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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

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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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