Monte-Carlo simulations of shock acceleration of solar energetic

particles in self-generated turbulence

Rami VainioDept of Physical Sciences, University of Helsinki, Finland

Timo LaitinenDept of Physics, University of Turku, Finland

COST Action 724 is thanked for financial support

Large Solar Particle Events

Reames & Ng 1998

Reames (2003)

1

0.1

0.01

Fract

ion

of

tim

e (

%)

10

0.001

GOES Proton flux 1986-1997

104 105 106 107 108

Hourly fluence (protons/cm2 sr)

104 105 106 107 104 105 106

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 scatteringv||VA

v' =

con

st.

v

dv/dt < 0 → wave growth

dv/dt > 0 → wave damping

vv = velocity insolar-windframe

Particle acceleration at shocks

Particles crossing the shockmany times (because of strongscattering) get accelerated

Vsh

W1 = u1+vA1

W2

v||ΔW = W2 - W1

v' =

con

st.

v 2 >

v 1

dv/dt > 0 → particle acceleration

v = particle velocity in the ambient AW frame

v1

upstream →downstream

downstream →upstream

Vshv

Self-generated Alfvén waves

Alfvén-wave growth rate

Γ = 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 requires

1 << ∫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)½

where A = cross-sectional area of the flux tubedN/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 = 1033 sr-1 (n/2·108cm-3)½ (vA /vA(r))

lowest in corona

Apply a simple IP transport model: radial diffusion → @ 1 AU,

dJ/dE|max = 15·(MeV/E)½/cm2·sr·s·MeV

for 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

p dN/dp [sr-1]

r [Rsun] 1 10 100

1033

1034

Vainio (2003)

solar-wind model with a maximumof vA in outer corona

most efficientwave growth

r

r

p dNp/dr

r

log P(r)

r

p Sp(r)Γ(r)

t = t1

t = t2 > t1

Γ(r)p Sp(r)

Coupled evolution of particles and waves

weak scattering (Λ > LB)

weak scattering

turbulenttrapping withgradual leakage

p dNp/dr

impulsive release of escaping protons

Protons Alfvén waves

weak scattering

weak scatteringlog 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 + 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 simultaneously

Γ = π2 fcp · pr Sp(r,pr,t)/nvA <(Δθ)2>/Δt = π2 fcp · fr P(r,fr,t)/B2

pr = fcp mpV/f fr = fcp mpV/p

u

B

Vs

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 keV

Λ0 = 1, 5, 30 Rsun

Injection rate

q = Ninj/tmax << qsw

where qsw = ∫ n(r)A(r) dr /tmax = 2.2·1037 s-1

Scattering center compression ratio of the shock,

rsc = 2, 4

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

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

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

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

Comparison with the theory of Bell (1978)

Qualitative agreement at the shock below cut-offGood agreement upstream behind escaping particles

Escaping particles (Λ0 = 1 Rsun)

threshold forwave-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 ~ 4·1033 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 !

Ninj [sr–1]1035 1036 1034

10–1

100

101

102

Ec [M

eV

]

simulation time = 640 s

log E

log f @shock

Bell (1978)

Bell/10

Ec

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

Vs = 2200 km/s, rsc = 4, t = 640 s,

q ~ 4.7·1032 s-1, Λ0 = 1 Rsun

protons waves