DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

Rρ = (β∂S/∂z)/(α∂/∂z) >1 (and stability
N 2 > 0).
diffusive shock acceleration Acceleration
due to repeated reflection of particles in the plasmas
converging at the shock front, also called
Fermi-acceleration. Diffusive shock acceleration
is the dominant acceleration mechanism at
quasi-parallel shocks because here the electric
induction field in the shock front is small, and
therefore shock drift acceleration is inefficient.
In diffusive shock acceleration, the scattering
on both sides of the shock front is the crucial
process. This scattering occurs at scatter centers
frozen-in into the plasma, thus particle scattering
back and forth across the shock can be
understood as repeated reflection between converging
scattering centers (first order Fermi acceleration).
Particle trajectory in diffusive shock acceleration.
With f being the phase space density, U the
plasma bulk speed, D the diffusion tensor, p
the particle momentum, and T a loss time, the
transport equation for diffusive shock acceleration
can be written as
∂f
∂t
∇U ∂f f
+ U∇f −∇(D∇f)− p +
3 ∂p T
+ 1
p 2
∂
p
∂p
2
dp
f = Q(r,p,t)
dt
with Q(r,p,t) describing an injection into the
acceleration process. The terms from left to
right give the convection of particles with the
plasma flow, spatial diffusion, diffusion in momentum
space (acceleration), losses due to particle
escape from the acceleration site, and convection
in momentum space due to processes
that affect all particles, such as ionization or
Coulomb losses.
© 2001 by CRC Press LLC
diffusive shock acceleration
In a first-order approximation, the last two
terms on the right-hand side (losses from the
acceleration site and convection in momentum
space) can be neglected. In addition, if we limit
ourselves to steady state, some predictions can
be made from this equation:
1. Characteristic acceleration time. With the
indices u and d denoting the properties of the
upstream and downstream medium, the time required
to accelerate particles from momentum
po to p can be written as
t =
3
p
uu − ud
po
dp
p ·
D u
+
u u
Dd
.
u d
Here D denotes the diffusion coefficient. Alternatively,
a characteristic acceleration time τ a
can be given as
τ a = 3r D u
r − 1 u2 u
with r = u u/ud being the ratio of the flow
speeds in the shock rest frame. For a parallel
shock, r equals the compression ratio. τ a then
gives the time the shock needs to increase the
particle momentum by a factor of e. Note that
here the properties of the downstream medium
have been neglected: It is tacitly assumed that
the passage of the shock has created so much turbulence
in the downstream medium that scattering
is very strong and therefore the term D d/ud
is small compared to the term D u/u u.
2. Energy spectrum. In steady state, diffusive
shock acceleration leads to a power law spectrum
in energy J(E) = Jo · E −γ . Here the
spectral index γ depends on the ratio r = u u/ud
of the flow speeds only:
γ = 1 r + 2
2 r − 1
in the non-relativistic case, or γ rel = 2γ in the
relativistic case.
3. Intensity increase upstream of the shock.
The spatial variation of the intensity around the
shock front can be described as
f(x,p)= f(x,0) exp{−β|x|}
with β = u u/D u. Ifβ is spatially constant, an
exponential intensity increase towards the shock