June 23, 2014
•
Expectations from Bell non-resonant instability:


Why do we need it?
Maximum energy
• Description of our toy-model:




•
Spectrum
Energetics
Comparison with KASKADE-GRANDE data
Comparison with ARGO data
Conclusions
RESONANT
INSTABILITY
(Skilling 1975)
Excitation of Alfvén waves
with
λ ≅ rL
NON
RESONANT
INSTABILITY
(Bell 2004)
Purely growing waves at
wavelengths non-resonant
with rL (λ << rL), driven by
the CR current jCR.
Growth rate dependence on the shock velocity (α vs2) and on
plasma density (α n⅙)
Very Young
SNRs
ρ = cost
ISM
Ze
xCR 2
EM (R) @
4pr
Rvsh (R)
10
cL
·
WIND
•
•
•
ρ α R-2
Ze M xCR 2
EM (R) @
vsh (R)
5 Vw Lc
Independent from the B field strength
Proportional to CR efficiency (ξCR)
Strong dependence on shock velocity
ED
phase
1
1 dt
N acc (E) = fesc r (t)vsh3 (t)4p Rsh2 (t)
2
E(t) dE
Caprioli et al. 2010,
Schure&Bell 2013
Acceleration
spectrum
éæ ölED ×a æ ölST ×a ù
t
t
R(t) = R0 êç ÷
+ç ÷ ú
êëè t0 ø
è t0 ø úû
STARTING
POINT
WIND
ρej α R-k
k=[7,9]
ì - 2-lED
ï E 2(1-lED )
ï
N acc (E) µ í
ï
E -2
ï
î
lED =
p=[3,4]
k -3
k -2
lST =
2
5
No sharp
cut-off
above EM!
Observed
Spectrum
-d
 H æ E ö æ X(E) ö
N obs (E) = N acc (E)´
÷
ç ÷ ´ ç1+
2
2p RD D0 è Ze ø è
XCR ø
Diffusion
Spallation
Fixed
Variables
Mej= 1 M
ESN= Supernova energy
dM/dt= 10-5 M/yrs
R = Explosion rate
Vw= 10 km/s
Rd= 10 kpc
H= 3 kpc
nd= 1 cm-3 kpc
D0= 3 x 1028 cm2 s-1
δ= 0.65
σsp= α(E) Αβ(E)
ξCR
EM
t0
V0
ESN= 1051 erg
R= 1/30 yrs
EM= 6.6 x 1014 eV
ξCR= 12%
t0= 96 yrs
v0=10.445 km/s
ESN= 1051 erg
R= 1/30 yrs
EM= 6.4 x 1014 eV
ξCR= 10 %
t0= 88 yrs
v0=11.339 km/s
ESN= 1051 erg
R= 1/30 yrs
EM= 6.3 x 1014 eV
ξCR= 8.5 %
t0= 84.5 yrs
v0=11.830 km/s
~2x1012
~3x1015
• τdyn> τpp≅ 103 s
• τdyn> τIC ≅ 106 s
• τdyn> τph ≅ 109 s
k=9
ESN= 1052 erg
R= 1/950 yrs
EM ≅ 9.4 x 1015 eV
ξCR ≅ 13 %
t0 ≅ 27 yrs
v0 ≅37.500 km/s
k=9
ESN= 1052 erg
R= 1/950 yrs
EM ≅ 9.4 x 1015 eV
ξCR ≅ 13 %
t0 ≅ 27 yrs
v0 ≅37.500 km/s
k=9
ESN= 1051 erg
R= 1/30 yrs
EM ≅ 6.3 x 1014 eV
ξCR ≅ 8.5 %
t0 ≅ 84 yrs
v0 ≅11.800 km/s
k=9
ESN= 4 x 1051 erg
R= 1/60 yrs
Uncertainty on
the data?
EM ≅ 1.2 x 1015 eV
ξCR ≅ 4.5 %
Additional
component?
t0 ≅ 42 yrs
v0 ≅23.700 km/s
 Bell non-resonant instability (NRI) predicts that very energetic SNRs
can reach PeV energies
 Our toy-model shows that the NRI leads to the release of a steep
power-law spectrum in the ejecta dominated phase
 The “knee” provided by our model is at E < 3x1015 eV for standard
energetics
 KASKADE Grande data can be fitted only by requiring a
challengingly large energetics of a SNR
 Our model can fit ARGO data of the light component but a fit to the
overall spectrum requires the existence of another population of
very energetic particles in addition to the SNR one.