![Page 1: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/1.jpg)
1
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Diffusive shock acceleration:a first order Fermi process
![Page 2: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/2.jpg)
2
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Shock waves
■ Discontinuity in physical parameters
shock front
n1, p1, T1
v1
n2, p2, T2
v2
(immaterial surface)upstream mediumdownstream medium
■ What can we tell from the purely macroscopic point of view?
![Page 3: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/3.jpg)
3
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Shock waves “jump conditions”
■ Conservationequations
shock front
n1, p1, T1
v1
n2, p2, T2
v2
(immaterial surface)upstream mediumdownstream medium
mass
momentumenergy
adiabatic index
![Page 4: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/4.jpg)
4
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Shock waves “jump conditions”■ Solve macroscopic conservation equations
shock front
n1, p1, T1
v1
n2, p2, T2
v2
(immaterial surface)upstream mediumdownstream medium
![Page 5: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/5.jpg)
5
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
■ Solution:
γ + 1
γ + 1/M12 ± (1 - 1/M1
2)v2 =v1
M1 = v1/c1
■ Trivial solution: v2 = v1 !
γ + 1γ - 1 + 2/M1
2=
v2
v1=
n1
n2 γ + 12γM1
2 - (γ - 1)=p2
p1
■ Shock wave solution:
■ NB: M1 can be either > 1 or < 1, but[ M1
2 - (γ - 1)/2γ ] × [ M22 - (γ - 1)/2γ ] = ((γ + 1)/2γ)
2
M1 > 1 ⇔ M2 < 1 But entropy must increase!
![Page 6: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/6.jpg)
6
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Astrophysical shocks■ Supernovae eject supersonic material
SN 1006 Tycho Kepler
< 35 pc
SNR
1-3 105 yr
~ 1051 erg
■ + gamma-ray bursts(relativistic fireballs)
■ Stellar mass blackholes emit plasma blobs
![Page 7: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/7.jpg)
7
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)3C 219
Astrophysical shocks■ Active galactic nuclei produce jets
with internal shocks and hugeshocks at the end (hot spots)
![Page 8: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/8.jpg)
8
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
■ SN shocks: V ~ 10 000 km/s ; E ~ 2 MeV/proton
■ Stopping length ~ 1 kpc !
■ But: interaction with B and E fields◆ RL = p/qB ~ 10-8 pc◆ Streaming with v > cA “impossible”.
What is a collisionless shock?!
■ In the interstellar or intergalactic medium, allthe shocks are “collisionless”!
![Page 9: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/9.jpg)
9
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
“Acceleration by change-of-frame”
Magnetic cloud, MHD wave, etc.:anything that is moving are carryingB fields able to scatter particles…
![Page 10: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/10.jpg)
10
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Second order, stochasticFermi acceleration
V BB
B
■ The energy gain is only due to the difference inboth collision rates, which is ∝V/c
■ More head-on collisions than overtaking collisions
■ Energy change at each “collision”: ∝ V/c
■ Resulting average energy gain: ∝ (V/c)2
![Page 11: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/11.jpg)
11
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Diffusive shock acceleration■ Shock wave (e.g. supernova explosion)
■ Magnetic wave production◆ Downstream : by the shock (compression, turbulence, hydro
and MHD-instabilities, shear flows, etc.)◆ Upstream : by the cosmic rays themselves !
■ → ‘isotropisation’ of the distribution (in local rest frame)
Vshock
Interstellar mediumShocked medium
![Page 12: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/12.jpg)
12
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
You’re always lucky!
■ At each crossing, the particle sees a‘magnetic wall’ at V = (1-1/r)
■ → only head-on collisions…
VshockVshock/ r
Interstellar mediumShocked medium
![Page 13: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/13.jpg)
13
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Shock acceleration■ 3 points of view...
Shock rest frame
n1, p1, T1
v1
n2, p2, T2
v2
Upstreamrest frame
v1 - v2
Downstreamrest frame
v1 - v2v2
As seen from upstream, thedownstream medium is coming closer
v1
As seen from downstream, theupstream medium is coming closer
head-on “collisions”at V = v1-v2⇒ energy gain
![Page 14: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/14.jpg)
14
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
New version of the same calculation
■ Magnetic cloud replacedby the plasma on the otherside of the shock front
Upstreamrest frame
v1 - v2
v1
![Page 15: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/15.jpg)
15
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
■ Angular distribution at the crossing of the shockUpstreamrest frame
v1 - v2
v1
relative velocity
■ Average crossing angles
and
(isotropy on either side of the shock front)
![Page 16: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/16.jpg)
16
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Average energy gain
![Page 17: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/17.jpg)
17
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Shock acceleration:
■ Acceleration process
■ Fermi process
■ First order process
■ Acceleration cycles
![Page 18: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/18.jpg)
18
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Shock acceleration cycles
■ Cf. stochastic, second order Fermi accelerationprocess: exponential growth of the energy
■ After n cycles
■ Acceleration rate:with τacc independent of E
■ Escape rate independent of E ⇒
![Page 19: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/19.jpg)
19
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Acceleration rate
■ Time to complete one cycle:◆ Confinement distance: κ/u◆ Average time spent upstream: t1 ≈ 4κ / cu1◆ Average time spent upstream: t2 ≈ 4κ / cu2
u2 u1
κ2/u2 κ1/u1
upstreamdownstream
diffusion coefficient
![Page 20: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/20.jpg)
20
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Acceleration rate
■ Finally, τacc ~ tcycle × Vs/c ~ 1 month !
■ Bohm limit: κ = rgv/3 ~ Eβ2/3qB◆ Proton at 10 GeV: κ ~ 1022 cm2/s◆ ⇒ tcycle ~ 104 seconds !
u2 u1
κ2/u2 κ1/u1
upstreamdownstream
![Page 21: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/21.jpg)
21
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Achtung !
u2 u1
κ2/u2 κ1/u1
upstreamdownstream
with
and
The acceleration timescale depends on E
But this was required to get a power-law spectrum (cf. Fermi miracle!)
![Page 22: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/22.jpg)
22
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Second ingredient: escape timescale
■ Flux of particles crossing fromupstream to downstream
Vshock
Vshock/ r
■ Flux of particles escapingdownstream
■ Escape probability
■ Escape timescale also depends on E!
![Page 23: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/23.jpg)
23
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
A power-law spectrum anyway!
■ Both τacc and τesc depend on E, but τacc/τesc does not!
■ So let’s forget about time, and think in terms of cycles…◆ Both the relative energy gain and the escape probability percycle are independent of E: this is enough!
where
![Page 24: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/24.jpg)
24
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Resulting energy spectrum
■ Return probability:
■ Energy after n cycles
■ Remaining number of particles after n cycles:
![Page 25: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/25.jpg)
25
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Resulting energy spectrum
■ Return probability:
■ Energy after n cycles
■ Remaining number of particles after n cycles:
![Page 26: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/26.jpg)
26
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Universal power-law index■ One obtains:
■ For a non-relativistic shock◆ Pesc << 1◆ ΔE/E << 1
with
■ … where ‘r’ is the shock compression ratio◆ r = γ+1/γ-1 for a strong shock
■ For a monoatomic or fully ionised gas, γ=5/3
x = 2, compatible with observations
![Page 27: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/27.jpg)
27
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Maximum energy achievable?■ Energy losses
◆ Friction with ambient medium◆ Synchrotron, IC◆ Pair production◆ photo-π
■ Escape◆ Keep the particle in
the accelerator◆ Diffusion, gyroradius
■ Destruction◆ Photo-disintegration
■ Age of theplayers!
![Page 28: Diffusive shock acceleration: a first order Fermi processMagnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v ... Cf. stochastic, second](https://reader033.vdocuments.us/reader033/viewer/2022060714/607b0bda739a82044c557998/html5/thumbnails/28.jpg)
28
jan.-fév. 2007 — NPAC , rayons cosmiques — E. Parizot (APC)
Fermi 1 vs Fermi 2
■ Acceleration timescale
■ Second order (stochastic) Fermi acceleration
with
■ First order (shock) Fermi acceleration
and
so
withand
so