Radio goniopolarimetry: Dealing with multiple or 1-D extended sources

Authors


Abstract

[1] The goniopolarimetric methods, also known as direction finding, are powerful methods which are used to obtain the intensity, the polarization, and the position of the low-frequency radio sources in the sky plane. These methods have been applied to several auroral sources and to the solar wind sources. The results obtained by these methods have allowed us to drastically improve our understanding of the processes occurring in the concerned regions. These methods are only valid for point source, or weak extended sources, whereas in many cases extended or multiple sources are expected. Some methods allow to obtain the characteristic size of the source, but not its shape or the intensity distribution inside it. In the present paper a method is proposed which can determine the intensity distribution inside an extended, but one-dimensional source and which can be applied to observations of two close sources. Application to simulated auroral and solar wind emissions are also presented.

1. Introduction

[2] Low-frequency (few kHz to few tens of MHz) radio astronomy is a way to explore the low-density regions of our solar system, i.e., the interplanetary medium and the magnetospheres of the magnetized planets, as these frequencies correspond to the eigen frequencies of the plasma of these regions (typically the electron cyclotron frequency ωc or the plasma frequency ωp). Unfortunately the wavelengths of these waves range from a few meters up to kilometers limiting the resolution that one can expect from the observations. Moreover, as the terrestrial ionospheric cutoff is about 10 MHz, most of these observations have to be conducted from spacecraft, whose antenna length is limited to a few tens of meters. The spatial resolution of the instruments are limited to ∼λ/D radians, with λ the wavelength and D the antenna length. For the wavelengths in which we are interested, this ratio is larger than 1, most of the time by many orders of magnitude. It implies that low-frequency (LF) radio astronomy is performed with almost no spatial resolution.

[3] However, one can note that the antenna beaming pattern is not isotropic. Assuming the short dipole approximation (λD) this beaming pattern varies as sin2θ, where θ is the angle between the source direction and the dipole direction [Kraus, 1966]. It is then possible to exploit this anisotropy to derive the source direction. This can be performed using several dipoles with different orientations, or by using the variation of the angular distance between the effective antenna direction and the direction of arrival of the observed wave, for example in the case of a spinning spacecraft. As the wave components associated with different frequencies can be separated, the position of the sources can be obtained for each frequency. The specific methods which exploit the beaming pattern anisotropy to obtain the source direction are commonly named direction-finding methods. However, the direction of arrival of the wave, using several antennas, cannot be found without determining of the source Stokes parameters which describe the wave flux and polarization. Thus Cecconi [2007] proposed the word “goniopolarimetry” to name this method, which more accurately describes it and henceforth will be used in the present paper.

[4] The goniopolarimetry methods can be split into two kinds: (1) The analysis of the modulations of the signal observed by a spinning spacecraft [Fainberg and Stone, 1974; Lecacheux, 1978; Manning and Fainberg, 1980; Reiner and Stone, 1988; Ladreiter et al., 1994] and (2) the analysis of the autocorrelations and cross correlations between two or three antennas on a three-axis stabilized spacecraft [Lecacheux, 1978; Ladreiter et al., 1995; Hanasz et al., 2000; Santolík et al., 2001; Panchenko, 2003; Vogl et al., 2004; Cecconi and Zarka, 2005; Cecconi, 2007]. In the present paper we will address only the second kind of methods, as they are used on the most recent spacecraft with goniopolarimetry capacity (Cassini [Gurnett et al., 2004] and STEREO [Bougeret et al., 2008; Cecconi et al., 2008]).

[5] The autocorrelation and cross-correlation methods have until now been used to determine the position of point sources only (at least for instantaneous measurements, over a longer time the spread of the results may define the extension of an extended source [Cecconi et al., 2009]). This is due to the limited number (up to nine) of quasi-simultaneous measurements, which limit the number of unknowns, when six are needed for each point source. This is an issue since numerous astrophysical sources of low-frequency radio emissions correspond to phenomena occurring on large spatial scales and are expected to be extended (along an auroral oval for example). Thus methods that are applicable to extended sources or for multiple sources emitting at the same time and frequency, have to be developed. Some methods have been proposed which can determine a characteristic scale of the sources for observations made by a spinning spacecraft [Fainberg and Stone, 1974; Steinberg et al., 1985; Manning and Fainberg, 1980]. However, these methods assume an isotropic, gaussian distribution for the intensity, which may not correspond to the actual distribution of the source intensity. Thus while these methods provide some information as to the spatial extent of the source, they do not provide the source shape or the actual distribution of intensity inside the source.

[6] In the present paper we present a new method with this capacity, although it has some limitations due to the mandatory hypothesis we have to make in order to reduce the number of unknowns to be computed. We restrict ourselves to sources with a linear (one-dimensional) extent. This method can be applied to most of the astrophysical cases under investigations. In particular we show in sections 4 and 5 its application to auroral and solar wind emissions. However, it should be noted that the present paper only proposes a new goniopolarimetric method in a general, theoretical case. This method has to be adapted for each source and each spacecraft. The examples of sections 4 and 5 deliberately ignore some specifics to stay in a more general point of view.

2. Goniopolarimetry Methods for a Single Source

[7] In the following we will deal with the case of the goniopolarimetric methods applied to space-based observations of radio emissions done with three antennas. In this case one can obtain nine measurements at each time and at each frequency: three autocorrelations on each antenna, three real parts of the antenna cross correlations and three imaginary parts of the cross correlations. In this section a brief description of the state of the art in retrieving the position and the Stokes parameters of the source is presented. For a more detailed description, including several analytical or numerical methods to solve the goniopolarimetric equations, see Cecconi and Zarka [2005] and Cecconi [2007, and references therein].

2.1. Point Source

[8] For a point source, the expression of either the autocorrelation or cross correlation PIJ of the measurements of the three antennas was derived by Ladreiter et al. [1995] and Cecconi and Zarka [2005]. It can be written as:

equation image

with S, Q, U, V the Stokes parameters of the incoming wave, hI (or hJ) the effective length of antenna I (or J), G the gain and Z0 the impedance of the free space. The functions ΩI and ΨI stand for the coordinates of antenna I projected on the wave plane. These functions depend on the direction of arrival of the wave in the spacecraft coordinate system. For the example of the present study we choose to use the cartesian coordinate system and the antenna geometry of the Cassini spacecraft orbiting Saturn, which are shown in Figure 1 and described in detail by Vogl et al. [2004]. In the present paper we choose to use a spherical coordinate system derived from the spacecraft cartesian system:

equation image

In this coordinate system the ΩI and ΨI functions are given by [Cecconi and Zarka, 2005]:

equation image
equation image

where θI and ϕI are the declination and the azimuth of the antenna I, respectively, and θ and ϕ are the colatitude and the azimuth of the source, respectively.

Figure 1.

The three RPWS electric antennas onboard the Cassini spacecraft and the reference coordinate system [from Cecconi and Zarka, 2005].

[9] As one can see from equations (1), (3), and (4), the system of equations to be solved is not linear. In order to solve the goniopolarimetric equation (equation (1)) several methods have been proposed. Cecconi and Zarka [2005] found a particular set of solutions for which the goniopolarimetry equation can be solved analytically. Other methods involve χ2 minimization methods (or gradient methods). However, the χ2 coefficient is not linear and neither is its first derivative. Solving the goniopolarimetry equation is thus not trivial and requires some higher-level methods. Ladreiter et al. [1995], Santolík et al. [2003, 2006], and Vogl et al. [2004] developed numerical algorithms able to solve the above equation.

2.2. Extended Source

[10] The case of the extended sources has been addressed by Cecconi [2007]. The author uses simple models of the intensity distribution (e.g., uniform, gaussian) and shows that the point source inversion is unable to deal with sources whose apparent spatial extension is larger than 5°. In the case of an extended source, the goniopolarimetry equation is slightly modified. Cecconi [2007] assumes a phase decorrelation inside the source, so that the antenna measurement results from the integration of equation (1) on the apparent surface of the source:

equation image

This assumption is based on the fact that emissions near the plasma eigen frequencies are generally due to resonant amplification of the background noise, by Cyclotron Maser Instability in the case of the auroral emissions [Zarka, 1998] or by weak-beam instability in the solar wind case [Smith, 1970]. Cecconi [2007] derives analytic expressions of the goniopolarimetric equations for specific cases but does not propose any means to derive either the source extent or the distribution of intensity.

3. Multiple Sources Within a Spatially Extended Region

3.1. Outline

[11] The single-source goniopolarimetry method is limited by its inability to deal with multiple or extended sources, whereas in many cases a non-point-like source is expected. UV and IR auroral sources have a large extent in longitude, which results in a spread of the single-source goniopolarimetry measurements [Cecconi et al., 2009]. The solar wind sources emit at both the fundamental and first harmonic [Alvarez et al., 1974; Dulk et al., 1984], which means that for a given frequency there can be several sources. Therefore a method investigating the source distribution must be developed. However, as six values are required to determine each source Stokes parameters and direction of arrival and nine simultaneous measurements can be made at most, additional hypothesis are needed.

[12] In the present paper the hypothesis of an extended but one-dimensional source is assumed. This describes well the auroral emissions, as for a given frequency the sources are along a nearly one-dimensional auroral arc.

[13] The simplification we make have to reduce the number of unknowns. We fulfill this requirement by assuming that the coordinates of the one-dimensional curve along which the sources are is known a priori. This can be done quite easily for the auroral sources, as they are located along the field line related to the UV emissions, at an altitude which is related to their frequency (the emissions occur at the electron cyclotron frequency [Zarka, 1998]). Then a magnetic model of the planet internal field and observations of the UV auroral oval are sufficient to know the sources coordinates. In the solar wind such computation is a lot more tricky and will be discussed in section 5.

[14] Assuming that the coordinates of the sources (i.e., the 1-D curve along which they are spread) are known, the next step is to determine the variation of the Stokes parameters along the curve. For that purpose we define:

C

the 1-D curve along which the sources are spread;

s ∈ [0,1]

the curvilinear coordinate along C;

S(s)

the intensity profile;

L

a scalar defining the linear polarization degree;

l(s)

the normalized linear polarization profile;

α(s)

the angle profile defining the linear polarization direction;

V(s)

the circular polarization profile.

[15] The usual Stokes parameters defining the linear polarization (Q and U) have been combined in three parameters (L, l(s) and α(s)). This has been done since the variation of the amplitude l(s) and the direction α(s) of the linear component are more straightforward to obtain from theoretical arguments than the Stokes parameters (see section 4 as example). The L parameter is used to scale the amplitude distribution of the linear component. The Stokes coefficients for the linear polarization are related to L, l(s) and α(s) by:

equation image
equation image

[16] In the following examples the profiles l(s) and α(s), related to the linear polarization of the wave, are assumed from theoretical arguments. In a more general case these profiles can be unknown (provided that we have enough measurements to determine them). Then we have to determine S(s), V(s) and L. The purpose of the method presented in this paper is not to determine the profiles, but as many of the Fourier coefficients characterizing them as we can:

equation image
equation image

with NS the number of Fourier orders used to describe the intensity profile S(s), and NV the number of Fourier orders describing the circular polarization profile V(s). As the circular polarization degree acts only on the imaginary part of the cross correlation between the antennas (three measurements at most) the maximum value for NV is 1 (three unknowns: V00, V10, V11).

3.2. Goniopolarimetry Equation and Solving for an Extended Source

[17] The goniopolarimetry equation for the Fourier decomposed source is:

equation image

with the functions Ω and Φ depending on the curvilinear coordinate s. In the following examples the dependance of the measurement PIJ on the distributions S(s) and V(s) has been decomposed into two separate sets of linear equations which are written, using Einstein's sum convention:

equation image
equation image

with PS the measurements related to the intensity profile determination and PV related to the circular polarization profile determination. The two matrix MS and MV depends on the linear polarization. Typically this split is possible by assuming one of the profiles to be known, or by setting the matrices MS to be the real parts of the measurements and MV to the imaginary ones.

[18] Note that for a fixed linear coefficient L the equation sets are purely linear and thus could be solved by inverting the matrix MS and MV. However, the measurements on the antennas are affected by a random noise. When taken into account, this noise can strongly modify the result of the direct inversion of the linear matrices, making it unusable in practice. Thus a least squares fit, taking into account the existence of a random noise should be used. The method used hereafter proceeds in two steps: First, we determine approximate Fourier coefficients assuming that the contribution of each Fourier order decreases as the order increases, allowing to determine them by a perturbation method. Then we use these approximate coefficients as starting point for a Levenberg-Marquart least squares fit. The resolution algorithm described here is written to solve the intensity distribution, but it applies the same way to the circular polarization distribution.

[19] The Fourier coefficients of each order k are determined sequentially, beginning with k = 0. When the coefficients of a given order are determined, the contribution of this order to the measurements P is subtracted. The higher order are thus determined from low-amplitude residuals (noted hereafter Pk, with k the order of the residual). This may be an issue for determining the highest-order coefficients, but we privilege a good estimation of the low orders, even if it implies that our resolution could be limited to these low-order scales.

[20] At each step the phase Θk = arctan(Sk1/Sk0) is determined by searching for the phase which maximizes the correlation coefficient between cos(Θk)Mk0 + sin(Θk)Mk1 and Pk−1. Then the amplitude Sk = equation image is obtained by searching for the amplitude which gives the minimum residual ∥Pk/σ∥ = ∥(Pk−1Sk(cos(Θk)Mk0 + sin(Θk)Mk1))/σ∥ (with P−1 = P). The vector σ describes the amplitude of the noise on each antenna. With this method we obtain an approximation of the Fourier coefficients which are then used as starting point for a least squares fit.

[21] There is no general analytical method to obtain the linear polarization degree L. However, if the total polarization is known it can be approximated by determining S0 with L = 0. We then determine L using S(s) = S0 (i.e., a uniform distribution of intensity) and the same method as for the Sk coefficients, i.e., by minimizing the residual. The higher orders Sk>0 are subsequently determined using this value of L. The determination of L is the most difficult part of the method we propose. This coefficient is strongly dependent on the characteristics of the particular case to which the method is applied and so is the method to determine it. As we present the method in a case as general as possible we choose to not address it in the present paper; however, it will have to be addressed before applying the method to observations. In the next two sections we present realistic examples of the application of this method.

3.3. Simulations

[22] Hereafter we apply of the proposed goniopolarimetric method to simulated observations. The first goal of these simulations is to determine the sources of errors that can occur. These sources have been listed by Cecconi et al. [2008]:

[23] 1. The first source is galactic background. In the low-frequency range the sky background is bright and has to be subtracted before any goniopolarimetric analysis.

[24] 2. The second source is receiver and digitization noise and calibration indetermination.

[25] 3. The third source is intrinsic source variability. In the present paper we assume that the source(s) characteristics do not vary over the measurement duration. However, it may be an issue when observing quickly varying sources.

[26] 4. The fourth source is geometrical configuration. The beaming pattern of the antennas has sharp nulls in the antenna directions. Thus when the source is near the direction of an antenna the goniopolarimetry methods gives inaccurate results. In order to quantify the effect of the geometrical configuration, the simulations are performed several times with a source position varying in the sky plane.

[27] 5. The fifth source is signal-to-noise ratio. The presence of a bright sky as well as the receiver noise generates fluctuations of the antenna measurements. A signal-to-noise ratio (SNR) larger than 20–30 dB is necessary to perform a goniopolarimetric analysis of an observation [Cecconi and Zarka, 2005]. In the present paper we assume a SNR of 26 dB. Since the noise is not coherent, we assume that it is present on the autocorrelation measurements only. In fact it also affects the cross-correlation measurements, but with a lower amplitude so we choose to neglect it. These assumptions are consistent with those in previous single-source studies [Cecconi and Zarka, 2005; Cecconi, 2007] and thus permit comparisons.

[28] 6. The sixth source is multiple or extended sources: the purpose of the present method.

4. Application to Auroral Emissions

4.1. Outline

[29] The polar auroras are phenomena which occur at all planets with an internal magnetic field. They appear in infrared, visible and UV as polar ovals or spots on the top of the ionosphere of the planet. In addition to these emissions there are low-frequency radio emissions, generally in the kilometric range (few hundreds of kHz) but reaching frequencies as high as 40 MHz in the case of Jupiter. The frequencies of these emissions correspond to the local electron cyclotron frequencies along the magnetic field lines related to the infrared and UV ovals or spots [Zarka, 1998]. Thus each frequency corresponds to an oval or a spot at a given altitude above the planet's ionosphere. Goniopolarimetric studies of these emissions have been performed in the past. Some have used goniopolarimetric methods based on the probe spin but the most recent used a two or three antennas direct inversion. The most complete studies using three-axis stabilized spacecraft have been performed using the Cassini spacecraft data at Saturn [Cecconi et al., 2009; Lamy et al., 2009]. These studies, involving single-source goniopolarimetry only, have confirmed that the emitting field lines, related to the auroral emissions of Saturn, have footprints which map the UV auroral oval. Moreover, the observed frequencies match quite well with the local electron cyclotron frequency. Previous studies show results consistent with this scheme at Earth [Calvert, 1985; Huff et al., 1988; Panchenko, 2003] and Jupiter [Stone et al., 1992; Ladreiter et al., 1994]. These ovals can be determined through the use of a model of the internal magnetic field of the planet and hence make perfect candidates for the method presented in this paper.

[30] In this case the location of the sources is known to be along a 1-D curve (C) which can be computed analytically. The polarization of the source is measured to be almost 1 for the largest signal-to-noise ratio, purely circular when the observer (i.e., the Cassini spacecraft) is located at low latitudes (<30°) [Lamy et al., 2008b]. This circular polarization is right-handed when the emission occurs in the northern hemisphere and left-handed in the southern. A linear component of the polarization appears for higher latitudes [Fischer et al., 2009]. This appearance is consistent with the cyclotron-maser instability theory which predicts a linear polarization component perpendicular to the magnetic field [Shaposhnikov et al., 1997]. The direction of this linear component can thus be computed analytically since it depends only on the projection of the magnetic field direction in the observer's field of view. Using the previously defined notations:

equation image
equation image
equation image

where eobs is a unit vector defining the observer's line of sight, e0 is the unit vector defining the angular origin and b is the magnetic field unit vector at the source.

[31] The unknowns are the intensity distribution along the curve C and the linear polarization degree L. As the total polarization is taken equal to one the circular polarization V can be deduced from the linear polarization and its sign from the observed hemisphere. Note that in the present examples we assume nine measurements whereas Cassini provides only seven of them (three autocorrelations and two complex cross correlations). This implies that for the Cassini case we should limit ourselves to NS = 3. However, as we study here the general case of the goniopolarimetric method we propose and hence use NS = 4.

[32] In the following examples we choose to simulate a smooth continuous distribution of intensity, whereas it has been shown by Lamy et al. [2008a] that the anisotropic pattern of radio emissions leads to the observation of shadow zones along the oval, in particular emissions cannot be observed in front of the spacecraft. As such features are beyond the subject of the present paper we choose to ignore them. However, these shadow zones can be predicted by numerical codes such as the PRES code [Hess et al., 2008; Lamy et al., 2008a]. Then one can use a discontinuous curve C to model the oval as there is no mathematical requirement to have a continuous curve. This kind of improvement to our method is too specific to the particular case we take as example and therefore we will not discuss it in the present paper.

4.2. Equation

[33] As there are nine observables we can solve nine unknowns, including the linear polarization coefficient. We can thus use eight unknowns to determine the distribution of intensity. These unknowns correspond to the Fourier decomposition:

equation image

where s ∈ [0, 1] is the curvilinear coordinate along the curve C. The goniopolarimetry equation becomes then:

equation image

4.3. Examples

[34] As a first example of the method, we choose emitting sources spread along a straight line in the angular space θ, ϕ with an extension of 10° both in declination and in azimuth. The intensity distribution is shown in Figure 2a (curve 0) and the emission is supposedly coming from the north (RH polarization). The antenna geometry is that of the Cassini spacecraft, as described by Cecconi and Zarka [2005] (Figure 1).

Figure 2.

Simulation of auroral sources spread along a straight line: (a) Averaged distribution of intensity for correlation coefficients comprised between 1–0.85 (curve 1), 0.85–0.75 (curve 2), 0.75–0.65 (curve 3), 0.65–0.55 (curve 4), and 0.55–0.45 (curve 5). Curve 0 stands for the simulated distribution of intensity. (b) Histogram of the correlation coefficients. Most of the obtained distributions of intensity fits the simulated one with a correlation higher than 0.7. (c) Correlation coefficient between the chosen intensity distribution and the one obtained by goniopolarimetry versus colatitude and azimuth of the source. Stars stand for the antennas directions, and crosses stand for the direction opposite to the antennas.

[35] We introduce additional gaussian noise on the autocorrelation measurements whose amplitude relative to the signal intensity is −26 dB. This corresponds to the typical noise level of the observation of Cassini at Saturn for which the single-source method gives accurate results [Cecconi and Zarka, 2005].

[36] The measurements PIJ are computed using equation (17). Then we perform the inversion of this equation using the method described previously. The method is applied for different positions of the arc in the (θ, ϕ) space. The result of the application of the proposed goniopolarimetry method is shown in Figure 2c, which shows the correlation coefficient between the chosen distribution of intensity and the one obtained by the goniopolarimetry method, as a function of the declination and the azimuth of the center of the source. It shows that the correlation coefficient is larger than 0.65 except when the emission is near the direction of an antenna. This dependence of the accuracy on the antenna direction is consistent with what is obtained for a single source method [Cecconi and Zarka, 2005].

[37] Figure 2b shows the histogram of the correlation coefficients. Most of the measurements have correlation coefficients above 0.7. In order to evaluate to what extent the coefficient of correlation is related to the accuracy of the measurements, Figure 2a shows the average distribution of intensity for correlation coefficients between 1–0.85, 0.85–0.75, 0.75–0.65, 0.65–0.55 and 0.55–0.45. Above 0.65 the curve obtained is quite similar to the one simulated (i.e., the relative position and amplitude of the intensity bumps and gaps are sufficiently well described to permit a study of the intensity distribution).

[38] Trying a more realistic example, we simulate an observation of the Saturn's northern auroral oval (chosen to fit the 70° northern parallel) at 200 kHz, with the spacecraft observing from eight Saturn radii. The spacecraft latitude is 30° north. The geometry of the oval is shown in Figure 3a, and the intensity distribution is shown in Figure 2c (curve 0). The oval is incomplete since we can observe only the emissions from field lines for which the angle between the magnetic field and the line of sight is lower than 90°.

Figure 3.

Simulation of auroral sources spread along the Saturn auroral oval: (a) The auroral oval as seen by the spacecraft (the origin is arbitrary). (b) Correlation coefficient between the chosen intensity distribution and the one obtained by goniopolarimetry versus colatitude and azimuth of the source. Stars stand for the antennas directions, and crosses stand for the direction opposite to the antennas.

[39] The histogram of the correlation coefficients is comparable to the one obtained before (although with slightly higher coefficients) and the distributions of intensity are the same as those obtained in the previous case. The major difference appears in the distribution of the correlation coefficients in the angular space (θ,ϕ) shown in Figure 3b. This is explained by the fact that, contrary to the single-source case, the accuracy of the goniopolarimetry method depends not only on the relative positions of the source and the antennas, but also on the shape of the source.

4.4. Impact of the Error on the C Curve Position Determination

[40] Although the C curve along which the goniopolarimetry equation is integrated appears easy to obtain (provided that accurate auroral imaging and magnetic field model exist), the precision with which it is determined is not infinite. Thus we investigate the dependence of the results on the accuracy of the C curve position determination. We use the situation described in the first example, i.e., an emission along a straight line in the sky plane. We generate the PIJ autocorrelations and cross correlations using a given C0 curve and we solve the goniopolarimetry equation (17) using another C1 curve. This latter curve is shifted with respect to the C0 curve by an angle Δ along the direction of the C0 curve and by a angle Δ perpendicular to it.

[41] For each combination of shifts (Δ, Δ) we apply our method for different positions of the curve in the sky plane. We retain the number of results, relative to the total number of results, for which the correlation coefficient between the chosen distribution of intensity and the one obtain by inversion is larger than 0.65 (noted hereafter N0.65). Figure 4 shows the value of N0.65 for different values of the shifts. Note that for random results (no fits at all) N0.65 tends toward 0.175 as the coefficient of correlations are uniformly spread between 1 and −1.

Figure 4.

Relative number of fits with a correlation coefficient larger than 0.65 (N0.65) for different errors on the C curve localization: Δ stands for errors perpendicular to the C curve (in degrees), and Δ stands for an error along the C curve. Statistically the fits correspond to random results if N0.65 ≃ 0.175.

[42] From the results shown in Figure 4 we can see that the goniopolarimetric analysis proposed in the present paper can be applied within an error of ∼1° on the C curve position determination. Note that the N0.65 coefficient decreases rapidly in the Δ direction, reaching values lower than 0.175. This is due to the fact that a shift along the direction of the C curve results in a phase difference (and a deformation) of the distribution of intensity obtained from goniopolarimetry relative to the one we assumed. The N0.65 coefficient reaches its lowest value for a shift Δ ≃ 5° which corresponds to a complete out of phase of the distribution of intensity along the 10° long C curve.

5. Application to Solar Type III

5.1. Outline

[43] Solar Type III bursts are emitted in the solar wind by beams of non thermal electrons whose velocities are a fraction of the light speed. The waves are emitted near the local plasma frequency AND near its harmonic frequency [Alvarez et al., 1974; Dulk et al., 1984]. Due to the beam extent we can simultaneously observe a fundamental and an harmonic emission produced by two separated sources at the same frequency. At high frequency (i.e., at the high density, close to the Sun) the duration of these bursts is very short and it is almost impossible to separate the fundamental from the harmonic emissions. A goniopolarimetric treatment which can spatially separate the two sources may allow the separation of the fundamental and harmonic Stokes parameters, in particular the respective intensities of each source.

[44] Goniopolarimetric methods using spinning spacecraft have been used to infer the characteristic size of solar wind sources. The mean half-width scale has been found to be ∼5–8° for sources emitting near 500 kHz [Reiner et al., 1998; Steinberg et al., 1985] and up to ∼50° for sources emitting near 100 kHz [Steinberg et al., 1985]. These sizes have been interpreted as the result of scattering in the interplanetary medium [Steinberg et al., 1985; Cairns, 1998]. These methods assume an isotropic and gaussian distribution of intensity, which may not be representative of the actual distribution of intensity in the sources. In particular it implies that if two sources emit at the same frequency the goniopolarimetric methods will see them as a single source with a large extent. Moreover, the size of the source is determined on one rotation of the spacecraft which can have a duration of several seconds. The variation of the intensity intrinsic to the source on this timescale may alter the estimation of the size of the source. In which case the size of the solar wind radio source is not accurately known. However, the results of these previous studies show that the source apparent size is not negligible compared to the separation of the fundamental and harmonic sources, probably even larger. Thus it may just be impossible to separate the fundamental and the harmonic sources when observing from 1 AU, whatever the method used. Of course in this case our method will not work. However, we suppose here, for the purpose of our example, that the fundamental and the harmonic sources have a width smaller than their separation.

[45] Type III bursts at low frequency are generally not (or have a very low) polarization; however, some Type III have been observed with weak circular polarizations (less than 10% at 1 MHz which decreases roughly as the logarithm of the frequency [Reiner et al., 2007]). We assume here that we simultaneously observe two sources emitting at the same frequency circularly polarized waves with no linear component (L = 0).

[46] In this case we must deal with eight unknowns: S1, V1, θ1, ϕ1 for the first source and S2, V2, θ2, ϕ2 for the second one. It could be solved without additional assumptions, as there are less unknowns than there are observables. Nevertheless, this “direct” method is unable to separate close sources (with a separation of few degrees) with signal to noise ratio of 26 dB. This is consistent with the fact that the typical resolution of the single-source goniopolarimetry method is 1–2 degrees for this signal to noise ratio [Cecconi and Zarka, 2005], forbidding the resolution of sources separated by less than few degrees. Moreover, each source may have an angular width which is not negligible compared to their separation [Steinberg et al., 1985; Cecconi et al., 2008]. Thus we apply our method to solve this case.

5.2. Method

[47] Unlike the auroral emission case the Type III bursts do not occur along any predefined oval. But for each event, characterized by the emission of an energetic electron beam from the Sun, sources are spread along the beam path. As the electron beam can be emitted in all directions, the position of the sources (the C curve in this study) cannot be defined a priori. At this point we will use the information that can be deduced from a standard single-source goniopolarimetric analysis of the emission. The single-source analysis gives one location of the sources for each frequency and time. It is then possible to deduce from a broadband single-source analysis the trajectory of the electron beam in the sky plane [Cecconi et al., 2008]. Moreover, we can associate a frequency with each point of this trajectory. This determination is affected by the error on the source location (a few degrees), and the frequency associated to each point is the barycenter of the fundamental and harmonic source. To take into account that we get only the location of the barycenter of the sources and the error on location, we use a curve C, centered on the barycenter, along the beam trajectory which has a length Δs larger than twice the angular separation between the considered frequency and its harmonic (Figure 5).

Figure 5.

Sketch showing the determination of the C curve in the solar wind case. The crosses stand for the position of the sources obtained from the single-sources method at different frequencies. From them the trajectory of the electron beam emitting the wave can be determined (dashed line). The position of the cross corresponding to the frequency f0 is determined by the position of the sources emitting at f0 the fundamental and harmonic of the plasma frequency (in the extreme cases this position corresponds to one or the other source). Thus the two sources are located in a range going from the position of the source emitting at f1 = f0/2 to the source emitting at f2 = 2f0. This range is located along the beam trajectory.

[48] Along the curve C we expect the presence of two sources in a range covering roughly half of the curve, and no emissions outside this range. It leads to slight modification of our scheme. The first orders (k = 0 and k = 1) are merged in a gate function γ(s) which defines the range in which there are emissions. Moreover, as each source may have a different circular polarization degree we have to split the goniopolarimetry equations into a system of equation giving the intensity distribution (autocorrelations and real part of the cross correlations) and a system of equations giving the circular polarization distribution (imaginary part of the cross correlations):

equation image
equation image
equation image

where s0 is the center of the range in which there are emissions and Δs is the length of the curve C. The intensity distribution S(s) appearing in the system of equations (19) is the one deduced from the solution of the system of equations (18). In this case the meaning of the different terms is clear: s0 defines the range in which the sources are located, S0 is the mean value of the intensity. S3l defines the position of the two sources in the range, S2l defines the relative intensities of the sources. V0 is the mean value of the circular polarization degree and V2l defines the difference of polarization of the sources. This set of equations is solved using the same method as in the previous case.

5.3. Example

[49] We take the example of two sources separated by 1.2°, with corresponds roughly to the expected separation of the fundamental and the harmonic sources for a fundamental emission at 10 Solar radii (providing that the sources are at the limb). The curve C is supposed to be a straight line, with a length Δs = 4° and an inclination in the angular space (θ, ϕ) of 45°. We choose an intensity distribution consisting in two gaussian peaks, with a intensity ratio of roughly 0.5 (Figure 6a, curve 0).

Figure 6.

Simulation of simultaneous fundamental and harmonic emissions in the solar wind: (a) Averaged distribution of intensity for correlation coefficients comprised between 1–0.9 (curve 1), 0.9–0.85 (curve 2), 0.85–0.8 (curve 3), 0.8–0.75 (curve 4), and 0.75–0.70 (curve 5). Curve 0 stands for the simulated distribution of intensity. (b) Histogram of the correlation coefficients. Most of the obtained distributions of intensity fits the simulated one with a correlation higher than 0.7. (c) Correlation coefficient between the chosen intensity distribution and the one obtained by goniopolarimetry versus colatitude and azimuth of the source. Stars stand for the antennas directions, and crosses stand for the direction opposite to the antennas.

[50] As we did for the auroral emissions, we solve the system of equations for different locations of the source in the sky plane (θ, ϕ). The results are summarized in Figure 6c; the contours indicate a correlation between the solution and the given intensity distribution of 0.85 and 0.75. Figure 6b shows the corresponding histogram of the correlation coefficients, and Figure 6a shows the averaged intensity profiles corresponding to several ranges of correlation coefficient values. Note that a correlation coefficient higher than 0.80 gives results sufficiently accurate to be used in a study of the Type III emissions, as the relative intensities of the sources are well described. This minimal value of the correlation for which the solution can be considered accurate is larger than in the auroral case. This may be due to the fact that we use only six observables to determine it, three of them being autocorrelations affected by a random noise.

[51] Our simulation shows that our method permits the separation of two unpolarized sources separated by only 1°; however, in the present example the source size was about 0.5° which is not negligible in regard of the source separation, but still smaller. The solar wind sources, emitting near 10 solar radii, have been estimated to have an apparent size of 8° when observed from the terrestrial orbit [Steinberg et al., 1985]. If this value is confirmed, this may be an issue for the STEREO observations. However, Solar Probe which should approach 10 solar radii and have goniopolarimetric capabilities should be able to separate the fundamental and harmonic emissions, even far from its perihelion.

6. Discussion

[52] The method presented here has the advantage, compared to the previously proposed methods, of being applicable for extended or multiple sources. In particular by applying this method one can get the correct one-dimensional distribution of intensity within an extended source region or between the multiple sources. However, these results can only be obtained by providing a good description of the source region (the C curve), i.e., with an error ∼ 1° in the case we simulate. The determination of the C curve is thus the major issue of the present method, which must be performed carefully in order to obtain reliable results.

[53] Another issue of the present method is to determine the degree of confidence of the results obtained. In the single-source methods it has been found quasi-empirically that the results can be trusted if the signal-to-noise ratio is higher than ∼23 dB and if the source is at more than 20° from the nearest antenna plane [Cecconi et al., 2008]. However, as we saw in the examples, the dependence of the accuracy of the results on the distance to the direction of the antenna varies with the emitting region shape. The best way to deal with this issue is, after having obtained observational results, to perform a simulation with the same C curve used to invert observations and the distribution of intensity obtained. If the result of the simulation is correct (correlation coefficient larger than 0.65 or 0.80 depending on the type of inversion needed) then goniopolarimetric inversion can be trusted. Another issue is the determination of the linear polarization degree. This point has to be addressed carefully before applying the method we propose to any observations.

[54] As a conclusion we note the new perspectives allowed by the present method. (1) Its application to several, realistic examples has shown that it is sufficiently robust to be applied to the spacecraft observations. (2) We showed that it permits the simultaneous observation of extended planetary auroral emissions and can provide a way to separate the fundamental and harmonic sources in the solar wind, assuming in the latter case that the apparent size of each source is smaller than the source separation. Many other applications may appear depending on the observed objects.

Acknowledgments

[55] The author thanks the reviewers for their fruitful comments. This work has been granted by the NASA STEREO/Waves program.