Diffusive shock acceleration: a first order Fermi process. jan.-fév NPAC, rayons cosmiques E. Parizot (APC)

Transcription

1 1 Diffusive shock acceleration: a first order Fermi process

2 2 Shock waves Discontinuity in physical parameters shock front n 2, p 2, T 2 n 1, p 1, T 1 v 2 v 1 downstream medium (immaterial surface) upstream medium What can we tell from the purely macroscopic point of view?

3 3 Shock waves jump conditions Conservation equations n 2, p 2, T 2 shock front n 1, p 1, T 1 v 2 v 1 downstream medium (immaterial surface) upstream medium mass momentum energy adiabatic index

4 4 Shock waves jump conditions Solve macroscopic conservation equations shock front n 2, p 2, T 2 n 1, p 1, T 1 v 2 v 1 downstream medium (immaterial surface) upstream medium

5 5 Solution: M 1 = v 1 /c 1 v 2 = v 1 γ + 1/M 1 2 ± (1-1/M 12 ) γ + 1 Trivial solution: v 2 = v 1! Shock wave solution: v 2 n 1 γ /M 2 1 2γM 2 = = 1 - (γ - 1) = v 1 n 2 γ + 1 p 1 γ + 1 NB: M 1 can be either > 1 or < 1, but [ M (γ - 1)/2γ ] [ M (γ - 1)/2γ ] = ((γ + 1)/2γ) 2 p 2 M 1 > 1 M 2 < 1 But entropy must increase!

6 6 Astrophysical shocks Supernovae eject supersonic material S N R < 35 pc yr ~ erg + gamma-ray bursts (relativistic fireballs) Stellar mass black holes emit plasma blobs SN 1006 Tycho Kepler

7 7 Astrophysical shocks Active galactic nuclei produce jets with internal shocks and huge shocks at the end (hot spots) 3C 219

8 8 In the interstellar or intergalactic medium, all the shocks are collisionless! What is a collisionless shock?! SN shocks: V ~ km/s ; E ~ 2 MeV/proton Stopping length ~ 1 kpc! But: interaction with B and E fields R L = p/qb ~ 10-8 pc Streaming with v > c A impossible.

9 9 Acceleration by change-of-frame Magnetic cloud, MHD wave, etc.: anything that is moving are carrying B fields able to scatter particles

10 Second order, stochastic Fermi acceleration 10 More head-on collisions than overtaking collisions V B B The energy gain is only due to the difference in both collision rates, which is V/c Energy change at each collision : V/c Resulting average energy gain: (V/c) 2 B

11 11 Diffusive shock acceleration Shock wave (e.g. supernova explosion) Shocked medium Interstellar medium V shock 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)

12 12 You re always lucky! Shocked medium V shock / r Interstellar medium V shock At each crossing, the particle sees a magnetic wall at V = (1-1/r) only head-on collisions

13 13 Shock acceleration 3 points of view... n 2, p 2, T 2 Shock rest frame n 1, p 1, T 1 v 2 v 1 v 1 - v 2 Upstream rest frame v 1 head-on collisions at V = v 1 -v 2 energy gain Downstream rest frame v 2 v 1 - v 2 As seen from upstream, the downstream medium is coming closer As seen from downstream, the upstream medium is coming closer

14 14 New version of the same calculation Magnetic cloud replaced by the plasma on the other side of the shock front Upstream rest frame v 1 - v 2 v 1

15 15 Angular distribution at the crossing of the shock (isotropy on either side of the shock front) v 1 - v 2 Upstream rest frame v 1 relative velocity Average crossing angles and

16 16 Average energy gain

17 17 Shock acceleration: Acceleration process Fermi process Acceleration cycles First order process

18 18 Shock acceleration cycles After n cycles Cf. stochastic, second order Fermi acceleration process: exponential growth of the energy Acceleration rate: with τ acc independent of E Escape rate independent of E

19 19 Acceleration rate downstream u 2 u 1 upstream κ 2 /u 2 κ 1 /u 1 Time to complete one cycle: Confinement distance: κ/u diffusion coefficient Average time spent upstream: t 1 4κ / cu 1 Average time spent upstream: t 2 4κ / cu 2

20 20 Acceleration rate downstream u 2 u 1 upstream κ 2 /u 2 κ 1 /u 1 Bohm limit: κ = r g v/3 ~ Eβ 2 /3qB Proton at 10 GeV: κ ~ cm 2 /s t cycle ~ 10 4 seconds! Finally, τ acc ~ t cycle V s /c ~ 1 month!

21 21 Achtung! downstream u 2 u 1 upstream κ 2 /u 2 κ 1 /u 1 with and The acceleration timescale depends on E But this was required to get a power-law spectrum (cf. Fermi miracle!)

22 Second ingredient: escape timescale 22 V shock / r V shoc k Flux of particles crossing from upstream to downstream Flux of particles escaping downstream Escape probability Escape timescale also depends on E!

23 23 A power-law spectrum anyway! Both τ acc and τ esc depend on E, but τ acc /τ esc does not! So let s forget about time, and think in terms of cycles Both the relative energy gain and the escape probability per cycle are independent of E: this is enough! where

24 24 Resulting energy spectrum Return probability: Remaining number of particles after n cycles: Energy after n cycles

25 25 Resulting energy spectrum Return probability: Remaining number of particles after n cycles: Energy after n cycles

26 Universal power-law index 26 One obtains: with For a non-relativistic shock P esc << 1 ΔE/E << 1 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

27 Maximum energy achievable? 27 Energy losses Friction with ambient medium Synchrotron, IC Pair production photo-π Age of the players! Destruction Photo-disintegration Escape Keep the particle in the accelerator Diffusion, gyroradius

28 28 Fermi 1 vs Fermi 2 Acceleration timescale Second order (stochastic) Fermi acceleration and with so First order (shock) Fermi acceleration and with so