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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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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