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Keywords:

  • HF radar;
  • radar cross section;
  • backscatter radar

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[1] High frequency (HF) radars are used to detect ionospheric irregularities, meteor trails, and moving targets. The Precision Expandable Radar Calibration Sphere (PERCS) is a simple radar target in space to help determine the operational parameters of ground HF radars. PERCS will have a known radar cross section that is independent of observation direction within 0.5 dB. The PERCS satellite can be launched in a stowed configuration that has about 1 m in diameter. After launch, the PERCS will expand to a diameter of almost 10 m. Upon expansion, a stable wire frame is formed to act as a radar scatter target in the form of a polyhedral sphere. The simplest version of the sphere has 60 vertices (V60) that are joined to 90 rigid segments. Each segment is hinged so that the PERCS can be folded into a compact package for launch. Analysis of the V60 wire frame with a 10 m diameter shows that the radar cross section (RCS) is nearly independent of viewing angle up to 30 MHz. Another design with 240 vertices produces even better performance. Radar systems will be calibrated using the radar echo data and the precise knowledge of the target RCS, position, and velocity. The PERCS can reflect radar signals from natural targets such as field aligned and current driven irregularities not presently accessible from ground-based radars. The wire frame structure has several advantages over a metalized spheroid “balloon” with (1) much less drag, (2) larger radar cross section, and (3) lower fabrication cost.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[2] In the radio frequency range 2 to 40 MHz, Medium Frequency (MF), High Frequency (HF), and Very High Frequency (VHF) radars are used by atmospheric researchers to study natural and artificial ionospheric irregularities, and backscatter from meteoric ionization. The network of SuperDARN radars tracks high latitude disturbances to measure convection electric fields used in space weather models. HF radars in Alaska support the Alaskan High Frequency Active Auroral Research Program (HAARP) transmitter by detecting backscatter off field aligned irregularities generated by high power radio waves. All of these HF radars lack a precise calibration target with both a known position and well defined radar cross section.

[3] The Precision Expandable Radar Calibration Sphere (PERCS) will provide an HF radar target with an isotropic radar cross section that can be used by all ground based HF radars. The PERCS target will make use of Hoberman sphere technology for deployment of large objects in space. The absolute accuracy for scatter from field aligned irregularities or meteor trails will be greatly improved after ground MF and HF radars are absolutely calibrated with the PERCS target.

[4] The PERCS is a deployable sphere that has <1 m diameter in stowed configuration and a 10 m diameter in deployed state. The PERCS can be designed to yield a radar cross section that is constant at any viewing angle within 0.5 dB. The fully deployed sphere has a precise radar cross section for frequencies less than 23 MHz and some frequencies up to 40 MHz. The sphere can be placed in a high inclination orbit to service (1) the SuperDARN radars located around the north and south polar regions as well as the new SuperDARN radars being setup in the United States and (2) the HF radar diagnostics that are part of the HAARP program. PERCS is a geodesic sphere with 60 or 240 vertices and 12 pentagonal faces. In chemistry, these structures are denoted as C60 and C240 fullerenes with the C60 specifically named Buckminsterfullerene or “Buckyball”. The next section describes spherical polyhedra that form a wire frame suitable for radar scatter. The calculated radar cross sections of these polyhedral structures are shown in section 3 to be suitable for radar calibration target in space over a restricted frequency range. The monostatic radar cross section is computed as a function of radar frequency and viewing orientation. In section 4, deployable polyhedra are presented using a technique developed by Chuck Hoberman. The PERCS target is launched into space with a relatively small stowed size and, with spring loaded hinges, which allows it to expand into a large wire frame. The final sections outline the construction of the radar target, effects of atmospheric drag, optical tracking of the sphere and potential uses of the PERCS for calibration of backscatter radar systems. A large, well calibrated radar target in orbit around the Earth will be available for many new research applications.

2. Spherical Polyhedra With Pentagon and Hexagon Faces

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[5] All of the spherical radar calibration targets will be based on wire frame approximations to a smooth spheroid. These wire frame representations of a nearly spherical object use conducting wires at the edges of spherical polyhedra. All of the geometric structures are derived from the icosahedron (Figure 1a) which has 12 vertices, 20 faces and 30 edges.

image

Figure 1. Geometric structures of potential radar calibration targets. (a) The icosahedron has each vertex truncated to yield a hexagonal face. The rest of the spherical surface is filled with hexagons to yield spherical polyhedra with (b) 60 vertices and (c) 240 vertices.

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[6] Truncation at each vertex of the icosahedron yields a well known Archimedean solid or “soccer ball” with 60 vertices (V60) as illustrated in Figure 1b. During the truncation process, each edge of the triangular face of an icosahedron is divided by 1/3rd and each corner is removed leaving a hexagon. All the vertices of this structure lie at a common distance from the center and all the 90 edges have the same length. The faces are thus composed of 12 pentagons and 20 hexagons. The radar cross section from the V60 truncated icosahedron will vary to some degree if the vertex, edge, hexagon, or pentagon face is rotated toward the radar beam.

[7] The polyhedron becomes more like a perfect sphere as the number of faces increases. Consider if the edges of the icosahedron are truncated at a distance 1/6th from the vertex. The remaining long edge is divided into two edges on either side of a hexagon. Using the procedure outlined in Appendix A and described by Wang and Chiu [1993], the resulting spherical V240 polyhedron is generated with 240 vertices, 360 edges, 12 pentagons, and 110 hexagonal faces (Figure 1c). As the number of edges increases, the target becomes more spherical and the variation of the radar cross section with viewing direction is reduced.

[8] The physical dimensions of the polyhedra determine their mechanical and radar scattering properties. The V60 truncated icosahedron has 90 edges of identical length meeting at 60 identical vertices. The higher order spherical polyhedra are comprised of hexagon and pentagon faces with several length edges. Length of a radius vector is defined as the constant distance to each vertex from the center of the sphere for each polyhedron. The edges at the vertices for the spherical polyhedra have come together with a finite number of angles in the plane perpendicular to the radius vector at each angle. These angles add up to 360°. The angles between the radius angle and edge at a vertex also have a small number of values which increase with the number of vertices. The properties of the V60 and V240 spherical polyhedra with a 5 m radius are given in Table 1.

Table 1. Distribution of Edges and Vertices on 10-m Spherical Polyhedra
Polyhedron Vertex DesignationVertex RadiusEdgesVertices
LengthNumberVertex Angle With Radius, degPlanar Angles, degNumber
V605.02.10779078.389111.4, 124.3, 124.360
V2405.00.952424084.535108.7, 126.6, 126.660
  1.08146083.793117.9, 117.9, 124.260
  1.13196083.500119.37, 119.40, 121.23120

[9] For simplicity of construction and reduction of atmospheric drag, the polyhedral surfaces shown in Figure 1 will be replaced by the wire frame targets of Figure 2. The targets are 10 m in diameter with 60 and 240 vertices joined with conducting wires. If the individual wire segments or edges are much smaller than an HF radar wavelength, they act together to form a good approximation to a perfectly spherical radar target. The polygons on the geodesic structure are open so the wire frame has much lower drag than a similar design using a spheroid such as metalized balloon due to the dramatic decrease in presented frontal area. Other polyhedral configurations using triangle faces could be used for the radar calibration target their consideration will be postponed to the future.

image

Figure 2. Wire frame configurations for HF radar calibration targets with (a) 60 vertices and (b) 240 vertices. The dimensions are shown for 10-m diameter spherical polyhedral structures.

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3. Radar Cross Section of Spherical Polyhedra

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[10] Monostatic radar cross section determines the amount of power reflected from an object back to the transmitter. The radar equation incorporates the radar cross section with transmitting system and receiving system parameters and range to the target

  • equation image

where P is received (r) and transmitted (t) power, G is gain, L is loss, R is range, λ0 is wavelength, σ is radar cross section and C0 is the radar system parameter. The directional antenna gain G(f, θ, ϕ) for a ground radar system is dependent on the radar frequency (f), on the zenith angle to the target (θ) and the azimuth angle to the target (ϕ). The system losses L(f) are only frequency dependent. The monostatic cross section σ(f) of a spherical radar target is dependent on frequency but is independent of θ and ϕ. In terms of incident (Ei) and scattered (Es) electric fields, the total bistatic radar cross section is defined as

  • equation image

where the incident field Eii, θi) is a plane wave propagating along a direction given in spherical polar coordinates by the angles ϕi and θi, and the scattered field Es(R, ϕs, θs) is a spherical wave in the far field with the form Es(R, ϕs, θs) = ess, θs) Exp(−jk0R)/R. For backscatter the incident and scattered directions lie along the same path with opposite directions.

[11] Measurements of the ratio Pr/Pt from (1) with a known σ over a range of look directions and frequencies permits estimation of the system parameter C0(f, θ, ϕ). The radar cross section for any scattering medium is found from

  • equation image

Another parameter that is affected by scatter from a target is the polarization. The complex electric field vector are completely determined by it components Eϕ and Eθ with the formula

  • equation image

With R = ±j, the waves are right-handed (RHCP) or left-handed (LHCP) circular polarized, respectively. Each electromagnetic wave E can be decomposed into these two circularly polarized waves according to

  • equation image

where equation image and equation image are unit vectors. An initial electric field that is right hand circular polarized (RHCP) can scatter into both RHCP and LHCP modes. The radar cross sections for scattering into the same and different modes are

  • equation image

A perfect sphere does not change the polarization of the reflected wave so for a RHCP incident wave, σRH = σTotal and σLH = 0. For the PERCS to be a useful radar calibration target, most of the energy radiated should come back with the same polarization. A measure of the amount of electromagnetic energy coupled into the other polarization is the ratio of the polarization RCS given by

  • equation image

where ρLH−RH = 0 for a perfect sphere and ρLH−RH = 1 if have the initial RHCP wave is scattered into the LHCP wave.

[12] A 10-m spheroid or metal balloon was selected as the baseline for radar cross-section (RCS) comparisons. A sphere with another dimension will have radar cross section that scales in frequency with ratio of sphere diameter to radio wavelength (2r/λ) and scales with cross section magnitude as the projected area of the sphere (πr2). This scaling applies to the Rayleigh, Mie, and geometric optics regions of scatter. The 10-m sphere should be large enough to provide usable echoes for ground HF radars. The Mie scattering for the radar cross section of a perfectly conducting sphere is given in Appendix B based on the theory of Ruck et al. [1970].

[13] The radar cross section (RCS) is computed for frequencies up to 50 MHz (Figure 3). Below 4 MHz, the cross section monotonically drops off as λ4 where λ, the radio wavelength, is greater less than the sphere radius r. This is called Rayleigh scattering. For high frequencies with λ ≪ r, the radar cross section is approximately πr2 in the asymptotic geometric optics limit. Strong localized minima in RCS for the 10-m sphere are found near 17 MHz and 29 MHz.

image

Figure 3. Radar cross section for a 10-m diameter sphere with a perfectly conducting surface. The solid red line is the analytic theory described in Appendix B, and the black dots are the numerical solutions using the 3-D electromagnetic computations with the WIPL-D model.

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[14] Next, the RCS is computed for a conducting polyhedron as a radar target. All of the 10-m diameter wire frames in Figure 2 will reflect HF radar signals. Their radar cross section (RCS), however, will fluctuate as the wire frame is rotated. The objective of the calibration target design is to produce a minimal variation in radar cross-section, 0.5 dB, as the target orientation is changed. This will first be studied using the V60 wire frame shown in Figure 2a. The computational frequency range will be from 0 to 50 MHz to match the operational frequencies for most HF and lower-band VHF radars on the ground.

[15] The estimations of RCS were obtained by applying the method of moments solutions to the electric field integral equations (EFIEs) using the WIPL-D 3-D Electromagnetic Solver [Koludzija et al., 2004]. The polyhedral wire frame is excited by a right-hand circular polarized (RHCP) electromagnetic wave. The incident and reflected waves are compared to yield the monostatic RCS. In the model, currents in both wire segments and plates are expressed as finite polynomial sums with orders between 1 and 9 automatically determined by the electrical length of the conducting component. If the wire becomes longer than two wavelengths, it is divided into small segments each shorter than λ/2. The diameter of each wire in the model was set to 4 mm to resemble the mechanical structure. The WIPL electromagnetic model was validated by comparing the analytic MIE theory for the conducting spheroid with the numerical RCS computations (Figure 3).

[16] Comparison of the V60 wire frame and conducting spheroid models show many differences in RCS (Figure 4). In the 9 to 24 MHz frequency range, the wire frame has a larger RCS than the metalized spheroid. Below 9 MHz, the wire frame RCS is less. Also, the dip in RCS near 17 MHz is less for the V60 wire frame. The RCS is illustrated for two different spherical coordinate directions (blue and green curves) in Figure 4. The incident wave makes an angle θ with the x-y plane and an angle ϕ with the x-axis in the x-y plane. If (θ, ϕ) = (0, 0), the wave is incident in the negative x-direction. If (θ, ϕ) = (π/2, π/2), the wave is incident in the negative z-direction. Only for frequencies below 23 MHz and between 30 and 34 MHz there minimal change in RCS for the two viewing angles. The variation of RCS near 20 MHz for the two view directions is less than 0.5 dB. Rotation of the wire frame produces large variations in RCS in the 23 to 30 MHz frequency range and for frequencies above 34 MHz. To better illustrate the dynamic range for backscatter variations with frequency and orientation, the RCS will be displayed as dB m2 rather than m2.

image

Figure 4. Comparison of the radar cross section (RCS) for a V60 wire frame (blue and green) with a conducting spherical surface with the same diameter (red). The RCS for the wire frame only significantly depends on observation direction for frequencies from 23 to 30 MHz and above 32 MHz.

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[17] The full range of RCS changes variations with radar observation direction is obtained by stepping the WIPL simulations through all target angles. The numerical values for RCS were computed over at total of 648 viewing directions using 10 degree steps in both θ and ϕ. The average value of RCS for all viewing angles is given in Figure 5. Between 9 and 33 MHz, the wire frame has a larger radar cross section than the spheroid with a continuous conducting surface.

image

Figure 5. Computed monostatic RCS averaged over all viewing directions from a V60 polyhedron with equal edge lengths.

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[18] The maximum RCS deviation based on the WIPL computations shows that the 10-m diameter V60 wire frame is well suited for a calibration target for frequencies less than 23 MHz where this deviation is less than 0.5 dB (Figure 6). Above 23 MHz, a local maximum in RCS variation with orientation is found to be 9.6 dB at 26 MHz. This makes the V60 of less use in the 26 to 29 MHz band unless the orientation of the sphere is known. Figure 6, however, shows that that in the 30 to 34 MHz the variation in RCS with is at a local minimum so that at this frequency the V60 conducting frame may be a useful radar calibration target.

image

Figure 6. Maximum change with orientation of the monostatic RCS for the V60 wire frame radar reflector with equal length edges.

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[19] The RCS behavior at different frequencies is explained in terms of resonances with the edges and faces of the pentagons and hexagons in the wire frame. Below 16 MHz, a spherical backscatter pattern is produced by the v60 wire frame. Between 16 and 20 MHz, the small lumps in the backscatter pattern (Figures 7a) are aligned with the faces of the 12 pentagons in the V60 sphere. Between 23 and 24 MHz, the backscatter bulges are associated with the hexagon faces. Near 30 MHz when the sphere is equation image wavelength in diameter, the backscatter has even contributions from the pentagons and hexagons on the surface of the V60 (Figure 7d). From 31 to 34 MHz the peaks in the RCS pattern corresponds to the locations of the edges between the hexagons (Figure 7e). From 35 to 40 MHz, backscatter from the faces of the hexagons dominates the pattern (Figure 7f). At higher frequencies, the edges and faces contribute to the radiation with increasingly complex patterns.

image

Figure 7. Radar cross section patterns for the V60 wire frame at selected frequencies.

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[20] Figure 7 shows that the change in RCS with look direction is associated with the transitions between hexagon and polygon faces of the wire frame. To reduce this difference, the wire frame was redesigned with an increase pentagon face area and a decrease in the hexagon face area to make all the polygons have approximately the same area. For the 10-m diameter sphere, this redesign yielded a spherical polyhedron with a radius of 5 m to each vertex, 30 Edges with length 1.53 m, 60 edges with length 2.29 m, pentagons with area 9.03 m2 and hexagons with area 9.34 m2 (Figure 8).

image

Figure 8. Redistribution of polyhedral edges to make all faces have approximately equal areas.

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[21] The revised V60 sphere has an aspect independent RCS out to 30 MHz. Figure 9 shows the average RCS and the maximum deviation in RCS with viewing direction. A comparison with Figures 5 and 6 shows that the revised design has much lower deviations in RCS with viewing direction and it meets the 0.5 dB specification out to 30 MHz whereas the old design met the specification only to 24 MHz. Figure 9 also shows that the total RCS for the wire frame is larger than the RCS for the reference spherical ball at all frequencies between 10 and 32 MHz.

image

Figure 9. HF RCS for the spherical polyhedron with equal area polygon faces. The useful frequency range extends to 30 MHz.

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[22] The nonspherical imperfections in the V60 wire frame may change the polarization by scattering of individual elements of the sphere. The definition of polarization coupling from RHCL to LHCP given by (7) was computed with the WIPL code for scattering equal-edge V60 from all directions. Figure 10 shows that the frequencies of low average (green) and maximum (blue) polarization coupling coincide with the frequencies of low variations of total RCS. The frequencies below 23 MHz which have low variation with radar view direction also have negligible change in polarization by scatter. Similar results are obtained for scattering of left hand circular polarization as well as with both the ϕ and θ directions of linear polarization from all of the wire frame structures.

image

Figure 10. Polarization conversion from right hand circular polar (RHCP) to left hand circular polar (LHCP) by scatter from the V60 radar target.

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[23] To reduce the viewing angle variations in RCS and the polarization changes above 23 MHz, the RCS properties of the V240 wire frame were examined. First consider the V240 sphere with the three lengths given in Table 1. The direction-averaged RCS for the V240 is closer to that of a conducting spherical surface (Figure 11) than for the V60. The deep RCS minimum near 18 MHz in the V240 RCS is close to that of the reference spheroid. There is another deep reduction in RCS near 38 MHz. For most of the frequencies above 20 MHz, the wire frame has about 5 dB larger cross section than the solid sphere.

image

Figure 11. Comparison of the RCS a for 10-m diameter V240 wire frame with a reference 10-m metalized solid sphere.

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[24] The V240 has better uniformity in RCS pattern. The RCS deviation is less than 0.5 dB for frequencies up to 26 MHz (Figure 12). The maximum RCS deviation is only 2.2 dB at 28 MHz. As expected, the addition of small hexagon facets to the spherical surface reduces the dependence of RCS on target orientation. The new V240 target meets the 0.5 dB design requirements between 30 and 36 MHz.

image

Figure 12. Maximum variation in monostatic RCS for the V240 wire frame radar reflector.

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[25] Samples of the RCS patterns are shown in Figure 13 for selected frequencies. There is a one-to-one correspondence with the features of the RCS pattern and the locations of the polygons on the surface of the sphere. For frequencies less that 30 MHz, the peak faces in the RCS pattern are collocated with the pentagon faces. The smooth RCS surface between 30 and 35 MHz shows small valleys in the RCS at the locations of the pentagons. For instance, near 42 MHz, the peak RCS faces are located directly over the hexagons located in the center of a triangle formed by three pentagon faces on the surface of the sphere.

image

Figure 13. Large variations with frequency in radar cross section for the V240 wire frame with relatively small variations with direction.

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[26] An attempt is made to reduce the RCS variations with look direction by redesigning the wire frame to make the area pentagon faces equal to most of the area of the hexagon faces. Figure 14 shows the new design.

image

Figure 14. Tuned V240 wire frame with 5-m radius to each vertex, 60 edges with length 0.45 m, 240 ledges with length 1.08 m, 60 edges with length 1.28 m, pentagon areas of 2.01 m2 and three types of hexagons with areas 1.81, 1.85, and 3.22 m2.

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[27] The average RCS for the two designs of the V240 are nearly identical (Figures 11 and 15a). The RCS for the revised V240 design does provides better RCS behavior at frequencies up to 35 MHz (Figure 15) but the deviations with viewing direction are larger above 28 MHz that shown in Figure 12. As discussed in the next section, the difficulties for deployment of the mechanical structure with a few extremely short sides may affect the decision of which wire frame design is used as a radar target.

image

Figure 15. HF RCS for the spherical V240 polyhedron with larger pentagon faces. The useful frequency range extends to 35 MHz.

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[28] For both versions of the V240, in the band for constant radar cross section below 26 MHz, the polarization conversion is less than 1%. Figure 16 illustrates the polarization change for the first version of the V240 sphere. In the frequencies above 30 MHz, the V240 has much less polarization change than produced by the V60 wire frame.

image

Figure 16. Polarization conversion from right hand circular polar (RHCP) to left hand circular polar (LHCP) by scatter from the V240 radar target.

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[29] As a practical consideration, the signal to noise ratio for a V60 wire frame flying over the SuperDARN radars is considered. The signal to noise ration determines the usable size for the radar target that can be detected at a large range with the ground based HF radar. The radar equation (1) divided by the sky noise power can be manipulated to give the signal-to-noise ratio as equation image where R is the range, k is the Boltzmann Constant, TSky is the radio sky temperature in Kelvin, Bn is the radar receiver bandwidth, PT is the transmitter power, G is the antenna gain, and λ is the radio wavelength. The radio sky noise that comes from atmospheric, galactic, and manmade sources is estimated by the standard International Radio Consultative Committee (CCIR) model. The RCS (σ0) is given in Figures 5 or 9a as a function of operating frequency. Using parameters representative of the SuperDARN radar (Bn = 3.3 kHz, PT = 10 kW). The antenna gain varies from G = 100 at 11 MHz to G = 250 at 17 MHz [Walker et al., 1987; Lester et al., 2004]. With target ranges of R = 300 to 1500 km, the required radar signal-to-noise for detection of the target is illustrated in Figure 17. The signal to noise ratio is larger than 0 dB with frequencies near 20 MHz for all ranges. Consequently, a 10-m sphere at an orbit of 450 km altitude is big enough to provide useful calibration for SuperDARN radar systems looking with low elevation angles at the higher frequencies where the sky noise drops and the antenna gain increases.

image

Figure 17. Signal-to-noise ratio (bottom) for SuperDARN radar detection of the PERCS 10-m sphere based on atmospheric, galactic, and manmade sky noise (top) given by the CCIR model.

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[30] In summary, wire frame polyhedra with 60 and 240 vertices provide excellent calibration targets for radar wavelengths up to the target diameter. The higher order spheres have regions near where, with radar wavelengths near 1.26 the target diameter, RCS is nearly independent of viewing direction. For use in space, the spheres must be deployed from a compact configuration. The mechanical scheme for stowing and deploying these spheres is discussed next.

4. Deployment of a Spherical Polyhedron in Space

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[31] Launching a fully deployed 10-m diameter wire frame in low Earth orbit (LEO) is not feasible. The polyhedral structure must be collapsed into a small package and then deployed after injection into orbit. For this purpose, hinges are introduced each vertex and along the edges allow the structure to be stowed for launch. Using the technique patented by Hoberman [1990, 1991], two concentric geodesic spheres are interleaved to form a stable structure with alternating vertices of each sphere folding either inward or outward. Hoberman Designs Inc. has marketed this structure as a Transforming Sphere©.

[32] The mechanically stable structures for both stowed and deployed configurations are produced by introducing “scissors” along each edge. As illustrated in Figure 18 adapted from Hoberman [1991], each scissors edge is an angled blue and red arm (10 and 20) that is connected by a common vertex point (12, 22). As the bracket pivots around the vertex, the end points (16, 26) and (14, 24) begin to separate. During this process, the left end points, the center pivot point, and the right end points are restrained to follow straight lines (40, 30, 50) that converge to one origin. When one of the end points of the red arm reaches this origin, the scissors are stowed and the polyhedron is fully collapsed. The minimum stowed size is determined by the distance between the ends of the scissor arms and the compacted spacing of the vertex hinges around the stowed center. The geometrical design of the scissor arms is discussed in Appendix C.

image

Figure 18. Structure of the Hoberman Scissors (adapted from Hoberman [1991]). The ends of the two scissor arms move along lines 40 and 50 while their pivot point stays on line 30.

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[33] The motion of a single set of scissor arms with equal length struts is shown in Figure 19. The dimensions of the arms are taken from Table 1 for the V60 wire frame. The computed length of the struts is 1.03 and the angle between the struts is 156.7 degrees. As the scissors ends and pivots move along radial lines, they form a collapsible frame for the wire frame.

image

Figure 19. Motion of the two scissor arms for a 5 m radius sphere with a distance between edges of 2.018 as given by Table 1 for the V60 wire frame.

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[34] Full collapse and full extension is never reached because of hinges located at each vertex. Table 1 gives the design parameters for all the wire frame hinges. Figure 20 illustrates on hinge configuration for a V60. Each of the vertex hinges and the edges for the V60 wire frame is identical. The angle at the pentagon face is smaller than the angle for the hexagon faces. The angles for the V60 hinges given by Figure 20 are in the plane normal to the radius vector from the center of the sphere. Each fully deployed edge makes a 78.4 degree angle with this radius vector for the V60. The V240 and higher order Hoberman sphere, besides having more elements, have a more complex design. As shown in Table 1 for the V240 wire frame, more than one type of hinge needed at different hexagon and pentagon vertices. In all designs, small torsion springs are added to each segment (Figure 21) so that the precision expandable radar calibration sphere (PERCS) will be self expanding after release into space.

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Figure 20. (a) Top view and (b) side view of vertex hinges used to fold the V60 wire frames into a small package.

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image

Figure 21. Detail of torsion spring inserted around scissors pivot to provide opening force.

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[35] PERCS can be considered for launch as a secondary small-satellite payload from a rocket if the diameter of the stowed configuration is about 1 m. To achieve this stowed diameter, the wire frame is converted into a Hoberman sphere with spring loaded edges. The limitation in stowed size is determined by the length of the edges and the gathering of the hinges at the center. Consider a V60 wire frame with extremely large vertex (33 cm) hinges. Figure 22 shows a comparison of the expanded and collapsed spheres with straight line approximations to the scissor arms. The central hinges come together into a small sphere which is structurally related to the dual of the original polyhedral structure. A polyhedron dual is constructed with edges between the center of each polygon face that forms the original polyhedron. The dual of each structure given in Figure 2 has triangles as faces. To obtain the six-sided vertex hinge shown in Figure 20, these triangles are converted into hexagon hinges by truncating each triangular vertex. The total diameter of the stowed configuration is the sum of the diameter of the dual sphere at the center and twice the length of the scissor arms between each vertex hinge. In Figure 21, where the hinge size of 33 cm has been chosen for the purpose of illustrating the internal sphere, the diameter of the stowed radially extended arms is 4.88 m.

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Figure 22. Comparison of stowed and expand configurations of a V60 Hoberman sphere with 10 m diameter and intentionally large 33 cm vertex hinges. For a 10 m diameter sphere, the internal vertex sphere has a diameter of 0.86 m and the outside diameter of the collapsed sphere is 4.88 m.

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[36] Greater compression for the stowed configuration is achieved with smaller vertex hinges and higher order polyhedra. Figures 23 and 24 illustrate the V60 and V240 wire frames collapsed using 3 cm hinges. As the order of the polyhedra increases, the stowed structure has a reduced diameter. With a 3 cm hinge, all of the spheres have stowed diameters greater than 1 m so further size reduction may be required for placement into orbit.

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Figure 23. Comparison of stowed and expand configurations of a V60 Hoberman sphere with 10 m diameter and 3 cm vertex hinges. For a 10 m diameter sphere, the internal vertex sphere has a diameter of 0.08 m and the outside diameter of the collapsed sphere is 4.10 m.

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image

Figure 24. Comparison of stowed and expand configurations of a V240 Hoberman sphere with 10 m diameter and 3 cm vertex hinges. For a 10 m diameter sphere, the internal vertex sphere has a diameter of 0.16 m and the outside diameter of the collapsed sphere is 2.08 m.

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[37] Additional reduction in size can be achieved by using multiple scissors on each edge. The V60S2, which designates a 60 vertex structure with two scissors per edge, is illustrated in several stages of deployment in Figure 25. The vertex hinges are deleted and the scissor arms are colored red and blue to indicate the two polyhedral spheres that are interlaced. The 10 m dual-scissors V60 would have a minimum stowed diameter of 2.1 m using 3 cm hinges. Table 2 lists the stowed diameters of all the wire frame polyhedra for both single and dual scissors implementations. The configurations near or under 1 m diameter can be accommodated as a small satellite.

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Figure 25. V60 radar reflector with two scissors per edge.

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Table 2. Minimum Stowed Diameters for the Hoberman Spheres Using 3 cm Hinges
ParameterEdge TypeV60V240
External Spokes Sphere DiameterSingle Scissors4.102.08
Dual Scissors2.101.12
Internal Hinge Sphere DiameterSingle and Dual Scissors0.080.16

5. Performance of the Hoberman Sphere Wire Frame as a Radar Target

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[38] When the wire frame is deployed as a Hoberman sphere, two factors must be considered. First the struts along the edges are joined together at mechanical hinges where electrical resistance may be a factor. Second, the Hoberman sphere does not fully deploy into a polyhedra because the internal and external hinges cannot merge together. To study these effects, the WIPL electromagnetic solver was used to calculate the RCS of practical mechanical configurations.

[39] The resistance between arms of the Hoberman sphere will be minimized by design with high conductivity (i.e., gold) surfaces with maximum pressure over maximum area at the hinge interfaces. The torsion springs on each scissors will be chosen to apply pressure at the hinges when the sphere is in the deployed state. To obtain an understanding of the effects of hinge resistance, the RCS for the wire frame was computed with the addition of fixed resistance inserted at the ends of 30 wires of the 90 edges in the V60 wire frame. A resistance of 1 Ohm in 30 places had insignificant effects on the RCS. With 10 Ohms, the total RCS was changed by about 0.5 dB for frequencies above 10 MHz (Figure 26) and the variation in RCS with viewing angle was changes by a maximum of 0.12 dB (Figure 27). From this study, it is concluded that if care is taken to keep the resistance at each contact point less that 10 Ohms, the Hoberman sphere radar target will meet a requirement to know the absolute RCS within 0.5 dB at each frequency. For this reason, all of the metal components of the wire frame should be gold plated to prevent corrosion at the conductive interfaces.

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Figure 26. Effect of contact resistance at 30 vertices on computed radar cross sections for the V60 radar target.

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Figure 27. Effect of contact resistance (30 vertices) on maximum variation in radar cross section.

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[40] The RCS calculations used wires with infinite conductivity. It may be profitable to introduce a small amount of resistance into a wire to reduce the “Q” of the sphere so that some of the frequency dependencies become broader.

[41] To identify the effects of “crinkle” in the surface of the Hoberman sphere, a partially deployed configuration was considered. Figure 28 shows the affects on RCS of a V60 sphere with 95% deployment (0.25 m out of 5 m radius). This is a substantial exaggeration of the actual surface imperfections. The actual deployed Hoberman sphere is expected to be deployed within 0.01 m of a full 5 m radius. The average RCS and the variations in RCS with viewing direction are affected by less than 0.2 dB with this large fluctuation in surface structure. In fact, this calculation shows that partial deployment of the sphere yields a structure with reduced absolute cross section with increased range to 25 MHz for less than 0.5 dB variations with viewing direction. In summary, for applications as a radar target, imperfections in the Hoberman sphere such as vertex hinge resistance and slight crinkle of the surface can be minimized so as to have negligible impact on the radar RCS.

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Figure 28. Radar cross section for a V60 wire frame with a 5% variation radius at the scissor hinges.

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6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[42] Precise orbit determination is needed for accurate validation of target location algorithms for ground HF radars. The primary perturbation on the orbit of the spherical radar target will be atmospheric drag. Atmospheric drag will ultimately limit the lifetime of the target in orbit. The satellite acceleration (or deceleration) due to drag is given by the equation [Montenbruck and Gill, 2000]

  • equation image

where r is geocentric radius, CD is the drag coefficient assumed to have a value of 2, A is the projected 2-D cross section area of the metal object, m is the mass, r is the neutral mass density of the atmosphere, vr is the speed of the sphere with respect to the medium and er is the unit vector in the direction of the orbit. The mass density model used for this study was obtained from the NRL MSIS Model described by Hedin [1987] and is shown in Figure 29. The actual neutral density profiles will vary with season and solar cycle but the one selected for the drag calculations is representative of minimum solar activity at equinox.

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Figure 29. Neutral density used for atmospheric drag calculations for the spherical radar target in space.

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[43] Numerically integrating (3) for circular orbits yield a time history of the orbiting sphere. The mass and drag area for a fully metalized target with a 30 micron thick spherical shell is estimated to be 43 kg and 78 m2, respectively resulting in an area to mass ratio of 1.84 m2/kg. When this solid-surface spheroid is injected to orbit at 450 km, it will have a lifetime of only 5 days (see red line in Figure 30). The V60 Hoberman sphere made with 8 mm diameter aluminum arms has an estimated mass of 49 kg and a frontal area of 2.3 m2. This corresponds to an area to mass ratio of 0.047 m2/kg, which is 40 times smaller than the value for the fully metalized target. Due to the dramatic reduction area to mass ratio, the spherical wire frame will remain in orbit for over 1/2 year (see blue line in Figure 30). For this model, the wire frame consists of gold plated aluminum struts with 8 cm thickness. If the wire frame were made from a more dense material like gold plated steel, the mass would increase to 143 kg and the sphere would stay in orbit almost 2 years (see green curve in Figure 30). The area to mass ratio in (3) will be similar to the V60 for the higher order wire frames. Consequently, the orbit lifetime should be greater than 600 days for orbits above 450 km altitude. Even though the details of the orbit lifetime will vary with number of vertices and background neutral density, the wire frame has an obvious advantage a metal spherical balloon.

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Figure 30. Limitations on the lifetime of a 10-m spherical radar target because of atmospheric drag.

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[44] The addition of small, optical corner-cube-reflectors to each vertex of the Hoberman sphere permits satellite laser ranging of the radar target. Corner reflectors provide a reflection cross section for visible light that depends on the incident angle (Figure 31). With one corner cube on each vertex, those that are facing toward a ground laser system will reflect visible light back to the source (Figure 32). The total intensity of the reflected light will vary as the spherical target rotates. Using the V60 Hoberman sphere and the tilt angle dependence on optical cross section, the fluctuations in total optical cross section were calculated for rotation around several axes on the sphere (Figure 32). The rotation rate for the target can be determined for the temporal fluctuations in the reflected light (Figures 33a and 33c). The lowest harmonic component of the frequency spectrum gives the rotation period (Figures 33b and 33d).

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Figure 31. Laser light reflection cross section for a small corner cube reflector with less than 1 cm diameter.

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Figure 32. Laser illumination of a V60 sphere with corner cube reflectors at each vertex.

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image

Figure 33. Illustration of fluctuations in optical reflection cross section (a and c) and frequency spectra (b and d) as the spherical polyhedra rotates about an axis with tilts of 60 degrees (a and b) and 18 degrees (c and d).

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[45] Satellite laser ranging sites can therefore provide both position and rotation information on the orbiting polyhedral sphere in space. This information can provide precise location of the target for HF radar calibration. Any small fluctuations in the HF radar return signal can be correlated with the independently measured rotation of the spherical target.

7. Applications Summary

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[46] The Precise Expandable Radar Calibration Sphere (PERCS) has many applications that would be of use to several scientific organizations. The National Science Foundation funds the operation of many ground radars for upper atmospheric studies. The Air Force and Navy jointly manage the High Frequency Active Auroral Research Program (HAARP). Jointly the Air Force, Navy and NSF are funding new ionospheric modification facilities in Arecibo, Puerto Rico and Jicamarca, Peru. PERCS will provide the first long range calibration target for HF radars and transmitter antennas associated with these research systems. All HF radars are affected by ionospheric refraction. The refraction effects are estimated using plasma density data from backscatter ionograms and other independent measurements. With radar scatter from the PERCS at a known location, the algorithms for target location will be tested and validated. The space weather community is developing both empirical and physics based models for the Global Assimilation of Ionospheric Measurements (GAIM). The High Latitude Data Assimilation Model developed by Utah State University uses high latitude convection from the National Science Foundation (NSF) SuperDARN radars as inputs to this physics based calculation that is under development as part of the GAIM program. A PERCS calibration sphere can improve the accuracy of convection radar data used in these and other space weather models.

[47] Several options for a orbiting calibration targets in space have been considered. A deployable Mylar sphere is a coating of gold was found to cost about $1M from commercial vendors. Atmospheric drag on a 10 m metalized “balloon” requires that the orbit altitude be 700 km to give a lifetime of 1 year. The alternative sphere using the Hoberman configuration should have a substantial lower cost. A 10 m PERCS will have a lifetime of over 5 years for altitudes above 600 km. The PERCS solution is cheaper, lighter and has a much longer lifetime than the Mylar balloon solution. Another alternative is to scatter radar off large orbiting satellites like the International Space Station. This provides precise target location but cannot give an monostatic radar cross section with better than 6 dB accuracy. PERCS is the best target solution to meet the HF radar community requirements.

[48] Once PERCS is in orbit, most of the applications will involve calibration of ground HF transmitter antennas and radars. Figure 34 illustrates the concept of operations where the sphere would fly over the ground radar for measurements of the radar system sensitivity. Both vertical and oblique antenna patters can be calibrated with PERCS. Most publications of HF radar results use “Signal to Noise Ratio – SNR” or “Arbritary Units” to describe the radar backscatter data. With calibration using PERCS, the units can be expressed a absolute cross section in dB m2. This is especially important in study of meteor head echoes where the strength of the radar backscatter is used to determine the number of electrons produced by meteor entering the atmosphere [Close et al., 2002]. High latitude applications of the HF SuperDARN radars to observe backscatter from auroral irregularities [Bristow and Lummerzheim, 2001] and from heater induced irregularities [Hughes et al., 2004].

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Figure 34. PERCS calibration sphere passing over a ground HF radar for calibration.

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[49] For other applications, nearly vertical radar signals can be deflected by the PERCS to permit radar measurements at directions not accessible from the ground. Figure 35 illustrates side scatter from the PERCS to view horizontally propagating electrostatic waves such as may be produced with the electrojet currents found near the Earth's equator. Both modified two-stream (i.e., Farley-Buneman) and gradient drift instabilities are used to explain irregularities driven by the equatorial electroject [Kelley, 1989]. Observing these irregularities with ground radar is difficult because the vertical or oblique radar beam is at an angle with the primary direction of the horizontal irregularity structure. By using the geometry illustrated in Figure 35, ground radar waves can be scattered into the horizontal direction with the PERCS. After backscatter off the irregularities, the radar signal is scattered downward into the aperture of the ground radar. Since the RCS for PERCS is accurately specified for bistatic scatter and the PERCS sphere has been used to calibration the ground radar system, the absolute intensity of the electroject instabilities can be determined. Similar applications can be used to study high latitude, field-aligned irregularities where the magnetic field is nearly vertical. High power transmitters such as available with the HAARP facility in Alaska, the SPEAR very high latitude heating facility in Norway, and the planned future HF facilities for Arecibo, Puerto Rico and Jicamarca, Peru will be required to give strong enough signals for the PERCS oblique-scatter mode to work.

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Figure 35. Radar oblique scatter to detect irregularities from two-stream and gradient drift instabilities.

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[50] In summary, the design of the wire-frame radar calibration target uses both mechanical and electromagnetic modeling. The PERCS mechanical design employs polygons to tile the surface of a sphere. Deployment of a large PERCS as a satellite with scissored hinges is based on the Hoberman transformable sphere design. Expansion after release into orbit is accomplished with torsion springs at the pivot points of the scissors. Once the PERCS is expanded into a 10-m sphere, it will provide a well calibrated radar target for high frequency radars. The WIPL EM code has shown that the radar cross section will be nearly independent of orientation for frequencies less than 23 MHz. At 600 km altitude, drag computations with the NRL MSIS atmosphere demonstrate that the sphere will remain in orbit for at least 5 years. Once in space, the PERCS target will provide ground HF radars the ability to routinely check system sensitivities and target location algorithms.

Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[51] To illustrate the construction of a high order polyhedral approximation to a sphere, the V240 design will be outlined. Each edge of an icosahedron is truncated at a distance of 1/6th. With an icosahedron's edge length = 2, the truncation yields line segments of length 1/3. For illustration, three of the pentagons resulting from the truncation are projected on to a flat surface (Figure A1).

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Figure A1. Truncation of an icosahedron to yield pentagons.

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[52] Next the remaining long edges are divided into a series of lines and hexagons. The lines between the pentagon vertices lie on a great circle between points 1 and 2 in Figure A2. The side b is the length of one pentagon edge. The side 2b′ is the diagonal across each new hexagon. The equations for the angles in Figure A2 are

  • equation image

where the quantities b, R1, L12 are defined by the initial truncated icosahedron. Because of the curved surface, the length b′ is greater than b.

image

Figure A2. Angles and spherical edges for construction of the hexagon surfaces on a sphere.

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[53] Equations (A1) are used to define the points A and B along the great circle. The red segments (1-A) and (B-2) are drawn both in the planar projections and the spherical surface of Figure A3. Additional line segments between the free ends of the red lines are added with the orange lines attached to single hexagons, yielding the segments shown in Figure A4. Finally all the undefined edges are obtained by joining the uncommitted end points of the orange and green lines to give the wire frame in Figures 2b. All of the red and orange edges are of the same length while the other edges will be longer because of the curved spherical surface.

image

Figure A3. Initial line segments between pentagons and yet undefined hexagons.

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image

Figure A4. Line segments obtained along great circle paths of lines and hexagons between existing red line-segment end points.

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[54] As a result of this procedure, a spherical polyhedral surface with 240 vertices is formed with 240 edges of equal length and two sets of 60 edges with slightly longer dimensions. Figure A5 shows the sphere edges color coded by length and the distribution of these lengths. Using a similar procedure, the edges for the higher order wire frames can be constructed [Wang and Chiu, 1993].

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Figure A5. Distribution of edge lengths for the V240 spherical polyhedron based on an icosahedron with an edge length of 2 units.

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Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[55] The general Mie scattering for radar echoes from spheres is given by Ruck et al. [1970]. Far field solutions exist when the distance to the sphere (R) is much larger than the radius of the sphere (r). For a perfectly conducting sphere with R ≫ r, the monostatic cross section looking in the direction of the incident polarization is given by

  • equation image

where

  • equation image

Jn(z) is the cylindrical Bessel Function of the first kind, Hn(z) is the Spherical Bessel function of the third kind and the wavelength is related by the speed of light c to the radar frequency by λ0 = c/f. The RCS is computed numerically from (A1) with a finite truncation of the infinite series. In the Rayleigh region with k0r < 0.4, the backscatter cross section is given by

  • equation image

In the high frequency (or short wavelength limit k0r > 20) the spherical radar cross section is simply the geometric projection area

  • equation image

Appendix C:: Design of the Hoberman Scissor Arms

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[56] The angles and length for each strut in the Hoberman scissor arms are determined by geometric considerations. Consider the diagram shown in Figure C1. Two identical scissor arms have lengths L1 ≥ L2 joined at a pivot with angle θ12. In addition, L3 is defined the distance between the end points of each arm. The pivot point moves along a vertical line with a radial distance Rb from the origin. The end points follow converging lines that have radial distances Ra, Rc, Rd, and Re from the origin. Using the sine law, a constant L0 of the scissors is given by

  • equation image

where the angles are defined in Figure C1. When the scissors are fully, Rb = RH where

  • equation image

When Rb = L1, the scissors are fully collapsed. Notice the at the internal angles (θ12, θ13, θ23) of the triangle with sides (L1, L2, L3), also determine the angles lines emerging form the origin. Using the Sine and Cosine laws yields the variation of the radial distances to the scissors end points as the pivot radius, Rb is varied

  • equation image

The choice of the dimensions and angles for the arms can be determined by the selection of three parameters R1 the fully deployed radius to an end point with the two scissor arms merge together, L3, the distance between neighboring vertices at deployment and L4 the minimum value of Rd when the sphere is stowed. With these given parameters all the other mechanical specifications are given by

  • equation image
image

Figure C1. Geometry for the design of Hoberman scissors.

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Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information

[57] The research was funded by the Office of Naval Research at the Naval Research Laboratory. The authors would like to thank Benjamin T Root of the Naval Research Laboratory and Robert M. Robinson of the National Science Foundation for helpful discussions on the PERCS concept. The original idea of using a large sphere in low Earth orbit for global calibration of ground HF radars was first suggested by Gordon Frazer of the Australian Defence Science and Technology Organisation. The authors thank one of the referees for suggesting the discussion on the SuperDARN radar signal-to-noise for scatter from PERCS.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Spherical Polyhedra With Pentagon and Hexagon Faces
  5. 3. Radar Cross Section of Spherical Polyhedra
  6. 4. Deployment of a Spherical Polyhedron in Space
  7. 5. Performance of the Hoberman Sphere Wire Frame as a Radar Target
  8. 6. Atmospheric Drag and Optical Tracking of the Spherical Radar Target
  9. 7. Applications Summary
  10. Appendix A:: Geometric Construction of Spherical Polyhedra With Hexagonal Faces From Truncated Icosahedra
  11. Appendix B:: Far Field Backscatter Cross Section for a Perfectly Conducting Sphere
  12. Appendix C:: Design of the Hoberman Scissor Arms
  13. Acknowledgments
  14. References
  15. Supporting Information
  • Bristow, W. A., and D. Lummerzheim (2001), Determination of field-aligned currents using the Super Dual Auroral Radar Network and the UVI ultraviolet imager, J. Geophys. Res., 106, 18,57718,588.
  • Close, S., M. Oppenhein, S. Hunt, and L. Dyrud (2002), Scattering characteristics of high-resolution meteor head echoes detected at multiple frequencies, J. Geophys. Res., 107(A10), 1295, doi:10.1029/2002JA009253.
  • Hedin, A. E. (1987), MSIS-86 thermospheric model, J. Geophys. Res., 92, 4649.
  • Hoberman, C. (1990), Reversibly expandable doubly-curved truss structure, United States Patent 4942700, 24 July .
  • Hoberman, C. (1991), Radial expansion/retraction truss structures, United States Patent 5024013, 18 June .
  • Hughes, J. M., W. A. Bristow, and R. T. Parris (2004), SuperDARN observations of spectral enhancements excited during an ionospheric heating experiment, Geophys. Res. Lett., 31, L08808, doi:10.1029/2004GL019613.
  • Kelley, M. C. (1989), The Earth's Ionosphere, Academic, San Diego, Calif.
  • Koludzija, B., J. Oganjanovic, T. Sarkar, M. Tasic, D. Olcan, B. Janic, and D. Sumic (2004), WIPLE-D Software Users Manual, WIPL-D Ltd., Belgrade.
  • Lester, M., et al. (2004), Stereo CUTLASS – A new capability for the SuperDARN HF radars, Ann. Geophys., 22, 459473.
  • Montenbruck, O., and E. Gill (2000), Satellite Orbits: Models, Methods and Applications, Springer, Berlin.
  • Ruck, G. T., D. E. Barrick, W. D. Stuart, and C. K. Krichbaum (1970), Radar Cross Section Handbook, Plenum, New York.
  • Walker, A. D. M., R. A. Greenwald, and K. D. Baker (1987), Determination of the fluctuation level of ionospheric irregularities from radar backscatter measurements, Radio Sci., 22, 689705.
  • Wang, B.-C., and Y.-N. Chiu (1993), Geometry and orbital symmetry relationships of giant and hyperfullerenes: C240, C540, C960, and C1500, Synthetic Metals, 55–57, 29492954.