The efficiency of the upstream plasma flow energy conversion into
nonthermal particles could be high enough providing a hard spectrum
of nonthermal particles up to some maximal energy
. If the
efficiency of ram energy transfer to
the energetic particles is high enough, an extended shock precursor
appears due to the incoming plasma flow deceleration by the fast
particle pressure. The precursor scale L is of the order of
(c / vsh)
- orders of
magnitude larger than the width of the shock transition region (see
Fig. 5). Here
is the
maximal mean free
path of a particle in the energy-containing part of the spectrum and
vsh is the shock velocity. We shall later refer to these
energetic particles as cosmic rays.
It has been shown that the front of a strong collisionless shock wave consists of an extended precursor and a viscous velocity discontinuity (subshock) of a local Mach number that is smaller than the total Mach number of the shock wave (see Fig. 5). The compression of matter at the subshock can be much lower than the total compression of the medium in the shock wave with allowance for high compression in the precursor. We shall refer later to such shocks as CR-modified.
The large scale ("macroscopic") structure of a CR-modified shock can be modelled by a two-fluid approach with a kinetic description of nonthermal particles (see e.g. Blandford & Eichler 1987, Berezhko et al. 1996, Malkov & Drury 2001, Blasi 2004 and references therein) or by a Monte Carlo method (e.g. Jones & Ellison 1991, Ellison et al. 1996). In both methods some suitable parameterisation of particle scattering process must be postulated a priori. Monte-Carlo simulations, however, have no assumption of isotropy for particle distributions, and that allows an internally self-consistent treatment of thermal particle injection. While the injection depends on the assumptions made for the particle pitch-angle scattering, these assumptions are applied equally to particles of all energies. The Monte Carlo technique eliminates a free injection parameter, which is present in all models based on the diffusion approximation and is used to set the injection efficiency. The strong feedback between injection, shock structure, and magnetic field amplification makes this property of the Monte Carlo technique particularly important. The Monte Carlo technique allows to iteratively obtain a shock velocity profile and particle distribution function conserving mass, momentum and energy fluxes taking into account the nonlinear feedback from the accelerated energetic particles.
In Fig. 6 Monte Carlo simulated proton spectra (multiplied by [p / (mpc)]4) are shown, in the downstream shock from Vladimirov et al. (2006). To illustrate the dependence of the maximal energy of an accelerated proton on the system scale size, a free escape boundary condition was applied at some distance from the subshock position in the shock rest frame. The heavy solid and dotted curves in the right panel correspond to the free escape boundary located at a distance 104rg1 (where rg1 = mp vsh c / e B1), the dashed curve has 103rg1, and the light solid curve has 105rg1. The simulations were done for a supernova shock in the interstellar medium with a shock speed vsh = 5000 km s-1 and an unshocked proton number density n1 = 1 cm-3. In the left panel the spectra are given for the same position of the free escape boundary, but for different prescriptions of the scattering model.
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![]() |
Figure 6. Spectra of protons accelerated by a strong shock. The spectra were simulated with a non-linear Monte-Carlo model which accounts for particle injection and magnetic field amplification by the shock (for details see Vladimirov et al. 2006). On the left panel the transitions from the thermal-like component to the high energy tail are marked by pinj. On the right panel different curves correspond to different locations of the free escape boundary (see in the text). |
5.1. Magnetic field amplification in CR-dominated shocks
An important predicted feature of strong shocks with efficient CR
acceleration is the possibility to amplify an initial seed magnetic
field by orders of magnitude (e.g.
Bell & Lucek
2001,
Bell 2004).
CR current and CR pressure gradient upstream of the strong shock
could drive magnetic fluctuations on the shock precursor scale
length. The CR-shock precursor scale L is ~ (c /
vsh)
which is
expected to be above a kpc, moreover, the width is L
100 kpc for a
shock of a size
comparable to that of a galaxy cluster. The precursor scale size L
is >> 109 times larger than the subshock transition region
where strong small scale magnetic field fluctuations are directly
produced by instabilities of super-Alfvénic bulk plasma flows
illustrated in Fig. 2. That small
scale fluctuations are
responsible for bulk plasma motion dissipation process and adiabatic
amplification of the transverse magnetic field in collisionless
shocks. At the same time the collisionless dissipation process is
thought to inject a minor fraction of incoming particles to be
accelerated to high energies by Fermi mechanism. Recent models of
diffusive shock acceleration allows a substantial fraction (say,
30%) of the MHD shock ram pressure to be converted to accelerated
particles filling a vicinity of the shock of the scale L. The large
scale current and density gradient of the accelerated CRs may
convert a fraction of the CR energy to magnetic field due to
multifluid instabilities of different kinds providing a way to
amplify the initial magnetic field by a factor larger than the
shock compression ratio.
Recent non-linear simulations of magnetic field amplification in
diffusive shock acceleration by a Monte-Carlo model
(Vladimirov et
al. 2006)
and a kinetic model
(Amato & Blasi
2006)
confirmed the possibility of a significant effect. The amplitude of
the fluctuating magnetic field energy density WB is of the
order of the shock accelerated CR pressure which is in turn a
substantial fraction of the shock ram pressure 0.5
1
vsh2. Here
1 is
the shock upstream ambient gas density.
For typical cluster parameters the discussed mechanism could
provide a µG range magnetic field amplitude in a hundred kpc
range scale of CR-modified shock precursor. The Faraday rotation
measure RM provided by a strong CR-dominated shock in a cluster
can reach values of
10 rad m-2 and even a few times
higher. For the case of the so-called Bohm diffusion model the
rotation measure RM is proportional to the maximal energy of the
ions in the energy-containing part of the CR-spectrum accelerated by
the shock. Radio observations, Faraday rotation and
synchrotron-Compton emission measurements are used to estimate the
magnetic fields in clusters (e.g.
Carilli &
Taylor 2002,
Newman et
al. 2002).
Large filaments of polarised radio emission
of scale size about 400 kpc were discovered by
Govoni et
al. (2005)
in the halo of the cluster of galaxies Abell 2255 and by
Bagchi et
al. (2006)
in Abell 3376 (see
Fig. 8). They
could be connected to large scale shocks due to accretion/merging
activity of the cluster.
5.2. Gas heating and entropy production in strong CR-modified shocks
An exact modelling of a collisionless shock structure taking into account the nonthermal particle acceleration effect requires the nonperturbative self-consistent description of a multi-component and multi-scale system including strong MHD-turbulence dynamics. Such a modelling is not feasible at the moment. Instead, a simplified description of a multi-fluid strong shock structure can be used with an appropriate parameterisation of the extended pre-shock and of the gas subshock. The predicted observable characteristics of the shocks can be confronted to the observational data. We will now discuss the effects of plasma heating by modified shocks and then make some specific predictions for possible observational tests.
In the shocks with efficient high energy particle acceleration the energy flux carried away by escaping energetic particles Qesc must be accounted for in the energy continuity equations. The energy loss results in a lower effective adiabatic index, but it allows to increase the total compression of the gas in the shock downstream.
The total compression ratio rtot of a strong MHD shock modified by an efficient nonthermal particle acceleration can be estimated as
![]() |
(17) |
assuming that the energy density in the shock upstream is dominated
by the ram pressure and that the CR escape is through the cut-off
momentum regime (e.g.
Malkov & Drury
2001).
Here is the
effective adiabatic exponent. In Fig. 7 we
illustrate the
dependence of the compression ratio on Qesc /
a
vsh3 for
= 4/3
and 5/3 assuming that the effective
adiabatic exponent is between the two values depending on the
spectrum of the accelerated relativistic particles.
The distribution function of nonthermal particles and the bulk flow
profile in the shock upstream region are sensitive to the total
compression ratio rtot. Thus, the exact calculation of the
escape flux Qesc can be performed only in fully nonlinear
kinetic simulations. Nevertheless, an approximate iterative approach
(e.g. in the Monte Carlo model discussed above) can be used to make
the steady-state distribution function consistent with the shock
compression assuming some diffusion model. The subshock is the
standard gas viscous shock of a Mach number
sub.
For that simplified two-fluid model of a strong CR-modified
shock the effective ion temperature in the downstream
Ti(2) can be estimated for a shock of a
given velocity, if rtot and rsub are
known:
![]() |
(18) |
Single fluid strong shock heating represents the limit
sub =
s >> 1,
since there is no precursor in that case, resulting in Eq. 9. In
single-fluid systems the compression ratio rtot =
rsub
(
g + 1) /
(
g -1) does not depend on the
shock velocity and Eq. 18 reduces to Eq. 9.
However, in multi-fluid shocks the total compression ratio depends
on the shock velocity and could be substantially higher than that in
the single-fluid case. This implies somewhat lower postshock ion
temperatures for the strong multi-fluid shock of the same velocity
and could be tested observationally. It is convenient to introduce
the scaling rtot(vsh)
vsh
to
describe the different cases of strong shock heating (see
Bykov (2005)
for details). Then from Eq. 18, Ti2
(
sub)
vsh2(1 -
). The
subshock Mach number
sub depends, in
general, on
s and
a. Thus, an
index
approximates the
velocity dependence of
(
sub)
vsh
. Finally, if Ti2
vsha, then the index a =
2(1 -
) +
. For the
case of shock precursor heating by CR
generated Alfvén waves, the index a
1.25
(Bykov 2005).
A distinctive feature of multi-fluid shocks is their high gas
compression rtot(vsh) that could be
well above the single fluid shock limit
(g + 1) /
(
g - 1)
(see Fig. 7). At the same time entropy
production for a
strong multi-fluid shock scales as
rtot(vsh) -
(
g + 1) and it is significantly reduced
compared to the single-fluid shock of the same velocity. The effects
are due to energetic particle acceleration and magnetic field amplification.
Energetic particles penetrate into the shock upstream region. They
are coupled with the upstream gas through fluctuating magnetic
fields (including the Alfvén waves generated by the energetic
particles). Magnetic field dissipation provides gas preheating and
entropy production in the extended shock precursor. Such a heated
pre-shock region of kT
0.5 keV would
appear as an extended
filament of width L ~ (c / vsh)
3 × 1014
B-6-1 cm. Here
(in GeV) is
the highest energy of the hard branch
of the accelerated particle spectrum. If B-6 ~ 0.1 in the
cluster outskirts and if the hard spectrum of energetic nuclei
extends to ~ 109 GeV (cf.
Norman et
al. 1995)
we have L ~ 1 Mpc and even wider. Projected on a hot X-ray
cluster, such filaments could produce a soft X-ray component "excessive"
to that produced by the hot cluster. A warm gas (~ 0.2 keV) emission
filament found with XMM-Newton in the outskirts of the
Coma cluster by
Finoguenov et
al. (2003)
could be an extended heated precursor of a
strong multi-fluid accretion shock. For a detailed review of the
soft X-ray/EUV excesses see
Durret et
al. 2008
- Chapter 4, this volume.
5.3. ICM entropy production by multifluid accretion shocks
Cold gas falling into the dark matter (DM) dominated gravitational
well passes through a strong accretion shock. The shock is a source
of gas entropy production in the intercluster medium (ICM) (e.g.
Knight & Ponman
1997,
Tozzi & Norman
2001,
Voit et al. 2003).
The post-shock entropy K = Kb T /
2/3
used in the ICM analysis and simulations (e.g.
Bialek et
al. 2001)
is related to the standard thermodynamic entropy s through
K
exp(s / cv). In the standard scenario with a
single-fluid accretion shock the post-shock entropy scales
Ksf
vsh2
1-2/3 (e.g.
Voit et al. 2003).
The multi-fluid nature of the collisionless accretion shock modifies the standard scaling relation to be
![]() |
(19) |
The compression ratio in CR-shocks is higher than in a strong
single-fluid shock of the same velocity resulting in reduced
post-shock entropy production. For example, in the case of Alfvén
heating the post-shock entropy of a multi-fluid shock reduces as
Kmf / Ksf ~ (15 /
a) for
a > 15 and
s2
>
a. Here
and below in numerical estimations we assume
g = 5/3,
though a non-thermal baryonic component could reduce the index
g.
Since rtot(vsh) and
(
sub) are
shock velocity dependent, the simple scaling K
vsh2
1-2/3 is not valid. In CR-modified
shocks Kmf
vsh
1(1 -
g) or
Kmf
T
/a, where
= 2 - (1 +
g)
+
. For the case of
Alfvén wave heating the index
is
1.25 and
Kmf is
T0.8 assuming
g = 5/3. Recently
Ponman et
al. (2003)
and
Piffaretti et
al. (2005)
found that the dispersion in the observed
cluster entropy profiles is smaller if an empirical relation K
T0.65 is used instead of the standard K
T (see also
Pratt et
al. 2006).
Consider the simple model of smooth accretion of cold gas through a
strong accretion shock by
Voit et al. (2003).
The gas of velocity
vac accretes at a rate
g through
the shock at a radius rac where
![]() |
(20) |
Here M(t) is the cluster mass and rta is
the matter turnaround radius. Then the entropy Kmf
just behind the multi-fluid shock is expressed through
Ti(2)(vac) and
2 =
rtot(vac)
1.
In the Alfvén wave
heating case Kmf(t)
(Mt)(1
+
) / 3, instead of
Ksf(t)
(Mt)2/3 in the single-fluid regime. A
multi-fluid shock results in a slower post-shock entropy production.
As we have noted above, the regime of CR-shock compression depends
on the plasma parameter
in the
infalling gas. The plasma
parameter
is currently poorly known because the intercluster
magnetic fields are not well constrained. The effects of shock
modifications are important for both the models of smooth accretion
of cold gas and for accretion of hierarchical structures.
Preheating of accreting gas by different physical processes (e.g. due to early star formation in a protocluster region) was suggested by Evrard & Henry (1991), as a possible reason for the breaking of the scaling relations for pure gravitational cluster compression by [Kaiser 1986]. The observed high metallicity of clusters at different redshifts indicates that strong starburst activity was highly likely at some stage. The preheating produces some initial level of gas entropy ("entropy floor", see e.g. extensive simulations by Bialek et al. 2001, Borgani et al. 2001, Borgani et al. 2005). Multi-fluid strong shocks provide a natural alternative way of preheating accreting gas. The non-thermal components are essential for detailed modelling of global properties of X-ray clusters, including the mass-temperature and luminosity-temperature relations (Ostriker et al. 2005).