If ts >>t0 +1 then at ts >> t >> t1 a single term remains in the right-hand part of Eq. 2.7.15 which describes the quasi-stationary distribution of particles in the frontal hemisphere. Using this value of F from Eq. 2.7.18, we obtain ts = x-1 >> 1. (2.7.19)
As results from the estimates, during the time interval ts = l/vxo a stream of CR has a specific structure: the frontal hemisphere is completely filled by particles and in the backward hemisphere there are few particles and abrupt gradient in the angular distribution exists near x = xo .
At v ^ 2 qualitative features of the isotropization process remain the same as in the case v = 2 . For the isotropization time the estimate is ts ~ xL~v , which holds true for x1a~v >> 1. A path for scattering at the angle n according to this estimate and Eq. 2.7.11 and Eq. 2.7.13 has the order of value
An additional increase of the free path A occurs if the regular field Ho is inhomogeneous and particles move in the direction of its decrease. Focusing of particles arising as a result of the conservation of the adiabatic invariant sin2#/Ho prevents particle penetration into backward of the angular space. An estimate of ts can be obtained in this case in a following way. In a weakly inhomogeneous field the Eq. 2.7.1 for a stationary case takes a form (x0 < x << i):
where 61 = (// 2)divh = const ; 61 > 0 if the particles move towards the decrease in Ho . The solution of Eq. 2.7.21 with the same boundary conditions as for Eq. 2.7.12
at G << vxV0 1 then gives the same result as in the case H0 = const, and at G >> vxV-1 we obtain
Estimating the isotropization time we obtain
Was this article helpful?