CALC-115

The 1978 downlink peak's frame settled from the document's own geometry, and the range it implies

Supports CLM-1092 and CLM-1093 and closes OI-019. A calculation is required: the claim states values computed here. Computed 2026-09-19; every headline figure was reproduced independently before this landed (recorded at the end). The three live records on the question (CLM-0865, CLM-0867, CLM-0870) and CLM-0020 were read whole first: CLM-0865 carries the 32 per cent as an unresolved limit, CLM-0867 the ellipse sensitivity (b/a), CLM-0870 the "requires about 37,000 km"; none carries the frame or the range.

Provenance. The project's working file beam_frame_slant_range.py, 11,374 bytes, sha256 E2EBCE3D5565B4976180CBE21A72EC4EACD165DD0134F7CA812CAEC1AD40FB45, standard library only (PIL and numpy for the optional --digitise re-measurement). Inputs: SRC-085 PDF p.37 (printed 25), p.39 (27, Figure 15), p.40 (28, Figure 15 continued), p.43 (31, Figure 16), p.45 (33, Figure 18), p.49, p.101 (79), p.138 (A-39), p.146, p.204, p.230 (B-28), p.235 (B-33, Table R-4 and Figure R-21); CLM-0020, CLM-0865, CLM-0867, CLM-0870, CALC-088. Pages 37, 39, 43, 45, 138, 230 and 235 were read at the render (zoom 2) and Figure 16's taper plot on page 43 at a zoom-4 crop. ASSUMED: the rectenna lies on the satellite's meridian (SRC-085 states a latitude and no longitude offset; section 5 sizes it). Read from a plot: the ten step radii of Figure 16, which the figure's table does not print (section 4 sizes that too).

Geometry: for a ground point at latitude phi on the satellite's meridian, with R = 6,378 km and r = R + h, the central angle is phi, the slant range d = sqrt(r^2 + R^2 - 2 r R cos phi), and the zenith angle z at the ground satisfies cos z = (r cos phi - R) / d. Density on a horizontal plane = beam-normal density x cos z. On-axis far-field density S0 = |sum_i sqrt(p_i) A_i|^2 / (lambda^2 d^2) over the ten rings (the rings add in amplitude on axis, not in power).

1. Geometry at the reference site. At 34 deg N (Rockwell's site, p.204 and p.230) the satellite is at elevation 50.5 deg, z = 39.5 deg, cos z = 0.771, d = 37,062 km; at 35 deg N (SRC-086 Figure 1-1, CLM-0020) elevation 49.3 deg, z = 40.7 deg, cos z = 0.759, d = 37,134 km. The slant range is 3.5 to 3.7 per cent above the 35,800 km altitude p.101 gives ("Geosynchronous altitude (35800 km) has been used in all studies", normalised), so (d/h)^2 = 1.072 to 1.076.

2. The frame is beam-normal, and the document says so four times over. (a) p.138 (A-39), read at the render: "The basic area of the rectenna is 78.54 km2 at the equator" and "at higher latitudes, the rectenna ground area will increase as much as 35 km2, or a factor of 50 percent. (Even though the rectenna is still only intercepting the circular power beam, since the beam arrives at the Earth at greater angles at the higher latitudes, the ground footprint is changed from a circle to an ellipse.)" 78.54 km2 is pi x 5^2 exactly: the 5 km radius of p.37 ("88% of its energy within a 5 km radius perpendicular to boresight", normalised) is the beam-normal circle, and the ground footprint is that circle over cos z. (b) The printed ellipse: p.204 and p.230 "a typical rectenna site located at 34 deg N latitude covers an elliptical area 13 km in the north-south direction by 10 km in the east-west direction" (normalised; the layer prints the degree sign as a stray character). 10 km / cos z at 34 N = 12.96 km, the printed 13; at 35 N, 13.18 km. And Table R-4 (p.235, render) prints "AREA AVAIL. (TOTAL) 102.1 km2": pi x 6.5 x 5 = 102.1, the 13 x 10 ellipse to the decimal, against 101.8 km2 from cos z at 34 N. (c) The panels: p.49 "the rectenna is a series of serrated panels perpendicular to the incident beam" and p.146 "rows of panels oriented normal to the incoming beam" (raw); p.230 "750 rows of rectenna panels tilted 40 degrees from the horizontal, providing an active intercept area of 78.2 km2". A panel tilted 40 deg from horizontal faces a satellite at zenith angle 40 deg; the geometry gives 39.5 deg at 34 N. The "active intercept area" 78.2 km2 is the beam-normal circle (78.54) to 0.4 per cent, not the 102 km2 of ground. (d) The number itself: p.37 gives the 23 mW/cm2 as "a peak power density limitation in the ionosphere" (render; the layer prints "23.1mW/cm2" with a stray dot), and the same 23 heads Figure 18's "Power Density at Rectenna" curve (p.45, raw) and Figure R-21's "Rectenna Incident Power Density" (p.235). An ionospheric limit is an in-beam quantity. If 23 were instead the horizontal ground figure, the in-beam peak would be 23 / 0.759 = 30.3 mW/cm2 at 35 N, above the limit the document says it is designed to. So: 23 mW/cm2 is the density in the plane normal to the beam, incident on the tilted panels, and equal to the on-axis in-beam value; the horizontal ground-plane equivalent is 23 x cos z = 17.7 mW/cm2 (177 W/m2) at 34 N and 17.4 (174 W/m2) at 35 N. The 32 per cent OI-019 names is 1/cos z - 1 = 29.6 to 31.8 per cent over 34 to 35 N. Figure 18's axis label "Ground radius, m" is exact along the ellipse's east-west (minor) axis, where ground radius and beam-normal radius coincide, and that is the only axis on which its 5 km reads as 5 km; Figure R-21 plots the other axis separately ("rectenna major semi-axis, km").

3. The document's own "35 km2, or a factor of 50 percent" is loose. 35 km2 over 78.54 is 44.6 per cent, not 50; 50 per cent would be 39.3 km2. The increase reaches 35 km2 at 40.0 deg N (cos z = 0.692) and 39.3 km2 at 41.7 deg N. "Higher latitudes" therefore means about 40 to 42 deg N, and at the 34 to 35 deg N site the increase is 23 to 25 km2 (30 to 32 per cent), which is what the printed 13 x 10 km ellipse carries.

4. The forward peak from Figure 16's taper, and the range the stated 23 implies. Figure 16 (p.43, printed 31) tabulates ten step densities, 22.14 down to 2.46 kW/m2, and plots them against antenna radius without tabulating the step radii. The step edges were digitised from the zoom-4 crop by locating each vertical riser as the longest dark run in its pixel column (radius 0 at the dashed y axis, 500 m at the outer riser, 2.681 px/m): 105.0, 178.9, 231.3, 273.8, 322.8, 361.8, 389.4, 418.5, 460.3 and 500 m, read to about +/- 3 m. Radiated power from the ten rings: 6.777 GW against the printed 6.72 GW (p.39, render), +0.8 per cent; moving every edge by -3 m gives 6.681 GW and by +3 m 6.873 GW, so the printed figure sits inside the digitisation band. Ideal on-axis density at 2.45 GHz: 24.87 mW/cm2 at 35,800 km (CALC-088's Route A printed 24.87 and Route B 248.2 W/m2 from codes sharing no lines with this one); 23.21 at the 34 N slant range and 23.12 at 35 N. The error budget p.39 prints, applied as peak factors: exp(-(10 deg)^2) = 0.970 for the "TOTAL RMS PHASE ERROR FOR EACH SUBARRAY = 10" (raw) and (1 - 0.02)^2 = 0.960 for the "FAILURE RATE OF DC-RF POWER CONVERTER TUBES = 2%" (render), product 0.932; the "+/- 1 DECIBEL" amplitude tolerance and the "2" deg mean edge phase error move the peak by under 0.1 per cent each and are not applied. Four combinations of range and errors: ideal at altitude 24.87 (+8.1 per cent on the stated 23); ideal at the slant range 23.1 to 23.2 (+0.5 to +0.9); error-loaded at the altitude 23.17 (+0.7); error-loaded at the slant range 21.5 to 21.6 (-6.0 to -6.4). Inverting the stated 23: the ideal beam needs 37,228 km (36,937 to 37,521 over the digitisation band), which is the slant range to a site at 36.3 deg N on the meridian; the error-loaded beam needs 35,932 km, the altitude to 0.4 per cent. The two explanations of the printed 23 are degenerate: (d/h)^2 = 1.072 to 1.076 against 1/0.932 = 1.073, and the document cannot discriminate between them. p.40 says of Figure 18 "ANTENNA PATTERN AT RECTENNA USING BASELINE ERROR PARAMETER: SEE FIGURE 18" (raw), which says the plotted 23 is error-loaded and therefore computed at the altitude, not at the site's slant range. On that reading the same beam at the 34 to 35 deg N site peaks at 21.5 to 21.6 mW/cm2. CALC-088's "requires a range near 37,000 km" holds for the ideal beam only; its "7.9 per cent" is 248.7/230 - 1 = 8.1 per cent here, the ideal-at-altitude excess, of which the site geometry accounts for 7.2 to 7.6 and the error budget for 6.8, not both.

5. The unstated longitude offset. On the meridian the slant range is at its minimum. At 35 N a 5 deg offset moves the density by 0.1 per cent, 10 deg by 0.5, 15 deg by 1.1, 20 deg by 1.9 and 30 deg by 4.1 per cent (z from 40.7 to 51.6 deg). A continental US site served from a satellite over its own longitude sector sits inside 15 deg, so the ASSUMED offset is worth about 1 per cent on the density and under 1 deg on the panel tilt.

Sweep inputs, each cited: latitude 34 (p.204, p.230) and 35 deg N (SRC-086 Figure 1-1 via CLM-0020); altitude 35,800 km (p.101); step densities and radii (Figure 16, p.43, table and plot); error budget (p.39); longitude offset 0 to 30 deg (ASSUMED, swept).

Limits. (1) The step radii are digitised from a plot, worth +/- 1.4 per cent on the radiated power, +/- 1.6 per cent on the peak density and +/- 0.8 per cent on the range that reproduces 23; the printed 6.72 GW is inside that band, so nothing here rests on the digitisation being better than the plot. (2) The error factors are the standard peak reductions for uncorrelated rms phase error and a random 2 per cent of dark elements; the document does not say how it applied its budget, and CALC-088's Route A reached 23.06 rather than 23.17 with its own treatment. Either lands within 1 per cent of the printed 23 and the degeneracy of section 4 stands with either. (3) Which of the two degenerate readings the contractors used is not settled here and cannot be from SRC-085; p.40's caption is evidence for the error-loaded one, not proof. (4) The frame finding is about the reference document; later designs that quote "23 mW/cm2" or "230 W/m2" inherit it only if they inherit the geometry. (5) Nothing here touches the 88 versus 95 per cent collection question (CLM-0870), which is a different term.

INDEPENDENTLY REPRODUCED ON LANDING, 2026-09-19, by a second code sharing no lines with the working file, standard library only, from the printed inputs. Geometry: d = 37,062 and 37,134 km, cos z = 0.7714 and 0.7587 (z = 39.5 and 40.7 deg), (d/h)^2 = 1.0718 and 1.0759 at 34 and 35 deg N; 10 / cos z = 12.96 and 13.18 km; pi x 6.5 x 5 = 102.10; pi x 25 = 78.54; 23 x cos z = 17.7 and 17.4 mW/cm2; 1/cos z - 1 = 29.6 and 31.8 per cent. Radiated power from the ten printed densities and the digitised edges 6.777 GW, and 6.681 and 6.873 GW with the nine inner edges moved -3 and +3 m (the outer riser held at 500 m, which is what the band above means). Ideal on-axis density 24.87, 23.21 and 23.12 mW/cm2 at 35,800, 37,062 and 37,134 km with lambda = c/f = 0.12236 m; at the rounded 0.1224 m the text states each is 0.05 per cent lower (24.86, 23.19, 23.10), so the working file's wavelength is c/f. Error factors 0.9700, 0.9604 and 0.9316; error-loaded 23.17 at the altitude and 21.62 and 21.53 at the two slant ranges; the printed 23 inverts to 37,228 km ideal (36,937 to 37,521 over the edge band) and 35,932 km error-loaded. Longitude offsets at 35 N: 0.12, 0.48, 1.08, 1.89 and 4.11 per cent at 5, 10, 15, 20 and 30 deg, z = 51.6 deg at 30. The footprint increase reaches 35 km2 at 39.96 deg N (cos z = 0.692) and 39.3 km2 at 41.72 deg N. All match. Two corrections to the draft: (1) the +/- 3 m edge band is +/- 0.8 per cent on the aperture amplitude sum and on the range that reproduces 23, but +/- 1.6 per cent on the peak density, so "+/- 0.8 per cent on the peak" understated the peak band by a factor of two; the degeneracy of section 4 is unaffected, both readings sitting inside either band. (2) 37,228 km is the slant range to a site at 36.3 deg N on the meridian, not 35.9 (35.9 deg N gives 37,200 km). Both are corrected above and in CLM-1093 as landed. The ten Figure 16 densities and the p.138 sentences were re-read at the renders and match the quotations. The working file's size and sha256 match the provenance line (checked from the bytes; the file was neither opened nor run by the reproduction).

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