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arXiv:astro-ph/0606592v1 23 Jun 2006
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 5 February 2008 (MN LATEX style file v2.2)
Non linear particle acceleration at non-relativistic
shock waves in the presence of self-generated
turbulence
E. Amato1⋆and P. Blasi1†
1INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi, 5, 50125, Firenze, Italy
Accepted —-. Received —–
ABSTRACT
Particle acceleration at astrophysical shocks may be very efficient if magnetic
scattering is self-generated by the same particles. This nonlinear process adds
to the nonlinear modification of the shock due to the dynamical reaction of
the accelerated particles on the shock. Building on a previous general solution
of the problem of particle acceleration with arbitrary diffusion coefficients
(Amato & Blasi (2005)), we present here the first semi-analytical calculation
of particle acceleration with both effects taken into account at the same time:
charged particles are accelerated in the background of Alfv´ en waves that they
generate due to the streaming instability, and modify the dynamics of the
plasma in the shock vicinity.
Key words: acceleration of particles - shock waves
1 INTRODUCTION
Soon after the pioneering papers by Krymskii (1977); Blandford & Ostriker (1978); Bell
(1978a,b), introducing the test particle theory of particle acceleration at collisionless shocks,
it became clear that the dynamical reaction of the accelerated particles on the plasmas in-
volved in the shock formation may not be negligible. It is now clear that such reaction may
in fact make shocks efficient accelerators and change quite drastically the predictions of the
⋆E-mail: amato@arcetri.astro.it
† E-mail: blasi@arcetri.astro.it
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E. Amato and P. Blasi
test particle theory. The main consequences of the shock modification induced by the accel-
erated particles can be summarized as follows: 1) a precursor, consisting in a gradual braking
of the upstream fluid, is created; 2) particles with different momenta feel different effective
compression factors, which reflects in the fact that the spectrum of accelerated particles is
no longer a power law, but rather a concave spectrum, as hard as p−3.2at high momenta; 3)
the shock becomes less efficient in heating the background plasma, so that the temperature
of the downstream gas is expected to be lower than predicted through the usual Rankine-
Hugoniot relations at an unmodified shock front (see Drury (1983); Blandford & Eichler
(1987); Jones & Ellison (1991); Malkov & Drury (2001) for reviews on different aspects of
the subject). The reaction of the accelerated particles has been calculated within differ-
ent approaches: the so-called two-fluid models (Drury & V¨ olk (1980, 1981)), kinetic mod-
els (Malkov (1997); Malkov, Diamond & V¨ olk (2000); Blasi (2002, 2004)) and numerical
approaches, both Monte Carlo and other simulation procedures (Jones & Ellison (1991);
Bell (1987); Ellison, M¨ obius & Paschmann (1990); Ellison, Baring & Jones (1995, 1996);
Kang & Jones (1997, 2005); Kang, Jones & Gieseler (2002)). In most of these calculations,
the diffusion properties of the plasma upstream and downstream are provided as an input to
the problem. This also results in fixing the value of the maximum momentum of the accel-
erated particles. However, one of the well known and most disturbing problems associated
with the mechanism of particle acceleration at shock fronts is that a substantial amount
of magnetic scattering of the particles is required (e.g. Lagage & Cesarsky (1983a,b)). In
the absence of it, the maximum energy of the accelerated particles is exceedingly low and
uninteresting for astrophysical applications (e.g. Blasi (2005)). Bell (1978a) proposed that
the streaming instability of cosmic rays could be responsible for the generation of pertur-
bations in the magnetic field of an amplitude necessary to provide pitch angle scattering
(and therefore spatial diffusion) of the accelerated particles. Lagage & Cesarsky (1983b)
used this argument to estimate the maximum energy of particles accelerated at shocks in
supernova remnants. In all previous works either the shock was considered unmodified, or
the diffusion coefficients were fixed a priori, because a comprehensive theory of particle ac-
celeration was missing. Recently, Amato & Blasi (2005) have found a general exact solution
of the system of equations describing the diffusion-convection of accelerated particles, and
the dynamics and thermodynamics of plasmas in the shock region, for an arbitrary choice of
the spatial and momentum dependence of the diffusion coefficient. In the present paper we
use the formalism proposed by Amato & Blasi (2005) and combine it with calculations of
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Non linear particle acceleration at shock waves
3
the perturbations created through streaming instability, so that the diffusion coefficient, as
a function of spatial location and momentum, is determined from the spectrum and spatial
distribution of the accelerated particles. This provides the first combined description of the
process of particle acceleration at collisionless shocks in the presence of particle reaction and
wave generation. In this approach, the spectrum of accelerated particles, their distribution
in the upstream plasma and the diffusion coefficient are outputs of the problem.
The paper is organized as follows: in Sec. 2 we summarize the findings of Amato & Blasi
(2005). In Sec. 3 we illustrate our treatment of the streaming instability and determine a
relation between the power spectrum of magnetic fluctuations and the diffusion coefficient
upstream. In Sec. 4 we describe the results of our calculations. In Sec. 5 we shortly discuss
how the results presented in the previous section change when the effects of turbulent heating
are taken into account. We conclude in Sec. 6.
2 CALCULATIONS OF THE SPECTRUM FOR ARBITRARY DIFFUSION
COEFFICIENT
In this section we briefly summarize the mathematical procedure proposed by Amato & Blasi
(2005) to calculate the spectrum and spatial distribution of particles accelerated at astro-
physical shocks, and their dynamical reaction on the shock structure, for an arbitrary dif-
fusion coefficient D(x,p). The reader is referred to the paper by Amato & Blasi (2005) for
more details.
The equation for the conservation of the momentum between upstream infinity and a
point x in the upstream region can be written as:
ξc(x) = 1 +
1
γgM2
0
− U(x) −
1
γgM2
0
U(x)−γg, (1)
where ξc(x) = PCR(x)/ρ0u2
0and U(x) = u(x)/u0and we used conservation of mass ρ0u0=
ρ(x)u(x) (here ρ0and u0refer to the density and plasma velocity at upstream infinity, while
ρ(x) and u(x) are the density and velocity at the location x upstream. M0is the sonic Mach
number at upstream infinity).
The pressure in the form of accelerated particles is defined as
PCR(x) =1
3
?pmax
pinj
dp 4πp3v(p)f(x,p), (2)
and f(x,p) is the distribution function of accelerated particles. Here pinjand pmaxare the
injection and maximum momentum. The function f vanishes at upstream infinity, which
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E. Amato and P. Blasi
implies that there are no cosmic rays infinitely distant from the shock in the upstream
region1. The distribution function satisfies the following transport equation in the reference
frame of the shock:
∂
∂x
?
D(x,p)∂
∂xf(x,p)
?
− u∂f(x,p)
∂x
+1
3
?du
dx
?
p∂f(x,p)
∂p
+ Q(x,p) = 0. (3)
2
The x axis is oriented from upstream infinity (x = −∞) to downstream infinity (x =
+∞), with the shock located at x = 0. The injection is introduced here through the function
Q(x,p). The diffusion properties are described by the arbitrary function D(x,p), depending
on both momentum and space3.
Amato & Blasi (2005) showed that an excellent approximation to the solution f(x,p)
has the form
f(x,p) = f0(p)exp
?
−q(p)
3
?0
xdx′u(x′)
D(x′,p)
?
, (4)
where f0(p) = f(x = 0,p) is the cosmic rays’ distribution function at the shock and q(p) =
−dlnf0(p)
The function f0(p) can be written in a very general way as found by Blasi (2002):
dlnp
is its local slope in momentum space.
f0(p) =
?
3Rtot
RtotUp(p) − 1
?
ηn0
4πp3
inj
exp
?
−
?p
pinj
dp′
p′
3RtotUp(p′)
RtotUp(p′) − 1
?
. (5)
Here we introduced the function Up(p) = up/u0, with
up= u1−
1
f0(p)
?0
−∞dx(du/dx)f(x,p) , (6)
where u1 is the fluid velocity immediately upstream (at x = 0−). We used Q(x,p) =
ηngas,1u1
4πp2
(x = 0−) and η the fraction of the particles crossing the shock which are going to take part
injδ(p − pinj)δ(x), with ngas,1 = n0Rtot/Rsub the gas density immediately upstream
in the acceleration process. In the expressions above we also introduced the two quantities
Rsub= u1/u2(compression factor at the subshock) and Rtot= u0/u2(total compression fac-
1This assumption implies that we are not considering any reacceleration of pre-existing seed particles.
2Since we will be using this equation in Sect. 4 for the case in which diffusion is due to a strongly amplified turbulent magnetic
field, a few comments are in order: rigorously, this equation describes the isotropic part of the distribution function, and as
long as the quasi-linear theory holds, the anisotropic part is expected to represent a small perturbation. It is not clear how the
equation would generalize to the strongly non-linear case, though it may be reasonable to assume that the anisotropy remains
rather small as long as the Alfven speed in the perturbed field is negligible compared with the fluid speed.
3In writing Eq. 3 in this form we are neglecting the velocity of the scattering centers uw with respect to the fluid velocity
upstream. This is always a good approximation for the cases considered in this paper.
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Non linear particle acceleration at shock waves
5
tor). If the heating of the upstream plasma takes place only due to adiabatic compression,
the two compression factors are related through the following expression (Blasi (2002)):
Rtot= M
2
γg+1
0
?(γg+ 1)Rγg
sub− (γg− 1)Rγg+1
2
sub
?
1
γg+1
, (7)
where M0is the Mach number of the fluid at upstream infinity and γgis the ratio of specific
heats for the fluid. The parameter η in Eq. 5 contains the very important information about
the injection of particles from the thermal bath. We adopt here the recipe proposed by
Blasi, Gabici & Vannoni (2005) that allows us to relate η to the compression factor at the
subshock as:
η =
4
3π1/2(Rsub− 1)ξ3e−ξ2.
Here ξ is a parameter that identifies the injection momentum as a multiple of the momentum
(8)
of the thermal particles in the downstream section (pinj= ξpth,2). The latter is an output of
the non linear calculation, since we solve exactly the modified Rankine-Hugoniot relations
together with the cosmic rays’ transport equation. For the numerical calculations that follow
we always use ξ = 3.5, that corresponds to a fraction of order 10−4of the particles crossing
the shock to be injected in the accelerator.
In terms of the distribution function (Eq. 4), we can also write the normalized pressure
in accelerated particles as:
ξc(x) =
4π
3ρ0u2
0
?pmax
pinj
dp p3v(p)f0(p)exp
?
−
?0
xdx′U(x′)
xp(x′,p)
?
, (9)
where for simplicity we introduced xp(x,p) =3D(p,x)
q(p)u0.
By differentiating Eq. 9 with respect to x we obtain
dξc
dx= λ(x)ξc(x)U(x), (10)
where
λ(x) =< 1/xp>ξc=
?pmax
pinjdp p3
?pmax
1
xp(x,p)v(p)f0(p)exp
?
−?0
xdx′U(x′)
xdx′ U(x′)
xp(x′,p)
?
pinjdp p3v(p)f0(p)exp
?
−?0
xp(x′,p)
?
, (11)
and U(x) is expressed as a function of ξc(x) through Eq. 1.
Finally, after integration by parts of Eq. 6, one is able to express Up(p) in terms of an
integration involving U(x) alone:
Up(p) =
?0
−∞dx U(x)2
1
xp(x,p)exp
?
−
?0
xdx′U(x′)
xp(x′,p)
?
,(12)
which allows one to easily calculate f0(p) through Eq. 5.
Eqs. 1 and 10 can be solved by iteration in the following way: for a fixed value of the
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E. Amato and P. Blasi
compression factor at the subshock, Rsub, the value of the dimensionless velocity at the shock
is calculated as U(0) = Rsub/Rtot. The corresponding pressure in the form of accelerated
particles is given by Eq. 1 as ξc(0) = 1 +
1
γgM2
0−Rsub
Rtot−
1
γgM2
0
?Rsub
Rtot
?−γg. This is used as a
boundary condition for Eq. 10, where the functions U(x) and λ(x) (and therefore f0(p)) on
the right hand side at the kthstep of iteration are taken as the functions at the step (k−1).
In this way the solution of Eq. 1 at the step k is simply
ξ(k)
c(x) = ξc(0)exp
?
−
?0
xdx′λ(k−1)(x′)U(k−1)(x′)
?
, (13)
with the correct limits when x → 0 and x → −∞. At each step of iteration the functions
U(x), f0(p), λ(x) are recalculated (through Eq. 1, Eqs. 12 and 5, and Eq. 11, respectively),
until convergence is reached. The solution of this set of equations, however, is also a solution
of our physical problem only if the pressure in the form of accelerated particles as given by
Eq. 1 coincides with that calculated by using the final f0(p) in Eq. 9. This occurs only for
one specific value of Rsub, which fully determines the solution of our problem for an arbitrary
diffusion coefficient as a function of location and momentum.
3 SELF-GENERATED TURBULENCE AND PARTICLE DIFFUSION
The streaming of cosmic rays at super-Alfv´ enic speed induces a streaming instability, which
has been discussed in previous literature (e.g. Bell (1978a)).
Let us define F(x,k) as the energy density per unit logarithmic band width of waves
with wave-number k. Neglecting the damping, and assuming a steady state, the following
relation holds (see e.g. Lagage & Cesarsky 1983):
u∂F(x,k)
∂x
= σ(x,k)F(x,k) ,(14)
where u = u(x) is the fluid velocity upstream of the shock and σ is the growth rate of waves
with given wavenumber k, which can be related to the distribution function of the resonant
cosmic rays, f(x,p(k)), through:
σ =4 π
3
vA
UMF
?
p4v∂f(x,p)
∂x
?
p=¯ p(k)
.(15)
In Eq. 15, v and p are the particle velocity and momentum respectively, and the latter is
related to the wave number k through the resonance condition ¯ p(k) = eB/kmc, UM is the
energy density of the background magnetic field B0(UM = B2
0/8π), while vAis the local
Alfv´ en velocity.
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Non linear particle acceleration at shock waves
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All these expressions have been actually obtained for shocks that are not modified by
the dynamical reaction of cosmic rays. In principle, the Fourier analysis used to obtain
the previous expressions and in fact used to reach the conclusion that there are unstable
modes, is not formally applicable, since all these calculations assume that the background
quantities (the fluid velocity u in particular) are spatially constant. However, provided that
1/k remains much smaller than the spatial extension of the precursor, the conclusions are,
in first approximation, still applicable. Clearly this condition is broken by definition at the
maximum momentum pmaxat least in those cases in which this is determined by the finite
size of the accelerator rather than by energy losses. Special care should be taken of the fact
that all quantities involved in the equations above depend on the location in the presursor.
It follows that for a cosmic ray modified shock, vAis not spatially constant since both the
upstream plasma density, ρ, and, in general, the background magnetic field, B0, are space
dependent. However all previous calculations apply to the case of a parallel shock, for which
the strength of the background magnetic field B0can be taken to be constant, since there
is no adiabatic compression of the magnetic field lines.
Using the equation for conservation of mass ρ(x) = ρ0/U(x), we can therefore write the
local Alfv´ en velocity as:
vA(x) =
B0
√4πρ0U(x)1/2. (16)
An additional warning should be issued in that Eq. 14 neglects the adiabatic compression
of waves in the shock precursor: this reflects in the absence of terms proportional to the
gradient of the velocity field. Unfortunately, to our knowledge, discussions of this problem
in the literature are limited to integrated quantities (e.g. total energy density and pressure
of the waves) while a description of the behaviour of the modes with different wave-numbers
is more complex. In fact, in principle even the concept of modes with given k becomes ill
defined in a background which has spatial gradients of the quantities to be perturbed.
Once F(x,k) is known, the diffusion coefficient is known in turn (Bell (1978a)):
D(x,p) =4
π3 F
From the latter equation, where rLstands for the Larmor radius of particles of momentum
rLv
. (17)
p, it is clear that the diffusion coefficient tends to Bohm’s expression for F → 1. On the
other hand, it is also straightforward to check what the expected saturation level for the
overall energy density of the perturbed magnetic field is. If we define
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E. Amato and P. Blasi
δB2
8π
= I =B2
0
8π
?dk
kF(k) ,(18)
from Eq. 14 and Eq. 15 we see that:
B2
0
8πk
4 π
3 dx
= vAdPCR
dx
udI
dx
=
?dk
vA
σ F(k,x) =
?
=
d
dp v(p) p3f(x,p) = (19)
.
Integration of the latter equation is straightforward when non-linear effects on the fluid are
neglected so that u and vAare both spatially constant. One obtains δB2/8π = (vA/u)PCR,
or, in terms of amplification of the ambient magnetic field:
?2
ρ0u2
0
with MA= u0/vAthe Alfv´ enic Mach number.
?δB
B0
= 2 MA
PCR
,(20)
It is worth stressing that for PCR/ρ0u2
0∼ 1 and MA≫ 1, the predicted amplification
of the magnetic field exceeds unity. In fact, this result was initially obtained in the context
of the so-called quasi-linear theory, therefore it should be taken with caution and checked
versus numerical calculations of the non-linear phase of amplification of the waves. It seems
clear, however, that the growth may well enter this non-linear regime and lead to turbulent
fields in the shock vicinity that exceed the pre-existing background magnetic field.
Let us now go back to Eqs. [14-17] with the aim of recasting the relation between the
diffusion coefficient and the cosmic ray distribution function in a more compact form. Using
Eqs. [14-16], we can write:
F(x,p) =8 π
where vA0= B0/√4πρ0is the Alfv´ en speed at upstream infinity and
dx′
U(x′)1/2
3
v p4Φ(x,p)
ρ0u0vA0
,(21)
Φ(x,p) =
?x
−∞
∂f
∂x′(x′,p) . (22)
With this definition of Φ, from Eq. 17, we obtain:
D(x,p) =
3
2π2DB0
n0
p3Φ(x,p)
vA0
c
u0
c
, (23)
with DB0= mpc3/3eB0a constant.
It is important to notice that since the constant DB0 is inversely proportional to the
strength of the background field B0, and the Alfven speed vA0is proportional to B0, the
diffusion coefficient in Eq. 23 turns out to be independent of B0. This result holds only
within the context of quasi-linear theory. Even in the context of a quasi-linear theory of the
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Non linear particle acceleration at shock waves
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development of magnetic perturbations, a dependence on B0could be introduced through
the quantity Φ, which is affected by the laws of conservation of momentum and energy in
the precursor. However, in the cases of interest for us we will show below that these effects
are fully negligible.
4 SPECTRA OF THE ACCELERATED PARTICLES AND
SELF-GENERATED DIFFUSION COEFFICIENT
Following the mathematical procedure outlined in Sec. 2 and in Sec. 3 we are able to de-
termine self-consistently the spectrum of accelerated particles and the diffusion coefficient.
Within the obvious limitation of using quasi-linear theory to calculate the diffusion coefficient
for the non-linear case, this is the first attempt at determining the space and momentum
dependence of the diffusion coefficient together with the spectrum of accelerated particles.
While in a time-dependent approach to the problem it would be possible to estimate the
maximum energy in a self-consistent way, here we assume for simplicity that the maximum
momentum is a given parameter. We chose to carry out the calculations presented in the
following for pmax= 105mc.
The spectra of the accelerated particles for Mach numbers at upstream infinity ranging
from M0= 4 to M0= 200 are shown in Fig. 1 for a background magnetic field at upstream
infinity B0= 1µG. As stressed above the result is however expected and actually found to
be independent of the strength of the background magnetic field. In the bottom part of the
same figure we plot the slope of the spectrum as a function of momentum.
It is evident that for low Mach numbers and at given pmaxthe modification of the shock
due to the reaction of the accelerated particles is small (see for instance the case M0= 4).
For the strongly modified case (e.g. M0= 200) the asymptotic spectrum of the accelerated
particles is very flat, tending to p−αwith α = 3.1 − 3.2 for p → pmax. The momentum at
which the spectrum becomes flatter than p−4, the prediction of linear theory, depends on the
level of shock modification: it is higher (10−20 mc ) for relatively low Mach numbers (namely
weaker modification) and approaches a few GeV for high Mach numbers and large shock
modification. The asymptotic spectrum is reached at p/mc > 102. These effects might be
important in the perspective of reconciling the concave shape of the instantaneous spectra of
accelerated particles with observations of the diffuse spectrum of cosmic rays in the Galaxy.
Most measurements, mainly related to the abundance of light elements are in fact limited to
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E. Amato and P. Blasi
Figure 1. Spectrum and slope at the shock location as functions of energy for pmax= 105mc and magnetic field at upstream
infinity B0 = 1µG. The curves refer to Mach numbers at upstream infinity ranging from M0 = 4 to M0 = 200: dotted for
M0= 4, dashed for M0= 10, dot-dashed for M0= 50, solid for M0= 100 and dot-dot-dashed for M0= 200.
relatively low energies, where the spectra predicted in this paper are compatible with power
laws softer than p−4. Serious work aimed at predicting the actual spectrum of cosmic rays
escaping the sources is urgently needed but still missing in the context of non-linear theories
of particle acceleration at shocks.
The diffusion coefficient associated with the self-generated waves is given by Eq. 23. We
plot this diffusion coefficient at the shock location in Fig. 2 for Mach numbers M0 = 10
(dashed lines) and M0 = 100 (solid lines). We fix B0= 1µG, but as stressed in the pre-
vious section, the diffusion coefficient obtained within the quasi-linear theory of magnetic
perturbations is independent of B0. For comparison, we also plot the corresponding Bohm
diffusion coefficient DB(p) ∝ v(p)p in the unperturbed magnetic field B0, for B0 = 1µG
and B0= 10µG. The comparison strikingly shows that for most momenta of the accelerated
particles the diffusion takes place at super-Bohm rates (namely the diffusion is slower than
predicted by the Bohm coefficient in the unperturbed magnetic field, as could be expected).
Moreover, the difference between the self-generated diffusion coefficient and the Bohm coeffi-
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Non linear particle acceleration at shock waves
11
cient increases at the highest momenta, which might suggest that somewhat higher energies
could be achieved if the self-generated turbulence were taken into account. In this respect
it is also important to notice that, as one might expect, the more modified the shock, the
slower the diffusion.
It is worth keeping in mind that the diffusion coefficient in the amplified magnetic
field, as obtained through our calculations, remains larger than the Bohm value in the
same field. The latter is in fact considered as a sort of lower limit to the diffusion rate
(Casse, Lemoine & Pelletier (2002)) even in the case of strong turbulence. The only region
in momentum space where this condition may be violated in our calculations is very close
to the maximum momentum pmax. It is clear however that a realistic determination of the
diffusion coefficient cannot be achieved in the context of quasi-linear theory and that even
numerical approaches to diffusion, such as those of Casse, Lemoine & Pelletier (2002) can
only suggest a general trend as long as the turbulent structure of the magnetic field is pre-
defined rather than determined by the diffusing particles themselves. In this sense, the limit
at the Bohm value in the amplified field should also be taken with caution.
As stressed above, the fact that the diffusion coefficient is smaller than the Bohm co-
efficient in the background field is the consequence of the fact that the fluctuations in the
magnetic field become strongly non linear, namely δB2/B2
0≫ 1, at least close to the shock
surface. In fact we find that δB/B0at x = 0 is exactly as predicted by Eq. 20. In these
conditions it is important to check that the dynamical role of the turbulent magnetic field
remains small. In Fig. 3 we plot δB2/8π normalized to ρ0u2
0(top panels) and the cosmic
ray normalized pressure ξc(x) and velocity U(x) (bottom panels). The curves refer to Mach
number M0= 10 (dashed lines) and M0= 100 (solid lines). The plots on the left (right) are
obtained for B0= 10µG (B0= 1µG). The x-coordinate is in units of x∗= −DB(pmax)/u0,
where DB(p) stands for the Bohm diffusion coefficient appropriate to the considered value
of B0.
The highest values of δB2/8πρ0u2
0, reached close to the shock front, are of the order
of 10−2− 10−3, confirming that even in the extreme non linear cases the dynamical effect
of the magnetic field remains unimporant. This result serves as a justification a posteriori
that we could neglect the pressure of the waves and their energy flux in the equations
of conservation of momentum and energy respectively. This result is very specific of the
resonant channel of production of Alfven waves, and is very likely not correct in the case of
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E. Amato and P. Blasi
Figure 2. The self-generated diffusion coefficient at the shock location x = 0−as a function of the particle momentum for
Mach numbers M0= 10 (dotted line), M0= 100 (dashed line) and M0= 200 (solid line). Also plotted is the Bohm diffusion
coefficient corresponding to B0= 1µG (solid line with triangles) and B0= 10µG (solid line with diamonds). The y-axis is in
units of cm2s−1.
Figure 3. Top panels: the energy density in magnetic field fluctuations δB2/8π normalized to the fluid ram pressure ρ0u2
at upstream infinity. Bottom panels: the cosimc ray pressure normalized to ρ0u2
the normalized velocity U (thin curves). All functions are plotted versus spatial location, with the x-coordinate in units of
x∗ = −DB(pmax)/u0, where DB(p) stands for the appropriate Bohm diffusion coefficient. The left and right panels refer to
different strengths of the background magnetic field B0, as specified in each panel, while the different line-types correspond
to different Mach numbers: dashed for M0 = 10 and solid for M0 = 100. In the upper panels we also plot for comparison a
dot-dashed curve corresponding to δB = 10B0.
0
0, ξc (thick curves), is plotted together with
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Non linear particle acceleration at shock waves
13
Figure 4. Effects of using the self-generated diffusion coefficient on the particles’ distribution functions. In the top and bottom
panels we plot the particles’ spectrum and spectrum slope respectively. The continuous curves are obtained for self-generated
D(x,p): dashed curves are for M0= 10 and solid for M0= 100. The symbols represent the results obtained adopting the Bohm
diffusion coefficient: diamonds are for M0= 10 and filled circles are for M0= 100.
non-resonant scenarios, such as the one proposed by Bell (2004), where larger amplifications
of the magnetic field could be achieved.
The shape of the spectra of accelerated particles is affected in a sizeable way by the
adoption of the self-generated diffusion coefficient: Fig. 4 illustrates this point. The continu-
ous lines are for self-generated diffusion (dashed for M0= 10 and solid for M0= 100) while
the symbols are for Bohm diffusion (diamonds for M0 = 10, almost perfectly superposed
on the dashed curve, and filled circles for M0= 100). While in the weakly modified cases
the spectrum is basically independent of the assumed diffusion coefficient, in the fully non
linear solution self-generated diffusion leads to steeper spectra at low momenta and harder
spectra at high momenta.
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E. Amato and P. Blasi
5 THE ROLE OF TURBULENT HEATING
A major uncertainty in all types of calculations of the non linear particle acceleration at
shock fronts is the effect of turbulent heating. This generic expression is used to refer to any
process that may determine non-adiabatic gas heating. The two best known examples of
this type of processes are Alfv´ en heating (McKenzie & V¨ olk (1982)) and acoustic instability
(Drury & Falle (1986)). Both effects are however very hard to implement in a quantitative
calculation: in the case of Alfv´ en heating, the mechanism was originally introduced as a way
to avoid the turbulent magnetic field to grow to non linear levels, while it is usually used
even in those cases in which δB/B0≫ 1.
Acoustic instability develops in the pressure gradient induced by cosmic rays in the
precursor and results in the development of a train of shock waves that heat the background
gas (Drury & Falle (1986)). The analysis of the instability is carried out in the linear regime,
therefore it is not easy to describe quantitatively the heating effect.
In both cases the net effect is the non-adiabatic heating of the gas in the precursor,
which results in the weakening of the precursor itself and in the reduction of the acceleration
efficiency compared with the case in which the turbulent heating is not taken into account.
In order to illustrate this effect, we adopt a phenomenological approach, similar to that
of Berezhko & Ellison (1999). We stress that this approach, developed for the case of weak
turbulence, is inadequate in principle for the case of interest here, where the turbulence
can, in principle, become strong. We adopt it here only for illustration of the main physical
effects.
The approach consists in redefining the equation of state of the gas taking into account
the heating induced by the accelerated particles (McKenzie & V¨ olk (1982)):
∂
∂x
?
Pg(x) ρ(x)−γg?
where u/vH= MHis the local Mach number of the turbulence relevant for the heating (for
= (γg− 1)vH(x)
u(x)
∂PCR
∂x
ρ(x)−γg, (24)
instance vH= vAfor Alfv´ enic heating). After defining τ(x) = Pg(x)/ρ0u2
0, we replace Eq. 1
in the set of equations presented in Sec. 2 with the two following equations:
ξc(x) + U(x) + τ(x) = 1 +
1
γgM2
0
, (25)
τ(x) =U(x)−γg
γgM2
0
?
1 + γg(γg− 1)M2
0
MH0
?1 − U(x)γg+s
γg+ s
+ Iτ(x)
??
(26)
where the latter is obtained by rewriting Eq. 24 in terms of the normalized pressures and
integrating between upstream infinity and a generic location x in the upstream medium,
Page 15
Non linear particle acceleration at shock waves
15
after expressing ξc(x) in terms of τ(x) and U(x) through Eq. 25. We have assumed a spatial
dependence of the turbulence characteristic velocity vH in the form of vH(x) = vH0U(x)s
(with s = 1/2 in the case of Alfv´ en heating, from Eq. 16), and used as a boundary condition
τ(−∞) = 1/(γgM2
0). The term Iτ(x) appearing in Eq. 26, finally, is defined as:
−∞U(x)γg−1+sdτ
Iτ(x) = −
?x
dx′dx′. (27)
The only other changes induced by the inclusion of turbulent heating in our initial set of
equations concern the relation between the compression ratios Rtotand Rsub(Eq. 7) and the
temperature jump between downstream and upstream infinity, that reflects on the minimum
cosmic ray momentum pinj. In both cases the changes can be summarized in the appearence
of a factor (1 + FH) with
FH= γg(γg− 1)M2
0
MH0
?
1
γg+ s
?
1 −
?Rsub
Rtot
?γg+s?
+ Iτ(0)
?
. (28)
With this definition of FHwe find:
Rtot= M
2
γg+1
0
?(γg+ 1)Rγg
sub− (γg− 1)Rγg+1
2(1 + FH)
sub
?
1
γg+1
, (29)
and
T2= T0
?Rtot
Rsub
?γg−1
(1 + FH)(γg+ 1) − (γg− 1)R−1
(γg+ 1) − (γg− 1)Rsub
sub
. (30)
The usual results (adiabatic heating) are recovered when MH0/M2
0→ ∞. In spite of the
apparent simplicity of these revised relations there are two complications arising in the
solution of the system of equations. A minor difficulty is that the equation relating Rsub
and Rtotnow cannot be solved analytically due to the presence of the ratio Rsub/Rtotin the
definition of FH. A more serious complication, instead, has to do with Iτ, which requires
the knowledge of the complete solution of the problem. However not even this is too severe
a problem to overcome within the framework of an iterative method, although sometimes it
results in an appreciable slowing down of the calculation. In fact, it can be seen a posteriori
that Iτis always negligible compared to (1 − Uγg+s)/(γg+ s).
In order to show how our results for particle acceleration may be affected by the inclusion
of turbulent heating we carried out the calculations for the case of Alfv´ en heating, namely
considering the Alfv´ en velocity as the characteristic turbulence velocity, vH = vA, which
also implies s = 1/2 in the equations above, according to Eq. 16. Results obtained with and
without inclusion of Alfv´ en heating are shown in Fig. 5, where for the background magnetic
field we have assumed the largest of the two values so far considered, B0= 10µG, with the
Page 16
16
E. Amato and P. Blasi
Figure 5. The plots on the left show: in the upper panel, the ratio between the turbulent and background magnetic field as a
function of space for two different values of the Mach number (10 and 100), with and without inclusion of the turbulent heating;
in the lower panel the corresponding normalized cosmic ray pressure. The plots on the right show the particles’ spectrum and
slope in the same cases. The continuous curves correspond to cases when the turbulent heating is not taken into account:
dashed for M0 = 10 and solid for M0 = 100. The symbols correspond to cases including the turbulen heating: diamonds for
M0= 10 and filled circles for M0= 100.
aim of minimizing the turbulence Mach number and hence maximizing the effects of the
heating.
In the left panel of Fig. 5 we show how the turbulent magnetic field strength and cosmic
ray pressure are now reduced, for both a mildly (M0= 10 diamonds versus dashed line) and
a strongly (M0= 100, filled circles versus solid line) modified case. In the strongly modified
case we find that, in the vicinity of the shock, the turbulent magnetic field strength is
decreased by 10%, while the cosmic ray pressure is decreased by 20%. Changes in both
quantities are of order few % in the weakly modified case. In the right panel of the same
figure we show, using the same notation for the different curves, how the particles’ spectra
are affected: while changes are negligible in the weakly modified case, for M0 = 100 the
concavity of the spectrum is appreciably reduced, namely, the spectrum becomes harder
toward the low energy end and softer at high energies.
It is worth stressing once more that this way of including turbulent heating, that is used
in many currect approaches to particle acceleration in supernova remnants, is far from self-
consistent and the results should only be considered as an indication of a trend. Sometimes,
in order to attempt a slightly more realistic approach, one substitutes the Alfv´ en speed
in the background field B0with the corresponding quantity in the amplified field B0+ δB.
Needless to say that such an attempt, though justified by the complete lack of any non-linear
theory of turbulent heating, is far from being realistic.
Page 17
Non linear particle acceleration at shock waves
17
6 CONCLUSIONS
We described the mathematical theory of particle acceleration at non-relativistic shock fronts
with dynamical reaction of accelerated particles and self-generated scattering waves. The
diffusion coefficient itself is an output of the calculations, though within the limitations
imposed by the usage of quasi-linear theory applied to the case of potentially strong magnetic
field amplification. The scattering in the upstream plasma is generated through streaming
instability, as discussed extensively in previous literature.
We determined the spectra of accelerated particles, their spatial distribution and the
space dependence of the fluid velocity, pressure and temperature. The diffusion coefficient
and the strength of the self-generated magnetic perturbations are also calculated, as a func-
tion of the distance from the shock front in the precursor. We confirm the general finding
that the spectra of accelerated particles are concave, an effect which is particularly evident
for strongly modified shocks, namely for large Mach numbers of the moving fluid. However,
the shape of the concavity is somewhat affected by the self-determined diffusion coefficient,
as visible in Fig. 4.
Having in mind the comparison between the predicted spectra at the sources and the
observed cosmic ray spectrum at the Earth, it is worth reminding the reader that what can
actually be measured is the combination of the diffusion time, the gas density along the
trajectory (responsible for the spallation) and the injection spectrum. In order to infer some
conclusions about the spectrum at the source, one has to make assumptions on the diffusion
coefficient in the interstellar medium. In alternative it would be a precious step forward if
we could measure unambiguously the spectrum of gamma rays generated by π0decays close
to the source itself, an evidence that unfortunately is still missing.
The asymptotic slope of the spectra for p → pmaxmay be as flat as ∼ 3.1 − 3.2, but
this conclusion is not strongly affected by the fact that the diffusion coefficient is calculated
self-consistently.
The most striking new result of our calculations is the energy dependence of the diffusion
coefficient and the strength of the amplified turbulent magnetic field. As could be expected,
the diffusion coefficient is not Bohm-like, and the turbulent component of the magnetic
field is amplified so efficiently that the diffusion coefficient is much smaller than the Bohm
coefficient in the background magnetic field. This is especially true at the highest momenta,
which leads to think that a full non-linear theory might predict higher values of the maximum
Page 18
18
E. Amato and P. Blasi
momentum than expected on naive grounds. Unfortunately a full, self-consistent calculation
of pmaxfor a strongly modified shock has never been carried out, the main difficulty being
in accounting for the spatial dependence of all the quantities involved.
When compared with the Bohm diffusion coefficient as calculated in the amplified mag-
netic field, our diffusion coefficient remains always larger, with the possible exception of a
narrow momentum region close to pmax.
While the calculation presented here is fully self-consistent in the determination of the
shock modification due to the reaction of the accelerated particles, the part related to the
amplification of the background field suffers from all the limitations related to the usage
of quasi-linear theory for the streaming instability. This approach, initially developed for
weakly amplified magnetic fields, is widely applied in the literature to situations that violate
this condition. Unfortunately at the present time this is the only way we have to achieve a
(at least partially) self-consistent picture of the process of particle acceleration at cosmic ray
modified shocks with self-generated turbulence. This problem is in fact even more serious for
those approaches that predict levels of magnetic field amplification which are much higher
than those found here (e.g. Bell & Lucek (2001); Bell (2004)).
The high acceleration efficiencies obtained in the context of all approaches to particle
acceleration at shocks are known to be reduced by the effect of turbulent heating. Any
non-adiabatic heating of the gas in the precursor leads to reducing the energy channelled
into non-thermal particles at the shock. This is a serious problem, because the effect of
turbulent heating depends dramatically on the type of mechanism that is responsible for the
heating: Alfv´ en heating, often used in the literature, is only one of these mechanisms, and
not necessarily the most efficient. For instance, the instability induced by the propagation
of acoustic waves in the precursor was shown to lead to the formation of weak shocks in the
precursor, which in turn heat the upstream plasma (e.g. Drury & Falle (1986)).
These non-linear effects can hardly be taken into account in a credible way. Most notably,
the phenomenological expressions proposed in the literature and used also in the present
paper, have originally been proposed as mechanisms to reduce the amount of magnetic field
amplification and remain in the context of small perturbations of the background magnetic
field. However, as shown in Fig. 5, even with the Alfv´ en heating taken into account, the
magnetic field can be amplified by a factor in excess of ∼ 10 with respect to the background
field. This means that a fully non-linear theory of the turbulent heating is required in order
to make fully reliable predictions.
Page 19
Non linear particle acceleration at shock waves
19
From the phenomenological point of view, the best evidences for both magnetic field am-
plification and efficient particle acceleration come from observations of supernova remnants
(see the reviews of Hillas (2005) and Blasi (2005) and references therein). In fact, it has been
argued that the amount of field amplification required to explain the thickness of the X-ray
bright rims in several remnants is of the order of ∼ 200−300µG (V¨ olk,Berezhko, & Ksenofontov
(2005)). An important role in explaining this level of amplification could be played by dif-
ferent versions of the streaming instability (Bell & Lucek (2001); Bell (2004, 2005)), not
requiring resonant interactions of particles and waves. A full non-linear theory including
these effects will be described elsewhere (Amato and Blasi, in preparation).
ACKNOWLEDGMENTS
This research was funded through grant COFIN2004-2005. We wish to acknowledge useful
conversations with D. Ellison, S. Gabici and M. Vietri. We are also grateful to an anonymous
referee for useful comments.
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