CALC-012

The finished calculation, on measured data

Supersedes every earlier range figure. This uses measured incoherent scatter radar pairs, the domain condition, and no independence assumption.

First: the coefficient discrepancy is resolved, and it was mine

CALC-011 flagged a factor of 1,034 between the Perkins and Goldman coefficient and Duncan's, and warned no figure could be published until it was resolved. It is resolved, and the error was in my renormalisation, not in either paper.

Duncan's own precursor, LA-UR-80-1562, obtained openly from OSTI, states the relation at a 2.4 GHz normalisation:

Pthresh(W/m2) = 6 W/m2 x (1e6/n)^3 x (T0/1000K)^4 x (f/2.4GHz)^3 x CF
renormalised:   6 / 2.4^3 = 0.434 W/m2 at 1 GHz     against the review's 0.43.  MATCH.

Working backwards, 0.434 W/m2 at 1 GHz implies 1.465 microwatts per square metre at 15 MHz. The OSTI abstract as retrieved reads "1.5 mW/m2". So the discrepancy is a lost micro prefix in the abstract as transcribed, not a disagreement between the papers. They agree to about 2 per cent.

*Postscript 2026-09-20.* The published abstract, read on the publisher's page, prints 1.5 uW/m2 for the F-region coefficient: the lost prefix is confirmed at the source, 1.5e-6 x (1000/15)^3 = 0.444 W/m2 at 1 GHz against 0.43.

Duncan's 0.43 W/m2 is traceable and correct, so every CALC-011 figure using it stands.

The measured data

Five files from the Madrigal database, extracted via the madrigalWeb API. Sondrestrom incoherent scatter radar at 67.0 N, spanning solar maximum 2000 and solar minimum 2008, across winter, equinox and summer, with day, night and twilight sampling within continuous multi-hour runs. Units confirmed from Madrigal's own parameter metadata rather than assumed: Ne in m^-3, Te in K.

43,002 raw F-region rows at 200 to 300 km. Quality filter, stated explicitly: 500 <= Te <= 5000 K, 1e9 <= Ne <= 1e13 m^-3, and relative uncertainty at most 25 per cent on both. That keeps 40,736 rows, 94.7 per cent. The filter matters: the unfiltered set contained a 32,724 K electron temperature, which is a fitting artefact, and Te enters to the fourth power.

The independence assumption was wrong, as CLM-0081 predicted. Density and temperature are positively correlated. And the specific pair that produced the extreme end of my earlier range, minimum density with maximum temperature, occurs at zero measured points. The opposite pair, maximum density with cool temperature, occurs at 855 points, so it is real.

The threshold over measured conditions, domain condition applied

Growth length 25 km, F region.

The headline result

At 2.45 GHz a 230 W/m2 beam exceeds the self-focusing threshold at 38.5 per cent of measured F-region conditions. At 5.8 GHz it exceeds it at 4.9 per cent.

That is a far more useful statement than a threshold ratio, because it is a duty cycle. The constraint is not a wall the design either clears or does not. It is a condition met roughly two days in five at 2.45 GHz and one day in twenty at 5.8 GHz.

How the estimate moved as the method improved

From 10.6 orders to 2.0. Two of those steps were corrections to my own errors: evaluating a threshold where the instability cannot occur, and pairing inputs that never co-occur.

Caveats, and one is substantial

1. The data is high-latitude only. Sondrestrom sits at 67 N, sub-auroral. Arecibo and Millstone Hill, the two preferred instruments, were unreachable: their host is unreachable over HTTPS from this environment and federated queries returned zero experiments even for well-sampled years, which looks like silent federation failure rather than absent data. The equatorial instrument returned only drift-mode runs with no measured Ne or Te. High-latitude F-region behaviour differs from mid-latitude, with auroral precipitation and polar cap effects, and a rectenna would not be sited at 67 N. So these percentages should not be read as applying to a UK or equatorial site until mid-latitude data is obtained. 2. The growth length of 25 km is stated as approximate in the source and moves the critical density proportionally. 3. Whether self-focusing, once triggered, matters at all is still unestablished. Nothing found quantifies the consequence, and the 1979-81 experiments did not confirm the instability.

Caveat 1 is now discharged by CALC-014, which obtains mid-latitude data. Caveat 3 is partly discharged by CALC-013.

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