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

Large Solar Particle Events Reames & Ng 1998

Reames (2003) Fraction of time (%) GOES Proton flux Hourly fluence (protons/cm 2 sr) Most of the IP proton fluence comes from large events N ~ F -0.41

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

Particle acceleration at shocks Particles crossing the shock many times (because of strong scattering) get accelerated V sh W 1 = u 1 +v A1 W2W2 v || ΔW = W 2 - W 1 v' = const. v 2 > v 1 dv/dt > 0 → particle acceleration v = particle velocity in the ambient AW frame v1v1 upstream → downstream downstream → upstream V sh vv

Self-generated Alfvén waves Alfvén-wave growth rate Γ = ½π ω cp · p r S p (r,p r,t)/nv A p r = m ω cp /|k| S p = 4π p 2 ∫ dμ vμ f(r,p, μ,t) = proton streaming per unit momentum Efficient wave growth (at fixed r,k) during the SEP event requires 1 > (2/π) nAv A /ω cp = sr -1 (v A  /v A ) (n   /2·10 8 cm -3 ) ½ where A = cross-sectional area of the flux tube dN/dp = momentum distr. of protons injected to the flux tube Vainio (2003) sr

Self-generated waves (cont'd) Threshold spectrum for wave-growth p dN/dp| thr = sr -1 (n   /2·10 8 cm -3 ) ½ (v A  /v A (r)) lowest in corona Apply a simple IP transport model: radial diffusion 1 AU, dJ/dE| max = 15·(MeV/E) ½ /cm 2 ·sr·s·MeV for p dN/dp = 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 p dN/dp [sr -1 ] r [R sun ] Vainio (2003) solar-wind model with a maximum of v A in outer corona most efficient wave growth

r r p dN p /dr r log P(r) r p S p (r) Γ(r) t = t 1 t = t 2 > t 1 Γ(r) p S p (r) Coupled evolution of particles and waves weak scattering (Λ > L B ) weak scattering turbulent trapping with gradual leakage p dN p /dr impulsive release of escaping protons ProtonsAlfvén waves weak scattering log P(r)

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 + v A = 400 km/s  parallel shock with constant speed V s and sc-compression ratio r sc  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 simultaneously Γ = π 2 f cp · p r S p (r,p r,t)/nv A /Δt = π 2 f cp · f r P(r,f r,t)/B 2 p r = f cp m p V/f f r = f cp m p V/p u B VsVs

Examples of simulation results Shock launched at R = 1.5 R sun at speed V s = 1500 km/s in all examples. Varied parameters:  Ambient scattering mean free r = 1.5 R sun and E = 100 keV Λ 0 = 1, 5, 30 R sun  Injection rate q = N inj /t max << q sw where q sw = ∫ n(r)A(r) dr /t max = 2.2·10 37 s -1  Scattering center compression ratio of the shock, r sc = 2, 4

r sc = 2, q ~ 4.7·10 32 s -1, Λ 0 = 1 R sun - 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

r sc = 4, q ~ 4.7·10 32 s -1, Λ 0 = 1 R sun - 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

r sc = 4, q ~ 1.9·10 33 s -1, Λ 0 = 5 R sun - 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

r sc = 4, q ~ 3.9·10 32 s -1, Λ 0 = 30 R sun - 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

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

Escaping particles (Λ 0 = 1 R sun ) threshold for wave-growth NOTE: Observational streaming- limited spectrum somewhat softer than the simulated one (~ E -1/2 ).

Cut-off energy 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): p dN/dp| esc ~ 4·10 33 sr –1 Power-laws cut off at an energy, which depends strongly on the injection rate q = N inj /t max E c ~ q a with a ~ 0.5 – 2 High injection rate leads to very turbulent environment → challenge for modeling ! N inj [sr –1 ] – E c [MeV] simulation time = 640 s log E log shock Bell (1978) Bell/10 EcEc

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 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

V s = 2200 km/s, r sc = 4, t = 640 s, q ~ 4.7·10 32 s -1, Λ 0 = 1 R sun protonswaves