Coupled ion acceleration
and
Alfven wave excitation at an expanding coronal shock
E. Berezhko and S. Taneev
Yu.G. Shafer Institute of Cosmophysical Research and Aeronomy
Yakutsk, Russia
Main question:
What is a role of selfexcited Alfven waves in
Solar Energetic Particles (SEP) production by CME driven shock?
Remark:
Solar Energetic Particles = Solar Cosmic Rays (SCRs)
SH 0125 11.08.12 17:30-19:30
SEP event characteristics
Miroshnichenko & Perez-Peraza, 2008
Diffusive shock acceleration of CRs
Krymsky 1977
Bell 1978
log nCR
Δp
p-
log p
scattering centers
 = ( + 2)/( -1)
shock compression ratio
Cutoff due to
finite shock size or
due to restricted
system age
Coronal Mass Ejection driven shock
w
Earth
Shock
Sun
CME
Vp
Vs
Intensity of energetic particles,
measured by distinct observer
Rs
I
freshly accelerated
particles, g < 0.1
flare
ε1 > ε2 > ε3
Solar wind
escaping particles
SCRs,
g > 0.1
RSVS
g ( ) 
 ( )
ε3
Berezhko, Krymsky (1987)
ε2
ε1
Modulation parameter:
shock
t
SCRs
At which distances from the Sun SCRs are produced?
Zank, Rice, Wu (2000): Solar wind (RS > 20RSun),”onion-shell” plane wave approach,
assumed efficient Alfven wave excitation leading to Bohm limit diffusion.
εmax ~ 100 MeV
Rice, Zank, Li (2003): Solar wind (RS > 20RSun),”onion-shell” plane wave approach,
upstream Alfven wave intensity according to quasilinear model (Gordon et al.1999)
εmax ~ (1- 10) MeV ( too small !)
Ng, Reames, Tylka (2003): Solar wind (RS > 10RSun ), focused upstream transport
of SCRs with their ad hoc source term,
Alfven wave excitation by SCRs in upstream region
Lee (2005): Solar wind (RS > 10RSun), plane wave steady state approach,
upstream Alfven wave intensity according to quasilinear model (Gordon et al.1999)
However according to observations SCR onset usually occur at ~ 0.2 – 1 hr after
flare at height of (2-4)RSun above photosphere (Kahler 1994; Krucker, Lin 2000)
Physical factors relevant for acceleration efficiency
NCR
CR energy spectrum
NCR = A ε−γexp(−ε/εmax)
ninj
 < εinj
εinj
ε -γ
vinj ~ VS− w
speed of injected ions
γ > γ
εinj
εmax ε
CR diffusion coefficient
κ(εmax) ≈ 0.1 RS(VS – w – cA)
maximal CR energy
A ~ RS3ρ(RS)(VS – w)3/(σeff -1)
γ = 0.5 (σeff +2)/(σeff
σeff = σ(1 – cA/VS)
- 1)
total number of accelerated CRs
power law index
effective compression ratio
Solar corona parameters
Alfven speed
solar wind
speed
Solar wind
number density
r/RSun
SCR acceleration efficiency within the region r < 1 AU
Vs, km/s
1500
1000
500
1
inefficient
A, AU
10-3
efficient
10-6
εmax, MeV
100
10
4
γ
3
2
1
1
10
100
RS / R
Alfven waves in solar corona
Alfven wave energy flux at r = RSun
Fw = 106 erg/(cm2 s)
P(ν)=dEw /dν~ r -δ ν -λ
νP
Alfven wave energy spectrum
δ ≈ 4.3
rR
λ = 1 at 10-3 < ν < 5×10-2 Hz
λ = 3/2 at ν > 5×10-2 Hz
Relevant
for SCRs
r = 1 AU
(e.g. Suzuki, Inusuka 2006)
(e.g. Tu, Marsh 1995)
10-3
10-2
10-1
ν, Hz
B = B0(r0/r)2
Alfven wave excitation by shock accelerated CRs
f0
CR anisotropy → wave excitation
f = f0 + f1 cosθ
p
VS
θ
f1 ~ - df0/dx ~ f0VS/v
shock
front
Bell (1978)
Lee(1982)
Gordon et al.(1999)
x
∂Ew/∂t = Γ Ew + …
wave energy
Γ ~ f1
wave growth rate
∂f/∂t = ∂(κ∂f/∂x)/∂x + …
κ ~ 1/Ew
τacc ~ κ/VS2
particle diffusion coefficient
acceleration time
Quasilinear model of SCRs acceleration
f
w f
 f  wf 
p Q
t
3
p
Q 
1u1
 ( p  pinj ) (r  Rs )
2
4 mpinj
 ( p )  pB / Ew (k  B1 )
Ew
Ew
 (VS  w)
 Ew
t
r
f
( k )    dp
r
CR transport equations (Krymsky, 1964)
source (injection) term
diffusion coefficient
Alfven wave transport equation
growth (damping) rate
(Gordon et al., 1999)
System of equations is solved numerically within the distance range of
efficient shock acceleration RS < R* = (2-5) RSun
At r > R* accelerated SCRs are assumed to propagate purely diffusively
Compared with the previous considerations
the model includes:
• SCR acceleration within corona (r < 5 RSun)
• Selfexcited Alfven waves
• The influence of Alfven speed on the effective compression
ratio, which determines SCR spectrum
• Consistent determination of SCR maximal energy
Compared with our previous study (Berezhko&Taneev 2003) the model incledes
selfexcited Alfven waves
Overall energy spectra of SCRs,
produced by CME driven shock in solar corona
linear
approach
quasilinear
approach
Selfexcited Alfven waves reduce the number of SCRs
( due to decrease of effective shock compression ratio)
and do not affect SCR maximal energy/
Proton energy spectrum at 1 AU in the event 1977 November 22
VS = 1000 km/s
η = 10-3
linear
quasilinear
Conclusions
• Efficient SCR production by CME driven shock takes
place within the region r < 2-5 RSun
• Selfexcited Alfven waves reduce the number of SCRs
due to the decrease of effective shock compression
• Alfven wave generation insignificantly influence SCR maximal energy
due to steep energy spectrum of accelerated SCRs
• Calculated spectra of shock accelerated SCRs in satisfactory way
agree with SCR spectra measured in gradual events
Alfven wave spectra near the shock front
VS = 1500 km/s
CR diffusion coefficient
VS = 1500 km/s
κL ≈ κQL
linear
approach
Bohm limit
qusilinear
approach
Solar Cosmic Ray spectrum at the Earth’s orbit
Berezhko, Taneev (2003)
VS = 2000 km/s
Solar corona/wind parameters
Vs = 1500 km/s
Vs = 600 km/s
RS / R
Overall particle energy spectrum
in/near acceleration region
  N inj / N
η > 10
-5
injection rate
→ efficient CR production
Expected injection rate
η ~ 10-3