Diffusive shock acceleration: an introduction – cont.

Diffusive shock acceleration I order acceleration where u = u1-u2 in the shock rest frame Compressive discontinuity of the plasma flow leads to acceleration of particles reflecting at both sides of the discontinuity: diffusive shock acceleration (I-st order Fermi)

INDEPENDENT ON ASSUMPTIONS ABOUT LOCAL CONDITIONS the phase-space Distribution of shock accelerated particles particles injected at the shock background particles advected from - INDEPENDENT ON ASSUMPTIONS ABOUT LOCAL CONDITIONS NEAR THE SHOCK

test particle non-relativistic Spectral index depends ONLY on the shock compression adiabatic index shock Mach number For a strong shock (M>>1): R = 4 and α = 4.0 σ = 2.0 Spectral shape nearly parameter free, with the index  very close to the values observed or anticipated in real sources. Diffusive shock acceleration theory in its simplest test particle non-relativistic version became a basis of most studies considering energetic particle populations in astrophysical sources.

Acceleration time scale at parallel shock for returning particles For a „cycle”: shock Minimum of tacc: Bohm

A few numbers for a (SNR-like) shock wave B ~ 10 μG , λ ~ rg , u = 1000 km/s (=108 cm/s) tSNR ~ 104 yr For a particle energy E = 1 MeV electron (rg ~ 108 cm , v ~ 1010 cm/s) tacc ~ 102 s proton (rg ~ 1011 cm , v ~ 109 cm/s) tacc ~ 104 s ~ 0.1 day E = 1 GeV rg ~ 1012 cm , v ~ 1010 cm/s tacc ~ 106 s ~ 0.1 AU ~ 1 month E = 1 PeV (= 1015 eV) rg ~ 1018 cm , v ~ 1010 cm/s tacc ~ 1012 s ~ 1 pc ~ 105 yr E= 1 EeV (=1018 eV) rg ~ 1021 cm , v ~ 1010 cm/s tacc ~ 1015 s ~ 1 kpc ~ 108 yr

perpendicular oblique parallel

Oblique magnetic fields Ψ≠0 reflection B2 > B1 transmission B1 shock For uB,1 << v the spectral index is the same as at parallel shocks ! However tacc can be substantially modified

Ψ1

The absolute minimum acceleration time scale (outside the diffusive approximation) „shock drift acceleration” at quasi-perpendicular shock waves with Ψ ≈ 90° Cross-field diffusion is required to form a power-law spectrum !

for Ew – energy density of Alfvén waves with k ~ 2π/rg(p) per log p Non-linear modifications of the acceleration process A. Self-induced scattering (Bell 1978) Wave generation due to streaming instability upstream of the shock for Ew – energy density of Alfvén waves with k ~ 2π/rg(p) per log p damping coefficient CR density growth rate V (decaying) (growing)

Streaming instability Accelerated particles generate waves upstream of the shock Wave generation leads to decreasing the particle diffusion coefficient thus the acceleration time scale diminishes leading to more rapid acceleration leading to more efficient wave generation etc., until the process saturates at B ~ B

(two fluid approximation: g + CR) B. Modification of the shock structure by CR precursor (two fluid approximation: g + CR) is included into the Euler equation: and the resulting velocity profile u(x) into CR kinetic equation Possible efficient acceleration: in this simple two fluid model up to 98% of the shock kinetic energy can be converted into CRs !

Simplification: hydrodynamical approach where is an effective spatial diffusion coefficient. Then, with PC=(C-1)EC and PG=(G-1)EG the stationary solution can be given as a set of first integrals:

PG U The resulting non-linear shock solutions can be explained in the „phase space” (PG , U) : PG U

M = 2 Pg u From Drury & Völk 1981 – weak shock (two fluid model) Pg precursor subshock Pcr Velocity profile

CR spectrum in the modified shock log n(E) subshock acceleration full shock transition log E

M = 13 Pg u Efficient acceleration in a strong shock (two fluid model) Pcr

c. Three fluid model gas + CRs + waves wave damping heats gas wave distribution defines  + nonlinearities of the two-fluid model All the above processes are described in approximate way, thus the considered class of multi-fluid models can be analysed only qualitatively.

Conclusions from non-linear computations: CRs can produce perturbations required for efficient acceleration possible efficient acceleration at high Mach shocks spectrum flattening at high CR energies a value of the upper energy cut-off important for shock modification (divergent energy spectra at high energies) - test particle spectra are only approximations for real shocks

The second-order Fermi acceleration in turbulent MHD medium V Diffusion in the momentum space second-order Scattering centres  MHD waves V ~ 10 km/s (interstellar medium, solar wind) V ~ 102-3 km/s (solar flares, large scale extragalactic jets)

If particle distribution function f is so slowly varying in the space that we can neglect /x terms, the transport equation takes the form: The momentum diffusion coefficient for ultrarelativistic particles interacting with isotropically propagating scaterers A typically considered power-law relation (p)  p leads to Dp p 2-

The shortest mean free path occurs in the „Bohm limit”: (p) = rg  pc/eB (this relation yields =1) . One should note that for the isotropic diffusion the spatial diffusion coefficient  = (1/3) c (p) and there exists a useful relation  Dp = V2 p2 /9

Acceleration time scale The escape time scale (the diffusion time scale to reach the system boundary at L) 2L

Comparison of tacc and tesc for 1 GeV particle in a few astrophysical scenarios (for these evaluations we assume the Bohm limit with  = rg ) Interstellar medium of our Galaxy: V=10 km/s, L=1 kpc, B=3 G, rg ~ 1013 cm tacc ~ 3(c/V) rg/c ~ 104 yr tesc ~ 3 (L/c) (L/rg) ~ 3 1012 yr but in real conditions  >> rg and spatial diffusion is highly anisotropic. A role of the CRs second-order acceleration is still disputed in the literature. b. Shocked plasma in the supernova remnant: V=10 km/s, L=0.01 pc, B=30 G, rg ~1012 cm tacc ~ 103 yr tesc ~ 3 103 yr so the second-order acceleration can be important. c. Large scale astrophysical jet: V=1000 km/s, L=Rj=1 kpc, B=300 G, rg ~ 1011 cm tacc ~ 3 103 yr tesc ~ 3 1014 yr but for electrons the radiative losses time scale can be also << tesc and the jet life-time ~107 yr

Diffusive shock acceleration in the presence of the second order Fermi acceleration It is a difficult mathematical and physical problem even for stationary conditions analysed in the linear approximation: a particle momentum distribution varies in space both upstream and downstream of the shock the spatial diffusion is coupled to the momentum diffusion through the diffusive terms with  = (x,p) , Dp = Dp(x,p) problems with background and boundary conditions, Below we consider the particle spectrum at the shock in the conditions allowing for the power law solutions.

u2 u1 II order acceleration I order acceleration where V – Alfvén velocity I order acceleration in the shock rest frame where u = u1-u2

I and II order acceleration at parallel shocks (with isotropic alfvénic turbulence) plasma beta (  Pg/PB ) Alfvén velocity (Ostrowski & Schlickeiser 1993)

Our knowledge of acceleration processes acting at non-relativistic shocks is still very limited. There are basic problems with energetic particle injection processes (electrons !) existence of stationary solutions for efficient shock acceleration description of processes forming or reprocessing MHD turbulence near the shock the time dependent solutions the upper energy cut-offs, when compared with measurements CR electron spectral indices observed in objects like SNRs etc.

Problems to be solved are usually difficult, often being highly non-linear and/or 3D and/or non-stationary. Progress in studies of the diffusive shock acceleration is very slow since an initial rapid theory development in late seventies and early eighties of last century.