Drury et al. (1999) note that 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 u^p), reflecting the existence of an extended upstream shock precursor region sampled on different length scales by particles of different energies. With a momentum-dependent u (p) the logarithmic slope of the spectrum is d ln f = — 3(u1 — 4opL + (p/3)((U1/dp)— op2 (dL1/dp)) (422 13) d ln p U1 — U2 — 3opL
with a pile-up criterion of
U1(p) — 4u2 — p(duxj dp)+ 3a1p2 (dL.1l dp )> 0 at p = pcr, (4.22.14)
where pcr is determined by Eq. 4.22.9. From Eq. 4.22.14 it can be seen 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 U1(pcr) larger they make it easier for pile-ups to occur. On the other hand, if the variation is more rapid than U1(p)— p, the derivative term dominates and inhibits the formation of pile-ups. If the electrons are test-particles in a shock strongly modified by proton acceleration, and if the Malkov (1998a,b)
scaling k(p )— p12 holds even approximately, then a strong synchrotron pile-up appears inevitable (unless the maximum attainable momentum is limited by other effects to a value less than pcr).
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