A&A 378, 309-315 (2001)
DOI: 10.1051/0004-6361:20011191
R. Vainio^{1,2,}^{} - R. Schlickeiser^{2}
1 - Space Research Laboratory, Department of Physics, 20014 Turku
University, Finland
2 - Insitut für Theoretische Physik, Lehrstuhl IV:
Weltraum- und Astrophysik, Ruhr-Universität Bochum, 44780 Bochum, Germany
Received 16 February 2001 / Accepted 27 August 2001
Abstract
Alfvén-wave transmission through a parallel shock in case of an
anisotropic gas pressure (
)
is considered from
the point of view of test particle acceleration in such shock waves. Treating
the shock's gas compression ratio and the firehose factor,
,
on both sides of the shock as given
parameters, we calculate the wave-transmission coefficients for
circularly-polarized Alfvén waves of finite amplitude. We also give an equation
for the shock's gas compression ratio including the effects of pressure
anisotropy and waves. We study wave transmission and test-particle acceleration
in a model, where the downstream plasma is isotropic and the upstream plasma
remains stable against the firehose instability (). The results can
be arranged in a following manner with respect to the upstream parallel plasma
beta,
(with B_{0}being the ordered magnetic field): (i) for shocks with
the results for particle acceleration are only quantitatively different for
anisotropic upstream plasma; (ii) for
,
large
anisotropies alter the wave-transmission and particle acceleration picture
completely relative to the isotropic case yielding much harder energy spectra
with spectral indices
for all shocks with 1<r<4; (iii) for
very hot and firehose-stable plasmas, the qualitative changes in particle
acceleration due to pressure anisotropies are limited to the weakest shocks and,
thus, should be analyzed in a non-linear manner to confirm the results.
Key words: acceleration of particles - shock waves - turbulence
Diffusive shock acceleration (Axford et al. 1977; Krymsky 1977; Bell 1978; Blandford & Ostriker 1978) is a universal mechanism for the acceleration of non-thermal particle populations (Drury 1983; Blandford & Eichler 1987). Since energetic particles can be confined near the shock by plasma turbulence, they are accelerated due to two effects: (i) first-order Fermi acceleration due to multiple crossings of the shock compression front and (ii) stochastic second-order Fermi acceleration in the turbulent fields near the shock. Assuming that the turbulence consists of low-frequency waves on both sides of the shock enables one to investigate the relative importance of these two mechanisms. In this picture, the presence of the second-order Fermi effect requires waves propagating in both directions relative to the mean magnetic field. This condition is always satisfied in the shock's downstream region since an upstream Alfvén wave, when transmitted through a fast-mode shock, converts to two downstream waves propagating in both directions relative to the field (McKenzie & Westphal 1969). However, a consideration of the acceleration time scales of the two mechanisms shows that (i) is much faster than (ii), and should dominate acceleration under the assumption that the upstream waves are self-generated by the cosmic rays streaming against the plasma flow and, therefore, propagating in the backward direction relative to the flow (Vainio & Schlickeiser 1998, 1999).
When the second-order Fermi acceleration is neglected one obtains the canonical
power-law (over momentum) test-particle energy spectrum,
Vainio & Schlickeiser (1998) studied the transmission of Alfvén waves through a parallel shock wave and calculated the resulting scattering-center compression ratio for a model neglecting the effect of waves on the shock's gas compression ratio. They showed that the scattering-center compression ratio tends to infinity at shocks with , where is the upstream plasma beta^{}, and P_{1} and B_{0} are the (isotropic) upstream gas pressure and the ordered magnetic field, respectively. At this limit, the Alfvénic Mach number of the shock, , and the downstream backward Alfvén waves become standing waves in the shock frame yielding formally infinite intensities for these waves. This singularity is not a physical one, and is removed by the inclusion of wave pressure and energy flux to the shock's Rankine-Hugoniot equations (Vainio & Schlickeiser 1999). The scattering-center compression ratios remain high also in the self-consistent wave-transmission model which, therefore, is able to produce test-particle spectra that are harder than the limiting value of occurring for r_{k}=r=4 in the original magnetostatic formulation of diffusive test-particle acceleration at non-relativistic shocks.
Anisotropic gas pressure modifies the dispersion relation of the low-frequency (Alfvén) waves. As noted by Schlickeiser & Vainio (1998), the important effects on particle acceleration in the model of Vainio & Schlickeiser (1998, 1999) were the result of the inclusion of finite phase speeds of the waves into the theory. Furthermore, pressure anisotropies also modify the jump relations at the shock in a non-trivial way. We know from observations that the pressure in the solar wind is anisotropic (see, e.g., Marsch 1991 for a review). Particle reflection from the shock also adds its contribution to the pressure anisotropy, so the gas pressure just upstream a shock is probably anisotropic even if the far upstream conditions were isotropic.
The purpose of this paper is to investigate the influence of pressure anisotropies on the wave transmission problem, shock compression ratios, and the resulting test-particle acceleration at parallel shocks. Although we will discuss briefly the pressure anisotropies associated with nonthermal particles, it is beyond the scope of the present study to develop a theory with self-consistent determination of pressure anisotropies around the shock. Thus, the effects of anisotropy are studied in a simplified way treating the anisotropy of the gas on both sides of the shock as a known parameter, and constraining its values so that the plasma stays stable against the upstream firehose stability criterion. Similar ideas have been employed previously by, e.g., Lyu & Kan (1986) in a study of plasma parameters at shocks measured in near-Earth space.
In a plasma with a non-zero magnetic field, the motion of the particles along
the magnetic field lines differs fundamentally from their motion perpendicular
to the field lines. This points at the necessity of allowing different amounts
of random energy for particle motions parallel and perpendicular to the field,
respectively. One expects the distribution function to have an axial symmetry
around the local magnetic field ,
if the Larmor radii of the
particles are very small compared to the density scale in the perpendicular
direction. Then the gas pressure is a tensor
We consider the conservation laws across a planar shock discontinuity. In
addition to the macroscopic conservation laws, we use Maxwell's equations
and
,
and Ohm's law in form
to
arrive at jump conditions (see also Lyu & Kan 1986)
Pressure anisotropies also modify the dispersion relation of Alfvén waves, which
at long-wave-length limit becomes (e.g., Stix 1962)
Combining Ohm's and Faraday's laws gives the relation
If
and
,
the flow around the
shock is entirely super-Alfvénic. Then all upstream Alfvén waves are convected
through the shock to become downstream Alfvén waves. Some waves are transmitted
retaining their propagation direction relative to the flow, but others get
reflected, i.e., assume the other propagation direction. Alfvén-wave
transmission through a parallel shock wave can be calculated by taking the
tangential magnetic-field and velocity components to be due to waves,
and
.
We shall first
investigate the transmission coefficients in a shock with a pre-described gas
compression ratio. Thus, by using the first four jump conditions,
Eqs. (8-9), together with
Eqs. (15-16) and assuming a
degenerate upstream cross helicity,
,
we find after an amount of straight-forward algebra
Let us, next, calculate the gas compression ratio for a shock propagating with
Mach number M into a gas with parallel plasma beta,
and firehose factor
One should also check for the physicality of the obtained shock solutions. We
will not analyze this in detail, but note only that if the analysis of the
entropy change across the shock is based on ideal gas law, we can write (Lyu &
Kan 1986)
The easiest way to analyze the compression ratio as a function of Mach number is
to solve for b or
in
Eqs. (22-23), regard it
as a function of two variables (r and M), and to contour plot it fixing
the values of other parameters. Thus, e.g.,
Figure 1: Gas compression ratio of a parallel shock with propagating into a plasma with (solid curves), 2.0 (dashed curves), and 10.0 (dot-dashed curves) with b=0.1, and (thin curves) or (thick curves). | |
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The Alfvén-wave transmission coefficients for the shocks in Fig. 1 are given in Fig. 2. We choose outwards-propagating (h=-1) upstream waves consistent with their generation by counter-streaming particles. For the colder upstream gas, the transmission coefficients are affected only a little by the pressure anisotropy, but for the hotter case we see substantial effects; especially the reflection coefficient is substantially increased in presence of anisotropies. Generally, T>R, for h=-1, but for a marginally firehose-stable upstream plasma ( ), the downstream state tends to an equipartition between forward and backward propagating Alfvén waves () as .
Figure 2: The transmission coefficients T and R for parallel shocks with propagating into a plasma with (solid curves), 2.0 (dashed curves), and 10.0 (dot-dashed curves) with b=0.1, and (thin curves) or (thick curves). The dotted line separates the curves for R and . | |
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The most important parameter controlling the energy spectrum of cosmic rays
accelerated by a shock wave is the compression ratio of the scattering centers
that are responsible for the isotropization of the energetic particles. In case
of Alfvén waves generated in the upstream region, we may write
Figure 3: Scattering-center compression ratio of a parallel shock with propagating into a plasma with (solid curves), 2.0 (dashed curves), and 10.0 (dot-dashed curves) with b=0.1, and (thin curves) or (thick curves). | |
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The results for the scattering-center compression ratio were also calculated for the interesting case of at several values of the firehose factor (Fig. 4) to see, how large anisotropies are needed for qualitative effects on particle acceleration from pressure anisotropies. It is evident that relatively small values, below , are needed to produce large effects on particle acceleration. But, for the scattering-center compression ratio is above 4 for all 1<r<4.
Figure 4: Scattering-center compression ratio of a parallel shock with propagating into a plasma with b=0.1 and , with (solid curves) and (dashed curve), 0.5 (dotted curve), and 0.25 (dot-dashed curve). | |
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Perhaps the most unexpected result of our calculation is the ability of weakly compressive ( ) shocks to accelerate particles efficiently in warm plasmas. This result, however, depends crucially on the assumption of the shocks ability to isotropize the fluid. If r differs considerably from unity, one can probably safely assume that the small-wavelength magnetic turbulence generated in the shock transition can be responsible for the isotropization. On the other hand, the transverse magnetic field attains large values across the shocks with regardless of the compression ratio. This can also add to the isotropization of the particles since the field changes direction across the shock at scales comparable to the ion Larmor radius. However, the shocks with small r do not contain large amounts of energy to be given for accelerated particles. When at the same time the test-particle spectral index is below 2, the pressure in accelerated particles diverges without a cutoff in the spectrum. The less there is energy available, the smaller value for the cutoff-momentum must be, and it may well turn out that the weak shocks can not be described by the test-particle picture at all. The confirmation of this requires non-linear analysis, which we shall undertake in the future.
We will briefly discuss the pressure anisotropies created by the shock-related
non-thermal particles assuming non-relativistic particle speeds for
simplicity. For particles accelerated at the shock to speeds clearly exceeding
the shock-frame scattering-center speed, V_{1}, particle scattering results in a
distribution function,
,
that is only weakly dependent on the pitch-angle,
.
In the upstream wave frame, the small anisotropic portion of the
distribution,
where
denotes angle averaging, is antisymmetric relative
to
if the net magnetic helicity of the waves is zero (e.g., Schlickeiser
1989), which we assume. Thus, the pressures due to these particles are
(29) | |||
(30) |
(31) |
Another contribution to the pressure anisotropy comes from the particles that
have just been reflected by the shock and subsequently been picked up by the
upstream waves. Let us, for simplicity, consider a cold background plasma, and
assume that the shock reflects a small fraction of the incoming
background-plasma ions back to the upstream region. We model those particles as
a beam with a density of
propagating along the
magnetic field away from the shock and include also the particles that return to
the shock after their isotropization by the upstream waves. Specifically, in the
upstream wave frame we take the distribution function for the reflected and
returning particles to be
(32) |
(33) |
We have considered the Alfvén-wave transmission and test-particle acceleration problem (Vainio & Schlickeiser 1998, 1999) for shocks with anisotropic pressure. In our model, the firehose factors on both sides of the shock are predetermined parameters. For a detailed analysis, we chose a model with isotropic downstream pressure, and upstream pressure anisotropy that is bounded by the requirement of firehose stability. We showed that the pressure anisotropies have only a minor effect on wave transmission and particle acceleration for plasmas with low . However, plasmas with upstream and seem to develop qualitative effects on both the wave transmission and particle acceleration relative to the isotropic-pressure case. Our study revealed the capability of weak shocks propagating into such plasmas to accelerate particles effectively by creating a large change in the average scattering-center speed across the shock though the increase of the phase speed of low-frequency waves across the shock. Plasmas with large values of beta, , can not develop very large anisotropies ( ) without becoming firehose unstable, which again prevents large deviations in our model between the anisotropic and isotropic cases except for the weakest shocks with M<M_{isotr}(r=1). The low-Mach-number shocks are weakly entropy increasing in the ideal gas model and should, therefore, be studied more carefully using kinetic analysis. In conclusion, the results of our study point out the importance of kinetic analysis of up- and downstream plasma to fully understand the physics of shock acceleration.
Acknowledgements
R. V. acknowledges the financial support of the Academy of Finland (project # 46331) and the PLATON Network (EC contract # HPRN-CT-2000-00153).