Physics of auroral phenomena : proceedings of the 40th annual seminar, Apatity, 13-17 March, 2017 / [ed. board: N. V. Semenova, A. G. Yahnin]. - Апатиты : Издательство Кольского научного центра РАН, 2017. - 143 с. : ил., табл.

Theory o fa receiving antenna applied to the spacecraft observations o fquasi-electrostatic whistler mode waves Here q = ( d p ! (l + [//(й^)]2) , // = cot , e = exx = and /7 = ^ are the transverse and longitudinal components of relative permittivity tensor calculated for the carrier frequency, respectively (z-axis is parallel to the local geomagnetic field), к is the wavenumber, у/ is the azimuth angle in k -space, p a^{k) is the spectrum (calculated on the resonance cone) of the trial charge distribution on the receiving antenna; Ar, to, and p„(k) are the length of a fictituous electric dipole source, its distance to the receiver along the group velocity resonance direction, and the spectrum (calculated on the resonance cone) of its charge distribution p tI(r) . The reason why we use this source model is that we need to specify the incident wave field. Since chorus generation is a complicated nonlinear process, it is quite difficult to write down an expression for its electric field. However, it is possible just to introduce a fictituous (or effective) electric dipole producing the detected field characteristics, and that is what has been done. Two main properties o f wave packets we want to model are (i) the field should have wave normal angles close to the resonance cone, and (ii) the wave number spectrum is assumed fairly broad which seems to be a natural property of resonant whistler-mode emissions. We make this simplifying assumption since we do not know actual wave number spectrum of quasi-electrostatic chorus waves. These two assumptions allowed us to specify a simple electric dipole model of effective source of the measured radiation field. Such a source does not need to coincide with an actual chorus source. According to the above discussion, we choose p„(r ) in the form that corresponds to a thin dipole of length /,r , directed along the z-axis: A r(r) = - 8? - z e x p 4r f 4z2^ l l j 8{x)S(y), (4) where QtI is the total half-dipole charge on the effective transmitter. Then in 'N v\o Г/ (5) res 16 The described shape of effective source is chosen for the sake of symmetry and simplicity. The charge distribution along z is smooth which means that the source region has no sharp boundaries. We limit ourselves by the dipole approximation and do not consider any multipoles of higher orders, because the dipole charge distribution relatively easily provides the wave field with measured parameters if this field corresponds to a quasi-electrostatic wave packet with a spread in wave vectors. Indeed, length /„ of this effective transmitter is determined by the wavenumber kobs that corresponds to the observed spectral maximum: ^oJtrCOS^cs = 2 / 2. (6) Let us now deal with the trial charge spectrum p ^ . If the receiving dipole consists of 2 thin straight rods with a gap between them, then the current distribution along it can be chosen as a triangular one [Chugunov et al., 2015], and 8 i Pok (*> И = - ■— exp[-z7ctf0 (y,)] sin2 / \ r / rec (7) Here —sin a sin вт cos(y/—/$) + cos a cos &KS, R4)(y/) = x0sin 6>rc5cos y/ +y a sin 6>ressin y/ +z 0 cos ; x0, y0, and z0 are the receiver Cartesian coordinates (see Fig. 1): x0 = r 0cos<9rescos (pobs, y0=T0cos вт sin <p obs, z0 = r0sin0res; (8) (Pobs is the azimuth angle of the incident wave, a and /3 are the receiver orientation angles: a is the angle between the geomagnetic field and the dipole axis, and [1 is the azimuth angle o f the dipole. If the receiver consists of 2 small, as compared to the distance between them, spherical conductors placed on the thin metal rod, then it may be represented as 2 point charges: YiwWrt PQk(k, Y ) = -2 i exp [~ikR0(y/)\ sin 2 (9) In the following, we will consider quasi-monochromatic (at each moment of time) wave packets, and therefore will choose o>=&>0 which simply means appropriate choice of a>0 for each spectral component. Importantly, this does not prevent к to vary in a wide range due to the resonance wave dispersion. Consequently, the only effective source parameters that determine the receiver effective length are ltT and r0 . Length /tr is determined by the wave and plasma parameters ( kobs and 0re8) according to (6), and r 0, generally speaking, is a free parameter. Let us discuss its choice. As it was shown in the previous studies using ray tracing [Chum axid Santolik, 2005], the chorus wave normal angle changes significantly due to refraction on the distance corresponding to the geomagnetic latitude Лт change of 1°. Therefore, in order to neglect the refraction effects on 59