CALC-048

The adopted 23 mW/cm2 against every central estimate in the studies cited for it

Feeds CLM-0045 and CLM-0066. SHARED - one calculation, two claims. Answers QST-010 and QST-009.

Why it was needed. CLM-0035 recorded an unexplained factor-of-two discrepancy between reference 26's 12 mW/cm2 and the adopted 23 mW/cm2, and that discrepancy had to be either resolved or confirmed as an error. It resolves - but resolving it raises the sharper question of *where inside the band* the adopted figure sits, which is CLM-0066's subject. Both claims work the same four numbers, so one record serves both.

Inputs.

Unit conversion, applied throughout: 1 mW/cm2 = 10 W/m2.

Working, transcribed.

Band from the stated uncertainty:
  lower   12 x 0.5 =  6 mW/cm2   (=   60 W/m2)
  central          = 12 mW/cm2   (=  120 W/m2)
  upper   12 x 2   = 24 mW/cm2   (=  240 W/m2)

Position of the adopted figure inside that band:
  23 / 24 = 0.958,  so about 0.96 of the upper bound

Ratios of the adopted figure against each central estimate:
  23 / 12 = 1.92 x    against Duncan and Gordon runaway
  23 / 15 = 1.53 x    against Raytheon runaway
  23 / 20 = 1.15 x    against the self-focusing figure both studies give

Result. There is no arithmetic discrepancy - 23 sits inside the 6 to 24 band. And the adopted figure exceeds every central estimate in the studies cited for it, by 1.15x to 1.9x, sitting at about 0.96 of the upper bound of the only stated uncertainty band.

Conclusion, and the direction is the point. A figure the whole field treats as a conservative ceiling was selected from the least conservative end of its own source's uncertainty range. Choosing the top maximises permitted intensity and therefore minimises rectenna size and cost. That may be sound engineering or it may be optimism, and no source found gives a reason for the selection.

Where this fails. The 0.5x-2x reading of "within a factor of two" is assumed, not defined by the source; a different reading moves the upper bound and therefore moves 0.96. Duncan and Gordon state their own simplifications bias the threshold low (CLM-0064), which argues the true value is higher than any of these central estimates, so exceeding them is not necessarily wrong. Reference 26's full report is not held. And after 1981 the question became moot in the other direction, since the experiments then suggested 55 to 60 mW/cm2.

PARTIAL OVERLAP WITH CALC-007, FLAGGED SO IT IS NOT DUPLICATED SILENTLY. CALC-007's "Result 2, the factor-of-two band made explicit" reaches 59 / 119 / 238 W/m2 at 2.45 GHz and states "the 230 W/m2 design figure sits at 0.97 of the upper bound ... That is CLM-0066's finding, now derived rather than inherited." That is the same quantity by a different route: CALC-007 evaluates the Duncan and Gordon relation at Tn = 200 K and applies the factor of two to the result; this record applies the factor of two to the reported 12 mW/cm2 directly. The two routes agree to about one per cent (120 against 119 W/m2 central; 240 against 238 upper), which is a real cross-check and worth keeping. What CALC-007 does not contain is the 23/12, 23/15 and 23/20 ratios, which are the whole of CLM-0066's support field. CALC-007 and this record must be read together. CALC-007 recovers the relation and evaluates it at Tn = 200 K; this one applies the factor of two to the reported 12 mW/cm2 directly. They agree to about one per cent, which is the cross-check - but they are ONE finding reached two ways, and a reader meeting them separately counts them as two.

POSTSCRIPT 2026-09-19. "Reference 26's full report is not held" is now false: both reports are held, SRC-587 (the September 1977 report that ER-0023 reference 26 cites, obtained 2026-09-19) and SRC-108 (its July 1978 successor). The band arithmetic above stands as arithmetic; its attribution moves to the 1978 report, where the 12 mW/cm2, the factor of two and the analytic relation first appear. Reference 26 itself states approximately 20 mW/cm2 for runaway, scaled from a 15 MHz, 1 MW Arecibo experiment, with no band (SRC-587 PDF p.12 / 3-3), so against the document ER-0023 cites the adopted 23 is 1.15x the reference's figure, not a point inside a 6 to 24 band. On the ASSUMED symmetric 0.5x to 2x reading: the 1978 report gives its factor of two as the accuracy of an analytic estimate, "about the same accuracy to which the cooling reaction rates are known" (SRC-108 PDF p.18), and states the direction of its own modelling bias twice, that neglecting O(3P) fine-structure cooling and a model ionosphere makes the estimate an underestimate, more so with altitude (pp.18 and 20); so the symmetric reading is the rate-coefficient uncertainty only, and the bias "Where this fails" carries from CLM-0064 is now supported at the page (CLM-1094, SRC-587, 2026-09-19).

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