Simultaneous observations of AKR emission using the four-spacecraft Cluster array were used to make the first direct measurements of the angular beaming patterns of individual bursts. By comparing the spacecraft locations and AKR burst locations, the angular beaming pattern was found to be narrowly confined to a plane containing the magnetic field vector at the source and tangent to a circle of constant latitude. Most rays paths are confined within 15° of this tangent plane, consistent with numerical simulations of AKR k-vector orientation at maximum growth rate. The emission is also strongly directed upward in the tangent plane, which we interpret as refraction of the rays as they leave the auroral cavity. The narrow beaming pattern implies that an observer located above the polar cap can detect AKR emission only from a small fraction of the auroral oval at a given location. This has important consequences for interpreting AKR visibility at a given location. It also helps re-interpret previously published Cluster VLBI studies of AKR source locations, which are now seen to be only a subset of all possible source locations. The observations are inconsistent with either filled or hollow cone beaming models.
 Determining the angular beaming characteristics of terrestrial auroral kilometric radiation (AKR) has been an important focus of AKR investigations since its discovery more than forty years ago. Gurnett  made the first analysis of AKR beaming using one year of Imp-6 and Imp-8 satellite observations of AKR emission. He found that the statistical power pattern formed a ‘distinct cone-shaped boundary’ at a large angle to the magnetic field direction. Subsequent statistical analyses of AKR radiation patterns were fitted to either frequency-dependent filled cone [Green et al., 1977; Green and Gallagher, 1985] or hollow cone [Calvert, 1981b, 1987] beaming models, the latter suggested by studies of Jovian decametric emission [e.g., Dulk, 1970], whose properties are consistent with a very thin (∼1°) hollow cone. More recently, Kasaba et al.  analyzed 38 months of AKR bursts recorded on the Geotail spacecraft. They could not distinguish between hollow and filled cone models, but found that the AKR illumination pattern was systematically modified both by geomagnetic activity and by season.
 These studies provide statistical descriptions of the overall sky pattern illuminated by AKR bursts, but they do not address the angular beaming pattern of individual AKR bursts. AKR emission is thought to consist of a large number of 'elementary radiation sources' with narrow frequency-time structures [Gurnett and Anderson, 1981; Pottelette et al., 2001; Mutel et al., 2006]. Hence, instruments with inadequate frequency and/or time resolution (e.g. swept-frequency receivers) receive the sum of contributions from many spatially separated elementary sources whose frequency-time structures lie within the time-frequency resolution window of the instrument. Also, since most beaming studies have utilized observations from single satellites, the observed power pattern on the sky is a statistical ensemble of beams from many individual radiating sources rather than the angular beaming pattern of an isolated source.
 The wideband (WBD) instrument on the four Cluster spacecraft provides a unique opportunity to determine the angular power pattern of individual AKR bursts for the first time. The WBD system records the received waveform directly, so that it has adequate time and frequency resolution to isolate emission from individual elementary AKR sources. In addition, since the Cluster constellation forms a 2-dimensional array on the sky, it can simultaneously sample the burst over a range of solid angles. Finally, by measuring the differential delays between pairs of spacecraft, the locations of individual AKR bursts can also be determined [Mutel et al., 2003]. This provides a spatial filter which isolates radiation from a single region and allow the array to sample individual burst power patterns.
2. AKR Beaming: Model Predictions
 The electron cyclotron maser instability (CMI) is widely believed to be the mechanism responsible for auroral kilometric radio emission [e.g., Treumann, 2006]. AKR sources are found in thin, low density cavities in the upward current region above the auroral zones [Calvert, 1981a; Ergun et al., 1998]. The cavities are generally oriented tangent to the auroral oval and aligned with the magnetic field. They have small latitudinal widths (10 km –100 km) compared with their longitudinal extent, which can extend several thousand km.
 The radiation is generated most efficiently in the extraordinary (X) mode close to the electron gyro-frequency. The growth rate is strongly peaked for propagation nearly perpendicular to the magnetic field, as seen in both in situ observations [Roux et al., 1993; Ergun et al., 1998] and in model calculations [Pritchett et al., 2002; Mutel et al., 2007]. Both Louarn and Le Quéau  and Pritchett et al. , using 2-dimensional simulation of AKR propagation in a thin cavity, found that the emission is highly beamed in the 'along track' or longitudinal direction. This prediction is strengthened by FAST in situ observations in the upward current region which found that the tangential power is 100 times stronger than the perpendicular power [Pritchett et al., 2002].
3. Geometry of Beamed AKR Emission
 We used the wideband plasma wave instrument (WBD) [Gurnett et al., 2001] on the four Cluster spacecraft in VLBI mode to determine the locations of individual AKR bursts for more than 12,000 individual AKR bursts over 39 epochs. The VLBI technique solves for AKR burst locations using measurements of differential delays between all six pairs of spacecraft [Mutel et al., 2003].
 Once a source position is determined, we calculate the tangent plane coordinates of each spacecraft as seen from the source location. This is done by constructing an orthonormal 3-dimensional coordinate system with origin at the source, aligned outward along the magnetic field direction, in the meridian plane pointed toward the local magnetic pole, and = × . The tangent plane latitude is the angle between the spacecraft vector and the tangent (xy) plane, while the longitude is the angle between the spacecraft vector projected onto the tangent plane and . The coordinate geometry is illustrated in Figure 1a.
3.1. Confinement to Tangent Plane
 Once a source location is determined, we calculate the tangent plane coordinates of the Cluster spacecraft at the time of reception of the individual burst. Histograms of the spacecraft tangent plane latitudes and longitudes for 12,000 individual AKR bursts are shown in Figure 2. The latitude distribution is strongly peaked near 0°, with more than 98% of solutions within 20° of the origin, and appears to be frequency-independent. The longitude distribution has been folded about the origin, since the tangent plane model is symmetric to reflection about the axis. The histogram for all three frequencies is peaked near the origin, indicating that the ray paths are strongly refracted.
 In this paper we estimate the expected refraction outside the auroral cavity with a simple ray tracing calculation using the cold plasma dispersion equation. Rays at 125, 250, and 500 kHz are launched perpendicular to the magnetic field at magnetic latitude 70° from heights corresponding to the local electron gyro-frequency. We assume a dipole magnetic field, and an average electron density profile
where re is the radius of the Earth. Although the electron density varies significantly with both magnetic latitude and geomagnetic activity, this profile represents an approximate long-term average [cf. Laakso et al., 2002, Figure 3a]. The resulting refraction angles versus coordinate position, along with the calculated ray paths, are shown in Figure 1b. The ray tracing solution shows that the expected refraction is very strong, with average refraction angle between 70° and 80°, consistent with observed ray paths. More realistic magnetospheric refraction models, including possible reflection from the plasmasphere, result in similar total refraction angles [e.g., Xiao et al., 2007].
3.3. Visibility Maps
 AKR emission that is beamed in a tangent plane and refracted upward will illuminate two symmetrically displaced azimuthal slices on a great circle belt that is the extension of the tangent plane on the sky. The belt will have a characteristic width given by the effective opening angle of the tangent plane and a longitudinal extent given by the (upward) refracted ray paths determined by the ambient magnetospheric plasma outside the cavity. For a given spacecraft location, only a fraction of all source locations on the polar cap are able to illuminate the spacecraft. Hence, for a given tangent plane opening angle and refraction angle range, we can calculate the locus of all hypothetical AKR source locations on the polar cap which can illuminate the spacecraft. We denote this locus of points as the visibility map for that spacecraft location.
Figure 3 (top) shows VLBI maps of AKR source positions for three epochs mapped onto a CGM coordinate grid. Joint visibility maps of the Cluster array (i.e., the visibility area common to all four spacecraft) are shown in yellow. We computed the visibility maps assuming a tangent plane beaming pattern with a −15° to 15° latitudinal range and 0° to 45° longitude range. Note that for each epoch the AKR burst locations lie almost entirely within the visibility map. Figure 3 (bottom) shows the tangent plane coordinates of each spacecraft as viewed from each AKR source location. This figure provides clear confirmation that the spacecraft are located close to the tangent plane equator of each source as expected from the tangent plane beaming model. We can also see that the longitudinal positions are close to the source zenith, i.e., the ray paths suffer substantial refraction.
3.4. Comparison With Hollow Cone Models
 Could the observations discussed in this paper also be compatible with hollow cone beaming models? In Figure 4 we compare visibility maps calculated for a tangent plane beaming model (yellow) with hollow cone models having 20° opening angle (red) and 40° opening angle (green). The tangent plane model has a width of 30° while the hollow cone models have widths of 20°. The visibility maps were computed for Cluster spacecraft positions on Nov 10, 2005 at 11:30 UT. The black x's (125 kHz) and squares (250 kHz) show AKR source locations determined by VLBI. It is clear that hollow cone models cannot fit the observed source locations, while the tangent plane model provides an excellent fit.
4. Implications of the Tangent Plane Beaming Model
 The observations reported in this paper demonstrate that individual AKR bursts do not radiate in either filled or hollow cones, as previously suggested, but rather are confined to a narrow plane tangent to the source's magnetic latitude circle and containing the local magnetic field vector. The rays are also directed upward, consistent with expected refraction as rays leave the auroral cavity. This geometry confirms the numerical models of Louarn and Le Quéau  and Pritchett et al.  predicting longitudinal propagation. It also implies that AKR observations from remote locations sample only a small part of the auroral oval from any given location. For example, the maps of AKR emission published by Mutel et al. [2003, 2004] likely represent only a small fraction of the total extent of AKR emission on the auroral oval.
 The University of Iowa acknowledges the support of NASA Goddard Space Flight Center under grant NNX07AI24G. We are grateful to Roman Schreiber and Jan Hanasz for several fruitful discussions.