S EED P OPULATION E NCOUNTERS WITH C ORONAL
S HOCKS OF VARIED G EOMETRY
M. Battarbee1,5, R. Vainio2, T. Laitinen3 and H.Hietala4
Introduction
Particle Acceleration at Coronal Shocks
Acceleration at shocks driven by Coronal Mass Ejections
(CMEs) is currently considered the primary source of large
solar energetic particle (SEP) intensities.
Diffusive Shock Acceleration (DSA) is the process of
repeated shock encounters and isotropizations, which
leads to 1st order Fermi acceleration.[1]
Particles return to the shock by scattering off upstream
turbulence, which can be excited by the particles
themselves. [2]
This bootstrapped process requires knowledge of the
whole injected particle population, and especially the initial
encounter between seed particles and the shock.
We present analytical and numerical results, which show
how parallel shocks can inject a significant portion of the
thermal seed population
Assessed shocks have shock-normal velocity
Vs = 1500 km s−1, shock-normal angle θn = 0 ... 60◦
Seed particles & the shock
Solar wind population depicted by κ-distribution
(Maxwellian + power-law tail)[3]
Sample composition: 99 % thermal (κ = 15) and 1 %
suprathermal (κ = 2)
Seed population, isotropic in the upstream plasma frame,
is convected to the shock
Pitch-angle distribution adjusted for v > u1 particles to
preserve continuity of flux
At the simplified step-model shock, particles experience
magnetic compression ∆B and cross-shock potential
∆Φ[4]
Particles isotropize in the downstream plasma frame (flow
speed u2, cross helicity Hc2 = 0).
Ratio of fluxes towards and away from shock give
probability of return:
Pret =
′
v − u2
v ′ + u2
2
Methods
Three methods were used:
Analytical assessment of velocity thresholds through
calculation of particle properties at shock encounter and/or
transmission
Reflection due to ∆B or ∆Φ
Return from downstream through scatterings (v ′ > u2)
Results: Semianalytical integration
Results (cont.)
Transmission parameters (some solved numerically):
Alfvénic Mach number MA
Upstream incident flow speed u1
Gas compressio ratio rg
Magnetic compression ratio rB
Downstream plasma flow speed u2 = u1rB rg−1
Transmission / reflection of flux
Represent seed particle distribution function f (p) with
1000 × 721 cells
Use Liouville’s theorem for shock transmission of particle
distribution and integrate flux
Graph flux density dF/(dvk dv⊥) to view downstream
speed distribution
Results intro
All three methods show similar θn-dependence
At θn ≤ 6◦, all particles can be injected
At θn ≥ 14◦, reflection becomes dominant
As θn →∼ 7.5◦, injection efficiency decreases by 2
magnitudes
If instant downstream isotropy is not assumed, the
θn-dependent weakening is amplified (by ∼ 3 magnitudes
at θn = 10◦)
Reflection at the shock front
Conservation of magnetic moment:
Figure 3: Flux density dF/(dvk dv⊥), κ = 2
(1 − µ22)v22 (1 − µ21)v12
=
≡M
B2
B1
Solving for downstream parallel kinetic energy:
Figure 5: Incident, reflected and injected fluxes
1 mµ2v 2 = 1 mµ2v 2 − 1 mM∆B − q∆Φ
2 2
1 1
2
2
2
If RHS < 0, the particle is reflected at the shock.
Figure 1: Reflection threshold velocities (upstream)
Incident (blue contours), reflected (red contours) and
transmitted flux (red curves) with 1 magnitude intervals, 0 %
and 25 % injection thresholds (dashed and dotted semicircles)
Returning flux
Assume particles isotropize instantly in the downstream
Integrate reflected flux with upstream values
Integrate resultant flux towards the shock from Pret
Solved reflection threshold speeds for seed particles. Within
the shaded area, reflection is not possible.
Injection after downstream scatterings
Solve particle downstream speed as function of upstream
properties:
2
q
v ′2 = u2 − u12 − 2µvu1 + v 2 − v 2(1 − µ2)rB − 2(q/m)∆Φ
2
2
+ (1 − µ )v rB
Threshold for possible injection is found at v ′ = u2
Figure 2: Injection threshold velocities (upstream)
Results: Monte Carlo
Numerical particle simulation scheme
Simulate flux with ∼ 106 suprathermal particles,
64 · 103 − 106 thermal particles
Pre-propagate fast (v > u1) particles to simulate
pitch-angle adjustment
Transmission or reflection as per analytical assessment
a) Particles isotropize instantly in the downstream at x = −0
b) or: propagated within isotropization region x = [−0 ... − 2λ]
calculate Pret at x = −0 or x = −2λ
Figure 4: Monte Carlo results for θn = 30◦
Distribution function f (p) transmitted across the shock
Numerical integration over momentum space to find total flux
Monte Carlo simulations
Representative particles transmitted and scattered
Two variations to find effect of downstream isotropization times
(instant vs. propagation to 2λ)
Model parameters[5]
2-D (guiding centre approximation) model
cs = 2.34 · 107 cm s−1
vA = 6.97 · 107 cm s−1
usw = 9.98 · 106 cm s−1
V = 7.97 · 107 cm s−1
United Kingdom
4Space & Atmospheric Physics Group, Department of Physics,
Imperial College London, United Kingdom
5e-mail: [email protected]
Analytical Assessment
Semianalytical integrations of flux
r = 3.17 R⊙
ρ = 3.48 · 105 cm3
B1 = 0.19 G
T = 2.0 · 106 K
1Department of Physics and Astronomy, University of Turku, Finland
2Department of Physics, University of Helsinki, Finland
3Jeremiah Horrocks Institute, University of Central Lancashire,
Solved injection threshold speeds for transmitted particles.
Within the shaded area, injection is possible.
Logarithmic quantity of injected particles: at θn = 30◦,
reflection dominates.
Comparisons of integrated fluxes: Semianalytical methods and
two Monte Carlo setups.
Conclusions
Obliquity & suprathermality
Fast parallel shocks are capable of injecting a large
portion of the thermal seed population
As shock obliquity increases, the injection of thermal
particles decreases rapidly
As DSA is a bootstrapped process, this effect of parallel
shocks can help to explain the injection problem
Specifics of downstream processes (e.g., isotropization
time) play a major role
At large θn reflection dominates injection
Presence of a cross-shock potential increases reflection,
decreases return from downstream
Further research
Analysis of expanded parameter space (e.g.,
Vs = 1000 kms−1 or Vs = 2000 kms−1)
Further assessment of the cross-shock potential ∆Φ
References
[1] A. R. B ELL, MNRAS 182, 147 (1978).
[2] R. VAINIO, A&A 406, 735 (2003).
[3] V. M. VASYLIUNAS, J. Geophys. Res. 73, 2839 (1968).
[4] A. J. H ULL, J. D. S CUDDER, R. J. F ITZENREITER, K. W.
O GILVIE, J. A. N EWBURY, and C. T. RUSSELL,
J. Geophys. Res. 105, 20957 (2000).
[5] M. B ATTARBEE, T. L AITINEN, R. VAINIO, and N. AGUEDA,
Twelfth International Solar Wind Conference 1216, 84
(2010).
A manuscript with extended analysis will be submitted shortly.