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4.18.8. The spatial diffusion in the electric field of the sheet in the case of three-dimensional geometry

The observed discrepancy in previous Section 4.18.7 is associated with the application of the two-dimensional geometry with pure anti-parallel field assuming that Hz = 0. This pattern is explicitly unreal, which can be seen from the difficulties arising due to the transverse electric and magnetic fields which, in case of the undoubted weaker electric field of the sheet, E < EDr ~ 10-(3"4)H, precludes the particle motion in the electric field everywhere in the sheet except for the narrow central zone (Pustil'nik, 1978): Aa ~ 10-(3"4)a . The situation will change if a quasi-homogeneous longitudinal magnetic field Hz ± 0 is inserted in the system. The magnetic field in the sheet will exhibit a shear, the force line will be deflected to the Z-axis (see Fig. 4.18.5), the electric field will be projected onto force line, and the acceleration by the regular field will be possible. Another important consequence of the existence of the longitudinal magnetic field is the increase of the acceleration path depending on the distance to the center of the sheet. In fact, the merging force lines are injected into the sheet at an angle a = X/R, where X is the distance to the center of the sheet, R is the curvature radius of the force lines, of the external field. The distance between the force lines in the sheet increases and the value of the transverse field decreases as H^ ~ Hoa~ Ho (X/R). At the same time, the longitudinal field is invariable, which steepens the inclination of the force lines to the Z-axis as 6~ HH// = X/R . In this case the acceleration path is effectively increased as l = aj6 = aR/X and reaches lmax = L ~ 1010 cm in the center of the sheet.

Fig. 4.18.5. The scheme of the sheet in the frame of three-dimensional geometry. By the thick curve B'A is shown the real 3-D magnetic force line, by broken curve BA is shown the 2-D projection of 3-D magnetic force line (usually used in the 2-D model of current sheet, see Fig. 4.18.4) According to Pustil'nik (1977b).

Fig. 4.18.5. The scheme of the sheet in the frame of three-dimensional geometry. By the thick curve B'A is shown the real 3-D magnetic force line, by broken curve BA is shown the 2-D projection of 3-D magnetic force line (usually used in the 2-D model of current sheet, see Fig. 4.18.4) According to Pustil'nik (1977b).

If the power spectrum Emax ^ 1/X with D(E) x E Y is formed in each sector A/, it can be easily shown (Pustil'nik, 1978) that the total spectrum D * (E) will also retain a power form but with y* = y+1 and the maximum particle energy

Emax = ZeEL = EoL/a ~ (l ^ 10) GeV. Thus, the following result is obtainable for the considered above case: the quasi-diffusion in the three-dimensional sheet generates the spectrum of ejection from the entire sheet D * (E E "3 with cut-off at high energies Emax observations.

(l ^ 10) GeV, which is in a good agreement with

4.18.9. Comparison of the quasi-diffusive acceleration and stochastic acceleration on the Langmuir plasmons

It can be seen from the comparison between the effectiveness of the quasi-diffusive acceleration (index qd) and stochastic acceleration (st) on the Langmuir plasmons (Pustil'nik, 1978):

A f r ElEDr

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