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Astron. Astrophys. 347, 370-374 (1999)

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5. Nonlinear effects

At 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 [FORMULA], 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 [FORMULA]; thus [FORMULA] must be a monotonically increasing function of p. Repeating the above analysis with a momentum-dependent [FORMULA] the logarithmic slope of the spectrum is in this case

[EQUATION]

with a pile-up criterion of,

[EQUATION]

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 p. By making [FORMULA] larger they make it easier for pile-ups to occur. On the other hand, if the variation is more rapid than [FORMULA], the derivative term dominates and inhibits the formation of pile-ups.

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 [FORMULA]. 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, [FORMULA], and the momentum flux, [FORMULA] are conserved. Here [FORMULA] is the pressure contained in energetic particles and the gas pressure is assumed to be negligible upstream. At a distance [FORMULA] upstream only particles with momenta greater than p remain in the acceleration region. This suggests that in the "box" model the reaction of the particles on the flow is described by the momentum flux conservation law

[EQUATION]

where [FORMULA] is the highest momentum particle in the system and v is the particle velocity corresponding to momentum p. Differentiating with respect to p gives

[EQUATION]

With no losses and for [FORMULA] we can now recover Malkov's spectral universality result for strong modified shocks (Malkov, 1998). In the limit of [FORMULA] and [FORMULA] the conservation equation reduces to the requirement than the upward flux in momentum space be constant (Eq. (9)),

[EQUATION]

When combined with Eq. (22) this gives

[EQUATION]

where we have used the elementary result from relativistic kinematics that the particle velocity v is the derivative of the kinetic energy T with respect to momentum. Integrating for relativistic particles, [FORMULA], we get the fundamental self-similar asymptotic solution found by Malkov,

[EQUATION]

If the electrons are test-particles in a shock strongly modified by proton acceleration, and if the Malkov scaling [FORMULA] 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 [FORMULA], all the kinetic energy dissipated in the "shock" is used in generating the upwards flux in momentum space [FORMULA] and there is no downstream advection. It is not clear that a stationary solution exists in this case. The problem is that as [FORMULA] so [FORMULA] 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 [FORMULA] negligible compared to [FORMULA]. 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 [FORMULA] 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 [FORMULA]. 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.

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© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999
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