## 5. Nonlinear effectsAt the phenomenological and simplified level of the "box" models it
is possible to allow for nonlinear effects by replacing the upstream
velocity with an effective momentum-dependent velocity
, reflecting the existence of an
extended upstream shock precursor region sampled on different length
scales by particles of different energies. Higher energy particles,
with larger diffusion length scales, sample more of the shock
transition and have larger effective values of
; thus
must be a monotonically increasing
function of with a pile-up criterion of, We see that whether or not the nonlinear effects assist the
formation of pile-ups depends critically on how fast they make the
effective upstream velocity vary as a function of In most cases the shock modification will be produced by the
reaction of accelerated ions, and the electrons can be treated as
test-particles with a prescribed .
However in a pair plasma, or if one applies the "box" model to the
ions themselves, the effective upstream velocity has to be related to
the pressure of the accelerated particles in a self-consistent way. We
require in the "box" model a condition which describes the reaction of
the accelerated particles on the flow. Throughout the upstream
precursor and in the steady case both the mass flux,
, and the momentum flux,
are conserved. Here
is the pressure contained in
energetic particles and the gas pressure is assumed to be negligible
upstream. At a distance upstream
only particles with momenta greater than where is the highest momentum
particle in the system and With no losses and for we can now recover Malkov's spectral universality result for strong modified shocks (Malkov, 1998). In the limit of and the conservation equation reduces to the requirement than the upward flux in momentum space be constant (Eq. (9)), When combined with Eq. (22) this gives where we have used the elementary result from relativistic
kinematics that the particle velocity If the electrons are test-particles in a shock strongly modified by proton acceleration, and if the Malkov scaling holds even approximately, then Eq. (20) predicts that a strong synchrotron pile-up appears inevitable. It is perhaps worth remarking on some peculiarities of Malkov's solution. Formally it has , all the kinetic energy dissipated in the "shock" is used in generating the upwards flux in momentum space and there is no downstream advection. It is not clear that a stationary solution exists in this case. The problem is that as so if a diffusion model is used for the downstream propagation. The solution appears to require some form of impenetrable reflecting barrier a finite distance downstream if it is to be realised in finite time. Also, although the accelerated particle spectrum at the shock is a universal power law, none of these particles escape from the shock region. From a distance the shock appears as an almost monoenergetic source at whatever maximum energy the particles reach before escaping from the system. The case of a synchrotron limited shock in a pure pair plasma is also interesting. Here the upper cut-off is determined not by a free escape boundary condition but by the synchrotron losses. If most of the energy dissipated in the shock is radiated this way, the shock will be very compressive and the downstream velocity negligible compared to . The same caveats about time scales apply as to Malkov's solution, but again we can, at least as a gedanken experiment, consider a cold pair plasma hitting an impenetrable and immovable boundary. In this case, if there is a steady solution, the upward flux due to the acceleration must exactly balance the synchrotron losses at all energies. In general it appears impossible to satisfy both this condition and the momentum balance condition for unless the diffusion coefficient has an artificially strong momentum dependence. However a solution exists corresponding, in the box model, to a Dirac distribution at the critical momentum . This steady population of high energy electrons has enough pressure to decelerate the incoming plasma to zero velocity and radiates away all the absorbed energy as synchrotron radiation. This extreme form of pile-up may be of interest as a means of very efficiently converting the bulk kinetic energy of a cold pair plasma into soft gamma-rays. © European Southern Observatory (ESO) 1999 Online publication: June 18, 1999 |