where d(ju) is the diffusion coefficient in angular space, N(z\Ek, t) is the isotropic part of the distribution function. The primed values correspond to the rest plasma system, Z is counted along magnetic lines of force. Since the distribution function is an invariant of the Lorentz transformation, it is sufficient to express the primed terms in the right hand part of Eq. 4.15.66 by those unprimed to the transition to the system of the rest shock front. The above assumptions u << avi, a<< 1, make it possible to write Ek = Ek - pu, |pu| << Ek . Expanding F over the small addition to the energy, we shall have
4n j—,i j—, X 1 „ X dN av dN J dj F (z, E ,u, t) = — N (z, E, t)-pu— -—— J ^
In the small anisotropic addition in Eq. 4.15.66 we have set Ek = Ek, J =U-Furthermore, we have included in one-dimensional case that the distribution function depends only on z so that dzy = dzja. Substituting Eq. 4.15.67 into Eq. 4.15.62 and integrating we have a2v 1 dN2
is the transport path along magnetic lines of force. We shall set below D(u) = D = const and use the expression A// = 2D/v.
The other fluxes are calculated in a similar way, in particular:
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