Polarimetry of auroral kilometric radiation with a triaxial nonorthogonal antenna system

Authors


Abstract

[1] The response of a triaxial nonorthogonal radio-polarimeter on board a spinning spacecraft has been analyzed in order to determine effective length vectors of its short electric antennas. In the ideal case all three antennas of a radio-polarimeter would be orthogonal to each other. In practice, their electric axes do not coincide with the physical ones. It is shown that effective length vectors of the antennas can be found by fitting theoretical temporal variations of the Stokes parameters to those determined from a spinning spacecraft. This idea has been applied with the Polrad radio-polarimeter on board the Interball-2 spacecraft, which frequently observed circularly polarized auroral kilometric radiation. Polrad was equipped with one monopole (Y) and one dipole (Z) in the plane perpendicular to the spacecraft spin axis, and one monopole (X) extended along the spacecraft spin axis. For the Y antenna the resulting components of the antenna effective length vector are hy/hz from 0.88 to 0.94 (depending on the source colatitude), colatitude 7.9° ± 0.1°, azimuth 5.5° ± 0.2° measured from its physical axis, and for the X antenna hx/hz from 1.1 to 0.85 and the tilt a negligible. The Z antenna is taken as a reference. These results have been applied to determine the transformation matrix (describing the instrumental polarization), used for rectification of the measured Stokes parameters to the orthogonal antenna system. The method can also be applied to determine complete polarization state and direction of arrival of any radio emission received by a triaxial polarimeter.

1. Introduction

[2] The terrestrial auroral kilometric radiation (AKR) discovered by [Benediktov et al., 1965] is generated close to the gyrofrequency of electrons in the auroral regions of Earth's magnetosphere in the frequency range from 20 to 900 kHz. The first inferences about AKR polarization were obtained from Hawkeye [Gurnett and Green, 1978] and ISIS-1 [Benson and Calvert, 1979; James, 1980]. The antenna systems of these spacecraft were not designed to provide direct measurements of the polarization parameters. Determination of the wave polarization modes was based only on an examination of the cutoff frequencies of the waves propagating in a magnetized plasma. The first direct determinations of the AKR polarization modes were made by Voyagers 1 and 2 [Kaiser et al., 1978] and DE-1 [Shawhan and Gurnett, 1982; Mellott et al., 1984] by measuring the phase difference between signals from two orthogonal electric antennas. With the Plasma Wave and Sounder experiment on board of the Akebono (EXOS-D) satellite [Morioka et al., 1990] determined the wave polarization and the Poynting vector. This measurement was done with two orthogonal electric and three orthogonal magnetic antennas. The MEMO experiment on the Interball-2 spacecraft provided estimations of the eccentricity of the polarization ellipse and the wave normal direction from three magnetic and one electric wave field components [Lefeuvre et al., 1998]. Information about the Poynting vector is essential for a definitive determination of the Stokes parameters. Contrary to ground-based radio astronomy, in spacecraft observations the direction of the Poynting vector is generally not a priori known. Most of these observations are consistent with the main theoretical predictions [Wu and Lee, 1979] that AKR is the circularly polarized emission usually of the extraordinary (R-X) wave mode.

[3] Polrad on Interball-2 was the first experiment that directly measured the spectra all of the Stokes parameters of AKR [Hanasz et al., 2000]. Equipped with the triaxial polarimeter, Polrad allowed to obtain the Stokes parameters in a plane perpendicular to the Poynting vector from measurements of the wave polarization in its three approximately orthogonal antenna planes [Hanasz et al., 1998].

[4] Polarization measurements from space are difficult because they are strongly dependent on various instrumental effects and the antenna geometry. The ideal case is a system where the antennas are exactly orthogonal and their effective lengths are equal. However, it is already known that electric antenna axes generally do not coincide with physical ones. This effect can be explained by distortion of the antenna directional characteristics caused by interaction of the antenna with other electric devices and spacecraft structures (booms and other antennas). These structures induce parasitic currents in the antennas and thereby cause the tilt in the antenna electric axis. As was shown in a paper by [Fainberg et al., 1985], short monopoles are more susceptible to such tilts. The authors have shown that when the ISEE − 3 receiver worked in the dipole configuration the tilt of the electric antenna axis was less than 1.5°. Whereas when the receiver was operated in the monopole mode the electric tilt was about 15°. In the case of Voyager it was shown that the electric plane defined by the two orthogonal monopoles was displaced by 23° from the plane of the physical antenna [Lecacheux and Ortega-Molina, 1987]. In addition, the tilt of this plane was different at different azimuths [Wang and Carr, 1994]. Rheometry measurements for a model of the Cassini spacecraft by [Rucker et al., 1996] have shown that the offsets of the electric axes of its monopoles were about 6° to 8° and effective antenna lengths were not equal.

[5] Since Interball-2 had many structures deployed, especially in the plane perpendicular to its spin axis, it was expected that the electric axes and effective lengths of the Polrad antennas (two of them being monopoles) were not the same as the physical ones. The goal of this paper is to analyze the response of the triaxial polarimeter of Polrad to a circularly polarized wave in order to determine the effective length vectors of the antennas. The resulting values for the tilt angle and effective length of the antennas are then used to rectify the polarization measurements to an orthogonal antenna system.

[6] The structure of the paper is as follows: The principle of Polrad polarization measurements with a triaxial antenna system is presented in section 2, simulations of the polarimeter response to the circularly polarized wave for the triaxial orthogonal and nonorthogonal antenna system are described in section 3, fitting of the theoretical variations of the Stokes parameters to those determined from Interball-2 observations is given in section 4, the results for the effective length vectors of the Polrad antennas obtained with the singular-value decomposition (SVD) technique are shown in section 5, mutual calibrations of the antenna effective lengths and rectification of the measured Stokes parameters to the orthogonal antenna system are shown in section 6, the results are discussed in section 7, and the conclusions are given in section 8.

2. Principle of Polarization Measurements With Polrad

[7] Interball-2 was launched on August 29th, 1996. One of its instruments, Polrad, was aimed at measuring the dynamic spectra all of the Stokes parameters of AKR [Hanasz et al., 1998]. Polrad was a step frequency analyzer which covered a frequency range from 4 to 1000 kHz with a resolution of 4 kHz and a sampling period of 6 or 12 s. Polrad was equipped with three electrical antennas (see Figure 1). The X antenna (11 m tip-to-center) was a monopole extended along the spacecraft spin axis in the antisolar direction. The spin axis was reoriented toward the Sun every 6 to 8 days.The spin period was 120 s. The Y antenna was constructed as a dipole with one arm shifted by 10° due to technical reasons. However, after the launch it appeared that the signal received from the Y antenna was roughly half of that received from the Z antenna. This led to the conclusion that the unshifted arm (+Y) of the Y antenna failed to work. The gain correction was introduced in the data processing. The Z dipole (22 m tip-to-tip) as well as Y antenna were designed to lie in the spin plane perpendicular to the X antenna. The configuration of the Polrad antennas was therefore as follows: (1) monopole X, 11 m, antisolar direction, along the spin axis; (2) monopole Y, 11 m, tilted by 10° from orthogonality in the plane YZ and lying in the spin plane; (3) dipole Z, 22 m, in the spin plane.

Figure 1.

Scheme of the Polrad antenna system. The X monopole (11 m tip-to-center) was extended along the spacecraft spin axis in the antisolar direction. The −Y monopole (11 m tip-to-center) and Z dipole (22 m tip-to-tip) were located in the spacecraft spin plane. The Y antenna was designed as a dipole. Its +Y arm was broken after launch. The −Y arm was shifted by 10° from orthogonality due to technical reasons.

[8] The scheme of the polarization measurements with Polrad is shown in Figure 2. Power densities of the nine signal combinations were measured simultaneously in separate polarimeter channels. Channels 1, 2 and 3 were designed to measure the spectral power densities (Px, Py, Pz) of wave components received with single antennas X, Y and Z. Channels 4, 5 and 6 were used to measure the spectral power densities of the sum of the antenna signals combined in phase (Px+y, Py+z, Pz+x). The last three channels (7, 8 and 9) measured the power densities of the sums of signal components, in which the phase of one component was delayed by 90° with special phase shifters (Px*+y, Py*+z, Pz*+x). All polarimeter channels were calibrated on the ground. The Stokes parameters for each frequency step were determined from outputs of these channels in coordinates of the spinning antennas [Hanasz et al., 1998, 2000]:

equation image

where the indices k and j denote the names of the antennas in such a way that k = X when j = Y, k = Y when j = Z, and k = Z when j = X.

Figure 2.

The scheme of the Polrad triaxial polarimeter. X, Y and Z represent the three antennas. The cross in the circle symbolizes the summation of the signals. The principle of Polrad polarization measurements is described in section 2.

3. Simulation of the Stokes Parameters Measured With Orthogonal and Nonorthogonal Antennas

[9] In this section the response of the Polrad triaxial polarimeter to a circularly polarized wave is simulated. The following assumptions have been made:

[10] 1. AKR is a circularly polarized wave. A linear component of the polarization of AKR has not been reported [see Kaiser et al., 1978; Shawhan and Gurnett, 1982; Mellott et al., 1984; Morioka et al., 1990], and the generally accepted theory of the AKR generation [Wu and Lee, 1979] predicts a circularly polarized emission.

[11] 2. AKR is propagating in free space, since at the altitudes of Polrad the AKR refraction index is close to 1.

[12] 3. The radiation is emitted from a point source. The data have been selected for the periods when Interball-2 was near apogee, far from the source regions.

[13] 4. Radiation patterns of the X and Y monopoles are assumed to be those of short dipoles. This follows for example from recalibration of the Voyager antennas [Wang and Carr, 1994].

[14] 5. The antennas are immersed in a free space. In case of Interball-2 the AKR is observed, as a rule, at frequencies high above the resonance frequencies of the local plasma (plasma and electron cyclotron frequencies), and plasma does not influence the antenna impedance [Balmain, 1969].

[15] 6. The effective length vector of the Z antenna is taken as reference.

[16] In a coordinate system of a spinning spacecraft (Figure 3) three orthogonal antennas are deployed along X, Y and Z axes. Effective length vectors equation imagex, equation imagey and equation imagez represent the directions of the electric axes of the antennas and their effective length. In the ideal case of a short dipole the magnitude of its effective length vector is equal to half of its physical lengths and its direction coincides with its rods.

Figure 3.

Geometry of a right-hand circularly polarized wave propagating from a point source in the spinning orthogonal coordinates of the spacecraft. X, Y and Z antennas are assumed to be deployed along coordinate axes; equation imagex, equation imagey and equation imagez are effective length vectors of the antennas; equation image is wave electric vector; ω is wave angular frequency; θ and ϕ are direction angles of the wave incidence;and Ω is angular frequency of the spacecraft spin.

[17] Let us consider the ideal case of three orthogonal short dipoles. This antenna system receives a right-handed circularly polarized wave propagating from a point source (counterclockwise for approaching wave, according to IRE definition 1942 [Krauss, 1966]). Then, the electric wave vector equation image can be described by two orthogonal components equation image1 and equation image2 (Figure 3):

equation image

where ω is a wave angular frequency, ϕ is the phase and ∣equation image1∣ = ∣equation image2∣ = E0 for a circularly polarized wave.

[18] In the antenna coordinate system equation image1 and equation image2 can be expressed as:

equation image

where θ and ϕ are the wave incidence angles.

[19] The input open-circuit voltages of the receiver channels are then:

equation image

where equation image is the wave electric vector given by equation (2) and i = equation image.

[20] The measured power outputs of the channels are:

equation image

where Z0 is the impedance of vacuum and T is the integration time of a single measurement.

[21] Introducing equations (4) into equation (5) the spectral power densities measured by each of the polarimeter channels then become:

equation image

For the example for the Px channel:

equation image

[22] The Stokes parameters can be determined from equations (1) and (6). The above description presents an ideal case when the effective length vectors of the antennas are orthogonal to each other (hereinafter called the orthogonal model).

[23] Consider the case where the effective length vectors equation imagex and equation imagey are tilted from orthogonality by an angle α and β for equation imagex and by an angle γ and ψ for equation imagey (see Figure 4). In a new nonorthogonal coordinate system related with equation imagex, equation imagey and equation imagez the components of the wave electric vector equation image can be expressed as:

equation image

where equation image1 and equation image2 are the components of the wave electric vector in the orthogonal antenna system given by equation (3) and equation image(α, β, γ, ψ) is the transformation matrix

equation image
Figure 4.

Scheme of transformation of the antenna coordinates. X, Y, Z determine the orthogonal and equation imagex, equation imagey, equation imagez determine the nonorthogonal coordinate systems.

[24] Equations (1), (6) and (7) are used for simulation of the Stokes parameters in a model of a nonorthogonal antenna system. This model has 8 input parameters: α, β, γ and ψ (which are the tilt angles of the antenna electric axes), hx/hz and hy/hz (which are ratios of the antenna effective lengths), θ and ϕ (which are the wave incidence angles).

[25] The solid lines in Figure 5 show simulated temporal variations of the normalized Stokes parameters determined in three orthogonal antenna planes for an oblique circularly polarized wave propagating from a point source. The dotted lines represent Stokes parameters calculated for a model of a nonorthogonal antenna system.

Figure 5.

Model variations of the normalized Stokes parameters for oblique incidence of a circularly polarized wave (point source) calculated for the orthogonal antenna system (solid line), and for the nonorthogonal antenna system with assumed tilts of the X and Y electric antenna axes to be α = 10°, β = 60°, γ = 10° and ψ = 15° (dotted lines). In both cases hx/hz = hy/hz = 1.0 and assumed wave incidence angles are θ = 56° (colatitude) and ϕ = π/2 + Ωt (azimuth), where spin angular frequency Ω = equation image s−1.

[26] The left panels of Figure 5 show the temporal variations of the Stokes parameters for the X and Y antennas. It is seen that every second maximum (when ϕ ≈ 2πn, n = 0, 1, 2,.) of the modulated Qxyn parameter (n, index marks the nonorthogonal model, dotted line in Figure 5) is higher than Qxyo (o, index marks the orthogonal model, solid line). The remaining maxima (ϕ ≈ π + 2πn) of the Qxyn are lower than corresponding maxima for Qxyo. Every second minimum (ϕ ≈ 3/4π + 2πn) of the Qxyn is shallower than Qxyo. Moreover, there are phase delays of every maximum of the Qxyn pattern and its every second minimum (when ϕ ≈ π/2 + 2πn). The modulation of the Uxyn pattern is delayed in phase and shifted down with respect to Uxyo. The minima and maxima of the Vxyn are delayed with respect to Vxyo.

[27] The middle panels of Figure 5 show that for the pair of Y and Z antennas every second minimum (ϕ ≈ 2πn, n = 0, 1, 2,.) of the Qyzn pattern is deeper than Qyzo and the remaining minima (ϕ ≈ π + 2πn) are shallower. Also the minima of Qyzn are phase delayed. Every second maximum of the Uyzn pattern (ϕ ≈ 7/4π + 2πn) is higher and every second minimum of Uyzn (ϕ ≈ 5/4π + 2πn) is shallower than for Uyzo pattern. The Vyzn pattern is only slightly modulated and Vyzo is constant.

[28] The right panels of this figure show that every second maximum (ϕ ≈ π + 2πn) of Qzxn is higher than Qzxo and every second minimum (ϕ ≈ 3/4π + 2πn) is shallower. The modulation pattern of the Uzxn is distorted and the minima of Vzxn are delayed with respect to Vzxo.

4. Fitting of the Modeled Stokes Parameters to the Observations

4.1. Nonlinear Least Squares Method

[29] A nonlinear least squares method was used to fit the Stokes parameters in the nonorthogonal antenna system to those determined from the observation. It is based on an iteration process in which the following system of equations is solved at each step [Press et al., 1986]:

equation image

where χ2 = equation imageσi−2(SobsiSmodi)2; σi is the uncertainty of measurements; Sobsi and Smodi are the observed and modeled Stokes parameters; X = (θ, ϕ, α, β, γ, ψ, hy/hz, hx/hz) are the parameters of the model; X0 is an initial guess for X; and the n and m indices symbolize the components of X. The best fit is obtained when simultaneously χ2 = min for all 9 Stokes parameters.

[30] The measurement errors σi in equation (9) can be derived from equations (1): σi = equation imageΔPkj, where Smod are the modeled Stokes parameters, Pkj are the modeled power densities from equations (6) and ΔPkj are the fluctuations of the measured power densities in each polarimeter channel. The fluctuations ΔPkj can be approximated as ΔPkj/Pkj = 1/equation image, where Δf is the receiver bandwidth and τ is the time constant. For each frequency step of the Polrad radio spectrometer ΔF = 4096 Hz, and τ = 0.006 s and ΔPkj/Pkj ≈ 0.2. Large signal fluctuations in Polrad measurements are mainly due to the short time constant. In this work the observations are averaged over the range of 10 frequency steps (40.9 kHz) and ΔPkj/Pkj = 0.2/equation image ≈ 0.06.

[31] For simplicity of calculations the fitting procedure has been applied in three steps:

[32] Step 1: Fitting the normalized Qzx and Qyz. They are used for determination of θ, ϕ, α, β, hx/hz (Qzx) and θ, ϕ, γ, ψ, hy/hz (Qyz) instead of the whole set of 9 Stokes parameters. The reason is that the parameters U and V are more fluctuating than Q, since three channel outputs are needed for their determination (Pk+j, Pk, Pj for the U and Pk*+j, Pk, Pj for the V).

[33] Step 2: Reduction of a large number of possible solutions. Since the equations which determine the modeled Stokes parameters are nonlinear, the inverse problem has 64 = 1296 sets of physical solutions (for six unknown angles, each having four possible values). The nonphysical solutions with negative hx/hz and hy/hz were rejected at the beginning. The large number of solutions can then be reduced in the following way: (1) Since the main aim of this investigation is finding the antenna geometry, we are not interested in exact determination of the source direction (θ, ϕ). Thus only those sets of solutions in which 0 < θ < π/2 and 0 < ϕ < π/2 have been chosen and the number of possible solutions is reduced to 44. (2) It is expected that the offsets of the antenna electric axes are small relative to π. Then the sets of solutions in which antenna offsets are π ± α, π ± γ or π ± ψ can be rejected. Furthermore, since the solutions with α and −α are the same, the sets of solutions with −α can also be rejected. Thus only 16 possible sets of solutions are remained. One of them will be selected in the next step.

[34] Step 3: Choice of a solution that describes the real geometry of the antennas. This large number of remaining sets of solutions is a consequence of using only two Stokes parameters for fitting. To find the solution that describes the real geometry of the antennas all 16 solutions are tested for fitting to the whole set of 9 Stokes parameters. The solution which fulfills the condition of the smallest deviations from the observations (equation imageχ2 = min) is selected.

[35] As an example an AKR event of 17 September 1997 will be analyzed. The power spectrum of the Iyz parameter is shown in Figure 6. The frequency range and time interval selected for the analysis is marked by the rectangle. The solid lines in Figure 7 show the temporal variations of the normalized Stokes parameters averaged over the frequency range from 200 to 300 kHz and smoothed with the Savitsky-Golay filter. The dotted lines in this figure represent variations of the normalized Stokes parameters calculated for the nonorthogonal model and fitted to the observations. The best fit for this AKR event is achieved with the following parameters of the model: α = 0.01° ± 0.26°, γ = 8.0° ± 0.5°, ψ = 15.6° ± 2.3°, θ = 56.1° ± 0.6°, ϕ = 81.4°(±1.6°) + Ωt, hy/hz = 0.91 ± 0.01 and hx/hz = 0.96 ± 0.01 (β is left undetermined due to the small value of α).

Figure 6.

Power spectrum of Iyz parameter for AKR event of 14 September 1997. The rectangle indicates the time-frequency range (0908:40 to 0917:00 UT and 200–300 kHz) of the analysis.

Figure 7.

Variations of the measured normalized Stokes parameters for the AKR event of 17 September 1997, 0908:40 to 0917:00 UT averaged over the frequency range from 200 to 300 kHz (solid lines). Variations of the Stokes parameters calculated for nonorthogonal model of the antenna system (dotted lines) and fitted to observations. The best fit is obtained for α = 0.01° ± 0.26°, γ = 8.0° ± 0.5°, ψ = 15.6° ± 2.3°, hx/hz = 0.96 ± 0.01, hy/hz = 0.91 ± 0.01 and wave incidence angles θ = 56.1° ± 0.6° and = 81.4° (±1.6°) + Ωt.

[36] The similarity of the modeled and observed temporal variations of the Stokes parameters is obvious. The best fit is achieved for the Q parameters. The reason that the U and V parameters are not fitted as good is that for determination of the Q parameter only two channel outputs (Pk, Pj) are sufficient, whereas for the U and V parameters all three channel outputs (Pk+j, Pk, Pj) or (Pk*+j, Pk, Pj) are needed (see equation (1)).

4.2. Application of Singular-Value Decomposition

[37] The system of equations (9) can be rewritten in the matrix form:

equation image

where equation image, the matrix equation image, and Δequation image ≡ ΔXm.

[38] Since in some ranges of source directions the system of equations (10) may be ill-posed, the singular-value decomposition (SVD) technique [see Connerney, 1981; Ladreiter et al., 1995, and references therein] was applied to examine and evaluate the uniqueness of the solution:

equation image

The matrices equation image and equation image are composed of the eigenvectors of equation imageTequation image and equation imageequation imageT. The diagonal elements of the matrix equation image denote the singular values (λi) of equation image. The solutions (11) are linearly independent when the number of significant singular values is equal to the number of unknowns in equation (10). The ratio of the largest singular value to the smallest one is C = λmaxmin, which is the condition number of the least squares equations that describes how well the system of equations is determined. If C is very large the system of equations (10) is ill-conditioned and has no unique solution when noise on the data is considered.

[39] Now the system of equations (10) is analyzed in the same way as described by Ladreiter et al. [1995]. First, the data are simulated using equations (1), (6), and (7) with addition of the normal random noise (ΔPk/Pk ≈ 0.06) to outputs of the polarimeter channels. The assumed model parameters are: α = 5°, β = 60°, γ = 10°, ψ = 15°, hx/hz = hy/hz = 1.0 and for source directions θ = 0°, ., 90°, ϕ = 0°, ., 180°. The SVD least squares procedure described above has been applied to these data. As noted in subsection 4.1, only the Qzx and Qyz parameters have been used. Since the Polrad coordinate system is spinning, the condition number of equations (10) is axisymmetric with respect to the spin axis X and does not depend on the source azimuth ϕ. Figure 8 shows the logarithm of the condition number of equations (10) for the Qzx and Qyz parameters averaged over the source azimuth ϕ as a function of the source colatitude θ. Analysis shows that the singular values λ, for which λmax/λ > Clim, can be set to zero without significantly increasing χ2 in equations (10). Clim is merely dependent on the uncertainty of the measurements and the assumed parameters of the model. In our case Clim ≈ 1000 for Qzx and Clim ≈ 100 for Qyz. Now, zeroing the column of eigenvalues in matrix equation image corresponding to λ = 0, the resolution matrix can be obtained as equation image = equation imageequation imageT (for more details, see Ladreiter et al. [1995]). Each diagonal element of equation image, named the parameter resolution, describes the ability of the equations (10) to obtain the linearly independent solution for the given parameters of the model. The determined parameter resolutions (R(α), .,R(hy/hz)) of all model parameters are shown in the bottom panels of Figures 9 and 10. For the range where R = 1 the solution of the equations (10) is stable.

Figure 8.

The logarithm of the condition number of the least squares equations (10) for the modeled Stokes parameters Qzx and Qyz averaged over the source azimuth ϕ as a function of a source colatitude θ. The horizontal dashed lines indicate the critical values of Clim = λmaxmin. Clim ≈ 1000 for Qzx and Clim ≈100 for Qyz for the given model parameters and uncertainty of the measurements (see section 4.2).

Figure 9.

The top panels show the scatterplots of the obtained tilt angles of the electric antenna axes versus source colatitude θ. β is not shown because its determination does not make sense for small α. Error bars are also shown. The bottom panels show the parameter resolutions of α, γ and ψ angles. They have been derived from resolution matrix for the modeled Qzx and Qyz Stokes parameters. The resolution parameters determine the range of the source colatitude θ, where the linearly independent solutions for α, γ and ψ exist.

Figure 10.

The top panels show the scatterplots of the relative effective lengths of the Polrad antennas. The solid lines present the best fit of hx/hz and hy/hz as a function of the source colatitude (hx/hz ≈ 1.1 (±0.01) − 0.25 sin2θ and hy/hz ≈ 0.88 (±0.01) + 0.06 sin2θ). The bottom panels show the parameter resolution of hx/hz and hy/hz.

5. Determination of the Polrad Antenna Geometry

[40] Now 25 sufficiently strong and relatively long (∼1 hour) events of AKR emission have been selected. The Stokes parameters simulated for the nonorthogonal antenna model are fitted to the measurements using the least squares method described in the previous section. The fitting procedure has been used in the following way. The normalized Stokes parameters Qzxobs and Qyzobs which are determined from the observations, are averaged over 10 frequency steps (40.9 kHz). Fitting of the model parameters Qzxmod and Qyzmod is limited to 10 minutes to avoid effects of the spacecraft orbital motion relative to the AKR source. The beginning of the next time interval is shifted by 2 min and the same fitting procedure repeated.

[41] The top panels of Figures 9 and 10 show scatterplots of the determined tilt angles of the electric axes of the X and Y antennas and their effective lengths as function of AKR source colatitude θ. The bottom panels of Figures 9 and 10 show the parameter resolution of the resulting parameters for the antenna geometry. The parameter resolution determines the range of θ, where the linearly independent solution of equation (10) α, γ, ψ, hx/hz, or hy/hz exists. It is well seen from the figures that in the range where the given parameter resolution is less than 1.0 (R < 1.0) the scattering of the obtained solutions as well as their errors are noticeably larger then in the range where R = 1.0.

[42] The solutions are averaged over the range of R(α) = 1.0, ., R(hy/hz) = 1.0 and their weighted mean values are shown in Table 1. It is found that the electric axis of the X antenna practically coincides with its rod. Its tilt appears to be 0.1° ± 1.1° (β is uncertain for small α). The tilt of the electric axis of the Y antenna from its rod is +5.5° ± 0.2° in the spin plane, and +7.9° ± 0.1° in the solar direction (see Figure 5).

Table 1. Components of the Effective Length Vectors of the Polrad Antennasa
AntennaPhysicalEffective
h/hzαβγψh/hzαβγψ
  • a

    The column headed “physical” contains the values which describe the relative length and direction of the antenna rods. The column headed “effective” contains the relative effective antenna lengths and directions of the electric antenna axes.

X1.0--1.1(±0.01) − 0.25 sin2 θ0.1° ± 1.1°---
Y1.0--10°0.88(±0.01) + 0.06 sin2 θ--7.9° ± 0.1°15.5° ± 0.2°
Z1.0----1.0----

[43] It is also well found that the effective lengths of the Polrad antennas are not equal (Figure 10). The top panels of Figure 10 show that relative effective lengths of the X and Y antenna depend on direction of the observed AKR according to the formulas: hx/hz ≈ 1.1(±0.01) − 0.25 sin2θ and hy/hz ≈ 0.88 (±0.01) + 0.06 sin2 θ. This effect can be explained by distortion of the real angular pattern of the Polrad antennas from an ideal dipole.

6. Rectification of the Polrad Polarization Measurements

6.1. Mutual Calibration of the Effective Antenna Lengths

[44] Let equation image0 = (Px, Py, Pz, Px+y, Py+z, Pz+x, Px*+y, Py☆+z, Pz*+x) be the power density vector, whose components are measured in the polarimeter channels with antennas of equal effective lengths (hx = hy = hz). Let equation image be the power density vector measured with hxhyhz. Then using equations (6):

equation image

where equation image is a transformation matrix:

equation image

[45] With hz = 1 the effective lengths of the X and Y antennas (see Table 1) are hx ≈ 1.1–0.25 sin2 θ and hy ≈ 0.88 + 0.06 sin2 θ. The source colatitude θ is estimated according to Panchenko [2003]: sin2θ ≈ 2Px/(Px + Py + Pz). Using equation (12) the measured power densities can be corrected.

6.2. Rectification of the Stokes Parameters to the Orthogonal Antenna System

[46] Consider a circularly polarized wave impinging on the system of three orthogonal antennas. Then the Stokes parameters determined in each antenna plane can be described by a vector equation imageort(Ixy, Iyz, Izx, Qxy, Qyz, Qzx, Uxy, Uyz, Uzx, Vxy, Vyz, Vzx). The vector equation imageort has 12 components. The Stokes parameters determined in the nonorthogonal antenna system are described by a vector equation imagen. It contains some instrumental components that originate from nonorthogonality of the antennas. The dependence between equation imageort and equation imagen can be described as:

equation image

where equation image2 is a 12 × 12 matrix. It transforms the Stokes parameters determined with the orthogonal antenna system to the nonorthogonal one. Since in the case of Polrad only the Y antenna appears to be significantly tilted from orthogonality (see previous section), the matrix equation image2 depends only on the tilt angles γ and ψ:

equation image

where a = sin γ, b = cos γ, c = sin ψ, d = cos ψ.

[47] The measured Stokes parameters can then be rectified to the orthogonal antenna system using equation (14).

[48] In Figure 11, the solid and dotted lines show the variations of the measured normalized Stokes parameters before and after calibration of the antenna effective lengths and after rectification into the orthogonal antenna system. It is seen that the details of the modulation patterns (discussed in section 3) arising from the antenna nonorthogonality have practically disappeared (compare with Figure 5). The corrected Stokes parameters for the ZX antenna plane are the same as the measured ones due to negligible tilt of the electric axis of X antenna.

Figure 11.

Rectification of the Polrad polarization measurements to the orthogonal antenna system with equal effective antenna lengths. Solid lines show observed normalized Stokes parameters for the AKR event 17 September 1997, 0908:40 to 0917:00 UT. Dotted lines show rectified Stokes parameters.

7. Discussion

[49] The analysis of the response of the triaxial polarimeter to the circularly polarized AKR allows us to determine the offsets of the electric antenna axes from orthogonality as well as their effective lengths. Their values can be used to correct the AKR measurements to the orthogonal antenna system.

[50] The main purpose of Polrad is determination of the AKR propagation modes. The handedness of the wave electric vector can be determined in the orthogonal antenna system. However, knowledge on the source direction is needed to establish unambiguously the wave mode. One possible way to determine the AKR source direction with three orthogonal electric antennas is described by Panchenko [2003].

[51] The source direction finding needs knowledge how the antenna electric axes are set against the spacecraft coordinate system. In section 5 the Polrad antenna geometry has been found with assumption that the electric and physical axes of the Z dipole cover each other. To check this assumption it is necessary to have a point radio source of a sufficiently constant intensity and an a priori known location. Solar type III bursts are strong enough but they cannot be used as reference sources due to their broad angular extent. Jupiter is the point radio source from the distance of Earth. However, it is not accessible due to low sensitivity of Polrad because of its short antennas. In this situation our assumption of the negligible tilt of the Z electric axis is based on results from other spacecraft. For ISEE − 3 [Fainberg et al., 1985] and the Cassini model [Rucker et al., 1996] the electric and physical axes of the dipole antennas practically coincide with each other. For ISEE − 3 the tilt was less than 1.5° and for the Cassini model it was ≈0.5°. Since the Z antenna of Polrad is a dipole one can expect that the offset is relatively small.

[52] It should also be noted that large tilt of the electric axis of the Y monopole (γ = 7.9° and ψ = 5.5°) is in accordance with results obtained for the monopoles antennas of ISEE-3, Voyager, and Cassini (see introduction). It is interesting to note that contrary to our expectations, the electric axis of the X monopole of Polrad appeared to be negligibly tilted. The possible explanation can be that the X antenna was located sufficiently far from most of spacecraft structures which could disturb the antenna electric properties.

[53] In our investigation we have also assumed that the directional characteristics of the antennas are identical with those of a short dipole. In fact the angular dependence of the relative effective antenna lengths hx/hz and hy/hz from the source colatitude (see Table 1 and section 5) can be a signature of distortions of these characteristics from the ideal dipole. Without a reference radio source these distortions cannot be determined directly from the Polrad measurements.

[54] In this study it was also assumed that the AKR is emitted from point sources. Although the source extension can influence the results it has not been taken into account since the angular size of the AKR sources cannot be determined directly from the Polrad measurements. It was calculated by Panchenko [2003] that the differences between the model of extended and point radio sources are less than the accuracy of the Polrad measurement for sources of half width less than 30°. This condition was fulfilled near the apogee of the spacecraft orbit.

[55] The geometry of the Polrad antenna system and source direction were obtained using only the one Stokes parameter Q, which was more accurately measured than others (U and V). With a more sophisticated analysis of all Stokes parameters it is possible to obtain the complete wave polarization state and wave direction without the assumption of circular polarization [Lecacheux et al., 1979]. This idea is also implemented to RPWS instrument on Cassini.

8. Conclusions

[56] 1. It is shown that effective length vectors of the antenna system can be determined by fitting the modeled Stokes parameters to the ones observed with triaxial polarimeter. The method has been implemented for the Polrad-Interball-2 electric antenna system.

[57] 2. For the Y monopole of Polrad the tilt of its electric axis has been determined to be +5.5° ± 0.2° in the spacecraft spin plane (15.5° from orthogonality) and +7.9° ± 0.1° in colatitude (antisolar direction). For the X monopole the tilt of its electric axis appears to be negligible (0.1° ± 1.1°).

[58] 3. For Polrad the determined relative effective antenna lengths are: hx/hz ≈ 1.1(±0.01) − 0.25 sin2 θ and hy/hz ≈ 0.88(±0.01) + 0.06 sin2θ. The dependence of hx/hz and hy/hz on the source colatitude θ can be explained by distortion of the antenna directional characteristic from an ideal dipole.

[59] 4. Having determined the effective antenna length vectors the measured Stokes parameters can be rectified to the orthogonal coordinate system.

Acknowledgments

[60] The author wishes to extend special thanks to J. Hanasz for many helpful discussions and assistance in editing this paper. He is very grateful to the referees for their valuable critical comments and suggestions. The Polrad data have been processed in CNES, Toulouse (France). This work was financially supported by the Polish Committee on Scientific Research through the grant 5T12E 001 22.

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