ~ 0.5 – 2 • High injection rate leads to very turbulent environment → challenge for modeling ! Bell (1978) Bell/10 Ec log E 102 101 Ec [MeV] 100 10–1 1034 1035 1036 Ninj [sr–1] simulation time = 640 s

  18. Summary and outlook • Large SEP events excite large amounts of Alfvén waves • need for self-consistent transport and acceleration modeling • quantitatively correct results require numerical simulations • Monte Carlo simulation modeling of SEP events: • qualitative agreement with analytical models of particle acceleration (Bell 1978) and escape (Vainio 2003) • modest injection strength (q < 10-4 qsw) can result in > 100 MeV protons and non-linear Alfvén-wave amplitudes • streaming-limited intensities;spectrum of escaping protons still too hard in simulations • The present model needs improvements in near future: • more realistic model of the SW and shock evolution • implementation of the full wave-particle resonance condition

  19. Vs = 2200 km/s, rsc = 4, t = 640 s,q ~ 4.7·1032 s-1, Λ0 = 1 Rsun protons waves

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