CALC-014

The same calculation at mid-latitude, and a denominator correction

Discharges the leading caveat on CALC-012, which was that all measured data came from 67 N. Millstone Hill data was obtained on a second attempt and is mid-latitude.

First: CALC-012 reproduces exactly, and its wording does not

The Sondrestrom result was recomputed from the same five files with the same filter, using the same code later applied to Millstone Hill. Everything reproduced: 40,736 rows, Ne 1.55e10 to 1.85e12 median 3.4e11, Te 502 to 4,864 median 1,650, r = +0.520, median threshold 364 W/m2 at 2.45 GHz and 1,823 W/m2 at 5.8 GHz. The calculation was right.

Its expression was not. CALC-012 says a 230 W/m2 beam "exceeds the self-focusing threshold at 38.5 per cent of measured F-region conditions". 38.5 per cent is the fraction of the in-domain subset, which is itself 61.7 per cent of measured conditions. Against all measured conditions the figure is 23.8 per cent, and at 5.8 GHz 1.5 per cent rather than 4.9.

Conditions outside the domain are ones where self-focusing cannot occur, which is a safe state. Excluding them from the denominator inflates the apparent exposure. Corrected in CLM-0095.

Implementation check

Before use, the code reproduced CALC-011's own internal consistency result: critical density 2.32e11 m^-3 at 2.45 GHz and 5.49e11 at 5.8 GHz, and a threshold at the critical density of 506.8 W/m2 at both frequencies, which is the frequency-independence that falls out of the algebra. Match to all quoted digits.

The two sites, identical method

Quality filter identical to CALC-012: 500 <= Te <= 5000 K, 1e9 <= Ne <= 1e13 m^-3, relative uncertainty at most 25 per cent on both.

The headline, on the all-conditions denominator

Fraction of all measured F-region conditions at which a 230 W/m2 beam is over threshold:

Result: 11.4 per cent at 2.45 GHz and 4.5 per cent at 5.8 GHz at mid-latitude, on the all-conditions denominator. The same denominator correction moves the Sondrestrom 2.45 GHz figure from 38.5 per cent to 23.8 per cent, and the 5.8 GHz figure from 4.9 to 1.5.

Mid-latitude is roughly half as exposed at 2.45 GHz, and three times more exposed at 5.8 GHz.

Why 5.8 GHz loses most of its advantage at mid-latitude

The domain condition admits only denser plasma as frequency rises: Ne above 5.49e11 m^-3 at 5.8 GHz against 2.32e11 at 2.45 GHz. Threshold power falls as Ne^-3. The mid-latitude F2 layer is denser than the sub-auroral one, with a measured maximum of 7.30e12 against 1.85e12, so the surviving in-domain points sit where the threshold is lowest.

Quantified: median in-domain density is 3.40e11 at 2.45 GHz against 1.255e12 at 5.8 GHz, a ratio of 3.69. Cubed that is 50.3, against the frequency term's (5.8/2.45)^3 = 13.3. So the predicted median ratio is 13.3/50.3 = 0.264, against 0.206 observed. The mechanism accounts for the inversion.

On the in-domain denominator 5.8 GHz is actually worse at mid-latitude, 55.8 per cent against 31.4 per cent. The two denominators point opposite ways, which is why CLM-0095 matters.

Applicability to the UK, which is the point of the exercise

The UK density range sits entirely inside the measured range, and the dip angles differ by half a degree. So the mid-latitude percentages transfer to the UK without extrapolation. That was the stated blocker on CALC-012 and it is now removed.

A data quality note

24 of 22,705 Millstone Hill points imply foF2 above 16 MHz, up to 24.3 MHz, which is not credible at mid-latitude. They are 0.11 per cent of the set, all from two solar-maximum days at 88 degrees elevation, and they move no median. But because threshold goes as Ne^-3 they do set the reported minimum, and they are 1.3 per cent of the 5.8 GHz in-domain subset. Medians and percentiles are sound; the minimum is not.

Limits

Four experiment days, two at solar maximum and two at solar minimum, so this samples rather than characterises the climatology. The 25 km growth length is approximate in the source and is applied at both frequencies. Whether self-focusing matters once triggered is still unestablished, though CALC-013 now bounds one consequence: it is not an energy loss.

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