CALC-144

The 1978 reference beam with its subarrays out of phase: the shape, size and peak of the ground patch

REGISTERED 2026-10-21. Drafted for CLM-1481 (lessons records draft G549, outputs/lessons/lessons-records-draft-G549-2026-10-17.md), checked by an independent reader (outputs/review/lessons-records-check-G549-2026-10-17.md (rounds 1 to 4 FIX, round 5 PASS)), landed by records session claw-2. Feeds CLM-1481; CLM-1479 and CLM-1480 give the two published descriptions it sizes.

Basis: 2.45 GHz; widths at half power, in degrees and km; power density in mW/cm2 on a surface facing the satellite. An estimate on the report's figures, not a design calculation.

Script. dephase-spread-check.py, run with python -I; numpy for step 8. Five gates assert the main results and exit 1 on any miss; --mutate freq (5.8 GHz) and --mutate area (4 x 108 m2) must each exit 1.

Why it was needed. CLM-0911 holds how far the peak falls when the pilot is lost (23 to 0.003 mW/cm2) but not how wide the spread beam is. RP-1076 says the pattern becomes one element's (CLM-1479); DOE/ER-0097 says 2 pi steradians (CLM-1480). The two give very different areas for the 1978 design, and a lesson asked which applies.

Inputs.

Working.

1. With random phases between subarrays and none within, the powers add, not the fields. Each subarray has the same beam shape, so the summed ground pattern has that shape and its peak is P G_sub / (4 pi R^2), whatever the taper. 2. A uniform square aperture of side D has a sinc^2 beam in each axis: half power where (sin x / x)^2 = 1/2, x = 1.3916, a full width of 0.8859 lambda / D; first nulls at plus or minus lambda / D. 3. Slant range R = sqrt(Re^2 + Rg^2 - 2 Re Rg cos(lat)); the ground width north to south is the east-west width over sin(elevation). 4. Power inside the first-null square: the 1-D integral of sinc^2 between the first nulls over pi, squared. 5. Depth: coherent peak over incoherent peak = (sum sqrt P_i)^2 / (sum P_i), which is N for equal subarrays and less for a taper; for a continuous Gaussian with power 10 dB down at the edge of a disc it is N (mean sqrt p)^2 / (mean p). 6. Sensitivity: the patch if phase were lost per tube (108 / 36 = 3 m2 and 108 / 4 = 27 m2). 7. The same power spread evenly over 2 pi steradians, for scale, and one subarray's beam solid angle, 4 pi / G. 8. Speckle: a square grid of 10.392 m subarrays inside the 1 km disc (the 7,220 cells nearest the axis, so the count and area match the design), uniform amplitude, one random phase per subarray, summed by a 2048-point 2-D FFT for 1000 fixed seeds; each point is compared with the average there (grid count times the subarray envelope).

Script output, transcribed (fix round 3 run, 1000 seeds; main run exit 0; `--mutate freq` and `--mutate area` each exit 1 with two gates failing; run 2026-10-21).

1. inputs: lambda 0.12236 m; subarray side sqrt(108) = 10.392 m (square ASSUMED); subarray gain 4 pi A / lambda^2 = 90,641 (49.57 dBi) 2. one subarray's beam: half-power full width 0.8859 lambda/D = 0.01043 rad (0.598 deg); first-null full width 2 lambda/D = 0.02355 rad (1.349 deg) 3. ground patch and average peak, rectenna on the satellite's meridian (N-S = E-W / sin elevation): lat slant km elev deg E-W half-power km N-S half-power km E-W null-to-null km N-S null-to-null km peak mW/cm2 vs 0.003 vs plot 0.00296 0 35786 90.00 373 373 843 843 0.00378 +1.01 dB +1.07 dB 35 37120 49.34 387 510 874 1152 0.00352 +0.69 dB +0.75 dB 50 38377 32.69 400 741 904 1673 0.00329 +0.40 dB +0.46 dB 4. share of the power inside the first-null square of one subarray's pattern: 0.9028 per axis, 0.8151 in all 5. depth: uniform 38.59 dB (10 log10 7220); with a continuous Gaussian 10 dB power taper over a disc of equal subarrays, ratio N x 0.9025 = 6,516 (38.14 dB); printed 23 to 0.003 is 38.85 dB, plot 0.00296 gives 38.90 dB 6. sensitivity, phase lost per tube instead of per subarray (RP-1076's 'power module'): 36 tubes per subarray: 3.0 m2 each, side 1.73 m, half-power width 3.59 deg, 2,240 km at nadir 4 tubes per subarray: 27.0 m2 each, side 5.20 m, half-power width 1.20 deg, 747 km at nadir 7. the same power spread evenly over 2 pi steradians (DOE/ER-0097): 8.35e-08 mW/cm2 at 35,786 km, 84.4 dB below 23 mW/cm2; one subarray's beam fills 4 pi / G = 1.386e-04 sr, 1/45,320 of 2 pi sr 8. fixed random phase sets, 1000 seeds; 7220 square subarrays of side 10.392 m, the cells nearest the axis (779,760 m2), uniform amplitude, 2048-point 2-D FFT; points inside the half-power square; grain about lambda / 1 km x R = 4.4 km at nadir: share above 3x mean: mean 4.97 per cent (range 4.57 to 5.40); above 5x: mean 0.67 per cent highest point over the mean peak: median 9.2, 5 to 95 per cent 7.6 to 11.6, range 6.8 to 16.5 at nadir: median 0.0346 mW/cm2, 5 to 95 per cent 0.0289 to 0.0438, range 0.0256 to 0.0625 COMPARISON whole-antenna gain in place of one subarray's: 27.5 mW/cm2 at nadir GATES: GATE half-power width 0.8859 lambda/D: PASS (0.885893) GATE first-null share 0.9028 per axis: PASS (0.902823) GATE nadir peak in CLM-0911's printed span plus 1.1 dB: PASS (0.00378 in [0.0025, 0.00451)) GATE taper depth 38.14 dB: PASS (38.1396) GATE 2 pi sr over the beam 45,300 within 1 per cent: PASS (45,320) RESULT all gates PASS

Result. With the subarrays out of phase, the 1978 beam spreads to one subarray's own beam, about 0.6 degrees wide: a patch about 370 to 400 km wide east to west at half power (more north to south away from the equator), holding about 82 per cent of the power inside 850 to 900 km east to west. Averaged over random phase sets, the computed peak, 0.0033 to 0.0038 mW/cm2, is 0.4 to 1.0 dB above the report's 0.003. A hemisphere (2 pi steradians) would be 45,000 times larger in solid angle and about 46 dB weaker again. These are averages over random phase sets: one fixed set speckles the patch in grains about 4.4 km across, and its highest point is typically about 9 times the average (8 to 12 in nine sets out of ten), about 0.03 mW/cm2 (0.03 to 0.04), below the 1981 US standard of 10 but above the USSR guideline of 0.01.

Where this fails. - The gain is an ideal aperture's. The 10 degree rms error within each subarray, the 2 per cent of failed tubes and any loss would lower it, which moves the peak towards 0.003; the phase error alone costs about 0.13 dB (exp(-0.1745^2) = 0.970), and the tubes and losses are not sized here. - The result is the average over random phase sets. One fixed set gives a speckled patch, with grains about 4.4 km across at nadir (the wavelength over the 1 km antenna, times the range). Step 8 sums the subarray grid directly: about 5 per cent of points lie above three times the average, and the highest point in the half-power square is typically about 9 times it (8 to 12 in nine sets out of ten), about 0.03 mW/cm2 (0.03 to 0.04). If the phases keep drifting, the average over time returns to the result above; the 1978 report does not say which happens. - The subarray is taken as square. A rectangular one would give different widths along its two sides. - The 1978 report does not say whether phase within a subarray survives a pilot loss; it says the subarrays are "no longer phased together". If each tube drifted on its own, step 6 applies. - Step 5's taper is continuous; the design's is a 10-step approximation, and its edge subarrays do not fill a disc exactly. The 0.45 dB taper effect is the size of the gap step 5 shows and is not a correction of CLM-0911. - The densities are on a surface facing the satellite. On flat ground multiply by sin(elevation). - No time is computed. How fast the spread happens is in CLM-1020.

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