Whistler mode resonance-cone transmissions at 100 kHz in the OEDIPUS-C experiment


  • Deceased: 8 September 2012.

Corresponding author: H. G. James, Communications Research Centre Canada, Ottawa ON K2H 8S2, Canada. (gordon.james@crc.ca)


[1] A radio transmitter was operated at one end of the tethered sounding rocket double payload OEDIPUS C, and a synchronized receiver at the other end. Both the transmitter and the receiver were connected to “double-V” dipoles. On the flight downleg after the tether had been cut, direct bistatic propagation experiments were carried out successfully with the transmitter-receiver pair. This paper addresses the transmission of 300-μs pulses at a carrier frequency of 100 kHz between the dipoles over distances of about 1200 m. The waves of interest propagate in the whistler mode close to its resonance cone, where the transmitter is situated in the cone apex. The radiated field under these conditions is computed as well as the resonance response of the receiving antenna, i.e., its effective length. In the whistler mode, the influence of the plasma is important and it results in qualitative changes in the structure of the radiated field and in the value of the receiving antenna effective length as compared to the free space case. Our main concern is the excitation and reception of a pulsed signal when time and space dispersion play important roles in both the delay and spreading of such a signal.

1. Introduction

[2] The OEDIPUS-C rocket experiment was launched at the Poker Flat Research Range in central Alaska, U.S.A. at 06:38 UT on 7 November 1995 [James and Calvert, 1998]. Among its principal results, this double-payload mission has provided new perspectives on plane, plasma-wave propagation in magnetoplasma between dipole antennas. The rocket payload reached an apogee altitude of 824 km over northern Alaska about 8.6 min after launch. During the early part of this flight its forward and aft subsections were aligned with the terrestrial magnetic field B and gradually separated as a 24-gauge conducting tether was unreeled between these two sections. Then, at about 1.8 min after apogee, the tether was cut at both ends in order to provide two free-flying subpayloads with a starting separation T of 1174 m aligned within about 5° of B for the remainder of the flight. The 100-kHz observations analyzed here were recorded in the latter, free-flying part of the flight.

[3] The forward subpayload contained a digitally controlled radio transmitter called the High Frequency Exciter (HEX) and the aft subpayload contained a corresponding Receiver for Exciter (REX) which was tuned to the same frequency. Both subpayloads used monopole antenna elements connected electrically as V dipoles. That is, pairs of adjacent orthogonal monopoles were connected together to produce the “V” shape connected to each antenna terminal. The V dipole thus formed was equivalent to a linearly polarized antenna perpendicular to the magnetic field at each end of the propagation path. At the instant of the subpayload separation, the two effective linear dipoles were parallel. Both subpayloads were spin-stabilized along the B axis. By the time of the observations under discussion close to the end of the flight, the total azimuth of the forward dipole led that of the aft by about 800°, because of a slightly higher forward spin rate (0.087 rps). For the observations reported in this paper, the HEX and REX were operated in a pulsed swept-frequency mode with a 300-μs pulse and a 3-ms listening period. The HEX pulse frequency and the REX band center frequency were swept synchronously from 0.1 to 8.0 MHz in 50-kHz steps.

[4] Further details about the OEDIPUS-C flight and the operation of the HEX-REX pair can be found inJames and Calvert [1998]. During the time after launch (TAL) period from 865 to 930 s, the HEX-REX physical geometry and plasma parameters conspired to make the group resonance cone of the whistler mode [Budden, 1985] centered at HEX contain the HEX-to-REX separation direction for some frequency in the REX receiver bandwidth centered at 100 kHz. That is, the so-called resonance cone frequencyfrc lay somewhere in 100 ± 33.3 kHz. Figure 1 shows the geometry of HEX and REX in relation to the group resonance directions, in broken line, for three different whistler mode frequencies, of which f2 = frc.

Figure 1.

The relative locations of the HEX and the REX with respect to the direction of the magnetic field B. The group resonance directions are shown in broken line for three different whistler mode frequencies. The frequency f2 = frcis the frequency that makes the HEX-REX separation direction lie in the group resonance cone.

[5] The evolution of the spectrum of the received 100-kHz pulses during the aforementioned period [James, 2006] is shown in Figure 2, overlaid with the locus of frc. Figure 2 is the graphical juxtaposition of 82 individual spectra recorded in 850 < TAL< 932 s. As expected, the frequency spectra of 300-μs pulses peak at 100 kHz with side lobes spaced at 3.33 kHz. Notice, however, that the signal amplitudes are significantly enhanced at frequencies close to frc. This paper analyzes the enhancement of wave amplitudes detected at REX after transmission from HEX along directions near the group resonance cone.

Figure 2.

Evolution of the spectrum of the transmitted 100-kHz pulse during the last 82 s of the OEDIPUS-C rocket mission. The thin oblique black line shows the position of the resonance cone frequencyfrc belonging to the line connecting the gaps of the transmitting and receiving antennas.

[6] The periodicity with time in Figure 2 arises from the spin of the emitting and receiving dipoles. So the periodicities of the spin modulation in TAL and of the pulse spectrum in frequency combine to give Figure 2 a “tartan” background pattern.

[7] What is new in the present paper compared with earlier work on OEDIPUS-C whistler mode transmissions is our taking into account the effects of dispersion on both the amplitudes and spectral shapes of the transmitted 100-kHz pulses. After a short review insection 2 of the oblique whistler mode resonance, section 3 sets up the formalism of the dipole radiated electric field for a general dipole, then section 4 shows how this works out in the cases of a distributed dipole and a point dipole. In Section 5we proceed to evaluate the radiation field for the OEDIPUS-C emitting dipoles.Section 6 concludes the paper with computations of the receiving dipole induced voltage, in which the reciprocity principle allows us to compute the total effects of both incident and scattered field at the receiving dipole.

2. Resonances in Magnetized Plasma

[8] An important factor that allows for the formation of singularities in the radiation field of antennas is the existence of proper electrostatic modes, called plasma resonances [Andronov and Chugunov, 1975]. A large part of the energy exciting the antenna can go to resonant excitation in some cases.

[9] Anisotropic plasmas generally support two electromagnetic modes, ordinary and extraordinary, and their wavelength depends on the angle θ between the direction of propagation and the external magnetic field. The whistler mode is one of the two modes for frequencies in a specific frequency range [Budden, 1985]. For the study of radiation of continuous-wave (CW) sources, it is convenient to use the surface of wave numbers ω inline image = const.

[10] As is well known, in some frequency bands the wave surfaces are open and have a hyperbolic shape. In the vicinity of the asymptotes, the wave number is high, and k(ω) → ∞ on the asymptote. At a given frequency the angle θ between the external magnetic field and the wave vector is given by ctgθ = inline image, where ε1(ω) and ε3(ω) are the diagonal components of the dielectric tensor across and along the magnetic field, respectively. In the vicinity of these asymptotes the wave electric field inline image is described by a scalar potential alone inline image = −φ, the wave magnetic field can be neglected and the wave is quasielectrostatic. In general, there are three such frequency intervals given by the condition and ε1(ω)ε3(ω) < 0. In the present experiment, ωωpe, ωHe, which are the electron plasma frequency and the electron gyrofrequency respectively, and ctgθinline image. If the characteristic dimension of the source is much lower than the electromagnetic wavelength, the source effectively excites the eigen oscillations of the plasma that propagate as quasielectrostatic modes; these conditions are referred to as resonance conditions.

3. Structure of the Quasielectrostatic Field Under the Resonance Conditions

[11] In a homogeneous magnetized plasma, the equation for the potential φ is

display math

where ρext is the externally induced charge on the antenna and inline image, in general, is an integro-differential space-time operator [Andronov and Chugunov, 1975]. Its Fourier transform is the dielectric tensor εαβ(ω, inline image. For the external charge, we take a pulsed CW signal given by the expression

display math

where ρ inline image is the charge distribution on the antenna, ΠT (t) is the unity rectangular-pulse function in the interval t ∈ [0, T], ω0 = 2πf0, and f0 is the carrier frequency.

[12] The solution of equation (1) can be written as a Fourier integral over space and time variables

display math

[13] Here ρext(ω, inline image = ρ( inline image inline image, ρ inline image denotes the Fourier spectrum of ρ inline image, and inline image is the longitudinal dielectric permittivity of the plasma. We will look for the field in the vicinity of the resonance surface r = μ0z in the coordinate space, where r2 = x2 + y2, μ0 inline image, and the z coordinate is in the direction of the external magnetic field. Following Mareev and Chugunov [1987] we rotate the coordinate system (x, y, z) by the resonance angle γ0π/2 − θ0 about the y axis, where γ0 is as shown in Figure 1. Without any restriction, we can generally assume that the direction of interest lies in the plane xz. In the new coordinate system (τ, y, ξ) the τ coordinate is directed along the resonance surface and the ξ coordinate is perpendicular to it.

[14] In this new coordinate system we have now

display math

where inline image, inline image is the dispersion equation [Mareev and Chugunov, 1987], and Γ(ω) = (exp(i(ω − ω0)T) − 1)/i(ω − ω0) is the Fourier spectrum of ΠT(t). Our dispersion equation is the generalization and decomposition of Appleton's equation near the whistler mode resonance cone taking into account thermal, electromagnetic and collisional corrections [Appleton, 1924; Helliwell, 1965].

[15] In the D(ω, inline image function the corrections that are of importance under resonance conditions are included. The correction proportional to Δμ ≡ μ(ω) − μ(ω0), inline image, is due to the finite pulse length. In the term inline image are included the thermal, electromagnetic and collisional corrections. In a resonance frequency band inline image can be written as

display math

where δ = VTe/ω, VTe is the electron thermal velocity, kx and kz are functions of kτ and kξ, κ2 = kx2(kτ,kξ) + ky2, A = −3ωpe2ω02/4ωHe4, B = ωpe2/ωHe2, C = −3ωpe2/ω02, ε1 = 1 + ωpe2/ωHe2, δε1 = pe2/ωHe2, ε3 = −ωpe2/ω02, δε3 = pe2/ω02, inline image and inline image, where νe is the collision frequency of electrons [Mareev and Chugunov, 1987].

[16] In (4) we integrate over kτ with the use of the residue theorem at D(ω, kτ, kξ, ky) = 0. The integration over ky uses the method of stationary phase as τL, (Lis the antenna half-length) and we take into account only the radiated field for which ∂ω/∂kτ > 0, i.e., the waves going away from the source (see Mareev and Chugunov [1987] for details).

[17] We introduce the following variables

display math

where factors λ and R depending on frequency take into account that in the 100 kHz frequency range they differ somewhat from those in Chugunov et al. [2003] because of higher values here of ωpe2/ωHe2. In addition, frequency dispersion plays an important role at 100 kHz in the radiation and reception of pulsed signals. The parameter q (ωPe2 + ωHe2)1/2/ωPeωHe makes a correction in phase of the order of ΔΨ ≃ 2πxqτf0(f/f0 − 1). This correction, proportional to the carrier frequency, is unimportant at 25 kHz [Chugunov et al., 2003] but not negligible here.

[18] The lengths ξ, λ, τ, and R are now normalized by L, and charge density is normalized to Q, where Q is the amplitude of the charge on the antenna. The variable x = kξL is introduced to get:

display math

where now, φ0 =  inline image, and Ω = ω − ω0. This shows that along the resonance surface the field E goes down slowly as τ−1/2 and its perpendicular structure is determined by the charge distribution on the antenna. Note also that this wave packet is a superposition of plane waves propagating essentially in one direction, their phase velocity is along the ξ coordinate perpendicular to the resonance surface, and the resulting electric field is wavelike.

[19] We now integrate over frequency to obtain for the electric field Eξ(τ,ξ,t) = − ∂φ/∂ξ:

display math

where Πt(t − qτx) is a unit function in the interval (t − qτx) ∈ [0,T], t > 0, x > 0, and E0 = − πiφ0/L.

[20] Consequently, the field can be written as

display math

where the integral I is inline image· exp(iΨ(x)).

[21] These expressions show that the formation of the radiated pulse depends on dimensionless time t/qτ. For example the steady state of a sinusoidal signal is reached for T → ∞, i.e., for t/ > 1 (formally for t/ ≫ 1) after switching on the transmitter at t = 0. The process of formation of the pulsed radiated field around the resonance cone is quite complex. It depends of course on the charge distribution on the antenna and on its shape. This reflects the fact that a number of plane waves with quite different wave numbers kξ arrive at a point near the resonance surface. The group velocity of these waves is Vgr = (kξq)−1 and their interference results in the field described by (9). The pulse of the resonance field is not rectangular even when we take here only the linear term in the development of the dispersion relation in frequency, i.e., the dispersion spreading is not taken into account. In a fixed point of the space (τ, ξ) at a time t the field is determined by a superposition of wave components with wave numbers from zero (in the quasistatic approximation kξ → 0, Vgr → ) to kξ = t/ when 0 ≤ t ≤ T and from kξ = (t − T)/ to kξ = t/ when t > T. As the maximum in the spatial spectrum of the source is at a component with characteristic wave number kξ ∼ L−1, the characteristic time of formation of the pulse of the resonance field at a point (τ, ξ) is equal to the group delay of these components /L.

4. Analysis of Radiation of Some Simple Sources

4.1. Dipole Antenna

[22] For a dipole of the length 2L with a triangular distribution of current making an angle α with the external magnetic field, the normalized charge density is ρ(0,0,x) = (1/Γx)sin2x/2), where Γ = Γ(α,ϑres,ϕ) = sin α sin ϑres cos ϕ + cos α cos ϑres [Chugunov, 2001], and the integrals in (9) are expressed by incomplete gamma functions if the thermal correction in the dispersion relation is neglected. Simple results are obtained when T →  and t/ ≫ 1. In such an approximation we obtain a CW electric field close to the resonance cone

display math

where F1 = (ξ + isτ)−1/2 exp[(4i/λ)(ξτ)1/2], inline image, and the term with collisions is retained only in the amplitude multipliers. These results are obtained when T →  and t/ ≫ 1.

[23] From these relations we can see that the CW resonance field of antenna radiation is made up of a superposition of three wave packets. The first of them is the resonance cone excited by the antenna gap, the second and third packets are the resonance cones from the tips. This reflects the singularities in the charge distribution on the dipole. In the gap the charge changes sign, whereas at the tips it goes down abruptly. Such singularities in the charge distribution manifest themselves also as “weak” singularities of the electric field for ξ = 0 and ξ ± ΓL = 0, that are removed either by collisions, as shown here, or by including the thermal correction in the dispersion, which allows for excitation of a plasma wave. The surfaces of a constant phase in coordinates (τ, ξ) are hyperbolas τξ = const.

4.2. Point Dipole

[24] The case of a point dipole may be also of interest. With conditions of L → 0 and definition of a point dipole P = QL = const (where P is the dipole moment), we have for the electric field (dimensional variables, and T → , t/ ≫ 1)

display math

where inline image, Ψ(x) = px + σ/x, p = ξ + isτ, and σ = /(4λ2).

[25] From (11) we obtain

display math

[26] It is easy to see that in case of a dipole oriented along the magnetic field axis (Γ = cos ϑres), the expression for the resonance field agrees with the azimuthal component of the electric field of a dipole in the uniaxial crystal (ε1 = 1, ε3 < 0) when the dipole is oriented along the optical axis [Felsen and Marcuvitz, 1972].

[27] Above we looked at resonance structures of the potential excited by smooth charge distributions and those with singularities. We will show now that the smoothness of the charge distribution determines the speed of decrease of potential with time. According to (9), when tT the electric field approximately is

display math

[28] After taking the derivative

display math

[29] From (14) we can see that the speed of decrease of the potential as a function of time is determined by the charge spectrum on the antenna. As the spectrum of discontinuous functions is richer than the spectrum of smooth ones we can conclude that the field at a given point in space excited by a smooth charge distribution goes down more quickly as a function of time than one caused by a discontinuous distribution.

5. Application of These Results to the OEDIPUS-C Experiment

[30] Short antennas, wherein the current distribution is assumed close to triangular, were used in this experiment. Its results make it possible to study the radiation, propagation and reception of the waves under resonance conditions at 100 kHz in pulses of 0.3 msec duration. The current distribution on the transmitting antenna is

display math

where x is a dimensionless variable [Chugunov et al., 2003]. Here the angle ϕtr gives the azimuthal position of the antenna arms in the plane perpendicular to the Earth's magnetic field.

[31] In the experiment both the spreading of the signal and its delay were observed as illustrated in Figure 3 for two TALs in the amplitude versus time plots shown. The time variable in (9) characterizes effects of the time dispersion and it lies in the interval ∼ (1–3)10−2 ms from TAL = 850 s to TAL = 930 s.

Figure 3.

Amplitude-versus-time scan of the voltage pulse on the receiving antenna (a) forTAL = 885.99 and (b) for TAL= 928.49. The vertical broken lines show the rising and the falling edges of the emitted 300-μs pulse.

[32] The integral (8) can be analyzed as follows. The function inline imageρ(x,ϕtr)exp(−sτx) in the integral goes to zero at the ends of integration interval and its maximum is at x ∼ x* > 1. Its neighborhood makes the principal contribution to the integral. When we neglect the thermal and electromagnetic corrections in the phase Ψ(x) we arrive at the following approximation for the electric field:

display math

[33] Here Π(tqτx*) is the rectangular pulse of T duration, in the time interval t ∈ (qτx*qτx* + T). In such approximation, the pulse form and duration do not change. During pulse propagation through the distance τ, its time delay is given by the time qτx*. In reality the situation is more complicated and numerical calculations are needed. These show that x* ∼ 5 − 7 in the time interval given above. As a result in the delay of the arrival of the signal at the receiving antenna tdelqτx ∼ (1–2)10−1 ms, which agrees with the profiles of amplitude versus time plots shown. The frequencies ωpe, ωHe which enter into the value of q, change along the trajectory of the payload, e.g., for TAL = 885.99 we have: ωpe = 4.21 · 106 s−1, ωHe = 8.42 · 106 s−1 and the electromagnetic wavelength λ = 130 m. As shown in Figure 3a, at a distance of the order of ten wavelengths the observed delay is about 10−4 s, and the spreading of the pulse for TAL = 885.99 s goes to twice its initial length. Approximately the same holds for TAL = 928.49, Figure 3b. But note that the shapes of the two pulses differ. For TAL = 885.99 s the pulse changes sharply but, all in all, keeps a compact form. For TAL = 928.49 s, the shape clearly has two maxima in a pulse of comparable length, the first at the beginning of the pulse, the second near its end.

[34] Figure 4 shows the pulse, as a function of the dimensionless time t/, (t ∈ [0,T]), computed according to equation (9), with the position of the receiver at ξ = 1, τ = 103 (both coordinates are normalized to the antenna length), ϕtr = π/3, and s = 0. Calculations are made for the same TALs as experimental data, namely Figure 4a shows the event at TAL = 885.99 s, and Figure 4b presents results for TAL = 928.49 s.

Figure 4.

Normalized amplitude-versus-time of the voltage pulse of radiation on the position of the receiving antenna as a function of dimensionless timeβ = t/q (a) for TAL = 885.99 and (b) for TAL= 928.49. The light vertical lines are the rising and falling edges of the 300-μs pulse.

[35] The group delay about 10−4–5 · 10−4 s of the signal is apparent. Moreover, the computed value of electric field shows nicely both the spreading of the 0.3-ms pulse and the changes of its form in agreement with ourFigure 3(a 0.3-ms emitted pulse length corresponds to 10 units of dimensionless time in theFigure 4a and to 30 such units in the Figure 4b). Near the resonance direction far away from the source (at a distance of about ten wavelengths) the signal spreads 3–4 times. Also, the pulse is no longer rectangular, which results from interference of a number of plane waves during the resonance excitation of quasi-potential waves.

[36] The time delays are close to those seen in experimental data. The pulse spreads because the different plane waves reach the receiver at different times (both the group and phase velocities depend on the wave vector) and interfere there. Moreover these waves have different excitation coefficients that are determined by the spatial spectrum ρ(x,ϕtr) of charge on the antenna. The shapes of observed and computed pulses are similar, but it should be kept in mind that for detailed comparison the response of the receiving antenna on the incoming signal must be taken into account. The receiving antenna serves here as a frequency-spatial filter with corresponding excitation coefficients and this contributes to the shape of the voltage recorded on the receiving terminal, which is treated in what follows.

6. Calculation of the Voltage on the Terminal of Receiving Antenna

[37] In the first part of the paper the resonance structure of the radiated pulse near the resonance characteristics was considered. Now, we will treat the response of the receiving antenna to such a signal. It should be kept in mind that the incident field near the resonance cone is a superposition of plane waves with wave vectors that are close to the resonance cone but with widely differing magnitudes. The group velocities of these waves are perpendicular to the wave number surface. In coordinate space the phase fronts of these waves advance in a direction perpendicular to the group-velocity resonance cone. All these results are in a narrow radiation diagram of resonant waves around the cone. Also the picture is completed by the existence of “illuminated” and “shadow” regions; see the later discussion afterequation (18).

[38] In this problem we must take into account that the receiver processes a signal (changes analog to digital) with a characteristic time approximately equal to the period of the carrier frequency. Consequently, to write correctly the expression for the voltage induced on the antenna terminals, it is necessary to follow the antenna response to the incident field in time. In other words it is not the root mean square value of the voltage. Recall that the 25 kHz CW case treated recently allowed for using the approach with the RMS value [Chugunov et al., 2003]. In case of 100 kHz pulses it is not so, and a time-dependent response is used.

[39] To proceed we use again the reciprocity theorem and then we compute the losses due to re-radiation of energy (diffraction) by the receiving antenna. This approach is followed here in the treatment of data on 0.3 ms pulses of 100 kHz wave trains of the OEDIPUS-C experiment.

[40] The reciprocity theorem is written as [Chugunov, 2001]

display math

and it follows that Q = 1. ΔU(t) is the time-dependent amplitude of the voltage induced on the receiving antenna (subscript “ant”), and inline image, so that inline image and

display math

[41] Notice that inline image is the electric field of the incident wave, and φ0( inline image,t) is the potential of the trial field in the plasma (subscript “pl”), that is, the field in the plasma from the receiving antenna acting as an emitter.

[42] Now we make use of the Green integral theorem [Morse and Feshbach, 1953]:

display math

Here n is the normal to the antenna surface Sa.

[43] Further we take into account that at every moment the trial field potential is constant on the surface of the conducting antenna (this is the dipole antenna with conductive surfaces of values S1a and S2a), so that

display math

[44] The quantity in square brackets equals zero because inline image is the normal component of the unperturbed electric field of the waves incident on the surface of the antenna conductor.

[45] Consequently,

display math

[46] Now let us take for the trial field a CW signal with a frequency ω from the spectrum of the incident field, i.e., q0(t) = exp(−iωt), inline image. Then

display math

[47] We develop inline image and U(t) into Fourier integrals in time, i.e. inline image, inline image, so that (18) gives now the spectral component of the incident field, namely

display math

[48] For a CW signal ∼ exp(−0t), inline image and consequently Uω = (ω − ω0). U is the amplitude of the voltage on the antenna terminals and (18) is rewritten as

display math

[49] For a rectangular pulsed signal of length T we obtain from the Fourier transform inline image, where Ω = ω − ω0, or if the time span goes from t = T / 2, we obtain

display math

[50] Here the dependence on Ω reflects the dispersion in the medium. We start with (19), and we develop inline image and inline image in Fourier r-space as inline image, and inline image. Then the integral in (19) equals

display math

[51] The Fourier component of the trial field at the frequency ω is [Andronov and Chugunov, 1975], inline image. Consequently

display math


display math

[52] Here, as in the previous parts of the work inline image is the Fourier component of the current density with the amplitude of unity on the receiving antenna, and inline image. The expression (24)shows that, in the “amplitude” of the current of the exciting probing field or in the response of the antenna to the received signal, there is the spectral (space-time) component of the incident field

display math

[53] If now in (24) we multiply and divide the integrand by εl*(ω, inline image) and sort out the part corresponding to the losses of the receiving antenna on re-radiation, we obtain

display math

[54] In the limiting case of weak absorption, Im(εl*(ω, inline image)) → 0, the integral (26)expresses the losses due to re-radiation of quasi-potential waves incident on the receiving antenna

display math

[55] Here ρ0k is the Fourier component of the charge distribution of the trial field on the receiving antenna (with the amplitude of the current equal to one). The integral in (27)is proportional to the losses of the quasi-electrostatic field with the effective “square” of the charge of the source, the spatial spectrum of which equals

display math

[56] As the resonance re-radiation of the incident quasi-potential wave packet takes place, we can limit ourselves to the vicinity of the resonance cone on which the line joining the source and the receiver lies, as we did in calculating the radiated field. The calculations follow the procedure used for expressing the resonant radiation field in space variablesτ and ξ, ξ being the coordinate perpendicular to the resonance cone surface (see section 3 on the radiated field). As the τ coordinate fixes the distance from transmitter to receiver along the resonance cone, we can write down the spectral voltage on the receiving antenna terminals as

display math

where inline image, Ψ(x) = x(ξ0 + Ω) − R2τx3, ξ0is the coordinate of the gap of the receiving antenna, and all values are normalized to the half-length of the transmitting dipoleLtr.

[57] The effective charge density reflects the excitation of the receiving antenna by the field of quasi-electrostatic waves running along it, as the spatial Fourier component of the incident field on a given frequency is

display math

[58] We write the electric field amplitude in SI system of units [cf. Chugunov et al., 2003] as inline image, where ω0 = 2πf0 = 2π100 kHz, τ is the normalized distance from the transmitter to the receiver, and ε0is the vacuum dielectric constant. We stress that the amplitude of the current of the excited probing field is proportional to the spectral (spatial-temporal) component of the incident field in which the vector inline image is replaced by inline image. This reflects the process of re-radiation of waves. Also, in the expression(29) the following abbreviations were used:

display math


display math

[59] In general we are interested in the absolute value of the voltage

display math

[60] The expressions (29), (30), (31), and (32) are used in numerical computations for interpretation of the experimental data.

[61] The amplitude-versus-time scans and the corresponding spectra for differentTALsreflect the specific features of the response of the receiving antenna to the resonant field of the pulses of radiated quasi-potential waves. This response is due in the present experiment to the re-radiation of the incident pulse by the receiving antenna. It depends substantially on the position of the antenna gap relative to the resonance cone on the given frequency. This position changes withTAL as the plasma frequency changes along the payload trajectory. The position of the resonance cone in the coordinate space is given as inline image. As a pulse is being radiated the illuminated region is given by angles (cf. Figure 4) inline image, where Δfis the width of the spectrum of the rectangular pulse. The spectral distribution of the intensity in the radiated pulse has, of course, its maximum at the carrier frequency. The receiving antenna, the coordinate of which is given by the angle that the separation line makes with the magnetic field, can be partly in the illuminated region as well as in the shadow. This means that the coefficient of excitation of the re-radiated field, that determines the response of the receiving antenna, also depends onTAL.

[62] It is to be noted that the excitation coefficient is small if the receiving antenna is in the illuminated region but sufficiently “far away” from the resonance cone on the carrier frequency, which is given by ϑsep = yrec0, where inline image.

[63] The coefficient reaches its maximum when ϑsepyrec0 and it falls quickly when the receiving antenna goes into the shadow region, ϑsepyrec0. In this case as the antenna approaches the resonance cone at the carrier frequency from the illuminated region, the carrier frequency f0 and frequencies f < f0 are excited. As the antenna enters the shadow region the excitation on the carrier frequency gets weaker but the coefficient of excitation of waves with f > f0 grows.

[64] This is clearly visible in spectra of received signal (Figure 2). This plot presents the evolution of the spectrum of the received 100-kHz pulses during the aforementioned period [James, 2006], overlaid with the locus of frc. Figure 2 is the graphical juxtaposition of 82 individual spectra recorded in 850 < TAL< 932 s. As expected, the frequency spectra of 300-μs pulses peak at 100 kHz with side lobes spaced at 3.33 kHz. Notice, however, that the signal amplitudes are significantly enhanced at frequencies close to frc. The periodicity with time in Figure 2 arises from the spin of the emitting and receiving dipoles. So the periodicities of the spin modulation in TAL and the pulse spectrum in frequency combine to give Figure 2 “tartan” background pattern. We deal with the enhancement of wave amplitudes detected at REX after transmission from HEX along directions near the group resonance cone.

[65] At low TALs (from 849.99 s to 868.49 s) the signal is weak, not far above the noise, and the pulse is nearly rectangular. This means that the antenna is relatively deep, well inside the illuminated region. Beginning with TAL = 870.99 s we can see that there are both CW waves close to f0 plus waves with frequencies f < f0, and that the received pulse gets wider and more intense. In fact, as the full radiated pulse is recorded, it widens and is no longer rectangular, in accordance with the first part of the paper. This evolution of the shape of the receiving signal is clearly seen on the spectra for TAL = 885.99 (Figure 5a).

Figure 5.

Two examples of comparison of computed and measured spectra and the corresponding antenna effective length: (a and c) for TAL = 885.99 s and (b and d) for TAL = 928.49 s.

[66] Beginning around TAL = 920 s, the antenna enters the shadow region, the signal on the carrier frequency gets weaker and it grows for frequencies f > f0. The pulse now shows two maxima, which reflects the excitation of high frequencies due to the sharp beginning and end of the radiated pulse. This is clearly seen at TAL = 928.49 s (Figure 5b), where the carrier frequency is absent altogether and the spectral maximum is at f ≈ 122 kHz.

[67] To compare the measured and computed spectra a comment on the presented data is necessary. In the treatment of measured data, use is made of a fast Fourier transform that returns transformed values in the same units as the input. Hence here the voltage spectrum is presented in volts instead of V/Hz, and the same applies to the spectrum of the current pulses that excite the radiated field. The computed voltage spectrum is plotted in Figures 5a and 5b as

display math

i.e., the spectrum of the current pulse injected into the transmitting antenna is inline image. The antenna impedance was computed according to Balmain [1964]. The expression (33) can be put also in the form

display math

which defines an “effective length” Leff. It should be borne in mind that it cannot be compared to such a notion for reception of a CW signal.

[68] Figure 5 shows the measured (black line) and computed (gray line) spectra for TAL = 885.99, in Figure 5a, when the payload was at a height ∼350 km, and for TAL = 928.49 in Figure 5b, at a height ∼195 km. Due to the difference in the payload altitudes and consequently in plasma parameters, the resonance cone frequency is different in the two cases. For TAL = 885.99 the resonance cone frequency is 86.09 kHz and the major part of wave spectrum is in the illuminated region quite close to the carrier frequency. The expected high amplitude of measured U is seen at local resonance cone frequency at Ω = −14 kHz. A second local maximum of the U, greater than the resonance cone value, is located at a carrier frequency of 100 kHz (Ω = 0) and is induced by the current pulse spectral distribution which has the maxima on this frequency. After this, the U value logically goes down, which is caused by a decrease of the wave amplitudes as waves move away from the local resonance cone. For the other event at TAL = 928.49 s, the resonance cone frequency is near 120.03 kHz and the carrier frequency is deep inside the shadow region. Both computed and measured voltage shapes show clear maxima near the frequency of resonance cone at Ω = 21 kHz and after this frequency both profiles decrease rapidly, especially the numerical result.

[69] Computed values of the voltage spectra are a bit larger than the measured ones, in both TAL cases. This property of computed shapes probably reflects the fact that in the actual ionospheric plasma there are some wave dissipative processes which we did not take into account in the simulation. Also we supposed in calculations that the transmitted signal propagates in a homogeneous medium to the receiver. In reality some (small) gradients do exist on the raypath. Calculations of the receiving antenna voltage amplitude versus time (not shown) produced pulse shapes in good agreement with the observations in Figure 3a. For Figure 3b, overall delays are reproduced in the computed pulse, but fluctuations faster than about 100 μs are not. We attribute the disparity in the time domain again to the lack of precise knowledge of dissipative processes and of the plasma density experienced by the waves during transmission, leading to errors in phase path at reception.

[70] The behavior of the Leff parameter depends strongly on the position of the resonance cone frequency frc and its separation from the carrier frequency. For TAL = 885.99 s (Figure 5c) the illuminated region contains a very wide frequency band and frc and the carrier frequency are in the illuminated region close to each other. In this situation, the radiating antenna generates waves with increasing amplitudes up to the carrier frequency. The receiving antenna measures waves with different values of the Leff which is seen well on the plot 5c. This explains well the form of the spectrum in the Figure 5a. For TAL = 928.49 s (Figure 5d) the carrier frequency is deep inside the shadow region and the resonance cone shows itself more clearly than at the other TAL. A structured frequency distribution of the Leff is not obtained; instead, there is one clear peak at the resonance cone frequency. This means that contribution of waves generated at other frequencies (including the carrier which is inside the shadow region) is negligibly small. This agrees with the theory: in the shadow region, effective radiation of the 100 kHz waves is impossible, in spite of the higher value of the generating current.


[71] This work was supported by Russian Fund for Basic Research, Region project 11-02-97088-р_povol'ge_а and Program of RAS 22 “Basic Problems of Solar System Research and Development” and by grant M100420904 of the Academy of Science of the Czech Republic. The OEDIPUS-C experiment was a collaboration of the Canadian Space Agency and the National Aeronautics and Space Administration.