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 Meteoroid head echoes are radar returns associated with scatter from the dense plasma surrounding meteoroids striking the Earth's atmosphere. Such echoes are detected by high power, large aperture (HPLA) radars. Frequently such detections show large variations in signal strength that suggest constructive and destructive interference. Using the ARPA Long-Range Tracking and Instrumentation Radar (ALTAIR) we can also observe the polarization of the returns. Usually, scatter from head echoes resembles scatter from a small sphere; when transmitting right circular polarization (RC), the received signal consists entirely of left circular polarization (LC). For some detections, power is also received in the RC channel, which indicates the presence of a more complicated scattering process. Radar returns of a fragmenting meteoroid are simulated using a hard-sphere scattering model numerically evaluated in the resonant region of Mie scatter. The cross- and co-polar scattering cross-sections are computed for pairs of spheres lying within a few wavelengths, simulating the earliest stages of fragmentation upon atmospheric impact. The likelihood of detecting this sort of idealized fragmentation event is small, but this demonstrates the measurements that would result from such an event would display RC power comparable to LC power, matching the anomalous data. The resulting computations show that fragmentation is a consistent interpretation for these head echo radar returns.
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 In its orbit about the Sun, the Earth sweeps up a steady rain of meteoroids. At altitudes near 100 km these particles rapidly decelerate, ablating material and creating a compact volume of plasma around the meteoroid. They also leave a lengthy trail of ionization in their wake. The radar frequencies used for detection are high enough that for most meteors the trail is largely transparent until it becomes turbulent. This work concerns radio wave scatter from the plasma near the meteoroid, known as a head echo. Usually, there is a clear time delay (about 3 ms) between the head echo and when the trail becomes detectable, offering a clear window to measure only the head plasma component. For the few very high SNR meteor trails, this delay, or absence of scattered power, may not be observed because of the high plasma density. Using the ARPA Long-Range Tracking and Instrumentation Radar (ALTAIR), we have recorded that a small fraction of the head echoes return mixed polarization, not consistent with scatter from a sphere. Particularly, the experiments presented here show that the fragmentation of the meteoroid particle during its collision with the Earth's upper atmosphere can cause mixed polarization in the radar return. Ceplecha et al.  give a thorough overview of meteoroid interaction with the atmosphere. We will focus on the small, sporadic particles that do not create a visible light streak. These particles have typical masses in the microgram range.
 The compact, dense plasma near the meteoroid is created by collisions between ablated meteoroid particles and the background neutral gas as described by Zinn et al. . Thus, the head echo can be modeled as scatter from a plasma blob whose density decreases with distance from the meteoroid. The plasma near the meteoroid may be sufficiently dense that the plasma frequency exceeds the radar frequency. Scatter from these overdense plasmas can be studied in a simplified way by considering scatter from a single, conducting sphere with radius equal to the critical plasma layer radius. Many head echoes are in fact not overdense, but for scatterers smaller than a wavelength, the distinction between overdense and underdense scatter is minor. As the meteoroid plasma becomes larger, one would expect the overdense plasma approximation to depart from the scatter from more complex models which include plasma physics. However, we would expect the polarization results to be quite similar with either model. For a more rigorous treatment of head echo scattering, see Close et al. [2004, 2007]. Meteoroid ablation and plasma creation is beyond the scope of this work. This fragmentation model can, however, offer proof-of-concept insight for determining scattering phenomena such as shown by Close et al. .
 This study is motivated by observations made at ALTAIR, a dual-frequency actively calibrated dual-polarization HPLA radar. ALTAIR transmits in the right-circularly (RC) polarized electric field orientation, and receives in both the right- and left-circular (LC) orientation. ALTAIR calibrates its polarization measurements passively through detection of calibration spheres and random dipole drops as well as actively, using a transponder to retransmit the incoming pulse with a well-known attenuation. A typical head echo at ALTAIR will be measured entirely in the LC channel, with an SNR of 20 dB. Polarization features of head echoes from the 1998 Perseid meteor shower data taken at ALTAIR have been reported by Close et al. . The authors note that the Polarization Ratio (PR), the ratio of LC received power to RC received power, of the 692 head echoes has a strong mean at 19 dB. This is consistent with the idea of scatter from a sphere-like object, taking into account known instrumental effects in PR dynamic range at ALTAIR. The data set showed spread in the PR histogram, with some meteor head echoes having small PR or even equal amounts of LC and RC power, a PR of 0 dB, which has not yet been explained. The accuracy of the PR measurement will be a function of the SNR in the LC and the RC channels for each particular head echo. Particular head echo data at ALTAIR show head echoes with low to 0 dB PR, with each LC and RC channel having SNR greater than 15 dB.
 Using a conducting sphere to model the critical plasma layer around the meteoroid itself, the PR in the far field is easily studied as a function of scattering angle and particle radius. Particularly examined is scatter when the critical plasma radius is comparable to a wavelength, such that 0.01 ≤ ka ≤ 4, where a is the spheroid radius and k is the wave number, 2π/λ. With ka values less than 1, Rayleigh scattering is a sufficient model. For larger ka values, the full Mie scattering solution is needed, which we describe in section 2.2. Figure 1 shows the ka value based on the radius of the critical plasma layer at the ALTAIR VHF (158 MHz) and UHF (422 MHz) frequencies. The extent of the plasma around the particle is dependent on ablation mechanics, background atmospheric density, and velocity. For these small meteoroids, we do not expect the critical plasma layer to be very large, but these mixed polarization head echo returns are anomalous, so we cover the case up to a radius of 0.45 m with the understanding that under normal conditions that radius may only be a few cm.
 From Mie Scattering theory, there should be no RC backscatter from RC illumination on a single perfect electric conducting spherical shell, thus the ratio of LC power to RC power, the PR, will be very large. However, radar backscatter from meteor head plasmas is not always consistent with scatter from the sort of spherical structure ablation theory suggests. In addition to the polarization data in the 1998 ALTAIR data set, data collected in 2007 from ALTAIR displays similar spread in PR, and also included PR values much different from those expected.
 There are several potential explanations for the unusual PR displayed by these outliers; some form of symmetry breaking must occur such as a nonspherical head plasma or multiple scattering (e.g. fragmentation). Another approach has been developed by Wannberg et al. , using plasma resonance effects in the region immediately behind the meteoroid to describe the polarization anomalies seen in head echoes at the EISCAT radar.
 Presented is the investigation of low PR radar returns as a result of a fragmentation event. If two scatterers were within the near field of each other, the electric field interaction between them may produce a more complicated return. Fragmentation indisputably occurs in the larger meteoroids and is verified by many optical measurements such as those seen by the Harvard Radio Meteor Project in the early 1960s presented by Southworth and Sekanina . On the other hand the possibility of small meteoroid fragmentation remains an open question. Compelling arguments have been made to describe a sinusoidal beating pattern observed in the returned radar power from meteor head echoes by Kero et al.  using the EISCAT UHF radar system, and by Mathews et al.  using the Arecibo Observatory. The authors point out that this behavior in the SNR would be expected from two point scatterers drifting apart over the course of several radar pulses, causing a sinusoidal variation in power caused by constructive and destructive interference. This points to meteor fragmentation as an explanation of the unusual SNR signature. Currently there is no realistic view of the differential velocity of the meteoroid fragments which would depend on the fracturing mechanism and the masses of the fragments.
 In addition to SNR, ALTAIR is able to observe the polarization signatures of such an event. This research addresses the following question: If a meteoroid fragments, how will it affect the polarization of the radar return? In order to model the event, the head echo scattering mechanism is simplified as scatter from a conducting spherical shell. Using the Numerical Analysis Code (NEC-2) [Burke and Poggio, 1981], a frequency domain analysis is performed with the achievement of a complete numerical solution in excellent agreement with the analytic single particle Mie Scattering solution. Angelakos and Kumagai  published measured results from two metallic spheres of size ka = 7.4 in several geometries. We adapted our model to match their experiment, only altering the scale of our spheres and changing our illumination mode to linear polarization. Our two-body scattering model is validated for the specific instances of end-on and broadside illumination using results presented in that paper. After the validation of the model, we present the results of two identically sized meteoroid fragments for varying ka size of the critical plasma layer and varying geometric orientation of the particles. In particular, the case of ka = 3 and ka = 0.25 are analyzed in detail, noting properties of their backscattered LC and RC power. Overall, significant structure arises in the RC return polarization when the particles are in the near field of each other. By examining the PR as a comparison of the RC and the LC power, it is clear that two-body scattering events can produce a significant amount of RC power. The results show a generalized SNR return which agrees with and offers insight into the sinusoidal fluctuation described in the works previously mentioned.
2. Computational Model
2.1. Model Assumptions and Overview
 The meteoroid model is composed of conducting planar patches tangent to a sphere, resembling a Mercator grid. An incident plane wave in free space is numerically imposed with RC polarization at 158 MHz. The far field numerical scatter is then compared to analytic Mie Scattering theory.
 A second sphere is then placed in the near field of the first scatterer, and again the far field scatter is examined, particularly noting changes in backscatter polarization. The scattered electric field is broken into its LC and RC polarizations in the far field to gain insight into how meteoroid fragmentation might affect the polarization features observed with ALTAIR.
2.2. Spherical Model for Validation
 Each spherical model is made of 192 patches of roughly equal area, such that there are at least 25 patches per λ2 of surface area, with respect to the largest sphere studied. A range of spheres are studied with radius from 3 mm to 0.45 m, and taking into account ALTAIR's VHF and UHF frequencies this corresponds to ka values from 0.01 to 4.
 The model spheres are initially constructed with some symmetry which could conceivably affect the numerical results. To control for this, the spheres are given random 3D rotations, each time recomputing the scatter. Using Mie Scattering theory, we can derive the theoretical PR for all scattering angles for a specific size ka for RC incident Polarization. The computational results of the PR are quite close to these theoretical results, as can be seen for ka = 0.25 in Figure 2 and for ka = 3 in Figure 3. For smaller spheres, such as shown in Figure 2, the analytical curve is fairly simple to follow, displaying clearly that for backscatter (180°) high PR is expected. As the sphere gets larger compared to a wavelength, a more complicated current pattern is driven on the surface of the sphere, including creeping waves that lead to a more complex radiation pattern. We are seeing the results of this in the PR. Regardless of sphere size, Mie Scattering tells us to expect an infinite PR in the backscatter direction. For a complete discussion of Mie Scattering theory, please refer to Ishimaru  or Tsang et al. . A summary of the residual error is found in Table 1. For the single sphere validation, the forward scatter and backscatter angles are excluded from the error analysis as the theoretical curve tends to infinity. It is expected that NEC-2 will produce less precision and more bias at large values of the PR ratio, where either the RC or LC scatter will be small, and more influenced by numerical noise.
Table 1. For 100 Trials, We Present the Average Residual Error and Range in the PR of the Simulation for Scatter From a Single Sphere, Excluding the Angles of 0° and 180° Because the Theoretical Result Is Infinite
Average Range of Error (dB)
 The single sphere validation will provide information about the numerical limits achievable with NEC-2, especially with respect to the Polarization Ratio. Upon adding the structural complexity of two scatterers, it is expected that the backscatter (180°) PR will be bounded so that we can retain the backscatter computation.
2.3. Two-Body Scattering Validation
 There have been many analytical approaches presented for the two-body scattering problem, all involving considerable computational complexity [Heng-yu and Lu, 2007; Bruning and Lo, 1971; Liang and Lo, 1967]. One can also approach an analytical solution from a geometric optics approximation using ray-tracing as described by Felsen and Marcuvitz . For the purpose of validating our numerical model for the two-body scatter case, we can compare directly to published scattering data shown by Angelakos and Kumagai . The particular experiments we recreate in our simulation involve looking at the backscattering cross-section from a pair of equally sized spheres as they are separated in space with incident linear electric field polarization. Specifically, the authors present the measured backscatter for end-on and broadside illumination, and include a region where the centers of the two spheres may be close enough that the spheres are merged. The scattering cross-section is measured for separation distances of 0 (a single sphere) through 2.4λ apart (the merged cases) through 6λ (separate spheres). The spheres used in the published experiment were of larger size than chosen to explore in this meteoroids model, ka = 7.4, but this validation holds for the smaller spheres as the numerical limitations of NEC-2 are less prevalent with smaller sized scatterers.
 The results of the recreation of these two experiments are presented in Figures 4 and 5 and compare directly with Figures 2 and 4 from Angelakos and Kumagai . These figures show five independent simulations for each separation distance. The randomness we create by giving each sphere a 3D rotation allows us to look beyond the bias of our surface mesh to see the underlying consistency of the results. In both the end-on and broadside simulations, there is less precision in the region where the spheres are merged (distances under 2.4λ). This may be due to the fact that we may need a more sophisticated mesh to accurately model the merged structure. Consistent in both figures, the region where the two spheres are truly separated in space, the simulated results and the Angelakos and Kumagai  results show agreement with the sinusoidal structure present in the original data. For the case of broadside illumination, the model results agree with the measured results in the publication within 2 dB. For end-on illumination, the model does even better, agreeing with the measured values within 1 dB. It is this region, where the two spheres are not merged and are separated in space, that we move on to present the full polarization results as pertaining to meteoroid fragmentation.
3.1. Two-Body Scattering Experimental Setup
 To proceed to the two-body fragmentation modeling, we start with two spheres of equal size. Figure 6 illustrates the geometry. The first sphere anchored at the origin, the second sphere is translated from the origin to a new location (x ≥ 0, z ≥ 0). Numerically, the model propagates a plane wave at 158 MHz in the +z-direction, and computes the far field backscattered total power, LC power, RC power, and PR. Simulations were repeated until we acquired backscatter data for the second sphere position densely populated in the x–z quadrant. The simulations were carried out until the spheres lay beyond the near-field zone, at which point the results were indistinguishable from that of simple scatter. This process is employed for several ka values from 0.01 to 4. The results are analyzed and compared in detail for ka = 3 and ka = 0.25.
3.2. Two-Body Fragmentation Results, ka = 3
 We will begin by reviewing the results from two larger scatterers with ka = 3, corresponding to a diameter of 0.95λ, illustrated in Figure 7. As discussed before, the plots reflect four measurements: scattering cross-section, LC (co-polar) cross-section, RC (cross-polar) cross-section, and PR, each as a function of the position in meters of the center of the second sphere, with respect to the center of the first sphere held at the origin. The plot extends 8.5 meters in each direction so we can examine the region where the spheres are in each other's near field and allow the electric field to interact between them. Outside of this region, the electric field will interact less between the spheres and the total returned power will be a superposition of each scatterer's individual interactions with the radar plane wave. The empty quarter circle near the origin denotes the region where the spheres would overlap. The closest measurement we can achieve in the simulation is when the spheres are just barely touching, so our first data begins exactly at 2 radii from the origin, or at 1.81 m. The plots show us how we would still be able to measure a fragmentation event even without being able to range resolve the scattering bodies.
Figure 7a shows the total scattering cross section. Strong horizontal lines of small scatter lay parallel to the incoming plane wave. When the second sphere is placed at a position on one of those lines, the total returned power drops to 0. Examining Figure 7b we see that most of the structure between the two plots is similar, with a scale difference for the LC cross-section. The first horizontal trough of small scatter (at the bottom of the plot) is λ/4 off the x-axis. All the striations above are spaced equally by λ/2.
 Examining the backscatter from the individual spheres, we expect to see entirely LC polarization. With the second sphere on the bottom horizontal trough, λ/4 further from the radar, it will return the same amount of LC power, but its scatter will arrive in antiphase. These two fields destructively interfere, canceling the power in the far field. If we displace the second sphere λ/2 further away from the radar (up the z-axis), we see that we have added a full wavelength λ to the path. The simple destructive interference model predicts the perfect structured striations we see in the results, and since the simulation gives the full numerical result, we see that this simple First Born Approximation is still quite dominant. We see the full result deviate in the form of the lines rippling slightly, and this is due to phase shifts as the electric field interacts with both scatterers. If we imagine the plot to extend further in space and the spheres leave the near field of each other, we expect the rippling to dissipate and to see the interference lines straighten out to match the Born Approximation.
 These results give a visual example of how we could achieve SNR curves as described and shown by Kero et al.  and Mathews et al. . If we are sensing a two-body fragmentation event and the particles are drifting apart, for the sake of the simulation, this is equivalent to looking at the total received power plot of Figure 7a with the second sphere drifting away from the origin as the pair interact with the radar. In effect, we are sensing a path, a set of relative positions between the spheres. For a radar with narrow beamwidth such as ALTAIR, we would not expect to see the angular orientation of the two spheres with the plane wave change. As the fragments drift over several radar pulses, enough pulses to adequately sample the structure of the SNR curve, we would see a squared sinusoidal pattern of power as predicted by the color values shown in the plot. In this sense, Figure 7a shows all possible paths the two meteoroids could drift, and how that would translate to the power received at the tip of the receiving antenna, unperturbed by channel effects, receiver noise, and receiver beam pattern. We note that we would only see a sinusoidal beating pattern of the SNR if the particles are drifting apart in a way that they pass through the interference lines. If they are drifting parallel to the incoming plane wave, parallel to the x-axis, it could appear as one meteoroid with large returned power, or worse be entirely lost to the radar as the geometry causes destructive interference over the whole sensing period. It is also worth noting that this destructive interference pattern is purely a function of λ, the relative drift, and the positions of the scatterers. The resulting patterns are not a function of size of the spheres, however the actual power will always be a function of scattering cross section. The rippling of the interference lines in the results, however, is directly linked to the size of the particles in addition to the other variables.
 We noted above that the LC polarization power return shown in Figure 7b resembles the total power return. We look now to Figure 7c to see how the RC power makes up the difference in the two graphs. First, we can see that the striation pattern of destructive interference does not play a significant role in the RC channel. We see the strong region of red where the RC power is measurable. According to the results, if the fragmented meteoroid passes through the radar beam where the centers of both particles are just less than 3 m apart and roughly oriented at a 45 degree angle to the plane wave we will receive a significant amount of power in the RC orientation. The power is still small compared to the LC power return, note the change in scale on the colorbars. As we extend the graph beyond the simulation, we expect this entire red region to fall off into the noise floor. If the two particles are far enough away that their interaction is negligible, there is no way to return RC polarized electric field.
 To compare the LC and the RC, we now turn to Figure 7d which shows the PR, the ratio of LC power to RC power. Large values (red) of PR mean that the LC power greatly exceeds the RC power. Low PR (blue) means that the RC power is comparable, or exceeding the LC power. The scale is in dB, so a PR of 0 dB means that there is equal LC and RC power. In the plot, we see that we have a significant amount of light blue, corresponding to small PR values around 0 dB. First we note that the light blue values outside of the region where the RC is strong (close to the z-axis) are negligible. These results are from the LC power going to zero as well as having zero RC power. This is an effect that would not be measurable with the radar due to small total power. However, we note that in the region where the RC is strong, the power is comparable. This shows that using this two-body fragmentation model, we would expect to be able to measure RC power as a result of the interaction of the particles in their near field. Further, the simulation shows an argument for when we could actually receive more RC power than LC power. In the central region where the RC is significant, if the second sphere passes through a trough which is unique to the LC power return, we know the LC power will drop to zero while the RC power remains consistent.
3.3. Two-Body Fragmentation Results, ka = 0.25
 Now we direct our attention to Figure 8 which shows the same experiment performed on two spheres each size ka = 0.25. In this case, the size of the scattering diameter is well under a wavelength, 0.08λ. Anything under ka = 1 we expect to be in the Rayleigh scattering regime and see much a much less complicated scattering pattern. We examine the distance between the particles out to half a meter, and as we can see that was enough to map the received polarization features.
 If we look at Figures 8a and 8b, we note that our plot extends to exactly λ/4, the first horizontal line where the scattered electric field destructively cancels. The overall scattering cross-section and the LC cross-section match well, displaying little deviation from expected results. This implies that when the size of the scattering diameter gets small compared to a wavelength, the effect we can measure due to the interaction of the scatterers is little compared to the simple superposition of the single-body scattering problem. In this case, with a radar looking exclusively at the LC channel, we see simply the wavelength-dependent pattern of interference, with no geometric deviations due to particle size.
 We now turn to the RC cross-section shown in Figure 8c. The quarter circle radius is equal to 15 cm, or 2 radii of the ka = 0.25 spheres. The area where we achieve notable RC power measurements does not extend much further than twice that distance. In order to get RC power back, scatterers this size need to be closer than λ/4, therefore eliminating the mechanism we observed in Figure 7 where the RC power could actually exceed the LC power when the second sphere sits in a dead zone of destructive interference.
 If we look to Figure 8d we compare the LC and RC power again in the PR. We have already mentioned how we do not expect to see our PR drop significantly below 0 dB. Compared to ka = 3, we see almost exclusively positive PR. The minimum PR value in the area with high RC power is as low as 10–11 dB. This is not particularly compelling in terms of being able to measure the RC signature. Lower PR values can be found at the very top edge of the plot, but those PR values coincide with low total power.
3.4. Overall Analysis
 Back in Figure 7d we noted how much area is traced by moving the two spheres that could generate considerable RC backscatter, and in turn a low PR. We extend that result to include the entire symmetry of the 3D two-body scattering geometry, and measure the extent of the volume that could exist between the scatterers to return a significantly low PR. In Figure 9 we show histograms of the volume in cubic meters that the scatterers can occupy to achieve the given PR for each size ka = 0.01 through ka = 4. For the largest test case, ka = 4 shown in black, we note that even though the experimental spheres are quite large (a diameter of 2.4 m at 158 MHz), they can also be positioned quite far away from each other to still produce a significant amount of RC backscatter and a low PR. The histogram gives a sense of a PR “density,” and perhaps more intuitive is a volume calculation that is cumulatively based on PR. That is, Figure 10 shows the amount of volume that the scatterers can occupy to achieve a PR less than or equal to a given PR. So we can see in that same example, that if we are interested in a fragmentation between large, ka = 4 particles that returns a PR of less than 10 dB, the scatterers can be anywhere in the specific scattering volume, which turns out to allow them just under 3000 m3 of space. For smaller, ka = 0.25 sized scatterers (diameter of 15 cm at 158 MHz), the center of the second scatterer is constrained to be within 0.25 m3 of volume. By this measure, we can see that the volume becomes much more restricted, as the size of the scatterers diminish. Of particular interest to the meteor community may be the smallest scatterers we examined: sizes ka = 0.1 and ka = 0.01, with diameters of 6 cm and 6 mm at 158 MHz, respectively. For the case of ka = 0.1, the scatters cannot be further apart than barely touching to produce a PR of 10 dB or lower. For the smallest case, the scatterers cannot get close enough, without touching, to produce enough RC backscatter to bring the PR below 10 dB.
4. Concluding Remarks
 In this study we have explored a simple model of scattering from a fragmented meteoroid. Each fragment is presumed to be surrounded by an overdense plasma, and the radar returned modeled as scatter from an idealized conducting spheres, representing the critical plasma layer. We looked at spheres of radius 3 mm up to 0.45 m, which translated to ka values from 0.01 to 4 at the ALTAIR VHF and UHF frequencies. We found that for the larger sized scatterers, it was possible to achieve a backscattered PR less than 0 dB, that is it is possible to receive more RC than LC power. This occurs because the LC returned power is subject to destructive interference based on the position of the spheres relative to each other and the radar. The LC can cancel itself out while the RC backscatter can be unaffected. In the smaller ka values we found that the amount of RC returned power is small, and while in some orientations of the spheres can produce a PR as low as 10 dB, it constrains the fragmentation to occur within a very small volume.
 A complete characterization of the set of head echo polarization outliers in the data is limited by the small number of detections in current data sets. The consequence to not understanding the mechanism behind this results in inaccurate estimates of meteoroid mass and density. While there may be other theories to describe the low PR head echoes, this model shows that scatter from two spheres will consistently produce low PR–a proof-of-concept that fragmentation could explain the polarization anomalies.
 We would like to acknowledge Los Alamos National Laboratory, specifically the LDRD program, project 20090176ER for funding portions of this work. In addition, the reviewers greatly contributed to the improvement of this article.