the theory of diffusive shock acceleration - dto. de fisicajarne/clases/stefano_gabici_three.pdf ·...
TRANSCRIPT
![Page 1: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/1.jpg)
The theory ofDiffusive Shock Acceleration
Stefano GabiciAPC, Paris
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Shock wavesSupersonic motion + medium -> Shock Wave
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Shock wavesSupersonic motion + medium -> Shock Wave
(SuperNova ejecta) (InterStellar Medium)
velocity of SN ejecta up to
sound speed in the ISM cs =
γkT
m≈ 10
T
104K
1/2
km/s
vej ≈ 30000 km/s
Mach number strong shocksM =v
cs>> 1
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Shock wavesThermodynamic quantities are discontinuous across a shock wave
dens
ity
->
<---
->high amplitude
c2s = γ
p
∝ γ−1
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this goes faster
this goes slower
Shock wavesThermodynamic quantities are discontinuous across a shock wave
dens
ity
->
<---
->high amplitude
c2s = γ
p
∝ γ−1
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Shock wavesThermodynamic quantities are discontinuous across a shock wave
dens
ity
->
<---
->high amplitude
c2s = γ
p
∝ γ−1
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Shock wavesThermodynamic quantities are discontinuous across a shock wave
dens
ity
->
<---
->high amplitude
c2s = γ
p
∝ γ−1
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shock!
Shock wavesThermodynamic quantities are discontinuous across a shock wave
dens
ity
->
<---
->high amplitude
c2s = γ
p
∝ γ−1
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Shock waves
ShockUp-stream Down-stream
u1 u2
1 , T1 2 , T2
-->
Shock rest frame
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Shock waves
u1 u2
1 , T1 2 , T2
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Shock waves
u1 u2
1 , T1 2 , T2
Mass conservation
1 u1 = 2 u2
u1
u2=
2
1= r
compression ratio
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Shock waves
u1 u2
1 , T1 2 , T2
Mass conservation
1 u1 = 2 u2
u1
u2=
2
1= r
compression ratioMomentum conservation
1u21 + p1 = 2u
22 + p2
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Shock waves
u1 u2
1 , T1 2 , T2
Mass conservation
1 u1 = 2 u2
u1
u2=
2
1= r
compression ratioMomentum conservation
1u21 + p1 = 2u
22 + p2
1u21
1 +
p1
1u21
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Shock waves
u1 u2
1 , T1 2 , T2
Mass conservation
1 u1 = 2 u2
u1
u2=
2
1= r
compression ratioMomentum conservation
1u21 + p1 = 2u
22 + p2
1u21
1 +
p1
1u21
= 1u
21
1 +
c2s,1
γu21
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Shock waves
u1 u2
1 , T1 2 , T2
Mass conservation
1 u1 = 2 u2
u1
u2=
2
1= r
compression ratioMomentum conservation
1u21 + p1 = 2u
22 + p2
1u21
1 +
p1
1u21
= 1u
21
1 +
c2s,1
γu21
= 1u21
1 +
1γM2
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Shock waves
u1 u2
1 , T1 2 , T2
Mass conservation
1 u1 = 2 u2
u1
u2=
2
1= r
compression ratioMomentum conservation
1u21 + p1 = 2u
22 + p2
1u21
1 +
p1
1u21
= 1u
21
1 +
c2s,1
γu21
= 1u21
1 +
1γM2
M >> 1
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Shock waves
u1 u2
1 , T1 2 , T2
Mass conservation
1 u1 = 2 u2
u1
u2=
2
1= r
compression ratioMomentum conservation
1u21 + p1 = 2u
22 + p2
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Shock waves
u1 u2
1 , T1 2 , T2
Mass conservation
1 u1 = 2 u2
u1
u2=
2
1= r
compression ratioMomentum conservation
1u21 + p1 = 2u
22 + p2 u1
u2= 1 +
p2
2u222u2
2 2u22
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Shock waves
u1 u2
1 , T1 2 , T2
Mass conservation
1 u1 = 2 u2
u1
u2=
2
1= r
compression ratioMomentum conservation
1u21 + p1 = 2u
22 + p2 u1
u2= 1 +
p2
2u222u2
2 2u22
u1
u2= 1 +
1γM2
2
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Shock waves
u1 u2
1 , T1 2 , T2
Mass conservation
1 u1 = 2 u2
u1
u2=
2
1= r
compression ratioMomentum conservation
1u21 + p1 = 2u
22 + p2 u1
u2= 1 +
p2
2u222u2
2 2u22
u1
u2= 1 +
1γM2
2
r = 1 +3
5M22
γ =53
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Shock waves
u1 u2
1 , T1 2 , T2
Mass + Momentum conservation
r = 1 +3
5M22
Energy conservation
121u
31 +
γ
γ − 1p1u1 =
122u
32 +
γ
γ − 1p2u2
enthalpy
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Shock waves
u1 u2
1 , T1 2 , T2
Mass + Momentum conservation
r = 1 +3
5M22
Energy conservation
121u
31 +
γ
γ − 1p1u1 =
122u
32 +
γ
γ − 1p2u2 M >> 1
enthalpy
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Shock waves
u1 u2
1 , T1 2 , T2
Mass + Momentum conservation
r = 1 +3
5M22
Energy conservation
121u
31 +
γ
γ − 1p1u1 =
122u
32 +
γ
γ − 1p2u2 M >> 1
enthalpy
122u3
2122u3
2
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Shock waves
u1 u2
1 , T1 2 , T2
Mass + Momentum conservation
r = 1 +3
5M22
Energy conservation
121u
31 +
γ
γ − 1p1u1 =
122u
32 +
γ
γ − 1p2u2 M >> 1
enthalpy
122u3
2122u3
2
r2 = 1 +3M2
2
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Shock waves
r = 1 +3
5M22
r2 = 1 +3M2
2
r = 4
M22 =
15
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Shock waves
r = 1 +3
5M22
r2 = 1 +3M2
2
r = 4
M22 =
15
What about down-stream temperature?
15
=M22 =
u22
c2s,2
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Shock waves
r = 1 +3
5M22
r2 = 1 +3M2
2
r = 4
M22 =
15
What about down-stream temperature?
15
=M22 =
u22
c2s,2
=u2
1
16 c2s,2
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Shock waves
r = 1 +3
5M22
r2 = 1 +3M2
2
r = 4
M22 =
15
What about down-stream temperature?
15
=M22 =
u22
c2s,2
kbT2 =316
mu21
=u2
1
16 c2s,2
=mu2
1
16 γ kbT2
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Shock wavesA (strong) shock:
compresses moderately the gas
makes the supersonic gas subsonic
converts bulk energy into internal energy
2
1= r = 4
M1 >> 1→M2 =1√5
< 1
kbT2 =316
mu21
Weak shock:
M >> 1
M 1
smaller compression and moderate gas heating r < 4 T2 T1
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Shock waves + Magnetic fields
ShockUp-stream Down-stream
u1u2
-->
B
B
B
B
B
B
Shock rest frame
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Diffusive Shock Acceleration
ShockUp-stream Down-stream-->
B
B
B
B
B
B
Up-stream rest frame
u1
--> relativistic particle of mas m (<< Mcloud) and energy E
u1 − u2
a
b
Ea = Eb
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Diffusive Shock Acceleration
ShockUp-stream Down-stream-->
B
B
B
B
B
B
Down-stream rest frame
--> relativistic particle of mas m (<< Mcloud) and energy E
u1 − u2
a
b
Ea = Eb
u2
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Diffusive Shock AccelerationSymmetry
u1 − u2 u1 − u2
Every time the particle crosses the shock (up -> down or down -> up), it undergoes an head-on collision with a plasma moving with velocity u1-u2
Up-stream Down-stream
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Diffusive Shock AccelerationSymmetry
u1 − u2 u1 − u2
Every time the particle crosses the shock (up -> down or down -> up), it undergoes an head-on collision with a plasma moving with velocity u1-u2
Asymmetry
(Infinite and plane shock:) Upstream particles always return the shock, while downstream particles may be advected and never come back to the shock
Up-stream Down-stream
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Diffusive Shock Acceleration
u1 − u2
xϑ
The particle has initial (upstream)energy E and initial momentum p
UP DOWN
The particle “sees” the downstream flow with a velocity:
and a Lorentz factor:
v = u1 − u2
γv
In the downstream rest frame the particle has an energy (Lorentz transformation):
E = γv(E + p cos(θ) v)
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Diffusive Shock Acceleration
E = γv(E + p cos(θ) v)
the shock is non-relativistic ----------------->
we assume that the particle is relativistic -->
γv = 1
E = pc
E = E +E
cv cos(θ)
∆E
E=
v
ccos(θ)
energy gain per half-cycle(up->down-stream)
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Diffusive Shock Acceleration
ASSUMPTION: particles up (down) - stream of the shock are rapidly isotropized by magnetic field irregularities
θ0 < θ <π
2
dθ
# of particles between and prop. to ---->
rate at which particles cross the shock prop. to --->
θ + dθθ sin(θ)dθ
c cos(θ)
c
p(θ) ∝ sin(θ) cos(θ)dθprobability for a particle to cross the shock:
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Diffusive Shock Acceleration
p(θ) ∝ sin(θ) cos(θ)dθ
The total probability must be equal to 1
A
π2
0dθ cos(θ) sin(θ) ≡ 1
![Page 39: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/39.jpg)
Diffusive Shock Acceleration
p(θ) ∝ sin(θ) cos(θ)dθ
The total probability must be equal to 1
A
π2
0dθ cos(θ) sin(θ) ≡ 1
sin(θ) = t
dt = cos(θ)dθ
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Diffusive Shock Acceleration
p(θ) ∝ sin(θ) cos(θ)dθ
The total probability must be equal to 1
A
π2
0dθ cos(θ) sin(θ) ≡ 1
sin(θ) = t
dt = cos(θ)dθ
A
1
0dt t
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Diffusive Shock Acceleration
p(θ) ∝ sin(θ) cos(θ)dθ
The total probability must be equal to 1
A
π2
0dθ cos(θ) sin(θ) ≡ 1
sin(θ) = t
dt = cos(θ)dθ
A
1
0dt t = A
t2
2
1
0
=A
2
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Diffusive Shock Acceleration
p(θ) ∝ sin(θ) cos(θ)dθ
The total probability must be equal to 1
A
π2
0dθ cos(θ) sin(θ) ≡ 1
sin(θ) = t
dt = cos(θ)dθ
A
1
0dt t = A
t2
2
1
0
=A
2
p(θ) = 2 sin(θ) cos(θ)dθnormalized probability:
![Page 43: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/43.jpg)
Diffusive Shock Acceleration
∆E
E=
v
ccos(θ)
(1) energy gain per half-cycle:(up->down-stream)
p(θ) = 2 sin(θ) cos(θ)dθ(2) probability to cross the shock:
![Page 44: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/44.jpg)
Diffusive Shock Acceleration
∆E
E=
v
ccos(θ)
(1) energy gain per half-cycle:(up->down-stream)
p(θ) = 2 sin(θ) cos(θ)dθ(2) probability to cross the shock:
<∆E
E>The average gain per half-cycle is (1) averaged over the
probability distribution (2).
<∆E
E> = 2
vc
π2
0dθ cos2(θ) sin(θ)
![Page 45: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/45.jpg)
Diffusive Shock Acceleration
<∆E
E> = 2
vc
π2
0dθ cos2(θ) sin(θ)
![Page 46: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/46.jpg)
Diffusive Shock Acceleration
<∆E
E> = 2
vc
π2
0dθ cos2(θ) sin(θ)
cos(θ) = t
dt = − sin(θ)dθ
![Page 47: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/47.jpg)
Diffusive Shock Acceleration
<∆E
E> = 2
vc
π2
0dθ cos2(θ) sin(θ)
cos(θ) = t
dt = − sin(θ)dθ
= −2vc
−1
0dt t2 = −2
vc
t3
3
−1
0
=2
3
vc
![Page 48: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/48.jpg)
Diffusive Shock Acceleration
<∆E
E> = 2
vc
π2
0dθ cos2(θ) sin(θ)
cos(θ) = t
dt = − sin(θ)dθ
= −2vc
−1
0dt t2 = −2
vc
t3
3
−1
0
=2
3
vc
full cycle: (up -> down) and (down -> up) : SYMMETRY
<∆E
E>up→down=<
∆E
E>down→up
![Page 49: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/49.jpg)
Diffusive Shock Acceleration
<∆E
E> =
4
3
vc
=
4
3
u1 − u2
c
Energy gain per cycle (up -> down -> up):
First-order (in v/c) Fermi mechanism
![Page 50: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/50.jpg)
Diffusive Shock AccelerationWhat happens after n cycles?
<∆E
E> =
4
3
vc
![Page 51: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/51.jpg)
Diffusive Shock AccelerationWhat happens after n cycles?
<∆E
E> =
4
3
vc
Ei+1 − Ei
Ei=
particle energy at i-th cycle
![Page 52: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/52.jpg)
Diffusive Shock AccelerationWhat happens after n cycles?
<∆E
E> =
4
3
vc
Ei+1 =
1 +
4
3
v
c
Ei
Ei+1 − Ei
Ei=
particle energy at i-th cycle
![Page 53: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/53.jpg)
Diffusive Shock AccelerationWhat happens after n cycles?
<∆E
E> =
4
3
vc
Ei+1 =
1 +
4
3
v
c
Ei
Ei+1 − Ei
Ei=
particle energy at i-th cycle
Ei+1 = β Ei
Energy increases by a (small) factor beta after
each cycle
![Page 54: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/54.jpg)
Diffusive Shock AccelerationWhat happens after n cycles?
<∆E
E> =
4
3
vc
Ei+1 =
1 +
4
3
v
c
Ei
Ei+1 − Ei
Ei=
particle energy at i-th cycle
UP DOWN
Particles can escape downstream!
Ei+1 = β Ei
Energy increases by a (small) factor beta after
each cycle
![Page 55: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/55.jpg)
Diffusive Shock AccelerationWhat happens after n cycles?
P -> probability that the particle remains within the accelerator after each cycle
after k cycles:
there are particles with energy aboveN = N0Pk E = E0β
k
![Page 56: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/56.jpg)
Diffusive Shock AccelerationWhat happens after n cycles?
P -> probability that the particle remains within the accelerator after each cycle
after k cycles:
there are particles with energy aboveN = N0Pk E = E0β
k
log
N
N0
= k logP log
E
E0
= k log β
![Page 57: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/57.jpg)
Diffusive Shock AccelerationWhat happens after n cycles?
P -> probability that the particle remains within the accelerator after each cycle
after k cycles:
there are particles with energy aboveN = N0Pk E = E0β
k
log
N
N0
= k logP log
E
E0
= k log β
log(N/N0)
log(E/E0)=
logP
log β
![Page 58: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/58.jpg)
Diffusive Shock AccelerationWhat happens after n cycles?
P -> probability that the particle remains within the accelerator after each cycle
after k cycles:
there are particles with energy aboveN = N0Pk E = E0β
k
log
N
N0
= k logP log
E
E0
= k log β
log(N/N0)
log(E/E0)=
logP
log β N(> E) = N0
E
E0
log Plog β
![Page 59: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/59.jpg)
Diffusive Shock Acceleration
N(> E) = N0
E
E0
log Plog β
n(E) ∝ E−1 + log Plog β
Integral spectrum Differential spectrum
We need to determine the value of P -> probability that the particle remains within the accelerator after each cycle
![Page 60: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/60.jpg)
Diffusive Shock AccelerationIt is easier to calculate the probability (1-P) that the particle leaves
the accelerator after each cycle
UP DOWN
![Page 61: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/61.jpg)
Diffusive Shock AccelerationIt is easier to calculate the probability (1-P) that the particle leaves
the accelerator after each cycle
UP DOWN
RinRin -> # of particles per unit time (rate) that begin a cycle
![Page 62: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/62.jpg)
Diffusive Shock AccelerationIt is easier to calculate the probability (1-P) that the particle leaves
the accelerator after each cycle
UP DOWN
Rin RoutRin -> # of particles per unit time (rate) that begin a cycle
Rout -> # of particles per unit time (rate) that leave the system
![Page 63: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/63.jpg)
Diffusive Shock AccelerationIt is easier to calculate the probability (1-P) that the particle leaves
the accelerator after each cycle
UP DOWN
Rin RoutRin -> # of particles per unit time (rate) that begin a cycle
Rout -> # of particles per unit time (rate) that leave the system
Rout
Rin= 1− P
![Page 64: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/64.jpg)
Diffusive Shock AccelerationLet’s calculate Rin...
n -> density of accelerated particles close to the shock
UP DOWN
n is isotropic: dn =n
4πdΩ
θ
velocity across the shock: c cos(θ)
![Page 65: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/65.jpg)
Diffusive Shock AccelerationLet’s calculate Rin...
n -> density of accelerated particles close to the shock
UP DOWN
n is isotropic: dn =n
4πdΩ
θ
velocity across the shock: c cos(θ)
Rin =
up→downdn c cos(θ)
![Page 66: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/66.jpg)
Diffusive Shock AccelerationLet’s calculate Rin...
n -> density of accelerated particles close to the shock
UP DOWN
n is isotropic: dn =n
4πdΩ
θ
velocity across the shock: c cos(θ)
Rin =
up→downdn c cos(θ) =
n c
4π
π2
0cos(θ) sin(θ)dθ
2π
0dψ =
1
4n c
![Page 67: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/67.jpg)
Diffusive Shock AccelerationLet’s calculate Rin...
n -> density of accelerated particles close to the shock
UP DOWN
n is isotropic: dn =n
4πdΩ
θ
velocity across the shock: c cos(θ)
Rin =
up→downdn c cos(θ) =
n c
4π
π2
0cos(θ) sin(θ)dθ
2π
0dψ =
1
4n c
...and Rout -> particles lost (advected) downstream Rout = n u2
![Page 68: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/68.jpg)
Diffusive Shock AccelerationThe probability of non returning to the shock (1-P) is:
1− P =Rout
Rin=
n u214 n c
=u1
c<< 1
most of the particles perform many cycles
![Page 69: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/69.jpg)
Diffusive Shock AccelerationThe probability of non returning to the shock (1-P) is:
1− P =Rout
Rin=
n u214 n c
=u1
c<< 1
most of the particles perform many cycles
Summarizing...
return probability -> P = 1− u1
c
energy gain per cycle-> β = 1 +4
3
u1 − u2
c= 1 +
u1
c
![Page 70: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/70.jpg)
Diffusive Shock Acceleration
n(E) ∝ E−1 + log Plog β
![Page 71: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/71.jpg)
Diffusive Shock Acceleration
n(E) ∝ E−1 + log Plog β
logP = log1− u1
c
∼ − u1
c
![Page 72: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/72.jpg)
Diffusive Shock Acceleration
n(E) ∝ E−1 + log Plog β
logP = log1− u1
c
∼ − u1
c
log β = log1 +
u1
c
∼ u1
c
![Page 73: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/73.jpg)
Diffusive Shock Acceleration
n(E) ∝ E−1 + log Plog β
logP = log1− u1
c
∼ − u1
c
log β = log1 +
u1
c
∼ u1
c
n(E) ∝ E−2
UNIVERSAL SPECTRUM
![Page 74: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/74.jpg)
Diffusive Shock AccelerationAssumptions made:
strong shock
isotropy both up and down-stream
test-particle (CR pressure negligible)
-> UNIVERSAL SPECTRUM n(E) ∝ E−2
shock velocity/Mach number
gas density/pressure
magnetic field intensity and/or structure
diffusion coefficient ...
It doesn’t depend on:
![Page 75: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/75.jpg)
Diffusive Shock AccelerationAssumptions made:
strong shock
isotropy both up and down-stream
test-particle (CR pressure negligible)
-> UNIVERSAL SPECTRUM n(E) ∝ E−2
shock velocity/Mach number
gas density/pressure
magnetic field intensity and/or structure
diffusion coefficient ...
It doesn’t depend on:
SNR shocks
turbulent B field
efficient CR acceleration
![Page 76: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/76.jpg)
Diffusive Shock Acceleration
E
E2N(E)
universal spectrum
∝ E−2
Emax
maximum energy
![Page 77: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/77.jpg)
Diffusive Shock Acceleration
E
E2N(E)
universal spectrum
∝ E−2
Emax
maximum energy
UP DOWN
ld
![Page 78: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/78.jpg)
Diffusive Shock Acceleration
E
E2N(E)
universal spectrum
∝ E−2
Emax
maximum energy
UP DOWN
ldld ≈
D td
![Page 79: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/79.jpg)
Diffusive Shock Acceleration
E
E2N(E)
universal spectrum
∝ E−2
Emax
maximum energy
UP DOWN
ld
while the particle diffuses the shock moves
ld = u1 td
ld ≈
D td
![Page 80: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/80.jpg)
Diffusive Shock Accelerationld ≈
D td
ld = u1 tdu1 td =
D td td ≈ D
u21
ld ≈ D
u1
![Page 81: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/81.jpg)
Diffusive Shock Accelerationld ≈
D td
ld = u1 tdu1 td =
D td td ≈ D
u21
ld ≈ D
u1
downstream: the same argument can be used to get the same result
remains a good order-of-magnitude estimate for the time of a cycletd
![Page 82: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/82.jpg)
Diffusive Shock Accelerationld ≈
D td
ld = u1 tdu1 td =
D td td ≈ D
u21
ld ≈ D
u1
downstream: the same argument can be used to get the same result
remains a good order-of-magnitude estimate for the time of a cycletd
D increases with E
E increases at each cycle the last cycle is the longest
![Page 83: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/83.jpg)
Diffusive Shock Accelerationld ≈
D td
ld = u1 tdu1 td =
D td td ≈ D
u21
ld ≈ D
u1
downstream: the same argument can be used to get the same result
remains a good order-of-magnitude estimate for the time of a cycletd
D increases with E
E increases at each cycle the last cycle is the longest
acceleration time
![Page 84: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/84.jpg)
Diffusive Shock AccelerationMaximum energy for the accelerated particles:
acceleration time td ≈ D
u21
![Page 85: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/85.jpg)
Diffusive Shock AccelerationMaximum energy for the accelerated particles:
acceleration time td ≈ D
u21
D = D0 Eα
= tage
Emax =
u21 tageD0
1α
![Page 86: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/86.jpg)
Diffusive Shock AccelerationMaximum energy for the accelerated particles:
acceleration time td ≈ D
u21
D = D0 Eα
= tage
Emax =
u21 tageD0
1α
The maximum energy: increases with time
depends on: age, shock speed, magnetic field
intensity and structure (through D), ...
it is NOT universal!
![Page 87: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/87.jpg)
Diffusive Shock Acceleration: weak shocksHomework: what happens if the shock is NOT strong?
n(E) ∝ E−α α =r + 2
r − 1Solution:( )
Examples: r = 4 --> alfa = 2
r < 4 --> alfa >2
r = 3 --> alfa = 3
Acceleration is less efficient at weak shocks
![Page 88: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/88.jpg)
x
u u0
low energy CRs high energy CRs
shock acceleration isintrinsically efficient cosmic ray
pressure is slowing down the upstream flow formation of a precursor
CR pr
essu
re
Non-linear Diffusive Shock AccelerationNon-linear DSA: what happens if the acceleration efficiency is high (~1)?
![Page 89: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/89.jpg)
x
u u0
low energy CRs high energy CRs
High energy CRsR > 4
slope < 4
Low energy CRsR < 4
slope > 4
Non-linear Diffusive Shock AccelerationNon-linear DSA: what happens if the acceleration efficiency is high (~1)?
![Page 90: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/90.jpg)
Diffusive Shock Acceleration at SuperNova Remnants and the origin of
Galactic Cosmic Rays
(1) Spallation measurements of Cosmic Rays suggest that CR sources have to inject in the Galaxy a
spectrum close to E-2. (2) Strong shocks at SNRs can indeed accelerate
E-2 spectra.-> Thus SNRs are good candidates as sources of
Galactic CRs.
![Page 91: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/91.jpg)
Diffusive Shock Acceleration at SuperNova Remnants and the origin of
Galactic Cosmic Rays
(1) Spallation measurements of Cosmic Rays suggest that CR sources have to inject in the Galaxy a
spectrum close to E-2. (2) Strong shocks at SNRs can indeed accelerate
E-2 spectra.-> Thus SNRs are good candidates as sources of
Galactic CRs.
![Page 92: The theory of Diffusive Shock Acceleration - Dto. de Fisicajarne/clases/Stefano_Gabici_three.pdf · Diffusive Shock Acceleration Symmetry u 1 − u 2 u 1 − u 2 Every time the particle](https://reader035.vdocuments.pub/reader035/viewer/2022071407/60fee20f792160638c34b9e7/html5/thumbnails/92.jpg)
Diffusive Shock Acceleration at SuperNova Remnants and the origin of
Galactic Cosmic Rays
(1) Spallation measurements of Cosmic Rays suggest that CR sources have to inject in the Galaxy a
spectrum close to E-2. (2) Strong shocks at SNRs can indeed accelerate
E-2 spectra.-> Thus SNRs are good candidates as sources of
Galactic CRs.E-2 is the spectrum at the shock, not the one released in the ISM!