# Collisionless shock wave

J. M. Laming (2009), Scholarpedia, 4(7):6071. | doi:10.4249/scholarpedia.6071 | revision #91138 [link to/cite this article] |

A **collisionless shock** is loosely defined as a shock wave where the
transition from pre-shock to post-shock states occurs on a
length scale much smaller than a particle collisional mean free path. The reason such a structure can exist is because particles
interact with each other not through Coulomb collisions, but by
the emission and absorption of collective excitations of the
plasma; plasma waves. In nonrelativistic collisionless shocks,
there is a belief, though not proved, that a pre-existing
magnetic field is necessary to allow the existence of such
plasma waves. The situation concerning relativistic shocks is
less clear, where even in the absence of pre-existing magnetic
field, at least in simulations, the colliding flows can
generate large scale magnetic field through the Weibel
instability (Medvedev & Loeb 1999). A similar phenomenon has been reported
for nonrelativistic shocks (Kato & Takabe 2008).

## Contents |

## Classifications

Given the apparent necessity of magnetic field, collisionless shocks are usually discussed within the framework of magnetohydrodynamics (MHD), even though this approximation clearly breaks down at the shock itself. Shocks are usually classified as "quasi-perpendicular" or "quasi-parallel" depending on whether the pre-shock magnetic field vector is aligned closer to the shock front (quasi-perpendicular) or to the shock velocity vector (quasi-parallel). In the exactly parallel case, the MHD shock reduces to a one dimensional hydrodynamic shock. Arbitrary magnetic field geometry shocks are further classified into slow-mode or fast-mode shocks, depending on whether the magnetic field along the shock front decreases or increases going through the shock. Slow mode shocks require an acoustic Mach number \(v_{shock}/v_{sound} > 1\ ,\) but an Alfven Mach number \(v_{shock}/v_{Alfven} < 1\) (i.e. magnetic energy density > gas pressure in the upstream medium). Fast mode shocks require \(v_{shock}/v_{Alfven} > 1\) in parallel propagation, \(v_{shock}/\sqrt{v_{Alfven}^2+v_{sound} ^2} > 1\) in perpendicular propagation. With these constraints, most shocks observed in astrophysics are fast-mode shocks.

One should add to this classification the shocks which occur in the absence of the the ambient field or when this field is very weak. Theories of such shocks (known as electrostatic shocks) involve Langmuir (electrostatic) waves or ion-acoustic waves, depending on plasma conditions, such as the electron and ion temperatures, for example. Early numerical simulations performed in 1D did indicate the formation of the electrostatic (Langmuir) shocks; however there are recent indications that they are not seen in higher dimension simulations.

## Kinetic Features

An important feature of collisionless shocks is the different ways electrons and ions respond to the shock transition. Since there are no collisions to enforce thermal equilibrium, naive application of the Rankine-Hugoniot jump conditions to electrons and ions separately would predict the electron-ion temperature ratio to be the same as the electron-ion mass ratio. Observations of shocks in supernova remnants have shown that this is to a large extent true (Ghavamian et al. 2007). The ions are indeed much hotter than the electrons, although the electrons appear to be heated by a small amount over the temperature predicted by the jump conditions. Similar results are found downstream of higher Mach number shocks in the solar wind (Schwartz et al. 1988). The lack of collisions enforcing thermal equilibrium also allows collisionless shocks to reflect particles out of the thermal pool from the shock ramp into upstream direction. Indeed, such a phenomenon may well be necessary to provide the dissipation required at supercritical shocks, where plasma resistivity alone is no longer sufficient to maintain the shock.

## Relativistic Shocks

Under certain conditions, the region of the main shock jump cannot be considered in the MHD approximation because no MHD waves exist to mediate the particle-particle interactions. These shocks are therefore intrinsically kinetic. A particularly interesting example of kinetic shocks are relativistic collisionless shocks which propagate at speeds comparable to the speed of light and the corresponding Lorentz factors, \(\Gamma >>1\ ,\) \( \Gamma=1/\sqrt{1-v^2/c^2}\ .\) Medvedev & Loeb (1999) have theoretically predicted that the Weibel instability plays a crucial role in their formation and propagation. Numerous subsequent studies via numerical modeling confirmed this prediction. These shocks produce strong magnetic fields, with energy density as high as ten percent of the shock kinetic energy density, from scratch, i.e., they can magnetize the initially only weakly or even completely unmagnetized and homogeneous medium.

An even more interesting feature of the Weibel-mediated shocks is that the produced magnetic fields are so small in scale that the particles (electrons) start moving along random paths rather than executing a regular Larmor rotation about a particular field line. This leads to an intrinsically different radiation process -- an alternative to the synchrotron emission -- namely, the jitter radiation (Medvedev 2000, 2006). The jitter radiation being produced by electrons deflected randomly in the Weibel fields probe the magnetic field distribution over small scales, and this information can, in principle, be deduced from observational data. Gamma-ray burst spectral observations show that more than one fourth of the spectra are inconsistent with the synchrotron radiation emission mechanism from collisionless relativistic shocks (Preece et al 2000, Kaneko et al 2006). In contrast, jitter radiation is not only consistent with the data but also explains why and how their spectra vary so fast.

## Low and High Alfven Mach Numbers

At low Alfven Mach numbers (\(1 < M_A < 2-3\)), a quasi-perpendicular fast-mode MHD shock may be represented as a laminar structure. A cross-B potential arises because of the different responses of ions and electrons to magnetic fields in traversing the shock. The potential is directed so as to decelerate ions and accelerate electrons (in the shock rest frame). At higher Mach numbers, the laminar solution no longer holds. The cross shock potential produced by steady state shock compression of the magnetic field (a factor of 4 in \(\gamma = 5/3\) gas) is no longer sufficient to provide the ion deceleration required to establish a shock. Another process must provide the necessary dissipation. At these higher shock velocities, the ion motion with respect to the upstream plasma and magnetic field is able to generate plasma waves through a two stream (or modified two stream) instability. Instability thresholds are typically the ion thermal (or ion acoustic) speed or Alfven speed, with the growth rate increasing as the relative velocity increases above this. The growing waves are able to interact with the unshocked plasma and decelerate the incoming ions. At quasi-parallel shocks, it is believed that no magnetic field compression takes place and hence no cross-shock potential exists. However, the physics is much more complicated by wave particle interactions. At relative velocities greater than the upstream Alfven speed, a firehose instability may occur that excites Alfven waves, which can then serve to decelerate the incoming unshocked ions.

## Diffusive Shock Acceleration

Ions in the high energy tail of the upstream Maxwellian distribution function with larger gyroradii are generally able
to penetrate further through the shock structure before
interacting with turbulence in being decelerated. In extreme
cases, they may even be reflected back upstream, whereupon they
either gyrate around field lines (in a quasi-perpendicular
shock) or generate more turbulence through a firehose
instability in a quasi-parallel shock, before returning back to
the shocked plasma. This process can repeat itself, the
energetic ions gaining yet more energy as they bounce back and
forth between two converging plasma flows. The process is known
as diffusive shock acceleration, or first order Fermi
acceleration (since the energy gain is first order in the
particle velocity, to distinguish it from Fermi's original
concept of stochastic acceleration in stationary conditions
where the energy gain is second order). The predicted particle momentum
spectrum in the test particle limit, \(f(p)\propto p^{-3r/(r-1)}\ ,\) where
\(r\) is the (nonrelativistic) shock compression ratio, is in good agreement with
observations for \(r\sim 4\) for strong shocks in monatomic gas. This process is
believed to be the mechanism by which cosmic rays are
accelerated, at least up to the "knee" (energies of order 10^{15}
- 10^{16} eV). If we identify this energy with that at which the
cosmic ray gyroradius becomes larger than the typical supernova
remnant dimension, a magnetic field of order 100 micro Gauss is
implied; significantly larger than that that would be inferred
from shock compression of a typical interstellar medium
magnetic field of 3 micro Gauss. A solution to this dilemma has
been proposed (Bell 2004) whereby the cosmic rays upstream of a supernova
remnant shock amplify the existing magnetic field. This magnetic
field appears to be predominantly generated in the plane of the
shock front, producing a quasi-perpendicular shock.

Diffusive shock acceleration at ultra-relativistic shocks appears to yield a universal power law for the accelerated particle distribution function, \(f(p)\propto p^{-4.23}\ ,\) independent of the shock compression ratio, at least for quasi-parallel shocks (Achterberg et al. 2001). Variations from this universal law for transrelativistic shocks, and for varying obliquities are investigated by Ellison & Double (2004).

## References

- Achterberg, A., Gallant, Y., Kirk, J. G., & Guthmann, A. W. 2001, MNRAS, 328, 393
- Bell, A. R. 2004, Monthly Notices of the RAS, 353, 550
- Ghavamian, P., Laming, J. M., & Rakowski, C. E. 2007, ApJ, 654, L69
- Ellison, D. C., & Double, G. P. 2004, Astroparticle Physics, 22, 323
- Kato, T. N., & Takabe, H. 2008, ApJ, 681, L93
- Kaneko, Y. 2006, ApJS, 166, 298
- Medvedev, M. V. 2006, ApJ, 837, 869
- Medvedev, M. V. 2000, ApJ, 540, 704
- Medvedev, M. V., & Loeb, A. 1999, ApJ, 526, 697
- Preece R. D., et al 2000, ApJs, 126, 19
- Schwartz, S. J., Thomsen, M. F., Bame, S. J., & Stansberry, J. 1988, J. Geophys. Res., 93, 12923

**Internal references**

- Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.

- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.

- Rainer Beck (2007) Galactic magnetic fields. Scholarpedia, 2(8):2411.

- Søren Bertil F. Dorch (2007) Magnetohydrodynamics. Scholarpedia, 2(4):2295.

- Peter Goldreich (2009) MHD turbulence. Scholarpedia, 4(2):2350.

- Eugene N. Parker (2008) Parker Wind. Scholarpedia, 3(9):7505.

- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

## Recommended Reading

- Blandford, R. A., & Eichler, D. 1987, Physics Reports, 154, 1, Particle Acceleration at Astrophysics Shocks: A Theory of Cosmic Ray Origin
- Drury, L. O'C., 1983, Reports on Progress in Physics, 46, 973, An Introduction to the Theory of Diffusive Shock Acceleration of Energetic Particles in Tenuous Plasmas
- Malkov, M. A., & Drury, L. O'C. 2001, Reports on Progress in Physics, 64, 429, Nonlinear Theory of Diffusive Acceleration of Particles by Shock Waves
- Tidman, D. A., & Krall, N. A. 1971, Shock Waves in Collisionless Plasmas (New York: Wiley-Interscience)