V. Krupař 1,2,3, O. Santolík 2,3, B. Cecconi 1, and M. Maksimović 1. Introduction

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1 WDS'10 Proceedings of Contributed Papers, Part II, 11 17, 010. ISBN MATFYZPRESS Estimation of the Apparent Source Size of Solar Radio Emissions Using SVD: An Application to Type III Radio Bursts at Long Wavelengths Observed by STEREO and Wind V. Krupař 1,,3, O. Santolík,3, B. Cecconi 1, and M. Maksimović 1 1 LESIA Observatoire de Paris/CNRS, Meudon, France. Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. 3 IAP/CAS, Prague, Czech Republic. Abstract. Type III radio bursts are intense solar radio emissions. They are frequently observed by the S/Waves instruments on-board the STEREO spacecraft. We describe a wave propagation analysis using the Singular Value Decomposition (SVD) which can be directly applied to spectral matrices measured by the High Frequency Receiver (HFR; a part of the S/Waves experiment). We have found an empirical relation between the decomposed spectral matrices and apparent source sizes for waves with a low degree of polarization. We present a joint observation of a type III radio burst by the STEREO and Wind spacecraft during small separation distances. We obtain consistent results for the apparent source size and k-vector direction using different analysis method for the measurements of the STEREO and Wind spacecraft. We demonstrate that SVD can be an effective tool for the wave analysis of radio emissions with very extended sources. Introduction Type III radio bursts belong among the most intense electromagnetic emissions in the heliosphere [Wild, 1950]. They are though to be generated by a beam of fast electrons ( 0.1c) accelerated near the Sun s surface streaming outward on the open magnetic field lines and connected with solar flares and/or coronal mass ejections driven shocks. An electron beam produces locally the bump-on-tail instability of the electron distribution function. This abrupt change of plasma parameters can lead to generation of electrostatic Langmuir waves (at the local plasma frequency f p, and/or the first harmonic f p ), which are afterwards converted by a nonlinear process into electromagnetic waves: type III radio bursts [Ginzburg and Zhelezniakov, 1958]. As an electron beam propagates outward from the Sun, emissions are generated at lower frequencies corresponding to a depression of f p which is proportional to the local plasma density: f p (khz) 9 n(cm 3 ). In comparison with other electromagnetic emissions, they have extended sources [Steinberg et al., 1984], which can be explained by either properties of an intrinsic beaming pattern and/or scattering by density fluctuations of the interplanetary medium. Hence an estimation of their apparent source size could yield important informations about density fluctuations in the solar wind considering scattering of a primary beam pattern. Although coronal type III radio bursts (f 00 MHz) can have up to 35% of the right-handed circular polarization, at long wavelengths (f 1 MHz) the degree of polarization is very low [Dulk, 000]. For comprehensive electromagnetic waves analysis we need instruments with multi-component measurements of auto-correlations and cross-correlations of the voltages induced by the wave electric and/or magnetic field. These instruments provide us with goniopolarimetric (GP) data, which can be used as an input for the direction finding and polarization analysis of an incoming wave [Lecacheux, 1978]. With such data we are able to compose a spectral matrix S ij containing the information about the incident wave: 11

2 S ij = E i Ej, (1) where E is the electric field vector of complex amplitudes, indices i and j represent three Cartesian components, means time averaging on an interval much longer than the observed wave period and corresponds to a complex conjugate. SVD is an efficient tool for wave analysis [Santolík et al., 003]. It can be directly performed on a 6 3 matrix A which contains separated real and imaginary parts of a 3 3 complex spectral matrix S ij. Applying SVD on this special real form of the complex spectral matrix we obtain real matrices U, W and V T : A = U W V T. () The SVD inversion retrieves information about the wave vector direction, ellipticity and directions of axes of the polarization ellipsoid and estimators of the planarity of polarization. Polarization properties and estimation of an angular source size are defined by the matrix W, which is a diagonal matrix 3 3 of three positive or null singular values that represent the axis lengths of the polarization ellipsoid. The 3 3 matrix V T with orthonormal rows contains directions of the axes of the polarization ellipsoid. This method was originally designed for 3 3 magnetic spectral matrices by assuming that B k = 0 [Santolík et al., 003], where k is the wave vector. It can also be used for electric spectral matrices if the condition E k = 0 is fulfilled. This is the case of radio waves when observed far from their propagation mode cutoff. The influence of extended sources (typical for type III radio bursts at long wavelengths) on the measured spectral matrix has been efficiently described by Cecconi [007] in equations (3 7): P ij = Z [ ( ( 0 Gh i h j S 0 Γ (1 + Q) A i A j + C ic j Γ 1 Γ )) ( ) ( ) Γ Γ +(U iv ) A i B j + (U + iv ) A j B i ( ( 1 +(1 Q) A i A j Γ 1 Γ + Γ ) ( 3 + Γ B i B j Γ 1 + Γ ) 3 + Γ ( Γ +C i C j Γ ))] 3 + Γ 1 4 (3) A k = sin θ k cos θ cos(φ φ k ) + cos θ k sin θ (4) B k = sinθ k sin(φ φ k ) (5) Γ a k(γ) = C k = sin θ k sin θ cos(φ φ k ) + cos θ k cos θ (6) 1 γ sin(kθ 1 cos γ M)dθ M = 1 cos(kγ) 0 k(1 cos γ). (7) The P ij matrix on the left hand side of equation (3) represents a modeled spectral matrix that we will compare with the matrix measured by the instrument. P ij is generally a nonorthogonal matrix unlike S ij defined in equation (1). The right hand side of equation (3) contains the impedance of free space (Z 0 ); parameters of used antennas: effective lengths (h k ), directions (θ k and φ k ) and gain (G); and incident wave properties: the k-vector directions (θ and φ), the Stokes parameters (the energy flux: S 0 ; the linear polarization degrees: Q and U; and the circular polarization degree: V )[Kraus, 1966] and the angular half aperture of the source (γ), as seen by a spacecraft contained in the Γ k coefficients. The shape of the source, that reflects a radial cut of a source brightness distribution, is considered to be uniform (equation 7). 1

3 Figure 1. The ratio of the smallest and largest components of the diagonal matrix W (w 1 /w 3 ) as a function of the apparent source size (γ). The 0.1% of the circular polarization and the uniform brightness of the source have been considered. Determination of the source direction and apparent size As mentioned above, SVD is an efficient tool for a wave analysis of multi-component measurements of the magnetic field with a point source [Santolík et al., 003], which can be also applied to the electric field of radio emissions by assuming E k = 0. Contrary of three-axial search-coil magnetometers, effective antenna directions and lengths are different from the physical ones due to their coupling with the spacecraft body. Hence we perform a transformation from the antenna frame into the orthonormal spacecraft system before we apply SVD (for the transformation of S/Waves antenna frame see Bale et al. [008]). We have used the effective antenna lengths and directions obtained by the in-flight calibration when STEREO (Solar TErrestrial RElation Observatory) observed the Auroral Kilometric Radiation (AKR) performed by M. Panchenko [personal communication]. SVD provides us with a unity vector κ = k/ k contained in a column of the V T matrix that corresponds to the minimum of the W matrix of singular values. In other words, SVD yields the direction where E has the minimum of its variance. Information about the apparent source size is hidden in the W matrix containing the axis lengths of the polarization ellipsoid. In an ideal case of an unpolarized radio wave with a point source, the minimum value of W (w 1 ) should be zero. Because type III radio bursts have very extended sources, the ratio w 1 /w 3, where w 3 is the maximum of W, yields informations about the apparent source size. Figure 1 shows us w 1 /w 3 as a function of the apparent source size. We have used equation (33) of Cecconi [007] for modeling spectral matrices that well reflect properties of type III radio bursts at long wavelengths. We have assumed a low degree of right-handed circular polarization. The shape of the source is considered to be uniform. For a point source (γ = 0 ) the polarization ellipsoid degenerates to an ellipse contained in a single plane and the ratio of its smallest and largest axis of the ellipsoid (w 1 /w 3 ) is zero. On the other hand the polarization ellipsoid of low polarized emissions propagating from a half-plane (γ = 90 ) becomes a sphere: w 1 /w 3 = 1. Analysis of a single type III radio burst observed simultaneously by STEREO and Wind The STEREO mission consists of two nearly identical spacecraft dedicated to stereoscopic investigation of solar processes with varying separation angles with respect to the Sun in the ecliptic plane. Whereas STEREO-A moves ahead the Earth in its orbit, STEREO-B trails be- 13

4 Figure. From :53 to 3:09 UT on the 8 May 007: the electric field spectral density, the apparent source size γ, the polar angle θ and the azimuthal angle φ for STEREO-A (65 khz), Wind (64 khz) and STEREO-B (65 khz). hind. Both spacecraft are three axis stabilized and embarking the S/Waves instrument [Bougeret et al., 008]. The most suitable receiver (a part of S/Waves) for investigation of the type III radio bursts is HFR that provides us with GP data [Cecconi et al., 008] in a frequency range from 15 khz up to 1975 khz in 38 separated bands with a 5 khz effective bandwidth. HFR is a sweeping receiver with two channels processing both auto- and cross-correlation products which are necessary for the GP analysis. Its time resolution is about 30 seconds per sweep between all frequencies and antenna configurations. Three mutually orthogonal monopole antenna elements (each 6 meters in length) form the sensor part of the S/Waves instrument [Bale et al., 008]. The effective antenna lengths and directions are different from the physical ones and can be modeled by computer simulations, estimated by rheometric measurements (e.g. Oswald et al. [009]) or by an in-flight calibration. We use the in-flight calibration using observations of AKR by STEREO-B. The effective antenna lengths and directions for STEREO-A are assumed to be the same. We have applied our methods to the HFR data set. As a reference spacecraft we have used the Wind spacecraft located roughly between the two STEREO during the observed event. The Waves instrument on-board Wind provides us with GP data in a frequency range from 0 up to 1040 khz with a 3 khz effective bandwidth [Bougeret et al., 1995]. For the GP analysis Waves uses methods dedicated for spinning spacecraft [Manning and Fainberg, 1980]. Wind provides us with highly accurate GP measurements using the spacecraft spin demodulation. All spacecraft observed an intense type III radio burst from :53 to 3:09 UT on 8 May 007. The distances between STEREO-A and STEREO-B, STEREO-A and Wind, STEREO-B 14

5 and Wind was 0.15, 0.09, 0.07 AU, respectively. Figure shows the event recorded at single frequency channels 65 and 64 khz from both STEREO and Wind, respectively. The plotted error bars for STEREO correspond to an uncertainty on the on receiver gain of 0.5 db. We have considered the uncertainty for Wind to be ± for the k-vector directions and ±4 for the apparent source size [Hoang, private communication]. The top panels contain the intensity in mv /m /Hz for STEREO and in Solar Flux Units (1 sfu=10 W /m /Hz at the Earth orbit) for Wind. The S/Waves calibration to sfu has not been performed yet. Albeit the Gaussian source shape better reflects real observations, Wind provides us with γ considering a uniform source shape. The second row contain the apparent source γ with the assumption of a uniform source brightness distribution (see equation 3). We have achieved a good agreement between STEREO and Wind: the source size γ is estimated to 5 on all three spacecraft during the maximum of the energy flux. For direction finding analysis we have used special spacecraft centered coordinates: the X axis is pointing to the Sun, the Z axis is contained in the plane that is perpendicular to the ecliptic, and points toward the North of the ecliptic hemisphere, the Y axis completes the right handed orthonormal frame. Last two rows contain the polar angle θ and the azimuth angle φ. Dashed lines indicate the direction to the Sun. We can observe a better correspondence in the estimation of the k-vector direction between Wind and STEREO- B, within their inaccuracy. This can be caused by the higher noise level on STEREO-A and/or a variance between effective antenna lengths and directions between two spacecraft (we apply the in-flight calibration obtained for STEREO-B to STEREO-A). The observed type III radio burst propagates roughly in the ecliptic plane. Discussion and conclusion The two STEREO spacecraft give us great opportunity for stereoscopic observations of solar radio emissions at long wavelengths. The Wind spacecraft, located roughly between them, is ideal for complementary measurements. The S/Waves instrument on-board STEREO provides us with GP data that allow to retrieve the k-vector direction and polarization properties. An analysis of observations of three type III radio bursts by both STEREO and Wind, using different methods than in this paper, has been done by Reiner et al. [009]. Preliminary results of an application of SVD to measurements of emissions with very extended sources has been recently published by Krupar et al. [010]. The main interest of this paper is a method for investigating type III radio bursts at long wavelengths, which have very extended sources and a low degree of polarization. We have obtained an empirical relation between the decomposed spectral matrices, that contain lengths of the polarization ellipsoid, and the apparent source sizes for low polarized emissions with the uniform source shape (Figures 1). This relation is almost linear for γ ranging between 40 and 90. For a point source (γ = 0 ) the polarization ellipsoid degenerates to an ellipse (w 1 /w 3 = 0), whereas for a very extended source (γ = 90 ) it becomes a sphere (w 1 /w 3 = 1). As an experimental confirmation of our method we present an observation of a single type III radio burst made by STEREO and Wind. Figure shows results from one frequency channel: 65 khz and 64 khz from both STEREO and Wind, respectively. Errors induced by the uncertainty of the HFR receiver gain has been considered. At a given frequency we can distinguish an intense peak at :58 (see the first row of Figure ). Albeit the wave power from Wind is calibrated into sfu, we have an output from STEREO in the spectral density of the electric field fluctuations only. The apparent source size is shown in the second row. During the maximum of intensity the source size calculated by SVD on STEREO is the same (γ = 5 ) as on Wind which is using a different method dedicated for spinning spacecraft. It confirms the validity of the obtained empirical relation for estimating the apparent source size. Other performed case studies of joint STEREO and Wind observations with small separation distances give us similar results. It indicates that the obtained empirical relation can be generally used for other type III radio bursts observations. This method can be either used for statistical studies 15

6 where we need robust and fast data processing or for estimation of initial values for a standard non-linear χ method (Levenberg-Marquardt, a gradient-expansion). The k-vector directions in the second and third row indicate that STEREO-B attain better agreement with Wind than STEREO-A as it has been expected due to higher noise levels and/or a variance of effective antenna lengths and directions between both STEREO. This leads us to the following conclusions: 1. We have found the empirical relation between decomposed spectral matrices and apparent source sizes for low polarized radio emissions with a special intention to type III radio bursts observed by STEREO;. This relation has almost a linear shape for the apparent source sizes between 40 and 90 ; 3. Its validity has been confirmed by joint observations made by the STEREO and Wind spacecraft during small separation distances. Acknowledgments. The authors thank Pierre-Luc Astier and Quynh Nhu Nguyen (from LESIA) for providing the S/Waves data and useful engineering insights on the receiver functions. We also thank M. Panchenko for the antenna parameters. The present work was supported by a scholarship from the French government, a CNRS PICS program and grants KONTAKT ME9107 and GAAV A The S/Waves experiment was built by LESIA with support from both CNES and CNRS. References Bale, S. D., Ullrich, R., Goetz, K., Alster, N., Cecconi, B., Dekkali, M., Lingner, N. R., Macher, W., Manning, R. E., McCauley, J., Monson, S. J., Oswald, T. H., and Pulupa, M., The Electric Antennas for the STEREO/WAVES Experiment, Space Science Reviews, 136, , 008. Bougeret, J.-L., Kaiser, M. L., Kellogg, P. J., Manning, R., Goetz, K., Monson, S. J., Monge, N., Friel, L., Meetre, C. A., Perche, C., Sitruk, L., and Hoang, S., Waves: The Radio and Plasma Wave Investigation on the Wind Spacecraft, Space Science Reviews, 71, 31 63, Bougeret, J. L., Goetz, K., Kaiser, M. L., Bale, S. D., Kellogg, P. J., Maksimovic, M., Monge, N., Monson, S. J., Astier, P. L., Davy, S., Dekkali, M., Hinze, J. J., Manning, R. E., Aguilar-Rodriguez, E., Bonnin, X., Briand, C., Cairns, I. H., Cattell, C. A., Cecconi, B., Eastwood, J., Ergun, R. E., Fainberg, J., Hoang, S., Huttunen, K. E. J., Krucker, S., Lecacheux, A., MacDowall, R. J., Macher, W., Mangeney, A., Meetre, C. A., Moussas, X., Nguyen, Q. N., Oswald, T. H., Pulupa, M., Reiner, M. J., Robinson, P. A., Rucker, H., Salem, C., Santolik, O., Silvis, J. M., Ullrich, R., Zarka, P., and Zouganelis, I., S/WAVES: The Radio and Plasma Wave Investigation on the STEREO Mission, Space Science Reviews, 136, , 008. Cecconi, B., Influence of an extended source on goniopolarimetry (or direction finding) with Cassini and Solar Terrestrial Relations Observatory radio receivers, Radio Science, 4, 003 +, 007. Cecconi, B., Bonnin, X., Hoang, S., Maksimovic, M., Bale, S. D., Bougeret, J.-L., Goetz, K., Lecacheux, A., Reiner, M. J., Rucker, H. O., and Zarka, P., STEREO/Waves Goniopolarimetry, Space Science Reviews, 136, , 008. Dulk, G. A., Type III Solar Radio Bursts at Long Wavelengths, in Radio Astronomy at Long Wavelengths, edited by R. G. Stone, K. W. Weiler, M. L. Goldstein, & J.-L. Bougeret, pp , 000. Ginzburg, V. L. and Zhelezniakov, V. V., On the Possible Mechanisms of Sporadic Solar Radio Emission (Radiation in an Isotropic Plasma), Soviet Astronomy,, 653 +, Kraus, J. D., Radio astronomy, Krupar, V., Maksimovic, M., Santolik, O., Cecconi, B., Nguyen, Q. N., Hoang, S., and Goetz, K., The apparent source size of type III radio bursts: Preliminary results by the STEREO/WAVES instruments, in American Institute of Physics Conference Series, edited by M. Maksimovic, K. Issautier, N. Meyer- Vernet, M. Moncuquet, & F. Pantellini, vol. 116 of American Institute of Physics Conference Series, pp , 010. Lecacheux, A., Direction Finding of a Radiosource of Unknown Polarization with Short Electric Antennas on a Spacecraft, Astronomy and Astrophysics, 70, 701, Manning, R. and Fainberg, J., A new method of measuring radio source parameters of a partially polarized distributed source from spacecraft observations, Space Science Instrumentation, 5, , Oswald, T. H., Macher, W., Rucker, H. O., Fischer, G., Taubenschuss, U., Bougeret, J. L., Lecacheux, A., Kaiser, M. L., and Goetz, K., Various methods of calibration of the STEREO/WAVES antennas, Advances in Space Research, 43, ,

7 Reiner, M. J., Goetz, K., Fainberg, J., Kaiser, M. L., Maksimovic, M., Cecconi, B., Hoang, S., Bale, S. D., and Bougeret, J.-L., Multipoint Observations of Solar Type III Radio Bursts from STEREO and Wind, Solar Physics, pp , 009. Santolík, O., Parrot, M., and Lefeuvre, F., Singular value decomposition methods for wave propagation analysis, Radio Science, 38, , 003. Steinberg, J. L., Hoang, S., Lecacheux, A., Aubier, M. G., and Dulk, G. A., Type III radio bursts in the interplanetary medium - The role of propagation, Astronomy and Astrophysics, 140, 39 48, Wild, J. P., Observations of the Spectrum of High-Intensity Solar Radiation at Metre Wavelengths. III. Isolated Bursts, Australian Journal of Scientific Research A Physical Sciences, 3, 541,

Visibility of Type III burst source location as inferred from stereoscopic space observations

Visibility of Type III burst source location as inferred from stereoscopic space observations M. Y. Boudjada, Patrick H. M. Galopeau, M. Maksimovic, H. O. Rucker To cite this version: M. Y. Boudjada, Patrick