1. Cosmic-ray acceleration by compressive plasma fluctuations in supernova shells Ming Zhang Department of Physics and Space Sciences, Florida Institute of Technology, Melbourne, Florida 32901, USA
Diffusive shock acceleration
Abstract
Supernova remnants have been considered as the main source of Galactic cosmic rays. In order for supernova shock fronts to accelerate cosmic rays to the knee energy of ~5 PeV, the interstellar magnetic field must be amplified to ~ mG range over a large distance upstream of the shocks. Theory for such strong magnetic field amplification is challenging, and strongly modified supernova shocks may result in a concaved cosmic ray spectrum, which does not agree with observations. In this paper, we present a theory of 2-stage acceleration of cosmic rays by supernova remnants. The first stage is done by the shock waves up to a certain cutoff energy, and it is followed by stochastic acceleration with compressive plasma fluctuations in the downstream region, which extends the cutoff energy. This situation happens as long as the rate of stochastic acceleration is faster than 1/5 of the adiabatic cooling rate. It is possible the amplified magnetic field and strong turbulence inside the supernova shells can yield a condition for cosmic rays to be accelerated to the knee. In this way, we can avoid a few prominent difficulties that we have encountered with the theory of shock acceleration alone.
Diffusive shock acceleration
Compression ration R
Shock spectrum cutoff energy Ec=cpc
Fig Shock spectrum cutoff energy from standard diffusive supernova shock acceleration calculation as a function of age.
Supernova remnants as a source of cosmic rays
There is rather convincing and circumstantial evidence that the bulk of cosmic rays are accelerated in Supernova remnants in our Galaxy. Direct evidence is based on several independent facts: (1) gamma rays unambiguously associated with production of neutral pions have been detected from several supernova remnants close to molecular clouds, (2) the gamma ray emission detected from the Tycho supernova remnant also appears to be most likely of hadronic origin, (3) the bright X-ray rims detected from virtually all young SNRs prove that electrons are accelerated to relativistic energies and the local magnetic field in the shock region has been substantially amplified, and (4) the energy input rate to replenish cosmic rays in the Galaxy, which eventually leaks out of the Galaxy every ~10 Myr, fits to the energy budget if a supernova explodes every 30 or so years. The standard theory describes that cosmic rays are energized at the supernova shock through diffusive shock acceleration. The composition of cosmic rays is consistent with the composition of interstellar medium.
Clearly the maximum cutoff energy is well below the knee. Other factors such as adiabatic cooling and limited shock radius can affect the maximum cutoff energy. They probably will reduce the maximum energy by another order of magnitude.
Knee energy
A solution to the maximum energy problem is the amplification of magnetic field. The amplification must happen to the upstream magnetic field in order to be effective to the particle acceleration by shocks. The amplified field must appear on scales roughly equal to or larger than the gyroradius of particles of highest energies. The upstream magnetic field needs to enhance by a factor of >10 or most possibly 102 after the consideration of energy loss mechanisms. It means the magnetic field ahead of suoernova shock should almost reach a few hundred μG over a few percent of a pc.
  Supernova shocks convert a significant fraction of the kinetic energies to cosmic rays and magnetic field in the form of turbulence. Nonlinear effects becomes dominant (1) The pressure exerted by accelerated particles on the plasma around the shock affects the shock dynamics as well as the acceleration process. The non-linearity appears through the modification of the shock compression ratio that in turn changes the spectrum of accelerated particles in a way that in general depends upon particle rigidity. (2) Sufficient number of cosmic rays can stream upstream as diffusion flux. The flux is strong enough to amplify interstellar magnetic field in the form of turbulence through plasma instabilities. The amplified magnetic field increases particle scattering or reduce particle diffusion coefficient, which speed up the particle acceleration process. The existence of magnetic field amplification is also the most likely explanation of the observed bright, narrow X-ray rims of non-thermal emission observed in virtually all young supernova remnants. (3) The magnetic fields required to explain the X-ray filaments are of the order 100 to 1000mG. If it is in the upstream region, the Alfven speed is much larger than the sound speed of upstream plasma, so it could affect the shock compression, which could in turn affect the particle acceleration.
  Even with the nonlinear diffusion shock acceleration, the calculated spectra of cosmic ray produced by supernova shocks still have major problems: (A) the maximum energy is still one order of magnitude lower than the knee energy ( ) even in the case of abnormally large shock speed of U1=20000 km/s and, (B) the spectral slope in the momentum range above is ~1.5, which is significantly flatter than 2 predicted from the nonlinear theory, and the entire spectrum shows a concaved shape.
Particle acceleration by compressive plasma fluctuations
Fig. 1 All cosmic ray energy spectrum. Bending of spectrum and change of composition occur at the knee energy ~3x106 GeV (from Horandel, 2006).
Model
Nonlinear diffusive shock acceleration
Various forms of Dpp(p) are derived from assumptions about the power spectrum of compressive plasma fluctuations.
Fig Schematics show particle heating and cooling in compressive wave trains.
Fig. 6 A comparison between the spectra from particle acceleration acceleration by a wave train and from momentum diffusion
Results of cosmic ray spectrum after a period of acceleration by compressive plasma fluctuations
Fig. 3 – Calculated downstream proton distribution functions scaled by p4. In the top panel, the shock speed, U10=u0, is varied; in the middle panel, the ambient density, n0, is varied; and in the bottom panel, the free escape boundary location, LFEB, is varied. The dots indicate the maximum momentum (from Bykov et al. 2014)
Conclusion
Fig Schematic viewing of a cosmic ray pressure modified shock wave in the shock frame.
For a fully developed compressive plasma turbulence, S(k) ~ k-4. It is the same as the spectrum for an ensemble of shocklets. This spectrum lead to a Dpp=D0p2 with a constant D0.
Stochastic acceleration can accelerate particles and increase the cutoff energy as long as D0 is faster than 1/5 the adiabatic cooling rate.
The spectrum slope of cosmic rays is the same as the diffusive shock acceleration slope for a constant D0.
Other forms of turbulence spectrum may modified the shape of cosmic ray spectrum if D0 is not a constant. If the S(k) is steeper than ~ k-4, cosmic ray spectrum may show a bump at the knee.
The cutoff energy is determine by k=VwL0, where L0 is the longest wavelength of compressive plasma fluctuations. It is possible that the knee energy is reached through stochastic acceleration inside the supernova shells.
Magnetic amplification in the upstream interstellar medium is not necessary.