Vn cos 6
where Q is the field-shock normal angle, the subscript 1 refers, as usually, to upstream of the shock, and 2 refers to downstream. The particle momentum increases at each acceleration event according to p = 1 + 4 u1cos61 -u2cos62 . (4.21.89)
The differential energy spectrum vs. kinetic energy Ek is calculated from j(Ek, t ) = P(n, t )/(Tn+1 - Tn). (4.21.90)
Fig. 4.21.3 shows results of Ruffolo and Channok (2003) for the time dependent energy spectrum of 4He for an oblique shock wave with u1 = 540 km/s, u2 = 140 km/s, 6= 45°, 62 = 75°, k = vX/3, and a parallel scattering mean free path X = 0.3
AU (based on Rankine-Hugoniot conditions; see Ruffolo, 1999). Note that this corresponds to injection at 0.01 MeV/nucleon; for an interplanetary shock the resulting spectrum would be the convolution of such a 'kernel' with the seed particle spectrum. We see that after a short time the particles receive only a small boost in energy. At intermediate times, there is a power law at low energy and a hump at a certain critical energy, Tc, followed by a drastic decline. The power law and hump correspond to the two terms on the right hand side of Eq. 4.21.84; in particular, the hump corresponds to the fraction of particles that have not yet escaped and have a Poisson distribution of acceleration events n, with (n) ~ rat. It is not clear whether a hump would be expected in observations, after convolution with the seed spectrum. The decline at high energy is qualitatively similar to that of Ellison and Ramaty (1985); however, Ruffolo and Channok (2003) obtain a different (Q/A) dependence, as shown below. At very long times the classic steady-state power law is recovered.
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