А. V. Artemyev et al qeq = qeq(\= ДЛ), П'я = /?«,(*=AX), mp is the proton mass, Tp is the proton temperature (in this study we use Tp -2110 C“), R e is the Earth radius, L is the shell number and 1д=5.78 is Roche limit, a/oade [ 0,1] is a filling factor (see details in ( Verigin et al., 2012)), c o s 4 = ^ £ ± ^ L (3) l r e and /7CI=1000 km is height of the exobase. Model profiles of the electron (or plasma) density along the field lines are shown in Fig. 1(a) with the approximations given by the IMAGE RPI experiment ( Denton et al., 2006). One can see that model (1) has a weaker gradient of the plasma density in the vicinity of the equator. Additionally, model (1) allows to consider the shift of the plasma density minimum away from the equator (AA^O). Statistics collected by the ALPHA-3 experiment indicate that ДА can be as large as 15°. In the following section we consider the effect of such shift of the plasma density minimum from the equator. 1- p -1—i—1 p*t—-1— —,—|—,—| 1 -45 -30 -15 0 15 30 45 -45 -30 -15 0 15 30 45 Figure 1. The distribution of the plasma density along the field lines for different system parameters (1=2.5). Ii. Wave-particle resonant interaction To investigate the efficiency of the wave-particle resonant interaction we calculate the pitch-angle diffusion coefficients describing relativistic electron scattering by whistler waves ( Trakhtengerts, 1966; Kennel and Petschek, 1966). The local (for fixed magnetic latitude) expression for pitch-angle diffusion coefficient Daa can be written as (Glauert & Horne, 2005): D max - I f « X T g ( X ) X d X y e2co2Bl(col) ' n d j у - sin2 a ^ 2 2 J l +X 2 4 - 4 л :Щ ) min v cos a vco s a - d c o ! dk^ (4) where Х=Хгп(в) and в is the angle between wavevector к and a background magnetic field, Qc is the electron local gyrofrequency, a is the electron pitch-angle, C 0 j is the resonant frequency given by the resonance condition co,- focos(d)cos(e)=-nQJy, v is the amplitude of electron velocity, and у is the electron gamma factor. In Eq. (4) the index / denotes the resonance number, while n is the harmonic number. The function Ф„* determines the relation between wave electric and magnetic fields for resonant waves (Lyons, 1974). To describe electron resonant interaction with hiss waves we use the simplified dispersion relation: Q c cosd 1 + ((Dpe/ k c ) (5) where is the plasma frequency. Functions g(X), and Щщ) determine wave spectrum, в distribution and its normalization: Bl = H As exP f ( Л2Л 8co] S = exp 5 X 2 2 \ N = - -^max g ( x ) x d x k 2d k 2 n 2 / (1 + X 2y n d a \ ( 6 ) 56

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