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![Page 1: Monte-Carlo simulations of shock acceleration of solar energetic particles in self-generated turbulence Rami Vainio Dept of Physical Sciences, University.](https://reader035.fdocuments.us/reader035/viewer/2022062500/56649ebd5503460f94bc6ce5/html5/thumbnails/1.jpg)
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
![Page 2: Monte-Carlo simulations of shock acceleration of solar energetic particles in self-generated turbulence Rami Vainio Dept of Physical Sciences, University.](https://reader035.fdocuments.us/reader035/viewer/2022062500/56649ebd5503460f94bc6ce5/html5/thumbnails/2.jpg)
Large Solar Particle Events
Reames & Ng 1998
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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
![Page 4: Monte-Carlo simulations of shock acceleration of solar energetic particles in self-generated turbulence Rami Vainio Dept of Physical Sciences, University.](https://reader035.fdocuments.us/reader035/viewer/2022062500/56649ebd5503460f94bc6ce5/html5/thumbnails/4.jpg)
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
![Page 5: Monte-Carlo simulations of shock acceleration of solar energetic particles in self-generated turbulence Rami Vainio Dept of Physical Sciences, University.](https://reader035.fdocuments.us/reader035/viewer/2022062500/56649ebd5503460f94bc6ce5/html5/thumbnails/5.jpg)
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
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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
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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
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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)
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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
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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
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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
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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
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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
![Page 14: Monte-Carlo simulations of shock acceleration of solar energetic particles in self-generated turbulence Rami Vainio Dept of Physical Sciences, University.](https://reader035.fdocuments.us/reader035/viewer/2022062500/56649ebd5503460f94bc6ce5/html5/thumbnails/14.jpg)
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
![Page 15: Monte-Carlo simulations of shock acceleration of solar energetic particles in self-generated turbulence Rami Vainio Dept of Physical Sciences, University.](https://reader035.fdocuments.us/reader035/viewer/2022062500/56649ebd5503460f94bc6ce5/html5/thumbnails/15.jpg)
Comparison with the theory of Bell (1978)
Qualitative agreement at the shock below cut-offGood agreement upstream behind escaping particles
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Escaping particles (Λ0 = 1 Rsun)
threshold forwave-growth
NOTE: Observational streaming- limited spectrum somewhat softer than the simulated one (~ E-1/2).
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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
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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
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Vs = 2200 km/s, rsc = 4, t = 640 s,
q ~ 4.7·1032 s-1, Λ0 = 1 Rsun
protons waves