CALC-112

Transmit-side waste heat and radiator area, reproduced from CR-134886 and bounded by Stefan-Boltzmann

The governing relationship for VC03 at L2 - converter efficiency, transmitted power, converter spacing and waste-heat flux - reproduced from a held source and then bounded, with the radiator area per kilowatt at the temperatures the register holds. Producer-computed 2026-09-17; the producer is barred from verifying its own work, and every headline figure was reproduced independently before this landed (recorded at the end).

Provenance. The project's working file radiator_area_bound.py, 6,400 bytes, sha256 3184314B7282A2E9, 101 lines, no dependencies. Inputs: P = 6.72 GW (RP-1076, SRC-107, p.50 Table III-1) and 7.0 GW (SRC-451 Vol IV p.10-3); eta = 0.70 and 0.85 (SRC-451 Vol III p.150) and 0.90 (the top of Brown's 70 to 90 per cent, SRC-417 via CLM-0933); F = 0.875, r0 = 500 m, the material flux limits and the column temperatures (SRC-451 Vol III pp.150-151, read from the render because the text layer on p.150 renders "3600, 3600" as "_00, 8600" and p.151 renders "87.5%" as "17.8%"); klystron collector 500 to 700 C (SRC-085 Table II); structure 200 to 300 C (SRC-450 pp.36-37). ASSUMED: emissivity 0.85 and a 250 K sink, alongside emissivity 0.5 and a 0 K sink, which are SRC-336 p.85's choices.

1. The relationship is held, not invented. SRC-451 Vol III p.150 gives the waste-heat flux at the antenna centre for converters spaced on a Gaussian of scale factor rho:

Q_max = 2 F (1 - eta) P / ( pi eta rho^2 [1 - exp(-2 (r0 / rho)^2)] ) W/m2

with F the fraction radiated toward the structure. Reading hazard: the render shows the exponent without its minus sign; the bracket must lie in (0, 1), so the minus is restored and flagged. The same page gives the flux the candidate materials tolerate - aluminium 3600, graphite/epoxy 3600, polyimide 8100 W/m2 - and states that 70 per cent "is now considered representative of the klystron performance rather than 75%". Fig 3.3-27 (p.151) plots the relation at F = 87.5 per cent for a 1 km antenna; Fig 3.3-26 gives column temperatures of 228 K (edge, no sun), 289 K (edge, full sun), 482 K (centre, no sun) and 600 K (centre, full sun).

2. Reproduced within 10 per cent across the plotted range, taking P = 7 GW from Vol IV because the figure does not state its P: at rho = 300, 600 and 1100 m the 70 per cent curve gives 18.6, 6.2 and 4.1 kW/m2 against the figure's 16, 6 and 3.7; the 85 per cent curve 7.7, 2.6 and 1.7 against 7, 3 and 1.7.

3. The document's own conclusion sits at the edge of its own plot. p.150, from the render: at 70 per cent and an aluminium structure "the microwave converters will have to be spaced according to rho >= 1100 meters, which produces a fairly flat Gaussian distribution, nearly a uniform distribution." 1100 m is where the x-axis ends, and at 1100 m the 70 per cent curve reads 3.7 to 4.1 kW/m2 - still above the 3.6 limit. Solving the formula for the crossing gives rho = 1,820 m at 7 GW (about 1,700 m at 6.72 GW), which is rho > 3 r0: not "fairly flat" but uniform. If the figure's unstated P is nearer 5.5 GW the document's reading holds; at either Vol IV's or RP-1076's power it does not. A qualification, not a contradiction, because P is unstated.

4. The waste heat a reader can check. Q_waste = P (1 - eta) / eta.

Dickinson's "approximately 1 GW" (SRC-450 p.36) and RP-1076's chain at 0.85 on 8.18 GW dc (1.18 GW) both sit at the 85 per cent row.

5. Radiator area per kilowatt, Stefan-Boltzmann: area = 1000 / (eps sigma (T^4 - Tsink^4)). At the centre column's 482 to 600 K, 0.16 to 0.70 m2/kW across emissivity 0.5 to 0.85 and sink 0 to 250 K. At a klystron collector's 773 to 973 K, 0.02 to 0.10 m2/kW. A 228 K edge column cannot radiate to a 250 K sink at all, which is why the sink assumption must travel with any figure.

6. The headline: the 1 km antenna face cannot shed its own waste heat at 70 per cent. Face area pi r0^2 = 0.785 km2. At the 3.6 kW/m2 material limit it sheds at most 2.83 GW. At eta = 0.70 the waste is 2.88 GW - over, even spread perfectly uniformly (3.67 kW/m2 average needed). At 0.85, 1.19 GW at 1.51 kW/m2 average - comfortable. The break-even converter efficiency for a self-radiating 1 km face at 3.6 kW/m2 and 6.72 GW is 70.4 per cent - and 70 per cent is what the 1975 study says the klystron delivers. The thermal limit and the achieved efficiency sit on the same number. Consistency check: a 600 K surface at emissivity 0.5 to a 0 K sink radiates 3.67 kW/m2 from one side, so the 3.6 kW/m2 limit is the centre column's full-sun temperature at that emissivity.

Limits. (1) The reproduction assumes P = 7 GW; the figure does not state its P, and the document's rho >= 1100 m conclusion depends on it. (2) Emissivity and sink are assumed; the area-per-kW band spans a factor of 4 across them. (3) The face-sheds-its-waste comparison is uniform flux against a material limit that is itself a 1975 judgement about 1975 materials. (4) CLM-0939's 85 per cent is a rectenna figure and is not used here; the transmit side is the klystron or amplitron. (5) The thermal constraint on rho pulls against the 10 dB Gaussian taper chosen for sidelobe control (SRC-085; SRC-450 Fig 1-9), which the document states in its own terms; that trade is not worked here.

INDEPENDENTLY REPRODUCED ON LANDING, 2026-09-17, without the script: face area 785,398 m2 and 2.827 GW at 3.6 kW/m2; waste 2.88 and 1.186 GW at 0.70 and 0.85 on 6.72 GW; break-even eta = 1 / (1 + 2.827 / 6.72) = 0.7039; 600 K at emissivity 0.5 to 0 K as 3,674 W/m2; 482 K at 0.85 to 0 K as 0.384 m2/kW; the p.150 formula at rho = 1100 m, F = 0.875, eta = 0.70, P = 7 GW as 4,080 W/m2, and the 3,600 W/m2 crossing at rho = 1,820 m. All match. The script runs clean at its declared digest. The p.150 material limits, the "rather than 75%" sentence and the rho >= 1100 m sentence were read from a 3x render because the text layer is class 11 on that page.

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