The Poynting vector applied to the complex refractive index in a hot plasma near the electron cyclotron frequency and to the cyclotron resonance observed on topside ionograms

Authors


Abstract

[1] The electron cyclotron resonance is the only fundamental resonance observed on topside ionograms that, up to the present, has not been understood. There is a solution to the hot plasma dispersion equations for the wave frequency f near the cyclotron frequency fH when the wave number k makes a small angle to the Earth's magnetic induction B. For this case, the magnitude of the imaginary part of k is more than half the real part. This means that the group velocity, /dk, where ω = 2πf, is complex. The real part of /dk is too large to explain the cyclotron resonance observations. In addition to the wave number, the dispersion relations for a hot magnetoplasma can be used to obtain the electric and magnetic fields of the cyclotron wave and hence the Poynting vector. Dividing the Poynting energy flux by the energy density of the wavefield, where the energy density is the sum of the electric, magnetic, and electron energy densities, gives the Poynting velocity VP, which is a good approximation to the actual wave energy velocity for f near fH. Cyclotron waves are strongly damped but can exist several milliseconds as observed. The waves radiate from the plane containing B and the dipole antenna L. The waves with f > fH beat with those with f < fH to produce the beat frequency observed on fixed-frequency ionograms.

1. Introduction

[2] The Alouette and ISIS topside-sounding satellites contained MF/HF radars that sounded the topside of the ionosphere. For characteristics of the sounders and the ionograms they produced, the reader is referred to Hagg et al. [1969] and to a special issue of the Proceedings of the IEEE on topside sounding, 57, 859–1179, 1969.

[3] An example of the electron cyclotron resonance observed on topside ionograms is shown in Figure 1. The left portion of the ionogram is for a fixed frequency of 0.25 MHz and covers a time period of about 5.0 s. For the right portion, the frequency sweeps upward from 0.10 MHz. The electron cyclotron frequency calculated from a magnetic field model and also from the higher cyclotron harmonics [Benson, 1972] is 0.268 MHz, which is slightly greater than the fixed frequency. Other resonances are indicated in Figure 1: the second, third and fourth cyclotron harmonics resonances (2 fH, 3 fH, 4 fH) [Muldrew, 1972a], the diffuse resonance (fD) [Benson and Osherovich, 1992], the plasma frequency resonance (fN) [McAfee, 1968], the upper hybrid resonance (fT) [McAfee, 1969], and the third and fourth f-Q resonances (fQ3, fQ4) [Muldrew, 1972b] which occur at the maximum frequencies of the cyclotron harmonic waves. The length of the square transmitter pulse is 98 μs and hence 0.268 MHz is within the second upper sideband of the pulse. The receiver bandwidth is about 60 kHz. The ordinate of the ionogram, called the apparent range, is one half the free-space velocity of light multiplied by the time after pulse transmission. The strongest cyclotron resonance signals have an apparent range less than about 1000 km or time delay of about 7 ms; however a few signals occur up to 2000 km or more. The variation of the beat pattern observed in the 5-s interval of fixed frequency is due to the rotation of the dipole antenna which has a period of about 20.4 s causing the beat pattern to repeat every 10.2 s. Figure 1 thus shows about one half of the beat pattern. Figure 1 is one of the best examples found of the cyclotron resonance on fixed-frequency ionograms. It is rare that the fixed frequency matches the electron cyclotron frequency. A large number of ISIS topside sounder ionograms have been digitized at the U. S. National Space Science Data Center [National Space Science Data Center (NSSDC), 2004]. A search of this database (R. F. Benson and N. Brown, private communication, 2003) for ISIS 2 ionograms recorded between 1972 and 1980 in which there was about a 25-s-long fixed-frequency ionogram of frequency approximately equal to fH followed by a swept-frequency ionogram resulted in 13 cases with a fixed frequency of 0.48 MHz and 9 cases with a fixed frequency of 1.00 MHz. The telemetry station, symbol, and recording time of 19 of these are listed in Table 1. The other 3 are of poor quality. The ionograms can be observed by going to the Internet address given in the reference [NSSDC, 2004]. Although most of these ionograms do show an interference pattern similar to that shown in Figure 1, the pattern is not as clearly defined. Many of these ionograms show a feature that is not observed in Figure 1 and this will be discussed briefly in section 7.

Figure 1.

ISIS 1 ionogram showing the cyclotron resonance on both fixed-frequency and swept-frequency ionograms [after Muldrew, 1972a]. Note the weak signals between 900 and 2400 km apparent range.

Table 1. Occurrences of the Cyclotron Resonance on ISIS2 Fixed-Frequency Ionogramsa
Telemetry StationYearDay NumberMonthDayUT
  • a

    Data are from NSSDC [2004]. A few to several ionograms on either side of the given times show the resonance. UT times shown are hours and minutes.

Fixed Frequency = 0.48 MHz
Quito (QUI)197307903200540
     1800
Faulklin Islands (SOL)197230310290246
Santiago (SNT)197325409110315
Ouagadougou (ODG)197435712231113
 08103220809
 18106301324
197307903201215
 08903301057
 15206010659
    1731
 
Fixed Frequency = 1.00 MHz
Brazzaville (BRZ)197313605161846
Johannesburg (BUR)197308903302122
Terre Adelie (ADL)197934212080553
 35012160511
198001001101726
Lauder (LAU)197310904191057
27910082115
33111270618

[4] The cyclotron resonance was discussed first by Lockwood [1963]. He suggests that the resonance is caused by bunching of the electrons circling the magnetic field because of the electric field gradient near the satellite antenna. Johnston and Nuttall [1964] suggest the bunching is due to the nonuniform antenna sheath. Fejer and Calvert [1964] attribute the resonance to electrostatic waves with near zero group velocity so that the waves can stay in the vicinity of the satellite for an extended time. Under their assumptions, they discovered that such waves could exist near the cyclotron frequency for wave normal directions near the magnetic field direction. However it will be seen below that although waves do exist near the cyclotron frequency at small angles to the magnetic field, they are not electrostatic in nature and they are highly damped. Benson [1972] found positive frequency shifts for the cyclotron and cyclotron harmonic resonances. He attributes these shifts to a positive instrumental frequency offset. Consequently, the observed frequency of the cyclotron resonance agrees closely with that calculated from the magnetic field model; that is, the Doppler shift is very small. Assuming propagation in the Bernstein mode, for wave frequencies slightly above the cyclotron frequency, results in a large Doppler shift that is not observed. It will be shown below that the Doppler shifts calculated on the basis of the present model are very small and can be neglected. Muldrew and Estabrooks [1972] calculated dispersion curves near the cyclotron frequency using the dispersion equations of Lewis and Keller [1962] for a hot Maxwellian magnetoplasma (see Figure 2). The refractive index has a large imaginary component. Muldrew [1972a] and Muldrew and Estabrooks [1972] discuss various dispersion curves near f = fH.

Figure 2.

Complex refractive index curves near the cyclotron frequency for homogeneous waves [after Muldrew, 1972a]. Only values for which the real part of the refractive index is greater than the imaginary part are shown.

[5] Higel and de Feraudy [1977] discuss experimental results from the EIDI 3 relaxation sounding rocket experiment in which the fH resonance was observed. They found the resonance frequency to be shifted from the cyclotron frequency by a factor of less than a few times 10−4; this agrees reasonably well with the theoretical calculations here. They concluded that the ordinary mode (i.e., whistler mode) was the most likely candidate to explain the cyclotron resonance. This also agrees with the model presented here in which dispersion curves are calculated for the extension of the whistler mode to frequencies near the cyclotron frequency in a hot plasma. They present ordinary-mode dispersion curves that show a zero group velocity with a finite refractive index at a frequency differing from the cyclotron frequency by a factor of about −3 × 10−7. This dispersion curve cannot be obtained using the dispersion equations for a hot plasma [Stix, 1992].

[6] If it is assumed that the waves responsible for the cyclotron resonance radiate only from the antenna, then the beat pattern observed in Figure 1 cannot be explained. The minimum beat frequency (observed at 28.0 s in the fixed-frequency portion of the ionogram) is too large and does not occur at the correct time. The minimum beat frequency (or maximum beat delay) from this model occurs when the angle between the long-antenna L and the Earth's magnetic induction B is a minimum and occurs slightly more than 2 s to the right of the observed minimum beat frequency. It will be shown below that the observed minimum beat frequency occurs when L0 (where L0 = L at the time of pulse transmission), B and the satellite velocity VS are coplanar. The reason that direct radiation from the antenna is not observed is uncertain but it could be due to extremely high electron temperatures near the antenna during pulse transmission that affect the propagation. Muldrew [1998] (hereinafter referred to as M98) found the minimum delay time of fixed-frequency proton echoes to occur for the same coplanar condition of these three vectors, implying that aspects of the proton echo theory are fundamental to the explanation of the electron cyclotron theory presented here. In the correction to Muldrew [1998], a slightly improved agreement between theory and observation, is presented.

[7] For proton echoes, the transmitter pulse energizes protons that pass near the antenna (over a range of transmitter frequencies). The dipole antenna tube is about one centimeter in radius so that the most highly energized protons probably pass within a few centimeters of the antenna. They then circle the magnetic field with various cyclotron radii but they all return to within a few centimeters of the plane containing L0 and B after one or more integral proton cyclotron periods after the transmitter pulse. They are energy modulated along their orbit by the transmitter phase and hence reproduce the transmitter pulse phase variation in this plane. Since the modulated positively charged protons are concentrated in this plane there is a potential variation, and hence electric current variation, at the transmitter frequency. If Bernstein waves can exist in the plasma at this frequency, they are generated and propagate to the new position of the antenna, producing a proton echo. A similar thing happens with the electrons. When transmission occurs near f = fH electrons energized by the antenna have different cyclotron radii but they all return to the (L0, B) plane at multiple cyclotron periods after energization. Since they are energy modulated at the cyclotron frequency, if a wave can exist in the plasma at this frequency, this wave is coherently generated. These electrons have a high velocity parallel to the magnetic field and are dispersed quickly after the end of the transmitter pulse of 98 μs. Hence, because of the finite pulse length, the generated waves will have a spectrum of wave frequencies about fH. Amplitude scans of the received signal strength as a function of time after transmission, indicate that the strength of the generated wave drops off rapidly with distance perpendicular to L0 in the (L0, B) plane. The scale length of this drop off is about 5 m.

[8] The author has been working on the cyclotron resonance off and on since about 1969 and has considered and rejected more ideas than he can possibly remember. One criterion for an acceptable theory was that it had to explain the interference pattern in Figure 1. The present theory does this with minor reservations. Another criterion is that an extremely small frequency shift must exist in the observed resonance from the actual cyclotron frequency as determined from a magnetic field model as discussed by Benson [1972] and by Higel and de Feraudy [1977].

[9] In a cold plasma with low damping, the wave energy velocity is given by the ratio of the wave energy flux (Poynting vector) to the wave energy density. In general, for a hot plasma the kinetic energy of the medium must be considered in addition to the electromagnetic energy flow and energy density of the wave to obtain the wave energy velocity [Stix, 1992; Allis et al., 1963]. Fortunately, for f near fH, these kinetic terms are small and can be neglected when calculating the Poynting vector and energy density. However, thermal terms cannot be neglected when calculating the dispersion relation. Good agreement is then obtained between theory and experiment. The beat pattern observed in Figure 1 is explained below as being due to the interference of two types of highly damped waves, one slightly above the electron cyclotron frequency and one slightly below the cyclotron frequency. These waves originate over an area of the (L0, B) plane and their integrated fields yield an interference pattern similar to the one observed.

2. Poynting Velocity

[10] A conservation principle for a warm plasma is given by equation (8.5) of Allis et al. [1963]. Following the nomenclature used in this paper and considering only electrons, they obtain

equation image

where E and H are the electric and magnetic fields of the wave, J and ρd are the induced electron current and density due to the wave (the subscript d is used because ρ is defined differently below), νth and N are the electron thermal velocity and ambient density, v is the induced electron velocity due to the wave, and the other symbols have their usual meaning. Allis et al. integrate equation (1) over a volume V bounded by a surface S to obtain a wave energy velocity equal to the group velocity ∂ω/∂k where ω = 2πf [see Allis et al., 1963, equation (8.103)]. For ffH the group velocity is complex and difficult or impossible to interpret. Also, the magnitude is too large to explain the cyclotron resonance. However, if equation (1) is examined for ffH using parameters corresponding to Figure 1, we find that the second term on the left hand side of equation (1) is much smaller than the first term; and the fourth term on the right hand side of equation (1) is very much smaller than the third term. For example, for f/fH = 0.999 and for the angle between the wave normal and B equal to 1° the second term on the left is about 10−4 times the first term and the fourth term on the right is about 10−11 times the third term. Hence it will be assumed that for ffH the time-averaged wave energy velocity, which will be called the Poynting velocity, is given by

equation image

where P, the time-averaged Poynting vector, is real and is the wave energy flow per unit area per unit time. P is given by

equation image

and W, the time-averaged wave energy density, is given by

equation image

It should be noted that although two thermal terms are neglected in equation (1), equation (2) is not a cold theory approximation. The values of H, E and v are obtained, as will be seen below, through the hot plasma dispersion relations and they are very different from the cold theory values. Equation (2) does not give the wave energy velocity for electrostatic waves such as Langmuir waves and Bernstein waves; in fact, the second term on the left hand side of equation (1) is usually dominant. For waves with f less than about 0.95 fH in the whistler mode, equation (2) agrees closely with the group velocity. H is given by

equation image

Stix [1992, equation (73), section 10-7] gives three equations for the components of the electric field of the form:

equation image

For a nontrivial solution, the determinant for the aij is equal to zero; solving this determinant yields the complex wave number k. Homogeneous plane waves are assumed; that is, the real and imaginary parts of k, kr and ki are parallel. The coefficients aij are functions of f/fH, fN/fH, Te, Te and β, where fN is the plasma frequency, Te, and Te are the electron temperatures parallel and perpendicular to B, and β is the angle between k and B. Once k has been determined, equation (6) can be solved for Ey/Exρ and Ez/Exζ.

[11] Following Stix [1992], in the ionosphere the Earth's magnetic induction B is in the z direction. The (x,z) plane contains the wave normal vector k. The components of k are kx = ksin β, ky = 0 and kz = kcos β. Equations (3) and (5) can be combined to give

equation image

where i, j, z1 are unit vectors along the x, y and z axes. The time-averaged electric field energy density of the wave is

equation image

H can be determined in terms of E from equation (5) and the time-averaged magnetic field energy density of the wave is

equation image

where c is the free-space velocity of light. The induced electron velocity v can be determined from the electric field force of the wave on the electron; the magnetic field force of the wave on the electron can be neglected [Budden, 1961]:

equation image

The gradient pressure term −∇p/N should be added to the right hand side of equation (10). However, for ωωH it can be shown to be negligible for the parameters corresponding to Figure 1. In equation (10)e is the electron charge and is negative. B is simply z1B where B is the magnitude of B, dv/dt can be replaced by −v since Stix [1992] takes the field variables to vary as exp[i(k · rωt)], where r is the location of a point in space and t is the time. Solving equation (10) for the x, y, z components of v yields

equation image

where ωN = 2πfN = Ne2/(ɛ0m) and ωH = 2πfH = ∣eB/m∣.

[12] The Poynting velocity can be obtained from equations (2), (4), (7), (8), (9), and (11). Note that ExE*x cancels out so that the Poynting velocity is only dependent on the ratio of the electric field components: Ey/Ex = ρ and Ez/Ex = ζ. A good approximation to Vp is

equation image

where g = ωH/ω.

3. Energization Plane

[13] The energization plane is the plane containing L0 and B that is energized by the electrons that return to this plane at integral multiples of the cyclotron period after being accelerated by the field near the antenna. The average energy gained by the electrons depends on the phase of the antenna potential. Hence if the transmitter frequency is close to the cyclotron frequency, these electrons will be energy modulated with approximately the cyclotron period. The (L0, B) plane thus acts as a transmitter that can radiate a signal into the plasma. Because of the high component of velocity of the electrons parallel to B, they disperse rapidly after the end of the transmitter pulse of 98 μs. For a pulse not much longer than 98 μs, waves with a frequency spectrum around fH are transmitted from the plane. At the time Figure 1 was recorded, the antenna was rotating with a period of about 20.4 s and hence the energization plane was also rotating with this period. The satellite velocity vector VS is in the energization plane once every 10.2 s. The angle between B and VS is 35°.

[14] Different magnetometers on the ISIS I satellite were used to record the three components of B in the spacecraft coordinate system. Magnetometer charts corresponding to Figure 1 enable the antenna orientation to be determined as a function of time. The time between magnetic field maxima on the two magnetometers looking perpendicular to the spin axis of the satellite is changing considerably because of the change in the magnetic field along the satellite path; this is shown in Figure 8 of M98. Unfortunately, there is a gap in the charts for about a minute on each side of the time that the fixed-frequency part of Figure 1 was recorded. Consequently, there is an error of about 0.3 s in determining, by extrapolation, the orientation of the antenna at the time of the fixed frequency of Figure 1. The magnetometer data can be used to determine when the long, or short, dipole antenna is perpendicular to B and the change in time between B maxima can determine codeclination and right ascension of the spin axis vector S (see Appendix of M98). The time VS is in the plane of L0 and B can then be determined. We wish to determine the orientation of the antenna at the time the beat frequency is a minimum (beat delay is a maximum) in the fixed-frequency portion of Figure 1, i.e., at 0804:28.0 UT. By extrapolating downward from the right of the data gap in Figure 8 of M98, the long-antenna L was found to be perpendicular to B at 0804:24.5 UT. By extrapolating upward from the left of the data gap, L was found to be perpendicular to B at 0804:24.65 UT. From the codeclination and right ascension of the spin vector, the time between when L was perpendicular to B and when VS was in the plane containing L0 and B was calculated to be 3.35 s. Thus the two extrapolations give times of 0804:27.85 and 0804:28.0 UT for when VS was in the plane containing L0 and B. One value agrees exactly with the time of the observed beat minimum and the other has a difference of 0.15 s that is within the estimated error due to extrapolation and spin vector accuracy. Hence, in this paper it will be assumed that the time of the minimum beat frequency occurs when VS is in the plane of L0 and B. Another example that supports this assumption is given in section 7.

[15] The amplitude as a function of time (A scan) for the minimum beat frequency recorded at 08:04:28.0 UT is shown in Figure 3. This A scan was recorded when VS is in the plane of L0 and B. The received signal strength S as a function of time can be calculated from the A scan and is found to be given quite accurately by S = S0 exp(−at) where a = 1.34 ms−1, t is the time, and S0 = S at t = 0. Note that the logarithmic ordinate of the A scan is very nonlinear.

Figure 3.

Amplitude scan for the minimum beat frequency in Figure 1 recorded at 0804:28.0 UT. The ordinate is a nonlinear dB scale relative to 1 mW into the receiver. The signal actually has almost a linear exponential decay with time.

[16] It should be pointed out here that there was a misunderstanding in the definition of right ascension in the work by M98. It was thought the right ascension was measured counterclockwise (west to east) from a line directed from the Earth away from the Sun at the vernal equinox. In fact, it is measured from a line directed toward the Sun. Consequently, 180° should be added to the values of right ascension in the work by M98. The results of M98 are unaffected because the value derived from the magnetometer charts and the value used in the calculations of the spin vector were consistent. The value of the satellite spin attitude obtained from the ISIS 1 charts prepared by the satellite controller then agrees exactly with one of the 4 values obtained using the magnetometer data (see M98, Appendix) but it is not the correct value.

4. Dispersion and Poynting Velocity Curves

[17] A dispersion curve for f/fH = 0.999 corresponding to Figure 1 is shown in Figure 4a. The plasma frequency fN = 0.74 MHz, fH = 0.268 MHz, and Te = Te = 10000 K. The high temperature is chosen because the plasma in the vicinity of the sounder for ffH is heated (see correction to M98); however the results are not strongly dependent on temperature unless it is extremely large, e.g., at the time of the transmitter pulse. In Figure 4a the real and imaginary parts of the refractive index for f/fH = 0.999 are plotted. The component parallel to B is nz and the component perpendicular to B is nx. The angle β between k and B is also shown. The real part of the refractive index is zero for β greater than about 14.2°. The Poynting velocity is shown in Figure 4b. The y component is perpendicular to the (k, B) plane; it is negative and decreases monotonically with β. The z component, parallel to B, has a maximum of about 1.3 km/s at β = 0° and falls to zero near β = 14.2°. The x component, in the (k, B) plane, has a minimum of about −3.7 km/s near β = 9°, and is zero at β = 0° and near β = 14.2°. If B remains the same and the direction of k is reversed, Vp is reversed. If the sign of β is changed, then the sign of the x and y components of Vp change but the sign of the z component remains the same. The y component is in the k × B direction, a result that agrees with Scott [1950]. For f/fH = 1.000, the Poynting velocity VP = 0 for all β. For f/fH = 1.001, VP is almost identical to that for f/fH = 0.999 shown in Figure 4b. Figure 5 shows the Poynting velocity for β = 0° as a function of f/fH; it is in the B direction. Note the almost perfect symmetry in VP between f/fH < 1 and f/fH > 1.

Figure 4.

(a) Dispersion curves for f/fH = 0.999, T = 10000 K, and (fN/fH)2 = 7.6. (b) Corresponding Poynting velocity components. For β greater than about 14.1°, the refractive index is purely imaginary, and the Poynting velocity of the evanescent wave is only in the y direction.

Figure 5.

The z component of Poynting velocity parallel to B as a function of f/fH for β = 0. The x and y components are zero.

[18] The group velocity, /dk, is complex near f/fH = 1. At f/fH = 1 and β = 0°, /dk = z1(37 − i1441) m/s. At f/fH = 0.9, 0.8 and 0.7, near β = 0°, the Poynting velocity is less than the real part of the group velocity by about 5%, 0.5%, and 0.1%, respectively.

5. Approximate Calculations

[19] Figures 4 and 5 illustrate that for values of f/fH ≈ 1 the magnitude of the Poynting velocity VP can be of the same order as the satellite velocity VS of about 6 km/s. Waves with a frequency spectrum are transmitted from the energization plane because of the pulse nature of the transmission. For f/fH = 1 + Δ, the Poynting velocity is almost the same as for f/fH = 1 − Δ when Δ is less than about 0.01. Hence, if for a given orientation of the antenna there is a preferred frequency or group of frequencies near f/fH = 1 of waves which can be transmitted from the energization plane and subsequently detected by the antenna because of the appropriate matching of Poynting and satellite velocity components, the received signal will have a beat frequency Δf ≈ 2fHΔ.

[20] The magnitude of the component of VS perpendicular to the energization plane varies from zero when VS is in this plane to a maximum of about 3 km/s. If f/fH = 0.999 (or 1.001) and β = 1.0°, then the component of VP perpendicular to this plane is 0.96 km/s for the parameters of Figure 1. If f/fH = 0.999 and β = 2.0°, then the component of VP perpendicular to this plane is 1.89 km/s. If the perpendicular component of VP matches the perpendicular component of VS and the antenna picks up waves only of frequencies 0.999 fH and 1.001 fH, the beat frequency would be 0.002 fH = 0.002(0.268 kHz) = 536 Hz, which corresponds to a beat of 280-km apparent range in the fixed-frequency ionogram of Figure 1. In reality, integration over all received frequencies must be carried out (see below). At the start of the fixed-frequency ionogram of Figure 1, the observed beat is about 150 km and at the minimum beat frequency or maximum beat period the observed beat period is either infinite or greater than about 700 km. Hence the calculated and observed beat frequencies are approximately in agreement for these β and f/fH values.

[21] In order to have zero beat frequency in the fixed-frequency ionogram, f/fH must equal one. If f/fH = 1, then VP = 0. A signal can then only be received if the satellite remains in the plane of (L0, B), i.e., if VS, L0 and B are coplanar. If VS is not in the plane of (L0, B), then there is a range of f/fH values which result in a beat pattern that is consistent with the beat periods observed.

6. Model for Received Signal Intensity

[22] Above it is shown that the Poynting velocities at frequencies close to the cyclotron frequency can be similar to the satellite velocity, that a beat frequency can be observed on fixed-frequency ionograms close to that which is observed, and that the beat frequency goes to zero when the satellite velocity is in the plane of (L0, B) as observed. Hence the study could be terminated here. However, by making a few not unreasonable assumptions, considerably more insight into the resonance is obtained.

[23] Since the free-space wavelength near 0.25 MHz is 1200 m and the signal in the antenna travels almost at the speed of light, the phase on the transmitting antenna is almost the same along the length of the antenna. Hence the phase over the (L0, B) plane at a given time is almost the same. Since antenna rotation in a few milliseconds is negligible, the receiving antenna is parallel to the (L0, B) plane and so the phase along the receiving antenna is about the same. Also, the wave number variation over the received frequency range of interest is negligible, hence the effective length of the antenna can be assumed constant over the frequency range and need not be considered in an approximate calculation of received signal intensity.

[24] From the above discussion and since the magnetic field gradient is negligible, it can be assumed that all of the energized electrons in the energization plane have the same phase at a given time after the transmitter pulse. Hence, if the plane were infinite in size, only waves with kr (real part of k) perpendicular to the plane would propagate. However, the plane is not infinite; the antenna tip-to-tip length of 72 m determines its size along L. It was pointed out above that the energy density of the energized electrons in the plane drops off with a scale length of about 5 m in the direction of VS that is roughly perpendicular to L. This 5-m length is much smaller than the wavelength of the cyclotron waves of roughly 60 m or more. Hence cyclotron waves can propagate at all angles to the energization plane and to B. The plane rotates with the antenna; hence its size does not change with rotation.

[25] Since the receiving antenna is only a fraction of a wavelength (about 60 m) from the (L0, B) plane, it might be argued that the receiving antenna is in the near field of the transmitted signal. The pulse in the (L0, B) plane radiates a signal that is not much longer in time than the transmitting pulse of 98 μs. Near-field effects disappear after the transmitting pulse. The signal is received up to several milliseconds after this. For an example, if we choose 5 ms, the transmitted waves near 268 kHz have undergone about 1500 cycles. The wavefronts near the (L0, B) plane are moving with a phase velocity of about 1.5 × 107 m/s even though the energy bundle is barely moving. Hence it will be assumed that the received signal is in the far field and that we can calculate the phases of the waves reaching the receiving antenna using plane wave propagation.

[26] To estimate the received signal strength as a function of time t after the transmission pulse we consider only those Poynting velocities in the plane perpendicular to L. Contributions will come from all directions but they will center on these values (see Figure 6). Justification for this assumption is given in Appendix A. In the few milliseconds during which the signal is received, L remains parallel to L0 since antenna rotation in a few milliseconds is negligible. The component of VS perpendicular to the (L0, B) plane is given by

equation image

and the required Poynting velocity component perpendicular to the (L0, B) plane must match the satellite velocity perpendicular to this plane in order for a signal to be received. The vectors VS, L0 and B are known in x′ (north), y′ (west), and z′ (vertically up) coordinates. The Poynting velocity VP and k are known in Stix coordinates, which are z (B direction), x (in the (k, B) plane) and y (right-hand system). For a given f/fH and β, possible values of k can lie anywhere on a cone with axis along B and having a half-angle β. Hence possible values of VP for a given f/fH and β also lie on a cone with axis along B. To calculate the component of the Poynting velocity perpendicular to the (L0, B) plane, the Poynting velocity which is in Stix coordinates (x, y, z) must be transformed to north, west, up coordinates (x′, y′, z′). B is in the z direction, hence

equation image

VP is assumed to be in the plane perpendicular to L0. Hence

equation image

and finally

equation image

These three equations can be solved for (VPx, VPy, VPz). The component of VP perpendicular to the (L0, B) plane is given by

equation image
Figure 6.

Sketch illustrating the assumption that only Poynting velocities perpendicular to L (rays like a, b, and c but not d) are considered.

[27] There are two solutions for (VPx, VPy, VPz) but one value of VP is just the negative of the other. A signal transmitted from the (L0, B) plane traveling in the plane perpendicular to L0 will arrive at L if

equation image

A value of β is chosen and, by trial and error, the value of f/fH which satisfies this condition is found. This process is repeated for several β values. The value of f/fH between two consecutive calculated values of f/fH can be found quite accurately using an empirical formula of the form

equation image

where a and b are constants. A quantity A(t) proportional to the received electric field is then estimated by integrating over β using an arbitrary weighting function of the form exp[−(β/β0)2] where β0 is a constant. This weighting function places more weight on the radiation with small β, and, since the radiation pattern is unknown, this is the major weakness in this derivation. For each value of β there are two contributions to the field, one from frequency ω(β) = ωH − Δω(β) and one from frequency ω(β) = ωH + Δω+(β) where Δω(β) and Δω+(β) are positive. The Doppler frequencies are −krequation imageVS, and −kr+equation imageVS. Hence the frequencies are ωH − Δω(β) − krequation imageVS and ωH + Δω+(β) − kr+equation imageVS. Since k±equation imager± = (kr± + iki±) equation imageVP±t, the received electric field is proportional to

equation image

where β1 is the smallest value of β for which solutions for VP±(β) (in x′, y′, z′ coordinates) can be found, β2 is a value of β for which the integrand becomes negligible because of the weighting factor, k(β) and k+(β) are the wave vectors corresponding to ωH − Δω(β) and ωH + Δω+(β), respectively. The real parts of k and k+ are kr and kr+; the imaginary parts are ki and ki+. The received signal strength is proportional to the square of the electric field. Before squaring the electric field, which is proportional to A(t), the real part of equation (20) must be taken. The common factor exp(−Ht) = cos(ωHt) − isin(ωHt) can be taken outside of the integral. Let the remaining integral be I. Then the real part of equation (20) becomes

equation image

[28] Squaring equation (21) and averaging over the fast time-varying factor ωHt yields a slowly varying quantity S(t) that is proportional to the received signal strength:

equation image

For a given β, the values of f±(β), and hence Δω±(β) for f < fH and for f > fH are obtained from the trial-and-error calculations and equation (19). Then k± and VP± are obtained as discussed in section 2. Finally, an arbitrary value of β0 is taken and integration over β yields S(t).

[29] S(t) maxima are calculated for a few antenna orientations, i.e., a few times after 0803:25 UT, and the results compared to the fixed-frequency portion of Figure 1. The results of the calculations are shown in Figure 7 for β0 = 1.0°. It can be seen that the agreement with Figure 1 is quite good although not perfect. At any time during the antenna rotation, good agreement could be obtained by choosing a different temperature or value of β0. If β0 were changed to 2.0° the apparent ranges in Figure 7 would be (39 ± 1)% greater. If the temperature were decreased to 4000 K (closer to ambient), the apparent ranges in Figure 7 would increase by (24 ± 2)%. Figure 1 shows a slight asymmetry before and after the time of maximum delay tm. Symmetry is expected since radiation from the (L0, B) plane should be symmetrical from both sides of the plane. The asymmetry might be due to varying electron temperature following the transmitter pulse in the vicinity of the satellite, as the antenna rotates. Calculations indicate that a wind or plasma drift would not cause asymmetry but only a shift in the whole interference pattern by up to a few tenths of a second. Doppler shifts are also symmetrical because the antenna is moving away from the (L0, B) plane at the same velocity at t = tm − Δt as it is at t = tm + Δt.

Figure 7.

Calculated signal maxima for β0 = 1.0 and T = 10000 K. There is also a maximum at an apparent range of zero. The minimum between this maximum and the one at the lowest apparent range shown in the figure does not show up in the ionogram because the minimum signal strength is too large and narrow in apparent range.

[30] Equation (22) can be used to calculate A scans (amplitude scans), that is, the strength of the signal as a function of time after the transmitter pulse. These calculated A scans can then be compared to the observed A scans. Using equation (20) with β0 = 1.0°, S(t) is calculated at tm − 1 s. The calculated and experimental amplitude scans are shown in Figures 8a and 8b. The agreement for delay times less than about 7 ms is good. However, for delay times greater than about 7 ms (1050 km), the calculated values indicate that an observed signal would probably not be expected. In fact from the ionogram very weak signals are observed up to delay times of about 15 ms. Although not illustrated clearly in Figure 8b, a slight maximum occurs near 1600 km. A scans recorded at t slightly less than tm − 1 s do show clear maxima.

Figure 8.

(a) Calculated signal strengths at time tm − 1 s and T = 10000 K. The ordinate is a linear dB scale with an arbitrary origin. (b) Observed A scan (amplitude scan) for tm − 1 s. Some of the A scans recorded near this time show a small minimum around 100 to 135 km apparent range.

[31] In Figure 1, weak signals can be seen between about 900 km and 2600 km apparent range. It is unlikely that these signals result from waves propagating with β values near 0° since the absorption would be too large and the beat delay too small. There is however a plausible explanation for these signals. For a given electron temperature and antenna orientation, the value of f/fH as a function of β for which equation (18) is satisfied can be determined. This is shown in Figure 9 for the electron temperature Te = 10000 K and at time tm − 1 s. Muldrew [1998] calculated an electron temperature near 10000 K when f was close to fH. Equation (18) cannot be satisfied for β less than about 0.17°. The real part of the refractive index falls to zero at about β = 14.1° and at this point ∂(ω/ωH)/∂β = 0. In fact, ω/ωH changes very little over a large range of β angles; at β = 8°, ω/ωH = 0.999568, at β = 10°, ω/ωH = 0.999589, and at β = 14.1°, ω/ωH = 0.999603. Hence almost the same beat frequency or beat apparent range occurs over a large range of β, resulting in an enhanced signal. For the case considered here the beat apparent range is about 705 km. For T = 10000 K and at time tm − 0.5 s, the beat apparent range is about 990 km. It can be seen from Figure 1 that these beat apparent ranges are of the right magnitude for the weak signals recorded between 900 and 2600 km apparent range. Note also that these signals appear to vary very little in strength over the apparent range in which they are observed. This can be verified from the A scans. Because the Poynting velocity and the imaginary component of the wave vector are almost at right angles to each other, the absorption, which in decibels is proportional to kiequation imageVP, is small. For the case considered above with β = 8°, the absorption is only about 1.3 dB at an apparent range of 2000 km.

Figure 9.

Frequency variation of the received signal as a function of transmitting location on the energization plane at time tm − 1 s, T = 10000 K. The location is determined from β, the angle between B and k, for VP perpendicular to L (η = 90°).

[32] It can be argued that the larger β propagation should be dominant over the lower β propagation for apparent ranges less than 1000 km, i.e., propagation times up to about 7 ms. However, to explain the beat delays of about 200 km or less, enormously high electron temperatures, corresponding to several keV electrons, are required. Although the sounder pulse can produce keV electrons, the whole electron population responsible for the propagation would not be this energetic. It thus seems necessary to conclude that the radiation from the energization plane prefers the smaller β values.

7. Discussion and Conclusions

[33] If section 6 (in which there are a few speculative assumptions) is ignored, it can still be concluded that the cyclotron resonance observed on topside ionograms can be explained using the Poynting velocity given by equation (2) and that the (L0, B) plane acts as the source of the detected radiation responsible for the resonance. The Poynting velocity appears to be valid for f/fH ≈ 1 even though the wave normal has a large imaginary component.

[34] Section 6 indicates that for radiation perpendicular to L0 and for an arbitrary weighting function that favors radiation with wave normal directions near the magnetic field direction, good agreement between the calculated and observed beat pattern is obtained, as the antenna rotates, for apparent ranges less than about 1000 km. For apparent ranges between about 900 and 2600 km weak signals are observed which are probably not generated by the radiation that makes an angle of less than a few degrees to the magnetic field direction. These signals appear to be generated by radiation that makes angles of several degrees or more to the magnetic field direction. There is a focusing effect due to the fact that for the received waves ω/ωH changes very little over a large range of angles. Also, there is little absorption since ki is almost perpendicular to VP.

[35] The beat pattern in Figure 1 is slightly asymmetrical about the time at which the beat frequency is zero. From the model presented here, it would be expected that radiation would be symmetrical from both sides of the plane of (L0, B) and consequently one would expect a symmetrical beat pattern. The most likely explanation for the observed result is that the time-averaged electron temperature that the cyclotron waves experience is changing as the antenna rotates. Calculations indicate that the asymmetry is not due to plasma winds or differing Doppler shifts.

[36] Fixed-frequency ionograms illustrating the cyclotron resonance are listed in Table 1 and can be viewed on the Internet [NSSCD, 2004]. One of the more interesting ones is illustrated in Figure 10. It clearly shows the interference pattern and it also shows a decrease in intensity corresponding to the time VS is in the (L0, B) plane, e.g., at 1758:58.8 UT. This type of decrease is seen on many of the ionograms listed in Table 1 but not in the ionogram of Figure 1. For some of the ionograms listed in Table 1, the decreased signals above about 1000 km apparent range, the decrease is fairly wide, but for the stronger signals at lower apparent ranges, the decrease is just a few sweep lines wide. A possible explanation for this could be as follows: The energized electrons converge on the (L0, B) plane and produce an excess charge density in the plane that oscillates at the cyclotron frequency. Since the oscillating charge density has the same phase over the plane, the resulting electric field will be perpendicular to the plane and hence perpendicular to L. Hence no current is generated in the antenna.

Figure 10.

ISIS 2 ionogram showing the cyclotron resonance on both fixed-frequency and swept-frequency ionograms. One of the maximum delays in the fixed-frequency resonance occurs at 1758:58.8 ± 0.1 UT.

[37] Figure 10 was used to check the basic assumption that the maximum delay of the cyclotron resonance occurs when VS is in the (L0, B) plane. To do this, the spin vector and the time the long antenna is perpendicular to B must be determined. The magnetometer data in PCM (phase code modulation) format for the pass containing the ionogram of Figure 10 were obtained from the Goddard Space Flight Center (R. F. Benson, private communication, 2006) and a program to display the data in readable format had been obtained earlier (R. F. Benson, private communication, circa 2003). L is perpendicular to B when the magnetometer measurement in the direction of the long antenna is zero. The spin attitude, i.e., right ascension (RA) and codeclination (CD), is available in chart form prepared by the satellite controller. However, M98 found that the spin attitude obtained from the chart was incorrect corresponding to an ISIS I pass, and hence it was decided to check the attitude in the present case using the magnetometer data. For each time or satellite position along the orbit, two value of RA can be calculated for each assumed value of CD. Repeating this for several locations along the orbit produces several curves of RA vs. CD. These all intersect in a relatively small region of RA and CD values. For the present case the values are CD = 32° ± 2° and RA = 17° ± 12°. These values agree with the chart values of CD = 30° and RA = 16°. The magnetometer values of CD, RA and the time L is perpendicular to B, yield a time 1758:59.6 ± 0.4 UT when VS is in the (L0, B) plane. The ionogram of Figure 10 gives a time of 1758:59.8 ± 0.1 UT. Hence it can be concluded with a high degree of certainty that the observed minimum beat frequency (maximum beat delay) occurs when VS is in the (L0, B) plane.

[38] James [1983] and James et al. [1999] found that sounder accelerated particles (SAP) of energy up to 20 keV occur from a transmitting antenna imbedded in a plasma. These are the particles that here are assumed to energize the plane containing the antenna and the magnetic field direction. The thickness of the plane depends on the size of the accelerating region around the antenna. The antenna tube diameter is about 2 cm.

[39] It is important to note that in a magnetoplasma, radiation is not just emitted from the antenna. Radiation is emitted from the plane containing the antenna and B. Evidence for this is given here and in the work by M98. James [2004] observed a somewhat similar effect; he deduced that slow-Z reradiation occurred from sounder-accelerated electrons.

[40] In the ionosphere, wave energy velocities near or less than the electron thermal velocity are usually associated with electrostatic waves, except for near-zero refractive index. For electrostatic waves the wave normal is parallel to the electric field of the wave and the magnetic field of the wave is negligible. However, the cyclotron waves considered here are not electrostatic. As an example, consider the case ω/ωH = 0.999, β = 1°. From (6) it is found that Ey/Exρ and Ez/Exζ are 1.453 + i0.990 and 0.536 + i0.832, respectively, whereas ky/kx = 0 and kz/kx = 57.3 + i0. Also the magnetic field of the wave is not negligible. Hence cyclotron waves with energy velocity much less than the electron thermal velocity are electromagnetic in nature.

[41] Extremely slow electromagnetic Poynting velocities also occur in the extension of the slow-Z mode for ωN < ωH when ω is slightly greater or slightly less than ωH. For example, for (ωN/ωH)2 = 0.5, Te = 10000 K, ω/ωH = 0.999, β = 1°, the refractive index is 6.84 + 3.65i and the Poynting velocity in m/s is (8908, 4689, 8209) using Stix coordinates. Using the same parameters except ω/ωH = 1.001, the refractive index is 6.83 + 4.05i and the Poynting velocity is (9282, 5433, 8167). For ω/ωH = 1, the refractive index is midway between the above two values and the Poynting velocity is zero. Hence a similar beat pattern to that corresponding to the extension of the whistler mode should be observed.

[42] The Doppler shifts, for the cyclotron waves considered here to calculate the beat pattern, are almost independent of ω/ωH and β. For ω/ωH = 0.999, β = 1°, and for ω/ωH = 1.001, β = 1°, the Doppler shift are −552.8 and −553.1 Hz, respectively. Since the Doppler shifts for ω/ωH > 1 are almost the same as for ω/ωH < 1, Doppler shifts have negligible effect on the beat pattern. Since the beat pattern of Figure 1 results from two groups of waves with equal frequency displacement above and below fH in the plasma, and both groups have the same Doppler shift, the observed frequency displacement would be just the Doppler shift of about −0.6 kHz. This is consistent with the findings of Benson [1972] and of Higel and de Feraudy [1977].

[43] The electrons accelerated by the antenna can occupy a region tens of meters in diameter as they gyrate in the magnetic field. These will constitute only a small fraction of the electrons in this region. The Earth's magnetic field energy density is at least 105 times the ambient electron thermal energy density. Hence any perturbation to the Earth's magnetic field is negligible and will not affect the phase of the energetic electrons.

[44] The work of Stix [1992], Allis et al [1963], and others leads to the conclusion that the integrated energy flux (electromagnetic plus acoustic) divided by the integrated energy density (electromagnetic plus thermal plasma) agrees with the group velocity /dk. This does not mean that the group velocity is the correct wave energy velocity. The group velocity is complex and difficult or impossible to interpret. In the case considered above for ω/ωH = 1 and β = 0°, the imaginary part is about 40 times greater than the real part. For ω/ωH = 0.999 and β = 1° the magnitude of the group velocity is about 1500 m/ms and the real part is about 410 m/ms. These velocities are far too great to explain the experimental results. However, by neglecting the apparently negligible terms in equation (1) a Poynting velocity is obtained which is real and has the right magnitude and other characteristics to explain the experimental observations. It appears that the two neglected terms play a role in deriving the complex group velocity from the integrated energy flux and energy density but that this velocity does not yield the true wave energy velocity near ω/ωH = 1. It is interesting that the group velocity begins to fail even when the imaginary part is relatively small [Muldrew and Gonfalone, Figure 8, 1974].

Appendix A:: Justification for Taking VP in the Plane Perpendicular to L

[45] In Figure A1, an antenna orientation is chosen corresponding to time tm − 1 s. The value of f/fH satisfying equation (18) is calculated as a function of η, the angle between VP and L, for β = 1° and β = 8°. Choosing appropriate values of β and η allows the calculation of ω/ωH corresponding to all points on the (L0, B) plane from which a signal transmitted from the plane reaches a point on the antenna L. No signal arrives at L from some areas of the plane because a match between VS and VP does not occur. It can be seen from Figure A1 that the curves for the various values of ω/ωH are very nearly symmetrical about η = 90° and that ∂(ω/ωH)/∂η = 0 near η = 90°. Thus the strongest signals would come from the region near η = 90°, i.e., from the plane perpendicular to L. It would be possible to integrate the signal over β and η, but this would be difficult and unnecessary for the purpose of this paper. Only integration over β is considered with η = 90°.

Figure A1.

Frequency variation of the received signal as a function of transmitting location on the energization plane at tm − 1, T = 10000 K. The location is determined from η, the angle between VP and L for β = 1° and 8°.

Acknowledgments

[46] I am grateful to H. G. James and L. R. O. Storey for helpful discussions and correspondence. I am also grateful to the National Space Science Data Center (NSSDC) and to R. F. Benson and student Nick Brown for compiling a list of the occurrences of ionograms (Table 1) for which the fixed frequency matched the cyclotron frequency using data from the National Space Science Data Center [NSSDC, 2004].